Answer Key

Answer Key 1.

2. 2 3

8

8 6 4 2

5 0

2

2 5

8 0

5 4

1

6

1

0

0

1

2

3

4

3

0

3.2

2 1 0

1

2 3 4

5

2

5 6

5.

3.7

0

2

7

4

6

4.5 6 4 2

0

9 2

4

4

5

3.2

5

4 3 2 1

6

2 0

1

8

6 5

0

12 7

5

9. Identity property of addition 10. Inverse property of addition 11. Distributive property

31. 1

27. 64

18. 30 1

32. 8

42. 46. 52.

37.

16

57. 3

61. d1

58. r

35. 17

4

5

6

7

8

9

59. h

V lw

60. d

3

7 6 5 4 3 2 1

65. x < 8 0

1

5

6

7

8

9

10 11

67. x ≥ 6.5

66. x < 3 2

I Pt

63. y ≥ 4

5 4 3 2 1

1

6x3

2A d2

64. x < 2

0

34. 62

33. 29

2x2

62. y > 6 3

30. 25

29. 25

10x 38. 2x2 13n 27 40. 7a 8b 41. 14a 16b 2b2 7b 43. 2 44. 10 45. 2 67 47. 2 48. 1 49. 8 50. 5 51. 2 3 23 5 2 53. 2 54. 6 55. 9 56. 4

36. 2485 39.

28. 25

4

5

6

7.5

7

6.5

6

5.5

0

1

2

2

C

18 20

4

2

0

2

81. x < 2 or x > 6 6 4 2

0

2

4

83. 0 ≤ x ≤

84. $9.45

0

6

8

5 2 5 2

6 10

4

3

82. x ≥ 6 or x ≤ 18

13. 17

19. 18 20. 10 21. 96 feet 22. 86 liters 23. $15.75 24. 2520 feet per minute 25. 64 26. 64

12 7

3 2 1

5

1

10

or x ≤ 16 7

16

17. 8

0

73.

7

12. Inverse property of multiplication

1

7

80. x ≥

8. Associative property of addition

16. 9

2

1

9

2 3 4

7. Commutative property of addition

15. 35

1

3 or 2 74. 14 or 2 2 19 75. 1 or 8 76. 36 or 18 77. 1 or 3 78. 9 < x < 7 79. x > 2 or x < 4 10

14. 11

0

71. 1 ≤ x ≤ 2 2

0

72. 9 or 3

8

6. 4 3

1

0.4

1

4 0

6

2

2

70. 0.4 < x < 0.4

5

4.

2

1

4.3

0.4

3.

69. 1 < x < 2

68. x > 1

Cumulative Review

10

85. 64%

1

86. 16

0

1

2

87. $24

3

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CHAPTER

NAME _________________________________________________________ DATE ____________

1

Cumulative Review For use after Chapter 1

Use a number line to order the real numbers. (1.1) 2

5

1. 5, 8, 5, 3

3

3. 0, 6, 2, 3.2

2. 4.3, 0, 8

5

4

4. 3.7, 4, 0, 7

4

6. 2, 5, 8, 3.2

5. 5, 4.5, 9, 3

Tell what property the statement illustrates. (1.1) 8. 3 4 5 3 4 5

7. 3 2 2 3 10. 5 5 0

9. 6 0 6

12. 66 1

11. 42 5 42 45

1

Select and perform an operation to answer the question. (1.1) 13. What is the sum of 25 and 8?

14. What is the sum of 5 and 6?

15. What is the difference of 26 and 9?


17. What is the product of 4 and 2?


19. What is the quotient of 6 and

13?

1

20. What is the quotient of 5 and 2?

Perform the given operation. Give the answer with the appropriate unit of measure. (1.1) 1

1

1

21. 42 feet 53 feet 23. 4.5 yards

1

22. 232 liters 153 liters

1$3.50 yard

24.

feet 60 seconds 142second 1 minute

Evaluate the power. (1.2) 25. 43

26. 43

27. 43

28. 52

29. 52

30. 52

Evaluate the expression for the given value of x. (1.2) 31. x 9 when x 8

32. 4xx 3 when x 2

33. x2 4 when x 5

34. x3 2 when x 4

35. 2x2 5x 1 when x 2

36. 4x4 3x when x 5

37. 4x2 3x 2x2 7x

38. 3x3 2x2 3x3 4x2

39. 42n 3 5n 3

40. 3a 2b 4a 6b

41. 4a b 52a 3b 5b

42. 4b2 b 32b2 b

43. 2x 3 7

44. 5x 30 20

45. 2a 8 4a 12

46. 3b 11 5 4b

47. 2.3a 1.8 2.8

Simplify the expression. (1.2)

Review and Assess

Solve the equation. (1.3)

49.

1 2m

4 2m 16

50.

1 5x

2 3

2 5x

1 3

48. 32a 7 5a 22 51. 6x 3 42x 5 45

Solve for y; find the value of y when x 3. (1.4) 52. 2x y 8

53. 5x 2y 8

54. 5x 6y 10

55. 4x 2y 6 0

56. x 4y 6

57.

118

Algebra 2 Chapter 1 Resource Book

2 3x

34 y 6

Copyright © McDougal Littell Inc. All rights reserved.

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CHAPTER

1

CONTINUED

NAME _________________________________________________________ DATE ____________

Cumulative Review For use after Chapter 1

Solve the formula for the indicated variable. (1.4) 58. Simple interest

59. Volume of a prism

Solve for r: I Prt

Solve for h: V lwh

60. Circumference of a circle

61. Area of a rhombus

1 Solve for d1: A 2d1d2

Solve for d: C d Solve the inequality. Graph its solution. (1.6) 62. 4y 24

63. 5y 6 26

64. 2x 8 5x 14

65. 2x 4 8

66. x 6 5x 12

67. 4.6 2x 8.4

Solve the compound inequality. Graph its solution. (1.6) 68. 2x 5 7 or 3x 9 12 2x

69. 6 6x 12

70. 0.5 5x 1.5 3.5

71. 0.7 2x 1.3 5.3

Solve the absolute value equation. (1.7)

7 2x 9

72. 2x 6 12

73. 4n 7 1

75.

76.

x 3 9 1 3

74.

x 3 4 1 2

77. 8 3x 11

Solve the inequality. Graph its solution. (1.7) 78.

x 1 8

79.

3x 3 9

80.

7x 2 14

81.

4 2x 8

82.

2 x 4

83.

4x 5 5

1 3

84. Driving Time You drive to school Monday, Wednesday, and Friday. The

school is 34 miles from your home on an interstate highway. The rest of your driving is in town. In a typical week, you drive 300 miles. Gasoline costs $1.28 per gallon, and your car’s fuel efficiency is 23 miles per gallon on the highway and 13 miles per gallon in town. How much do you spend on gasoline when you drive in town? (1.5) 85. Consumer Debt Last year 1.4 million Americans sought help from credit

counseling agencies. Five hundred four thousand of these people, with total debts of $2.3 billion, got into formal debt management or “workout” programs. What percent chose not to go into a formal program? (1.5) 86. Travel Services A local travel service advertised a round trip to Toronto Review and Assess

by motorcoach to see a popular stage show for $205. The same trip was available to attend a concert for $195. The travel service sold 14 tickets to the stage show. How many tickets to the concert were sold if the total sales were $5990? (1.5) 87. Buying Slacks A local store is advertising slacks for $31.99, which is 20%

off the original price. You purchase 3 pairs of slacks. How much did you save from the original price? (1.5)



119

Answer Key Practice A 1. 2.

1

0

1

2

3

4

5

1

0

1

2

3

4

5

4

5

6

7

8

9

10

3

2

1

0

1

2

3

3

2

1

0

1

2

3

10

9

8

7

6

5

4

3. 4. 5. 6.

7. 3 < 5 10. 4 > 2

8. 8 > 11 11. 6 < 0

9. 1
to show the relationship. 8. 8, 11

4 3 2 1

0

1

2

3

4

5

6

13 12

7

1

9. 1, 2

11

10

9

8

7

2

3

4

5

6

Lesson 1.1

7. 3, 5

10. 4, 2 1 2

2

1

0

1

2

0

11. 6, 0

1

12. 4, 2.7 2.7

7 6 5 4 3 2 1

0

5 4 3 2 1

1

0

1

2

3

Identify the property shown. 13. 3 5 5 3 5 5 16. 6 11 11 6 19. 34 2 3

432

14. 9 9 0

1 4 3 5 4 3 5

17. 6 20.

15. 37 73

1 6

18. 5 0 5 21. 41 4

Select and perform an operation to answer the question. 22. What is the sum of 4 and 6?








Give the answer with the appropriate unit of measure. 1

30. 6 inches 34 inches 32.

1000 meters 1 minute 301kilometers minute 1 kilometer 60 seconds

34. Filing Cabinet

A cabinet has 4 drawers. Each drawer is 13 inches tall. How tall is the cabinet?


1

3

31. 112 ounces 48 ounces 33. 10 miles

$8 1 mile

35. Touchdown

A football team scored 18 of their 27 points from touchdowns. If a touchdown is worth 6 points, how many touchdowns did the team score?


13

Answer Key Practice B 1.

11

10

9

8

7 3 4

2.

2

1

0

7 4

1

0

1

2

4

5.

6.

7.

8.

5

14 12 10 8

4

2

1

0

1

0

1

1

3

2

2.5 3

17 4

0

1

3

2

1

1

0

1

; 3 < 2.5

2

5

1

2 ; 2 < 5

4.1 3.2 4

; 3.2 > 4.1

7

3 3

0.9

2

2

2

5

10 3

25 14 ;2 < 5

0

0.8

2

8

9.

6

0.9

3

;

5

7 4

14

2

4.

17 4

3

25

3 ;4
10

6

5 2

3 1

8 7 ; 3 < 5

1

10. 4, 2, 2, 1, 2

3 7

11. 3, 0, 4, 2

1

12. 3, 7, 5, 2.1

13. 10, 2.9,

15 2,

8

5

14. 2, 3, 1, 2, 3 1 7 13 4

15. 6, 5, 2, 3,

16. Identity property of multiplication 17. Commutative property of addition 18. Inverse property of addition 19. Associative property of multiplication 20. Associative property of addition 21. Identity property of addition 22. 5 27. 27

23. 20 28. 4

31. 3 touchdowns

24. 4 29. 18

25. 3

30. 52 in.

32. 3 pieces

33. 2 or 2 under par

26. 24

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LESSON

NAME _________________________________________________________ DATE ___________

1.1

Practice B For use with pages 3–10

Plot the numbers on a number line. Decide which number is greater and use the symbol < or > to show the relationship. 3 7

1. 7, 10 25

3.

14

5. 0.8, 0.9

6. 3, 2.5

25

8. 3.2, 4.1

9. 3, 5

4. 2 , 5

Lesson 1.1

7. 2,

10 17 3, 4

2. 4, 4

8

7

Write the numbers in increasing order. 3

1

7 3

10. 2, 2, 4, 1, 2 13. 8,

15 2,

2.9, 10

1

1

11. 2, 4, 3, 0

12. 5, 2.1, 7, 3

5 14. 2, 3, 2, 3, 1

15. 3, 5, 6,

17. 3 1 2 1 3 2

18. 7 2 2 7 0

20. a b c a b c

21. a 0 a

7

13 4,

12

Identify the property shown. 16. 61 6 19. a

b c a b c

Select and perform an operation to answer the question. 22. What is the sum of 8 and 3?








2

30. Filing Cabinet

A cabinet has 4 drawers. Each drawer is 13 inches tall. How tall is the cabinet?

31. Touchdown

A football team scored 18 of their 27 points from touchdowns. If a touchdown is worth 6 points, how many touchdowns did the team score?

32. Eating Pizza

Eight friends buy 4 pizzas. Each pizza is cut into 6 pieces. Each person eats the same number of pieces. How many pieces does each person eat?

33. Playing Golf

The following table shows how many strokes over or under par Susan shot when she played nine holes of golf on Saturday. How far over or under par was her final score? Hole

1

Score 2

14

2

3

4

5

1

0

2

1 1 1


6

7

8

9

0 2


Answer Key Practice C ;5 > 1

1. 0

1

2

3

4

2.1

2.

4

0

1

1

0

3

; 1.3 > 2.1

3 0

3

4.

2

2

5 2

6

1.3

3 2 1

3.

5

1

1

2

3

4

; 45 < 23

7 2

; 3 < 7

4

2 3

5.

2

1

7

6.

0

2

1

2

; 43 > 2

0

2.8 3

1 4

7. 9, 8, 3, 2

; 7 < 2.8

4

3

1

8. 1.5, 4, 0, 5

9. 2.9. 8, 3, 2 10. Commutative Property of Addition 11. Commutative Property of Multiplication 12. Associative Property of Addition 13. Distributive Property 14. Commutative Property of Multiplication 15. Identity Property of Addition 1

17. 412 in.

18. $45

20. 70 miles per hour 22. 158.4 feet per hour

7

16. 688 lb

revolutions minute 21. 30.4 points per game

19. 120

23. Yes

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LESSON

NAME _________________________________________________________ DATE ___________

1.1

Practice C For use with pages 3–10

Graph the numbers on a number line. Then decide which number is greater and use the symbol < or > to show the relationship. 1. 5, 1

2. 1.3, 2.1

4. 3, 7

5. 3, 2

4

2

3. 5, 3

4

6. 7, 2.8

Write the numbers in increasing order. 2

3 1

8. 0, 1.5, 4, 5

9. 3, 8, 2.9, 2

Lesson 1.1

1

4

7. 3, 2, 8, 9

Identify the property shown.

b c b c a bc a b c b a ab0ab

10. a b c a c b

11. a

12. a b 3 a b 3

13.

14. ca b a bc

15.

Perform the given operation. Give the answer with the appropriate unit of measure. 3

1

1

16. 564 pounds 128 pounds 18. 15 ounce

1

17. 63 inches 24 inches

$3 1 ounce

19. 1

60 seconds 2 revolutions second minute

1

A cheetah can run 172 miles in 4 hour. What is the speed of a cheetah in miles per hour?

20. Cheetah’s Speed 21. Basketball

During the 1995–96 season, Michael Jordan scored 2491 points in 82 games. Find his average number of points scored per game. Give your answer to 3 significant digits.

22. Snail’s Speed

A snail can travel about 0.03 miles per hour. Convert this speed into feet per hour. Note that there are 5280 feet in 1 mile. Give your answer to 4 significant digits.

23. First Down

A football team must move 10 yards from its original position to gain a first down. In three plays a team ran for 6 yards, lost 8 yards due to a quaterback sack, and passed for 12 yards. Did the team make a first down?



15

Answer Key Practice A 1. 23 2. 56 3. 3x2 4. 75 5. 93 6. x4 7. 36 8. 81 9. 64 10. 7 11. 15 12. 42 13. 10 14. 12 15. 1 16. 5 17. 1 18. 24 19. 81 20. 32 21. 8 22. 8 23. 3 24. 3 25. 13 26. 1 27. 18 28. 15 29. 7 30. 28 31. 7 32. 9 33. 12 7 34. 6 35. 11 36. 1 37. 4 38. 3 39. 25 40. 2 43.

1 2 x4x

41. 2

1; 75

42. 2yx y; 30

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LESSON

1.2

NAME _________________________________________________________ DATE ____________

Practice A For use with pages 11–17

Write the expression using exponents. 1. 2

22

2. 5

4. 7 to the fifth power

55555

3. 3x

3x

5. 9 to the third power

6. x to the fourth power

8. 34

9. 26

Evaluate the expression. 7. 62

5 432

365 9321 3 23

10. 3 2

11. 11 8 2

12. 4

13.

14. 16 16 4

15.

46

16. 4 4 2 1

17. 3 2

19. 34

20. 25

18.

21. 14 7 5 1

Evaluate the expression for the given value of x. 23. 2x 3 when x 0

24. 4 x when x 7

25. 3x 1 when x 4

26. 2 3x when x 1

27. 4 7x when x 2

28. x2 x when x 5

29. 3x 0.5x 1 when x 3

30. x2 3x when x 4

31. x2 3 when x 2

32. 11 2x when x 4

33. 6x 3 when x 5

1

Evaluate the expression for the given values of x and y. 34. 2x 3y when x 3 and y 4

35. x2 5y when x 2 and y 3

36. x 5y when x 4 and y 1

37. 4x y3 when x 1 and y 2

38. 4

xy when x 5 and y 3

40. 2x y3 when x 3 and y 2

39. x2 y2 when x 4 and y 3 41. 2y3 5x when x 2 and y 1

Write an expression for the area of the figure. Then evaluate the expression for the given value(s) of the variable(s). 42. x 2 and y 3

43. x 6 x 4x 1

xy

2y



27

Lesson 1.2

22. x 5 when x 3

Answer Key Practice B 1. a5 2. 94 3. x3 4. 2y3 7 5. 4b2 2a2 6. 8n 7. 81 8. 64 9. 32 10. 5 11. 8 12. 24 13. 15 14. 32 15. 3 16. 15 17. 7 18. 28 5 19. 2 20. 9 21. 12 22. 2 23. 2 24. 2 27.

25. 4

1 2 x4x

26. 2yx y; 30

1; 75

28. 8.95x 29.95; $65.75

29. 6.95x 24.995 x; $70.83 30. $270; $415

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LESSON

1.2

NAME _________________________________________________________ DATE ___________


Write the expression using exponents.

aaaa 2y 2y 2y 7

1. a

2. 9999

4.

5. 4b

3. xxx

4b 2a 2a

6. 8 to the nth power

Evaluate the expression. 7. 34

9. 25

8. 26

2 1 4 5 23 3 4

10. 3

11. 14 7 5 1

13.

14. 2

12. 1 3 22

3 12

15. 5 23 9 6

Evaluate the expression for the given value of x. 16. x2 x when x 5

17. 3x 0.5x 1 when x 3

18. x2 3x when x 4

19. 2x 1 x when x 5

20. 25x 3 8 when x 3

21. 6 x3 x when x 2

Evaluate the expression for the given values of x and y.

Lesson 1.2

22. 2x y3 when x 3 and y 2 24.

3x y when x 3 and y 1 2x 1

23. 2y3 5x when x 2 and y 1 25.

y 23 when x 1 and y 4 2x y

Write an expression for the area of the figure. Then evaluate the expression for the given value(s) of the variable(s). 26. x 2 and y 3

27. x 6 x 4x 1

xy

2y

28. Photography Studio

A photography studio advertises a session with a sitting fee of $8.95 per person. The standard package of pictures costs $29.95. Write an expression that gives the total cost of a session plus the purchase of one standard package. Evaluate the expression if a family of four purchases this package.

29. Books

You want to buy either a paperback or hard covered book as a gift for 5 friends. Paperbacks cost $6.95 each and hard covered books cost $24.99 each. Write an expression for the total amount you must spend. Evaluate the expression if 3 of your friends get a paperback.

30. Weekly Earnings

For 1980 through 1990, the average weekly earnings (in dollars) for workers in the United States can be modeled by E 14.5t 270, where t is the number of years since 1980. Approximate the average weekly earnings in 1980 and 1990.

28



Answer Key Practice C 1. x3y2 2. 45 3. 34 4. 3x3 x2 1 5. 68 6. x y3 7. 112 8. 2 9. 16 5 10. 1 11. 64 12. 81 13. 66 14. 7 15. 43 16. 2 17. 4x 20 18. 9x2 12x 19. x 5y 20. x y 21. 2x3 3x2 2 22. 10x 2 23. 13.99 0.10x 0.08y; $20.99 24. 21.82 0.06x; $22.72.

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LESSON

NAME _________________________________________________________ DATE ____________

1.2


Write the expression using exponents. 1. x

x x y y

4. 3x3x3x x

x

333

2. 44444

3. 3

5. 6 to the 8th power

6. the sum of x and y to

the third power

Evaluate the expression. 7. 4 31 52 10. 3 1

8. 8 6 3 12

2 35

11. 25 3 55

9. 41 32 5 1 12. 4 20 42

Evaluate the expression for the given values of x and y. 13. 7x 3y 3 when x 6 and y 2 15. 4

x 3x 2y when x 4 and y 12 y

14.

5x y when x 2 and y 5 2xy 1

16.

x y2 3 1 3 when x 2 and y 2 xy

Simplify the expression. 18. 6x2 x 32x x2

19. 4x y 3x y

20. 4x y 3 y x

21.

x3

2

x2

1

x2

x 1

Lesson 1.2

17. 10x 3 25 3x

22. 0.52x 8 32 3x

23. Phone Bill

A phone company charges a basic rate of $13.99 per month. In addition the user is charged $0.10 per minute for all long distance calls made during the week and $0.08 per minute for all long distance calls made during the weekend. Write an expression that gives the total monthly bill. Evaluate the expression if you talk long distance for 30 minutes during the week and 50 minutes during the weekend.

24. Engraving

A gift shop advertises that they will engrave any gift purchased in their store at a rate of $0.06 per letter and the first three letters are free. A desk plate sells for $22. Write an expression for the total cost of buying the desk plate and having it engraved. Evaluate the expression if you wish to engrave a name that has 15 letters.



29

Answer Key Practice A 1. 3 2. 9 7. 4 8. 3 12. 18.

1 3 5 6

13. 5 7

19. 3

3. 1

4. 24

9. 18

10. 6

14. 4

15.

1 12

21.

20.

10 3 2 5

5. 3

6. 8 1

11. 3 1

16. 8 22. 3

17. 23. 4

24. 5

x 6 17 x 11 not added, from the right side of the equation. x 12 2 26. Twelve should be added, x 14 not subtracted, to the right side of the equation. 5x 10 27. The right side of the equation x2 should be divided, not multiplied by 5. 2x 1 7 28. One should be subtracted, 2x 6 not added, from the right side x3 of the equation. 29. The right side of the 3x 2 7 equation should be divided, 3x 9 not multiplied, by 3. x3 2x 3 8 30. The distributive property 2x 6 8 leads to 2x 6 on the left side 2x 2 of the equation. x1 31. 2x should be subtracted, 3x 3 2x 1 x4 not added, from the left side of the equation. 1 32. The right side of the 2x 4 2 1 equation should be multiplied, 2x 6 x 12 not divided, by 2 in the last step. 3 33. The distributive property 2 2x 1 5 3 3 leads to 3x 2 on the left side 22x 1 5 3x 72 of the equation. x 76 34. 9 in. 9 in. 35. 13 in. sides 36. $22 25. Six should be subtracted,

37. 3 tickets

1 2

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LESSON

NAME _________________________________________________________ DATE ___________

1.3


Solve the equation. Check your solution. 1. x 3 0

2. x 7 2

3. 5 x 4

4. 18 x 6

5. 5 x 2

6. 3 x 11

7. 6x 24

8. 5x 15

9.

43x

10.

8

13. 3 x 8 16. x 19. x

1 2 2 3

3 8

3

22. 5x 2 13

1 3x

6

11. 3 9x

12. 4 12x

14. 3x 12

15. 6x 20

17.

1 3

x

20. 4x

5 6

18. x

1 3

3 4

21. 7x

23. 9 3x 3

1 12 14 5

24. x 4 2x 9

Describe the error. Then write the correct steps. x 6 17 x 23

26. x 12 2

2x 1 7 2x 8 x4

29.

31. 3x 3 2x 1

32.

25.

28.

27. 5x 10

x 10

5x 4 x 45

3x 2 7 3x 9 x 27 1 2x

42 1 2x 6 x3

x 50 30. 2x 3 8

2x 3 8 2x 5 5 x2 33.

3 2 2x

1 5 3x 1 5 3x 4

Lesson 1.3

x 43 34. Perimeter

The perimeter of a square is 36 inches. Find its dimensions.

35. Perimeter

36. Sales Tax

37. Movie Tickets

The state sales tax in Pennsylvania is 0.06 (or 6%). If your total bill at the music store included $1.32 in tax, how much did the merchandise cost?

42


An equilateral triangle has sides of equal length. Find the dimensions of an equilateral triangle with a perimeter of 39 inches.

A ticket to the movies costs $7. You have $21. How many tickets can you buy?


Answer Key Practice B 1. 20 2. 2 3. 3 8. 5 13.

1 7

18.

27 2

9.

5 2

14. 13 19.

29 3

4. 2

10. 3 15.

4 7

20. 1

5. 4

6.

11. 20 4 16. 3 7 21. 6

1 4

7.

14 3

12. 24

17. 0

22. 2x 3 11 ft; 3x 5 16 ft;

15 x 8 ft 23. 15 2x 9 ft; x 7 10 ft 24. $22 25. 3 tickets 26. 7.5 hours 27. 2.75 hours 28. 4.2 hours 29. 3 children

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LESSON

NAME _________________________________________________________ DATE ___________

1.3


Solve the equation. Check your solution. 1. x 8 12

2. 2x 3 7

3. 5x 2 13

4. 6 x 4

5. 9 3x 3

6. 8x 3 5

7. 3x 5 9

8. x 4 2x 9

9. 3x 1 x 4

1 x 2

10. 4 5x x 8

11.

13. x 1 23x 1

14. 3x 2 54 x

16. 3x 4 38 2x 19.

3 2 x

5 7

17. 20.

6 4

1 4x 2 1 4x 2

12.

2 x 3

1x7

15. 27 x 61 2x

10 5 3x

18.

3 34x

21. 52x 2 4 2x

1 x 3

1 13x 8

Find the dimensions of the figure. 22. The perimeter of the figure is 35 feet.

23. The perimeter of the figure is 38 feet.

3x 5 2x 3

15 2x

15 x x7

The state sales tax in Pennsylvania is 0.06 (or 6%). If your total bill at the music store included $1.32 in tax, how much did the merchandise cost?

25. Movie Tickets

26. Weekly Pay

27. Plumbing Bill

You have a summer job that pays $5.60 an hour. You get $8.40 an hour for overtime (anything over 40 hours). How many hours of overtime must you work to earn $287?

28. Travel Time

You want to visit your aunt who lives 255 miles away. The interstate is 10 miles from your house and once you get off the interstate, you must travel 14 miles more to get to your aunt’s house. If you drive 55 miles per hour on the interstate, how many hours will you travel on the interstate?


A ticket to the movies costs $7. You have $21. How many tickets can you buy? The bill from your plumber was $134. The cost for labor was $32 per hour. The cost for materials was $46. How many hours did the plumber work?

29. Babysitting Rate

You charge $2 plus $.50 per child for every hour you babysit. You earn $3.50 an hour when you watch the Crandell children. How many children are in this family?


43

Lesson 1.3

24. Sales Tax

Answer Key Practice C 14 1 1. 3 2. 1 3. 3 4. 5 5. 0 6. 8 2 7 7. 2 8. 21 9. 9 10. 5 11. 4 1 12. No solution 13. 1 14. 3 15. 0.8 16. No solution 17. Identity 18. Identity 13 39 19. No solution 20. 3 2 and 2 4 21. 5 10 8 and 4 11 8 1 2 22. 83 in. 133 in. 23. 50 ft

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LESSON

NAME _________________________________________________________ DATE ___________

1.3


Solve the equation. Check your solution. 1. 18 6x

2. 3x 7 10

3. 4x 1 x 8

4. 3x 2 2x 4

5. 43 x 6 2x 3

6. 2x 1 4 32x 1

7. 31 x 3 x 8

8. 2x 1 3x 7 2

9. 62x 1 3 62 x 1 11. 13.

1 5 5 2x 2 3 3x

10.

3 4 2x

8 5 x

10 4x 3

12. 3.6x 3.1 35.2 1.2x

65 15 5x 1

14. 5x 23 x 4 x

15. 1.54x 2 20.5x 3.5

Determine whether the following equations have no solution or are identities. 16. 3x 2 35 x

18. 6x 2 4x 32x 1 2 2x

1 2

17. 5x 2 22x 1 x 19. 52x 3 24 3x 4x

Find the dimensions of the figures. 20. The two rectangles shown have the same

area.

21. The two triangles shown have the same

perimeter. 2

x

x1

x3

3 2x 3

Lesson 1.3

x5

22. Photo Frame

2x 1 x3

x8

You want to mat and frame a 5 7 photograph. The perimeter of the outside of the mat is 44 inches. The mat is twice as wide at the top and bottom as it is at the sides. Find the dimensions of the mat.

23. Garden Fencing

Your garden has an area of 136 square feet. You want to put a fence around the entire garden. How much fencing do you need?

8

3x 2

44



Answer Key Practice A 1. 2 2. 3 7. 2 8. 2

3. 2 9.

13 2

11. y 2x 12; 18

17. 19. 23. 26. 28. 30.

10. y

5. 12 1 2

6.

1 3

32x; 7 1

12. y 1 2x; 4

3x 4 ; 1 x 2 8 2x y ; 2 16. y x 1; 3 3x 3 3 22 y 2x 2; 4 18. y 83x 10 3; 3 I d d I 20. r 21. r 22. t t r t Pt Pr A 2 A 25. b s h 24. w l h 3 P 5 27. C F 32 s 3 9 P 2A 29. s ; 11 cm h b1 b2 4 A l ; 3 ft w

13. y 6x 12; 0 15.

4. 4

14. y

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LESSON

NAME _________________________________________________________ DATE ___________

1.4


Find the value of y for the given value of x by first substituting the value of x into the equation and solving for y. 1. 2x 3y 12; x 3

2. 4x 5 3 2y; x 1

3. xy x 3; x 3

4. 4x 3xy 16; x 2

5. 2y 5x 1; x 5

6. x 3y 1 2; x 0

7. 3x 7y 8; x 2

8. x 12 xy; x 4

9. 5x 2y 8; x 1

Find the value of y for the given value of x by first solving for y and then substituting the value of x into the equation. 10. 3x 2y 1; x 5

1

11. y 2x 12; x 3

12. 2x y 1; x 6

13. 2x 3y 4 0; x 2

14. xy 3x 4; x 2

15. 2x 3xy 8; x 2

16. 6x 9y 9; x 3

17. 3x 7 2y 3; x 4

18. 8x 3y 10; x 4

1

Solve the formula for the indicated variable. 19. Distance

20. Distance

Solve for t: d rt

Solve for r: d rt

21. Simple Interest

22. Simple Interest

Solve for r: I Prt

Solve for t: I Prt

23. Height of an Equilateral Triangle

Solve for s: h

3

2

24. Area of a Rectangle

Solve for w: A lw

s

25. Area of a Parallelogram

26. Perimeter of an Equilateral Triangle

Solve for b: A bh

Solve for s: P 3s

27. Celsius to Fahrenheit

28. Area of a Trapezoid

9 Solve for C: F C 32 5

h Solve for h: A b1 b2 2

Solve the formula for the indicated variable. Then evaluate the rewritten formula for the given value(s). (Include units of measure in the answer.) 29. Perimeter of a Square: P 4s

Solve for s. Find s when P 44 cm.

30. Area of a Rectangle: A lw

Solve for l. Find l when A 24 ft2 and w 8 ft.

Lesson 1.4

s

w s

s

s

56



Answer Key Practice B 13 1. 2 2. 2 3. 2 4. 8 5. 4 2 6. 1 7. y 3x 1; 3 3

8. y 2x 2; 4

14. 17. 19. 22. 24.

10 3;

22 3

21 5x 2 21 ; 7 11. y x ; 21 x 16 2 2 117 27 ; 99 13. s y x h 10 2 3 P 3V 5 15. h 16. C F 32 s 2 3 r 9 2A 2A 18. b2 h b1 b1 b2 h P S V 20. h 21. s ; 11 cm r 2 2 h r 4 A 3V l ; 3 ft 23. r 3 ; 4m w 4 8 m 25. 64 m 2 201.1 m 2

10. y 12.

8

9. y 3x

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LESSON

NAME _________________________________________________________ DATE ___________

1.4


Find the value of y for the given value of x by first substituting the value of x into the equation and solving for y. 1. 3x 7y 8; x 2 4.

4 5x

3 2y

2. x 12 xy; x 4

4; x 10

5.

2 3x

1 2y

3. 5x 2y 8; x 1

6; x 6

6. 2x 3y 1; x 1

Find the value of y for the given value of x by first solving for y and then substituting the value of x into the equation. 7. 6x 9y 9; x 3 10. 2 xy 5x; x 1

8. 3x 7 2y 3; x 4 11.

3 4x

4 7y

6; x 8

9. 8x 3y 10; x 4 3

2

12. 5x 9y 13; x 15

Solve the formula for the indicated variable. 13. Height of an Equilateral Triangle

Solve for s: h

3

2

14. Perimeter of an Equilateral Triangle

Solve for s: P 3s

s

15. Volume of a Right Circular Cone

16. Celsius to Fahrenheit

r 2h Solve for h: V 3

9 Solve for C: F C 32 5



h Solve for h: A b1 b2 2

h Solve for b2: A b1 b2 2

19. Lateral Surface Area of a

20. Volume of a Right Circular Cylinder

Solve for h: V r 2h

Right Circular Cylinder Solve for r: S 2rh

Solve the formula for the indicated variable. Then evaluate the rewritten formula for the given value(s). (Include units of measure in the answer.) 21. Perimeter of a Square: P 4s

22. Area of a Rectangle: A lw

Solve for s. Find s when P 44 cm.

Solve for l. Find l when A 24 ft2 and w 8 ft.

Hot Air Balloons In 1794, the French Army sent soldiers up in hot air balloons to observe enemy troop movements. One such balloon, the L’Entrepenant, had a 256 volume of cubic meters. 3 4 23. Solve the formula for the volume of a sphere V r 3 for r3. Then use 3 this formula to calculate the radius of the L’Entrepenant balloon.

Lesson 1.4

24. What was the diameter of the L’Entrepenant balloon? 25. Use the formula for surface area of a sphere S 4r 2 to approximate the

surface area of the L’Entrepenant balloon.



57

Answer Key Practice C 25 13 1. 0 2. 2 3. 1 4. 1 5. 19 6. 3 4x 5 1 5 10 7. y 8. y x ; 5 ; 3x 3 3 3 5 1 3x 8 14 9. y 10. y ; ; x3 2 2x 1 5 4x 7 2x 7 1 11. y ; 3 12. y ; 3x 1 4 3x 5 2A S 13. b1 b2 14. R r h s 15. T 7x 15y 16. 3; T represents the total amount earned, x represents the number of regular washes, y represents the number of washes and 2 waxes. 17. 35 customers 18. V r2h 3 r3 3V 2 r3 19. h 20. 12 6 18 ft 3 r 2

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LESSON

1.4

NAME _________________________________________________________ DATE ___________


Find the value of y for the given x by first substituting the value of x into the equation and solving for y. 1. xy 3x 6; x 2

2. 5x 2y 13 x; x 3

3. yx 3 2x 1; x 4

4. y2x 1 5; x 3

5. 6x y3 x 7; x 2

6. yx 2 7x 3 1; x 1

Find the value of y for the given x by first solving for y and then substituting the value of x into the equation. 7. 4x 3xy 5; x 1 9. yx 3 5; x 7 11. y3x 1 4x 2 1; x 2

8.

2 1 1 x y ;x2 3 5 3

10. y2x 1 3x 8; x 2 12. y4 3x 2x 1 9; x 3

Solve the formula for the indicated variable. 13. Area of a Trapezoid

h Solve for b1: A b1 b2 2

14. Lateral Surface Area of a Frustrum of

a Right Circular Cone Solve for R: S sR r

Fundraising The high school girls softball team is holding a car wash to raise money for new uniforms. At the car wash they offer a regular wash for $7 and a wash and wax for $15. 15. Write an equation that represents the total amount of money they earned. 16. How many variables are in the equation? What do they represent? 17. The softball team earned $365. If they washed and waxed 8 cars, how

many customers only wanted a wash? Silo The silo pictured at the right is a cylinder with half of a sphere on top. The silo can hold 576 cubic feet of grain. The radius of the sphere is 6 feet. 18. Given that the volume of a cylinder is V r2h and the volume of a

sphere is V

4 3 r , write a formula for the volume of the silo. 3

Lesson 1.4

19. Solve the formula you found in Exercise 18 for h. 20. Find the height of the silo.

58



Answer Key Practice A 1. Total Cost $120 Price per case $5.99 Number of cases x 2. 120 5.99x 3. About 20.03 4. 20 cases 5. Distance traveled 168 miles Rate of travel r Time traveled 312 hours 7 6. 168 2r 7. 48 8. 48 miles per hour 9. Total cost $80.96 Cost for first 8 books $1 Cost of a book $19.99 Number of books x 10. 80.96 1 19.99x 11. 4 12. 4 books 13. Yard size 27,500 square feet Coverage for one bag 5000 square feet Number of bags x 14. 27,500 5000x 15. 5.5 16. 6 bags

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Lesson 1.5

LESSON

1.5

NAME _________________________________________________________ DATE ___________


Party Supplies In Exercises 1–4, use the following information. You have $120 to purchase juice for a party. Each case of 24 bottles costs $5.99. Assuming there is no sales tax, how many cases can you purchase? Use the following verbal model. Total cost Price per case

Number of cases

1. Assign labels to the parts of the verbal model. 2. Use the labels to translate the verbal model into an algebraic model. 3. Solve the algebraic model. 4. Answer the question.

Vacation Trip In Exercises 5–8, use the following information. On a 1 trip to the Grand Canyon, you drove 168 miles in 32 hours. What was your average speed? Use the following verbal model. Distance Rate

Time


Book Club In Exercises 9–12, use the following information. A book club promises to send 8 books for $1, if you join the club. After you receive the 8 books, you may select more books at a rate of $19.99 per book. If you spend a total of $80.96, how many extra books did you purchase? Use the following verbal model. Total cost Cost for first 8 books Cost of a book

Number of books


Lawn Fertilizer In Exercises 13–16, use the following information. A bag of lawn fertilizer claims that it will cover 5000 square feet of grass. If your yard is 27,500 square feet, how many bags of fertilizer will you need? Use the following verbal model. Yard size Coverage for one bag

Number of bags

13. Assign labels to the parts of the verbal model. 14. Use the labels to translate the verbal model into an algebraic model. 15. Solve the algebraic model. 16. Answer the question. 68



Answer Key Practice B 1. Distance traveled 100 miles Rate of travel 763 miles per hour Time traveled t 2. 100 763t 3. About 0.131 4. About 0.131 hour or about 7.86 minutes 5. Total cost Price per

square yard

Number of square yards

6. Total cost $450 Price per square yard x Number of square yards 30 square yards 7. 450 30x 8. 15 9. $15 per square yard 10.

Distance traveled

Your speed

Your Friend’s time speed

Friend’s time

11. Distance traveled 300 miles

Your speed r Your time 3 hours Friend’s speed 52 miles per hour Friend’s time 3 hours 12. 300 3r 156 13. 48 14. 48 miles per hour 15.

Total Time time per trial

Number Time to of trials write report

1

16. Total time 12 hours or 90 minutes

Time per trial 5 minutes Number of trials x Time to write report 30 minutes 17. 90 5x 30 18. 12 19. 12 trials

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LESSON

Practice B

Lesson 1.5

1.5

NAME _________________________________________________________ DATE ___________

For use with pages 33–40

Land Speed Record In Exercises 1–4, use the following information. The land speed record was broken in 1997 by a British car called the Thrust SSC. The Thrust SSC traveled at a rate of 763 miles per hour. This was accomplished by using a jet engine. How long would it take the Thrust SSC to travel 100 miles? Use the following verbal model. Distance Rate

Time


New Carpeting In Exercises 5–9, use the following information. You just added a family room to your home. You have budgeted $450 for carpeting. If you need 30 square yards of carpeting, how much can you spend per square yard? 5. Write a verbal model. 6. Assign labels to the parts of the verbal model. 7. Use the labels to translate the verbal model into an algebraic model. 8. Solve the algebraic model. 9. Answer the question.

Sharing the Driving In Exercises 10–14, use the following information. You and a friend share the driving on a 300 mile trip. Your friend drives for 3 hours at an average speed of 52 miles per hour. How fast must you drive for the remainder of the trip if you want to reach your hotel in 3 more hours? 10. Write a verbal model. 11. Assign labels to the parts of the verbal model. 12. Use the labels to translate the verbal model into an algebraic model. 13. Solve the algebraic model. 14. Answer the question.

Time Management In Exercises 15–19, use the following information. You need to do an experiment at home for your science class and write a lab report on your findings. The experiment involves trials that take 5 minutes each to perform. 1 You want to watch a basketball game that starts in 12 hours. If it takes about 30 minutes to write the lab report, how many trials can you perform before the game starts? 15. Write a verbal model. 16. Assign labels to the parts of the verbal model. 17. Use the labels to translate the verbal model into an algebraic model. 18. Solve the algebraic model. 19. Answer the question. Copyright © McDougal Littell Inc. All rights reserved.


69

Answer Key Practice C 1. Distance Rate Time 2. Distance 2198 miles Rate 2 miles per hour Time t hours 3. 2198 2t 4. 1099 5. 1099 hours 6. 9.141 meters per second 7. $138,000 8. 0.5 hour 9. $175.92 10. $250 11. 3.1 ft

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LESSON

1.5

NAME _________________________________________________________ DATE ___________


Wagon Trains In Exercises 1–5, use the following information. In the 1800s settlers traveled across the country in wagon trains. A wagon train consisted of a group of families who traveled together. Each family had its own wagon and oxen or mules to pull the wagons. The wagons followed each other in a long line called a wagon train. Wagon trains traveled at a rate of approximately 2 miles per hour. The distance between Buffalo, New York and Los Angeles, California is 2198 miles. How long would it have taken for the wagon trains to travel from Buffalo to Los Angeles? 1. Write a verbal model. 2. Assign labels to the parts of the verbal model. 3. Use labels to translate the verbal model into an algebraic model. 4. Solve the algebraic model. 5. Answer the question 6. 100-Meter Dash

In 1996 Gail Devers won the 100-meter dash in the Olympic Games. Her time was 10.94 seconds. What was her speed in meters per second? Round your answer to 4 significant digits.

7. Commission

A salesman’s salary is $18,500 per year. In addition, the salesman earns 5% commission on the year’s sales. Last year the salesman earned $25,400. How much was sold that year?

8. Visiting Friends

Your friend’s family moved to a town 300 miles from where you live. You and your friend decide to meet halfway between the two towns to visit. Your friend averages 50 miles per hour on his trip. You average 60 miles per hour on your trip. If you and your friend leave at the same time, how much earlier do you arrive at the same meeting place?

9. Wallpaper Project

You want to wallpaper a room that will require 320 square feet of wallpaper. The wallpaper you selected costs $21.99 per roll. Each roll will cover 40 square feet. How much will your project cost?

10. Soccer Trophies

After winning the league title, a soccer team receives a team trophy as well as individual trophies. The table gives the cost of trophies at a local store. Trophy Total cost

Team $40

1 $50

2 $60

3 $70

4 $80

5 $90

Determine the total cost of giving trophies to a team with 21 members. 11. Area Rug

A circular rug covers about 30 square feet. Use the guess, check, and revise method to approximate the radius of the rug to the nearest tenth of a foot.

70



Answer Key Practice A 1. E 2. C 3. B 4. F 5. A 6. D 7. no 8. yes 9. yes 10. yes 11. no 12. yes 13. x < 2 14. x ≥ 7 15. x ≥ 3 16. x > 3 17. x ≤ 10 18. x > 3 19. x > 4 20. x < 9 21. x ≥ 21 22. 8 < x < 3 23. 3 < x < 2 24. 4 ≤ x ≤ 11 25. x < 6 or x > 2 26. x < 3 or x > 10 27. x ≤ 4 or x ≥ 1 28. x > 4 29. x ≥ 1 0

1

2

3

4

5

6

30. x ≤ 5 0

1

2

32. x ≤

3 2 1

0

1

2

3

4

5

6

6 5 4 3 2 1

0

31. x < 6 3

4

5

6

2 3

0

1

2

3

33. x < 3 2 3

3 2 1

0

1

2

3

34. 221,463 ≤ x ≤ 252,710 35. 80 ≤ x ≤ 98

36. 0.4 ≤ x ≤ 8

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LESSON

NAME _________________________________________________________ DATE ___________

1.6


Match the inequality with its graph. 1. x ≤ 0

2. 2 < x < 3

3. x < 2 or x > 3

4. x > 2

5. x < 3

6. x ≥ 0

A.

B. 0

1

2

3

4

3 2 1

0

1

2

3

4

D.

C. 3 2 1

0

1

2

3

4

3 2 1

0

1

2

3

4

E.

3 2 1

0

1

2

3

4

F. 3 2 1

0

1

2

3

4

Decide whether the given number is a solution of the inequality. 9. 2x 3 ≤ 3; 4

7. 3x 2 < 5; 1

8. 5x 9 > 4; 4

10. 5 3x ≥ 7; 4

11. 6x 2 < 14; 2

12. 2 ≤ x 2 ≤ 5; 3

14. x 5 ≥ 2

15. 4 ≤ 7 x

Solve the inequality. 13. x 3 < 1

1 2x

18. 3x > 9

≤ 5

16. 2x > 6

17.

19. 3x < 12

20. 2x > 18

21. 3x ≤ 7

22. 3 < x 5 < 2

23. 4 < 2x < 6

24. 0 ≤ x 4 ≤ 7

25. x 1 < 5 or x 1 > 3

26. x 2 < 1 or x 2 > 8

27. 7x ≤ 28 or 7x ≥ 7

1

Solve the inequality. Then graph the solution. 28. 2x 3 > 11 31.

1 3x

3 < 5

29. 3 2x ≤ 5 32. 7

3 2x

≥ 6

30. 3 x ≥ 2 33. 1 2x > x 10

34. Moon’s Orbit

As the moon orbits Earth, the closest it ever gets to Earth is 221,463 miles. The farthest away it ever gets is 252,710 miles. Write an inequality that represents the various distances of the moon from Earth.

35. January Temperatures

The highest January temperature in the United States was 98 F in Laredo, Texas in 1954. The lowest January temperature in the United States was 80 F in Prospect Creek, Alaska in 1971. Write an inequality that represents the various temperatures in the United States during January.

36. Bird Eggs

The largest egg laid by any bird is that of the ostrich. An ostrich egg can reach 8 inches in length. The smallest egg is that of the vervain hummingbird. Its eggs are approximately 0.4 inch in length. Write an inequality that represents the various lengths of bird eggs.



83

Lesson 1.6

3 2 1

Answer Key Practice B 1. 1

2. 0

1

2

3

4

4 3 2 1

5

3. 0

1

2

8

9

10

5 4 3 2 1

0

1

5

6

7

6. 1

2

3

7. x < 6

4

5

6

8. x < 7

10. x < 2 11. x ≥ 1 14. x ≥ 2

15. x < 2

17. x ≤ 0

18. x ≤ 3

3 2

9. x > 2 12. x ≤ 6

23. 8 ≤ x ≤ 12

5

16. x > 9

22. x < 2 or x > 4

24. x < 7 1

26. x ≤ 3

25. x > 9 8

9

13. x ≤ 1

20. 4 < x < 6

< x < 4

21. x ≤ 6 or x ≥ 16

7

2

4

5.

19.

1

4.

4 3 2 1

0

0

10 11 12 13

5 4 3 2 1

28. x ≤

27. x < 1 4 3 2 1

0

1

0

1

2 3 2 3

2 3 2 1

0

1

2

3

4 3 2 1

0

1

2

30. x ≤ 1

29. x < 3 6 5 4 3 2 1

1 3

0

31. 1855 ≤ x ≤ 7123 32. x 444 ≥ 540; x ≥ 96 33. d ≤ 652; d ≤ 130 35. 0.4 ≤ x ≤ 8

34. 80 ≤ x ≤ 98

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LESSON

NAME _________________________________________________________ DATE ___________

1.6


Graph the solution of the inequality. 1. x < 4

2. x > 3

3. x ≤ 1

4. x ≥ 7

5. 3 < x < 5

6. x ≤ 4 or x ≥ 1

7. x 8 < 14

8. 11x > 77

9. 2x 1 > 5

10. 3x 2 < 8

11. 5x 8 ≥ 3

12. 2x 4 ≤ 7

13. x 5 ≥ 6

14. 4 2x ≤ 0

15. 3x 5 > 1

16. 7 9x < 12

17. 5x 1 ≥ 1

18. 3x 1 ≤ 2x 2

19. 2 < 2x 5 < 3

20. 4 < 2 x < 6

21. x 4 ≤ 2 or x 4 ≥ 12

Lesson 1.6

Solve the inequality.

22. x 1 < 3 or x 1 > 3

23. 3 ≤

1 2x

1 ≤ 5

1

24. 2x 3 < 8

Solve the inequality. Then graph the solution. 2

25. 3x 5 > 1 28. 7

3 2x

≥ 6

26. 6 3x ≤ 5

27. 3 x > 2

29. 1 2x > x 10

30. 24 x ≥ 6

31. Extreme Points

The northernmost point of the United States is Point Barrow, Alaska. It lies on the 7123 latitude line. The southernmost point of the United States is Ka Lae, Hawaii. It lies on the 1855 latitude line. Write an inequality that represents the various latitudes of locations in the United States.

32. Exam Grades

The grades for a course are based on 5 exams and 1 final. All six of the tests are worth 100 points. In order to receive an A in the course, you must earn at least 540 points. Your grades on the 5 exams are as follows: 87, 95, 92, 81, and 89. Write an inequality that represents the various grades you can earn on the final and still get an A. Solve the inequality.

33. Speed Limit

The speed limit on a certain stretch of highway is 65 miles per hour. Write an inequality that represents the distances you can travel if you obey the speed limit for 2 hours. Solve the inequality.

34. January Temperatures

The highest January temperature in the United States was 98 F in Laredo, Texas in 1954. The lowest January temperature in the United States was 80 F in Prospect Creek, Alaska in 1971. Write an inequality that represents the various temperatures in the United States during January.

35. Bird Eggs

The largest egg laid by any bird is that of the ostrich. An ostrich egg can reach 8 inches in length. The smallest egg is that of the vervain hummingbird. Its eggs are approximately 0.4 inch in length. Write an inequality that represents the various lengths of bird eggs.

84



Answer Key Practice C 1 1. x < 1 2. x ≥ 2 3. 12 ≤ x ≤ 18 17 1 4. x < 2 or x > 2 5. x > 2 6. x > 8 13 7. x ≤ 8 8. x ≤ 3 9. 3 < x < 1 10. 0.84 < x < 1.76 11. 2 ≤ x ≤ 2.8 2 8 12. 10 < x < 16.5 13. x < 3 or x > 3 3 14. x ≥ 6 15. x ≤ 2 16. No solution 17. No solution 18. All real numbers 19. 57.9 < d < 5900 20. Calm 0≤S < 1 Light Air 1 ≤ S ≤ 3 Light Breeze 4 ≤ S ≤ 7 Gentle Breeze 8 ≤ S ≤ 12 Moderate Breeze 13 ≤ S ≤ 18 Fresh Breeze 19 ≤ S ≤ 24 Strong Breeze 25 ≤ S ≤ 31 Near Gale 32 ≤ S ≤ 38 Gale 39 ≤ S ≤ 46 Strong Gale 47 ≤ S ≤ 54 Storm 55 ≤ S ≤ 63 Violent Storm 64 ≤ S ≤ 72 Hurricane S > 72 21. 1.50 0.50x ≤ 4.25; x ≤ 5.5

You can play 5 games.

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LESSON

1.6

NAME _________________________________________________________ DATE ___________


Solve the inequality. 1 x1 ≤ 8 2

1. 4 2x > x 1

2. 5x 7 ≤ 7x 6

3. 5 ≤

4. 1 2x < 3 or 3 x > 5

5. 3x 5 > x 2

6. 4 3x < 5x 1

7. 2x 1 ≤ 62 x 3

8. 7 3x ≥ 2x 4

9. 4 < 3x 2 1 < 2

13.

11. 3.2 ≤ 2.5x 1.8 ≤ 5.2

3 3 x 1 < 0 or x 1 > 5 2 2

14.

2 x 8 ≤ 3x 2 3

12. 2.5 < 0.2x 0.5 < 3.8 15.

5 1 3 x ≥ 4 6 2

Decide which inequalities have no solution and which inequalities are true for all real numbers. 16. 2x 7 < 2x 3

17. 3x 2 4x > x 2x 8

18. 54 x ≤ 4x 20 x 19. Distance from the Sun

Mercury is the closest planet to the sun. Mercury is 57.9 million kilometers from the sun. Pluto is the farthest planet from the sun. Pluto is 5900 million kilometers from the sun. Write an inequality that represents the various distances from a planet to the sun.

20. Beaufort Scale

The Beaufort Scale is a system for describing the speed of wind. The table below shows the 13 descriptions of the Beaufort Scale. Write an inequality for each of the descriptions. Description Calm Light Air Light Breeze Gentle Breeze Moderate Breeze Fresh Breeze

Speed, S under 1 mph 1–3 mph 4–7 mph 8–12 mph 13–18 mph 19–24 mph

Description Strong Breeze Near Gale Gale Strong Gale Storm Violent Storm Hurricane

Speed, S 25–31 mph 32–38 mph 39–46 mph 47–54 mph 55–63 mph 64–72 mph over 72 mph

21. Video Arcade

You have $4.25 to spend at a video arcade. Some games cost $0.75 to play and other games cost $0.50 to play. You decide to play 2 games that cost $0.75. Write and solve an inequality to find the possible number of $0.50 video games you can play.



85

Lesson 1.6

10. 2.2 < 5x 2 < 6.8

Answer Key Practice A 1. x 2 7, x 2 7 2. 2x 1 5, 2x 1 5 3. 5x 11 6, 5x 11 6 1 1 4. 2t 3 1, 2t 3 1 5. 5 t 3, 5 t 3 6. 1 4t 9, 1 4t 9 7. 5x 4 6, 5x 4 6 8. 3x 4 8, 3x 4 8 9. 2x 3 7, 2x 3 7 10. 3x 7 5, 3x 7 5 1 1 11. x 2 9, x 2 9 12. 2.3 5.7x 11.4, 2.3 5.7x 11.4 13. 9, 9 14. 25, 25 15. 4, 4 16. 8, 2 20. 3,

13 3

17. 2,

10 3

1

21. 1, 7

18. 10, 4 19. 6, 10 22. 3 < x 7 < 3

23. 10 ≤ 2x 4 ≤ 10 24. 7 < 5 3x < 7 25. x 4 < 5 or x 4 > 5 26. 5x 1 ≤ 4 or 5x 1 ≥ 4 27. 2 x < 9 or 2 x > 9 1

28. 3 ≤ 3x 5 ≤ 3

29. 9 < 2 8x < 9

30. 3.5 2.1x ≤ 1.5 or 3.5 2.1x ≥ 1.5 3

31. 4x 1 ≥ 2 or 32. 33. 35. 37. 39. 41. 43.

3 4x

1 ≤ 2 3.3 < 2.3x 1.7 < 3.3 54 ≤ 23 14x ≤ 54 34. 8 < x < 8 x < 6 or x > 6 36. 3 ≤ x ≤ 3 4 < x < 6 38. 3 ≤ x ≤ 53 1 < x < 9 40. x ≤ 11 or x ≥ 5 x < 2 or x > 3 42. x < 73 or x > 5 x ≤ 27 44. x 58 ≤ 15

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LESSON

NAME _________________________________________________________ DATE ___________

1.7


Rewrite the absolute value equation as two linear equations.

1. x 2 7 4.

1 2t

3 1

3x 7 5

7. 5x 4 6 10.

5 t 3 3x 4 8

1 4t 9 2x 3 7 2.3 5.7x 11.4

2. 2x 1 5

3. 5x 11 6

5.

6.

8.

9.

9

11. x

1 2

12.

Solve the equation.

x 3 5

3x 2 8 11 3t 2

2x 6 14 7t 3 4

13. x 9

14. x 25

15. t 4

16.

17.

18.

19.

1 2t

4 1

20.

21.

Lesson 1.7

Rewrite the absolute value inequality as a compound inequality. 22. x 7 < 3

23. 2x 4 ≤ 10

24. 5 3x < 7

25.

26.

5x 1 ≥4 2 8x < 9 2.3x 1.7 < 3.3

27. 2 x > 9

3x 2 ≤ 7 2x 1 > 5

28. 31.

x 4 > 5

x 5 x 1 1 3

≤ 3

29.

3 4

≥ 2

32.

30. 33.

3.5 2.1x ≥ 1.5

2 3

14x ≤

5 4


x 5 < 1 x 8 ≥ 3

4 x < 5 11 3x > 4

34. x < 8

35. x > 6

36. x ≤ 3

37.

38.

39.

40.

41.

43. Touring a Ship

The diagram below shows the water line of a large ship. The ship extends 27 feet above the water and 27 feet below the water. Suppose you toured the entire ship. Write an absolute value inequality that represents all the distances you could have been from the water line.

42.

44. Water Temperature

Most fish can adjust to a change in the water temperature of up to 15 F if the change is not sudden. Suppose a lake trout is living comfortably in water that is 58 F. Write an absolute value inequality that represents the range of temperatures at which the lake trout can survive.

27 ft

0 ft

27 ft

98



Answer Key Practice B 1. yes 2. yes 3. no 4. no 5. no 6. yes 10 7. 8, 2 8. 2, 3 9. 10, 4 10. 6, 10 13 1 11. 3, 3 12. 1, 7 13. 0, 7 14. 12, 15 2 5 15. 5, 2 16. 4 < x < 6 17. 3 ≤ x ≤ 3 18. 1 < x < 9 19. x ≤ 11 or x ≥ 5 7 20. x < 2 or x > 3 21. x < 3 or x > 5 22. 4 ≤ x ≤ 16 23. x ≤ 24 or x ≥ 36 1 24. 2 < x < 1 25. x ≤ 27 26. x 58 ≤ 15 27. x 12.25 ≤ 8.75 28. x 35 ≤ 5

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LESSON

NAME _________________________________________________________ DATE ___________

1.7


Decide whether the number is a solution of the equation.

5 3x 8;

1. 5x 4 6; 2 4.

1

2. 3x 4 8; 4 5.

1 2x

2 4; 1

3. 2x 3 7; 2

6. 3

4; 28

1 4x

Solve the equation.

7. x 3 5 10.

1 2t

4 1

13. 2x 7 7

11 3t 2

8. 3x 2 8 11.

2

14. 1 3x 9

7t 3 4 4 5x 6

9. 2x 6 14 12. 15.


x 8 ≥ 3

2x 1 >5

4x 1 < 3

16. x 5 < 1

17. 3x 2 ≤ 7

18. 4 x < 5

19.

20.

21. 11 3x > 4

22.

1 2

≤ 5

1

23. 2 3x ≥ 10

25. Touring a Ship

The diagram below shows the water line of a large ship. The ship extends 27 feet above the water and 27 feet below the water. Suppose you toured the entire ship. Write an absolute value inequality that represents all the distances you could have been from the water line.

24.

26. Water Temperature

Most fish can adjust to a change in the water temperature of up to 15 F if the change is not sudden. Suppose a lake trout is living comfortably in water that is 58 F. Write an absolute value inequality that represents the range of temperatures at which the lake trout can survive.

27 ft

0 ft

27 ft

27. Hours of Daylight

According to the Old Farmer’s Almanac, the hours of daylight in Fairbanks, Alaska, range from approximately 1 32 hours in mid-December to approximately 21 hours in mid-June. Write an absolute value inequality that represents the hours of daylight in Fairbanks.


28. Elephant Longevity

On average an elephant will live from 30 to 40 years. Write an absolute value inequality that represents the typical ages of an elephant.


99

Lesson 1.7

x 3

Answer Key Practice C 18 11 5 11 13 1. 1, 2 2. 2, 5 3. 8 , 8 4. 3 , 3 4 28 5. 2, 10 6. 5, 5 7. a, a 8. b a, b a 9. b a, b a b b 10. , 11. ab, ab 12. 2a, 0 a a 13. x < 1 or x > 7 14. 1 ≤ x ≤ 2 15 3 15. x < 2 or x > 2 34 38 16. No solution 17. x ≤ 5 or x ≥ 5 2 18. All reals 19. x 5 20. No solution 21. All reals 22. D x ≤ 0.001; 12.999 ≤ x ≤ 13.001; 8.999 ≤ x ≤ 9.001; 5.999 ≤ x ≤ 6.001 23. x 2978.95 ≤ 2921.05 24. x 10 ≤ 3

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LESSON

NAME _________________________________________________________ DATE ___________

1.7


Solve the equation.

1. 6x 3 9

4. 3x 4 1

2. 4 5x 14

5. 24 x 12

3. 6.

3 2x 2 4 4 1 3 x 5 4 5

Solve for x. Assume that a and b are positive numbers.

ax b

7. x a 10.

8. x a b 11.

x b a

12.

Solve the inequality. If there is no solution, write no solution.

16. 5x 2 < 4

17. 6

1 5 x ≥ 6 3

19.

20. x 7 < 0

Lesson 1.7

13. 4 x > 3

2 5x ≤ 0

14. 8x 12 ≤ 4

|x a a

9. x a b

15.

2 x3 > 2 3

18. 1

3 x > 1 4

21. 2x 3 ≥ 0

22. Machine Shop

Three circles have to be cut into a piece of metal. The specifications state that each of the diameters must be within 0.001 centimeter of the given measurements. Let D represent the given measurement and let x represent the actual diameter of the circle. Write an absolute value inequality that describes the acceptable diameters of the circle. If the circles are to be 13 centimeters, 9 centimeters, and 6 centimeters, describe the acceptable diameters of each circle.

23. Distance to the Sun

The distance to the sun from the nine planets ranges from 57.9 million kilometers to 5900 million kilometers. Write an absolute value inequality that describes the possible distances from a planet to the sun.

24. Distance

Your house is 10 miles away from your school. Your friend’s house is 3 miles from your school. Write an absolute value inequality that describes the possible distances from your house to your friend’s house.

100



Answer Key Test A 1.

2.

y

y

1 x

1

1 x

1

The relation is a function. 3. 0 4. 0 5. 3 6.

The relation is not a function. 7.

y

y 1

1

1

8. y 2x

x

9. y 2x 1

11. y x 1 13.

14.

1

y

1 1

x

16.

y

1

1

x

1

x

y

1 1

17.

10. y x 1

12. y 2x 3

y

15.

x

1 1

x

18.

y

2

(0, 1) 1

x

19. 2s 4a 2800

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CHAPTER

NAME _________________________________________________________ DATE ____________

2

Chapter Test A For use after Chapter 2

Graph the relation. Then tell whether the relation is a function. 1. x

y

0 1

1 2

2 3

1 0

2 1

Answers 1.

Use grid at left.

2.

Use grid at left.

y

1 x

1

3. 4. 2. x

y

3 3

4 4

5 5

0 1 0 1

3 6

5.

y

x

1

7.

Use grid at left.

9. 10.

Evaluate the function for the given value of x.

Use grid at left.

8.

1

3. f x x 3 when x 3

6.

4. f x x2 3x when x 3

5. f x x 2 when x 5

11. 12.

Graph the equation. 2

6. x 1

7. y 3x 2 y

y

1

1 1

x

1

x

Write an equation of the line that has the given properties. 1

y-intercept: 0

9. slope: 2,

point: 1, 3

10. points: 2, 1,

Review and Assess

8. slope: 2,

3, 2

11. Write an equation of the line that passes through 4, 3 and is

parallel to the line y x 1.


perpendicular to the line y 12x 1.



119

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CHAPTER

NAME _________________________________________________________ DATE ____________

2

Chapter Test A

CONTINUED

For use after Chapter 2

Graph the inequality in a coordinate plane. 13. y ≥ 2

13.

Use grid at left.

14. y ≥ 2x 1

14.

Use grid at left.

y

15.

Use grid at left.

16.

Use grid at left.

17.

Use grid at left.

18.

Use grid at left.

y

1

1 1

x

1

x

19.

Graph the function. 15. f x

2,0, ifif xx >≤ 00

16. f x

xx 2,2,

y

if x ≥ 0 if x < 0

y

1

1 1

x

17. f x x 1

1

18. f x

y

1 2

x

x 4

y

1 1

1 1

x

x

Student tickets for a football game cost $2 each. Adult tickets cost $4 each. Ticket sales at last week’s game totaled $2800. Write a model that shows the different numbers of students and adults who could have attended the game.

Review and Assess

19. Ticket Prices

120



Answer Key Test B 1.

2.

y

y

1 1

x

1

x

1

The relation is a function. 3. 17 4. 3 5. 7

The relation is not a function.

6.

7.

y

y

1

1 1 x

8.

9.

y 4

y

1

x

4

x

1

x

1

1

10. y 2 x 2

11. y x 5

13. y x 5 y 15.

14. y 2 x 2 16.

12. y x 1

1

2

2

2

17.

y

x

18.

y

y

1

1 1

19.

x

2

x

1 x

20.

y

1

y

1 1

x

1

(0, 3)

21. n ≥ 300

22. (a)

3 4

(b) 20 feet

x

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CHAPTER

NAME _________________________________________________________ DATE ____________

2

Chapter Test B For use after Chapter 2

Graph the relation. Then tell whether the relation is a function. 1. x

3 y 2

1 0

0 1

2 2

2. x

3 3

y

3 2

3 3

4 1

0 1

Answers 3 4

2 2

1.

Use grid at left.

2.

Use grid at left.

y

y

3.

1 1

x

1

4. x

1

5.

Evaluate the function for the given value of x. 3. f x 25 2x when x 4 5. f x

x2

4. f x x 5 when x 2

5x 1 when x 1

Graph the equation. 6. y

1 2x

6.

Use grid at left.

7.

Use grid at left.

8.

Use grid at left.

9.

Use grid at left.

10.

3

7. y 2

11.

y

y

12.

1

1 1 x

8. 4x y 8

x

1

9. x 2

y

y

4

4

1

x

1

x

Review and Assess


10. slope: 2,

y-intercept: 2


11. slope: 1

point: 2, 3

12. points:

3, 4, 1, 0


121

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CHAPTER CHAPTER

2

Chapter Test B

CONTINUED



parallel to the line x y 4.



Graph the inequality in a coordinate plane. 15. y ≥ 1 16. y > 2x 1 y

y

2

2

2

x

x

2

13. 14. 15.

Use grid at left.

16.

Use grid at left.

17.

Use grid at left.

18.

Use grid at left.

19.

Use grid at left.

20.

Use grid at left.

21. 22.

17. x 2y ≤ 0

18. x ≥ 2

(a) (b)

y

y

1

1 x

1

Graph the function. 1, if x > 0 19. f x 1, if x < 0

1 x

20. f x 4 x 3

y

y 4

1

4

Review and Assess

1

x

x

21. Profit

The sophomore class needs to raise money. They sell boxes of holiday cards at a profit of $2 per box. How many boxes must they sell to make a profit of at least $600? Express your answer as an inequality.

22. Roofs

A roof rises 3 units for every 4 units of horizontal run. (a) What is the slope of the roof? (b) If the roof is 15 feet high, how long is it?

122



Answer Key Test C 1.

2.

The relation is a function. 3. 5 4. 1 5. 7 6.

The relation is a function. 7.

3

8. y 4x 2 10. y x 5

1

9. y 2x 5 11. y x

12. y 2x 8

13.

14.

15.

16.

17.

18.

19. 8w 10x 3100; 150

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CHAPTER

NAME _________________________________________________________ DATE ____________

2

Chapter Test C For use after Chapter 2

Graph the relation. Then tell whether the relation is a function. 1.

x y

1 2

2 2

3 2

4 2

2.

5 2

x 3 2 y 3 3

y

1 3

Answers 0 3

1 3

1.

Use grid at left.

2.

Use grid at left.

y

1

3.

1 x

1

1

x

4. 5.


3. f x x when x 5

4. f x x2 x 1 when x 1

6.

Use grid at left.

7.

Use grid at left.

8. 9.

5. f x 2x2 4 1 when x 0

10.

Graph the equation. 11.

2

6. y 0

7. y 3x 1

12.

y

y

1 1

x 1 x

1


8. slope: 4,

y-intercept: 2

1

9. slope: 2,

point: 4, 3

10. points: 0, 5,

5, 0

11. Write an equation of the line that passes through 5, 5 and is Review and Assess

parallel to the line 2x 2y 3.





123

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CHAPTER

NAME _________________________________________________________ DATE ____________

2

Chapter Test C

CONTINUED


Graph the inequality in a coordinate plane.

13.

Use grid at left.

13. x ≥ 3

14.

Use grid at left.

15.

Use grid at left.

16.

Use grid at left.

17.

Use grid at left.

18.

Use grid at left.

14. x > 2y 1 y

y 1 1

x

1 1 x

19.

Graph the function.

x,

if

15. f x 2x, if

3x, if

2 < x < 2 x < 2 16. f x x > 2

1 2x 3 4x

2, if x ≥ 0 3, if x < 0 y

y

1 2

1 2

17. f x

1 2

x

x

x 2

18. f x

xx 11,,

y

if x > 0 if x < 0

y

1 1

1 1

x

x

Review and Assess

19. Car Wash

A local car wash charges $8 per wash and $10 per wash and wax. At the end of a certain day, the total sales were $3100. Write a model that shows the different numbers of the two types of car washes. Then find the number of wash and waxes there were if 200 were washes only.

124



Answer Key Cumulative Review 1. inverse property of addition 2. associative property of addition 3. distributive property 4. 7 5. 31 6. 32 7. 6 8. 6 1 7 9 10. 71 11. 49 12. 2 13. 2 14. 4 16. 6

15. 2

17. 9

C 2 20. x < 2 or x > 1

45.

2

0

2

6

4

2

0

2

x

1 1

3

47. y 2x 6 8

49. y 5 x

21. 3 < x < 5 3

4

52. y x 5

5

0

1

2

3

4

32 5

50. y 1

53. y

54. y 2x; 10

6

56. y

1 4 x;

54

51. x 4

7 35 3 x; 3

55. y 8x; 40 57. y x; 5

58. y 4x; 20 59.

5

7 4

48. y 4 x

60. y

or 83 25. 28 or 4 26. 10 or 3 2 27. x > 4 or x < 8 28. 3 ≤ x ≤ 2 16 29. 2 < x < 5 30. 2 31. 10 32. 4 33. 1 34. 7 35. 16 36. The relation is a function. 37. The relation is a function. 38. The relation is not a function. 39. parallel 40. perpendicular 24.

y 1

16 3

41.

x

1

9. 7

23. x < 1 or x > 4 4

1

1

18. b1 2A b2

2

22. 4 ≤ x ≤ 2

y

y

19. r

4

46.

42.

x

2

Sample answer: y 87x 37 61.

62. y

y

x

1

2

y

y

1 1 1 x

x

1

1

1

1

x

x 1

43.

63.

44.

64. y

y

y

y

1 1 1

1 x

1

x

1

1 1

x

x

Answer Key 65.

66. 1

y

67. 24

x

1

1

69. 3

68. 5

70. 2

71. 19 72. y

2 x

2

0, 7; up; same width 74. y

1 1

x

2, 1; down; narrower 76.

77. y

y 1 1

x

1 1

0, 2; up; wider

x

0, 4; up; same width

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CHAPTER

NAME _________________________________________________________ DATE ____________

2

Cumulative Review For use after Chapters 1–2

Tell what property the statement illustrates. (1.1) 1. 4 4 0

2. 2 5 7 2 5 7

3. 32 4 32 34





2

Evaluate the expression for the given value of x. (1.2) 9. x 2 2 when x 3

8. x 8 when x 2

11. 3x 4 x 1 when x 2

10. 3x 2 x 1 when x 5

Solve the equation. (1.3) 12. 2x 1 8

13. 2a 1 4a 8

14. 6x 4 10x

15. 4.5a 1.7 7.3

16. 32a 8 8a 12

17.

1 3x

4 79 x

Solve the formula for the indicated variable. (1.4) 18. Area of trapezoid

Solve for b1. A

19. Circumference of circle 1 2 b1

b2

Solve for r. C 2r

Solve the compound inequality. Graph its solution. (1.6) 20. 3x 7 > 10 or 2x > 4

21. 15 < 5x < 25

22. 0.3 ≤ 0.2x 0.5 ≤ 0.9

23.

3x 1 < 11 or 5x 2 < 7

Solve the absolute value equation or inequality. (1.7)

x 2 > 6

24. 3x 4 12

25.

27.

28.

12 x 6 8 6x 4 † 8

3 5x < 13

26. 7 2x 13 29.

Evaluate the function when x 2. (2.1) 30. f x x

31. gx 5x

32. rx x 2

33. gx 3x 5

34. hx 2x 2 1

35. jx x3 2x 2

Review and Assess

Use the vertical line test to determine whether the relation is a function. (2.1) 36.

37.

y

(2, 6) (2, 4) (5, 4)

y

(6, 5) 1

2 2

130

38.

y

x

(5, 1)


1

1 1

x

x


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CHAPTER

NAME _________________________________________________________ DATE ____________

2

Cumulative Review

CONTINUED

For use after Chapters 1–2

Tell whether the lines are parallel, perpendicular, or neither. (2.2) 40. Line 1: through 3, 4 and 5, 8

39. Line 1: through 1, 8 and 7, 9

Line 2: through 2, 5 and 10, 6


Graph the equations. (2.3) 2

42. y 3 x 5

41. y 3x 4 44. x 5

45. y

43. 3x 2y 4

23

46. 5x 10y 20

Write the equation of the line that passes through the given points. (2.4) 47. 4, 2 and 7, 8

48. 5, 2 and 3, 4

49. 4, 0 and 1, 8

50. 2, 1 and 3, 1

51. 4, 5 and 4, 9

52. 0, 5 and 5, 0

The variables x and y vary directly. Write an equation that relates the variables. Then find y when x 5. (2.4) 1

53. x 3, y 7

54. x 2, y 4

55. x 2, y 4

56. x 8, y 2

57. x 6, y 6

58. x 0.2, y 0.8

Draw a scatter plot of the data. Then approximate the best-fitting line for the data. (2.5) 59.

x y

3 7

3 3

0 1

1 1

1 5

2 1

4 5

Graph the inequality in a coordinate plane. (2.6) 60. 5x 2y > 10 63. y >

1 3x

3

61. 4x < 20

62. 8y > 10

64. 0.25x 1 > 2

65. 3x < 2 y

1

Evaluate the function for the given value of x. (2.7)

{

f(x) 3x, if x > 5 x 2, if x ≤ 5 66. f 3

67. f 8

68. f 3

69. f 5

70. f 0

71. f 3 19

y x 32

72. y x 7 75.


x 2

73. y x 8 76. y

1 2

74. y 2 x 2 1

77. y x 4


131

Review and Assess

Graph the function. Then identify the vertex, tell whether the graph opens up or down, and tell whether the graph is wider, narrower, or the same width as the graph of y x . (2.8)

Answer Key Practice A 1. domain: 1, 0, 2; range: 3, 6, 16 2. domain: 3, 4, 9; range: 9, 0 3. domain: 1, 2; range: 12, 6, 24 4.

5.

y

14.

y 1

1

1

x

1

x

1

x

x

1

y

16. 1

15.

y

17.

y

y 1

2 x

1

1

x

2

x

1

The relation is a function. 6.

The relation is not a function. The relation is a function.

y

18.

19.

y

y 1

2 2

x

1

x

1 x

1

7. The relation is a function. 8. The relation is not a function. 9. The relation is a function. 10.

x 2 1 0 y 1 1 3

1 5

20.

y

y

2 7

1 2 x

21.

3 1 1

12.

13.

y

x

y

1

0

Year 1

x

1 1

x

1994 1995 1996 1997

7 2

1993

4

y

1991 1992

9 2

5

2

1990

y

1

1988 1989

x 2 1 0

Scores

11.

U. S. Open Champion Scores 286 285 284 283 282 281 280 279 278 277 276 275 274 0

1986 1987

2

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LESSON

NAME _________________________________________________________ DATE ___________

2.1


Identify the domain and range. 1. Input

Output

2 1 0

2. Input

3 6 16

3 9 4

Output

3. Input

9 0

Output 12 24 6

1 2

4.

x 0 y 3

1 5

2 3

3 1

5.

4 0

6.

x 3 3 4 5 9 y 0 1 2 3 11

x 2 1 0 y 1 2 3

1 4

Lesson 2.1

Graph the relation. Then tell whether the relation is a function. 2 5

Use the vertical line test to determine whether the relation is a function. 7.

8.

y

9.

y

y

1 1

1

1

x

x

1

x

1

Complete the table of values for the given function. Then graph the function. 1

10. y 2x 3

x 2 1 0 y

11. y 2x 4

1

x 2 1 0 y

2

1

2

Graph the function. 12. y x 2

13. y x 3

14. y 3x 4

15. y 6x 2

16. y 4x 3

17. y 3x 5

18. y 8x

19. y 2

20. y 2x 5

1

21. U.S. Open Champions

The table shows the golf scores of the U.S. Open Champions from 1986 to 1996. Use a coordinate plane to graph these results. Year 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 Score 279 285 281 283 278 277 275 277 279 274 276



13


2.

y

y 2 x

2

1 1

x

The relation is a function.

The relation is not a function.

3. The relation is a function. 4. The relation is not a function. 5. The relation is a function. 6.

7.

y

y 2 2

x

1 1

8.

x

9.

y

y

1 1

1

10.

x

11.

y

1

x

1

x

1

x

y 1

1 1

12.

x

13.

y

1

1 1

14.

y

x

15. linear; 4 16. not linear; 2

y

17. linear; 2 x 1

1

18. not linear; 14 19. not linear;

1 2

20. linear; 5 21. 54; S3 represents the surface area of a cube with sides of length 3. 22. 6, 7, 9, 10 23. 6, 7, 9, 10

24. yes

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LESSON

NAME _________________________________________________________ DATE ___________

2.1


Graph the relation. Then tell whether the relation is a function. 1. x 2

y

1 0 5 6

0

1 0

2. x 2

2 3

y

4

1 0 1 3

2 2 1 8

1 2

Use the vertical line test to determine whether the relation is a function. 3.

4.

Lesson 2.1

y

5.

y

y

3

3

1

x 1

x

1

1

x

Graph the function. 6. y 5x 1

7. y 3x 7 10. y

9. y x 2 12. y 2x 3

1 2x

3

8. y 2x 11. y 3x 5 1

14. y 3x 1

13. y 2

Decide whether the function is linear. Then find the indicated value of f x. 15. f x x 7; f 3

16. f x x3 x 2; f 1

18. f x 3x 1 ; f 5

19. f x

3 ; f 4 x2

17. f x 4 3x; f 2

3 4

20. f x x 1; f 8

21. Geometry

The surface area of a cube with side length x is given by the function Sx 6x2. Find S3. Explain what S3 represents.

Statistics

In Exercises 22–24, use the following information. The table below shows the number of games won and lost by the teams in the Eastern Division of the NFL’s National Football Conference for the 1996 season. Team Dallas Cowboys Philadelphia Eagles Washington Redskins Arizona Cardinals New York Giants

Won, x 10 10 9 7 6

Lost, y 6 6 7 9 10

22. What is the domain of the relation? 23. What is the range of the relation? 24. Is the number of wins a function of the number of losses? 14



Answer Key Practice C 1. The relation is not a function. 2. The relation is a function. 3. The relation is not a function. 4. First quadrant 5. Second quadrant 6. Third quadrant 7. Fourth quadrant 8.

9.

y

y 1 x

1 1 1

10.

x

11.

y

y

1

1 x

1

12.

1

13.

y

x

y

1 1

1

x

x 1

15. not linear; 1

14. linear; 16

16. not linear; 0 18. not linear;

56

17. not linear; 49 19. not linear; 2

20. Domain 8.3, 8.4, 8.6, 8.7, 8.9

21.

Deaths (thousands)

Range 1530, 2000, 2990, 5000, 10,700, 20,000, 28,000, 100,000, 200,000 y 200 150 100 50 0

0

8.3

8.5 8.7 Magnitude

8.9 x

22. No. For each input there is not exactly one out-

put. For example 200,000, 28,000, and 5000 are all outputs for the input 8.3.

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LESSON

NAME _________________________________________________________ DATE ___________

2.1


Tell whether the relation is a function. 1. Input

2.

Output 1 5 5 6

3 2 5 4

x 1 y 0

2 0

4 0

7 0

3.

0 0

y

1 1

x

Lesson 2.1

State the quadrant in which each point lies. Assume that a and b are positive numbers. 4. a, b

5. a, b

6. (a, b

7. a, b

Graph the function. 8. y 3x 5

9. y 3

1 x2 2

12. y 4

11. y

10. y 4 7x

3 x 4

13. y

Decide whether the function is linear. Then find the indicated value of f x. 14. f x 7x 2, f 2

15. f x x2 3x 1, f 3

17. f x x 32, f 4

18. f x

x7 , f 2 3x

3 x 5

16. f x x x, f 5 19. f x 2x3 4, f 1

Earthquakes In Exercises 20–22, use the table below which shows 10 of the worst earthquakes of the 20th century. Location (Year) Chile (1960) India (1950) Japan (1946) Chile (1939) India (1934) Japan (1933) China (1927) Japan (1923) China (1920) Chile (1906)

Magnitude, x 8.3 8.7 8.4 8.3 8.4 8.9 8.3 8.3 8.6 8.6

Deaths, y 5000 1530 2000 28,000 10,700 2990 200,000 200,000 100,000 20,000

20. Identify the domain and range of the relation. 21. Graph the relation. 22. Is the number of deaths a function of the magnitude of an earthquake? Explain. Copyright © McDougal Littell Inc. All rights reserved.


15

Answer Key Practice A 1. 2 2. 0 3. 1 4. 2 5. 3 6. 8 4 1 1 7. 2 8. 3 9. 3 10. rises 11. is horizontal 12. falls 13. rises 14. falls 15. is vertical 16. neither 17. neither 18. perpendicular 19. parallel 20. perpendicular 21. neither 3 1 22. 12.5 quarts per hour 23. Yes, 200 < 64

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LESSON

NAME _________________________________________________________ DATE ____________

2.2


Estimate the slope of the line. y

1.

y

2.

1

y

3.

1 1

x

1 1

x

x

1

Find the slope of the line passing through the given points. 4. 2, 3, 5, 9

5. 1, 4, 3, 2

6. 2, 7, 3, 1

7. 5, 1, 7, 5

8. 11, 0, 4, 5

9. 3, 4, 0, 0

Decide whether the line with the given slope rises, falls, is horizontal, or is vertical.

13. m

11. m 0

2 3

14. m

12. m 7

4 5

Lesson 2.2

10. m 2

15. m is undefined.

Tell whether the lines with the given slopes are parallel, perpendicular, or neither. 16. Line 1: m 2

18. Line 1: m

17. Line 1: m 5

Line 2: m

1 5

Line 2: m

8 3

20. Line 1: m

1 3

21. Line 1: m

2 3

Line 2: m 2 19. Line 1: m 4

Line 2: m 4

Line 2: m 3

Line 2: m

3 8

2 3

22. Picking Strawberries

One afternoon your family goes out to pick strawberries. At 1:00 P.M., your family has picked 3 quarts. Your family finishes picking at 3:00 P.M. and has 28 quarts of strawberries. At what rate was your family picking?

23. Ramp

The specifications of a ramp that leads onto a loading dock state 1 that the slope of the ramp must be no steeper than 64. If the ramp begins 200 feet from the base of the loading dock and the dock is 3 feet tall, does the ramp’s slope meet the specification? 3 ft 200 ft



27

Answer Key Practice B 1 3 3 1. 2 2. 1 3. 3 4. 2 5. 1 6. 2 7. Line 1: m 1; Line 2: m 4; Line 2 is steeper than Line 1. 8. Line 1: m 2; Line 2: m 3; Line 2 is steeper than Line 1. 1 9. Line 1: m 2; Line 2: m 1; Line 2 is steeper than Line 1. 1 1 10. Line 1: m 3; Line 2: m 8; Line 1 is steeper than Line 2. 11. m 1; rises 12. m 0; is horizontal 1 13. m 4; falls 14. m is undefined; is 5 vertical 15. m 3; falls 16. m 0; is horizontal 17. perpendicular 18. perpendicular 19. neither 1 20. perpendicular 21. 12 22. 0.85 ticket per minute

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LESSON

2.2

NAME _________________________________________________________ DATE ____________


Find the slope of the line passing through the given points. 1. 4, 5, 2, 9

2. 1, 4, 5, 0

3. 3, 5, 6, 2

4. 2, 7, 4, 4

5. 0, 8, 3, 5

6.

12, 34, 32, 94

Tell which line is steeper. 7. Line 1: through 2, 1 and 3, 6



Line 2: through 4, 5 and 2, 3 9. Line 1: through 0, 3 and 2, 4




Lesson 2.2

Find the slope of the line passing through the given points. Then tell whether the line rises, falls, is horizontal, or is vertical. 11. 4, 2 and 3, 3

12. 9, 2 and 3, 2

13. 3, 5 and 5, 3

14. 7, 5 and 7, 8

15. 10, 5 and 4, 15

16. 0, 4 and 3, 4

Tell whether the lines are parallel, perpendicular, or neither. 17. Line 1: through 3, 2 and 1, 5






21. Mountainside

The halfway point of a tunnel through a mountain is 1 miles from either end of the tunnel. The mountain is 660 feet 8 mile high. Find the slope of the side of the mountain. 3 2

1 mi 8

3 mi 2

22. Prom Tickets

You volunteered to take a shift selling prom tickets during your morning study hall. When your shift began at 11:00 A.M., 50 tickets had been sold. At 11:40 A.M., when your shift ended, 84 tickets had been sold. At what rate did you sell prom tickets?

28



Answer Key Practice C 1.

2 5

2.

7. rises 11. Line 1

3 2

17

1

3. 11

4. 4

8. is vertical 12. Line 2

14. They are equal.

5. 5

9. falls

10. Line 2

13. Line 1

15. They are negative reciprocals of each other. 16. vertical lines 964 180 1000 55 17. horizontal lines 18. 21 ; 17 19. 755; 151 y 20. This is true for an equilateral triangle m 3 m 3 of any size.

m0

x

16

6. 17

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LESSON

2.2

NAME _________________________________________________________ DATE ____________


Find the slope of the line passing through the given points. 1. 6, 3, 4, 1 4.

2. 5, 3, 7, 6

1, 13, 3, 23

5.

35, 3, 65, 0

3.

15, 35, 34, 14

6. 5, 2, 12, 14

Decide whether the line passing through the given points rises, falls, is horizontal, or is vertical. 7. 9, 11, 5, 5

8. 1, 6, 1, 7

9. 7, 0, 1, 12

Determine which line is steeper. 10. Line 1: through 3, 7 and 6, 2




Line 2: through 4, 3 and 1, 3 Lesson 2.2

14. Parallel Lines


If two nonvertical lines are parallel, what do you know about their slopes?

15. Perpendicular Lines

If two nonvertical lines are perpendicular, what do you know about their

slopes? 16. Vertical Lines

All vertical lines are parallel to what type of line?

17. Vertical Lines

All vertical lines are perpendicular to what type of line?

18. Washington Monument

The Washington Monument is 555 feet tall. The monument is composed of a 500-foot pillar topped by a 55-foot pyramid. The base of the pillar is 55 feet wide. The base of the pyramid is 34 feet wide. Approximate the slope of the sides of the pillar and the slope of the pyramid.

19. Pyramids of Egypt

The sides of the base of the largest pyramid, Khufu, has length 755 feet. The height of Khufu was originally 482 feet, but now is approximately 450 Feet. Find the slope of a side of the pyramid at its original size and at its present size.

20. Equilateral Triangles

An equilateral triangle has the same side lengths and angle measures. Draw an equilateral triangle on a coordinate plane such that one of the vertices is the origin. Approximate the slopes of the sides of your triangle. What are the slopes of the sides of any equilateral triangle in this position?



29

Answer Key Practice A 1.

18. 2.

y

y

19.

y 1

y

(0, 2) x

1

(2, 0)

1

x

(5, 0)

1 1

(0, 4)

1 1

x

x

1

20. 3.

4.

y

21.

y

y

y

(

1

1 2

,0

)

1

1 x

1

1

1

(2, 0)

1

x

x

(

(0, 3)

3 0, 2

)

x

1

22. 5.

6.

y

y

23.

y

y

1 1

1

x

1

x

1

x

1

x

1 1

7.

x

1 1

x

24. 8.

y

25.

y

y

y

1

1

1 x

1

1

x

1 x

1

26. 10. m 3; b 1

y

9.

27.

y

y

1 1

x

1

1 1

x

28. 11. m 4; b 7 13. m 8; b 2 15. m

15;

16.

y

12. m 6; b 4 10

x

(0, 1) (3, 0)

1

C

50 50

y

(4, 0)

(0, 2)

y

10

b 3

1

30. 100

5

14. m 3; b 1 17.

1

29. 3 31.

F

1

x

x

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LESSON

NAME _________________________________________________________ DATE ____________

2.3


Draw the line with the given slope and y-intercept. 1. m 2, b 3

2. m 3, b 1

3. m 1, b 4

4. m 2, b 1

5. m 0, b 6

6. m , b 2

2 3

7. m , b 3

1 3

3 2

4 5

9. m , b 0

8. m , b 4

Find the slope and y-intercept of the line. 10. y 3x 1

11. y 4x 7

13. y 8x 2

14. y x 1

12. y 6x 4

1 5

5 3

15. y x 3

17. x-intercept: 3

18. x-intercept: 2

Draw the line with the given intercepts. 16. x-intercept: 4

y-intercept: 2 19. x-intercept: 5

y-intercept: 4

y-intercept: 1 20. x-intercept:

y-intercept: 2

1 2

21. x-intercept: 2

y-intercept:

y-intercept: 3

3 2

Graph the equation. 22. y 2x 1

23. y 6x 4

24. y 3x 1

25. y x 1

26. y 3x

27. y 2x Lesson 2.3

28. Temperature

The formula which converts degrees Celsius to degrees 9 Fahrenheit is given by F 5C 32. Graph the equation.

Simple Interest In Exercises 29–31, use the following information. If you deposit $100 into an account that pays 3% simple interest, the amount of money in your account after t years is modeled by y 3t 100. 29. What is the slope of the line? 30. What is the y-intercept of the line? 31. Graph the line.



41

Answer Key Practice B 1. m 8; b 7 2. m 10; b 0 1 3 1 1 3. m 4; b 2 4. m 2; b 4 3 5 2 5. m 7; b 7 6. m 3; b 2 1 7. x-intercept: 3; y-intercept: 1 8. x-intercept: 6; y-intercept: 6 9. x-intercept: 3; y-intercept: 2 10. x-intercept: 12; y-intercept: 3 12 11. x-intercept: 5 ; y-intercept: 4 6 12. x-intercept: 7; y-intercept: 3 13. x-intercept: 2; y-intercept: 4 14. x-intercept: 4; y-intercept: 3 8 15. x-intercept: 5; y-intercept: 4 4 16. x-intercept: 4; y-intercept: 3 8 17. x-intercept: 4; y-intercept: 5 1 18. x-intercept: 2; y-intercept: 3 19.

20.

y

27.

2

30. y 7x 2

22.

1

y

1

x

1

x

y

1 x

1

23.

24.

y

y

1 1

1 x

1

25.

26.

y

x

y 1

1

1 1

x

31. 0.10d 0.25q 50

32. 0.03x 0.04y 250

x

y

29. 2

1 x

1

21.

2 7

1

3

1

28.

y

x

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Page 42

LESSON

NAME _________________________________________________________ DATE ___________

2.3


Find the slope and y-intercept of the line. 1. y 8x 7

2. y 10x

3. x 4y 6 0

4. 2x 4y 1 0

5. 3x 7y 5 0

6. 2x 3y 6 0

8. y x 6

9. y x 2

5 3

Find the intercepts of the line. 7. y 3x 1

1 4

2 3

7 2

10. y x 3

11. y x 4

12. y x 3

13. 2x y 4 0

14. 3x 4y 12 0

15. 5x 2y 8 0

16. x 3y 4

17. 2x 5y 8

18. 6x y 3

19. y 4x 3

20. y 3x 2

21. x 6y 3 0

22. 7x 2y 6 0

23. 4x 8y 20 0

24. 6x 9y 18

25. 2x y 2

26. 8x 2y 6

27. 3x 5y 15 0

Graph the equation.

Teeter-Totter In Exercises 28–30, use the following information. The center post on a teeter-totter is 2 feet high. When one end of the teetertotter rests on the ground, that end is 7 feet from the center post. 28. Find the slope of the teeter-totter. 29. Assume the base of the center post is at (0, 0) with the ground along the Lesson 2.3

x-axis. Find the y-intercept of the teeter-totter. 30. Write an equation of the line that follows the path of the teeter-totter. 31. Saving Change

Each time you get dimes or quarters for change, you throw them into a jar. You are hoping to save $50. Write a model that shows the different numbers of dimes and quarters that you could accumulate to reach your goal.

32. Commission Sales

A salesperson receives a 3% commission on furniture sold at a sale price and a 4% commission on furniture sold at the regular price. The salesperson wants to earn a $250 commission. Write a model that shows the different amounts of sale-priced and regular-priced furniture that can be sold to reach this goal.

42



Answer Key Practice C 1 1. m 4; b 2 2. m 3; b 2 2 3. m 3; b 4 4. m 2; b 3 4 1 5. m 0; b 6 6. m 3; b 3 9. m 11. m

7 8 5 ; b 5 8 3; b 0 3 1 7; b 7

13. x-intercept: 4 15. 17. 19. 21.

1 1

3

2

10. m 5; b

7 5

1

12. m 2; b

31.

5 2

14. x-intercept: 4

y-intercept: 8 x-intercept: 0 y-intercept: 0 x-intercept: 13 y-intercept: none 20. x-intercept: 1 y-intercept: 14

23.

y

24.

25.

x

27.

y 1

1

x

y 1

x 1

28.

x

1 1

26.

1

y

1

1

29.

y

x

y

1 1

1 1

x

100

0

100

200 300 T-shirts

400

x

100,001 2

50,000.5; The slope represents the decrease in value per year. b.

x

y

200

7x 15y 3000 Sample answers: 210, 102 300, 60 390, 18

32. a. 300,000; The V-intercept represents the initial value of the equipment.

y

1

y

0

1

1

x

8. m 2; b 2

y-intercept: 3 5 x-intercept: 3 16. y-intercept: 25 x-intercept: 34 18. y-intercept: 3 x-intercept: none y-intercept: 34 x-intercept: 12 y-intercept: 14

22.

y

Sweatshirts

7. m

30.

x

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LESSON

NAME _________________________________________________________ DATE ____________

2.3


Find the slope and the y-intercept of the line. 1 2

2 3

3. y x 4

1. y 4x 2

2. y 3x

4. y 3 2x

5. y 6

6. 4x 3y 1 0

7. 7x 5y 8 0

8. 3x 2y 4 0

9. 8x 3y 0

10. 2x 5y 7 0

11. 3x 7y 1 0

12. x 2y 5 0

13. 3x 4y 12 0

14. 2x y 8 0

15. 3x 2y 5 0

16. 5x 2y 0

17. 4x y 3

18. x 13 0

19. 4y 3 0

20. 2x 3y 3x y 1

21. x 5y 3 3x y 4

Find the intercepts of the line.

Graph the equation. 22. y 3x 5 25. x

4 3

28. x 2y 8 0

23. y 2x

1 2

26. 2x 3y 6 0 29.

1 x 2y 3 0 2

24. y

3 x1 4

27. 3x 4y 10 30. 4x

3 y10 2

31. Fund Raiser

Lesson 2.3

The marching band holds a fund raiser each year in which they sell t-shirts and sweatshirts with the school’s name and mascot on it. The t-shirts sell for $7 and the sweatshirts sell for $15. The band needs to raise $3000. Write a model that shows the number of t-shirts and sweatshirts that must be sold. Then graph the model and determine three combinations of t-shirts and sweatshirts that satisfy the model.

32. Linear Depreciation

A business purchases a piece of equipment for $300,000. The value, V, of the machine after t years is represented by the model 2V 100,001t 600,000. a. Find the V-intercept of the model. What does the V-intercept represent? b. Find the slope of the model. What does the slope represent?



43

Answer Key Practice A 1. y 3x 2 2. y 4x 5 3. y 6x 1 4. y x 9 5. y 2x 6. y 7 7. y 5x 3 8. y 3x 2 9. y 2x 4 10. y 4x 11 11. y 6 12. y x 5 13. y x 3 14. y 2x 1 15. y x 10 16. y 2x 1 17. y 8x 17 18. y 4x 8 19. y x 1 20. y 3x 20 3 21. y 4x 3 22. y 5x 11 1 17 23. y 2x 22 24. y 2x 2 25. y 3x 15 26. y 2x 3 1 1 27. y 3x 3 28. The data do not show direct variation. 29. The data do show direct variation, and the direct variation equation is y x. 30. y 0.06x

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Page 55

LESSON

NAME _________________________________________________________ DATE ____________

2.4


Write an equation of the line that has the given slope and y-intercept. 1. m 3, b 2

2. m 4, b 5

3. m 6, b 1

4. m 1, b 9

5. m 2, b 0

6. m 0, b 7

Write an equation of the line that passes through the given point and has the given slope. 7. 0, 3, m 5

8. 0, 2, m 3

9. 1, 2, m 2

10. 3, 1, m 4

11. 2, 6, m 0

12. 4, 1, m 1

13. 5, 2, m 1

14. 3, 7, m 2

15. 8, 2, m 1

Write an equation of the line that passes through the given points. 16. 1, 1, 5, 9

17. 2, 1, 3, 7

18. 1, 4, 2, 16

19. 3, 2, 1, 0

20. 5, 5, 8, 4

21. 0, 3, 4, 0

22. 2, 1, 1, 6

23. 7, 8, 2, 18

24. 9, 4, 1, 8

26.

27.

Write an equation of the line. 25.

y

y

y

2 2 1

x

1 x

1

x

1

Tell whether the data show direct variation. If so, write an equation relating x and y. 28.

x 1 y 2

2 3

3 4

4 5

29.

5 6

x 2 1 0 1 2 y 2 1 0 1 2

30. Sales Tax

Price, x (dollars) Tax, y (dollars)


10 20 30 0.60 1.20 1.80

40 2.40

Lesson 2.4

The amount of sales tax in Pennsylvania varies directly with the price of merchandise. Use the given tax table to write an equation relating the price x and the amount of sales tax y. 50 3


55

Answer Key Practice B 1. y 4x 4 2. y 6x 3 4

1

4. y 2x 4

3. y 3x 6 6. y 5

7. y 2x 5

9. y x 12 3

18. 20. 22. 24.

12. y 4x

8 3

14. y 2x 1

1 16. y 2x 17. y x 4 y 2x 1 19. y 3x 19 y 37x 57 21. y 32x 19 y 3x; 30 23. y 5x; 50 y 52x; 25 25. y 0.25x; 2.5

15. y

1 2x

5 2 1 2

8. y 5x 23

10. y 3x 7

11. y 8x 8 13. y 4x

5. y 8x

1

5

26. y 4x; 2

27. y

27 10 x;

27 28. y 29. Yes, you are traveling about 88.5 km/hr. 30. y 0.2t 14.7 31. 16.7 pounds 103 64 x

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LESSON

NAME _________________________________________________________ DATE ___________

2.4


Write an equation of the line that has the given slope and y-intercept. 1. m 4, b 4

1 2

4. m , b 4

4 3

2. m 6, b 3

3. m , b 6

5. m 8, b 0

6. m 0, b 5

Write an equation of the line that passes through the given point and has the given slope. 7. 2, 1, m 2 10. 1, 10, m 3

8. 4, 3, m 5 11.

12, 4, m 8

9. 7, 5, m 1 12.

23, 0, m 4


14. 1, 3, 1, 1

15. 3, 1, 3, 2

16. 4, 2, 6, 3

17. 1, 5, 4, 0

18. 3, 7, 2, 3

19. 6, 1, 5, 4

20. 3, 2, 4, 1

21. 10, 4, 6, 10

The variables x and y vary directly. Write an equation that relates the variables. Then find y when x 10. 22. x 2, y 6

23. x 1, y 5

24. x 4, y 10

25. x 1, y 0.25

26. x 8, y 2

27. x , y

1 3

9 10

Measuring Speed

In Exercises 28 and 29, use the following information. The speed of an automobile in miles per hour varies directly with its speed in kilometers per hour. A speed of 64 miles per hour is equivalent to a speed of 103 kilometers per hour. 28. Write a linear model that relates speed in miles per hour to speed in

kilometers per hour. 29. You are driving through Canada and see a speed limit sign that says the

speed limit is 80 kilometers per hour. You are traveling 55 miles per hour. Are you speeding?

Lesson 2.4

Fish and Shellfish Consumption

In Exercises 30 and 31, use the following information. For 1992 through 1994, the per capita consumption of fish and shellfish in the U.S. increased at a rate that was approximately linear. In 1992, the per capita consumption was 14.7 pounds, and in 1994 it was 15.1 pounds. 30. Write a linear model for the per capita consumption of fish and shellfish

in the U.S. Let t represent the number of years since 1992. 31. What would you expect the per capita consumption of fish and shellfish

to be in 2002? 56



Answer Key Practice C 1 16 1. y 9x 19 2. y 7 x 7 3. x 1 8 34 4. y 11 x 11 5. y 8 6. y x 1 7 1 11 7. y 2 x 2 8. y 4 x 4 3 11 9. y 2x 3 10. y 2 x 2 11. x 7 12. y 2 13. y 2x 5 1 11 14. y x 15. y x 2 16. y 2 x 2 3 9 17. y 4 x 2 18. y 9 19. y 0.49t 31.4 20. Yes. The model gives 41.2%, and the actual number was 41.1%. 21. No. The model gives 46.1%, and the actual number was 47%. 22. y 1420t 15,500 23. y 1420t 2,810,300 24. $29,700; yes.

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LESSON

NAME _________________________________________________________ DATE ____________

2.4



2. 5, 3, 2, 2

3. 1, 6, 1, 2

4. 7, 2, 4, 6

5. 3, 8, 1, 8

6. 6, 6, 2, 2

Write an equation of the line that passes through the given point and is perpendicular to the given line. 7. 1, 3, y 2x 1

2 3

10. 3, 1, y x 4

8. 3, 2, y 4x 3 11. 7, 3, y 8

9. 1, 1, y

1 x7 2

12. 5, 2, x 2

Write an equation of the line that passes through the given point and is parallel to the given line. 13. 2, 1, y 2x 5 16. 3, 4, y

Labor Force

1 x8 2

14. 1, 1, y x 3

3 4

17. 10, 12, y x 1

15. 3, 5, y 12 x 18. 4, 9, y 14

In Exercises 19–21, use the following information.

From 1840 to 1850, the rate at which the percent of the labor force in nonfarming occupations increased was approximately linear. In 1840, 31.4% of the labor force held nonfarming jobs. In 1850, 36.3% of the labor force held nonfarming jobs. 19. Write a linear model for the percentage of the labor force in nonfarming

occupations. Let t 0 represent 1840. 20. In 1860, the percent of the labor force in nonfarming occupations was

41.1%. Is the model for the percentage of nonfarming occupations from 1840 to 1850 still an appropriate model? 21. In 1870, the percent of the labor force in nonfarming occupations was

47.0%. Is the model for the percentage of nonfarming occupations from 1840 to 1850 still an appropriate model? College Tuition


The rate of increase in tuition at a college from 1990 to 1995 was approximately linear. In 1990, the tuition was $15,500 and in 1995 it was $22,600. 22. Write a linear model for the tuition from 1990 to 1995. Let t 0 Lesson 2.4

represent 1990. 23. Write a linear model for the tuition from 1990 to 1995. Use the actual

years as the coordinates for time. 24. Although the models in Exercises 22 and 23 are different, use both

models to approximate the tuition in 2000. Do both models yield the same result?



57

Answer Key Practice A 1. positive correlation 2. negative correlation 3. relatively no correlation 4.

5.

y

1

1 x

1

x

1

negative correlation

6.

y

relatively no correlation positive correlation

y

1 x

1

3

7. Answers may vary. Sample: y 4x 1

8. Answers may vary. Sample: y 4x 9. Computers per 1000 people

Computers per Capita 380 360 340 320 300 280 260 240 0

0

1

2 3 4 5 Years since 1990

positive correlation

6

3 2

7 2

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LESSON

NAME _________________________________________________________ DATE ____________

2.5

Lesson 2.5


Tell whether x and y have a positive correlation, a negative correlation, or relatively no correlation. 1.

2.

y

3.

y

y

1 x

1

1

1 x

1

1

x

Draw a scatter plot of the data. Then tell whether the data have a positive correlation, a negative correlation, or relatively no correlation. 4.

x 1 y 6

2 6

3 5

4 5

5 6

6 5

7 4

8 3

5.

x 1 y 4

2 5

3 1

4 6

5 6

6 3

7 1

8 6

6.

x 1 y 2

2 2

3 4

4 6

5 8

6 7 8 8 10 10

Approximate the best fitting line for the data. 7.

8.

y

y

4

4

3

3

2

2

1

1

1

2

3

4 x

1

2

3

4 x

9. Computers Per Capita

The table shows the number of computers per 1000 people in the U.S. from 1991 through 1995. Draw a scatter plot of the data and describe the correlation shown. Year Computers per 1000 people


1991

1992

1993

1994

1995

245.4

266.9

296.6

329.2

364.7


69


2.

y

1

1 x

1

x

1

positive correlation 3.

y

y

relatively no correlation negative correlation

1 x

1

2

4. Answers may vary. Sample: y 5x

11 4

1

5. Answers may vary. Sample: y x 4 y 6. Answers may vary.

Sample: y 13x 73

1 x

1

7.

Answers may vary. Sample: y 14x 15 4

y

1 1

8.

x

Broccoli Consumption b Pounds

4 3 2 1 0

0

1

2

3 4 5 6 7 Years since 1980

8

9

t

9. Sample

answer: b 0.3t 1.5 10. Sample answer: 8.1 pounds

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LESSON

NAME _________________________________________________________ DATE ___________

2.5

Practice B

Lesson 2.5


Draw a scatter plot of the data. Then tell whether the data have a positive correlation, a negative correlation, or relatively no correlation. 1.

x y

3 2.5 2 1.75 1.5 1 0.5 0.25 0.5 1 1.5 1.25 2 2.5

2.

x y

0 2.75

3.

x 2 1 y 1 1.25

0.5 3

1 1.25 1.5 2.5 2 1.75 0.5 0.5

0 0

2 2.5 1 1.25

3 1.5

0 2.5

0.5 0.75 1 3 3.25 3.5

1.5 3.75

3.25 2.5

3.5 3

4 3.25

4.25 3

0.25 1 1.5 2.5 1 1.25 2 2.25

2.75 2

3.5 4 4.5 3 3.25 3.5

Approximate the best-fitting line for the data. 4.

5.

y

y

4

4

3

3

2

2

1

1

1

2

3

4 x

1

2

3

4 x

Draw a scatter plot of the data. Then approximate the best-fitting line for the data. 6.

x 2 1.5 1 0.5 y 3 2.5 3 2.4

7.

x 5 y 3

4 2.5

3 2 1 2.8 3.2 3

0 2.2

0.5 2

1 1.5 2 2.1 1.8 1.5

0 4

1 4.2

2 4.3

3 4.5

Broccoli Consumption

In Exercises 8–10, use the following information. The table shows the per capita consumption of broccoli, b (in pounds), for the years 1980 through 1989. Year, t 1980 1981 1982 1983 1984 Pounds, b 1.6 1.8 2.2 2.3 2.7

1985 1986 1987 1988 1989 2.9 3.5 3.6 4.2 4.5

8. Draw a scatter plot for the data. Let t represent the number of years

since 1980. 9. Approximate the best-fitting line for the data. 10. If this pattern were to continue, what would the per capita consumption of

broccoli be in 2002? 70



Answer Key Practice C 1. y 7.6x 4.5 2. y 1.2x 3.5 3. Answers may vary. 4. Answers may vary. 3 31 Sample: y 5 x 5 Sample: y x 2.5 y

y

2

2 x

5. Answers may vary.

Sample: 3.3x 0.87 y

2 2

x


Sample: y 16.05t 319.44 Home runs

9.

y 55 50 45 40 0

0 1 2 3 4 5 6 t Years since 1990

2

x

6. African-American officials

2

y 500 400 300 0

0 2 4 6 8 10 12 t Years since 1980


Sample: approximately 640 10. Answers may vary. Sample: y 0.43x 45

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LESSON

NAME _________________________________________________________ DATE ____________

2.5

Lesson 2.5


Approximate the best-fitting line for the data. 1.

2.

y

y

1 1

x

5 x

1

Draw a scatter plot of the data. Then approximate the best-fitting line for the data. 3.

x y

3 8

2 7.2

1 6.4

0 6

1 5.5

2 5

3 4.8

4.

x y

0 4

1 3.5

2 3.8

3 4.6

4 6

5 7.8

6 10

5.

x y

0 0.6

0.5 3.2

1 4.4

1.5 5.8

2 7

2.5 8.2

3 12

African-American Elected Officials

In Exercises 6–8, use the following information. The table shows the number of African-American elected officials in U.S. and state legislatures for the years 1984 to 1993. Year 1984 Officials 396

1985 1986 1987 407 420 440

1988 436

1989 448

1990 447

1991 484

1992 1993 510 571

6. Draw a scatter plot for the data. Let t 4 represent 1984. 7. Approximate the best-fitting line for the data. 8. If this pattern continues, how many African-American officials will be in

the U.S. and state legislatures in 2000?

Home Run Champions In Exercises 9 and 10, use the following information. The table shows the number of home runs hit by the American League Home Run Champion from 1990 to 1996. Year Home Runs

1990 1991 1992 1993 1994 1995 1996 51 44 43 46 40 50 52

9. Draw a scatter plot for the data. Let t 0 represent 1990. 10. Approximate the best-fitting line for the data.



71

Answer Key Practice A 1. yes; no 2. no; yes 3. no; yes 4. yes; yes 5. yes; no 6. yes; no 7.

8.

y

y 3

1 x

1

9.

3 x

10.

y

y

1

2 x

1

11.

2 x

12.

y

y 2

3

2

x

1

x

3 x

13.

14.

y 1

y

x

1

1

15. D

16. F

17. C

21. 2x 3y ≥ 34

18. E

22. no

19. B

20. A

23. Answers may

vary. Sample: 13 2-point and 3 3-point or 17 2-point and 0 3-point.

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LESSON

NAME _________________________________________________________ DATE ___________

2.6


Check whether the given ordered pairs are solutions of the inequality. 1. x y < 5; 1, 2, 7, 2

2. x > 3; 0, 4, 5, 1

3. y ≤ 1; 1, 3, 2, 1

4. y x ≥ 1; 5, 6, 3, 1

5. x < 2y 5; 4, 0, 4, 5

6. y ≥ x 7; 2, 4, 8, 3

Lesson 2.6

Graph the inequality in a coordinate plane. 7. x > 3 11. y > 1

8. x < 1

9. x ≥ 5

12. y < 6

13. y ≤ 2

10. x ≤ 7 14. y ≥ 4

Match the inequality with its graph. 15. 3x y > 1

16. 2x y ≤ 3

17. 4x y < 1

18. 2x y ≤ 0

19. 5x > 2

20. 3y < 6

A.

B.

y

1

1 1

D.

C.

y

1

x

1

E.

y

1

x

F.

y

1 1

x

y

1

x

1

x

y

1 1

x

Basketball Stats In Exercises 21–23, use the following information. In order for this year’s star basketball player to break the school record for most points (excluding free throws), he must score at least 34 points. The points may be scored by two-point shots and three-point shots. 21. Write an inequality that represents the number of two- and three-point

shots he needs to break the record. 22. In the first game he scored 13 two-point shots and 2 three-point shots.

Did he break the record? 23. Give two possible combinations of two- and three-point shots that will

give him the record.

82



Answer Key Practice B 1. no; yes 2. no; yes 3. no; yes 4. yes; no 5. yes; no 6. no; yes 7.

19.

3

y

x 1

1 1

8.

y

20.

y 1

x

y

1 3

x

1 x

21.

y

1 x

1

9.

10.

y

y

1

22. t ≤ 5p

x

1

1

Defrosting Meat x

12.

y 1 1

Time (hours)

1

11.

23. 2, 12

y

x 2 x

2

t 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 p Weight (pounds)

24. Yes, a 2-pound roast takes at most 10 hours to 14.

y

y

1 1

x 1 1 x

15.

16.

y

y

1

defrost, so it will be completely defrosted before 12 hours passed. 25. 3x 5y ≤ 800 26. 50, 150 27. No, 50, 150 is not a Fundraiser solution of 300 3x 5y ≤ 800. Number of T-shirts

13.

200 100

0 0

100

200

300

Number of caps 1

x 1 1

17.

18.

y

1

x

y

1 1

x

1

x

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LESSON

2.6

NAME _________________________________________________________ DATE ____________


Check whether the given ordered pairs are solutions of the inequality. 1. 2x 3y ≤ 2; 0, 1, 3, 2

2. x 2y > 4; 2, 1, 3, 6

3. 5x y ≥ 3; 3, 6, 2, 5

4. 3x 10y < 8; 6, 3, 4, 2

5. 4y 2x < 5; 2, 0, 3, 1

6. 2y x ≥ 3; 1, 2, 1, 1

Graph the inequality in a coordinate plane. 8. x <

10. y < 4

11. y ≥ 5

13. y < 2x 1

14. y ≥

16. x 2y > 4

17. 5x 5y > 1

18. 3x y ≤ 7

19. 2x 4y > 8

20. 6x 3y ≥ 1

21. 12x 4y < 8

Lesson 2.6

1 2

7. x ≥ 1

9. 2x > 6 12.

1 x5 2

1 y ≥ 2 3

15. 4x y ≤ 2

Defrosting Meat

In Exercises 22–24, use the following information. According to one cookbook, you should always defrost meat in the original wrappings on a refrigerator shelf. You should allow 5 hours for each pound, less for thinner cuts. 22. Write and graph an inequality that represents the time t (in hours) and the

number of pounds p of meat being defrosted. Use t on the vertical axis and p on the horizontal axis. 23. What are the coordinates of a 2-pound roast that has been defrosting for

12 hours? 24. Is it possible that the roast in Exercise 23 is completely defrosted? Explain

your answer.

Fundraiser

In Exercises 25–27, use the following information. An environmentalist group is planning a fundraiser. The group wants to purchase caps and T-shirts with their logo on them and sell them at a profit. They can buy caps for $3 each and T-shirts for $5 each. They have $800 to spend. 25. Write and graph an inequality that represents the numbers of caps x and

T-shirts y that the group can buy. 26. Suppose the group purchased 50 caps and 150 T-shirts. What point on

the coordinate plane represents this purchase? 27. Is the point in Exercise 26 a solution of the inequality?



83


13. 2.

y

y x

1

1

1 1

x

15. 3.

4.

y

x

1

x

16. 4x 2y ≤ 92

y

y 1

1

y

1

2 2

14.

y

x

1 1 x

1

x

1

5.

6.

y

y 1 1

1

x

True/false

17.

x 1

y 50 (0, 46) 40 30 20 10 (23, 0) 0 x 0 10 20 30 40 50 Multiple choice

18. No.

19. 3T 3.50S ≥ 47.50 8.

y 1

y Stewarts

7.

x

1

1 1

9.

10.

y

x

20. Sample answers: 10 hours at Thompson’s and 1

x

1

x

1

11.

x

12.

y

2

y

1

2

x

0 3 6 9 12 15 18 T Thompsons

y 1

1

S 18 15 12 9 6 3 0

10 hours at Stewart’s, 15 hours at Thompson’s and 5 hours at Stewart’s, 5 hours at Thompson’s and 10 hours at Stewart’s 3 2 1 21. y < 5 x 2 22. y ≥ 3 x 2

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LESSON

NAME _________________________________________________________ DATE ___________

2.6


Lesson 2.6

Graph the inequality in a coordinate plane. 1. x 3 < 5

2. y 2 > 3

3. 3x 2y ≥ 0

4. 4x 7y > 0

5. 2x 3y ≤ 6

6. 4x 3y > 12

7. 3x 2y ≥ 9

8. 5x 3y < 10

9. 7x 4y ≤ 8

10. 6x 5y > 10

11. 4x 3y ≥ 2

12. 8x 9y ≤ 3

13. 2x 3y < 5

14. 4x 3y > 1

15. 3x 5y ≤ 8

Test Scores

In Exercises 16–18, use the following information. A history exam included multiple choice questions that were worth 4 points each and true/false questions that were worth 2 points each. The highest score earned by a person in your class was 92. 16. Write an inequality that represents the number of multiple choice

questions and true/false questions that could have been answered correctly by any member of your class. 17. Graph the inequality. 18. Is it possible that someone answered 20 multiple choice questions and 7

true/false questions correctly?

Babysitting Wages

In Exercises 19 and 20, use the following information. You earn $3 per hour when you babysit the Thompson children. You earn $3.50 per hour when you babysit the Stewart children. You would like to buy a $47.50 ticket for a concert that is coming to town in 5 weeks. 19. Write and graph an inequality that represents the number of hours you

need to babysit for the Thompson’s and Stewart’s to earn enough money to buy your concert ticket. 20. Give three possible combinations of babysitting hours that satisfy the

inequality.

Visual Thinking 21.

Write the inequality represented by the graph. 22.

y

y

1 1

1

x

x 1

84



Answer Key Practice A 1. 2 2. 3 3. 3 4. 2 5. 13 6. 5 7. 4 8. 8 9. 6 10. 5 11. 5 12. 9 13. E 14. B 15. D 16. F 17. C 18. A 0, 0 ≤ x ≤ 5 5, 5 < x ≤ 12 19. f x 12, 12 < x ≤ 18 18, x > 18

y 18 16 14 12 10 8 6 4 2 2

4

6

8 10 12 14 16 18 x

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Page 96

LESSON

NAME _________________________________________________________ DATE ___________

2.7


Evaluate the function for the given value of x. f x

3, if x ≤ 0 2, if x > 0

gx

x 5, if x ≤ 3 2x 1, if x > 3

hx

3 2x,4, ifif xx ≤> 2 2 1 2x

12

1. f 2

2. f 4

3. f 0

4. f

5. g7

6. g0

7. g1

8. g3

10. h2

11. h1

12. h6

9. h4

Match the piecewise function with its graph.

x3x, if4,x if>x1≤ 1 2x 3, if x ≥ 0 f x x 4, if x < 0

x2x4,4,ififxx≤>00 3x 1, if x ≥ 1 f x 5, if x < 1

3xx 2,2,ififxx>≤11 3x 1, if x ≤ 1 fx 5, if x > 1

13. f x

14. f x

15. f x

16.

17.

18.

Lesson 2.7

A.

B.

y

C.

y

y

4 2 4

D.

2

x

2

E.

y

2

x

F.

y

2 2

x

19. Amusement Park Rates

2

x

2

x

y

2 2

x

The admission rates at an amusement park are as

follows. Children 5 years old and under: free Children over 5 years and up to (and including) 12 years: $5.00 Children over 12 years and up to (and including) 18 years: $12.00 Adults: $18.00 Write a piecewise function that gives the admission price for a given age. Graph the function.

96



Answer Key

Practice B 1. 7 2. 1 3. 2 4. 16 5. 2 6. 7 7. 14 8. 7 9. 21 10. 9 5 11. 5 12. 3 13.

14.

y

2x 2, if x < 1 if x 1 x 3, if x > 1

24. f x 1, 25.

y

26. x > 0; C > 0

C 75 45

1

1

15 x

1

x

1

2

15.

16.

y

27. C

y

1 x

1

1 x

1

17.

18.

y

y

2 1

x

2

1

x

19. 20.

y

y 2 2

1 x

1

21.

y

1 1

x

x2x 3, 2, 2x 2, 23. f x x 3, 22. f x

if x ≤ 1 if x > 1 if x < 1 if x ≥ 1

x

6

10

0.05x, 0.08x,

x

if 0 < x ≤ 100,000 if x > 100,000

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Page 97

LESSON

NAME _________________________________________________________ DATE ____________

2.7


Evaluate the function for the given value of x. f x

3x 7, if x ≤ 2 6 2x, if x > 2

gx

3x 5, if x < 5 x 3, if x ≥ 5

hx

2x 1,3, ifif xx >≤ 3 3 2 3x

1. f 0

2. f 2

3. f 4

4. f 3

5. g5

6. g4

7. g3

8. g10

10. h3

11. h6

9. h9

12. h1

Graph the function. 3, if x ≤ 4 1, if x > 4 2x 3, if x ≥ 1 f x 3x 1, if x < 1

x2x, if3,x if>x0≤ 0 x, if x > 5 f x x, if x ≤ 5

13. f x

14. f x

16.

17.

18.

1 2

x 1, if x < 0 2x, if x ≥ 1 2, if x ≤ 3 19. f x x 1, if 0 ≤ x ≤ 2 20. f x 3x, if 2 < x < 1 21. f x 1, if 3 < x < 3 x 1, if x > 2 x, if x ≤ 2 3, if x ≥ 3 Write equations for the piecewise function whose graph is shown. 22.

23.

y

1

24.

y

1 1

x

y

1 1

x

1

x

Tour Bus

In Exercises 25 and 26, use the following information. A company provides bus tours of historical cities. The given function describes the rate for small groups and the discounted rate for larger groups, where x is the number of people in your group. C

8.95x, if 0 < x ≤ 10 7.50x, if x > 10

25. Graph the function. 26. Identify the domain and range of the function. 27. Commission Rate

You are employed by a company in which commission rates are based on how much you sell. If you sell up to $100,000 of merchandise in a month, you earn 5% of sales as a commission. If you sell over $100,000, you earn 8% commission on your sales. Write a piecewise function that gives the amount you earn in commission in a given month for x dollars in sales.



97

Lesson 2.7

2 5

x3 4,x, ifif xx 0 f x 2x 3, if x ≤ 0

15. f x

Answer Key Practice C 1. 7 2. 2 3. 6 4. 6 5. 3 6. 1 7. 3 8. 7 9. 8 10. 14 11. 2 12. 7 13.

14.

y

x

1

15.

16.

y

1 1

17.

x

1 x

18.

y

y

2 2

x 1 1

19.

20.

y

x

y

3

1

x

1 1

21.

x

y 1 1

22. f x

Price

23.

y 40 38 36 34 32 0

26. f x

x,x,

27. f x

xx 5,1,

x

y

1

y 1 1

1

25. f x

x

33.80 0.20x35

0 2 4 6 8 10 12 14 x Letters

if 0 ≤ x ≤ 6 if x > 6 24. $36.60

if x ≥ 0 if x < 0 if x < 3 if x ≥ 3

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LESSON

NAME _________________________________________________________ DATE ___________

2.7



3x 5, if x < 2x 1, if x ≥

f x

1 2 1 2

gx x

hx 3x 2 1

f 12

f 13

4. f 2

1. f 3

2.

5. g3.2

6. g1.8

7. g2.4

8. g6.9

9. h1.8

10. h3.1

11. h0.4

12. h3.1

3.

5

Graph the function.

Lesson 2.7

2x, if x < 2 14. f x x2, if 2 ≤ x ≤ 2 2x, if x > 2

15. f x

16. f x x 1

17. f x 3x 2

18. f x 4x

19. f x 2x 3

20. f x 23x 1 4

21. f x 2x 1 3

13. f x

x 3, if x < 2x 1, if x ≥

1 2 1 2

x3x1,1, ifif xx >< 11

Engraving In Exercises 22–24, use the following information. A gift shop sells pewter mugs for $35. They are currently running an engraving promotion. The first six letters are engraved free. Each additional letter costs $0.20. 22. Write a piecewise model that gives the price of the mug with x engraved

letters. 23. Graph the function. 24. What is the price of a mug with the name Jamie Lynn Krane engraved? 25. Commission Sales

A company pays its employees a combination of salary and commission. An employee with sales less than $100,000 earns a $15,000 salary plus 3% commission. An employee with sales of $100,000 to $200,000 earns an $18,000 salary plus 4% commission. An employee who earns more than $200,000 in sales earns a $20,000 salary plus 5% commission. Write a piecewise model that gives the pay of an employee with x in annual sales.

26. Absolute value

Write the function f x x as a piecewise function.

27. Absolute value

Write the function f x x 3 2 as a piecewise function.

y

y

1

1 1

98

x


1 x


Answer Key Practice A 1. E 2. B 3. C 4. F 5. D 6. A 7. down 8. up 9. up 10. up 11. down 12. down 13. 0, 3 14. 1, 2 15. 3, 5 16. 7, 2 17. 1, 9 18. 3, 0 19. same 20. narrower 21. wider 22. narrower 23. wider 24. wider 25.

Swimwear Swimsuits sold

600

(6, 540)

500 400 300 200 100 0

(12, 0)

(0, 0) 0

2

4

6

8

10

Month

26. $540; month number six

12

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LESSON

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2.8


Match the function with its graph.

f x x 4

1. f x x 4

2. f x x 4

4.

5. f x 4 x

6.

B.

C.

A.

y

1 f x x 4

3. f x x 4

y

y

1 x

1

1

1 x

1

D.

E.

y

x

1

F.

y

y 1

x 1

3 1 3 x

1 x

Tell whether the graph of the function opens up or down.

7. y 3 x

8. y 3 x 1

10. y 4 x 1 3

9. y x 1 10

11. y 2 x 1 7

12. y x 2 4

Lesson 2.8

Identify the vertex of the graph of the given function.

13. y 2 x 3

14. y x 1 2

17. y 2 x 1 9

16. y x 7 2

15. y x 3 5

18. y 5 x 3

Tell whether the graph of the function is wider, narrower, or the same width as the graph of y x .

19. y x 8

20. y 2 x 1

22. y 3 x 1 7

23. y

2 x6 3 3

21. y

1 x3 2 2

24. y

9 x 13 10

Swimwear

In Exercises 25 and 26, use the following information. A sporting goods store sells swimming suits year round. The number of suits sold can be modeled by the function S 90 t 6 540, where t is the time in months and S is the sales in dollars.

25. Graph the function for 0 ≤ t ≤ 12. 26. What is the maximum sales in one month? In what month is the

maximum reached? 110



Answer Key

11.

y 1

24. y

1 2

23. y x 2 3 25. 0, 22

26. The home is 22 feet high. 27.

y

x 1 1 x

1

People Served Number of people served

10.

x 2 1

22. y 2 x 3 1

Practice B 1. up 2. down 3. up 4. 13, 6 5. 4, 7 6. 2, 11 7. wider 8. narrower 9. narrower

120 100 80 60 40 20 0 0

2

4

6

8

10

12

Hours since noon

12.

13.

y

y

1

28. 6.5, 105; the restaurant serves the greatest

1 x

number of people, 105, at 6:30 P.M. 1 x

1

14.

15.

y

y 1 x

1 x

1

16.

17.

y

y

1 x

1

1 x

1

18.

19.

y 2 2 x

y

1

x 1

20.

21.

y

y

1 1

x

1

x

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LESSON

NAME _________________________________________________________ DATE ____________

2.8


Tell whether the graph of the function opens up or down.

1. y x 3 5

2. y 4 x 1 6

3. y

2 x2 9 3

6. y

1 x 2 11 5


4. y 2 x 13 6

5. y 3 x 4 7

Tell whether the graph is wider, narrower, or the same width as the graph of y x . 7.

3 y x 3 7 5

8. y 8 x 9 12

9. y

5 x1 3 2

Graph the function.

y x 1 3 y x 4 5 1 y x 2 2

12. y x 2 3

10. y x 4

11. y x 4

13.

14. y 2 x 3

15. y x 5

17.

18. y 2 x 7 4

16. 19.

20.

y 3x 1 2 2 y x 2 1 3

1 21. y x 1 2 2

Write an equation of the graph shown. 22.

23.

y

24.

y

y

1 1

1 x 1 x

x

1

Lesson 2.8

1

A-Frame Home In Exercises 25 and 26, use the following information. 11 The roof line of an A-frame home follows the path given by y 6 x 22. Each unit on the coordinate plane represents one foot.

y

5

25. Find the vertex of the graph.

5

26. What does the y-value of the vertex tell us about the home?

x

Fine Dining

In Exercises 27 and 28, use the following information. An exclusive restaurant is open from 3:00 P.M. to 10:00 P.M. Each evening, the number of people served S increases steadily and then decreases according to the model S 30 t 6.5 105 where t 0 represents 12:00 P.M.

27. Graph the function. 28. Find the vertex of the graph. Explain what each coordinate of the vertex represents. Copyright © McDougal Littell Inc. All rights reserved.


111

Answer Key Practice C 1. down 2. up 3. down 4. 2, 5 3 2 5. 8, 1 6. 3, 6 7. wider 8. wider 9. narrower 10.

11.

y

20.

1 x

1

1 x

y

y3x

1

x

y

2

1 1

21.

y

x

1

22.

y2x3 23.

y

y

1

12.

13.

y

1

x

1

y

x

1

1

1

1 x

y5x4

x

1

24.

y4x 2 25.

y

y

1

14.

15.

y

1 x

1

y

1 x

1 1 x

1

16.

17.

y 1

x

26.

y2x4 1 27.

y

y

y 1

x

1

y2x 3

1

2 x

1

2 x

1 x

1

18.

19.

y

2

29.

1 1

y2x

f x 1.2 x 450

y4x5 1

50 t 2 200, if 0 ≤ t < 3 28. f x 100 t 5 350, if 3 ≤ t < 6 50 t 9 400, if 6 ≤ t ≤ 10

y

1 x

y3x3 2

x

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LESSON

NAME _________________________________________________________ DATE ___________

2.8


Tell whether the graph of the function opens up or down. 1. y

1 x2 4 3

2. y 3

1 x1 2

3. y 4 2 x 3


4. y 3 x 2 5

5. y

1 3 x 1 3 8

6. y 6

4 2 x 5 3

Tell whether the graph is wider, narrower, or the same width as the graph of y x .

2 7. y x 1 4 3

8. y

5 x3 1 6

9. y 4

7 x3 6

Graph the function.

1 y x 3 1 2

1 y x 2 3 3

10. y 2 x 1 4

11. y 3 x 3 2

12. y 4 5 x 2

13.

14.

15. y 2 x

16. y x

2 1 3

1 3 2

17. y 2.5 x 1.3 2.4

18. y 1.8 x 2.2 1.6

Graph the function by making a table and plotting points. Then write a function of the form y a x h k that has the same graph.

y 5x 20 y 2x 8 1

19. y 2x 22.

28. Company’s Profit

y 4x 2 y 3x 9 2

112

P 450 400 350 300 250 200 150 100 50 0

y 2x 3 y 22x 10 1

20. y 3x

21. y 2x 6

23.

24.

26.

The profit for a company from 1988 to 1998 is modeled by the graph. The profit is measured in thousands of dollars and t 0 corresponds to 1988. Write a piecewise function that represents the profit. Profit (thousands of dollars)

Lesson 2.8

25.

27.

29. Pyramids of Egypt

The largest pyramid included in the first wonder of the world is Khufu. It stands 450 feet tall and its base is 755 feet long. Imagine that a coordinate plane is placed over a side of the pyramid. In the coordinate plane, each unit represents one foot and the origin is at the center of the pyramid’s base. Write an absolute value function for the outline of the pyramid.

450 ft

0 1 2 3 4 5 6 7 8 9 t Years since 1988


755 ft


Answer Key Review and Assessment

z

z

(0, 0, 6)

Test A 1.

(0, 4, 0)

2.

y

y

(0, 6, 0)

(3, 1)

1

x

1

x

x

1

x

y

1

y

(2, 0, 0)

(0, 0, 4) (6, 0, 0)

16. f x, y x y 9; 1 17. 2, 3, 4

3, 1 3.

infinitely many solutions 4. 2, 0 5. 3, 9

y 4

4

x

no solution 6. 6, 0

7.

x3

y

y2 1 x

1

8.

9.

y

y

yx2

yx4

y4 1

x1 2

1 x

x0 1

x

10. minimum of 0 at 0, 0;

maximum of 4 at 4, 0 11. minimum of 16 at 4, 1; maximum of 14 at 2, 6 12.

13. z

z

(3, 4, 2)

y x

14.

y

(1, 0, 1)

x

15.

18. 1, 2, 1 19. 5

20. 14

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CHAPTER

3

NAME _________________________________________________________ DATE ____________


Graph the linear system and tell how many solutions it has. If there is exactly one solution, estimate the solution and check it algebraically. 1. x y 4

xy2

1.

Use grid at left.

3. y 2x 2

2.

Use grid at left.

y 12x 2

3.

Use grid at left.

1

2. y 3x

Answers

2y 6x

4. y

y

y

5. 4

6.

1 1

1

x

4

x

x

1

Solve the system using any algebraic method. 4. x y 2

5. y 3x

3x 2y 6

Use grid at left.

8.

Use grid at left.

9.

Use grid at left.

10

6. 5x 2y 30

x y 12

7.

11.

x 2y 6

Graph the system of linear inequalities. 8. y > x 4

7. x > 3

y ≤ 2

9. x y ≤ 2

x ≤ 1

y

x ≥ 0 y ≥ 4 y

y

1 1

x 1

1 1

x

1

x

Find the minimum and maximum values of the objective function subject to the given constraints. 10. Objective function: C x y Review and Assess

Constraints: x ≥ 0 y ≥ 0 1 y ≤ 2x 2 11. Objective function: C 5x 4y

Constraints: x x y y

≤ 2 ≥ 4 ≥ 1 ≤ 6



93

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CHAPTER

NAME _________________________________________________________ DATE ____________

3

Chapter Test A

CONTINUED


Plot the ordered triple in a three-dimensional coordinate system. 12. 1, 0, 1

13. 3, 4, 2

z

12.

Use graph at left.

13.

Use graph at left.

14.

Use graph at left.

15.

Use graph at left.

z

16. 17. y

y

18. x

x

19. 20.

Sketch the graph of the equation. Label the points where the graph crosses the x-, y-, and z-axes. 14. x y z 6

15. 2x y z 4

z

z

y y

x

x

16. Write the linear equation x y z 9 as a function of x and y.

Then evaluate the function when x 3 and y 5.

Solve the system using any algebraic method.

Review and Assess

17. 2x 3y 2z 3

18. x 2y 3z 8

2y 3z 6

2x 3y z 3

z4

2x y 2z 2

19. Compact Discs

At a music store, compact discs cost $14.95 each, but are now on sale for $12.95 each. If you bought ten compact discs in the past month, and spent a total of $139.50, how many did you buy on sale?

20. Ages

You are 4 years older than your brother. Two years ago, you were 1.5 times as old as he was. What is your present age?

94



Answer Key Test B

14.

1.

2.

y

15.

z

(0, 0, 5)

z

(0, 0, 4)

y

(0, 10, 0) (1, 3)

y

(1, 0, 0)

(1, 2) 1 x

1

1, 3

(10, 0, 0) x

1

3.

17. 1, 2, 3 18. (7, 6, 3)

4. 1, 3

y

19. daytime $6; evening $8 20. 360

5. 2, 1

1

6. 1, 6

x

1

infinitely many solutions 7.

8.

y

y

y 2

1

y 2 2x

1

1

x

1 x

1

yx3

9.

x 3

y

y 4 2x 1

y0 x

1

x0

10. minimum of 0 at 0, 0;

maximum of 30 at 0, 6 11. minimum of 0 at 0, 0; maximum of 17 at 3, 4 12.

13.

z

z

(3, 4, 2) y x

x

16. f x, y 2x 3y 6; 7

1, 2

(4, 1, 2) x

y

(0, 2, 0) y

x

1

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CHAPTER

NAME _________________________________________________________ DATE ____________

3


Graph the linear system and tell how many solutions it has. If there is exactly one solution, estimate the solution and check it algebraically. 1. y 3x

2. y x 1

y x 4

1.

Use grid at left.

2.

Use grid at left.

3.

Use grid at left.

3. x 2y 2

y x 3

y

Answers

3x 6y 6

y

y

1 1

1

1 1

x

1

x

4.

x

5. 6.

Solve the system using any algebraic method. 4. x y 2

5. y 2x 5

y 2x 5

6. 2x y 8

y x 3

2x y 4

Graph the system of linear inequalities. 7. x 2y ≥ 4 xy ≤ 3

8. y ≤ 2 x > 3

9. x ≥ 0 y ≥ 0

y

Use grid at left.

8.

Use grid at left.

9.

Use grid at left.

10.

2x y ≤ 4

y

7.

11.

y

1 1

1 1

x

x 1 1

x

Find the minimum and maximum values of the objective function subject to the given constraints. 10. Objective Function: C 4x 5y

Constraints:

Review and Assess

x ≥ 0 y ≥ 0 xy ≤ 6

11. Objective Function: C 3x 2y

Constraints:


x ≥ 0 y ≥ 0 x 3y ≤ 15 4x y ≤ 16


95

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CHAPTER

NAME _________________________________________________________ DATE ____________

3

Chapter Test B

CONTINUED



13. 4, 1, 2 z

z

12.

Use grid at left.

13.

Use grid at left.

14.

Use grid at left.

15.

Use grid at left.

16. y

y

17. x

x

18. 19.

Sketch the graph of the equation. Label the points where the graph crosses the x-, y-, and z-axes. 14. x y 2z 10

20.

15. 4x 2y z 4 z

z

y y

x x

16. Write the linear equation 2x 3y z 6 as a function of x and y.

Then evaluate the function when x 2 and y 3.

Solve the system using any algebraic method. 17. x 4y z 12

18. x y 2z 5

y 3z 7

x 2y z 8

z3

2x 3y z 1

You work at a grocery store. Your hourly wage is greater after 6:00 P.M. than it is during the day. One week you work 20 daytime hours and 20 evening hours and earn $280. Another week you work 30 day time hours and 12 evening hours and earn a total of $276. What is your daytime rate? What is your evening rate?

Review Review and and Assess Assess

19. Earning money

20. Telethon

During a recent telethon, people pledged $25 or $50. Twice as many people pledged $25 as $50. Altogether, $18,000 was pledged. How many people pledged $25?

96




14. 2.

y

15. z

y

z

1

1 x

1

2

(4, 0, 0)

(0, 4, 0)

x

(2, 1)

y

(4, 0, 0)

(0, 4, 0)

x

2, 1 3.

no solution infinitely many solutions

y

4. 0, 5 5. infinitely many solutions

6. no solution

8.

y

y 3

y 2x 2 1

1

y0 x

1

2

x

y x 1 x0 x 2

10. minimum of 0 at

y

0, 0; maximum of 12 at 3, 0

1 1

x

y 2x 3

11. minimum of 46 at 3, 4; no maximum 12.

13. z

z

(2, 1, 4)

y x

(0, 0, 4)

16. f x, y 2x 3y 12; 1 19. 30 postcard and 20 letter

x

1

7.

x

17. 2, 3, 4 18. 1, 2, 2

1

9.

(0, 0, 4)

y x

(3, 4, 4)

20. 36

y

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CHAPTER

NAME _________________________________________________________ DATE ____________

3


Graph the linear system and tell how many solutions it has. If there is exactly one solution, estimate the solution and check it algebraically. 1. 2x y 5

2. 2x 3y 6

x y 3

2y 2 6x

y

2.

Use grid at left.

3.

Use grid at left.

1

x

1

Use grid at left.

y

1

1

1.

3. 3x y 1

3y 2x 3

y

Answers

1

x

x

1

4. 5. 6.

Solve the system using any algebraic method. 4. 3x 2y 10 5. 2x 4y 6 6. 3x 5y 10 0 5x 3y 15

x 2y 3

9x 15y 30

Graph the system of linear inequalities. 7. x ≤ 0 8. x y > 1 3x 2y > 4

y ≥ 0 y

9. y ≤ 2x 3

x2 ≤ 0

y

1

7.

Use grid at left.

8.

Use grid at left.

9.

Use grid at left.

10. 11.

y

1 1

x

1

x

1 1

x

Find the minimum and maximum values of the objective function subject to the given constraints. 10. Objective function: C 4x y

Review and Assess

Constraints: x ≥ 0 y ≥ 0 xy ≤ 3 11. Objective function: C 6x 7y

Constraints: x ≥ 0 y ≥ 0 4x 3y ≥ 24 x 3y ≥ 15



97

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CHAPTER

NAME _________________________________________________________ DATE ____________

3

Chapter Test C

CONTINUED



13. 3, 4, 4 z

12.

Use grid at left.

13.

Use grid at left.

14.

Use grid at left.

15.

Use grid at left.

z

16. 17. y x

y

18.

x

19. 20.


15. 3x 3y 3z 12

z

z

y x

y x

16. Write the linear equation 2x 3y z 12 as a function of x and

y. Then evaluate the function when x 4 and y 1.

Solve the system using any algebraic method.

Review and Assess

17. 3x 4y 6

18. 3x 2y 2z 3

5x 3z 22

2x 3y 3z 2

3y 2z 1

3x 5y z 9

19. Stamps

Postcard stamps are 20¢ each, while letter stamps are 33¢ each. If you have 50 stamps worth $12.60, how many of each type do you have?

20. Numbers

The sum of the digits of a two-digit number is 9. If the digits are reversed, the new number is 27 more than the original number. Find the original number.

98



Answer Key Cumulative Review 1. commutative property of multiplication 2. inverse property of multiplication 3. associative property of multiplication 4. 47 5. 6 6. 45 7. 10 8. 10x 2 3x 16 9. 13x 54 10. 10x 2 x 28 11. 22x 32y 12. 3 13. 4 14. 2 15. 2 16. 4 17. 15 8x 3x 6 x 18 18. 19. 20. x 2 3 30 6x x8 x 12 21. 22. 23. 5 x x 11 24. n < 7 25. x < 9

51.

52. y

y

1 2

x

1 2

x

53.

54. y

y

1

1 x

1

11

x

1

9 5

6

7

8

9 3

26. x ≥ 8 6

7

2

1

0

1

27. x ≥ 1

8

9

1

10

0

1

2

3

55. 7, 1; down;

56. 3, 2; up;

same width

same width

y

y

29. 3 < x < 7

28. x < 2 or x > 7

2 1

2

3

4

30. yes

5

6

7

4 2

8

31. yes

34. Line 2

32. no

35. Line 1

0

2

4

6

8

1 x

2

x

1

33. Line 2

36. Line 2 2

37. m 4, b 6 38. m 3, b 5

57. 0, 2; down; same width

3

39. m 0, b 10 40. m 2, b 7 1 8,

b 2 42. m 9, b 0 43. y 5x 7 44. y 4 2 7 2 5 45. y 3x 3 46. y 3 x 3 47. y 4x 20 48. y 3x 7 41. m

49.

1 x

1 1

y 1 1

x

y

1

y

1

same width

y

50.

1

58. 2, 0; up;

x

x

Answer Key 59. 0, 2; up;

60. 0, 4; down;

narrower

wider

y

y

1

1 1

x

61.

x

1

62. y

y

1 1

1

x

x

1

63.

64. y

y

1

2 1

x

65.

x

2

66. y

y

2 2

x

1 1 x

67.

infinitely many solutions of the form x, 2, x 2 68. infinitely many solutions of the form x, 2 x 5, x 5 69. infinitely many solutions of the form 3z 17, 4z 27, z 70. 34,509 sq ft 71. 3460 sq mi

MCRB2-0311-CR.qxd

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CHAPTER

NAME _________________________________________________________ DATE ____________

3


Tell what property the statement illustrates. (1.1) 1. 3

443

2. 4

14 1

3. 2

3 5 2 3 5





3

Simplify the expression. (1.2) 9. 3x 8 52x 6

8. 7x 2 5x 9 3x 2 2x 7

11. 84x 2y 25x 8y

10. 4x 2 x 7 32x 2 x

Solve the equation. Check your solution. (1.3) 12. 5x 7 22

13. 3a 5 7a 21

15. 32x 8 4x 2 4

16.

9 2x

14. 2x 8 2x 12

2 3x 4

17.

1 2x

53 23 x 56

Solve the equation for y. (1.4) 18. x xy 8

19. 6x 4y 12

20. x 3y 18

21. 6x 5y 30 0

22. xy 8 x

23. x 12 xy

Solve the inequality. Then graph the solution. (1.6–1.7) 24. 3n 4 < 9

25. 4 4x > 53 x

26.

27. 3x 7 ≥ 10

28. 4x 2 < 6 or 3x 1 > 22

29. 5 < 2x 1 < 15

1 2x

8 ≥ 12

Use the vertical line test to determine whether the relation is a function. (2.1) 30.

31.

y

(1, 6)

32.

y

(5, 5) 1

(3, 5) 2

y

(1, 1) 2

1 x

(3, 4)

1

1

x

x

Review and Assess

Tell which line is steeper. (2.2) 33. Line 1: through 3, 5 and 0, 1



104






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NAME _________________________________________________________ DATE ____________

3

Cumulative Review

CONTINUED


Find the slope and y-intercept of the line. (2.3) 2

37. y 4x 6

38. y 3 x 5

39. y 10

40. 3x 2y 14

41. x 8y 16

42. 9x y 0

Write an equation of the line that passes through the given point and has the given slope. (2.4) 43. 0, 7, m 5 46. 4, 1, m 3 2

44. 6, 4, m 0

45. 5, 1, m

47. 5, 0, m 4

48. 2, 1, m 3

2 3

Graph the inequality in a coordinate plane. (2.6) 49. x ≤ 3

50. 2y > 10

51. y ≥ 3x 2

52. y < 4 2x

53. 3x 4y > 12

54.

2 3x

12 y > 1

Graph the absolute value function. Then identify the vertex, tell whether the graph opens up or down, and tell whether the graph is wider, narrower, or the same width as the graph of y x . (2.8)

1 f x 2 x 4

55. f x x 7 1

56. f x x 3 2

57. f x x 2

58. f x x 2

59. f x 2 x 2

60.

Graph the system of linear inequalities. (3.3) 61. y ≥ 5

x ≤ 2 64. y > x 5

y < 2x 1

62. x y ≥ 4

63. 5x 3y ≤ 6

2x 4y > 8

2x y ≤ 3 65. x y ≥ 5

66. x > 6

3x y ≤ 8

xy ≥ 0

Solve the system using either the linear combination method or the substition method. (3.6) 67. x 2y z 2

2x 3y 2z 10 x 3y z 4

68. x y z 0

69. x y z 10

5x 3y z 10 x y z

2x y 2z 7 6x 4y 2z 6

70. Size of House

Review and Assess

In 1997, a house was reported to have sold for $98.8 million. At $2,863 per square foot, it was the world’s most expensive house. How big was the house to the nearest square foot? (1.1)

71. Surface Area

Lake Superior, the largest of the Great Lakes, has a surface area of 20,600 square miles. This is 3300 square miles larger than five times the size of Lake Ontario, the smallest. What is the surface area of Lake Ontario? (1.5)



105

Answer Key Practice A 1. no solution 2. infinitely many solutions 3. one solution 4. 1, 5 is not a solution. 5. 2, 3 is not a solution. 6. 3, 4 is a solution. 7. 1, 3 is a solution. 8. 2, 1 is not a solution. 9. 0, 4 is a solution. 10.

11.

y

18.

19.

y 1

y

x

(3, 4)

5

1

(2, 6)

x

1

2, 6

3, 4

20.

21.

y 1 1

y

y

x 2

1

1 1

x

(1, 4)

2

1, 4 no solution

one solution

12.

no solution

22.

13.

y

23.

y

1

1 1

x

1 1

x

14.

x

no solution

24.

y x

1 1

(1, 2)

x

4 x

4

infinitely many solutions 16.

one solution 17.

y

y 1

(1, 2)

1

1, 2

x

1

1

(5, 2)

x

5, 2

1, 2

2, 0

25. x y 42

y 1

15.

y

(2, 0) 1

infinitely many solutions infinitely many solutions

y

y

1 1

x

x

16x 12y 568 16, 26

x

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The graph of a system of two linear equations is shown. Tell whether the linear system has infinitely many solutions, one solution, or no solution. 1.

2.

y

3.

y

1

Lesson 3.1

1

y

1

x

1

1

x

1

x

Check whether the ordered pair is a solution of the system. 4. 1, 5

5. 2, 3

6. 3, 4

7. 1, 3

8. 2, 1

9. 0, 4

3x y 2 4x 2y 5

x 2y 5 2x y 1

3x 5y 2 2x 3y 13

2x 5y 1 3x 2y 4

4x 7y 16 6x y 14

3x 4y 16 2x y 4

Graph the linear system and tell how many solutions it has. 10. 2x y 1

11. 4x y 3

12. 5x y 2

13. x 2y 6

14. 2x 3y 3

15. 3x y 2

4x 2y 8 3x 6y 2

2x y 1

6x 9y 9

10x 2y 4 5x 2y 2

Graph the linear system and estimate the solution. Then check the solution algebraically. 16. x y 3

17. x y 7

18. y 3x

19. x 3

20. y 4

21. x y 4

22. x y 1

23. 2x y 4

24. 3x 2y 1

2x y 4

xy3

xy7

2x y 2

2x 2y 2

3x 6

x 2y 14 xy5

x y 3

25. Amusement Park

A group of 42 people go to an amusement park. The admission fee for adults is $16. The admission fee for children is $12. The group spent $568 to get into the park. How many adults and how many children were in the group? Use the verbal model to write and solve a system of linear equatoins. Number Number Total in of adults of children the group Price for adults

14

Number Price for of adults children


Total cost Number of children of admission Copyright © McDougal Littell Inc. All rights reserved.

Answer Key 17. R 5600t 18. C 3800t 110,000

Practice B 1. 2, 1 is a solution. 2. 3, 5 is not a solution. 3. 1, 2 is not a solution. 4. B; one solution 5. C; no solution 6. A; infinitely many solutions. 8.

y 1

y

500 400 300

Cost

200 100 0 0

Revenue 10 20 30 40 50 60 70 80 90 x Time (months)

x

1

(4, 2)

1

(3, 0)

Cost and Revenue y Thousands of dollars

7.

19.

x

1

20. Somewhere between 61 and 62 months.

3, 0

4, 2

9.

10.

y

y

1 1

5

x

x

5

(2, 5)

(3, 1)

3, 1

2, 5

11.

12.

y

y

(0, 4) (2, 2) 1

1 1

x

2, 2

2

x

0, 4

13.

14.

y

(1, 1)

1

1 3 x

1, 1 15.

1

x

infinitely many solutions no solution

y

1

y

1

x

16. There were 340 $38 tickets sold and 200 $56

tickets sold.

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2. 3, 5

x 2y 4 3x y 5

3. 1, 2

3x 7y 34 5x 2y 5

4x 5y 6 7x y 5

Match the linear system with its graph. Tell how many solutions the system has. 5. 3x 2y 4

6. x 2y 6

A.

B.

C.

3x y 1

6x 4y 10

y

Lesson 3.1

4. 2x y 9

2x 4y 12

y

y

1 1

1

x

1

2

x

x

2

Graph the linear system and estimate the solution. Then check the solution algebraically. 7. x 2y 3

7x 3y 21

10. 3x 5y 19

5x 2y 20

8. 2x 3y 2

9. 3x y 8

5x 2y 16

2x 5y 1

11. 4x 3y 14

12. x 7y 28

2x 5y 14

9x 2y 8

Graph the linear system and tell how many solutions it has. If there is exactly one solution, estimate the solution and check it algebraically. 13. 3x 5y 2

4x 2y 2

1

15. 7x 2y 1

14. 2x 2y 5

x 4y 10

16. Ballet Performance

14x 4y 8

A ballet company says that 540 tickets have been sold for its upcoming performance of Swan Lake. Tickets for the Orchestra Center and Front Balcony seats are $56. Tickets for the Left and Right Orchestra and Balcony seats are $38. The company has sold $24,120 in tickets. How many $56 and $38 seats were sold?

Orchestra Orchestra Orchestra Left Center Right Front Balcony Balcony

Break-Even Analysis In Exercises 17–20, use the following information. You purchase a skateboard shop for $110,000. You estimate that monthly costs will be $3800. The monthly revenue is expected to be $5600. 17. Let R represent the revenue you bring in 18. Let C represent your costs, including the during the first t months. Write a linear purchase price, during the first t months. model for R. Write a linear model for C. 19. Graph the revenue and cost equations on the

same coordinate plane. Copyright © McDougal Littell Inc. All rights reserved.

20. How many months will it take until revenue

and costs are equal (the “break-even point”)? Algebra 2 Chapter 3 Resource Book

15

Answer Key Practice C 1 1. 1, 2 is a solution. 3.

2.

is not a solution.

is a solution

14.

15.

y

13, 38

5.

y

(

1

12, 32

4.

y

1 11 , 5

2 x

5

)

y

1 1

115, 25

(2, 5)

(1, 3) 2

2 2

16.

1, 3

infinitely many solutions

x

x

4

17.

y

y 1

2, 5

6.

y

1

1 1

(5, 0)

(1, 1)

(3, 1)

5, 0

x

1, 1

18.

3, 1 9.

y

1

y

1

0

x

1

x

x

( ) 1 , 2

infinitely many solutions 19. consistent and dependent 20. inconsistent 21. consistent and independent

y

1 1

x

1 1 x

1

8.

x

1

7.

y

x

(0, ) 3 4

no solution

0, 34

10.

11.

y

(

1 1 2, 4

y

)

1

1

(

2 3,

x 1

12, 14

13.

y

1

1

x

26.

y

( , ) 8 9 5 5

(, ) 1 3 2 2

1

)

23, 12

12.

1 3 2, 2

1 2

x

x 1

In Exercises 22–24, sample answers are given. 22. x y 6 23. x y 4 2x 2y 8 2x 2y 8 24. x y 4 2x y 6 25. The lines look parallel, but one has a slope of 9 19 10 and the other has a slope of 20 . So, they are not parallel and therefore intersect. P C 30, C 50 P 0.70C,, C 50 You must buy less than $100.

85, 95

Comparing Department Store Sales P 140 Sale price (dollars)

12, 0

120 100 80 60

P 0.70C

40 20 0 0

P C 30 20 40 60 80 100 120 140 C Regular price (dollars)

27. 4x 8y 100

x y 20 15 multiple-choice, 5 essay

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2. 3, 8

1

3. 2, 2

1 3

3x 4y 5 4x 6y 1

1 3

3x 8y 2 9x 16y 9

2x 6y 8 5x y 4

Lesson 3.1

Graph the linear system and estimate the solution. Then check the solution algebraically. 4. x 2y 7

5. 2x 3y 11

6. 4x 5y 9

7. 3x 5y 4

8. 4x 5y 2

9. 3x 4y 3

10. 4x 4y 3

11. 3x 2y 3

12. 4x 2y 1

3x y 6

3x 2y 16

x 2y 1 2x 8y 1

2x 3y 1

8x y 4

x 8y 6

6x 2y 3

3x 3y 3

Graph the linear system and tell how many solutions it has. If there is exactly one solution, estimate the solution and check it algebraically. 13. 4x 3y 1

14. 2x y 4

15. x y 6

16. 3x 4y 15

17. 2x y 4

18. 8x 2y 10

3x 6y 6

2x 3y 10

x 2y 3

2x 2y 12

4x 2y 8

4x y 5

Determine whether the following systems are consistent and independent, consistent and dependent, or inconsistent. 19. 3x 7y 5

6x 14y 10

20. 3x y 3

12x 4y 1

21. 4x 3y 6

6x 8y 8

22. Write a system of equations that has no solution. 23. Write a system of equations that has infinitely many solutions. 24. Write a system of equations that has exactly one solution. 25. The graph of the system

9x 10y 3 19x 20y 34 is shown to the right. Explain why there is a solution to this system. 26. Bargain Hunting

A local department store is having a coupon sale in which you receive $30 off any purchase over $50. A competing store is offering 30% off all purchases over $50. Write and graph two equations that describe the prices at both stores. When does the store offering the coupon sale have a better deal than their competitor?

16


27. Test Questions

A history test is to have 20 questions. The teacher uses multiple choice and essay questions. The multiple choice questions are worth 4 points each. The essay questions are worth 8 points each. The test has a total of 100 points. Write a system of equations to determine how many of each type of question appears on the exam. Copyright © McDougal Littell Inc. All rights reserved.

Answer Key Practice A 1. 2, 1 2. 3, 0 3. 1, 1 4. 2, 5 17 9 5. 7, 5 6. 3, 4 7. 7 , 7 23 6 97 5 8. 7 , 7 9. 11, 11 10. 1, 3 11. 0, 4 12. 3, 2 13. 2, 1 1 1 14. 6, 2 15. 2, 3 16. 1, 4 17. 2, 2 18. 5, 4 19. 3, 7 20. 0, 0 6 80 4 23. 31, 31 61 23 , 23 4637, 2137 25. 2, 27 26. 4, 0

21. 1, 2 22. 24.

27. 3, 18 28. 70x 8y 226,

280x 70y 980; The boosters should rent 3 buses and 2 vans.

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LESSON

NAME

3.2

DATE


Solve the system using the substitution method. 1. x 3y 5

2. 2x y 6

3. 3x 7y 10

4. 5x 2y 20

5. x y 12

6. 4x y 8

7. 3x y 6

8. 4x 6y 8

9. x 7y 12

2x 3y 1

3x 5y 9

6x y 7

2x 3y 1

2x 4y 10

3x y 9

x 4y 5

x 3y 9

2x 8y 14

Solve the system using the linear combination method. 10. 4x 2y 2

11. 7x 3y 12

12. 6x 7y 4

13. 6x 3y 15

14. x 2y 7

15. 2x y 2

5x 2y 11 6x 5y 7

7x 2y 8 x 2y 5

x 7y 17

2x 5y 16

Lesson 3.2

Solve the system using any algebraic method. 16. x 2y 7

17. x 3y 8

18. x y 9

19. x y 4

20. 3x 4y 0

21. 2x y 0

22. 2x 5y 4

23. 5x 7y 12

24. 3x 4y 6

25. 4x 7y 10

26. 2x 3y 8

27. 6x y 0

3x 5y 17 x 2y 17 3x 4y 9

4x 3y 2 9x 4y 0 3x 2y 8

3x 7y 4

x 5y 4

xy1

2x y 4 4x 7y 1

15x 2y 9

28. Band Competition

The band boosters are organizing a trip to a national competition for the 226-member marching band. A bus will hold 70 students and their instruments. A van will hold 8 students and their instruments. A bus costs $280 to rent for the trip. A van costs $70 to rent for the trip. The boosters have $980 to use for transportation. Use the verbal model below to write a system of equations whose solution is how many buses and vans should be rented. Solve the system. Students per bus Price per bus

28

Number Students of buses per van Number Price of buses per van


Number Students of vans on trip

Cost of Number of vans transportation


Answer Key Practice B 1 1. 2, 1 2. 3, 2 3. 2, 4 4. 5, 3 5 1 1 5. 6, 9 6. 2, 2 7. 2, 4 8. 2, 1 2 7 9. 3, 7 10. 2, 2 11. 3, 2 2 12. 1, 5 13. 2, 7 14. 4, 0 5 1 15. 3, 18 16. no solution 17. 3, 2 18. infinitely many solutions 19. 4, 2 35 8 20. 2, 7 21. 11, 11 22. 1993 23. Forty full size bags and 84 collapsible bags. 24. You drove 3 hours and your friend drove 2 hours.

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LESSON

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DATE


Solve the system using the substitution method. 1. 2x 5y 9

2. 3x 4y 1

3. 6x 2y 11

4. x 2y 1

5. 4x 3y 3

6. 10x 16y 17

3x y 7

x 2y 1

5x 7y 4

2x y 3

4x y 6 xy3

Solve the system using the linear combination method. 7. 5x y 6

5x 3y 22

10. 2x 7y 10

3x 2y 10

8. 2x 3y 4

9. 4x y 5

11. 3x 4y 12

12. 5x 2y 15

8x 3y 1

4x 3y 9

6x 2y 11

7x 5y 18

Solve the system using any algebraic method. 13. 4x 7y 10

14. 2x 3y 8

15. 6x y 0

16. 6x 3y 1

17. 3x 8y 1

18. 4x 16y 4

19. 2x 8y 8

20. 5x y 17

21. 3x 9y 3

3x 7y 4

6x 2y 11

3x 2y 16

3x 2y 8

15x 2y 9 3x 12y 3

Lesson 3.2

4x 2y 7

x 5y 4

x 8y 9

22. CDs and Cassettes

For 1990 through 1998, the manufacturer’s shipments for audio cassettes, A (in millions), and compact discs, C (in millions), can be modeled by the equations A 31.8t 322 C 42.8t 110

Audio cassette shipments Compact disc shipments

where t is the number of years since 1990. In what year did the number of compact discs shipped surpass the number of audio cassettes shipped? 23. Golf Bags

A sporting goods store receives a shipment of 124 golf bags. The shipment includes two types of bags, full-size and collapsible. The full-size bags cost $38.50 each. The collapsible bags cost $22.50 each. The bill for the shipment is $3430. How many of each type of golf bag are in the shipment?

24. Vacation Trip

You and a friend share the driving on a 280 mile trip. Your average speed is 58 miles per hour. Your friend’s average speed is 53 miles per hour. You drive one hour longer than your friend. How many hours did each of you drive? Use the following verbal model. Your speed

Your time Friend’s speed

Friend’s time Total distance

Your time Friend’s time 1 hour



29

Answer Key Practice C 1. 4, 3 2. 2, 5 3. 1, 3 4. 1, 5 5. no solution 6. infinitely many solutions 7. 3, 2 8. 1, 4 9. 2, 3 10. infinitely many solutions 11. 6, 2 1 1 12. 2, 3 13. 10, 20 14. 8, 5 3 1 15. 5, 6 16. 3, 4 17. 5, 4 18.

23, 52

19.

76, 23

21. 2.3, 0.4 22. 1876

20.

143, 0

23.

24. Sample answer: U 20.51t 104.4; N 24.96t 83.1 25. 4.8, 202.6 In 1994 the

United Kingdom and Netherlands both had about 202 computers per 1000 people.

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LESSON

3.2

NAME

DATE


Solve the system using the substitution method. 1. 2x y 5

2. 4x 2y 2

3. x 3y 8

4. 6x 2y 4

5. x 3y 18

6. 4x y 6

4x 3y 7

x 3y 13

8x y 3

2x 3y 7

x 3y 12

43x 13y 2

Solve the system using the linear combination method. 7. 2x 3y 12

8. 7x 2y 1

3x 4y 1

8x 4y 8

10. 2x 5y 1

11.

x 52y 12

1 2x 1 3x

3y 9

9. 3x 4y 6

2x 5y 19

12. 4x 6y 4

6x 3y 2

y4

Solve the system using any algebraic method. 13. 0.25x 0.5y 12.5

14. 0.75x 0.3y 4.5

15. 0.2x 1.4y 9.4

16. 0.8x 2.1y 10.8

17. 5x 4y 4

18. 5x y

Lesson 3.2

0.3x 0.5y 13

0.125x 0.4y 1

0.5x 0.7y 1.7

7 2x 2y 10

1.6x 0.7y 7.6 19. 6x 9y 1

20. 0.3x 0.2y 1.4

2x 4y 5

0.12x 0.8y 0.56

5 6

3x 4y 8 21. 4.2x 2.1y 10.5

1.4x 1.3y 2.7

22. Labor Force

From 1840 to 1990 the percent of the labor force in farming and non-farming occupations can be modeled by the following equations where t is the number of years since 1840. y 0.48t 67.2

farming occupations

y 0.48t 32.9

nonfarming occupations

In what year was the labor force split equally into farming and nonfarming occupations? Round your answer to the nearest year. Computers Per Capita Use the table below of the number of computers per 1000 people in the United Kingdom and Netherlands from 1991 through 1995. Years since 1990, t United Kingdom, U Netherlands, N

1 125.7 109.7

2 144.8 131.1

3 164.8 156.9

4 187.4 184.3

5 216.5 214.8

23. Use a graphing calculator to make scatter plots for the data. 24. For each scatter plot, find an equation of the line of best fit. 25. Find the coordinates of the point of intersection. Describe what this point

represents.

30



Answer Key Practice A 1. 0, 1 is a solution. 2. 3, 2 is a solution. 3. 5, 2 is not a solution. 4. 0, 0 is not a solution. 5. 1, 1 is not a solution. 6. 1, 2 is not a solution. 7. Answers may vary. Sample: 0, 0 8. Answers may vary. Sample: 1, 2 9. Answers may vary. Sample: 1, 1 10. B 11. A 12. C 13.

14.

y

y

1 1

1 1

15.

x

x

16. t 0

y

t2 d 65t

1 1

x

17. No; To drive 200 miles at 65 miles per hour,

you would need to drive over 3 hours.

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LESSON

NAME

3.3

DATE


Tell whether the ordered pair is a solution of the system. 1. 0, 1

2. 3, 2 y

3. 5, 2 y

y 1

1

x

1

2

1 x

x

2

Tell whether the ordered pair is a solution of the system. 5. 1, 1

4. 0, 0

xy ≥ 2 x ≥ 0

6. 1, 2

2x y < 4 xy < 1 x > 0

2x y < 1 xy ≥ 2

Give an ordered pair that is a solution of the system. 7. 2x 3y < 5 x < 12

8. x 3y > 3 y < 8

9. 5x ≤ 2y x < 0 y > 0

Match the system of linear inequalities with its graph. 10. y ≤ x y ≥ 2 x ≤ 3

Lesson 3.3

A.

11. y ≥ x y ≥ 2 x ≤ 3 B.

y

12. y ≤ x y ≤ 2 x ≤ 3 C.

y

y 2

2

2 2

x

2

x

2

x

Graph the system of linear inequalities. 13. x > 4 y < 2

14. x ≥ 0 y ≤ x2

15. 2x y < 1 y ≥ 2

Distance Traveled

In Exercises 16 and 17, use the following information. You are taking a trip with your family. You are going to share driving time with your dad. You are only allowed to drive for at most two hours at one time. The speed limit on the highway on which you are traveling is 65 miles per hour. 16. Write a system of inequalities that describes the number of hours and

miles you might possibly drive. 17. Is it possible for you to have driven 200 miles?

42



Answer Key Practice B 1. B 2. C 3. A

16.

17.

y

y

2

4.

5.

y

y

x

2

2 4 x

1 x

1

1 x

1

18. 6.

1

7.

y

y

y 1

1

2 x

1

x

6

19. x y 44 8.

9.

y

1

x

1 x

1

11.

y

y 3

1

3

x

2

x

x

1

12.

13.

y

y

2 2 x

2

14.

15.

y 4

y 2 x

4

x

2

20. y 0

y3

yx

x0

1 y x 30 2 y 3x 105

y 1

10.

x

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LESSON

NAME

3.3

DATE


Match the system of linear inequalities with its graph. 1. x y > 2

2x 3y < 1 A.

3. x y > 2

2. x y ≥ 2

2x 3y > 1 B.

y

2x 3y > 1 C.

y

1

y

1 1

x

1 1

x

x

1

Graph the system of linear inequalities. 4. x > 2

y ≤ 4 7. x y < 3

5. y < 2

y > 3 8. y ≤ 2x

2x y > 5

x < 3

10. x 2y > 4

11. y ≤ 5

x 3y < 1 1

13. y > 2x 4

y ≤ x 3 y ≤ 2x 16. 2x y < 3

x < 5 9. 2x y ≤ 1

y > 3x 12. x ≥ 3

x > 3 y ≤ 2x 2 14. x y < 1

x ≤ 4 y < x5 15. y ≥ 3x 4

y ≤ 12x 3 x > 2

2x y < 4 x ≥ 2 17. x 2y ≤ 10

18. 2x y > 1

2x y ≤ 8 2x 5y < 20

x 2y < 4 x 2y > 4

Your class has rented buses for a field trip. Each bus seats 44 passengers. The rental company’s policy states that you must have at least 3 adult chaperones on each bus. Let x represent the number of students on each bus. Let y represent the number of adult chaperones on each bus. Write a system of linear inequalities that shows the various numbers of students and chaperones that could be on each bus. (Each bus may or may not be full.)

Lesson 3.3

x y > 6 y ≥ 0

6. y ≥ 0

19. Field Trip

The diagram at the right shows the cross section of an iceberg. Write a system of inequalities that represents the portion of the iceberg that extends above the water.

y

20. Iceberg

(20, 20) (30, 15) 10

(0, 0)

(35, 0) 10

x

Water level



43

Answer Key 15. y x 1

Practice C 1.

2.

y

1 1

3.

x

x

1

4.

y

16. C 900

C 1700 F0 F 0.30C No

y

1

x

4

6.

y

y

2 1

x

2

1

7.

8.

y

1

x

y 3

1

9.

1

x

4

x

x

10.

y

y

4 4 x

8

11.

12.

y

y 1 1

2

2

13. x 4

x2 y 3 y1

0 900

x

2

14. y 3x 3

y x y2

1300 1700 2100 C Total calories

x

17. 5.

F 510 470 430 390 350 310 270 0

2

x

1 x 5 1 5x

3 10 y 80 3 10 y

90 x0 x 200 y0 y 200

y 350 Test score

1

Fat calories

1

yx1 y 2

y

250 150 50 0

0 50

150 250 Quiz score

350 x

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LESSON

3.3

NAME

DATE


Graph the system of linear inequalities. 1. x 2y < 4 3x y > 1

2. 2x 3y ≥ 6 x 4y ≤ 8

3. 3x y < 0 3x 4y > 8

4. 2x y < 3 xy < 0 x > 3

5. x y ≤ 2 3x y ≥ 4 y ≥ 4

6. 2x y < 1 x 3y < 6 x > 0

7. x y > 3 x 2y > 4 xy < 4

8. 2x y ≥ 2 2x y ≤ 1 x 3y ≥ 3

9. 3x 6y > 4 3x 4y > 4 xy < 5

10. 2x y < 3 x y > 1 x ≥ 0 x < 2

11. x y ≤ 1 x 2y ≥ 2 xy ≤ 4 x ≥ 1

12. x y < 2 x y > 3 2x 3y > 0 2x 3y < 9

Write a system of linear inequalities for the shaded region. 13.

14.

y

15.

y

y

2 2 1

x 1

1 1

x

x

Each day the average toddler needs to consume 900 to 1700 calories. At most 30% of a toddler’s total calories should come from fat. Write and graph a system of linear inequalities describing the number of fat calories F and total calories C for the diet of a toddler. According to your model, is a toddler following a healthy diet if he or she consumes 1200 calories a day and 372 of those calories are from fat?

Lesson 3.3

16. Toddler Nutrition

17. Weighted Averages

To determine your grade in science class, your teacher uses a weighted average. Your grade is a combination of quiz and test scores. There are a total of 200 quiz points and 200 test points. 3 Your grade is calculated by adding 51 of your quiz points to 10 of your test points. To receive a B your weighted total must be less than 90 and at least 80. Write and graph a system of inequalities describing the possible combination of quiz and test points that you can earn to receive a B.

44



Answer Key Practice A 1. minimum 6; maximum 5 2. minimum 4; maximum 10 3. minimum 15; maximum 5 4. minimum 0; maximum 18 5. minimum 3; maximum 17 6. minimum 11; maximum 21 7. minimum 0; maximum 8 8. minimum 0; maximum 8 9. minimum 9; maximum 1 10. P 40x 55y

11. 2x 6y 150

5x 4y 155 x0 y0 12. Granola bars (cases)

Production Hours 40 35 30 25 20 15 10 5 0

(0, 25) (15, 20)

(31, 0) 0 5 10 15 20 25 30 35 40 Breakfast bars (cases)

13. Fifteen cases of breakfast bars and 20 cases of

granola bars should be made to maximize profit.

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LESSON

NAME _________________________________________________________ DATE ___________

3.4


The feasible region determined by a system of constraints is given. Find the minimum and maximum values of the objective function for the given feasible region. 1. C x y 2. C x 2y 3. C 2x y y

y

y

(2, 7)

(0, 6)

(0, 5) 2

(5, 0)

(0, 0)

(3, 3) x

2

y

(3, 3)

5. C 3x 4y

y

(0, 4)

6

(1, 3) (3, 2)

(1, 2)

(4, 0) x

(2, 1)

(2, 3)

(1, 0)

1

(0, 0)

(6, 3)

6. C 3x 5y

y

(3, 5)

x

2

2

(5, 1)

(0, 2)

4. C x 3y

(6, 1)

(3, 1)

x

2

(0, 3)

2

1

(3, 0) x

1

1

(6, 0)

6 x

Find the minimum and maximum values of the objective function subject to the given constraints. 7. Objective function:

8. Objective function:


C 2x y

Cxy

Cxy

Constraints: x ≥ 0 y ≥ 0 xy ≤ 4

Constraints: x ≥ 0 x ≤ 3 y ≥ 0 y ≤ 5

Constraints: x ≤ 0 y ≤ 4 x y ≥ 1

Breakfast Bars In Exercises 10–13, use the following information. Your factory makes fruit filled breakfast bars and granola bars. For each case of breakfast bars, you make $40 profit. For each case of granola bars, you make $55 profit. The table below shows the number of machine hours and labor hours needed to produce one case of each type of snack bar. It also shows the maximum number of hours available. Breakfast bars 2 5

Granola bars 6 4

10. Write an equation that represents the profit

(objective function). 12. Sketch the graph of the constraints found in

Exercise 11 and label the vertices. Copyright © McDougal Littell Inc. All rights reserved.

Maximum hours 150 155 11. Write a system of inequalities that represents

the constraints. 13. How many cases of each product should you

make to maximize profit? Algebra 2 Chapter 3 Resource Book

57

Lesson 3.4

Production Hours Machine hours Labor Hours

Answer Key Practice B 1. minimum 6; maximum 5 2. minimum 0; maximum 24 3. minimum 4; maximum 26 4. minimum 6; maximum 9 5. minimum 6; maximum 26 6. minimum 2; maximum 20 7. minimum 9; maximum 20 8. minimum 8; no maximum 9. minimum 6; maximum 30 10. two batches of bread and 16 batches of muffins 11. zero long distance calls (0 minutes) and 24 local calls (240 minutes)

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LESSON

NAME _________________________________________________________ DATE ___________

3.4


The feasible region determined by a system of constraints is given. Find the minimum and maximum values of the objective function for the given feasible region. 1. C x y

2. C 2x 4y

y

y

(2, 7)

(0, 6)

y

(0, 3)

(5, 0) 2

x

(6, 4)

(0, 2)

(4, 4) (6, 1)

2

(0, 0)

3. C x 5y

1

1

x

1 1

(1, 1)

(6, 0) x

(4, 1)

(0, 0)




C 3x y

C 2x 4y

C x 5y

Constraints: x ≥ 0 y ≥ 0 2x y ≤ 6

Constraints: x ≤ 3 xy ≥ 3 2x 3y ≥ 9

Constraints: 3x 2y ≤ 8 2x y ≥ 4 x 4y ≤ 2




C 4x 3y

C 2x 3y

C 5x 2y

Constraints: x ≥ 0 x ≤ 5 y ≥ 0 2x 5y ≥ 15

Constraints: x ≥ 0 y ≥ 1 4x y ≥ 6 x 2y ≥ 5

Constraints: x ≤ 4 2x y ≥ 3 x 3y ≤ 2 x 2y ≤ 6

10. Bakery A bakery is making whole-wheat bread and apple bran muffins. For

each batch of bread they make $35 profit. For each batch of muffins they make $10 profit. The bread takes 4 hours to prepare and 1 hour to bake. The muffins take 0.5 hour to prepare and 0.5 hour to bake. The maximum preparation time available is 16 hours. The maximum baking time available is 10 hours. How many batches of bread and muffins should be made to maximize profits? 11. Phone Bill On a typical long distance call you talk for 30 minutes. On a Lesson 3.4

typical local call you talk for 10 minutes. Your phone company offers a special low rate of $.08 per minute for long distance calls and $.03 per minute for local calls for customers who spend at least 240 minutes on the phone per month. Your parents have set a limit of no more than 15 long distance calls per month and 30 local calls per month. How many minutes of long distance and local calls should you make to qualify for the special rate plan and minimize your phone bill?

58



Answer Key Practice C 1. min. of 12 at (0,4); max. of 8 at (4,0) 32 8 8 2. min. of 4 at (1, 1); max. of 3 at 3, 3 3. min. of 0 at (0, 0); max. of 12 at 4, 0 4. min. of 6 at (0, 3); max. of 20 at (4, 0) 15 3 5. min. of 0 at (0, 0); max. of 2 at 3, 2 6. min. of 6 at 0, 2; max. of 27 at (6, 5) 7. min. of 2 at (0, 2); max. of 8 at (4, 4) 8. min. of 3 at 3, 5; max. of 42 at (3, 8) 9. min. of 30 at (0, 6); max. of 16 at 1, 3 10. 7 paperbacks and 1 hard cover book 11. 0.625 servings of pork, 3.75 servings of

potatoes

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LESSON

3.4

NAME _________________________________________________________ DATE ___________





C 2x 3y

C x 3y

C 3x 2y

Constraints: x ≥ 0 y ≥ 0 xy ≤ 4

Constraints: x 2y ≤ 8 xy ≥ 0 y ≥ 1

Constraints: x ≥ 0 y ≥ 0 xy ≤ 4 x y ≥ 3




C 5x 2y

C 2x y

C 2x 3y

Constraints: x ≥ 0 y ≥ 0 2x y ≤ 8 x 3y ≤ 9

Constraints: x ≥ 0 x ≤ 3 3 2x y ≥ 0 3x 2y ≤ 12

Constraints: x ≤ 6 y ≤ 5 2x 3y ≤ 6 x 3y ≥ 6




C 3x y

C 6x 3y

C x 5y

Constraints: y ≤ 4 xy ≥ 2 2x y ≤ 4 x y ≤ 2

Constraints: x ≥ 3 xy ≥ 0 2x y ≤ 11 x y ≤ 11 2x y ≥ 2

Constraints: x ≥ 3 y ≥ 3 y ≤ 6 x y ≤ 6 3x y ≤ 6 x y ≥ 3

10. Gift Basket You want to make a gift basket for your mother who is an

avid reader. You decide to include hard cover books and paperbacks in the basket. You have $80 to spend on books. Each hard cover costs $24 and each paperback costs $8. The basket will hold at most 3 hardcover books or 7 paperbacks. Find the maximum number of books you can include in the basket. 11. Nutrition You are planning to have roast pork and twice baked potatoes

carbohydrates fat protein

Pork 8g 6g 23 g


Lesson 3.4

for dinner. You want to consume at least 250 grams of carbohydrates, but no more than 60 grams of fat per day. So far today you have consumed 170 grams of carbohydrates and 30 grams of fat. The table below shows the number of grams of carbohydrates, fat, and protein in a serving of roast pork and twice baked potatoes. How many servings of each can you eat to fulfill your daily requirements for carbohydrates and fat while maximizing the amount of protein you consume? Potatoes 20 g 7g 5g Algebra 2 Chapter 3 Resource Book

59


2.

z

z

(0, 1, 2) (1, 3, 1) y

y

x

x

3.

4.

z

z

(2, 2, 4)

(2, 0, 1) y

y

x

5.

6.

11. x-intercept: 3

y-intercept: 4 z-intercept: 4 12. x-intercept: 4 y-intercept: 3 z-intercept: 2 14. x-intercept: 2 y-intercept: 7 2 z-intercept: 3

y-intercept: 6 z-intercept: 2 13. x-intercept: 10 y-intercept: 4 z-intercept: 20 3 15. x-intercept: 4 y-intercept: 9 z-intercept: 3

16. 13

17. 18

21. 27

22. 1

18. 0

19. 3

20. 17

23. 26 24. 2

25. f x, y 12 2x 3y

x z

10. x-intercept: 4

26. f x, y 1 3x 2y

z

27. f x, y x 3y 8 28. f x, y 5x 2y 4

y

29. f x, y 9 7x 8y

y

(1, 1, 0)

x

30. f x, y 6x y

(2, 5, 0)

x

31. A: 2, 0, 0

7.

z

8.

(1, 2, 4)

z

(3, 2, 5)

y x

y x

9.

z

y

(2, 3, 1) x

32. A: 0, 0, 2

B: 2, 4, 0 B: 0, 6, 2 C: 0, 4, 5 C: 0, 6, 0 D: 2, 0, 5 D: 4, 6, 0 33. z 18x 12y 50; Answers may vary. Sample: x 1 0 1 2 1 y z

0 68

1 62

1 80

1 98

2 92

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Lesson 3.5

LESSON

NAME _________________________________________________________ DATE ___________

3.5



2. 1, 3, 1

3. 2, 0, 1

4. 2, 2, 4

5. 1, 1, 0

6. 2, 5, 0

7. 1, 2, 4

8. 3, 2, 5

9. 2, 3, 1

Find the x-intercept, y-intercept, and z-intercept of the graph of the linear equation. 10. x y z 4

11. 2x y 3z 6

12. 3x 4y 6z 12

13. 2x 5y z 20

14. 7x 2y 21z 14

15. 12x y 3z 9

Evaluate the function for the given values. 16. f x, y 3x 2y, f 1, 5

17. f x, y x 6y, f 0, 3

18. f x, y 3x 2y, f 2, 3

19. f x, y x y, f 1, 4

20. f x, y 5x y, f 3, 2

21. f x, y 7x 2y, f 3, 3

22. f x, y 3x 4y, f 1, 1

23. f x, y 4x 3y, f 5, 2

24. f x, y 8x 3y, f 2, 6

Write the linear equation as a function of x and y. 25. 2x 3y z 12

26. 3x 2y z 1

27. x 3y z 8

28. 5x 2y z 4

29. 7x 8y z 9

30. 6x y z 0

31. Geometry

Write the coordinates of the vertices A, B, C, and D of the rectangular prism shown, given that one vertex is the point 2, 4, 5.

32. Geometry

Write the coordinates of the vertices A, B, C, and D of the rectangular prism shown, given that one vertex is the point 4, 6, 2. z

z

C D

A

(4, 6, 2)

B

(2, 4, 5) C y

A

x

y

D

B

x

33. Music Club

A music club requires an initial purchase of $50 worth of merchandise. After this initial fee, compact discs may be purchased for $18 and audio cassettes may be purchased for $12. Write an equation for the amount that you will spend as a function of the number of compact discs and audio cassettes that you buy. Make a table to show the different cost for several different numbers of compact discs and audio cassettes.

72




11. 2.

z

12.

z

z

(0, 0, 3)

z

( , 0, 0) 3 2

y

(0, , 0) 3 2

(1, 0, 0)

x

(3, 5, 2)

(1, 3, 1)

(0, 3, 0)

y

(0, 0, 3)

y

y

x

x

x

13. 3.

4.

z

14.

z

(

z

5 2

0, 0,

y

)

(0, 1, 0)

1 6

(0, 5, 0)

(4, 1, 3)

x

( , 0, 0) y

y

x

z

x

y

x (1, 0, 0)

(0, 0, 1)

(0, 0, 5)

15. 5.

6.

z

16.

z

(

0, 0,

(0, 4, 0)

z

z

(4, 0, 0)

)

1 2

(0, 6, 0) y

y

(3, 0, 0)

x

x

(0, 3, 2)

(0, 0, 3) y y

x

(1, 3, 2)

x

7.

8.

z

17.

z

(0, 0, ) 6 5

z

(0, 0, 8)

(0, 0, 4)

(3, 0, 0)

(0, 2, 0)

y

x

(0, 8, 0)

(0, 2, 0)

y x

y

(8, 0, 0)

x

(4, 0, 0)

18.

z

(0, 14, 0)

(4, 0, 0) y

9.

(0, 0, ) 14 3

z x

(0, 12, 0) y

(6, 0, 0)

19. f x, y 5 4x y; 4

(0, 0, 4)

x

20. f x, y 3 3x 2y; 1 21. f x, y 5x 3y 7; 21

10.

22. f x, y 3x y 2; 11

z

(0, 0, 5)

23. f x, y 3 x 2y; 2 1

(0, 15, 0) y

(6, 0, 0) x

24. f x, y 4x 2y 6; 4 3

1

33

25. f x, y 3 5x 5y; 0 2

1

26. f x, y 2 8x 4y; 2 3

1

5

27. f x, y 4x 5y 2; 4 1

28. 30

1

1 3

29. 48

30. f x, y 35x 60y 200; $1500 31. f x, y 2x 3.5y 12; $25

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LESSON

NAME _________________________________________________________ DATE ___________

3.5

Lesson 3.5



2. 3, 5, 2

3. 0, 0, 5

4. 4, 1, 3

5. 0, 3, 2

6. 1, 3, 2


8. x 2y z 4

9. 2x y 3z 12

10. 5x 2y 6z 30

11. 2x y z 3

12. 3x 2y z 3

13. 5x y 2z 5

14. 6x y z 1

15. 4x 3y 24z 12

16. 3x 2y 4z 12

17. 2x 3y 5z 6

18. 7x 2y 6z 28

Write the linear equation as a function of x and y. Then evaluate the function for the given values. 19. 4x y z 5, f 1, 5

20. 3x 2y z 3, f 0, 2

21. 5x 3y z 7, f 2, 6

22. 3x y z 2, f 2, 3

23. 2x y 2z 6, f 4, 2

24. 3x 2y 4z 24, f 1, 3

25. 2x y 5z 15, f 6, 3

26. 3x 2y 8z 16, f 0, 2

27. 5x 4y 20z 10, f 1, 5

Use the given point 3, 5, 2 to find the volume of the rectangular prism.

28. Geometry

Use the given point 4, 2, 6 to find the volume of the rectangular prism.

29. Geometry

z

z

(3, 5, 2) (4, 2, 6) y y

x x

30. Yearbook Advertisements

The yearbook club’s bank account has $200 remaining from last year’s advertising campaign. You are now trying to sell advertisements to local businesses for this year’s yearbook. A quarter page ad costs $35. A half page ad costs $60. Write an equation for the total amount of money you may spend as a function of the number of quarter and half page ads that you sell. Evaluate the model if you sell 20 quarter page ads and 10 half page ads.

31. Baseball Game

You and a group of your friends go to a professional baseball game. Your ticket costs $12. Bottled water costs $2 and hotdogs cost $3.50. Write an equation for the cost of going to the game as a function of the number of bottled waters and hotdogs you purchase. Evaluate the model if you buy 3 bottled waters and 2 hotdogs.



73


15. 2.

z

( , 0, 0) (0, ,0) 4 3

z

z

(2, 1, 4)

4 5

y y y

x

(1, 3, 2)

x

(0, 0, 2)

x

3.

16. f x, y 4 2x 3y; 6 2

4.

z

z

17. f x, y

( , 3, ) 1 2

3 2

18. f x, y

y

(2, 4, 1)

y

x

19. f x, y

x

5.

(

5 3

) y

x 2

(1, 3 , 3)

x

8.

(0, 5, 0)

z

y

z

(0, 0, 1)

(0, 7, 0)

y

x

(2, 0, 0) x

(0, 0, 10)

9.

10.

z

(0, 0, 1) (0, 3, 0)

( , 0, 0) 1 2

(0, , 0) 2 3

z

y

y

(

x

11.

3 , 2

0, 0

)

(0, 0, ) 2 5

x

12.

z

( , 0, 0) 2 3

(0, 0, ) 1 4

z

(0, 0, 2)

(0, , 0) 4 3

(0, 2, 0) y

y

x

( , 0, 0) 4 5

x

13.

14. z

z

(0, 0, ) 3 2

(0, 0, ) 3 4

x

(

3

)

2 , 0, 0

16x 13y; 23 3

y

(4, 0, 0)

34x 12y; 0

( , 0, 0) 1 2

(0, 1, 0)

(0, 3, 0) y

y x

21 2

21. f x, y 7 2x 2y; 6

z 1

2 , 2,

7.

12x 14y; 14

20. f x, y 2 5x 3y;

6.

z

1 4 3 4 3 2

7

19

In Exercises 22–24, sample answers are given. 22. 3x 2y 3z 6 23. 10x 15y 6z 30 24. 10x 5y 2z 5 25. f x, y 3x y ; 7 points 26. f x, y 14.95 6.95x 24.95 10y 2 19.90 6.95x 10y; $73.80 27. 3x 2y z 38,387; 4129 3-point shots

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Lesson 3.5

LESSON

NAME _________________________________________________________ DATE ___________

3.5


Plot the ordered triple in a three-dimensional coordinate system. 1. 2, 1, 4 4.

1 2,

3,

3 2

2. 1, 3, 2

5. 1,

2 3,

3. 2, 4, 1

3

6. 2, 2, 3 1

5

Sketch the graph of the equation. Label the points where the graph crosses the x-, y-, and z-axes. 7. 5x 4y 2z 20

8. 7x 2y 14z 14

9. 2x y 3z 3

10. 4x 3y 5z 2

11. 3x y 8z 2

12. 5x 3y 2z 4

13. 2x y 4z 3

14. 6x 3y 2z 3

15. 3x 5y 2z 4

Write the linear equation as a function of x and y. Then evaluate the function for the given values. 16. 6x 2y 3z 12, f 0, 3

17. 2x y 4z 1, f 1, 2

18. 3x 2y 4z 3, f 1, 0

19. x 2y 6z 9, f 3, 4

20. 5x 3y z 2, f 2, 2

21. 3x 7y 2z 14, f 1, 3

1

2

Write an equation of the plane having the given x-, y-, and z-intercepts. Explain the method you used. 22. x-intercept: 2

y-intercept: 3 z-intercept: 2

23. x-intercept: 3

24. x-intercept:


1 2


25. Place-kicker

In football a placekicker is responsible for kicking field goals worth 3 points and extra points after touchdowns worth 1 point. Write a model for the total number of points that a placekicker can score in a game. In Super Bowl XXXII, Jason Elam kicked 4 extra points and 1 field goal for the Denver Broncos. Use the model to determine the total number of points scored by Elam.

26. Photography Studio

A photography studio charges a $14.95 sitting fee. A sheet of pictures can consist of one 8 x 10, two 5 x 7’s, four 3 x 5’s, or twenty-four wallets. The studio charges $6.95 for a sheet of pictures. Holiday cards with your photo may be purchased. Twenty holiday cards cost $24.95 plus $10 for each addition 10-card order. Write a model for the total cost (not including tax) of buying pictures if you intend to purchase at least 20 holiday cards. Evaluate the model if you buy 40 holiday cards and two sheets of pictures.

27. N.B.A. Lifetime Leader

Kareem Abdul-Jabbar is the N.B.A. lifetime leader in points scored with 38,387. Today, a player can score a threepoint shot worth 3 points, a field goal worth 2 points, or a free throw worth 1 point. Write a model for the types of points needed to match Abdul-Jabbar’s record. How many three-point shots are needed in a career to match the record if 12,000 field goals and 2000 free throws are scored?

74



Answer Key Practice A 1. 1, 1, 1 is a solution. 2. 0, 3, 1 is a solution. 3. 2, 1, 6 is not a solution. 4. 3, 2, 1 5. 4, 1, 3 6. 3, 5, 4 7. 9, 4, 2 8. 2, 2, 0 9. 1, 3, 2 10. 1, 1, 3 11. 2, 1, 4 12. 1, 0, 5 13. infinitely many solutions 14. no solutions 15. 2, 1, 3 16. x y z 22 3y 4z 54 xz 17. Six are under age 5, 10 are ages 5–16, and 6 are ages 16 and up.

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LESSON

NAME _________________________________________________________ DATE ___________

3.6


Decide whether the given ordered triple is a solution of the system. 1. 1, 1, 1

2. 0, 3, 1

Lesson 3.6

xyz3 2x y 4z 5 x 4y 2z 3

3. 2, 1, 6

x 2y z 7 4x y 3z 0 2x y 5z 2

x y z 3 2x y z 9 4x y z 15

Solve the system using the substitution method. 4. x 2y 3z 4

5. x 3y 1

yz3 z 1 7. x 2y z 1

6. x 5y 7z 6

y 2z 5 z3

y 3z 7 z 4

8. 4x y 2z 6

yz2 4z 8

9. x 2y z 3

y 4z 2 2y 4

x 2y 5 x 1

Solve the system using the linear combination method. 10. x y z 5

11. x 2y 3z 8

2x y z 4 3x y 2z 8 13. x 2y 4z 2

12. 2x y z 7

2x y 3z 17 x 3y 3z 11

2x y 3z 17 2x 3y 2z 12

14. 2x 3y z 4

x 2y 4z 2 x 2y 4z 2

15. x y z 6

4x 6y 2z 6 2x y z 2

xyz0 xyz4

Pool Admission

In Exercises 16 and 17, use the following information. A public swimming pool has the following rates: ages under 5 are free, ages 5–16 are $3, and ages 16 and up are $4. The pool also has a policy that every child under age 5 must be accompanied by an adult. The families in your neighborhood decide to go to the pool as part of a summer party. There are 22 people in your group and an equal number of children under age 5 as people 16 years old and older. The total admission cost was $54. Use the model below. Number of people Number of people Number of people Total number ages 5–16 ages 16 and up of people under age 5 Rate for under age 5

Rate Number of for ages people 5–16 under age 5

Rate for Number of people ages 16 and over ages 5–16

Number of Total people ages cost 16 and over

Number of Number of people ages people 16 and over under age 5 16. Write a system of linear equations in three

variables to find the number of people in each age category in your group. 84


17. How many people in your group are in the

different age categories designated by the pool? Copyright © McDougal Littell Inc. All rights reserved.

Answer Key Practice B 1. 0, 0, 3 is a solution. 2. 1, 2, 5 is a solution. 3. 0, 0, 0 is not a solution. 4. 1, 3, 2 is a solution. 5. 5, 7, 1 is not a solution. 6. 4, 8, 9 is not a solution. 7. 2, 8, 1 8. 3, 5, 2 9. 1, 0, 2 10. 2, 3, 5 11. 1, 1, 2 12. 6, 5, 3 13. 3, 2, 5 14. 0, 2, 3 15. infinitely many solutions 16. 0.6x 0.5y 0.5z 1770 0.25x 0.35y 0.45z 1165 0.15x 0.15y 0.05z 365 There were 1200 pounds of pet food in the first shipment, 800 pounds of pet food in the second shipment, and 1300 pounds of pet food in the third shipment. 17. 0.55x 0.65y 0.60z 3405 0.25x 0.10y 0.20z 1070 0.20x 0.25y 0.20z 1225 There are 2000 comedies, 1700 dramas, and 2000 action movies at the store.

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LESSON

3.6

NAME _________________________________________________________ DATE ___________


Decide whether the given ordered triple is a solution of the system. 1. 0, 0, 3

2. 1, 2, 5

3x 4y z 3 2x 7y 2z 6 10x 12y z 3 4. 1, 3, 2

2x y 5z 29 6x 4y z 19 x y 2z 7 5. 5, 7, 1

xyz0 2x 3y z 0 3x y 4z 1 6. 4, 8, 9

x y z 13 2x 7y 5z 34 3x y 4z 25

Lesson 3.6

x 5y 6z 2 3x y 8z 16 4x 2y 7y 4

3. 0, 0, 0

x 2y 3z 39 2x y 7z 63 3x y z 13

Use any algebraic method to solve the system. 7. x y 2z 4

8. x y z 6

x 3z 1 2y z 15

2y 3z 4 y 2z 1

10. x 2y z 3

11. 2x 3y 2z 1

x y 2z 9 2x 3y z 0

x 4y z 7 3x y 3z 2

13. 8x 2y z 25

3x 3y 5z 10 5x 6y 2z 17

14. x 5y 2z 16

x 7y 3z 23 3x 10y 5z 5

9. x 2y z 3

x 3y z 1 x y 3z 5 12. x 2y 3z 7

4x 5y z 4 x y 2z 5 15. 3x 2y 8z 4

6x 4y 16z 8 12x 8y 32z 16

16. Pet Store Supplies

A pet store receives a shipment of pet foods at the beginning of each month. Over a three month period, the store received 1770 pounds of dog food, 1165 pounds of cat food, and 365 pounds of bird seed. Write and solve a system of equations to find the number of pounds of pet food in each of the three shipments. Pet food Dog food Cat food Bird seed

1st shipment 60% 25% 15%

2nd shipment 50% 35% 15%

3rd shipment 50% 45% 5%

17. Movie Rental Store

The table below shows the percent of comedies, drama, and action videos available at a video store. Write and solve a system of equations to find out how many comedies, dramas, and action movies are at the store. Assume that the store has a collection of 3405 general videos to be rented, 1070 children’s videos to be rented, and 1225 videos for sale. Store section General rental Children’s rental Videos for sale


Comedy 55% 25% 20%

Drama 65% 10% 25%

Action 60% 20% 20%


85

Answer Key Practice C 1 3 1. 5, 2, 6 2. 2, 1, 2 3. no solution 4. 1, 4, 2 5. 1, 1, 1 4 34 7 14 6. All points of the form 13 z 13, 13 z 13, z 7. 10.

12, 0, 2 8. 13, 23, 13 9. 765 , 383 , 49 76 1 2 5 1 1 2, 3, 1 11. 3, 3, 4

12. All points of the form z, z 2, z 13.

7, 8, 174, 94

14. 2, 1, 3, 2

15. a b c 3 16. 4a 2b c 12 17. a b c 3, a b c 3,

4a 2b c 12, a 2, b 3, c 2 18. y 2x2 3x 2

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LESSON

3.6

NAME _________________________________________________________ DATE ___________


Solve the system using the linear combination method. 1. 2x 3y z 10

2x 3y 3z 22 4x 2y 3z 2

2. 2x y 4z 4

3. 5x 3y 2z 3

3x 2y z 2 5x 2y 3z 0

2x 4y z 7 x 11y 4z 3

Lesson 3.6

Solve the system using the substitution method. 4. 3x y 2z 3

2x 3y 5z 4 2x y z 4

5. x 3y 5z 3

6. 2x 3y z 2

4x 5y 2z 7 3x 2y 4z 9

x 5y 3z 8 5x y z 12

Solve the system using any algebraic method. 7. 2x 5y 4z 7

4x 2y 3z 8 2x 8y 5z 11 10. 6x 6y 2z 5

12x 3y 4z 0 4x 9y 2z 6

8. 6x 3y 9z 7

9. 2x 3y 6z

2x 2y 9z 1 5x y 6z 3 11. 3x 3y 4z 3

7 2

3x 4y 7z 4 11 5x 2y 4z 4 12. x 2y z 4

x 2y 8z 1 6x 9y 4z 12

3x y 4z 2 6x 5y z 10

Solve the system of equations. 13. w x y z 1

2w x y z 4 w x 2y 2z 2 3w 2x y z 7

14. w 2x y 3z 3

w x 2y 2z 3 2w 2x 2y z 6 3w x y 4z 12

Polynomial Curve Fitting In Exercises 15–18, use the following information. You can use a system of equations to find a polynomial of degree n whose graph passes through n 1 points. Consider a polynomial of degree 2, y ax2 bx c. Suppose 1, 3, 1, 3, and 2, 12 lie on the graph. Using the point 1, 3, the following equation can be derived: y ax2 bx c 3 a12 b1 c 3 a b c. The equation a b c 3 becomes the first equation in the system. 15. Write the equation in the system that corresponds to the point 1, 3. 16. Write the equation in the system that corresponds to the point 2, 12. 17. Write a system of equations for the coefficients of a polynomial of degree

2 that passes through 1, 3, 1, 3, and 2, 12. Solve the system.

18. Write the polynomial.

86



Answer Key Review and Assessment Test A 2 1. 8

4

8 2. 3 0 3. 2 1 1 9 10 5 4. 5. 6. 10 6 10 6 1 7. 8. 3 9. 107 10. 10 11. 20 2 2

13. 6, 9

12. 15

2

14. 2, 1

2

15. 4, 5, 8

5 2 3 17. 3 3 4 4 37 5 19. 1 14 2 1 21. 4, 1 22. 2, 0 2 23. SEND MONEY 24. 46

1 9 18. 2 6 20. 14 16.

2

1

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CHAPTER

NAME _________________________________________________________ DATE ____________

4


Perform the indicated operation(s). 1.

03

2 2 5 5

0 3

3.

7

4 6 1 5

3 0

8

3 5. 4

Answers

2. 1

4.

2 0

43

0 6

1 2

1.

5 1

1 3

2.

4 5 1 3 0

2 6

3.

Solve the matrix equation for x and y.

2 6. 3

3 4

x 8 y 9

0 7. 1

4.

1 4

x 2 y 9

5.

Evaluate the determinant of the matrix. 8.

1 2

2 1

9.

9 7

5 8

10.

1 2 3

2 0 4

3 1 4

6. 7.

Find the area of the triangle with the given vertices.

8.

11. A3, 4, B2, 1, C6, 3

9.

12. A4, 2, B2, 2, C2, 5

Use Cramer’s rule to solve the linear system.

10.

13. x y 15

15. x 2y 14

11.

y 2z 11

12.

x y 3

14. 2x 3y 7

3x y 5

2x z 16

13. 14. 15.

Review and Assess



81

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CHAPTER

NAME _________________________________________________________ DATE ____________

4

Chapter Test A

CONTINUED


Find the inverse of the matrix. 16.

1 3

5 2

17.

4 3

16.

3 2

18.

2 1

4 9

17.

Solve the matrix equation. 19.

5 13 3 X 2 5 4

1 0

20.

2 11

5 4 X 1 2

1 0

18.

Use an inverse matrix to solve the linear system. 21. 5x 6y 14

22. 3x 2y 6

4x y 17

19.

xy2 20.

23. Decoding

A

Use the inverse of

12

2 3

to decode the message below. 9, 23, 6, 16, 26, 39, 13, 12, 45, 65 Solve using any method. In a certain two digit number, the units digit is 24 less than 3 times the sum of the digits. If the digits are reversed, the new number is 18 more than the original number. Find the two digit number.

22. 23. 24.

Review and Assess

24. Numbers

21.

82



Answer Key Test B 2 1. 1

3 9 18 71 2. 27 3 12 2 6 4 1 3. 4. 13 3 16 7 10 5. 6. x 2; y 1 10 7. x 3; y 24

11. 17

12.

15. 1, 1, 2

1012 16.

8. 5

14. 5, 1

13. 3, 1

3 2

2

12

1

17.

18.

2 3

7 16

10. 12

9. 1

35 47 1

72

3

1

19.

2 6 21. 1, 2 22. 4, 4 3 10 23. AN APPLE A DAY 24. A $3000; B $2000; C $4000 20.

107

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CHAPTER

NAME _________________________________________________________ DATE ____________

4


Perform the indicated operation.

Answers

1.

31 21 12 50

2. 3

3.

9 5 4 8

4.

5.

4

2

2

3 2 4 6

5 3

8

3 1

19

6 4

4 3 0 1

2

4

1 1

1.

2.

4 1 0

3.

4.

Solve the matrix equation for x and y. 6.

2x4

0 4 y 4

6x

0 1

7. 4

2 12 8 2 y 8

5.


2 1

3 4

9.

5 6

1 1

1 10. 3 2

2 3 1

1 1 0

6. 7.

Find the area of the triangle with the given vertices.

8.

11. A4, 2, B3, 4, C1, 2

9

12. A5, 2, B0, 0, C3, 3

10. 11. 12.

Review and Assess



83

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CHAPTER

NAME _________________________________________________________ DATE ____________

4

Chapter Test B

CONTINUED


Use Cramer’s Rule to solve the linear system. 13.

x 5y 8 4x 2y 10

14.

2x y 9 5x 2y 27

13. 15. 3x 5y 8

4x 7z 18 yz3

1 3 2

4

17.

15. 16.

Find the inverse of the matrix. 16.

14.

4 3

5 7

2 2

67

18.

17.


2 1x 1 5

2

4

3 2

20.

1 1x 3 3

2

1

0 1

2 4

Use an inverse matrix to solve the linear system. 21.

3x y 5 5x 2y 9

22.

2 23. Decoding Use the inverse of A 3 below.

18.

19.

3x 7y 16 2x 4y 8 1 to decode the message 1

20.

21.

44, 15, 3, 1, 80, 32, 39, 17, 3, 1, 12, 4, 77, 26

22.

24. Stock Investment You have $9000 to invest in three Internet

23. 24.

Review and Assess

companies listed on the stock market. You expect the annual returns for companies A, B, and C to be 10%, 9%, and 6%, respectively. You want the combined investment in companies B and C to be twice that of company A. How much should you invest in each company to obtain an average return of 8%?

84




1 1

7. 4.

11.

0 4 2. 16 56 3. cannot 32 16 0 10 4 1 7 2 5. 6. 10 6 4 5 9 2 1 8. 1 9. ab cd 10. 16 4 31 12. 26 13. 3, 2 14. 4, 5, 10 9 2

15. 4, 1, 4 17. cannot

16.

18.

5 14

3 14

1 7

27

1 15 17 19. 3 26 29 1 9 2 20. 21. 1, 1 22. 7, 2 1 22 4 23. 16, 50, 4, 7, 11, 22, 1, 2, 7, 9, 7, 14 24. Macadamia nuts: 6 oz; Peanuts: 10 oz; Cashews: 4 oz

2 5

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CHAPTER

NAME _________________________________________________________ DATE ____________

4


Perform the indicated operation(s), if possible.

0 2. 8 2 4

1 4 8 5 1. 5 6 3 7

6 0 7 1 3 2

1 3. 9 5 5.

4 1

3 4 5

6 7 8

0 6 2 5 3 4

4.

2 1 5 8

Answers 1 2

7 2

1.

2.

7 2 2 5

9 3 2 2

3.

3 1

5 8

4.

Solve the matrix equation for x and y. 6.

52

3 8

10 xy 26

7.

2 1

2 3

xy 9 14

5.

Evaluate the determinant of the matrix. 3 8. 7

1 2

9.

a d

c b

10.

1 2 3

2 0 4

Find the area of the triangle with the given vertices. 11. A8, 6, B0, 0, C5, 4

12. A3, 2, B5, 5, C1, 8

1 0 2

6. 7. 8. 9. 10.

Use Cramer’s Rule to solve the linear system. 13. 3x 5y 1

2x 3y 12

14. 5x 10y 70

5x 25z 270 10y 25z 300

15. 4x 3y z 9

3x 2y 5z 10

11. 12. 13. 14. 15.

2x 4y 3z 8

Review and Assess



85

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CHAPTER

NAME _________________________________________________________ DATE ____________

4

Chapter Test C

CONTINUED


Find the inverse of the matrix, if it exists. 16.

3 5

2 4

1

2

17.

4 2 2 3

16. 18.

5 3

1 2

17.


5

20.

12

7 2 X 3 3

6 3

1 2

3 9 X 1 4

12 5

18.

0 2

19.


22. 2x 3y 8

3x y 4 23. Encoding

A

x 2y 3

Use the matrix

11

20.

2 3

to encode the message BREAK A LEG.

21. 22. 23. 24.

Macadamia nuts cost $.90 per ounce, peanuts cost $.30 per ounce, and cashews cost $1.30 per ounce. You want a 20-ounce mixture of macadamia nuts, peanuts, and cashews that costs $.68 per ounce. If the combined weight of the macadamia nuts and cashews equals the weight of the peanuts, how many ounces of each nut should be used?

Review and Assess

24. Mixed Nuts

86



Answer Key Cumulative Review 9 1 1. 13 2. 11 3. 11 4. 2 5. 4 6. 4 1 7. 2 8. 5 9. x ≥ 1 10. 3 x 2 11. x ≤ 2 or x ≥ 5

12. x ≥

14. x 4 15. no 19. yes 20. yes 21.

16. yes

5 2

13. x

17. yes

5 2

18. no

22.

y

y

(2, 1)

1

1 x

1

(6, 1)

1

(0, 5)

x

(0, 5)

23.

24. y

y

(0, 4)

1

(4, 0) 1

1

(2, 0)

x

x

1

(0, 3)

25. y 5x 2

26. y 5x 3

4 28. y 23x 6 2 29. y 5x 3 30. y 3 31. y 3x 2 32. y 3x 5 33. y 5x 6 34. 5, 1 27. y

35. 39. 41. 42.

1 2x

4, 2 36. 1, 0 37. 12, 1 38. 13, 23 10, 6 40. f x, y x 6y 12; 20 f x, y x 2y 8; 9 f x, y x 32y 4; 16

43. f x, y 3x 3y 2; 8 2

5

44. f x, y 2x 3y 5; 14 3

45. f x, y 3x 3y 3; 11 2

4

46. 0, 1, 2 47. 3, 4, 7 48. 2, 0, 3 49. 5, 9 50. 3, 5 51. 1, 5 52. 7, 4 53. 5, 1 54. 0, 2 56. 236 57. 48 1

55. 5, 1, 0

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CHAPTER

NAME _________________________________________________________ DATE ____________

4


Evaluate the expression for the given values of x and y. (1.2) 1.

x2 when x 3 and y 5 3y 2

2. x3 2y3 when x 3 and y 2

3.

xy when x 5 and y 6 xy

4. x 2 2x 3y when x 4 and y 2

Solve the equation. (1.3) 5. 3x 2 4x 7 18

6.

7. 1.5x 8 11 0.5x 1

8. x 2 3x 3x 4 15

1 2x

13 x 73

Solve the inequality. (1.6) 9. 3x 1 8x 4

10. 10 3x 1 7

11. 2x 8 or 2x 3 7

12. 4x 2 8

13. 2x 3 8

14.

1 2x

46

Tell whether the relation is a function. (2.1) 15.

x 1 y 2

2 3

4 3

5 4

5 6

16.

x 2 y 0

17.

x 0 y 0

1 1

4 2

9 3

16 4

18.

x 0 y 0

19.

x 2 y 4

1 1

0 0

1 1

20.

x 2 y 6

2 4

1 0

3 0

1 1 1 1 1 3

4 0 4 4 2 2

0 1

1 3

2 6

Draw the line with the given information. (2.3) 2

21. m 3, b 5

22. m 3, b 5

23. x-intercept is 2, y-intercept is 4

24. x-intercept is 4, y-intercept is 3

Write an equation of the line that has the given slope and y-intercept. (2.4) 25. m 5, b 2 2

2

28. m 3, b 6 Review and Assess

1

27. m 2, b 4

26. m 5, b 3 29. m 5, b 3

30. m 0, b 3

Write an equation of the line. (2.4) 31.

32.

y

33.

y

(2, 6)

y

(2, 4)

(0, 5) 2

2

(0, 0) 2

(6, 1)

2 x

x

2

x

2

(0, 6)

92



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CHAPTER

NAME _________________________________________________________ DATE ____________

4

Cumulative Review

CONTINUED


Solve the system using the substitution method or the linear combination method. (3.2) 34. x 5y 10

35. 3x 4y 20

2x 3y 13

36. 5x 4y 5

2x 3y 14

37. 4x 3y 5

2x 5x 2

38. 3x 6y 5

6x 2y 5

39. 0.2x 0.3y 3.8

xy1

0.5x 0.7y 0.8

Write the linear equation as a function of x and y. Then evaluate the function for the given values. (3.5) 40. x 6y z 12, f 4, 2

41. x 2y z 8, f 3, 2

42. 2x 3y 2z 8, f 3, 6

43. 2x 5y 3z 6, f 9, 0

44. 3x 6y 2z 10, f 2, 4

45. 2x 4y 3z 9, f 3, 9

Use any algebraic method to solve the system. (3.6) 46. 5x 2y 2z 6

47. x 2y 3z 10

3x 3y z 1 5x 5y z 7

48. 2x 3y z 7

2x 2y z 9 4x y 3z 5

2x 5y 3z 13 3x 3y 2z 12

Solve the matrix equation for x and y. (4.1) 49.

83x 32

51. 3x

9 15 7 8

4 9 12 y 6 15

y 7

50.

24x

52.

31

3 4 8 3

2 16 1 y

1 9

2 5 3 3

x 8 7 2

9 y

Use Cramer’s rule or an inverse matrix to solve the system. (4.3, 4.5) 53. 2x y 11

3x 8y 7

54. 3x 6y 3

55. x 5y 2z 10

5x 8y 4

2x 8y 3z 2 xyz6

56. Tickets

Tickets to the Spring Concert cost $3 for students and $5 for adults. Sales totaled $1534. Twice as many adult tickets as students tickets were sold. How many adult tickets were sold? (3.2)

According to kinetic theory, 273 degrees Celsius is the temperature at which gas molecules would cease to move; this is called the absolute zero of temperature. In practice all gases, on cooling, liquefy or solidify before that temperature is reached. This temperature, 273 C, is taken as the zero point on the Kelvin scale, so Kelvin temperature is 273 higher than Celsius temperature. If the Kelvin temperature of a gas is 33 more than six times the Celsius temperature, what is the temperature of the gas in degrees Celsius? (4.3)

57. Kelvin

Review and Assess



93

Answer Key Practice A 1. 4 3 2. 3 2 3. 2 1 4. 3 4 5. not equal 6. not equal 7. equal 3 3 1 8. 9. 10. 6 4 6 5 6 11. The operation is not possible because the matrices do not have the same dimensions. 2 3 1 0 0 12. 13. 14. 1 3 11 0 0 15. The operation is not possible because the matrices do not have the same dimensions. 13 2 2 12 16. 17. 1 3 6 4 3 0 10 18. 19. 9 18 25 24 20. 12 24 4 21. 0 40 2 5 3 22. 6 1 7 23. x 2; y 4 0 0 9 24. x 5; y 1 25. x 7; y 3 435 562 26. 525 3

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Page 14

LESSON

NAME

4.1

DATE


Determine the dimensions of the matrix.

3 1 1. 2 4

5 2 6 3

7 9 1 5

2.

4 9 5 1 2 6

3.

4 3

4.

1 6 3

2 3 4

4 2 8

5 0 1

0

Lesson 4.1

Tell whether the matrices are equal or not equal. 5.

3 7

4 3 , 1 7

4 1

6. 2

2 6, 1 6

1

7.

1 4

0 , 3

2 2 8 2

31

Perform the indicated operation, if possible. If not possible, state the reason. 8.

12 34 24 01

9.

11.

72 3

14.

51 48 51 48

4

25 31

12.

30 41 42

15.

12

1 3 1

0 2 4

10. 4

1 2

5

13.

43 47

16.

00 04 131

2 7

Perform the indicated operation. 17. 2

3 2 1

6

20. 43

6

18. 3

3 6 1

0

19. 5

2

3 0 21. 8 5

1

5

2 22. 1 6 0

5 1 0

3 7 9

Solve the matrix for x and y. 23.

x 3 2 5 y 5

3 4

2x 10 24. 3 3 4 4y

25. 3x

21 21

7y

26. Endangered and Threatened Species The matrices below show the num-

ber of endangered and threatened animal and plant species as of June 30, 1996. Use matrix addition to find the total number of endangered and threatened species. (Source: 1997 Information Please Almanac) ENDANGERED

THREATENED

U.S. Foreign Animal 320 521 Plant 431 1

U.S. Foreign Animal 115 41 Plant 94 2

14



Answer Key Practice B

5 12 3 1 7 1. 2. 5 6 3. 3 6 1 0 10 4 0 3 4. The operation is not possible because the matri-

6 5

1 3

ces do not have the same dimensions.

5.

12 16 9

7. 13

9.

10.

3

2 0 13

15 9 3

1 6 4

0 8 3

4 2 10

50

20 25

10 5

10 8.

2

8 1 9 9 4 5 0 0 0 3 12 9 6

11 3

6.

15 20

11.

61

1 2 2 12 2 0 13. 19 4 3 2 0 4 8 14. 15. $1704 16. $1753 2 6 6 12.

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Page 15

LESSON

NAME

4.1

DATE


Perform the indicated operation, if possible. If not possible, state the reason. 1.

3.

3 5 4 7

4 1 2

3 5 7

2 1

2 1 4 4 6 6

2 1 7

5 3 9

8 3 9 4 4 12 1 12 5. 4 6 8 5

1 12 7

7. 6

3

2

1

8 7

5

6 10 11

4 2

3 4 5 7 4 2

4. 3

0

1 7 11

8

10

2 7 10

7 6. 1 8

9 1 6

5 1 3 10 13 8

3 11 10

4 8 13

2

Perform the indicated operation. 1 8. 3 3

1 2. 2 2

9. 2

12

0

2

3

4

1

2

3 2

5

10. 5

10

2 1

4 5

3 4

Perform the indicated operations. 11.

1 0

2 3 1 2

13

4 1 5 8 2

13. 3

4 5

0 4

2 3

1 12. 2 5

5 9

14. 2

3 0 6

20

8 2

9 4 3

2 7 1 0 1 5

1 2 6 0 1 1

3 3

3 1 3

Health Club Membership In Exercises 15 and 16, use the following information. A health club offers three different membership plans. With Plan A, you can use all club facilities: the pool, fitness center, and racket club. With Plan B, you can use the pool and fitness center. With Plan C, you can only use the racket club facilities. The matrices below show the annual cost for a Single and a Family membership for the years 1998 through 2000. 1998 Single

Plan A 336 Plan B 228 Plan C 216

1999

Family 624 528 385

Single


2000

Family 720 576 432

Single

Family


792 672 528

15. You purchased a Single Plan A membership in 1998, a Family Plan B

membership in 1999, and a Family Plan A Membership in 2000. How much did you spend for your membership over the three years? 16. You purchased a Family Plan C membership in 1998, and upgraded to the

next highest plan each year. How much did you spend for your membership over the three years? Copyright © McDougal Littell Inc. All rights reserved.


15

Lesson 4.1

1 2

Answer Key Practice C 1 3 4 1. 3 0 4

4.

18 18 9 15

2.

40

5.

4 14

3.

16 14

26 3 43 3

6. Not possible. The matrices cannot be added

because they do not have the same dimensions. 7.

9.

71 12 143 16

1 2 16

8.

45

13 6

33 70

13 70

10.

83

11.

13 2 2

13.

1 6 7 3 4 3

2

11 6

0

0 11

1 1

34

1

54

2

1

52

12.

5 4

12 12

12 6

12

60 7

3-point 2 1 3 0 0 0

field goals rebounds 10 3 6 3 5 1 4 6 4 5 3 5

3-point Patrick 30 Mark 15 45 15. Joe Craig 0 Daryl 0 Mike 0

field goals rebounds 150 45 90 45 75 15 60 90 60 75 45 75

Patrick Mark 14. Joe Craig Daryl Mike

16.

26 13

32 43

11 ; New York 25

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LESSON

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Perform the indicated operations, if possible. If not possible, state the reason. 1.

4. 3 Lesson 4.1

3 2

24

1 4

1 1 5 7 2 1 6 5 2

3 1

0 2

2 0

10. 3

1 3 1 4

1 8 1 9

3

43 12 310 14

5.

1

1 2 7. 4 2

2.

2 3

13 514 61

Basketball

4 0

3

2 2 7 7 7 1

1 2 1 10 2 5 8. 10 5 7 10 1 1

2

12 01 40 23 31 12

1 12. 3 2 3

14 326

3. 2

14 7

11.

0 3

1 6 2 4

5 3 15 7

6.

1 3 3 0

1 1

2 3 4 1

1 6

2 8

3 1 1 9. 3 0

2 2 1 4 3 2 2 0

3 1 4 2 4 12

0 4

8 4

1 2 4

1 13. 2 2

4 1 3 2 1

2 3 1

0 1 3 1 3 2 4 2

4 6 2

In Exercises 14 and 15, use the following information.

A high school basketball coach helps the six seniors on the team to set goals for the season. The goals per game for each senior are as follows. Patrick: 2 3-pointers, 10 field goals, 3 rebounds

Craig: 4 field goals, 6 rebounds

Mark: 1 3-pointer, 6 field goals, 3 rebounds

Daryl: 4 field goals, 5 rebounds

Joe: 3 3-pointers, 5 field goals, 1 rebound

Mike: 3 field goals, 5 rebounds

14. Write a matrix that represents the game goals for the six seniors. 15. If there are 15 games in a season, write a matrix that represents their

season goals. 16. World Series

The New York Yankees won the 1998 World Series in four games. The matrices below show the statistics for runs, hits, and RBIs for each team in each game. Write a matrix that gives the series statistics for runs, hits, and RBIs for each team. Which team had the most hits for the series? Game 1 R H RBI San Diego 6 8 5 New York 9 9 9

Game 3 R H RBI San Diego 4 7 3 New York 5 9 5

16





Answer Key Practice A 1. AB is not defined. 2. AB is defined; AB: 3 3 3. AB is defined; AB: 2 1 4. AB is defined; AB: 3 2 5. AB is not defined. 6. AB is defined; AB: 3 5 7. AB is not defined. 8. AB is defined; AB: 2 4 9. AB is defined; AB: 1 1 10. 42 23; 41 22; 40 24 11. 24; 26; 34; 36 12. 21 14; 01 34 13. 14 14. 1 0 15. 2 1 3 12 2 3 16. 17. 1 4 4 6 18. The matrices cannot be multiplied because 2 the number of columns in does not equal the 3 1 2 number of rows in . 3 1 1 3 3 2 19. 20. 13 21. 2 5 1 4 Opening night $2220 22. Second night $2525 Final night $2972.50

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LESSON

NAME

4.2

DATE


State whether the product AB is defined. If so, give the dimensions of AB. 1. A: 2 2, B: 3 2

2. A: 3 4, B: 4 3

3. A: 2 5, B: 5 1

4. A: 3 2, B: 2 2

5. A: 4 1, B: 4 1

6. A: 3 4, B: 4 5

7. A: 3 5, B: 3 3

8. A: 2 4, B: 4 4

9. A: 1 6, B: 6 1

Complete the next step of the matrix multiplication. 10.

34

11.

12.

20

1 2

23

1 2 4 3 1 3

6

0 32 13 31 12 30 14 4 ? ? ? 14 16

1 2

? ?

? ?

1 22 11 ? 4 02 31 ?

21

Lesson 4.2

Find the product. If it is not defined, state the reason. 13. 2

3

4 1

14. 1 17.

16.

31 1

4

19.

1 2

1 0

0 1

3 5

0

2 1 1

2 1 2

20. 1

2

0

3

3 4 2 5

15. 1

1

1 1 1

2 1

2

18.

23 13

21.

13 24 10 01

22. Senior Play The senior class play was performed on three different

evenings. The attendance for each evening is shown in the table below. Adult tickets sold for $3.50. Student tickets sold for $2.50. Use matrix multiplication to determine how much money was taken in each night. Performance Opening night Second night Final night

28

Adults 420 400 510


Students 300 450 475


Answer Key Practice B 1. A: 3 2; B: 1 3; AB is not defined. 2. A: 2 3; B: 3 3; AB is defined; AB: 2 3 3. A: 4 1; B: 1 2; AB is defined; AB: 4 2 4. A: 4 2; B: 3 4; AB is not defined. 5. 11 1 3 1 0 7 3 6. 7. 0 0 8. 0 1 38 14 1 3 4 6 5 7 3 11 9. 7 5 1 10. 2 6 18 0 16 4

1 3 2

0 11. 6 12. 6 8

6 12 28

8 6 8

18 11 2 15. 27 4 14 20 6 20 20 16. 45 42 17. 16 16 10 15 4 4 Opening night $2220 18. Second night $2525 Final night $2972.50 14.

22 33

10 4 13. 11 14

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LESSON

NAME

4.2

DATE


State the dimensions of each matrix and determine whether the product AB is defined. If it is, state the dimensions of AB.

2 4 , B 4 1

1 1. A 3 4

5 2 , B 1 3. A 3 1

9

1 2. A 3

3

4 1

2 , B 0

2 3 4. A 1 0

7

4 1 2 , B 6 6 5 3

3 5 3

1 2 6

7 2 4

3 6 8

2 4 7 4 1 0

3 0

Find the product. If it is not defined, state the reason. 2 5 1

5. 3

8.

2 4

11. 3

4

3 5

1

1 2

1 2 2 5 3

9.

2 3 0

1 1

0 1 0

1 2 4

1 12. 3 2

2 0 4

1 2

2 1 0

21

1 0 1 7. 1

1 1

1 0 4

1 0

4 5

2 3 1 2 3

10.

4 0 13. 3 6

3

12 32 21

1 2 1 2

4 5

Lesson 4.2

1 6

6.

2 1

32

Simplify the expression. 14. 4

16.

1 2 34

2 0 1

1 3 2

4 3 5

0 6 1

3 4

5 2

1 3 5

2 3 0 0 1 4

1 2 5

15.

2 0 1

17.

1

3

5

2 1 1 2

3 2 1 5 0 1

3 0

3 2

5 2

4 4

18. Senior Play The senior class play was performed on three different

evenings. The attendance for each evening is shown in the table below. Adult tickets sold for $3.50. Student tickets sold for $2.50. Use matrix multiplication to determine how much money was taken in each night. Performance Opening night Second night Final night


Adults 420 400 510

Students 300 450 475


29

Answer Key Practice C

12 8 1. 2. 0 3 3 5 11 16 9 16 2 3. 24 16 4. 3 12 6 9 20 2 2 6 5. Not defined. The number of columns of the first matrix does not equal the number of rows of the second matrix. 16 2 45 1 32 35 6. 7. 8. 114 26 40 17 1 7 93 4 12 40 15 2 9. 96 12 220 10. 20 52 68 16 136 11. x 2, y 3 12. x 5, y 1 0 2 5 ; reflection across x-axis 13. 1 3 1 14. Rebecca: 380; Craig: 370

4 3

8 11

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LESSON

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4.2

DATE


Find the product. If it is not defined, state the reason. 1.

2 1

4 3

4 0 2

1 4. 3 2

2 3

0 1 1 2

1 4 2. 6

2 1

4 3

5.

4 1

1 3

3 2 1 0 2

4 2

2 1 1 3 3 2 1 2

0 2 1

1 0 3. 3

2 4 1

3 1 0 4

2 1 6. 2 1

4 5

61 84 4 3 0 0

1 2 2 3 5 1 3 1

Simplify the expression. 7.

3 1

1 4

0 2 1

Lesson 4.2

2 9. 4 3 1

2 1

0 4

1 2 6 0 5 3

3 6

1 0 1

1 2 1 5 8

8.

10. 2

3 1

2 1

1 2 3 4

32

2 1

3 4

2 1 4 2

1

3

2

1 2

2

3

4 2 5

3 0

3 1

1 3

6

1

3

Solve for x and y.

1 11. 2 3

1 1 0

3 2 5 4 1 y 2 x 2

12.

1 1 3 5 4

x 15 1 31 2

9 y

13. Geometry

Matrix B contains the coordinates of vertices of the triangle shown in the graph. Calculate AB and determine what effect the multiplication of matrix A has on the graph. A

10

0 1

2 3

5 1

0 B 1

y

(2, 3)

1

(5, 1) (0, 1) 1

x

14. Class Election

Rebecca and Craig are running for student council president. After attending a debate, some students change their minds about the candidate for whom they will vote. The percent of students who will change their support is shown in the given matrix. Rebecca estimated that prior to the debate she would lose the election 350 votes to 400 votes. After the debate, how many votes will Rebecca and Craig receive?

Students who change their support To

From 30

Rebecca Craig

Rebecca Craig 0.80 0.20 0.25 0.75



Answer Key Practice A 1. 20 2. 19 3. 0 4. 7 5. 12 6. 16 7. 15 8. 14 9. 10 10. 20 11. 20 12. 24 13. 4 14. 8 15. 4 16. 1, 4 17. 2, 1 18. 3, 2 19. 0, 3 20. 4, 0 21. 1, 3 22. 0, 0 23. 3, 4 24. 2, 5 25. x 1926; y 1928

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LESSON

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4.3

DATE



13 54

5.

4

4 1 4 1

8.

62

1.

16 23

4.

7

7.

0

1 8

2

1 4

4 5

3.

6 4 2 3

6.

35 53

9.

02

5 9


2 0 0

2 13. 0 0

0 0 5 0 0 2 1 2 0

1 1 1

11.

1 0 0

2 14. 1 2

0 4 0

0 0 5 0 2 3

0 0 2

12.

2 0 0

1 1 15. 0

0 2 0

0 0 6

2 2 4 1 0 2

Use Cramer’s rule to solve the linear system. 16. x y 5

17. 2x 3y 1

18. 2x y 8

19. 4x 2y 6

20. 2x 5y 8

21. 2x y 5

22. 2x 5y 0

23. x 4y 19

24. 3x y 1

2x y 6 x 3y 9

Lesson 4.3

3x 7y 0

x 5y 3

x y 4

2x y 2

3x 2y 5 3x 2y 3

3x 2y 16

25. Children’s Literature A. A. Milne (1882–1956), an English author,

became famous for his children’s stories and poems. One of Milne’s most famous works, Winnie-The-Pooh, is based on his son Christopher Robin, and the young boy’s stuffed animals. Two years after the first book was published, the Pooh stories continued in the book The House at Pooh Corner. Solve the linear system given below to find the year that each of these books were published. (Use Cramer’s rule.) x y 2 1 1 x y 80 6 8

42



Answer Key Practice B 1. 44 2. 32 3. 8 4. 215 5. 222 6. 20 7. 129 8. 357 9. 15 3 1 10. 2, 5 11. 1, 1 12. 3, 6 13. 4, 4, 0 14. 1, 1, 2 15. 1, 2, 0 16. 3 17. 4 18. 8.5 19. x y z 538 20. x y 110 21. x z 255 22. 301, 191, 46

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LESSON

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4.3

DATE



3 4

5 8

2.

1 5

7 3

3.

2 3 1 2

4

1 0 1

20 5 15

9


5 4 1

13 1 7. 0

8 4 2 1 1 5 4 3 1

7 4 2

0 10 1

3 14 1

2 2 6

10 4 8. 3

7 2 2

8 5 5

5.

6.

1 0 9. 5

1 1 4

2 0 5

4 3 0

Use Cramer’s rule to solve the system of equations. 10. 2x y 9

11. 6x 11y 5

12. x 7y 39

13. 2x 2y 5z 1

14. x y 2z 6

15. 2x y 3z 0

2x 3y 19

6x 5y 1

8x z 6 x y 2z 1

2x 9y 48

2x 3y z 7 3x 2y 2z 5

3x 2y z 7 2x 2y z 2

Use a determinant to find the area of the triangle. 16.

17.

y

18.

y (2, 3)

(3, 1)

1

(0, 0)

y

(1, 3)

(0, 2)

1

(3, 2)

1

x

1

(4, 1)

1

x

(2, 1)

(2, 0)

1

x

Lesson 4.3

Electoral Votes In Exercises 19–22, use the following information. In the 1968 presidential election, 538 electoral votes were cast. Of these, x went to Richard M. Nixon, y went to Hubert H. Humphrey, and z when to George C. Wallace. The value of x is 110 more than y. The value of y is 145 more than z. (Source: 1997 Information Please Almanac) 19. Write an equation involving the variables x, y, and z, that represents the

total number of electoral votes. 20. Write an equation that relates the number of electoral votes received by

Nixon, x, to the number of electoral votes received by Humphrey, y. 21. Write an equation that relates the number of electoral votes received by

Nixon, x, to the number of electoral votes received by Wallace, z. 22. Use Cramer’s rule to find the values of x, y, and z.



43

Answer Key Practice C 1. 10 2. 7 3. 10 4. 36 5. 104 6. 28 7. 3, 2 8. 1, 4 9. 4, 15 10. 2, 1, 3 11. 1, 0, 2 12. 2, 4, 1 13. 1, 3, 2 14. 3, 2, 4 15. 1, 1, 5 16. det AB det BA 17. det A 0 18. Carbohydrates contain 4 calories per gram. Fat contains 9 calories per gram. Protein contains 4 calories per gram.

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DATE



4 3

4.

6 0 0

2 1 0 3 0

0 0 2

2.

3 2

5.

1 2 1

5 1

5 0 4

2 3. 1 2 3 0 8

6.

0 4 2

1 1 3

1 3 4

4 3 0

2 1 2

Use Cramer’s rule to solve the linear system. 7. 4x 2y 16

1

2

8. 3x 2y 11

9. 2x 3y 8

5x 6y 19

x 15y 7

3x y 7 10. 2x 4y z 11

11. x 2y z 1

12. 3x y 5z 3

13. x y z 6

14. 2x 2y 7z 30

15. 3x y 2z 12

x 3z 7 2y 4z 14

x 3y 2z 5 x y z 3

2x y 3z 1 3x 2y z 1

2x y z 9 x 4y 3z 15

3x 4y 2z 9 5x y z 9

16. Determiniant Relationships

Let A

1 3 and B 2

4

1 3

x 4y z 0 x y 3z 17

1 . How 2

is det AB related to det BA? 17. Determinant Relationships

Explain what happens to the determiniant of any matrix that includes a row of zeros. For lunch you eat a peanut butter sandwich on wheat bread and carrot sticks. The nutritional content for the peanut butter, wheat bread, and carrots is shown in the table. Use Cramer’s rule to determine how many calories are in a gram of carbohydrates, fat, and protein.

Lesson 4.3

18. Nutrition

Serving Peanut butter Wheat bread, 2 slices Carrots

44

Carbohydrates per serving 7g 26 g

Fat per serving 16 g 1g

Protein per serving 8g 6g

Calories per serving 204 137

8g

0g

1g

36



Answer Key Practice A 1. yes 2. no 9 5 6. 7 4 1 1

9.

3. yes

5

3

2

7.

4. yes

4 3

3 2

5. no 8.

1

0

0

1 2

10. The matrix does not have an inverse.

14.

17. 15.

19.

2

15 2 3 5 12. 3 4 3 4 5 5 3 4 5 7 The matrix does not have an inverse. 2 7 3 8 16. 3 4 4 10 15 17 15 7 40 18. 26 29 13 6 34

13. 11.

19 10 7 10

21. 13

3 10 9 10

5, 5

20.

75

20, 0

9 6 13, 5

0 1

0, 1

20,

0 19, 21 14, 19 5, 20 0 22. 75 44, 65 35, 26 13, 25 15, 45 23, 38 19, 133 77, 105 62, 100 60 23. 75, 44, 65, 35, 26, 13, 25, 15, 45, 23, 38, 19, 133, 77, 105, 62, 100, 60 1 3 24. 25. NOT TONIGHT 2 5

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LESSON

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4.4

DATE


Tell whether the matrices are inverses of each other. 1.

53

3 , 2

4 0 1

1 4. 3 3

3 5

23

2 5 5 , 3 4 3

2.

18 20 10 11 11 12

1 2

12 11, 11

3.

1 0 5. 2

0 3 4

1 4

47 12, 72

2 0 6 , 1 8 2

2 2 4

3 1 3


47 59

9.

65

12.

3 3

34 21

7.

32

4 3

10.

36

4 8

13.

75 43

8.

10 02

11.

43

3 2

14.

44 33


12 23X 45

17.

125 73X 23

19.

4 6

1 2

2 2 X 2 3

1 2

2 8 1 6

2 4

4 1 0 X 9 3 2

16.

115

18.

65 76X 13

20.

4 3

0 1

2 4

4 1 X 6 2

3 0

4 6

Encoding Messages In Exercises 21–25, use the following information. The message, MEET ME AT SUNSET, is to be encoded using the matrix A

52

3 . 1

21. Convert the message into 1 2 uncoded row matrices. 22. Multiply each of the uncoded row matrices found in Exercise 20 by A to

obtain the coded row matrices. 23. Write the message in code. 24. Find the inverse of A. 25. You receive the following response: 100, 57, 100, 60, 130, 75, 88,

51, 51, 29, 100, 60. Use the inverse of A to decode the response.

Lesson 4.4



57

Answer Key Practice B 1. No inverse exists. 3. Inverse exists.

5.

4.

1

0

12

1 4

2. Inverse exists.

6.

2 11 1 11

1 17

5 11 3 11

5 17 2 17

3 17

7.

1

1

53

2

8. The matrix does not have an inverse.

9.

11.

3

3 4

1

2

1 5 25 2 5

2 1

13.

23 15

15.

1 0

3 1

17. 13

10.

1 2 18 1 2

1 2

12

0

1 4

0

0

12

11 2

1

0

3

1 2

0

1

14 8

2 5 1 5 15

74

12.

14.

46 25

11 8

16.

19 10 7 10

3 10 9 10

5, 5 20, 0 13, 5 0, 1 20, 0 19, 21 14, 19 5, 20 0 18. 75 44, 65 35, 26 13, 25 15, 45 23, 38 19, 133 77, 105 62, 100 60 19. 75, 44, 65, 35, 26, 13, 25, 15, 45, 23, 38, 19, 133, 77, 105, 62, 100, 60 1 3 20. 21. NOT TONIGHT 2 5

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LESSON

NAME

4.4

DATE


Tell whether the matrix has an inverse.

8 1. 4 0

2 1 0

0 1 4

2 2. 1 3

1 0 2

2 1 1

5 3. 2 1

1 3 3

0 2 1


13 52

7.

65

3 3

5.

12 04

8.

36

4 8

6.

23 51

9.

43

3 2

2 4 1

6 1 2

Use a graphing calculator to find the inverse of the matrix. 10.

2 0 0

4 4 0

1 1 2

11.

1 0 2

3 1 1

5 1 0

12.

0 2 0


48 21X 16 22

15.

41 72X 32 74 26 23

6 13

14.

41 72X 49

12 5

16.

64

2 8 1 6

2 2 X 2 3

0 2

2 4

Encoding Messages In Exercises 17–21, use the following information. The message, MEET ME AT SUNSET, is to be encoded using the matrix A

2 5

3 . 1

17. Convert the message into 1 2 uncoded row matrices. 18. Multiply each of the uncoded row matrices found in Exercise 20 by A to

obtain the coded row matrices. 19. Write the message in code. 20. Find the inverse of A. 21. You receive the following response: 100, 57, 100, 60, 130, 75, 88,

Lesson 4.4

51, 51, 29, 100, 60. Use the inverse of A to decode the response.

58




8 13 1 13

3 13 2 13

47

2.

3 14 1 14

1 7

3. The matrix does not have an inverse. 4.

0.2 0.4 0.2

0.12 0.44 0.08

0.1875 0.625 0.25

0.8125 5. 1.375 0.75 0.2 6. 0.2 0.2

0.32 0.16 0.12

0.1 0.1 0.6

0.875 1.25 0.5

0.6 0.1 0.6

65

3 4

8.5 11

8.

15

3 2

5 15 8 22

10 12 4

11.

7.

10.

6 5

114

9.

10 3

9 8

11 10

13. A11 A

15.5 20.5 7.3 2.6 12. 4.5 6.5 3.5 4.5 3 14. AB 1 B1 A1

15. LIVE LONG AND PROSPER

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LESSON

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2 1

3 8

2.

1 2

3 8

3.

1 3

2

1 6

Use a graphing calculator to find the inverse of the matrix, if it exists.

1 4. 2 3

1 1 1

4 4 1

0 4 5. 2

2 4 1

5 3 4

6 0 2

1 2 0

4 1

2 3

8 7

0 6. 2 2


42 31X 24

9.

11.

1 4

0 2

4 2 8

8.

4 2 X 12 1

2 1 X 2 3 4 2

2 12. 2 4

1 6

0 7 3 3 3 0

1 3 4 0

2 1 4 8 X 6 9 6 2 1

13. Inverse Properties

2 1

Let A

21 11X 34

1 10. 0 2

1 5

1 3

1 7 1 6 4 3

2 1 2

1 4 6 2 X 1 19 1 3 3

2 9 4

1 5 6

52 32. How is A related to A

1 1?

1 . Calculate 4 AB, A1, B1, and AB1. How are A1, B1, and AB1 related?

14. Inverse Properties

Let A

10 34 and B 21

1 1 1 15. Cryptography Use the inverse of A 0 1 1 to decode 12, 65, 87, 0 2 3 5, 29, 41, 15, 43, 50, 0, 29, 43, 4, 36, 52, 18, 71, 90, 16, 57, 75.

Lesson 4.4



59

Answer Key Practice A 1 1 x 3 1. 2 1 y 4 3 1 x 1 2. 4 1 y 15 6 3 x 39 3. 5 9 y 25 1 1 1 x 2 4. 2 0 3 y 4 0 3 1 z 7 3 1 2 x 1 5. 1 2 1 y 12 1 4 0 z 18 5 3 1 x 6 6. 2 2 3 y 1 1 5 4 z 9 2 1 3 x 4 7. 3 1 5 y 9 2 1 4 z 1 5 3 1 x 6 8. 2 2 4 y 6 3 2 4 z 1 5 3 1 x 3 9. 6 1 1 y 7 10. 3, 5 3 5 3 z 5 11. 4, 2 12. 1, 7 13. 2, 1 14. 6, 2 15. 11, 6 16. 2, 1 17. 5, 0 18. 1, 3 19. 1, 3, 2 20. 5, 0, 3 21. 9, 12, 6 22. x y 10,000 0.1x 0.06y 800 1 1 x 10,000 23. 0.1 0.06 y 800 24. 5000, 5000 25. Invest $5000 in Stock A and $5000 in Stock B.

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Lesson 4.5

LESSON

NAME

4.5

DATE


Write the linear system as a matrix equation. 1. x y 3

2. 3x y 1

3. 6x 3y 39

4. x y z 2

5. 3x y 2z 1

6. 5x 3y z 6

7. 2x y 3z 4

8. 5x 3y z 6

9. 5x 3y z 3

2x y 4

4x y 15

2x 3z 4 3y z 7

5x 9y 25

x 2y z 12 x 4y 18

3x y 5z 9 2x y 4z 1

2x 2y 3z 1 x 5y 4z 9

2x 2y 4z 6 3x 2y 4z 1

6x y z 7 3x 5y 3z 5

Use an inverse matrix to solve the linear system. 10. x y 2

11. 3x 2y 8

12. 5x 2y 9

13. 4x 5y 13

14. 3x 7y 4

15. 2x y 16

16. 5x 3y 7

17. 4x y 20

18. x 2y 7

2x y 1

4x 3y 10

3x 4y 10

7x 3y 14

x 3y 0

3x 2y 4

6x 2y 78

7x 2y 35

2x 3y 11

Write the linear system as a matrix equation. Then use the given inverse of the coefficient matrix to solve the linear system. 19. 2x y z 3

20. x y z 2

3x z 5 5x 2y 2z 5 A1

2 11 6

0 1 1

1 5 3

9x 6y 7z 24 6x 4y 5z 15

A1

2 3 0

1 1 2

1 2 3

21. x y 2z 9

2x y z 0 x 2y 6z 21 1

A

8 13 3

2 4 1

3 5 1

Stock Investment

In Exercises 22–25, use the following information. You have $10,000 to invest in two types of stock. The expected annual returns for the stocks are shown in the table below. You want the overall annual return to be 8%. Investment Stock A Stock B

Expected return 10% 6%

22. Write a linear system of equations that represents the given information. 23. Write the system as a matrix equation. 24. Use an inverse matrix to solve the system. 25. How much should you invest in each type of stock?

70



Answer Key Practice B 2 4 x 7 1. 3 1 y 12 1 9 x 20 2. 2 4 y 15 1 2 3 x 14 3. 2 1 2 y 16 4. 51, 18 3 5 9 z 36 5. 1, 3 6. 2, 1 7. 8, 4 8. 4, 4 44 26 9. 1, 5 10. 5 , 5 11. 5, 0 12. 9, 8 13. 1, 2, 0 14. 1, 4, 3 15. 2, 3, 1 16. 5, 0, 3 17. 1, 3, 1 18. 6, 4, 4 19. x y z 20,000 0.12x 0.10x 0.06z 1800 3x y z 0 1 1 1 x 20,000 20. 0.12 0.10 0.06 y 1800 3 1 1 z 0 21. 5000, 7500, 7500 22. Invest $5000 in Stock X, $7500 in Stock Y, and $7500 in Stock Z. 23. You can make 300 pounds of alloy X, 700 pounds of alloy Y, and 400 pounds of alloy Z.

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LESSON

DATE

Practice B

Lesson 4.5

4.5

NAME


Write the linear system as a matrix equation. 1. 2x 4y 7

2. x 9y 20

3x y 12

3. x 2y 3z 14

2x 4y 15

2x y 2z 16 3x 5y 9z 36


5. 2x y 5

6. 2x 6y 2

7. 5x 3y 52

8. 3x 7y 16

9. 2x 3y 13

x 3y 3

x 4y 11

3x 2y 32

x 5y 3

2x 4y 8

10. 2x 3y 2

11. 2x 7y 10

x 4y 12

3x 4y 15

xy4

12. x y 1

2x 3y 6

Use an inverse matrix and a graphing calculator to solve the linear system. 13. 2x 6y 4z 10

14. 5x 4y z 14

15. x y 2z 7

16. x y z 2

17. x 2y 7

18. 2x y 3z 4

3x 10y 7z 23 2x 6y 5z 10

5x 2y 3 2x 5y 2z 24

x 2y z 8 y z 3

3x 5y z 11 5x 2y z 0

2x z 5 9x 2y z 25 x 2y z 2 x 3y 4z 10

Stock Investment In Exercises 19–22, use the following information. You have $20,000 to invest in three types of stocks. You expect the annual returns on Stock X, Stock Y, and Stock Z to be 12%, 10%, and 6%, as respectively. You want the combined investment in Stock Y and Stock Z to be three times the amount invested in Stock X. You want your overall annual return to be 9%. 19. Write a linear system of equations that represents the given information. 20. Write the system as a matrix equation. 21. Use an inverse matrix and your graphing calculator to solve the system. 22. How much should you invest in each type of stock? 23. Pewter Alloys Pewter is an alloy that consists mainly of tin. It also

contains small amounts of antimony and copper. Three pewter alloys contain percents of tin, antimony, and copper as show in the matrix below. You have 1296 pounds of tin, 69 pounds of antimony, and 35 pounds of copper. How much of each alloy can you make? PERCENTS ALLOY BY WEIGHT X Tin 0.90 Antimony 0.08 Copper 0.02


Y 0.94 0.03 0.03

Z 0.92 0.06 0.02


71


12

4 x 3 3 y 1 2 1 1 x 4 2. 3 1 4 y 2 1 1 1 z 6 1 1 1 1 w 4 2 1 3 1 x 8 3. 1 1 1 2 y 3 1 2 4 1 z 6 4. 2, 1 5. 3, 6 6. 25, 50 7. 6, 12, 6 8. 10, 3, 2 9. 3, 6, 12 10. 10, 5, 0 11. 6, 2, 1, 3 12. 150, 300, 300, 600 13. 3 lb ham, 3 lb turkey, 2 lb roast beef, 4 lb cheese 14. f s j s 690 ; f s j s 10 0.05f 0.05s 0.1j 0.16sn 61 0.1f 0.15s 0.12j 0.08sn 78 180 freshmen, 160 sophomores, 200 juniors, 150 seniors 1.

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Lesson 4.5

LESSON

4.5

NAME

DATE


Write the linear system as a matrix equation. 1. x 4y 3

2x 3y 1

2. 2x y z 4

3. w x y z 4

3x y 4z 2 xyz6

2w x 3y z 8 w x y 2z 3 w 2x 4y z 6


4x 4y 4

5. x 3y 21

6. 2x 9y 400

x 2y 15

3x y 25

Use an inverse matrix and a graphing calculator to solve the linear system. 7. 2x y 4z 48

x 2y 2z 6 x 3y 4z 54

10. 2x y 2z 15

3x 3y z 15 x 3y z 5

8. x y z 9

9. x y z 3

2y z 4 3y z 7

x 2z 27 x y 2z 21

11. w x y z 10

w 2y z 7 w x 3y z 8 w x 4y 4z 16

12. w 2x 4y 3z 1350

w 2x y 4z 3150 2w 3x y z 900 2w x y 3z 2100

13.

Deli Platter You want to order a deli platter for a sports banquet. You need 12 pounds of meat and cheese. You want twice as much meat as cheese on the platter and the same amount of ham and turkey. The price per pound is $4.95 for ham, $6.99 for turkey, $7.99 for roast beef, and $4.36 for cheese. How many pounds of each should you order if you plan to spend $69.24?

14.

72

School Population Six hundred ninety students attend your high school. There are 10 more upper classmen (juniors and seniors) than under classmen (freshmen and sophomores). Five percent of the freshmen, 5% of the sophomores, 10% of the juniors, and 16% of the seniors are members of the student government. The student government has 61 members. During the last grading period 78 students were named to the honor roll. Ten percent of the freshmen, 15% of the sophomores, 12% of the juniors, and 8% of the seniors made the honor roll. Write and solve a system of equations to find the number of students in each class.




2.

y

y

1

1 1

3.

x

2

4. 0, 4

y

x

5. 7, 7

2 x

2

6. 4, 3

7. 12, 12

9. 3, 5

10. 3 2i

8. 2 2, 2 2 11. 4 2i

5i 13. 3i, 3i 14. i 6, i 6 7 15. 5 16. 13 17. 3, 1 18. 2 ± 7 3 29 3 29 19. 1, 9 20. , 2 2 21. 100; 2 real solutions 22. 81; 2 real solutions 12.

23.

24.

y

1

y

1 1

25. 1.84 seconds

x

1

x

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CHAPTER

NAME _________________________________________________________ DATE ____________

5


Graph the quadratic function.

Answers

1. y x2

2. y x2 1 y

y

1.

Use grid at left.

2.

Use grid at left.

3.

Use grid at left.

4. 1

1 x

1

5. 1

x

6. 7.

3. y x2 10x 25

8.

y

9. 10 11.

2

12.

x

2

13.

Solve the quadratic equation by factoring. 4. x2 4x 0

5. x2 49 0

6. 3x2 21x 36 0

Solve the quadratic equation using any appropriate method. 7. x2 144

8. x2 8 0

9. 4x 12 64

11. 7 8i 3 6i 12. 5 7i

16. 17.

19. 20.

Solve the equation. 13. x2 9

15.

18.

Simplify the expression. 10. 3 4

14.

14. 2y2 6 y2

Find the absolute value of the complex number. 15. 2 i

16. 3i 2 Review and Assess

Solve the equation by completing the square. 17. x2 4x 3 0

18. x2 4x 3 0

Use the quadratic formula to solve the equation. 19. x2 10x 9 0


20. x2 3x 5 0


119

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CHAPTER

NAME _________________________________________________________ DATE ____________

5

Chapter Test A

CONTINUED


Find the discriminant of the equation and give the number and type of solutions of the equation. 21. x2 6x 16 0

22. 2x2 5x 7 0

Graph the quadratic inequality. 24. y ≤ 2x2 1

23. y > x2 y

21. 22. 23.

Use grid at left.

24.

Use grid at left.

25.

y

1

1 1

x

1

x

25. Ball Toss

You toss a ball into the air at a height of 5 feet. The velocity of the ball is 30 feet per second. You catch the ball 6 feet from the ground. Use the model 6 16t2 30t 5

Review and Assess

to find how long the ball was in the air.

120




2.

y

y

1

1 2

3.

x

4. 0, 8

y

x

1

5. 3, 3

1 x

1

6. 3, 5

7. 9, 9

9. 0, 4

10. 4 3i

12. 15. 18. 20. 21. 22.

8. 2 3, 2 3 11. 1 i

21 3i 13. 3i, 3i 14. 2i, 2i 50 2 5 16. 26 17. 3, 4 2 2, 2 2 19. 7, 3 3 i 11 3 i 11 , 2 2 19; two imaginary solutions 84; two real solutions

23.

24.

y

y

1 2

1 1

25. 1.84 seconds

x

x

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CHAPTER

NAME _________________________________________________________ DATE ____________

5


Graph the quadratic function. 1. y x2 1

Answers 2. y 2x2

y

y

1.

Use grid at left.

2.

Use grid at left.

3.

Use grid at left.

4. 1

1 1

x

5. 1

x

6. 7.

3. y x2 4x 4

8.

y

9. 10 11.

1 1

12.

x

13.

Solve the quadratic equation by factoring. 4. x2 8x 0

5. 3x2 27 0

6. 2x2 4x 30 0

Solve the quadratic equation using any appropriate method. 7. x2 81 0

8. 4x2 48

9. 4x 22 16

12.

15. 16. 17. 18.

Simplify the expression. 10. 4 4 i

14.

11. 9 7i 10 6i

3 7i

Solve the equation. 13. x2 1 8

14. 4y2 8 2y2 Review and Assess

Find the absolute value of the complex number. 15. 2 4i

16. i 5



18. x2 4x 2 0


121

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CHAPTER

NAME _________________________________________________________ DATE ____________

5

Chapter Test B

CONTINUED


Use the quadratic formula to solve the equation.

19.

19. x2 10x 21 0

20.

20. x2 3x 5 0

Find the discriminant of the equation and give the number and type of solutions of the equation. 21. x2 7 3x

22. 4x2 2x 5 0

Graph the quadratic inequality. 24. y < 2x2 3

23. y ≥ x2 y

21. 22. 23.

Use grid at left.

24.

Use grid at left.

25.

y

1

1 1

x

1

x

25. Vertical Motion

An object is released into the air at an initial height of 6 feet and an initial velocity of 30 feet per second. The object is caught at a height of 7 feet. Use the vertical motion model, h 16t2 vt s,

Review and Assess

where h is the height, t is the time in motion, s is the initial height, and v is the initial velocity, to find how long the object is in motion.

122




2.

y

y x

1

1 2

1

3.

x

5

5

4. 2, 2

y

5.

2 3

6. 2, 3

2 x

2

7. 2 5, 2 5

8. 1 6, 1 6

9. 1 i 19, 1 i 19

10. 3 3i

11. 1

3 4i 13. 2i 2, 2i 2 5 14. 8 2, 8 2 15. 65 16. 10 12.

17. 1 3i, 1 3i

9 105 9 105 , 4 4 3 i 11 3 i 11 19. 20. 2 2i, 2 2i , 2 2 21. 121; 2 real solutions 22. 23; 2 imaginary solutions 18.

23.

24.

y

1

y

1 1

25. 1.84 seconds

x

1

x

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Page 123

CHAPTER

NAME _________________________________________________________ DATE ____________

5


Graph the quadratic function. 1. y x2 1

Answers 2. y x2 2x 5 y

y 1

1.

Use grid at left.

2.

Use grid at left.

3.

Use grid at left.

x

1

4. 1

5. x

1

6. 7.

3. y x 22 4

8.

y

9. 10 11. 2

12. x

2

13. 14.

Solve the quadratic equation by factoring. 4. 4x2 25 0

5. 9x2 12x 4 0

6. 6 x2 x

Solve the quadratic equation using any appropriate method. 7. 5x2 100

8. 3x 12 4 22

9.

x2 x 20 10 5

15. 16. 17. 18.

Simplify the expression. 10. i 3 4 12.

11. 5 8i 4 8i

2i 2i

Solve the equation. 14.

1 2 4x

1 33

Review and Assess

13. 2x2 1 15


16. 5 i5



18. 2x2 9x 3


123

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CHAPTER

NAME _________________________________________________________ DATE ____________

5

Chapter Test C

CONTINUED


Use the quadratic formula to solve the equation.

19.

19. x2 3x 5

20.

20. 2x2 8x 16

Find the discriminant of the equation and give the number and type of solutions of the equation. 21. 6x2 4 5x

22. 2y2 3y 4

Graph the quadratic inequality. 23. y ≤ 2x2 1

24. y ≥ x2 5x 6

y

21. 22. 23.

Use grid at left.

24.

Use grid at left.

25.

y

1

1 1

x

1

x

25. Vertical Motion

An object is released into the air at an initial height of 9 feet and an initial velocity of 30 feet per second. The object is caught at a height of 10 feet. Use the vertical motion model, h 16t2 vt s,

Review and Assess

where h is the height, t is the time in motion, s is the initial height, and v is the initial velocity, to find how long the object is in motion.

124



Answer Key Cumulative Review 1. 24 2. 25 3. 25 4. 14 5. 12 16 7. 45 8. 2 4 9. x < 2 10. x ≤ 3 3 2 1

0

1

2

3

34.

35.

6. 12

y

y

1 x

1

4 3

1

3 2 1

0

1

2

3 x

1

3 7

11. x
3 or x < 2

16. 4 < x < 8

17. 23

x

1

18. 27

21. 0 22. undefined 23. perpendicular 24. neither 2 26. y 4 27. y 3 x 2

39. 2, 1, 4

1

19. 9

20. 4

28.

25. y 5x 3 40.

29.

41. y

y

y 1

(0, 5)

y

(2, 2) x

1

1

(3, 3)

1

x

1

1

1

x

x

1

(0, 6)

1

31. y 3 x

30.

8 3

42.

43.

y

y

(1, 5)

1

(0, 0) 1

1 x

32. y 2x 12

1

33. y x 1

x

8 12

12 2

Answer Key 44. 1, 3

45. 2, 2

47. 3, 4

46. 3, 1, 0

48. 1, 2

50.

49. 1, 0, 3 51.

y

y

x0 x3

2

(1, 0) (2, 6)

(1, 0)

(4, 6)

2

(3, 5) 2

(0, 3)

y (x 3)2 5

y 3(x 1)(x 1)

x

2

53. x 8, 2

52. y

x 1

1 x

1

(2, 2)

(0, 2)

(1, 5) y 3x2 6x 2

54. 6 33, 6 33 56. 2 8i

x

57. 17 i

59. 2 23i

60.

10 17

55. 4, 2

58. 3 3i 11 17 i

61.

15 17

8 17 i

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CHAPTER

NAME _________________________________________________________ DATE ____________

5


Evaluate the expression. (1.1) 1. 3 32 52

2. 52

3. 52

Simplify and evaluate the expression for the given value of the variable. (1.2) 4. x 2 8 x when x 2

5. 3a2 a 2a2 when a 3

6. 2n 1 4n 2 when n 1

Solve the equation. (1.3) 7.

1 2x

8. 32x 1 4x 1 5

13 2x 15

Solve the inequality and draw its graph. (1.5) 9. 3x 1 < 2x 3

10. 2x 3 ≥ 5x 1

11. 4x 3 > 3x

Solve the compound inequality. (1.6) 13. 4 < 2x 4 < 12

12. 3x 1 < 2x 9 or 5x 3 < 53

Solve the absolute value equation or inequality. (1.7)

14. 3x 5 10

15. 4x 2 > 10

16. x 2 < 6

Evaluate the function when x 5. (2.1) 17. gx x 2 2

18. f x x2 2

19. f x x 32 5

Find the slope of the line passing through the points. (2.2) 20. 4, 3 and 6, 5

21. 2, 0 and 8, 0

22. 5, 8 and 5, 14

Tell whether the two lines are parallel, perpendicular, or neither. (2.2) 23. Line 1: through 5, 3 and 8, 4




Write the equation with the given slope and y-intercept. (2.3) 25. m 5; b 3

2

26. m 0; b 4

27. m 3; b 2

29. y 4x 6

30. y 5x

Graph the equation. (2.3) Review and Assess

28. y

2 3 x

5

Write the equation of the line that passes through the given point and has the given slope. (2.4) 31. 5, 1; m

1 3

32. 6, 0; m 2

33. 4, 5; m 1

35. y < x 5

36. 2x y < 4

Graph the inequality. (2.6) 34. y ≥

130

2 3x

3



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CHAPTER

NAME _________________________________________________________ DATE ____________

5

Cumulative Review

CONTINUED


Solve the linear system. (3.2, 3.6) 37. 2x 3y 14

38. 3x 5y 14

x 5y 19

39. 2x 3y z 11

2x 3y 16

4x y 2z 1 3x 2y 2z 0

Graph the system of linear inequalities. (3.5) 40. y < x 2

41. y > 3x 2

y > 3x 1

42. 3x y ≥ 5

y > 2x 1

2x y ≤ 3

Perform the indicated operation. (4.1) 43.

16

3 3 3 2 2

5 0

Use Cramer’s Rule to solve the system. (4.3) 44. 2x 3y 11

45. 2x 2y 0

x 4y 11

46. 4x 2y 3z 14

5x 3y 4

2x y 5x 5 3x 2y 5z 7

Use matrices to solve the linear system. (4.5) 47. 2x 4y 22

48. 3x 2y 7

3x y 13

49. x 2y 3z 10

5x 4y 3

2x 3y 4z 10 2x 3y 5z 13

Graph the quadratic function. Label the vertex and the axis of symmetry. (5.1, 5.3) 50. y x 32 5

51. y 3x 1x 1

52. y 3x2 6x 2

54. x 2 12x 3 0

55. x 2 6x 8 0

Solve the equation. (5.3, 5.5) 53. 3x 52 27

Write the expression as a complex number in standard form. (5.4) 56. 4 3i 2 5i

57. 7 3i2 i

59. 3 2i4 5i

60.

3 2i 4i

58. 6 2i 3 5i 61.

4i 4i Review and Assess



131

Answer Key Practice A 1. y 2x2 x 1; opens up 2. y x2 x 3; opens down 3. y 5x2 3x 4; opens down 4. y x2 2x 1; opens up 5. y 3x2 4; opens down 6. y 9x2 x; opens up 7. y x2 5x 3; opens up 8. y 3x2 4x 1; opens down 9. y 2x2 3x 3; opens down 4 10. x 1 11. x 1 12. x 3 1 13. x 3 14. x 2 15. x 0 16. 1, 2 17. 2, 5 18. 3, 17 19. 0, 5 20. 0, 4 21. 1, 2 22. B 23. A 24. C 25. y 3x2 12x 13 26. y x2 2x 3 27. y 2x2 12x 19 28. y 2x2 4x 6 29. y x2 9x 18 30. y 4x2 12x 8 31.

32.

y x=1

x = 2

y

3

(1, 3) 1

1

(2, 1) 1

33.

x

34. 14 ft

y 1 1

(2, 1)

x=2

x

x

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Page 15

LESSON

NAME _________________________________________________________ DATE ___________

5.1


Write the quadratic function in standard form. Determine whether the graph of the function opens up or down. 1. y 2x2 x 1

2. y 3 x x2

3. y 4 3x 5x2

4. y 2x 1 x2

5. y 4 3x2

6. y x 9x2

7. y x2 3 5x

8. y 3x2 1 4x

9. y 3x 2x2 3

10. y 2x2 4x 1

11. y x2 2x 5

12. y 3x2 8x 2

13. y x2 6x

14. y 2x2 2x 3

15. y 3x2 5

16. y x2 2x 1

17. y 2x2 8x 3

18. y x2 6x 8

19. y x2 5

20. y x2 4

21. y 2x2 4x

Lesson 5.1

Find the axis of symmetry of the parabola.

Find the vertex of the parabola.

Match the quadratic function with its graph. 23. y x 3x 5

22. y x2 2x 15 A.

B.

y 2

2

x

24. y x 12 12 C.

y 2

2

y

x

2

x

2

Write the quadratic function in standard form. 25. y 3x 22 1

26. y x 12 2

27. y 2x 32 1

28. y 2x 3x 1

29. y x 3x 6

30. y 4x 1x 2

Graph the quadratic function. Label the vertex and axis of symmetry. 31. y x 12 3

32. y x 22 1

33. y x 22 1

34. Maximum Height The path that a diver follows is given

y

by y 0.4x 42 14 where x is the horizontal distance (in feet) from the edge of the diving board and y is the height (in feet). What is the maximum height of the diver?

2 2


x


15

Answer Key Practice B 1. y x2 2x 3; opens down 2. y 3x2 3x 4; opens up 3. y 4x2 5; opens down 1 1 1 4. 2, 4; x 2 5. 6, 12 ; x 6 1 15 1 6. 2, 4 ; x 2 7.

17.

(2, 5)

x=3 y

19.

20.

y

x = 2

(2, 3)

1 x

(0, 1) x

1

x=0

10.

y

x

1

1

1 x

y

1

3

9.

x

1

x

1

(2, 1)

(0, 0)

x=2

(3, 2)

1

x=0

1

y

1

8.

y

18.

y

x = 2

y x=1

21.

(0, 2)

22.

y

3

1

y

x

3

x=1

1 x

1

(1, 4)

x

1

1

(1, 1)

1 x

(

x=0

11.

12.

y 3

y

(4, 18)

15 x

23.

24.

y

15

x=3

1

)

x=

7 2

x=4

21 9

7

2 , 4

3

y

x=3

x

1

9 1 3

(3, 18)

3

x=

15 x

3

3 2

3 2

x

1

(3, 1)

25 4

( , ) 13.

x = 7

14.

y

30

10

y 10

x = 1

(1, 9)

30 x

25.

30

(7, 58)

26.

y

5 9 2 4

( , )

y

x=1 (1, 8)

1 2

50

x

1 x

2 x

2 5

15.

x = 2

16.

y 3

y

27.

1

x = 2

28.

y

(1, 3)

y

(1, 3) 1

(

1 , 7 2 4

1

x 1

)

1 1

x = 1

(20, 700)

x

x

x = 1

100 4

29. $700

30. 20

31. March 16

x

32. $1.26

MCRB2-0501-PA.qxd

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Page 16

LESSON

5.1

NAME _________________________________________________________ DATE ___________


Write the quadratic function in standard form. Determine whether the graph of the function opens up or down. 1. y 3 2x x2

2. y 3x 3x2 4

3. y 5 4x2

Find the vertex and axis of symmetry of the parabola.

Lesson 5.1

4. y x2 4x 8

5. y 3x2 x

6. y x2 x 4

Graph the quadratic function. Label the vertex and axis of symmetry. 7. y x2

8. y x2 1

9. y x2 2

10. y x2 2x

11. y 2x2 12x

12. y x2 8x 2

13. y x2 14x 9

14. y 2x2 4x 7

15. y 3x2 3x 1

Graph the quadratic function. Label the vertex and axis of symmetry. 16. y x 12 3

17. y x 32 2

18. y x 22 5

19. y x 22 1

20. y 2x 22 3

21. y 3x 12 4

Graph the quadratic function. Label the vertex and axis of symmetry. 22. y x 3x 4

23. y x 4x 1

24. y x 2x 4

25. y x 4x 1

26. y 2x 3x 1

27. y 3xx 2

Minimum Cost A manufacturer of lighting fixtures has daily production costs modeled by y 0.25x2 10x 800 where y is the total cost in dollars and x is the number of fixtures produced. 28. Sketch the graph of the model. Label the vertex. 29. What is the minimum daily production cost, y? 30. How many fixtures should be produced each day to yield a

minimum cost? Price of Gasoline The price of gasoline at a local station throughout the month of March is modeled by y 0.014x2 0.448x 2.324 where x 1 corresponds to March 1. 31. On what day in March did the price of gasoline reach its maximum? 32. What was the highest price of gasoline in March?

16




13. 2.

y

(0, 0)

x0

1 x

1

y

y

x0

1

14.

y

x

1

(3, 2)

1

1

(1, 1)

x

1

x

1

x = 1

x=3

(0, 5)

15. 3.

4.

y

y

x

2

7 2

2

(2, 3)

x

7 49 , 4 2

( 6.

y

18.

y

x

y

x0 (0, 1)

1

2

( y

(

5 1 , 24 12

6

)

10.

x

4

1

3 2

,

25 8

)

1

23. y 2x2 2x 24.

1x 23

x = 10

1

x

3

x = 2 5 49 , 8 8

23 , 319 10 10

)

(

y

(

1 81 , 4 8

)

x=

1 4

12.

26. y 3x2

y

28.

1 2

20 3x

41 6

x

(

3 5 , 4 2

1

5 8

)

x 1

3

x = 2

1

29. 1996

3

27. y 2x2 9x

31 2

As a increase the graph becomes more narrow.

y

1

2

1

25. y 8x2 4x

) 5

11.

3 4

2 y x2 13 15 x 15

1 5

3

22. y 2x2 2x 2

y

(

y 5

x

x = 8

(

15

5 12

21.

1

x

)

x=4

x=2

x

x=

) y

2

y

15

1

1

x

15 169 , 4 8

1

9.

20.

(2, 36)

x

1

1

(1, 4)

(

5 4

5 49 , 8 4

)

8.

y 1

(

23 25 , 24 12

) 19.

7.

1

x

x

x

2 4 ,3 3

y

1

2

(

)

x = 1

23 = 12

1

1

17 529 , 20 10

) 17.

2 3

x

1

(

2

x

17

x = 10

x0

2

x

y 5

x

1 x

(0, 8)

5.

16.

y 1

x = 2

x

MCRB2-0501-PA.qxd

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LESSON

NAME _________________________________________________________ DATE ___________

5.1


Graph the quadratic function. Label the vertex and axis of symmetry. 1. y 3x2

2. y x2 5

3. y 2x2 8

4. y x2 7x

5. y 3x2 4x

6. y 2x2 1

7. y 2x2 5x 3

8. y 6x2 5x 1

9. y 8x2 10x 3

11. y 2x2 x 10

12. y x2 3x 1

13. y 2x 12 1

14. y 3x 32 2

15. y 4x 22 3

16. y 5x 3x 4

17. y 3x 72x 3

18. y 3x 13x 5

19. y x 4x 8

20. y 2x 1x 7

21. y 2x 4x 1

2

5

1

Write the quadratic function in standard form. 22. y 2x 4x 1

23. y 2x

25. y 8x 12

26. y 3x 22

1

1

1 2

5

1 2

x 34 1 6

24. y x

2 3

x 15

27. y 2x 32 2 3

Use your graphing calculator to graph y ax 32 1 where a 2, 3, and 4. Use the same viewing window for all three graphs. How do the graphs change as a increases?

28. Visual Thinking

29. Poultry Consumption

From 1990 to 1996, the consumption of poultry per capita is modeled by y 0.2125t 2 2.615t 56.33, where t 0 corresponds to 1990. During what year was the consumption of poultry per capita at its maximum?



17

Lesson 5.1

10. y 10x2 46x 21

Answer Key Practice A 1. x 2x 3 2. x 5x 1 3. x 3x 1 4. x 3x 2 5. x 6x 3 6. x 3x 1 7. x 4x 2 8. x 4x 1 9. x 4x 1 10. x 4x 4 11. x 2x 2 12. x 3x 3 13. x 1x 1 14. x 3x 3 15. x 2x 2 16. x 8x 8 17. x 4x 4 18. x 8x 8 19. 2x 1x 1 20. 3x 2x 2 21. 2x 1x 1 22. 3x 1x 1 23. x 3x 3 24. 2x 4x 4 25. 2x 1x 2 26. 3x 1x 2 27. x 2x 3 28. 1, 3 29. 2, 1 30. 4, 5 31. 2 32. 1 33. 3 34. 5 35. 4, 4 36. 9, 9 37. 5, 2 38. 6, 6 39. 7 40. 12 ft by 3 ft 41. 17 ft by 3 ft 42. 2 seconds 43. 1 second 44. 3 seconds

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LESSON

5.2

NAME _________________________________________________________ DATE ___________


Factor the expression. If the expression cannot be factored, say so. 1. x2 x 6

2. x2 6x 5

3. x2 4x 3

4. x2 5x 6

5. x2 9x 18

6. x2 4x 3

7. x2 6x 8

8. x2 3x 4

9. x2 3x 4

10. x2 16

11. x2 4x 4

12. x2 6x 9

13. x2 2x 1

14. x2 6x 9

15. x2 4

16. x2 64

17. x2 8x 16

18. x2 16x 64

19. 2x2 4x 2

20. 3x2 12

21. 2x2 2

22. 3x2 6x 3

23. x2 6x 9

24. 2x2 16x 32

25. 2x2 2x 4

26. 3x2 9x 6

27. x 2 5x 6

28. x2 2x 3 0

29. x2 3x 2 0

30. x2 9x 20 0

31. x2 4x 4 0

32. x2 2x 1 0

33. x2 6x 9 0

34. x2 10x 25 0

35. x2 16 0

36. x2 81 0

37. x2 3x 10

38. x2 36

39. x2 14x 49

Factor the expression.

Lesson 5.2

Solve the equation.

Find the dimensions of the figure. 40. Area of rectangle 36 square feet

41. Area of rectangle 51 square feet

x9

x x 14 x

Find the time (in seconds) it takes an object to hit the ground when it is dropped from a height of s feet. Use the falling-object model h 16t2 s. 42. s 64

28


43. s 16

44. s 144


Answer Key Practice B 1. x 7x 3 2. cannot be factored 3. x 3x 5 4. x 7x 2 5. x 7x 4 6. x 6x 4 7. cannot be factored 8. 2x 1x 3 9. 3x 2x 1 10. 3x 1x 2 11. 2x 3x 1 12. 2x 15x 1 13. 6x 1x 2 14. 5x 33x 1 15. cannot be factored 16. x 8x 8 17. x 3x 3 18. x 7x 7 19. 2x 12x 1 20. 3x 23x 2 21. 3x 13x 1 22. cannot be factored 23. cannot be factored 24. 2x 53x 1 25. 2x 5x 3 26. x 3x 7 27. 3x 4x 1 28. 22x 1x 3 29. 32x 1x 5 30. 22x 3x 4 31. 32x 12x 1 32. 22x 32x 3 33. 53x 13x 1 34. 6, 5 1 1 35. 9, 1 36. 4, 8 37. 4, 2 38. 3, 3 2

40. 1, 3

7

45. 2, 3

39. 1, 5 44. 3, 2

5

5 1

2

41. 5 46.

1 2

42.

ft

1 7

4 4

43. 5, 5

47. $2.65

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Page 29

LESSON

5.2

NAME _________________________________________________________ DATE ___________


Factor the expression. If the expression cannot be factored, say so. 1. x2 4x 21

2. x2 6x 2

3. x2 8x 15

4. x2 9x 14

5. x2 11x 28

6. x2 10x 24

7. x2 3x 1

8. 2x2 5x 3

9. 3x2 x 2

10. 3x2 7x 2

11. 2x2 5x 3

12. 10x2 3x 1

13. 6x2 13x 2

14. 15x2 14x 3

15. 2x2 7x 1

16. x2 16x 64

17. x2 6x 9

18. x2 49

19. 4x2 4x 1

20. 9x2 12x 4

21. 9x2 1

22. x2 8x 5

23. 4x2 7

24. 6x2 17x 5

25. 2x2 4x 30

26. x2 10x 21

27. 3x2 15x 12

28. 4x2 14x 6

29. 6x2 33x 15

30. 4x2 10x 24

31. 12x2 3

32. 8x2 24x 18

33. 45x2 30x 5

34. x2 x 30 0

35. x2 10x 9 0

36. x2 12x 32 0

37. 2x2 7x 4 0

38. 3x2 8x 3 0

39. 5x2 3x 2 0

40. 3x2 8x 5 0

41. 25x2 20x 4 0

42. 49x2 14x 1 0

43. 25x2 16

44. 2x2 x 21


Lesson 5.2

Solve the equation.

45. 8x2 5x 4 2x2 8x 1 46. Furniture Manufacturing You are making a coffee

table with a glass top surrounded by a cherry border. The glass is 3 feet by 3 feet. You want the cherry border to be a uniform width. You have 7 square feet of cherry to make the border. What should the width of the border be?

x

3 ft

x x

3 ft

x

47. A magazine has a circulation of 140 thousand per month when they charge

$2.50 for a magazine. For each $.10 increase in price, 5 thousand sales are lost. How much should be charged per magazine to maximize revenue?



29

Answer Key Practice C 1. x 9x 10 2. x 5x 11 3. x 7x 13 4. x 13x 15 5. x 17x 12 6. x 11x 14 7. 2x 7x 6 8. x 33x 4 9. x 95x 2 10. 3x 72x 5 11. 2x 14x 3 12. 3x 15x 2 13. 2x 54x 1 14. x 82x 11 15. 3x 2x 15 16. Cannot be factored 17. 2x 135x 12 18. Cannot be factored 19. 3x 72 20. 5x 22 21. 2x 82 22. 4x 32 23. 3x 9x 9 24. 5x 11x 11 25. 23x 7x 2 26. 62x 1x 5 27. 2x3x 42x 1 28. 2x5x 32x 7 29. x33x 12 3 30. x28x2 2x 1 31. 10, 12 32. 7, 2 1 5 1 3 9 11 33. 3, 5 34. 2 35. 3, 2 36. 2, 3 1 7 37. 4, 3 38. 2, 5 39. 5, 7 40. No solution 41. 2 42. 1, 1 43. $350; $306,250

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Page 30

LESSON

5.2

NAME _________________________________________________________ DATE ___________


Factor the trinomial. If it cannot be factored, say so. 1. x2 19x 90

2. x2 16x 55

3. x2 6x 91

4. x2 28x 195

5. x2 29x 204

6. x2 3x 154

7. 2x2 5x 42

8. 3x2 13x 12

9. 5x2 43x 18

10. 6x2 29x 35

11. 8x2 10x 3

12. 15x2 x 2

13. 8x2 18x 5

14. 2x2 5x 88

15. 3x2 43x 30

16. 3x2 7x 2

17. 10x2 89x 156

18. 4x2 11x 13

19. 3x2 42x 147

20. 5x2 20x 20

21. 2x2 32x 128

22. 4x2 24x 36

23. 3x2 243

24. 5x2 605

25. 6x2 26x 28

26. 12x2 54x 30

27. 12x3 10x2 8x

28. 20x3 58x2 42x

29. 9x5 6x4 x3

30. 8x4 2x3 x2

31. x2 22x 120 0

32. 2x2 17x 21 0

33. 5x2 14x 3 0

34. 4x2 20x 25 0

35. 6x2 7x 3

36. 6x2 5x 99

37. 4x2 10x x2 x 4

38. 3x2 x 40 x2 2x 5

39. x 32 411 x

40. x 22 xx 3 1

41. x 12 3x 12 6

42. 2x 12 x 22

Lesson 5.2


Solve the equation.

43. Business

If a gym charges its members $300 per year to join, they get 1000 members. For each $2 increase in price they can expect to lose 5 members. How much should the gym charge to maximize its revenue? What is the gym’s maximum revenue?

30



Answer Key Practice A 1. 4 2 2. 2 3 3. 3 5 4. 5 5 5. 12 7 10 1 2 3 6. 36 7. 8. 9. 10. 2 3 11 5 2 2 6 10 11. 12. 13. 3, 3 14. 12, 12 3 5 15. 8 2, 8 2 16. 6, 6 17. 1, 1 18. 2 2, 2 2 19. 1, 1 20. 3, 3 21. 8, 8 22. 2, 2 23. 5, 5 24. 5, 5 25. 2, 2 26. 9, 9 27. 4, 4 28. 2.24 seconds 29. 3.16 seconds 30. 4.47 seconds 31. 8.06 32. 7.21 33. 1986

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LESSON

5.3

NAME _________________________________________________________ DATE ___________


Simplify the expression. 1. 32

2. 12

4. 125

5. 218

494 12 10. 25

1009 72 11. 5

1211 2 4 12. 3 3

13. x2 9

14. x2 144

15. x2 128

16. x2 36 0

17. x2 1 0

18. x2 8 0

19. 2x2 2

20. 4x2 36

21.

22. x2 3 1

23. x2 2 7

24. 16 x2 9

25. 3x2 1 5

26.

7.

3. 45

2

6. 54

8.

26

9.

Solve the equation.

1 2 3x

5 32

1 2 2x

32

27. 2x2 11 x2 5

Find the time it takes an object to hit the ground when it is dropped from a height of s feet. Use the falling-object model h 16t2 s. 28. s 80

29. s 160

30. s 320

Use the Pythagorean theorem to find x. Round your answer to the nearest hundredth. 31.

32.

4

x

Lesson 5.3

x

14

12 7

33. Cost of a New Car From 1970 to 1990, the average cost of a new car, C

(in dollars), can be approximated by the model C 30.5t 2 4192, where t is the number of years since 1970. During which year was the average cost of a new car $12,000?



41

Answer Key Practice B 1. 7 3 2. 2 15

3. 3 7

4. 96 6

15 7 2 5 7 8. 9. 17 3 2 10. 18, 18 11. 9, 9 12. 6, 6 13. 12, 12 14. 6, 6 15. 2 5, 2 5 16. 7, 7 17. 1, 1 18. 2, 2 7 5 19. 5, 1 20. 2, 6 21. 3, 3 22. 1, 4 23. 0, 8 24. 9, 7 25. 2.5 seconds; 3.54 seconds; no; doubling the height increases the time by the factor 2 . 26. 1992 27. 7 28. 49 ft 29. 14 ft 5. 120

6. 5 6

7.

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Page 42

LESSON

NAME _________________________________________________________ DATE ___________

5.3


Simplify the expression. 1. 147 4. 418 7.

2. 60

248

225 289

18 54

5. 8 8.

3. 63

73 143

6. 10

15

9. 15

3512


11. x2 81 0

13. 3x2 100 332

14.

16.

x2

1

3x2

13

2 2 3x

17. 2

8 16

x2

15.

4 10

19. 2x 32 8

20. 3x 22 4 52

22. 2x 32 25

23.

1 2 x

12. 5x2 180 0 1 2 2x

55

18. 3

1 9

x2

21. 3x 12 36 0

42 8

24.

1 4 x

12 16 0

25. Falling Object Use the falling-object model h 16t 2 s where t is

measured in seconds and h is measured in feet to find the time required for an object to reach the ground from a height of s 100 feet and s 200 feet. Does an object that is dropped from twice as high take twice as long to reach the ground? Explain your answer. 26. Truck Registrations From 1990 to 1993, the number of truck registra-

tions (in millions) in the United States can be approximated by the model R 0.29t2 45 where t is the number of years since 1990. During which year were approximately 46.16 million trucks registered?

Lesson 5.3

Short Cut Suppose your house is on a large corner lot. The children in the neighborhood cut across your lawn, as shown in the figure at the right. The distance across the lawn is 35 feet. 27. Use the Pythagorean theorem to find x. 28. Find the distance the children would have to travel if they did

4x

35

not cut across your lawn. 29. How many feet do the children save by taking the “short cut?” 3x

42



Answer Key Practice C 1. 7 6 2. 6 7 3. 270 4. 6 14 4 5 1 4 3 4 3 5. 6. 7 105 7. 8. 9. 9 3 27 3 10. 17, 17 11. 13, 13 12. 5, 5 13. 5, 5 14. 2 3, 2 3 15. 10, 10 16. 5, 5 17. 11, 11 18. 1, 3 19. 4 3, 4 3 5 5 20. 1 , 1 10 10 21. 3 2 3, 3 2 3 22. 0, 4 6 6 23. 5 , 5 12 12 2 2 24. 3, 3 3 3 25. 3 26, 3 26 7 7 26. 1 2, 1 2 27. 2 ,2 2 2 28. a > 0 29. a > 2 30. a > 1 31. a < 2 32. a > 4 33. a < 5 34. 6%

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Page 43

LESSON

5.3

NAME _________________________________________________________ DATE ___________


Simplify the expression. 1. 294 4. 21

7.

24

1227 14

2. 252

3. 312

160 162 20 8 8. 54 45

6. 35

5.

9.

527

21 7

3215 125 7572

Solve the equation. 1

10. x2 289 0

11. x2 13 0

13. 2x2 10 0

14. 3x2 4 8

15. 5x2 3 7

16. 2x2 7 x2 12

17. 3x2 4 2x2 1

18. 2x 12 8

19. 3x 42 9

20. 5x 12

1 4

21. 4x 32 5 2

22. 2x 22 4 2

23. 2x 52

2 3

3

25.

x 32 1 6 4 2

12. 5x2 5 0

1

26.

2

1

24. 3x

34

x 12 1 5 3 6 6

27.

2 2 3

4 13

2x 22 1 3 5 10 5

Find the values of a for which the equation has two real-number solutions. 28. x2 a

29. x2 2 a

30. 3x2 1 a

31. x2 a 2

32. a x2 4

33. 2x2 a 5


The formula A P 1

Lesson 5.3

r nt gives the amount of n money in an account, A after t years if the annual interest rate is r (in decimal form), n is the number of times interest is compounded per year, and P is the original principle. What interest rate is required to earn $1 in two months if the principle is $100 and interest is compounded monthly?

34. Compound Interest


43

Answer Key Practice A 1. 4i, 4i 2. 9i, 9i 3. 12i, 12i 4. i, i 5. 2, 2 6. 2i, 2i 7. A 2 3i, B 4 i, C 1 3i 8. A 4i, B 3 3i, C 3 i 9. A 2 4i, B 2i, C 4 10. 7 7i 11. 4 i 12. 4 i 13. 2 9i 14. 11 4i 15. 2 2i 16. 1 4i 17. 6 3i 18. 28 12i 19. 4 2i 20. 3 11i 3 3 21. 6 17i 22. 2 2 i 23. 2 i 24. 1 i 25. 2 26. 5 27. 37 28. 5 29. 5 30. 41 31.

32. imaginary

imaginary

1

1 1

real

33.

1

real

34. imaginary

imaginary

1

1 1

real

35.

1

real

36. imaginary

imaginary

1

1 1

real

1

real

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LESSON

NAME _________________________________________________________ DATE ___________

5.4



2. x2 81

3. x2 144 0

4. x2 5 4

5. x2 1 3

6. x2 7 4x2 5

Identify the complex numbers plotted in the complex plane. 7.

8.

9.

imaginary

B

A

imaginary

A

1

1 real

1

imaginary

A 1 1

C C

C

real

1

real

B

B

Write the expression as a complex number in standard form. 10. 5 3i 2 4i

11. 3 2i 1 i

12. 7 2i 3 3i

13. 5 i 3 8i

14. i 11 5i

15. i 6 i 4 2i

16. i4 i

17. 3i1 2i

18. 4i3 7i

19. 1 3i1 i

20. 5 i1 2i

21. 2 3i3 4i

22.

3 1i

23.

5 2i

24.

3i 2i


26. 2 i

27. 6 i

28. 1 2i

29. 3 4i

30. 5 4i

Plot the numbers in a complex plane. 31. 2i

32. 3

33. 1 3i

34. 4 3i

35. 1 2i

36. 2 4i

Lesson 5.4



55

Answer Key Practice B 1. 8i, 8i 2. i, i 4. 2i 3, 2i 3 6. 3i 3, 3i 3 1

1

9. 3 i, 3 i

19. 2 10i

3. 3, 3

5. 4i 3, 4i 3 7. 3i, 3i 8. 2i, 2i

10. 1 6i, 1 6i

1

1

11. 2 2 i, 2 2 i 13.

12. 5 7i, 5 7i 14.

imaginary

imaginary

1

1 1

real

15.

1

real

16. imaginary

imaginary

1

1 1

real

17.

1

real

18. imaginary

imaginary

1

1 1

real

1

real

22. 5 3i

23. 3 17i

25. 39 18i 12 13

18 13 i

20. 1 10i

3

21. 1 2 i 1

24. 6

26. 21 20i

5 12 i

27. 80

29. 1 i 2 3 1 2 3 i 31. 4 30. 4 4 201 73 i 33. 3 12i 34. 5 32. 34 34 3 36. 5 37. i 38. 1 39. i 35. 40. 1 41. i 42. 1 43. i 44. 1 45. If the exponent of i is a factor of 4, the expression can be reduced to 1. Therefore, to simplify i raised to any natural number, factor out the multiples of 4 in the exponent and simplify the remaining expression; i 231 i 228 i 3 1i 3 i. 28.

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LESSON

5.4

NAME _________________________________________________________ DATE ___________



2. x2 1 0

3. x2 5 14

4. x2 12

5. x2 48 0

6. x2 3 24

7. 2x2 9 3x2

8. x2 16 5x2

9. 11x2 1 2x2

10. 2x 12 72

11. 4x 22 1

12. 3x 52 147 0

Plot the number in a complex plane. 13. 3i

14. 2

15. 2 4i

16. 3 4i

17. 2 i

18. 4 3i

Write the expression as a complex number in standard form.

13 12 i 23 2i 12 23 i 23 14 i

19. 3 2i 5 8i

20. 2 4i 3 6i

21.

22. 4 2i 1 5i

23. 5 8i 2 9i

24.

25. 5 4i3 6i

26. 2 5i2

27. 4 8i4 8i

28.

6 2 3i

31. 22 i 1 i2

29.

3i 2 i

32.

1 6 2i 3 5i

30.

2i 3 i

33. 1 5i2 i i3 4i

Find the absolute value of the complex number. 34. 4 3i

35. 2 i

36. 3 2i

Write the complex number in standard form. 37. i

38. i2

39. i3

40. i4

41. i5

42. i6

43. i7

44. i8

45. Pattern Recognition Using the information from Exercises 37–44, write

Lesson 5.4

a general statement about the standard form of in where n is a positive integer. Use this statement to write i 231 in standard form.

56



Answer Key Practice C 14 14 1. i, i 2. 1 3i, 1 3i 2 2 2 2 3. 4 i, 4 i 4. 2i, 2i 2 2 21 21 5. 6 i, 6 i 3 3 6 6 6. 1 i, 1 i 7. a > 0, b > 0 2 2 8. a < 0, b > 0 9. a < 0, b < 0 10. a > 0, b < 0 11. a > 0, b 0 12. a 0, b < 0 13. 1 3i, 10 14. 8 7i, 113

15. 3 i, 10

1 3i, 10

17. 1 , 1

16.

18. 14 2i, 10 2 19. 4 7i, 65

11 10 221 i 170 13 i, , 21. 17 17 17 34 34 34 22. z0 0, z1 3, z2 6, z3 33; Not a member. 23. z0 0, z1 5, z2 5 2, z3 2605; Not a member. 24. z0 0, z1 2, z2 2 5, z3 2 85; Not a member. 25. real 26. pure imaginary 27. real 28. imaginary 20.

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LESSON

5.4

NAME _________________________________________________________ DATE ___________


Solve the equation. 1. 3x2 5 x2 2

2. 4x 12 12 0

3. 6x 42 3 0

4. 2x2 7 x2 10

5. 3x 62 7 0

6. 2x 12 3 6

Determine whether a and b are greater than zero, less than zero or equal to zero for the given complex number a bi . 7. a bi lies in the first quadrant

8. a bi lies in the second quadrant

of the complex plane


9. a bi lies in the third quadrant

10. a bi lies in the fourth quadrant



11. a bi lies on the positive real

12. a bi lies on the negative imaginary

axis of the complex plane

axis of the complex plane

Perform the given operation and find the absolute value of the complex number. 13. 3 2i 4 i

14. 5 2i 3 5i

15. 6 i 3 2i

16. 3 2i 4 5i

17. 2 i 3 4i 3i

18. 2 6i1 2i

19. 2 3i1 2i

20.

2 3i 4i

21.

1 2i 3 5i

Determine whether the complex number c belongs to the Mandelbrot set. Use absolute value to justify your answer. 22. c 3

23. c 2 i

24. c 2i

Determine whether the complex number is real, imaginary, pure imaginary, or neither. 25. The sum of a complex number

and its conjugate. 27. The product of a complex number

and its conjugate.

26. The difference of a complex number

and its conjugate. 28. The quotient of a complex number

and its conjugate.

Lesson 5.4



57

Answer Key Practice A 1. x 12 2. x 22 3. x 82 3 2 1 2 5 2 4. x 2 5. x 2 6. x 2 7. 9; x 32 8. 49; x 72 1 1 2 9. 16; x 42 10. 4; x 2 11. 121; x 112 12. 36; x 62 9 3 2 13. 100; x 102 14. 4; x 2 49 7 2 15. 4 ; x 2 16. 1 3, 1 3 17. 2 5, 2 5 18. 3 7, 3 7 19. 6 33, 6 33 20. 1 3, 1 3 21. 4 17, 4 17 22. 1, 15 1 5 1 5 23. 2, 1 24. , 2 2 2 2 2 25. y x 4 11; 4, 11 26. y x 52 18; 5, 18 27. y x 12 4; 1, 4 28. 16.675 ft by 10.675 ft 29. 4.782 ft by 8.782 ft 30. 8 ft by 6 ft 31. 3.662 ft by 19.662 ft

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LESSON

NAME _________________________________________________________ DATE ___________

5.5

Lesson 5.5


Write the expression as the square of a binomial. 1. x2 2x 1 4. x2 3x

2. x2 4x 4

9 4

5. x2 x

3. x2 16x 64

1 4

6. x2 5x

25 4

Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial. 7. x2 6x c

8. x2 14x c

9. x2 8x c

10. x2 x c

11. x2 22x c

12. x2 12x c

13. x2 20x c

14. x2 3x c

15. x2 7x c


17. x2 4x 1 0

18. x2 6x 2 0

19. x2 12x 3 0

20. x2 2x 2 0

21. x2 8x 1 0

22. x2 16x 15 0

23. x2 x 2 0

24. x2 x 1 0

Write the quadratic function in vertex form and identify the vertex. 25. y x2 8x 5

26. y x2 10x 7

27. y x2 2x 3

Find the dimensions of the figure. Round your answer to the nearest thousandth. 28. Area of rectangle 178 square feet

x

29. Area of triangle 21 square feet

x

x6 x4

30. Area of rectangle 48 square feet

31. Area of triangle 36 square feet x x 16

x

x2



69

Answer Key Practice B 1 2 1. x 42 2. x 52 3. x 4 1 2 4. 2x 32 5. 3x 12 6. 3x 4 7. 144; x 122 8. 225; x 152 25 5 2 81 9 2 9. 4 ; x 2 10. 4 ; x 2 11. 4; 2x 22 12. 1; 3x 12 13. 7, 1 14. 7 4 3, 7 4 3 15. 1, 4 9 57 9 57 16. 17. 1, 2 , 2 2 2 2 18. 4, 1 19. 1 6, 1 6 20. 1 5, 1 5 33 33 1 3 1 3 22. 2 , ,2 2 2 2 2 3 3 23. 1, 5 24. 2 7, 2 7 25. y x 42 5; 4, 5 26. y 2x 12 9; 1, 9 27. y 3x 12 5; 1, 5 28. 6.275 ft by 1.275 ft 29. 4.490 ft by 10.245 ft 30. x2 2x 72 442 ⇒ 5x2 28x 1887 0 31. 16.828, 22.428 32. 33.656 ft

21.

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Lesson 5.5

LESSON

5.5

NAME _________________________________________________________ DATE ___________


Write the expression as the square of a binomial. 1

1 16

1. x2 8x 16

2. x2 10x 25

3. x2 2x

4. 4x2 12x 9

5. 9x2 6x 1

6. 9x2 2 x

3

1 16

Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial. 7. x2 24x c 10. x2 9x c

8. x2 30x c

9. x2 5x c

11. 4x2 8x c

12. 9x2 6x c


14. x2 14x 1 0

15. x2 3x 4 0

16. x2 9x 6 0

17. 2x2 2x 4 0

18. 3x2 9x 12 0

19. 2x2 4x 10 0

20. 5x2 10x 20 0

21. 4x2 4x 2 0

22. 3x2 12x 1 0

23. 2x2 8x 10 0

24. x2 4x 3 0


26. y 2x2 4x 7

27. y 3x2 6x 8

Find the dimensions of the figure. Round your answer to the nearest thousandth. 28. Area of rectangle 8 square feet

29. Area of triangle 23 square feet

x x x5 1 x 2

No Passing Zone A “No Passing Zone” sign has the shape of an isosceles triangle. The width of the sign is 7 inches greater than its height. The top and bottom edges of the sign are 44 inches. 30. Use the Pythagorean theorem to write an equation that relates x,

2x 7

x

2x 7, and 44. 31. Solve the equation in Exercise 30 by completing the square.

(Hint: Use decimal representations and a calculator to simplify your work.)

8

x

NO PASSING ZONE 44 in.

32. What is the height of the sign?

70



Answer Key Practice C 1. x 4. 3x

1 2 3 2 12

2. 2x 52

5. x 0.82

3. 2x

1 2 5

6. 0.3x 0.42

7. 4 19, 4 19 8. 5 19, 5 19 9. 10. 11. 12. 14. 15. 16.

5 21 5 21 , 2 2 2 2 7 65 7 65 , 2 2 2 2 5 57 5 57 , 2 2 2 2 1 5, 1 5 13. 5, 1 3 21 3 21 , 2 2 2 2 2 15 2 15 , 1 1 5 5 3 33 3 33 17. 2, 1 , 2 2 2 2 5

3

5

3

18. 2 2 i, 2 2 i

1 3 1 3 5 5 5 5 i, i 20. , 2 2 2 2 2 2 2 2 21. 2 2, 2 2 22. y x 82 62; 8, 62 23. y 2x 32 23; 3, 23 19.

24. y 3x

25. y 2x

5 2 2 3 2 4

5 71 71 4 ; 2, 4

18; 34, 18 26. y x 22 3; 2, 3 27. y 4x

1 2 4

28. 10 ft; 30.03 ft

1 11 11 4 ; 4, 4

29. 204.96 ft/s

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LESSON

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5.5

Lesson 5.5


Write the expression as the square of a binomial. 2

1. x2 3x

1 9

4. 9x2 3x

1 4

4

1 25

2. 4x2 20x 25

3. 4x2 5x

5. x2 1.6x 0.64

6. 0.09x2 2.4x 0.16


8. x2 10x 6 0

9. x2 5x 1 0

10. x2 7x 4 0

11. 2x2 5x 3 x2 5

12. 4x2 2x 1 3x2 4x 5

13. 2x2 8x 10 0

14. 3x2 9x 4 5

15. 5x2 2x 3 10 8x

16. x2 3x 6 0

17. 2x2 6x 4 0

18. 2x2 10x 17

19. 3x2 4x 2 x2 6x

20. x2 3x 7 8x 2

21. 3x2 2x 1 x2 6x 3


23. y 2x2 12x 5

24. y 3x2 15x 1

25. y 2x2 3x 1

26. y x2 4x 1

27. y 4x2 2x 3

28. Biology

The impala is the most powerful jumper of the antelope family. When an impala jumps, its path through the air can be modeled by y 0.0444x2 1.3333x where x is the impala’s horizontal distance traveled (in feet) and y is its corresponding height (in feet). How high can an impala jump? How far can it jump?

29. Falling Object

An object is propelled upward from the top of a 500-foot building. The path that the object takes as it falls to the ground can be modeled by y 16t2 100t 500 where t is time (in seconds) and y is the corresponding height (in feet) of the object. The velocity of the object can be modeled by v 32t 100 where t is time (in seconds) and v is the corresponding velocity of the object. With what velocity does the object hit the ground?



71

Answer Key Practice A 1. 3x2 4x 3 0; a 3, b 4, c 3 2. x2 3x 2 0; a 1, b 3, c 2 3. 3x2 3x 4 0; a 3, b 3, c 4 4. 2x2 4x 5 0; a 2, b 4, c 5 5. x2 9x 2 0; a 1, b 9, c 2 6. 6x2 3 0; a 6, b 0, c 3 7. 11 8. 0 9. 28 10. 21 11. 24 12. 0 13. 16; 2 14. 17; 2 15. 11; 0 16. 1; 2 17. 31; 0 18. 41; 2 19. 0; 1 20. 0; 1 21. 11; 0 22. 12; 0 23. 84; 2 1 5 1 5 24. 0; 1 25. , 2 2 3 13 3 13 26. , 2 2 27. 3 7, 3 7 28. 0, 7 29. 3, 0 30. i 6, i 6 31. 6, 6 3 i 11 3 i 11 32. , 2 2 1 i 55 1 i 55 33. , 2 2 34. x2 2x 4 0; 1 5, 1 5 35. x2 2x 1 0; 1 36. x2 2x 15 0; 3, 5 37. x2 6x 11 0; 3 i 2, 3 i 2 1 1 38. x2 x 0; 4 2 39. x2 3x 0; 3, 0 40. 3.28 in. 41. 5.38 in.

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LESSON

5.6

NAME _________________________________________________________ DATE ___________


Write the equation in standard form. Identify a, b, and c. 1. 3x2 4x 3 0

2. x2 3x 2

3. 3x 4 3x2

4. 3x2 5 x2 4x

5. 2 x 8x x2

6. 5x2 2 x2 1

Find the discriminant of the quadratic equation. 7. x2 x 3 0

8. x2 2x 1 0 11. x2 2x 7 0

12. x2 6x 9 0

Lesson 5.6

10. x2 5x 1 0

9. x2 2x 6 0

Find the discriminant and use it to determine the number of real solutions of the equation. 13. x2 2x 3 0

14. x2 5x 2 0

15. x2 3x 5 0

16. x2 5x 6 0

17. 2x2 x 4 0

18. 2x2 x 5 0

19. x2 18x 81 0

20. x2 4x 4 0

21. x2 3x 5 0

22. x2 3 0

23. x2 21 0

24. 5x2 0

Use the quadratic formula to solve the equation. 25. x2 x 1 0

26. x2 3x 1 0

27. x2 6x 2 0

28. x2 7x 0

29. x2 3x 0

30. x2 6 0

31. x2 36 0

32. x2 3x 5 0

33. x2 x 14 0

Write the equation in standard form. Use the quadratic formula to solve the equation. 34. x2 5 2x 1

35. 3x2 2x 2x2 1

37. x2 11 6x

38. x2 1 x

36. x2 2x 15

3 4

39. x2 3x 2x2

Find the value of x. Round your answer to the nearest hundredth. 40. Area of rectangle 24.5 square inches

41. Area of parallelogram 63.9 square inches

x

x

x 4.2


x 6.5


83

Answer Key Practice B 1. 11 2. 25 3. 0 4. 76 5. 49 6. 100 7. 1; 2 8. 0; 1 9. 8; 0 10. 47; 0 11. 37; 2 12. 9; 2 13. 4, 5 1 1 33 1 33 14. , 2 15. , 2 4 4 1 1 51 1 51 16. , 2 17. , 4 10 10 7 113 7 113 18. , 16 16 2 19. x 4x 2 0; 2 6, 2 6 20. x2 3x 4 0; 1, 4 1 57 1 57 21. 2x2 x 7 0; , 4 4 3 69 3 69 22. 3x2 3x 5 0; , 6 6 5 33 5 33 23. 2x2 5x 1 0; , 4 4 3 41 3 41 24. 4x2 3x 2 0; , 8 8 15 89 15 89 25. 2x2 15x 17 0; , 4 4 2 26. 2.4x 3.5x 2.2 0; 1.933, 0.474 27. 4.2x2 6.8x 2 0; 0.386, 1.233 28. Yes; Your garden should be approximately 12.93 ft by 27.07 ft. 29. No; The area of the room can be expressed as x8 x. The equation x8 x 20 has no real solution. 30. 10.22 seconds 31. h 16t2 27t 6 32. 1.89 seconds

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LESSON

5.6

NAME _________________________________________________________ DATE ___________


Find the discriminant of the quadratic equation. 1. x2 3x 5 0

2. 3x2 x 2 0

3. 4x2 12x 9 0

4. 5x2 2x 4 0

5. 2x2 3x 5 0

6. 3x2 2x 8 0

Lesson 5.6

Find the discriminant and use it to determine the number of real solutions of the equation. 7. x2 3x 2 0

8. 4x2 20x 25 0

10. 3x 2 x 4 0

11. x2 3x 4 2x2 3

9. 3x2 2x 1 0 12. 4x2 3x 0

Use the quadratic formula to solve the equation. 13. x2 x 20 0

14. 2x2 3x 2 0

15. 2x2 x 4 0

16. 4x2 9x 2 0

17. 10x2 2x 5 0

18. 8x2 7x 2 0

Write the equation in standard from. Use the quadratic formula to solve the equation. 19. 3x2 4x 2x2 2

20. x2 5 3x 1

21. 4 2x2 x 3

22. x2 3x 2 4x2 3

23. 9x x2 x2 4x 1

24. 6x2 5 2x2 3x 7

25. 2x 32 3x 1

26. 2.4x2 3.5x 2.2

27. 6.8x 2 4.2x2

28. Fencing Your Garden It takes 80 feet of

fencing to enclose your garden. According to your calculations, you will need 350 square feet to plant everything you want. Is your garden big enough? Explain your answer.

29. New Carpeting You have new carpeting

installed in a rectangular room. You are charged for 20 square yards of carpeting and 16 yards of tack strip. Do you think these figures are correct? Explain your answer. Tack strip

x x 40 x

8x

Throwing an Object on the Moon An astronaut standing on the moon throws a rock upwards with an initial velocity of 27 feet per second. The astronaut’s hand is 6 feet above the surface of the moon. The height of the rock is given by h 2.7t 2 27t 6. 30. How many seconds does it take for the rock to fall to the ground? 31. Suppose the astronaut had been standing on Earth. Write a vertical motion

model for the height of the rock after it is thrown. 32. Use the model in Exercise 31 to determine how many seconds it takes for

the rock to fall to the ground on Earth.

84



Answer Key Practice C 1. 13 2. 25

3. 169

4. 24

5. 31

5 37 5 37 , 2 2 2 2 3 17 3 17 8. , 4 4 4 4 1 9. 6 3 3, 6 3 3 10. 2, 1 5 37 5 37 11. , 2 2 2 2 12. 5 19, 5 19 19 19 1 1 13. i, i 14. 20 10 10 10 10 15. 19.11, 1.39 16. 0.71, 0.51 17. 0.08 0.53i, 0.08 0.53i 1 87 1 87 18. i, i 19. 4, 4 2 6 2 6 20. 2 6, 2 6 21. No solution In Exercises 22–24, answer may vary. Sample answers are given. 22. 1, 2 23. 1, 2 24. 2, 3 25. Object launched downward 26. h 16t 2 10t 100 27. h 16t2 100 28. h 16t 2 10t 100 29. launched upward: 2.8 s, dropped: 2.5 s, launched downward: 2.2 s The object launched downward reaches the ground first. 6. 24

7.

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LESSON

5.6

NAME _________________________________________________________ DATE ___________


Find the discriminant of the quadratic equation. 1. x2 3x 1 0

2. 3x2 7x 2 0

3. 8x2 3x 5 0

4. 2x x2 5 0

5. 10 3x x2 0

6. 6x 3 5x2 0

Use the quadratic formula to solve the equation. 7. x2 5x 3 0

9.

10. 3x2 2x x2 5x 1

11. x2 3 2x2 5x

12.

13. 5x2 9 x 8

14.

16. 4.5x2 1.2x 2.1 1.3x2

17. 7.3x2 2.1 1.1x

1 2 16 x

52x 25

1 2 3x 1 2 2x

4x 3 0 2x 5 2 3x

15. 2.3x2 4.1x 2.1x2 5.3 18.

1 5 x

12 15x 13

Find all values of b for which the equation has one real solution. 19. x2 bx 4 0

20. 2x2 bx 3 0

21. 3x2 bx 5 0

Give two examples of values of b for which the equation has two imaginary solutions. 22. x2 bx 5 0

Vertical Motion

23. 2x2 bx 1 0

24.

2 2 5x

bx 10 0


Three objects are launched from the top of a 100-foot building. The first object is launched upward with an initial velocity of 10 feet per second. The second object is dropped. The third object is launched downward with an initial velocity of 10 feet per second. 25. Without doing any calculations, which object do you think will hit the

ground first? 26. Write a height model for the object launched upward. 27. Write a height model for the dropped object. 28. Write a height model for the object launched downward. 29. Use the quadratic formula to verify your answer in Exercise 25.



85

Lesson 5.6

8. 2x2 3x 1 0

Answer Key Practice A 1. 1, 2 is a solution 2. 2, 1 is not a solution 3. 4, 4 is not a solution 4. 3, 6 is a solution 5. 1, 1 is not a solution 6. 2, 3 is a solution 7. C 8. A 9. F 10. E 11. B 12. D 13. y

y

1

1 x

15.

1

x

16. y

y

1

1 x

1

17.

1

x

18. y

y 1 1 x

3

19.

1

x

1

x

20. y

y

1

1 1 x

y

1 1

14.

1

22. B

21.

x

23. C

24. A

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5.7


Determine whether the ordered pair is a solution of the inequality. 1. y < x2 2x 4, 1, 2

2. y > 2x2 x 5, 2, 1

3. y ≤ 2x2 5x 6, 4, 4

4. y ≥ 3x2 4x 1, 3, 6

5. y < 2x2 3x 4, 1, 1

6. y ≥ x2 3x 5, 2, 3

Match the inequality with its graph. 7. y ≥ x2 4x 3

8. y ≤ x2 4x 3

9. y ≤ x2 2x 3

10. y < x2 4x 3

11. y > x2 4x 3

12. y > x2 2x 3

A.

B.

C.

y

y

1

1 1

1

x

1

x

E.

y

1 x

F.

y

Lesson 5.7

D.

y

y

1

1 1

x

1

1

x

x

1

Graph the inequality. 13. y ≤ 2x2 1

14. y ≥ x2 2x

15. y < x2 3

16. y > 3x2 2

17. y < x2 5x

18. y > x2 2x

19. y ≥ x2 5x 6

20. y ≤ x2 2x 1

21. y ≤ x2 6x 8

Match the system of inequalities with its graph. 22. y < x2 1

23. y < x2 1

y > x2 1 A.

24. y > x2 1

y > x2 1 B.

y

y < x2 1 C.

y

2

y 2

2 2

x

2 2


x

x


97

Answer Key Practice B 1. 1, 1 is not a solution 2. 1, 6 is not a solution 3. 2, 7 is a solution 4. 3, 3 is a solution 5.

13.

14. y

y

1 1

x

6. y

x

1

1 y

12

x 5 x

5

15.

16.

12

y

y 4

7.

14

x

8. y

y

4

16

x

1

17.

x

1

1

18. y

x

1

y

1

9.

1

10. y

x

1

x

x

1

y

1

1 x

1

1 1

11.

x

19.

20. y

y

1

12. y

1 x

1

y

1 1 x

1 1

x

21.

22. y

y

1

1 1

x

1

x

Answer Key 23. 3 < x < 5

24. x < 2 or x > 8

25. 4 ≤ x ≤ 1

28. x ≤ 3 or x ≥ 6

27. 4 < x < 7

12

29.

26. x ≤ 4 or x ≥ 3 2

≤ x ≤ 3

30. x ≤ 3 or x ≥ 4

31. 3 ≤ x ≤ 7 32. x ≤ 2 2 or x ≥ 2 2 33. 1 ≤ x

35. a.

b.

y

y

1

1 x

1

36. b

5 2

37. a

38. y ≥ 0.33x2 2x 4,

y ≥ 0.33x2 2x 4

1x

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LESSON

5.7

NAME _________________________________________________________ DATE ___________


Determine whether the ordered pair is a solution of the inequality. 1. y < 2x2 2x 5, 1, 1 3. y ≤

1 2 2x

3x 1, 2, 7

2. y > 5x2 7x 4, 1, 6 4. y ≥ 3 3 x2, 3, 3 2

Graph the inequality. 5. y < x2 10x 9

6. y > x2 4x 21

8. y < x2 2x 1

9. y ≤ x2 6x 7

7. y > 3x2 6x 10. y ≥ 3x2 6x 2

11. y > x2 6x 9

12. y ≥ 2x2 4x 2

13. y < 2x2 8x 5

14. y > 3x2 5x 2

15. y ≤ 4x2 16

16. y < 12 3x2

18. y ≥ 2x2 4

19. y ≤ x2 4

Graph the system of inequalities. 17. y ≥ x2

y ≤ x2 3

Lesson 5.7

20. y ≤ x2 4

y ≥ x2 2x 1

y ≤ x2 1

y ≥ x2 2x 1

21. y > x2 4x 1

22. y ≥ 2x2 12x 16

y ≤ x2 2x 1

y < x2 2x 3

Solve the inequality algebraically. 23. x2 2x 15 < 0

24. x2 6x 16 > 0

25. x2 5x 4 ≤ 0

26. x2 7x 12 ≥ 0

27. x2 11x 28 < 0

28. x2 9x 18 ≥ 0

29. 2x2 5x 3 ≤ 0

30. 3x2 ≥ 10x 8

31. x2 4x ≤ 21

32. 2x2 ≥ 8x 4

33. 3x2 4 < 7x

34. 2x2 > 5x

Gift Shop Logo You are using a computer to create a logo for a gift shop called On the Wings of a Dove. The logo you have designed is shown at the right. 35. Sketch the intersections of the graphs of the inequalities. a. y ≥ 0.33x2 2x 4

y ≤ 0.09x2 1.3x

b. y ≥ 0.33x2 2x 4

y ≤ 0.09x2 1.3x

36. Which region in Exercise 35 represents the dove’s left wing? 37. Which region in Exercise 35 represents the dove’s right wing? 38. Which two inequalities (when intersected) make up the dove’s tail?

98



Answer Key 13. 10 ≤ x ≤ 7

Practice C 1.

2.

y 5 10

15.

y

17.

x 2

72 14

< x < 4 ≤ x ≤

x

2

19. x
6 3

18. x ≤ 2 or x ≥ 5 3

5 89 4

1 2

5 89 or 4

20. x ≤

21. No solution

1 13 1 13 or x 6 6 2 5 2 5 < x < 23. 24. All real numbers 5 5 1 y 25. 3, 2 26. 22. x
6x2 x 2

5. y ≥ x2 3x 1

6. y < x2 4x 6

7. y ≥ 2x2 3x 1

8. y > 2x2 5x 4

9. y ≤ 4x2 x 6

Graph the system of inequalities. 10. y ≥ x2 3x 4

11. y > 2x2 5x

y ≤ x2 4x 5

12. y > x2 3x 1

y < 12x2 x 2

y < x2 2x 3

Solve the inequality algebraically. 13. x2 3x 70 ≤ 0

14. x2 15x 36 ≥ 0

15. 2x2 x 28 < 0

16. 3x2 26x 48 > 0

17. 12x2 25x 7 ≤ 0

18. 12x2 12x 9 ≥ 0

19. 9x2 30x 25 < 0

20. 2x2 5x 8 ≥ 0

21. x2 2x 6 ≥ 0

22. 3x2 x 1 < 0

23. 5x2 4 > 0

24. 2x2 x 1 > 0


The area of a region bounded by two parabolas is given by Area

a 3 dB

3

A3

b 2 eB

2

y

A2 c f B A

y dx2 ex f

where y ax2 bx c is the top parabola, y dx2 ex f is the bottom parabola, and A and B are the x-coordinates of the intersection points of the parabolas with A < B.

y ax2 bx c x

25. To find the x-coordinates of the intersection points of

two parabolas, set the two quadratic equations equal to each other and solve for x. Find the x-coordinates of the intersection points of y x2 3x 1 and y x2 2x 4. 26. Graph the system of inequalities

y ≥ x2 3x 1 y ≤ x2 2x 4 27. For the region in Exercise 26, which parabola is the top boundary? 28. For the region in Exercise 26, which parabola is the bottom boundary? 29. Find the area of the region from Exercise 26. 30. Find the area of the region.

y ≥ x2 4x 3 y ≥ 2x2 5x 3



99

Lesson 5.7

Geometry

Answer Key Practice A 1. y x 22 1 2. y x 12 2 1 3. y x 32 4. y 3 x 3x 3 5. y x 3x 2 6. y xx 4 7. y x2 2x 5 8. y x2 2x 3 9. y x2 2x 2 10.

11. y 0.52x2 2.84x 7.07 12.

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NAME _________________________________________________________ DATE ___________

5.8


Write a quadratic function in vertex form for the parabola shown. 1.

2.

y

(2, 1)

3.

y

1

y

1 1

x

1

x

1

(0, 1) (0, 3)

(4, 1) (3, 0) x

1

(1, 2)

Write a quadratic function in intercept form for the parabola shown. 4.

5.

y

6.

y 1

1

y

x

1

(3, 3)

x

1

(2, 4) (0, 3)

1

1 x

Write a quadratic function in standard form for the parabola shown. 7.

8.

y

4

(1, 8)

Lesson 5.8

y 1

(2, 3)

(1, 0)

(3, 1) 1

(0, 5) (1, 4)

9.

y

x

4

x

(1, 1) (2, 2)

2

(4, 5) 2

x

Australia’s Unemployment Rate The following table shows the percentage of people who were unemployed in Australia from 1990 to 1995. Assume that t is the number of years since 1990. Year, t Percentage of people unemployed, y

0

1

2

3

4

5

6.9

9.6

10.8

10.9

9.7

8.5

10. Use a graphing calculator to make a scatter plot of the data. 11. Use a graphing calculator to find the best fitting quadratic model for the data. 12. Use a graphing calculator to check how well the model fits the data.

110



Answer Key Practice B 1. y x 22 3 2. y x 12 4 3. y x 22 1 4. y x 42 2 5. y x 32 1 6. y x 12 5 7. y x 32 1 8. y x 42 5 9. y x 62 10. y x 2x 4 11. y x 3x 5 12. y x 1x 4 13. y x 2x 6 14. y x 5x 4 15. y x 1x 7 16. y 2xx 5 1 17. y 4xx 3 18. y 2x 8x 2 19. y 2x2 x 2 20. y x2 x 7 21. y 2x2 x 3 22. y x2 x 4 23. y 3x2 x 1 24. y 2x2 4x 5 25. y x2 2x 4 26. y x2 3x 2 27. y x2 3 28. P 0.23t2 2.03t 22.93 29. V 0.03t2 1.17t 70.30

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5.8

NAME _________________________________________________________ DATE ___________


Write a quadratic function in vertex form whose graph has the given vertex and passes through the given point. 1. vertex: 2, 3

2. vertex: 1, 4

3. vertex: 2, 1

4. vertex: 4, 2

5. vertex: 3, 1

6. vertex: 1, 5

7. vertex: 3, 1

8. vertex: 4, 5

9. vertex: 6, 0

point: 0, 7

point: 1, 8

point: 3, 3

point: 1, 10

point: 2, 0

point: 2, 0

point: 1, 1

point: 1, 4

point: 3, 9

Write a quadratic function in intercept form whose graph has the given x-intercepts and passes through the given point. 10. x-intercepts: 2, 4

11. x-intercepts: 3, 5








point: 1, 3

point: 2, 3

point: 3, 3

point: 3, 28

point: 3, 8

point: 1, 12

point: 5, 12

point: 1, 8

point: 4, 12

Write a quadratic function in standard form whose graph passes through the given points. 19. 1, 1, 0, 2, 2, 8

20. 1, 7, 1, 5, 2, 1

21. 1 2, 0, 3, 1, 0

22. 1, 4, 1, 6, 2, 10

23. 0, 1, 1, 3, 2, 11

24. 2, 11, 1, 1, 1, 7

25. 1, 7, 1, 3, 2, 4

26. 1, 2, 1, 4, 2, 4

27. 1, 2, 2, 1, 3, 6

28. Population Model The table shows the population of a town from 1990

Year, t Population, P

0 23.2

1 24

2 26.5

3 27.2

4 27.1

5 27.3

6 26.8

7 25.9

Lesson 5.8

through 1998. Find a quadratic model in standard form for the data. Assume that t is the number of years since 1990 and that P is measured in thousands of people. 8 24.4

29. Voter Turn-out The table shows the percentage of eligible voters that

participated in presidential elections from 1964 through 1992. Find a quadratic model in standard form for the data. Assume that t is the number of years since 1964. Year, t Percent voted, V


0 69.3

4 67.8

8 63.0

12 59.2

16 59.2

20 59.9

24 57.4

28 61.3


111

Answer Key Practice C 1. y 2x 12 3 2. y 3x 62 2 3. y 3x 22 5 1

4. y 5x 32 2

5. y x 3 5 6. y 7. y 2x 7x 6 1 2

2 3

1 2

x 122 32

8. y 3x 4x 2 1

x 12 x 3 y 65 x 34 x 12

9. y 10.

1 4

11.

y 4x 35 x 58 12. y 73x x 27 13. y 7x2 21x 27 14. y x2 3x 1 1

15. y 2x2 2x 5

3

16. y 4x2 x

1

17. y 2x2 3x 6 18. y 3x2 5x 20. y 3x2

2 3

3 8

19. y x2 6x 9

21. y 2x2 3x

22. F 0.498t2 22.103t 768.941 23. A 3.142r 2; 3.142

1 4

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5.8


Write a quadratic function in vertex form whose graph has the given vertex and passes through the given point. 1. vertex: 1, 3

2. vertex: 6, 2

4. vertex: 3, 2

5. vertex:

point: 2, 5

3. vertex: 2, 5

point: 4, 14

1

9 point: 2, 10

point:

point: 5, 2

13, 5 1, 419

6. vertex: 2, 2 1 3

point: 1, 3

Write a quadratic function in intercept form whose graph has the given x-intercepts and passes through the given point. 7. x-intercepts: 6, 7


point: 2, 80

point: 5, 1

1 3

point: 5, 11

5 3


point:

1

9. x-intercepts: 3, 2 2



12, 103

77 point: 2, 10

point:

57, 53

Write a quadratic function in standard form whose graph passes through the given points. 13. 2, 13, 3, 27, 4, 55 16. 19.

14. 2, 9, 0, 1, 1, 3

1 1 2, 21 4 , 2 , 16 , 1, 0 1 23 3 9 2 , 4 , 1, 2, 2 , 4

17. 3, 13, 0, 6, 3, 11

20. 1,

11 3

, 13, 1, 23, 2

15. 2, 3, 2, 11, 4, 21

18. 8, 64, 1, 8 , 4, 16 1 67

13

5

19

21. 2, 2, 2, 1, 2, 0 1

1

3

The table shows the average fuel consumption (in gallons) of a passenger car between 1970 and 1996. Use a system of equations to write a quadratic model for average fuel consumption F as a function of time t, where t is the number of years since 1970. Check your model using the quadratic regression feature of a graphing calculator.

Lesson 5.8

22. Average Fuel Consumption

Year, t Average Fuel Consumption, F

0 760

5 695

10 576

15 559

20 520

25 530

26 531

23. Geometry

The table shows the areas of a circle with a given radius. Use the quadratic regression feature of a graphing calculator to write a quadratic model for the area of a circle A as a function of its radius r. Round the values for a, b, and c to three decimal places. Using A r 2, what is a three decimal approximation of ? Radius, r Area, A

112

2 12.5664

3 28.2743


4 50.2655

5 78.5398


Answer Key Test A 1 1. 2. 8x3y3 3. y6 4. 5x 2y x 5. f x → as x → and f x → as x → ; x 2 1 y 5 7

0 1

1 5

y

2

2 7

6

6. f x → as x →

x

y

and f x → as x → ; x 2 1 0 y 0 3 0

1 3

7. 2x 2 2x 2

8. 4x 2 y 2

10. 5x 15x 1

x

1

9. x3 1

11. x 1x 2 x 1

12. 4x 2y3x 2y 2 5y 6 14. 2, 2, 3, 3

1

2 0

13. 4, 4

15. 1, 1, 4

2x 3 17. 2x 2 8x 8 18. Possible zeros: 1, 1, 3, 3; Zeros: 1, 3 19. Possible zeros: 1, 1, 2, 2, 4, 4, 8, 8 Zeros: 4, 1, 2 20. f x x3 2x 2 11x 12 21. f x x 2 7x 12 22. 1.29, 2, 3.24 23. x-intercepts: 3, 0, 3, 0; local max: 0, 27; local mins: 3, 0, 3, 0 The graph rises to the right and to the left. 16.

x2

24.

f 1 1

f 2 0 1

f 3 1 1

2

f 4 4 3

2

f 5 9 5

2

f 6 16 7

2

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x5 x6

Answers

2. 2xy3

3.

y3 y3

4.

25x3y 2 5xy

1. 2.

Describe the end behavior of the graph of the polynomial function. Then evaluate for x 2, 1, 0, 1, 2. Then graph the function. 5. y 3x3 9x 1

6. y x3 4x

x y

x y

3. 4. 5.

Use grid at left.

6.

Use grid at left.

7. y

y

8. 9.

2

1 2

x

10. 1

x

11. 12. 13. 14.

Perform the indicated operation. 7. x 2 x 1 x 2 x 1

8. 2x y2x y

15.

9. x 1x 2 x 1

Factor the polynomial. 10. 25x 2 1

11. x3 1

12. 12x 4y3 20x 2y 2 24x 2y

Solve the equation. 13. x 2 16

14. x 4 13x 2 36 0

Review and Assess

15. x3 4x 2 x 4 0

130



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6

CONTINUED

NAME _________________________________________________________ DATE ____________


Divide. Use synthetic division if possible.

16.

16. x3 7x 6 x 2

17.

17. 2x3 6x 2 8 x 1

List all the possible rational zeros of f using the rational zero theorem. Then find all the zeros of the function. 18. f x x 2 4x 3

19. f x x3 x 2 10x 8

18. 19. 20.

Write a polynomial function of least degree that has real coefficients, the given zeros, and a leading coefficient of 1.

21.

20. 4, 1, 3

22.

21. 4, 3

22. Use technology to approximate the real zeros of

f x 0.25x x 2. 3

23.

2

24.

23. Identify the x-intercepts, local maximum, and local minimum of the

graph of f x 13 x 32x 32. Then describe the behavior of the graph.

24. Show that the nth-order finite differences for the function

f x x 2 4x 4 of degree n are nonzero and constant.

Review and Assess



131

Answer Key Test B y 1 1. 2. 6 9 3. x8y8 4. 1 x xy 5. f x → as x → and f x → as x → ; x 2 1 y 8 1

0 0

1 2 1 8

6. f x → as x →

and f x → as x → ; x 2 1 0 1 y 0 4 9 3

y

1 x

1

y 6

2 0

6

7. x3 2x 2 2

8. x 2 7xy 12y 2

9. 2x3 x 2 1

10. 10x 3y10x 3y

11. y 1

x

y 1 12. 5xy3x y 2xy 1 13. 9, 9 14. 6, 0, 1 15. 0, 20 16. x 2 4x 12 17. 2x 2 5x 3 18. Possible zeros: 1, 1, 5, 5; Zeros: 1, 5 19. Possible zeros: 1, 1, 2, 2, 4, 4, 8, 8 Zeros: 4, 1, 2 20. x 2 x 20 21. x3 3x 2 4x 12 22. 4.66, 2.80, 1.75 23. x-intercepts: 2, 0, 2, 0; local max: 0, 4 local mins: 2, 0, 2, 0 The graph rises to the right and to the left. y2

2 2

24.

f 1 3

f 2 f 3 f 4 f 5 f 6 15 48 105 192 0 3 15 33 57 87 12 18 24 30 6 6 6

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x3y 2 x 4y

Answers

2. x 2y33

3.

x 4y 4 x y

4.

4 4

xy 1

xy1

1. 2.

Describe the end behavior of the graph of the polynomial function. Then evaluate for x 2, 1, 0, 1, 2. Then graph the function. 5. y x3

6. y 2x3 x 2 8x 4

x y

x y

3. 4. 5.

Use grid at left.

6.

Use grid at left.

7. y

y

8. 9.

2 1

2 1

x

10.

x

11. 12. 13. 14.

Perform the indicated operation. 7. 3x3 x 2 4 2x3 x 2 2

8. x 3yx 4y

15.

9. x 12x 2 x 1

Factor the polynomial. 10. 100x 2 9y 2

11. y3 1

12. 15x3y3 10x 2y 2 5xy

Solve the equation. 13. x 2 81

14. 5x3 30x 25x 2

Review and Assess

15. xx 5x 4 x3

132



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Chapter Test B

CONTINUED



16.

16. x3 28x 48 x 4

17.

17. 2x 11x 18x 9 x 3

18.

3

2

List all the possible rational zeros of f using the rational zero theorem. Then find all the zeros of the function. 18. f x x 2 6x 5

19. f x x3 x 2 10x 8

19. 20. 21.


22.

20. 4, 5

23.

21. 2, 2, 3


24.

f x 0.35x3 2x 2 8.

23. Identify the x-intercepts, the local maximum, and local minimum of

the graph of f x 14 x 22x 22. Then describe the behavior of the graph.

24. Show that the nth-order finite differences for the function

f x x3 4x of degree n are nonzero and constant.

Review and Assess



133

Answer Key Test C 1 1. 3 2 xy

2. x5y5

3. x3y3

4. x 4y14

5. f x → as x →

y

and f x → as x → ; x 2 1 0 1 2

y 0 6 4 0 0

1 x

3

6. f x → as x →

y

and f x → as x → ; x 2 1 0 1 2

y 4 0 6 4 0

1 x

1

7. x3 6x 2 2x 6

8. x 2y 2 xy 12

9. 2x3 3x 2y 3xy 2 y3 10. 42x y2x y 11. 2y 14y 2 2y 1 12. 4c dc dc 2d 14. 0, 3, 4

15.

1 3 2, 2

13. 6, 6

16. x 2 x 3

17. x3 3x 2 x 1 18. Possible zeros: 1, 1, 19. Possible zeros:

1 1 1 , ; Zero: 2 2 2

1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 12, 12; zeros: 4, 1, 3 20. x3 6x 2 11x 6 21. x4 5x2 36 22. 9.67, 1.93, 1.61 23. x–intercepts: 4, 0, 4, 0; local max: 0, 16; local min: 4, 0, 4, 0 The graph rises to the right and to the left. 24.

f 1 0

f 2 12 12

f 3 40 28

16

f 4 90 50

78 28

22 6

f 5 168

6

112 34

6

f 6 280

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Simplify the expression. 1. x3y 21

2.

Answers

x3y3 x y

3.

2 2

1 xy3

4.

x 2y3 y4

y4

x2y3

2.

Describe the end behavior of the graph of the polynomial function. Then evaluate for x 2, 1, 0, 1, 2. Then graph the function. 6. y x 1x 2x 2 3

5. y x3 x 2 4x 4

x y

x y

1.

3. 4. 5.

Use grid at left.

6.

Use grid at left.

7. y

y

8. 9. 10.

1

1 1

x

11. 1

x

12. 13. 14.

Perform the indicated operation. 7. 4x3 3x 2 x 2 5x3 3x 2 x 4 8. xy 4xy 3

15.

9. 2x yx 2 xy y 2

Factor the polynomial. 10. 16x 2 4y 2

11. 8y3 1

12. 4c3 8c 2d 4cd 2 8d 3

Solve the equation. 13. 2x 2 72

14. 4y3 48y 2 4y 4

Review and Assess

15. 2x 2 32 4xx3 6

134



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Chapter Test C

CONTINUED



16.

16. x3 2x 2 9 x 3

17.

17. x 10x 2x 3 x 3

18.

4

2

List all the possible rational zeros of f using the rational zero theorem. Then find all the zeros of the function. 18. f x 2x3 x 2 2x 1

19. f x x3 2x 2 11x 12

19. 20. 21.


22.

20. 1, 2, 3

23.

21. 3, 3, 2i, 2i


24.

f x 0.2x3 2x 2 6.

23. Identify the x-intercepts, local maximum, and local minimum of the 1 graph of f x 16 x 42x 42. Then describe the behavior of the graph.

24. Show that the nth-order finite difference for the function

f x x3 2x 2 x 2 of degree n is nonzero and constant.

Review and Assess



135

Answer Key Cumulative Review 1. 14 2. 5 3. 14 4. 26 5. 27 19 23 6. 14 7. 6 8. 2 9. 11 10. 6 11. 8 12. 10 13. x ≥ 2 14. x > 2 3 2 1

0

3 2

15. x >

1

2

3 2 1

3

or x < 3

0

1

2

5

3 2 1

0

1

2

2.5 0

1

2

55.

6

y

y

0.5

6

3 3 2 1

19. Line 2

4

18. 2.5 ≤ x ≤ 0.5

17. 2 < x < 3 3 2 1

2

39.

32. x 2

32 x

54.

3 0

36.

24 5

2 34. y 23 x 73 35. 2 6 37. 3 38. 0 infinitely many solutions 40. none 1 one 42. 3, 2 43. 2, 4 44. 1, 0 13, 4 46. 2, 5 47. 0.5, 0.3 48. 8 6 50. 7 51. 65 52. 14 53. 1

33. y

49.

5

6 4 2

3

45.

16. x > 5 or x ≤ 5

30. y 5 x

31. y 4 x 5

41.

3

2

29. y 3x 5

20. Line 2

0

1

2

21. Line 1

3 1

22. Line 2

x

1

x

x

1

23.

2

24. y

y

(0, 8)

56.

1 2

(3 , 0)

1

x

1

57. y

y

2 x

2

(2 , 0) 2 3

1

(0, 7)

x

1

25.

1

26. y

y

58.

59. y

(0, 4)

(

2 5 ,

)

0

1

(0, ) 1

1

y

1 2

x

1

(3, 0) x

1

1 1

27.

28. y

y

(0, 8) 2

(6, 0) 2

2 2

x

1

x

x

x

Answer Key 60. ± 3 64. ± 3 69. 26

61. 3 ± 6 2

65. 3, 5 70. 10

62. ± 4

66. 5

3

63. 2

67. 25

68. 2

71. 11

1 ± 41 9 ± 41 73. 74. 4 ± 7 4 20 4 75. 3, 1 76. ± 32 77. 2 ± 3 78. 3x2 13x2 1 79. x2 2x2 1 80. 3x2x2 1x2 2 81. 3xx 3x2 3x 9 82. 4x2 1x2 9 83. 3x2 12x 3 15 84. 2x2 4x 5 x2 4 38 85. x 1 86. 3x 8 x1 x4 5 87. x3 x2 3x 10 x3 72.

88. 8 in. on each side

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Evaluate the expression for the given values of the variables. (1.2) 1. 3x 5 when x 3

2. 4x 8x 4 when x

4. x3 4x 2 x when x 2

5. x3 2x 3 when x 3

6. x 2 5x when x 2

8. m 15 3m 4

9. 43x 5 x 3

1 4

3. x 2 5x when x 2

Solve the equation. (1.3) 7. 4x 8 32 10.

3 2x

4 2x 1

11.

1 2x

3 4

3 2x

29 4

2 5x

12.

1 23 10 x 11 3

Solve the inequality. Then graph the solution. (1.6) 13. 2x 5 ≥ 9

14. 5 2x < 15 3x

15. 4x 2 > 8 or 4x 2 < 10

16. 3x 7 > 8 or 2x 1 ≤ 9

17. 5 < 3x 1 < 10

18. 0.25 ≤ 0.5x 1 ≤ 0.75

Tell which line is steeper. (2.2) 20. Line 1: through 0, 5 and 3, 8








Graph the equation using standard form. Label any intercepts. (2.3) 23. 3x y 8

24. 2x y 7

25. 4x 3y 12

26. 5x 4y 2

27. y 8

28. x 6

Write an equation of a line using the given information. (2.4) 29. The line passes through the point 0, 5 and has a slope of 3. 30. The line passes through the point 2, 4 and has a slope of

2 5.

3

31. The line has a slope of 4 and a y-intercept of 5. 32. The line passes through the point 2, 4 and is parallel to x 7. 33. The line passes through the point 2, 1 and is perpendicular to the line y 3 x 5. 2

34. The line passes through the point 4, 5 and is parallel to the line y 3 x 7. 2

Evaluate the function for the given value of x. (2.7) 3x, if x ≤ 2 x 1, if x > 2

35. f 3

36. f 2

37. f 1

38. f 0

Review and Assess

f x

Tell how many solutions the linear system has. (3.1) 39. 4x 2y 8

8x 4y 16


40. 3x 2y 6

6x 4y 8

41. 5x 6y 7

2x 3y 5


141

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Cumulative Review

CONTINUED


Solve the system using an algebraic method. (3.2) 42. 3x 4y 17

43. 4x y 6

2x y 8

44. 3x 8y 3

8x 2y 4

45. 6x 2y 6

46. 3x 1 2x

9x 3y 15

1 5y 2 5y

7

2x 5y 2 47. 2x 5y 0.5

1

4x 3y 1.1

Evaluate the determinant of the matrix. (4.3) 48.

4 4

5 51. 2 4

3 1 4 1 2

49.

3 0 5

0

1 3

1 1 1

2

0 52. 5 6

50.

3 2 7

3 1 2

2 2

4 3 53. 2

2 3 5

1 1 2

Graph the quadratic function. (5.1) 54. y x 2 2x 3

55. y x 2 4x 5

56. y 2x 2 8x 3

57. y x 12 3

58. y 2x 42 2

59. y 2 x 52 2 1

Solve the quadratic equation. (5.2, 5.3) 1 3 x

32 2

60. 2x 2 5 11

61.

63. 4x 2 12x 9 0

64. x 2 9 0

62. x 2 2 14 65. 3x 2 13x 10 0

Find the absolute value of the complex number. (5.4) 66. 3 4i

67. 4 2i

68. 1 i

69. 1 5i

70. 3 i

71. 2 7i

Use the quadratic formula to solve the equation. (5.6) 72. 2x 2 x 5 0

73. 10x 2 9x 1 0

74. x 2 8x 9 0

75. 3x 2 x 4 0

76. x 2 18 0

77. 2x 2 4x x 2 1

78. 9x 4 1

79. x 4 3x 2 2

80. 6x5 15x3 6x

81. 3x 4 81x

82. 4x 4 37x 2 9

83. 6x3 9x 2 2x 3

Review and Assess

Factor using any method. (6.4)

Divide using synthetic division. (6.5) 84. 2x3 3x 5 x 2

85. x 2 2x 3 x 1

2 86. 3x 4x 6 x 4

87. x 4 2x3 x 25 x 3

An open box with a volume of 32 in.3 is made from a square piece of metal by cutting 2-inch squares from each corner and then folding up the sides. Find the dimensions of the piece of metal required to make the box. (5.5)

88. Dimensions of a box

142



Answer Key Practice A 1 1. 19,683 2. 256 3. 1024 4. 64 5. 4 1 1 1 6. 5 7. 216 8. 243 9. 128 10. 243 1 1 11. 25 12. 32 13. 25 14. 343 15. 1 1 16. 32 17. 3 18. 15,625 19. 256 1 1 27 4 20. 729 21. 81 22. 8 23. 25 24. 64 81 1 1 16 25. 1 26. 1 27. 9 28. 64 29. 9 30. 16 x4 31. x8 32. x12 33. x24 34. 27x3 35. 16 1 9 16 36. x5 37. 6 38. 2 39. 2 x x x 40. 2.47 1010 mi2 42. 5.59 107 mi2

41. 1.88 108 mi2

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LESSON

6.1

NAME _________________________________________________________ DATE ___________


Use the properties of exponents to evaluate the expression.

35 41 44 62 61

22 25 23 32 33

42 57 58 24 23

1. 34

2. 26

3. 43

4.

5.

6.

7.

10. 3233

8.

9.

11. 5658

12. 2322

54 52

14.

76 79

15.

16.

28 23

17.

33 34

18. 523

19. 242

26.

0

25. 130

29.

2

28. 43

3 2

3

2 5

4 5 3 4

4

21.

24.

2

23.

22.

20. 323

35 35

1 3

Lesson 6.1

13.

1 4

3

27. 32 30.

2 3

4

Simplify the expression. 31. x3

x5

34. 3x3 37.

x3 x9

32. x 4

x8

x 2

4

35.

2

38.

3 x

33. x 46 36.

x7 x2

39.

x 4

2

Surface Area In Exercises 40–42, use the formula S 4r 2 to find the surface area of each planet. 40. The radius of Jupiter is approximately 44,366 miles. Find the surface area

of Jupiter. 41. The radius of Earth is approximately 3863 miles. Find the surface area of

Earth. 42. The radius of Mars is approximately 2110 miles. Find the surface area of

Mars.



13

Answer Key Practice B 1. 9 2. 15,625 7. 9 13.

y3 8

8. 16

3.

9. 1

14. 256x12

8 27

4.

10. x5 15.

1 y2

1 64

5. 1

6.

1 16

2 12. 9x2 y2 5x3 16. 2y2 11.

y5 3 18. 19. x5 20. 3x4 2 3x 2x3 21. 7.02 101 peoplemi2 22. 1.08 105 mih 23. 580.52 computers/1000 people 17.

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LESSON

6.1

NAME _________________________________________________________ DATE ___________


Use the properties of exponents to evaluate the expression.

Lesson 6.1

1. 3432

3

2. 523

3.

6.

4 43 46

9.

56 532

4.

84 86

5. (7676

7.

325 38

8.

11.

2y3 y5

1 2

4

2 3

Simplify the expression. 10. x3

x2 3

13.

16.

5x2y 2x1y3

y 2

12. 3x2

14. 4x3 4 17.

15. x0y2

3xy 9x3y4

19. Geometry

Find an expression for the area of the triangle.

x2

18.

3x2 6x5

20. Geometry

Find an expression for the area of the circle.

πx 2

2x 3

21. Population per Square Mile

In 1996, the population of the United States was approximately 265,280,000 people. The area of the United States is approximately 3,780,000 square miles. Use scientific notation to find the population per square mile in the United States.

22. Speed of Mercury

Mercury travels approximately 226,000,000 miles around the sun. It takes Mercury approximately 2100 hours to revolve around the sun. Use scientific notation to find the speed of Mercury as it revolves around the sun.

23. Computers per 1000 People

The population of the United States is approximately 265,280 thousand people. It is estimated that by the year 2000, there will be 154,000,000 computers in the United States. How many computers will there be per 1000 people?

14



Answer Key Practice C 81 27 1. 16 2. 2187 3. 1 4. 8 5. 4096 6. 243 x3 4 5y 7. 8. 2 9. 1 10. 2 11. 12x 2 2y 3y 4x 2 10 2 4x y z 1 12. 4 13. 16x5 14. 15. 48 16. 2 y 4x2 x 17. 6 18. 4 19. 3 20. 5 21. 3 22.

in.

4 7 3 4

3

3

43 74

3

343 3 in. 48

6517 203 38,400 24. about 53.32% 25. 345,600 in.3 23.

190

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LESSON

NAME _________________________________________________________ DATE ___________

6.1


Use the properties of exponents to evaluate the expression. 1.

4.

24 25 23

23

323 31

5.

38

8.

2x2 3xy3

3

3.

2421 23

6.

13 13

9.

x4 y2

2 2

1 3

3

4

2

Lesson 6.1

2.

Simplify the expression. 7. 10. 13.

xy 4

5x2y 8

2x2 y3

2x1y x3y

2x3 4x5 2x3

2x1 y1

11.

2x23 15x2 2x3 5

14.

x2y 5z 2x3

12.

y2

x4

x3y2 2x4

2

2

15. (x 46 2

Use the properties of exponents to simplify the left side of the equation. Then solve the equation as demonstrated below. 4x1 42 ⇒ x 1 2 ⇒ x 3 16. 2x23 25 19.

43 40 4x

17.

3x 34 32

18. 5x3 512

20.

2xy2 25 y2

21. 2x0323x 31

Class Project In Exercises 22–25, use the following information. Your class project is to design a piece of playground equipment for an elementary school. You design a romper room that will contain small plastic balls for the children to roll around in. The room will be 10 feet by 10 feet. The plastic balls will cover the entire floor to a depth of 2 feet. A toy distributor can ship you 190 balls (each with a radius of 134 inches) in a cubic box, 20 inches on a side. 22. Find an expression for the volume (in cubic inches) of one ball. 23. Find an expression that represents the ratio of the volume of 190

balls to the volume of the cubic box. 24. What percent of the volume of the cubic box is filled with plastic

balls? 25. Find the volume of the region in the romper room that will contain

plastic balls. Give your result in cubic inches.

10

2 10



15

Answer Key Practice A 1. yes 2. yes 3. no 4. yes 5. no 6. yes 1 7. 5; 3 8. 7; 2 9. 4; 8 10. 2; 3 11. 2; 5 12. 9; 3 13. f x 2x3 3x2 5 14. f x 5x2 2x 3 15. f x 2x3 5x2 3x 3 16. f x 5x2 3x 14 17. f x 5x4 6x 2 18. f x 5x3 7x2 x 3 19. 1 20. 1 21. 34 22. 14 23. B 24. C 25. D 26. A 27. C 0.99x2 14.93x 75.32 28. 2; 0.99 29. 324

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LESSON

NAME _________________________________________________________ DATE ___________

6.2


State whether the following function is a polynomial. 1. f x 3x2 7x 3

2. f x 5 3x 4

3. f x 2x 3x 1

4. f x 9

5. f x 2x 5x 8

6. f x x3 x2

State the degree and leading coefficient of the polynomial. 7. f x 3x5 3x2 8

8. f x 2x7 3x

9. f x 8x 4

10. f x 24 4x 3 x2 1

11. f x 3x 5 x25

12. f x 4x5 4x 7x8 3x9

Lesson 6.2

Write the function in standard form. 13. f x 3x2 5 2x3

14. f x 3 2x 5x2

15. f x 3x 5x2 3 2x3

16. f x 14 3x 5x2

17. f x 6x 5x 4 2

18. f x x 3 5x3 7x2

Use direct substitution to evaluate the polynomial function for the given value of x. 19. f x 3 x2 4x x3, x 2

20. f x 3x2 5x 2x5 x 4, x 1

21. f x 7x 2x2 5, x 3

22. f x x2 5x 22, x 4

Use what you know about end behavior to match the polynomial with its graph. 23. f x 2x 4 2x 1

24. f x 2x3 x2 3x 3

25. f x x2 3x 2

26. f x 2x3 x2 1

A.

B.

C.

y

D.

y

1

y

y

1 1

x

1 1

x

1

1 2

Computers

x

x


From 1990 to 1995, the number of computers per 1000 people in Germany can be modeled by C 75.32 14.93t 0.99t 2 where C is the number of computers per 1000 people and t is the number of years since 1990. 27. Write the model in standard form. 28. State the degree and leading coefficient of the model. 29. Estimate the number of computers per 1000 people in the year 2000.

26



Answer Key Practice B 1. yes; f x 2x3 3x2 4x; 3; 2 2. no 3. no 4. yes; f x 2x5 7x2 3; 5; 2 1 1 2 1 5. yes; f x 6x2 3x 3; 2; 6 6. yes; f x 5 x4 2x2 x 7; 4; 5 7. 7 8. 3 9. 23 10. 30 11. 8 12. 52 13. 6 14. 84 15. 98 16. 86 17. 5 18. 74 19. 37 20. 0 21. 72 22. 6 23.

29.

30. y

y

2 1 2

x

32. $1.02

31.

24.

33. $491,662.20

y

y

y 1 2 2

1 x

1

25.

1

x

1

x

26. y

y

2 1 x

1

27.

28. y

y

1 2

x

1 2

x

1

x

x

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LESSON

NAME _________________________________________________________ DATE ___________

6.2


Decide whether the function is a polynomial function. If it is, write the function in standard form and state the degree and leading coefficient. 1. f x 3x2 2x3 4x

2. f x 3x3 2x 1

3. f x 4x 2x 7x3 1

4. f x 2x5 3 7x2

5. f x 3 x

6. f x x 5 x 4 2x2 7

1

2 3

16 x2

Use direct substitution to evaluate the polynomial function for the given value of x. 7. f x 3x 2, x 3

8. f x 2x3 3x2 5x 1, x 1

9. f x 4x2 5x 2, x 3

10. f x 3x 4 2x2 3x 4, x 2

11. f x 3x7 2x6 5x 8, x 0

12. f x 6x3 2x2 5x 2, x 2

13. f x 2x5 3x3 2x 5, x 1

14. f x x 4 2x3 4x2 6x 3, x 3

15. f x 2x3 3x2 4x 2, x 4

16. f x 2x 4 3x3 5x2 2x 6, x 2

17. f x 5x4 3x2 2x 5, x 1

18. f x x6 3x 4, x 2

19. f x 2x2 4x 7, x 3

20. f x x 4 3x3 2x2 8x, x 4

21. f x 4x3 2x2 6x, x 3

22. f x 3x3 5x2 6x 8, x 1

Graph the polynomial function. 23. f x x3 2

24. f x 2x 4 1

26. f x 3 x2

27. f x x3 2x 3

28. f x x 4 2x3 5x 1

29. f x 1 x2 x3

30. f x 2 x2 x 4

31. f x x3 x2 2

25. f x 2x3 1

32. Value of the Dollar

From 1988 to 1998 the value of a dollar in 1998 dollars can be modeled by V 0.002t2 0.06t 1.37 where V is the value of the dollar and t is the number of years since 1988. What was the value of a dollar in 1996 in terms of 1998 dollars?

33. Preakness Stakes

From 1990 to 1998, the money received by the winning horse can be modeled by W 6266.2t 3 79,306.8t2 295,834.9t 157,544.5 where W is the winnings and t is the number of years since 1990. How much did Silver Charm win in 1997?



27

Lesson 6.2

Use synthetic substitution to evaluate the polynomial function for the given value of x.

Answer Key Practice C 35 1. 3 2. 15 3. 16 4. 8 5. 10 61 6. 13 52 7. 0 8. 45 9. 3 9 215 11. 2 12. 27 13.

21. y

10.

103 10

1 2

x

14. y

y 1 x

1

22. Sample answer: f x 3x4 2x 1 23. Sample answer: f x 2x3 3x2 x 5

1 1

x

24. 20.48 years

15.

16. y

y

1

1 1

x

17.

2

x

18. y

y

2

2

1

x

19.

1

x

1

x

20. y

y

1 1

x 1

25. older

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LESSON

NAME _________________________________________________________ DATE ___________

6.2


Use direct substitution to evaluate the polynomial function for the given value of x. 1. f x 3x3 4x2 x 7, x 2

2. f x 2 x2 x 3, x 4

3. f x 2 x3 4 x2 3x 1, x 2

4. f x 2x 4 3x2 5, x

5. f x x 4 2x2 5, x 5

6. f x 2x6 x 4 5x 1, x 2

3

1

1

1 2

Use synthetic substitution to evaluate the polynomial function for the given value of x. 7. f x 2x5 3x 4 x3 x2 6x 3, x 1 8. f x 3x 4 2x2 5, x 2 9. f x 3 x3 4x2 2 x 2, x 2 2

1

11. f x 4x3 2x2 x 3, x 2 1

10. f x 5 x2 3x 2, x 3 1

1

12. f x x3 3x 7, x

1 3

Lesson 6.2

Graph the function. 13. f x 4 x3

14. f x 3x2 5

15. f x x3 3x 1

16. f x 2x 4 3x 1

17. f x 2x7 1

18. f x

20. f x 3 x 4 5

21. f x 2x3 7

19. f x

x3 2x 1 4

2 3

x2

22. Critical Thinking

Give an example of a polynomial function f such that f x → as x → and f x → as x → .


Give an example of a polynomial function f such that f x → as x → and f x → as x → .

First-time Brides


The median age of a female when she gets married for the first time in the United States from 1890 to 1996 can be modeled by A 0.001t2 0.098t 22.763 where A is the age and t is the number of years since 1890. 24. What was the median age of first time brides in 1950? 25. Describe the end behavior of the graph. From the end behavior, would

you expect first time brides in 2000 to be older or younger than the brides in 1996?

28



Answer Key Practice A 1. 3x2 5x 6 2. 2x3 5x2 x 4 3. 4x4 3x3 2x2 4x 8 4. 4x2 5x 5. 5x5 3x4 2x3 x2 3x 8 6. 2x2 3x 8 7. 4x3 6x2 2x 2 8. x 4 9. 2x3 2x2 2x 2 10. x 2 11. x 3 12. x2 2x 7 13. 2x2 7x 3 14. x3 5x2 8x 5 15. 2x5 5x2 9 16. x12 5x8 5x 4 17. 2x3 4x2 15x 4 18. x3 2x2 10x 7 19. 14x2 9x 18 20. x2 x 12 21. x2 8x 12 22. x2 5x 6 23. 2x2 5x 3 24. 2x2 11x 5 25. 3x2 x 2 26. x2 16 27. x2 49 28. x2 6x 9 29. x2 12x 36 30. x2 16x 64 31. x2 8x 16 32. x2 5x 33. 2x2 5x 3 34. x2 10x 25 35. M 3052.04t 515,887.88

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LESSON

NAME _________________________________________________________ DATE ___________

6.3


Find the sum. x2 2x 5

1.

2x2 3x 1 3.

6x3 2x2 x 3

2.

4x3 3x2 2x 1

3x 4 x3 x2 x 9 x 4 2x3 3x2 5x 1

4. 3x2 2x 5 x2 3x 5

5. 2x5 3x 4 2x3 x2 x 8 3x5 2x

6. x2 7x 1 3x2 10x 7

7. 5x3 2x2 x 3 x3 4x2 3x 1

Find the difference. x2 4x 3

8.

x2 3x 1

2x 1

3x3

9.

5x 3

10.

6x 1

x3 2x2 4x 3

11. x 7 2x 4

12. x2 3x 1 2x2 x 6

13. 3x2 2x 1 x2 5x 2

14. 2x3 4x2 3x 7 3x3 x2 5x 2

15. 4x5 3x2 8 2x5 2x2 1

16. 7x12 3x8 2x 1 8x12 2x8 3x 5

Find the product. 17. 2x2 4x 1

Lesson 6.3

18. x2 3x 7

x4

7x 6

19.

x1

2x 3

20. x 4x 3

21. x 6x 2

22. x 3x 2

23. x 12x 3

24. 2x 1x 5

25. 3x 2x 1

26. x 4x 4

27. x 7x 7

28. x 32

29. x 62

30. x 82

31. (x 42

Write the area of the figure as a polynomial in standard form. 32.

33. x x5

34. x1 2x 3

x5

x5

35. Education

For 1990 through 1996, the number of bachelor degrees D earned by people in the United States and the number of bachelor degrees W earned by women in the United States can be modeled by D 12829.86t 1117893 W 9777.82t 602005.12

where t is the number of years since 1990. Find a model that represents the number of bachelor degrees M earned by men in the United States from 1990 through 1996. 40



Answer Key Practice B 1. 2x2 2x 2 2. 3x3 3x2 x 4 3. 2x2 6x 4 4. 4x2 2x 4 5. 7x3 3x2 2x 1 6. 2x3 4x2 3x 1 7. 3x4 2x2 x 5 8. 2x5 x3 x2 7x 4 9. 2x5 3x4 x2 5x 4 10. x3 3x2 8x 5 11. 10 12. 8x2 13. 3x2 x 14. 2x3 6x2 15. 3x3 x2 5x 16. x2 3x 10 17. x2 4x 3 18. x2 5x 4 19. 2x2 11x 5 20. 3x2 11x 4 21. 2x2 5x 3 22. 6x2 13x 5 23. 8x2 10x 3 24. 15x2 17x 4 25. x3 2x2 1 26. x3 7x 6 27. x3 3x2 10x 28. x2 81 29. 4x2 25 30. x2 20x 100 31. 16x2 24x 9 32. x2 24x 144 33. 9x2 48x 64 34. x2 42x 360 35. v 16.2t3 183t2 1352.5t 11504.1

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LESSON

NAME _________________________________________________________ DATE ___________

6.3


Find the sum or difference. 1. x2 2x 3 x2 5

2. 3x3 2x2 x 1 x2 2x 3

3. 4x2 x 3 2x2 5x 1

4. x2 2x 7 5x2 3

5. 4x3 2x 3x3 3x2 1

6. 2x3 3x2 x 3 x2 2x 4

7. 2x2 5 x 4x2 3x 4

8. 1 3x x2 x3 3 2x5 4x

9. 4x5 3x 4 5x 1 2x5 x2 3

10. 6x3 3x2 5x 1 7x3 3x 6

11. 3x3 2x2 7x 5 3x3 2x2 7x 5 12. 6x2 3x 7 2x2 3x 7

Find the product. 13. x3x 1

14. 2x2x 3)

15. x3x2 x 5

16. x 5x 2

17. (x 3x 1

18. x 4x 1

19. 2x 1x 5

20. 3x 1x 4

21. 2x 3x 1

22. 2x 53x 1

23. 4x 12x 3

24. 5x 43x 1

25. x 1x2 x 1

26. x 3x2 3x 2

27. x 2x2 5x

28. x 9x 9

29. 2x 52x 5

30. x 102

31. 4x 32

32. x 122

33. 3x 82

34. Floor Space

Find a polynomial that represents the total number of square feet for the floor plan shown below.

12 ft

Lesson 6.3

x ft x 6 ft

24 ft

35. Advertising

For 1980 through 1990, the amount of money A (in millions of dollars) spent on television and newspaper advertising can be modeled by A 16.2t3 153t2 3609.5t 26,265.9 where t is the number of years since 1980. The amount of money n (in millions of dollars) spent on newspaper advertising can be modeled by n 30t2 2257t 14,761.8 where t is the number of years since 1980. Write a model that represents the amount of money v (in millions of dollars) spent on television advertising.



41

Answer Key Practice C 1. x3 3x2 5x 4 2. x3 3x2 2x 3 3. 5x2 5x 6 3 11 4. 3x4 4x2 6x 5 5. 2x2 3 x 2 2 3 6. x2 5x 3 7. 10x3 3x2 2x 1 3 1 8. 8x2 6x 5 9. 6x2 25x 25 10. 5x2 32x 21 11. 24x2 35x 4 12. x3 x2 x 1 13. 2x3 5x2 x 4 14. 2x3 5x2 x 2 15. 2x3 x2 7x 3 16. x4 6x3 5x2 18x 6 17. x5 3x4 x3 6x2 12x 9 18. x6 x5 2x4 6x3 x2 10x 5 19. 2x7 6x6 3x5 3x4 x 20. 36x2 25 16 40 1 21. 4x2 49 22. 9 x2 3 x 25 1 4 4 23. 25x2 20x 4 24. 9x2 9x 9 25. x3 6x2 12x 8 26. x3 9x2 27x 27 27. 8x3 12x2 6x 1 28. 27x3 135x2 225x 125 29. 8x3 36x2y 54xy2 27y3 30. 16x2 9y2 31. 36x2 12xy y2 32. x2 8xy 16y2 33. x3 4x2 x 6 34. x3 4x2 11x 30 35. 2x3 9x2 10x 3 36. 4x3 20x2 31x 15 37. I 814,536.25t3 4,028,984.354t2 17,858,746.41t 560,699,692.4

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LESSON

NAME _________________________________________________________ DATE ___________

6.3


Find the sum or difference. 1. 2x3 3x2 5x 2 3x3 6x2 2

2. 4x2 5x 1 x3 x2 3x 4

3. 2x2 7x 7 3x2 2x 1

4. 2x3 6x 4 3x 4 2x3 4x2 1

5.

6.

7.

12 x2 3x 1 x2 23 x 3 15 x3 3x2 43 12 x3 2x 13

8.

25 x2 2x 1 35 x2 7x 13 38 x2 23 x 5 34 x2 12 x

Find the product. 9. 3x 52x 5

10. x 75x 3

11. 3x 48x 1

12. x 1x2 2x 1

13. x 12x2 3x 4

14. 2x 1x2 3x 2

15. 2x 1x2 x 3

16. x2 3x2 6x 2

17. x3 x2 3x2 4x 3

18. x3 2x 1x3 x2 5

19. 2x3 xx 4 3x3 2x2 1

20. 6x 56x 5

21.

1 2x 1 3x

7 3

1 2x

7

2 2

22.

43 x 52

23. 5x 22

25. x 23

26. x 33

27. 2x 13

28. 3x 53

29. 2x 3y3

30. 4x 3y4x 3y

31. 6x y2

32. (x 4y2

24.

Find the product of the binomials. 33. x 3x 2x 1

34. x 5x 3x 2

35. 2x 1x 3x 1

36. 2x 32x 5x 1

The principal source of collections by the IRS include individual income and profit taxes, corporation income and profit taxes, employment taxes, estate and gift taxes, and other taxes. From 1992 through 1996, the amount of taxes collected in each of these categories can be modeled by

Lesson 6.3

37. IRS Collection

T 7,810,103.714t2 61,813,629.34t 1,116,758,213

(Total collected)

C 18,508,265.4t 116,419,459.8

(Corporate income and profit)

E 23,846,333.7t 394,945,983.6

(Employment)

G 133,820.25t3 881,998.57t2 2,915,045.54t 11,328,112.36

(Estate and gift tax)

O 948,356.5t3 4,663,117.93t2 1,314,761.71t 33,364,964.86

(Other taxes)

where T, C, E, G and O are in thousands of dollars and t is the number of years since 1992. Write a model that represents the individual income and profit taxes I (in thousands of dollars) from 1992 to 1996.

42



Answer Key Practice A 1. D 2. C 3. E 4. F 5. A 6. B 7. G 2 8. x 1x x 1 9. x 3x2 3x 9 10. x 5x2 5x 25 11. x 1x2 x 1

2 12. x 2x 2x 4 2 13. x 4x 4x 16

2 14. x 3x 2

2 15. x 1x 4

2 16. x 5x 1

2 17. x 6x 1

2 18. x 4x 3

2 19. x 5x 2

20. 2, 0

22. 4, 1

23. 3, 2

25. 10, 10 28. 3, 1, 3

24. 7, 7

26. 2, 1, 1 29. C

21. 0, 3

30. A

27. 2, 1, 2 31. B

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LESSON

NAME _________________________________________________________ DATE ___________

6.4


Match the polynomial with its factorization. 1. x 4 16

A. x 3x2 3x 9

2. x3 2x2 6x 12

B. 2x 14x2 2x 1

3. x3 3x2 4x 12

C. x 2x2 6

4. 5x3 5

D. x 2x 2x2 4

5. x3 27

E. x 3x 2x 2

6. 8x3 1

F. 5x 1x2 x 1

7. 16x 4 81

G. 4x2 92x 32x 3

Factor the sum or difference of cubes. 8. x3 1

9. x3 27

11. x3 1

10. x3 125

12. x3 8

13. x3 64

14. x3 3x2 2x 6

15. x3 x2 4x 4

16. x3 5x2 x 5

17. x3 6x2 x 6

18. x3 4x2 3x 12

19. x3 5x2 2x 10

Factor the polynomial by grouping.

Find the real-number solutions of the equation. 20. x2 2x 0

21. x3 3x2 0

22. x2 3x 4 0

23. x2 5x 6 0

24. x2 49 0

25. x2 100 0

26. x3 2x2 x 2 0

27. x3 x2 4x 4 0

28. x3 x2 9x 9 0

Match the equations for volume with the appropriate solid. 29. V x3 4x

30. V x3 4x2 4x

A.

31. V x 4 16

B. x2

x2 x

C. x2

x2 4

x2 x x2

x2

Lesson 6.4



53

Answer Key Practice B 1. x 4x2 4x 16 2. x 6x2 6x 36 3. x 10x2 10x 100 4. x 7x2 7x 49 5. 2x 14x2 2x 1 6. 2x 54x2 10x 25 7. 3x 29x2 6x 4 8. 2x 104x2 20x 100 9. 3x 89x2 24x 64 10. 4x 316x2 12x 9 11. 10x 1100x2 10x 1 12. 5x 425x2 20x 16 13. x 3x2 5 14. x 4x2 2 15. x 2x2 7 16. x 43x2 2 17. x 15x2 1 18. x 62x2 5 19. x 22x 2 20. x 5x 3x 3 21. x 1x 4x 4 22. x 12x 32x 3 23. x 34x 14x 1 24. x 23x 23x 2 25. 2x 32x 34x2 9 26. x2 3x2 3 27. x2 3x2 2 28. x2 3x2 2 29. x2 3x2 8 30. x2 5x2 2 31. 2x2x 10x 10 32. 2x22x 32x 3 33. 3x23x 13x 1 34. 3x 1x 1x2 1 35. 2x2 2x2 6 36. x2 7x2 3 1 3 37. 6, 2, 2 38. 2 39. 2 40. 3 41. 7 42. 3, 3, 5 43. 2, 2 44. No real solutions 45. 3, 1, 1, 3 46. 5, 5 47. 6, 2, 2, 6 48. 22, 2, 2, 22 49. 750 ft3 50. x3 15x2 50x 750 51. 15 52. 10 ft by 15 ft by 5 ft

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LESSON

NAME _________________________________________________________ DATE ___________

6.4


Factor the sum or difference of cubes. 1. x3 64

2. x3 216

3. x3 1000

4. x3 343

5. 8x3 1

6. 8x3 125

7. 27x3 8

8. 8x3 1000

9. 27x3 512

10. 64x3 27

11. 1000x3 1

12. 125x3 64

13. x3 3x2 5x 15

14. x3 4x2 2x 8

15. x3 2x2 7x 14

16. 3x3 12x2 2x 8

17. 5x3 5x2 x 1

18. 2x3 12x2 5x 30

19. x3 2x2 4x 8

20. x3 5x2 9x 45

21. x3 x2 16x 16

22. 4x3 4x2 9x 9

23. 16x3 48x2 x 3

24. 9x3 18x2 4x 8

25. 16x 4 81

26. x 4 9

27. x 4 5x2 6

28. x 4 x2 6

29. x 4 5x2 24

30. x 4 7x2 10

31. 2x 4 200x2

32. 8x 4 18x2

33. 27x 4 3x2

34. 3x 4 3

35. 2x 4 16x2 24

36. x 4 10x2 21

Factor the polynomial by grouping.

Factor the polynomial.

Find the real-number solutions of the equation. 37. x3 6x2 4x 24 0

38. x3 8 0

40. 3x3 x2 3x 1 0

41. x3 7x2 4x 28 0

42. x3 5x2 9x 45 0

43. x 4 x2 12 0

45. x 4 10x2 9 0

46. x 4 4x2 5 0

47. x 4 10x2 24 0

48. x 4 10x2 16 0

Aquarium

39. 8x3 27 0 44. x 4 6x2 5 0


The aquarium shown at the right holds 5610 gallons of water. Each gallon of water occupies approximately 0.13368 cubic feet. 49. How many cubic feet of water does the aquarium hold?

(Round the result to the nearest cubic foot.)

x5 x

x 10

50. Use the result from Exercise 49 to write an equation that Lesson 6.4

represents the volume of the aquarium. 51. Find all real solutions of the equation in Exercise 50. 52. What are the dimensions of the aquarium?

54



Answer Key Practice C 1. x 9x2 9x 81 2. 4x 516x2 20x 25 3. 2x 2x2 2x 4 4. 52x 14x2 2x 1 5. 22x 34x2 6x 9 6. 2x 32x 34x2 9 7. x2 8x2 8 8. 3x 2x 2x2 4 2 9. x 2x 7x 7 10. x 1x 3 11. x 62x 12x 1 12. x 2x 1x2 x 1 13. x 12x 14x2 2x 1 14. x 5x 1x2 x 1 15. 8x 1x 2x2 2x 4 16. 3x 4x 1x 1 17. 3x 2x 2x2 2x 4 18. xx 3x 2x 2 19. x3x 12x 1 20. 6xx 2x2 2 2 21. x2x 1x4 1 22. 11 23. 3 24. 3 1 1 25. 2, 2 26. 2, 2 27. 2, 2, 3 28. 8 2 2 2 29. 3, 3, 3 30. 2, 5 31. 1, 3 32. 3 33. 1, 5 34. 6, 3, 0, 3 35. 1, 0, 1 36. 3, 0, 2 37. 2, 0, 1, 2 38. 1, 0, 1, 3 39. 3, 3, 2 40. 4, 7, 7 41.

52, 52, 1

43. 10, 0, 10 46. 99

in.3

42. 6, 6 44. 2, 2

45. 2, 2

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LESSON

6.4

NAME _________________________________________________________ DATE ___________


Factor the polynomial. 1. x3 729

2. 64x3 125

3. 2x3 16

4. 40x3 5

5. 16x3 54

6. 16x 4 81

7. x 4 64

8. 3x 4 48

9. x3 2x2 49x 98

10. x3 x2 3x 3

11. 4x3 24x2 x 6

12. x 4 2x3 x 2

13. 8x 4 8x3 x 1

14. x 4 5x3 x 5

15. 8x 4 8x3 64x 64

16. 3x3 12x2 3x 12

17. 3x 4 6x3 24x 48

18. x 4 3x3 4x2 12x

19. x6 x5 x 4 x3

20. 6x 4 12x3 12x2 24x

21. x7 x6 x3 x2

Find the real number solutions of the equation. 22. x3 1331 0

23. 27x3 8 0

24. 4x3 108

25. x 4 16 0

26. 32x 4 2

27. x3 12 3x2 4x

28. x3 8x2 6x 48 0

29. 9x3 27x2 4x 12

30. x 4 5x3 8x 40

31. 27x 4 8 27x3 8x

32. 2x3 6x2 2x 6 0

33. 2x 4 10x3 2x 10 0 34. x 4 6x3 9x2 54x

35. x6 x5 x 4 x3

36. x5 3x 4 8x2 24x

37. 5x 4 5x3 20x2 20x 0

38. x8 3x7 x6 3x5

39. x3 2x2 3x 6

40. x3 4x2 7x 28

41. 2x3 2x2 5x 5

42. x 4 36 0

43. x5 100x 0

44. 3x 4 12 0

45. x8 256 0

46. Manufacturing

A tool shop is hired to make a metal mold in which plastic is injected to make a solid block. (See diagram below.) The finished plastic block should have a length that is 8 inches longer than the height. It should also have a width that is 2 inches shorter than the height. Each plastic block requires 96 cubic inches of plastic. If the sides of the mold are to be 12 inch thick, how much metal is required to make the mold? Plastic injection

Lesson 6.4



55

Answer Key Practice A 1. Dividend: x3 2x2 14x 5, Divisor: x 5, Quotient: x2 3x 1, Remainder: 0 2. Dividend: 2x3 3x2 3x 17, Divisor: x 2, Quotient: 2x2 x 5, Remainder: 7 3. Dividend: x3 x 2, Divisor: x 3, Quotient: x2 3x 10, Remainder: 28 8 3 4. x 2 5. x 3 x1 x2 4 5 6. x 2 7. x 6 x3 x1 5 8. x 2 9. x 2 10. x 2 x5 5 3 11. x 4 12. x 3 x1 x2 8 13. x 4 14. x 1 x3 15 8 15. x 5 16. x 6 x2 x1 6 17. x 1 18. x 2 x2 2 19. x 5 20. x 6 x1 3 21. x 1 22. x 7 23. x 2 x4 24. x 3

6 1 10 3 9 x 3 . x3 1 3 10 The denominator of the remainder is x 3, not x 3. 26. As written, synthetic division cannot be used because the divisor does not have the form x k. 3 1

25.

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LESSON

NAME _________________________________________________________ DATE ___________

6.5

Lesson 6.5


Write the polynomial form of the dividend, divisor, quotient, and remainder represented by the synthetic division array. 1 1. 5

1 1 3. 3

1

2 14 5 15 3 1 0 3 3

1 9 10

5 5 0

2 2. 2

2

3 4 1

3 17 2 10 5 7

2 30 28

Divide using polynomial long division. 4. x2 3x 6 x 1

5. x2 x 3 x 2

6. x2 5x 2 x 3

7. x2 7x 1 x 1

8. x2 5x 6 x 3

9. x2 3x 5 x 5

10. x2 2x 8 x 4

11. x2 3x 1 x 1

12. x2 5x 3 x 2

Divide using synthetic division. 13. x2 7x 4 x 3

14. x2 2x 1 x 1

15. x2 3x 2 x 2

16. x2 7x 9 x 1

17. x2 3x 8 x 2

18. x2 7x 10 x 5

19. x2 6x 3 x 1

20. x2 5x 6 x 1

21. x2 3x 1 x 4

You are given an expression for the area of the rectangle. Find an expression for the missing dimension. 22. A x2 10x 21 x3

23. A x2 2x 8

24. A x2 8x 15

? ?

? x4

x5

Find the error in the example and correct it. 25. x2 6x 1 x 3

3

1 1

x3

26. x2 4x 5 2x 3

6 1 3 9 3 10

3

10 x3

x1


1 1

4 3 1

5 3 2

2 2x 3


67

Answer Key Practice B 1. x 5

21 x3

2. 2x 3

3. x2 x 1

5 2x 1 1 16 25 x2 x 3 9 93x 1 23 7 4x 2 22x 3 1 x2 2 x 3x 1 26x 11 x4 2 x x4

4. 2x2 x 3 5. 6. 7. 8.

9. 2x2 x 3 11. x3 4

10. x2 3x 7

9 x3

3 x5

43 x2 6 13. 5x3 3x2 5 x1 12. 3x2 8x 21

14. 3x3 10x2 40x 160 15. x2 x 1

635 x4

3 x1

16. 3x3 6x2 12x 24 1

19. 2, 1

47 x2

18. 2

21. x 2

22. Px 50x 5x3

23. about 0.3 million

2

20. 3, 1

17. 8, 2

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Lesson 6.5

LESSON

6.5

NAME _________________________________________________________ DATE ___________


Divide using polynomial long division. 1. x2 2x 6 x 3

2. 2x2 x 3 x 1

3. x3 x2 x 2 x 2

4. 4x3 7x 8 2x 1

5. 3x3 2x2 5x 1 3x 1

6. 8x2 5x 1 2x 3

7. x3 5x2 5x 3 x2 3x 1

8. x3 3x2 4x 6 x2 x 4

Divide using synthetic division. 9. 2x3 7x2 x 12 x 4

10. x3 2x 12 x 3

11. x 4 5x3 4x 17 x 5

12. 3x3 2x2 5x 1 x 2

13. 5x 4 2x3 3x2 5x 1 x 1

14. 3x4 2x3 5 x 4

15. x3 2 x 1

16. 3x 4 1 x 2

Given one zero of the polynomial function, find the other zeros. 17. f x x3 3x2 34x 48; 3

18. f x x3 2x2 20x 24; 6

19. f x 2x3 3x2 3x 2; 2

20. f x 3x3 16x2 3x 10; 5

21. Geometry

The volume of the box shown below is given by V 2x3 11x2 10x 8. Find an expression for the missing dimension.

?

2x 1 x4

Company Profit


The demand function for a type of portable radio is given by the model p 70 5x2, where p is measured in dollars and x is measured in millions of units. The production cost is $20 per radio. 22. Write an equation giving profit as a function of x million radios sold. 23. The company currently produces 3 million radios and makes a profit

of $15,000,000, but would like to scale back production. What lesser number of radios could the company produce to yield the same profit?

68



Answer Key Practice C 37x 4 x2 3x 1 28 2x2 4x 9 2x 3 5x 7 2x2 4x 6 2 x 2x 1 16x 19 4x 6 2 x 4 4 4x 15 2x 3 33x2 x 1 2 8 61 149 x x 3 9 27 273x 2

1. x 6 2. 3. 4. 5. 6.

1 x 10 x 2 22x2 1 1 5 38x 5 8. x 2 4 44x2 2x 1 7.

9. 5x3 8x2 23x 52

96 x2

192 x3 3 11. 2x2 2x 1 x1 10. 6x2 20x 65

12. 4x2 10x 20

39 x2 769 x3 3 1 16. , 2 3

13. 3x4 9x3 29x2 87x 256 14. 2x2 5x 1

15. 1, 7

17. 2 5, 2 5

5 17 5 17 , 19. 5, 1 2 2 3 3 5 3 5 , 20. 21. , 1 2 2 2 1 10 1 10 , 22. 23. 1 i, 1 i 3 3 18.

24. 5 i, 5 i 25. A 0.004t3 0.082t2 0.268t

3.206 2.61t 247; 0.0118 26. 810 yearbooks

quadrillion Btu million people

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LESSON

Practice C

Lesson 6.5

6.5

NAME _________________________________________________________ DATE ___________


Divide using polynomial long division. 1. x3 3x2 2x 6 x2 3x 1

2. 4x3 2x2 6x 1 2x 3

3. 2x 4 3x 1 x2 2x 1

4. 4x3 6x2 5 x2 4

5. 6x3 2x2 5 3x2 x

6. x3 2x2 5x 1 3x 2

7. x3 5 2x2 1

8. 2x3 4x2 3x 5 4x2 2x 1

Divide using synthetic division. 9. 5x 4 2x3 7x2 6x 8 x 2

10. 6x3 2x2 5x 3 x 3

11. 2x3 3x 4 x 1

12. 4x3 2x2 1 x 2

13. 3x5 2x3 5x 1 x 3

14. 4x 2x3 7x2 1 x 1

Given one zero of the polynomial function, find the other zeros. 15. f x x3 8x2 5x 14; 2

16. f x 12x3 8x2 13x 3;

17. f x x3 x2 13x 3; 3

18. f x 2x3 11x2 9x 2; 2

1 2 1

Given two zeros of the polynomial function, find the other zeros. 19. f x x 4 6x3 4x2 54x 45; 3, 3

20. f x x 4 3x3 3x 1; 1, 1

21. f x 2x 4 9x3 4x2 21x 18; 2, 3

22. f x 3x 4 2x3 12x2 6x 9; 3, 3

23. f x x 4 2x3 14x2 32x 32; 4, 4 24. f x x 4 3x3 7x2 15x 10; 2, 1 25. Hydroelectric Power

The amount of conventional hydroelectric power (in quadrillion Btu) consumed from 1990 to 1997 can be modeled by P 0.004t3 0.082t2 0.268t 3.206 where t is the number of years since 1990. For the same years, the U.S. population (in millions) can be modeled by P 2.61t 247 where t is the number of years since 1990. Find a function for the average amount of energy consumed by each person from 1990 to 1997. What was the per capita consumption of conventional power in 1992?

26. Yearbook Sales

If the school charges $15 for a yearbook, 800 students will buy a yearbook. For every $.50 reduction in price two more books are sold. It costs $10 to produce each book. How many books must be sold to earn a profit of at least $2000?



69

Answer Key Practice A 1. ± 1 2. ± 1, ± 7 3. ± 1, ± 2, ± 3, ± 6 4. ± 1, ± 3, ± 9 5. ± 1, ± 2, ± 3, ± 4, ± 6, ± 12 6. ± 1, ± 2, ± 4, ± 5, ± 10, ± 20 7. ± 1, ± 2, ± 3, ± 4, ± 6, ± 8, ± 12, ± 24 8. ± 1, ± 2, ± 4, ± 5, ± 8, ± 10, ± 20, ± 40 9. ± 1, ± 2, ± 4, ± 5, ± 10, ± 20, ± 25, ± 50, ± 100 10. 1 11. 1 12. neither 13. 1 14. neither 15. 1 and 1 16. 1 and 1 17. neither 18. 1 19. 3, 2, 4 20. 1, 1, 2 21. 3, 2, 2 22. 1 6, 1, 1 6 23. 5, 5, 3 24. x3 5x2 4x 84 25. ± 1, ± 2, ± 3, ± 4, ± 6, ± 7, ± 12, ± 14, ± 21, ± 28, ± 42, ± 84 26. 3 in. by 4 in. by 7 in.

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LESSON

NAME _________________________________________________________ DATE ___________

6.6


List the possible rational zeros of f using the rational zero theorem. 1. f x x3 2x2 4x 1

2. f x x2 3x 7

3. f x x3 2x2 5x 6

4. f x x 4 6x 9

5. f x x3 3x2 12

6. f x x8 2x5 x 4 3x 20

7. f x x5 2x 4 3x 24

8. f x x2 6x 40

Use synthetic division to decide which of the following are zeros of the function: 1, 1. 10. f x x2 6x 5

11. f x x3 x2 9x 9

12. f x x2 10x 21

13. f x x3 2x2 2x 1

14. f x x3 3x2 4x 12

15. f x x3 x2 x 1

16. f x x3 5x2 x 5

17. f x x3 6x2 8x

18. f x 2x2 x 1

Find all the rational zeros of the function. 19. f x x3 x2 14x 24

20. f x x3 2x2 x 2

y

y

10 2

x 1 x

1

Find all the real zeros of the function. 21. f x x3 3x2 4x 12

22. f x x3 3x2 3x 5

23. f x x3 3x2 5x 15

Geometry


The volume of the box shown at the right is given by V x3 5x2 4x. 24. Write an equation that indicates that the volume of the box is 84 in.3. 25. Use the rational zero theorem to list all possible rational zeros of the

equation in Exercise 24. 26. Find the dimensions of the box.


x

x4 x1


79

Lesson 6.6

9. f x x3 5x2 2x 100

Answer Key Practice B 1 3 1. ± 1, ± 2, ± 4 2. ± 1, ± 2, ± 3, ± 6, ± 2, ± 2 1 2 4 8 3. ± 1, ± 2, ± 4, ± 8, ± 3, ± 3, ± 3, ± 3 8 1 3 1 4. 2, 2, 3 5. 3, 1, 1 6. 2, 4, 2 7. 2, 1, 5 8. 3, 1, 2, 2 1 5 3 1 9. 3, 1, 2, 1 10. 2, 2, 2 11. 2, 2, 3 12. 3, 3, 3

1

13. 2, 2, 2, 2

14. 3, 2, 1, 2

1 17 1 17 , 4 4 3 2 16. t 13t 65t 105 0 17. ± 1, ± 3, ± 5, ± 7, ± 15, ± 21, ± 35, ± 105 18. 1, 3, 5, 7 19. 1983 15. 3, 1,

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LESSON

NAME _________________________________________________________ DATE ___________

6.6


List the possible rational zeros of f using the rational zero theorem. 1. f x x 4 2x3 3x 4

2. f x 2x3 x2 5x 6

3. f x 3x5 2x 8

Use the rational zero theorem and synthetic division to find all rational zeros of the function. 4. f x 2x3 3x2 11x 6

5. f x 3x3 8x2 3x 8

Lesson 6.6

y

y 9

5 x

1

3

x

6. f x 8x3 6x2 23x 6

7. f x x3 4x2 7x 10

8. f x x 4 4x3 x2 8x 6

9. f x 2x 4 5x3 5x2 5x 3

Find all real zeros of the function. 10. f x 2x3 5x2 4x 10

11. f x 4x3 8x2 15x 9

y

y 5 1

x

2 4

x

12. f x x3 3x2 3x 9

13. f x 2x 4 3x3 6x2 6x 4

14. f x x 4 2x3 5x2 4x 6

15. f x 2x 4 5x3 6x2 7x 6

European College Students


Many students from Europe come to the United States for their college education. From 1980 through 1990, the number S (in thousands), of European students attending a college or university in the U.S. can be modeled by S 0.07t3 13t2 65t 339 where t 0 corresponds to 1980. 16. Write an equation with a leading coefficient of 1 that represents the year that 31.08 thousand

European students attended a U.S. college or university. 17. Use the rational zero theorem to list all possible rational zeros of the equation in Exercise 16. 18. Which of the rational zeros listed in Exercise 17 are valid values of t? 19. In what year did 31.08 thousand European students attend a U.S. college or university?

80



Answer Key Practice C 2 1 5 3 1 1. 2, 2, 4 2. 4, 1, 3, 1 3. 1, 5, 2, 3 3 1 1 7 1 4. 4, 2, 3, 2 5. 3, 2, 3, 1 5 21 5 21 5 3 1 , 6. 2, 2, 2 7. 7, 2 2 1 8. 4 14, 6, 4 14 4 9. 1 6, 3, 1 6 10. 3 22, 1, 3 22, 3 5 17 5 17 , 1, 23, 11. 4 4 5 1 13 1 13 1 , , 12. , 2 6 6 2 9 5 7 5 1 13. 2, 3, 3, 4 14. 2, 3, 2, 2 15. a0 cannot be 0. 16. xx3 x2 24x 36 1 17. 3, 2, 6 18. 2, 3, 0, 2 19. 2, 1, 1 20. The zeros of f x are also the zeros of af x. 21. To apply the rational zero theorem, the coefficients must be integers. 22. 2f x x3 19x 30; 3, 2, 5

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LESSON

6.6

NAME _________________________________________________________ DATE ___________


Use the rational zero theorem and synthetic division to find all rational zeros of the function. 1. f x 4x3 8x2 29x 12

2. f x 3x 4 10x3 11x2 10x 8

3. f x 10x 4 43x3 11x2 79x 15

4. f x 24x 4 26x3 45x2 x 6

5. f x 6x 4 35x3 35x2 55x 21

6. f x) 8x3 28x2 14x 15

7. f x x3 2x2 34x 7 9. f x 3x3 10x2 7x 20

Lesson 6.6

Find all real zeros of the function. 8. f x 6x3 49x2 20x 2 10. f x x 4 4x3 14x2 20x 3

11. f x 6x 4 25x3 32x2 15x 2

12. f x 12x 4 28x3 11x2 13x 5

13. f x 6x 4 31x3 64x2 489x 540

14. f x 8x 4 68x3 178x2 103x 105

Critical Thinking In Exercises 15–18, consider the function f x x 4 x 3 24x 2 36x. 15. Explain why the rational zero theorem cannot be directly applied to this

function. 16. Factor out the common monomial factor of f. 17. Apply the rational zero theorem to find all other rational zeros of f. 18. Find all the real zeros of f x 3x5 x 4 12x3 4x2.

Critical Thinking In Exercises 19–22, consider the functions f x x 3 2x 2 x 2, g x x 3 2x 2 x 2, h x 2x 3 4x 2 2x 4, and j x 5x 3 10x 2 5x 10. 19. Use the rational zero theorem to find all rational zeros of each function. 20. Note that gx f x, hx 2f x, and jx 5f x. What can you

conclude about the zeros of f x and af x?

21. Explain why the rational zero theorem cannot be directly applied 1 19 to f x 2 x3 2 x 15.

22. Use the conclusion from Exercise 20 to find the rational zeros of the

function in Exercise 21.



81

Answer Key Practice A 1. 3 2. 5 3. 4 4. 6 5. 3 6. 2 7. 1 8. 5 9. 2 i 10. 3 5i 11. 6 2i 12. 7 3i 13. 2 i 14. 5 4i 15. 3 2i 16. 2 3i 17. 3 5 i 18. yes 19. no 20. no 21. yes 22. yes 23. no 24. f x x 6 25. f x x2 x 6 26. f x x2 6x 5 27. f x x3 3x2 x 3 28. f x x3 7x2 12x 3 2 29. f x x 8x x 42 30. x 3x 1x 2 31. Length: x 3; Width: x 1; Height: x 2 32. 10

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LESSON

6.7

NAME _________________________________________________________ DATE ___________


Determine the total number of solutions (including complex and repeated) of the polynomial equation. 1. x3 3x2 4x 2 0

2. x5 3x2 4x 1 0

3. 2x4 3x3 2x2 x 5 0

4. 3x6 2x5 x4 3x3 2x2 x 8 0

5. 5x3 2x2 3x 1 0

6. 6x2 3x 1 0

7. 5x 7 0

8. 2x5 3x4 x 9 0

Given that f x has real coefficients and x k is a zero, what other number must be a zero of f ? 9. k 2 i

10. k 3 5i

11. k 6 2i

12. k 7 3i

13. k 2 i

14. k 5 4i

15. k 3 2i

16. k 2 3i

17. k 3 5i

Decide whether the given x-value is a zero of the function. Lesson 6.7

18. f x x3 2x2 4x 7, x 1

19. f x x3 3x2 2x 1, x 2

20. f x x3 x2 4x 3, x 3 21. f x 2x4 x3 x2 4x 4, x 1 22. f x x3 2x2 2x 3, x 3

23. f x x4 2x3 6x2 5x 2, x 2

Write a polynomial function of least degree that has real coefficients, the given zeros, and a leading coefficient of 1. 24. 6

25. 2, 3

26. 1, 5

27. 1, 1, 3

28. 0, 3, 4

29. 2, 3, 7

Room Dimension Riddle information.

In Exercises 30–32, use the following

One of the bedrooms of a house has a volume of 1144 cubic feet. The volume of the bedroom is given by y x3 2x2 5x 6, where x is the number of rooms in the house. 30. Factor the polynomial that represents the volume of the bedroom. 31. The factors in Exercise 30 represent the length, width, and height of the

bedroom. Which do you think represents the length, the width, and the height? 32. How many rooms does the house have?

94



Answer Key Practice B 1. 3 2. 6 3. 5 4. 4 5. yes 6. no 7. yes 8. yes 9. x 3, x 1, x 2 10. x 4, x 1, x 2, x 11. x 6, x 2, x 1, x 1 12. x 3, x i, x i 13. x 4, x 5, x 2i, x 2i 14. x 3, x 2 i, x 2 i 15. f x x3 5x2 2x 8 16. f x x3 5x2 7x 3 17. f x x3 x2 6x 18. f x x3 2x2 x 2 19. f x x4 x3 9x2 9x 20. f x x4 3x3 3x2 3x 2 21. f x x4 10x2 9 22. f x x3 10x2 33x 34 23. f x x4 2x2 8x 5 24. 1, i, i 1 25. 3, 2, 2 26. 4, 3i, 3i 1 27. 1, 2, i, i 28. 2, 2, i, i 1 29. 4, 3, 2i, 2i 30. 1996 31. 5

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LESSON

6.7

NAME _________________________________________________________ DATE ___________


Determine the total number of solutions (including complex and repeated) of the polynomial equation. 1. 4x3 7x2 5x 9 0

2. 8x6 3x4 11x3 2x2 4 0

3. x5 2x3 4x2 7x 12

4. 3x 4 2x 3 15x 2 x 1 8

Decide whether the given x-value is a zero of the function. 5. f x x4 2x3 5x2 8x 4, x 1

6. f x x4 x3 8x2 2x 12, x 2

7. f x x3 4x2 x 4, x i

8. f x 2x3 x2 8x 4, x 2i

Identify the factors of a polynomial function that has the given zeros. 9. 3, 1, 2 12. 3, i, i

10. 4, 1, 2, 0

11. 6, 2, 1, 1

13. 4, 5, 2i, 2i

14. 3, 2 i, 2 i

Write a polynomial function of least degree that has real coefficients, the given zeros, and a leading coefficient of 1. 16. 3, 1, 1

17. 3, 2, 0

18. 2, i, i

19. 0, 1, 3i, 3i

20. 1, 2, i, i

21. i, i, 3i, 3i

22. 2, 4 i

23. 1, 1, 1 2i

Lesson 6.7

15. 1, 2, 4

Find all of the zeros of the polynomial function. 24. f x x3 x2 x 1

25. f x 2x3 3x2 11x 6

26. f x x3 4x2 9x 36

27. f x 2x4 x3 x2 x 1

28. f x x 4 3x2 4

29. f x 3x4 11x3 8x2 44x 16

30. Preakness Stakes

For 1990 through 1998, the value of a horse winning the Preakness Stakes can be modeled by V 2553x3 25,200.56x2 64,026.95x 428,075.56

where x is the number of years since 1990. Use a graphing calculator to determine in what year the winnings were $488,150. 31. NBA Standings

There are seven teams in the Atlantic Division of the Eastern Conference of the NBA. During the 1997-98 season the winning percentage of the teams in this division can be modeled by W 0.0051x3 0.063x2 0.260x 0.862

where x is the team’s rank within the division. Orlando’s winning percent was 0.500. Use a graphing calculator to estimate their ranking in the division.



95

Answer Key Practice C 1. 4 2. 3 3. 5 4. no 5. no 6. no 7. no 8. f x 2x4 4x3 32x2 64x 9. f x 2x4 58x2 200 10. f x 2x3 12x2 2x 68 11. f x 2x3 12x2 50x 12. f x 2x4 14x3 38x2 46x 20 13. f x 2x6 26x5 120x4 200x3 118x2 174x 14. 7, 5 i, 5 i 15. 1, 1, 2 3, 2 3 16. 3 17. 2 i3, 2 i3, 2, 0 18. i, i, 1 2, 1 2 19. 1 4i, 1 4i, 3, 3 20. R 315.035t 5 5562.592t 4 1832.426t 3 708,818.278t 2 6,449,569.245t 49,245,170.73; 1993 21. The fifth solution must be a repeated solution because complex solutions come in conjugate pairs.

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LESSON

6.7

NAME _________________________________________________________ DATE ___________


Determine the total number of solutions (including complex and repeated) of the polynomial equation. 1. 2x2 3x x 4 5x 1

3. 4 7x x2 3x5

2. 3 2x2 x3 0

Decide whether the given x-value is a zero of the function. 4. f x x3 5x2 4x 6, x 1 i

5. f x x3 2x2 3x 10, x 2 i

6. f x x 4 5x3 5x2 5x 6, x 2

7. f x x5 4x 4 10x3 4x 2 9x 36,

x 3i

Write a polynomial function of least degree that has real coefficients, the given zeros, and a leading coefficient of 2. 8. 4, 0, 2, 4 11. 3 4i, 0

9. 2i, 2i, 5i, 5i 12. 2 i, 1, 2

10. 4 i, 4 i, 2 13. 5 2i, i, 0, 3

Lesson 6.7

Find all the zeros of the polynomial function. 14. f x x3 17x2 96x 182

15. f x x 4 4x3 4x 1

16. f x x 4 12x3 54x2 108x 81

17. f x 2x5 4x 4 2x3 28x2

Find all the zeros of the polynomial function using the given hint. 18.

f x x 4 2x3 2x 1 Hint: i is a zero

19. f x x 4 2x3 14x2 6x 51

Hint: 1 4i is a zero

20. College Tuition

For 1990 through 1997 the enrollment of a college can be modeled by E 29.881t 2 190.833t 4935 where t is the number of years since 1990. For the same years, the cost of tuition at the college can be modeled by T 10.543t3 118.826t2 921.032t 9978.758 where t is the number of years since 1990. Write a model that represents the total tuition brought in by the college in a given year. In what year did the college take in $62,638,006 in tuition?


The graph of a polynomial of degree 5 has four distinct x-intercepts. Since the total number of solutions (including complex and repeated) must be 5, is the fifth solution a complex solution or a repeated solution? Explain your answer.

96



Answer Key Practice A 1. 4 2. 5 3. 3 4. 2, 5 local maximum; 1 2, 3 local minimum 5. 1, 2 local minimum 1 6. 1, 4 local minimum; 2, 1 local maximum; 1, 1 local minimum 7.

8. y

y

1 x

2

1 x

1

9.

10. y

y 1 2

x

2 x

2

11.

12. y

y 2 1

1

13.

Sales (millions of dollars)

2

x

300 250 200 150 100 0 1 2 3 4 5 6 7 8 9 Years since 1990

14. 1996

15. $244 million

x

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LESSON

NAME _________________________________________________________ DATE ___________

6.8


Determine the lowest-degree polynomial that has the given graph. 1.

2.

y

2

3.

y

y

1 x

2

1

2

x

2

x

1

x

Estimate the coordinates of each turning point and state whether each corresponds to a local maximum or a local minimum. 4.

5.

y

6.

y

y

1 2 2

x

1 1

x

Graph the function. 7. f x x 2x 4 10. f x x 1x 2x 3

Lesson 6.8

Sales

8. f x x 1x 3

9. f x x 2x 4

11. f x x 3x 1x 1 12. f x x 2x 22


From 1990 to 1999, the annual sales S (in millions of dollars) of a certain company can be modeled by S 0.4t3 4.5t2 9.2t 202 where t is the number of years since 1990. 13. Use a graphing calculator to graph the polynomial function. 14. Approximate the year in which sales reached a low point. 15. If this polynomial function continues to model the sales of the company in

the future, what can the expected sales be in 2000?

108



Answer Key Practice B 1. 3 2. 2 3. 5 4. 1 5. 6 6. 8 7. B 8. C 9. A 10. 3, 2, 5 11. 4, 6, 8 12. 3, 2 13. 5, 1, 7 14. 6, 2 15. 8 16.

17. y

24.

y

1 1

x

y

1 x

1

5 x

25. 8.78, 745.80 is a P 700 local maximum; 600 19.95, 652.46 is a local 500 400 minimum; During the 300 1980 games the gold 200 100 medal winner scored 0 more points than in 0 3 6 9 12 15 18 21 t Years since 1972 previous years and that was the record for several years following 1980. Starting in 1992, the number of points started to increase after several years of declining scores. 26. 8.5, 122,069.35 is C 140,000 a local maximum. 120,000 29.3, 96,068.15 is a local 100,000 80,000 minimum. In 1973, the 60,000 number of cattle on farms 40,000 20,000 reached a maximum of 0 0 5 10 15 20 25 30 35 t 122,069.35 thousand. This Years since 1965 number decreased to 96,068.15 in 1994 and then started to rise again. Points

1

18.

19. y

y

4 2 x

3

20.

Cattle (thousands)

x

1

21. y

y 5 x

1

1 2

x

22.

23. y

y

60 2

x 1 1

x

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LESSON

6.8

NAME _________________________________________________________ DATE ___________


State the maximum number of turns in the graph of the function. 1. f x x 4 2x2 4

2. f x 3x3 x2 x 5

3. f x 2x6 1

4. f x 4x2 5x 3

5. f x 3x7 6x2 7

6. f x 2x9 8x7 7x5

Match the graph with its function. 7. f x 2x 4 3x2 2 A.

8. f x 2x6 6x 4 4x2 2 B.

y

9. f x 2x 4 3x2 2 C.

y

1

y

1 x

1

1 1

x

x

1

Determine the x-intercepts of the graph of the function. 10. f x x 3x 2x 5

11. f x x 4x 6x 8

12. f x x 32x 2

13. f x x 5x 1x 7

14. f x x 63x 2

15. f x x 85

Graph the function. 16. f x x 4x 1

17. f x x 3x 4x 1

18. f x x 32x 2

19. f x x 6x 1x 2

20. f x x 2 x 1

21. f x x 12x 1x 4

22. f x x3x 3)x 5

23. f x x 1x2 x 1

2

24. f x x 2x2 2x 2

P 0.134t 3

5.775t2

Lesson 6.8

25. Olympic Platform Diving

The polynomial function 70.426t 481.945

models the number of points earned by the gold medal winner of the platform diving event in the summer Olympics, where t is the number of years since 1972. Graph the function and identify any turning points on the interval 0 ≤ t ≤ 24. What real-life meaning do these points have? (Hint: The Olympics only take place every four years.) 26. Livestock

The polynomial function

C 0.03t 4 3.53t3 271.40t2 3788.76t 107,148.79 models the number of cattle (in thousands) on farms from 1965 to 1998, where t is the number of years since 1965. Graph the function and identify any turning points on the interval 0 ≤ t ≤ 33. What real-life meaning do these points have?



109

Answer Key Practice C 1. 3 2. 2 3. 4

local minimum 0.86, 129.88, local maximum 1.72, 130.20; In 1996 you could buy more women’s and girls’ 0 1 2 3 x apparel with your money Years since 1995 than in previous years, but by 1997 prices were higher than in recent years.

4.

CPI

23. 5.

y

y

2 x

2

1 x

1

6.

7. y

y 2 x

1

5 x

1

8.

9. y

y

7 x

1

2 x

2

10. 1

11. 2, 4

12. 1, 3, 5

13. If n is even there is a turning point. If n is odd

the graph passes through the x-axis. y

y

1 1

1 x

1

14. 2, 3; 3

x

15. 1, 7; none

17. 4, 3; 4

18. 1, 3; 1

20. 3, 2, 5; none 22. 8, 1, 4; 1

16. 3, 5; 3, 5 19. 4, 0, 2; 4

21. 3, 2, 3; 3, 3

y 140 135 130 125 120 0

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6.8

NAME _________________________________________________________ DATE ___________


State the maximum number of turns in the graph of the function. 1. f x x 4 3x3 2x 5

2. f x 4 2x2 5x3

3. f x 2x 3x5 2x2 5

4. f x x 32x 1

5. f x x 22x 52

6. f x x 13x 32

7. f x x 23x 13

8. f x x 22x 3x 1

9. f x x 106

Graph the function.

Critical Thinking Consider the graphs f x x 1n where n 1, 2, 3, 4, and 5. 10. What is the x-intercept for all of the functions? 11. For what values of n does the graph have a turning point at the x-intercept? 12. For what values of n does the graph not have a turning point at the x-intercept? 13. Generalize your findings in Exercises 11 and 12. Test your theory for

f x x 16 and gx x 17.

Find all x-intercepts and identify the x-intercepts that are also locations of turning points for the graph of the function. 14. f x x 32x 2

15. f x x 73x 13

16. f x x 56x 32

17. f x x 48x 35

18. f x x 35x 12

19. f x xx 2x 42

20. f x x 3x 2x 5

21. f x x 32x 2x 34

22. f x x 85x 12x 43

Lesson 6.8

23. Consumer Economics

The consumer price index of women’s and girl’s apparel from 1995 to 1998 can be modeled by P 1.02t3 3.95t2 4.53t 131.5,

where t is the number of years since 1995. Graph the function and identify any turning points on the interval 0 ≤ t ≤ 3. What real-life meaning do these points have?

110



Answer Key Practice A 1. B 2. A 3. C 4. f x x 1x 1x 3 5. f x x 3x 2x 1 6. f x x 3x 1x 4 7. f x x 3x 1x 6 8. f x x 2x 3x 5 9. f x x 3x 4x 5 10. f x x 2x 1x 6 11.

f 1

f 2

f 3

f 4

f 5

f 6

f 7

0

0

2

6

12

20

30

0

2

4

6

8

10

2

2

2

2

2

f 1

f 2

f 3

f 4

f 5

f 6

f 7

2

11

34

77

146

247

386

12.

9

23 14

43 20

6

69

101

26 6

32 6

139 38

6

13.

f 1

f 2

f 3

f 4

f 5

f 6

f 7

8

28

74

158

292

488

758

20

46 26

84 38

12

134 50

12

14. y 0.33x

196 62

12

270 74

12

8.77x 10.64 15. y 0.22x 2.51x2 8.98x 20.43 16. y 0.58x3 5.07x2 19.20x 53.43 17. y 0.28x3 2.10x2 5.56x 1.79 3

3

3.25x2

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LESSON

NAME _________________________________________________________ DATE ___________

6.9


Match the cubic function with its graph. f x x 1x 2x 4

1. A.

y

2. f x 2x 1x 2)x 4

1 3. f x 2 x 1x 2x 4

B.

C.

4

y

y

2

1

x

1

x

1

x

1

Write the cubic function whose graph is shown. 4.

5.

y

6.

y

y 5

2

2

1

x

x

2 x

2

Write the cubic function whose graph passes through the given points. 7. 3, 0,1, 0,6, 0,0, 18

8. 2, 0, 3, 0, 5, 0, 0, 30

9. 3, 0,4, 0,5, 0,0, 60

10. 2, 0, 1, 0, 6, 0, 0, 12

Show that the nth order differences for the given function of n are nonzero and constant. 11. f x x2 3x 2

12. f x x3 x2 x 1

13. f x 2x3 x2 3x 2

Use a graphing calculator to find a cubic function for the data. 14.

x y

0 11

1 15

2 20

3 16

4 14

6 18

15.

x y

0 20

1 15

2 10

3 9

4 12

5 10

6 9

16.

x y

0 53

1 40

2 30

3 24

4 5 6 23 10 5

17.

x y

0 2

1 5

2 7

3 7

4 9

5 11

6 20

5 16

Lesson 6.9



121

Answer Key 13. f x x3 2x2 x 1

Practice B 1. f x x 2x 1x 2 2. f x 2x 1x 1x 3 1 3. f x 2x 2x 3x 4 4. f x 2x 1x 3x 2 1 5. f x 2x 4x 1x 5 6. f x x 2x 4x 6 7. f x 2x 1x 3x 4 8. f x 2xx 1x 8 1 9. f x 4xx 3x 9

14. f x x3 3x2 x 4 15. f x x2 3x 2 16. f x x3 x2 3x 2 17. M 0.000127t 3 0.00330t 2 0.0158t

9.77; 13.3 thousand miles

10.

f 1

f 2

f 3

f 4

f 5

f 6

f 7

1

1

11

35

79

149

251

2

10 8

24 14

6

44 20

6

70

102

26 6

32 6

11.

f 1

f 2

f 3

f 4

f 5

f 6

f 7

3

5

29

75

149

257

405

8

24 16

46 22

6

74 28

6

108 34

6

148 40

6

12.

f 1

f 2

f 3

f 4

f 5

f 6

f 7

3

6

33

90

189

342

561

9

27 18

57 30

12

99 42

12

153 54

12

219 66

12

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6.9



2.

y

3.

y

y

1

3 1

x

x

1

2 2

x

Write a cubic function whose graph passes through the given points. 4. 1, 0, 3, 0, 2, 0, 0, 12

5. 4, 0, 1, 0, 5, 0, 0, 10

6. 2, 0, 4, 0, 6, 0, 0, 48

7. 1, 0, 3, 0, 4, 0, 0, 24

8. 0, 0, 1, 0, 8, 0, 2, 24

9. 0, 0, 3, 0, 9, 0, 1, 4

Show that the nth order differences for the given function of degree n are nonzero and constant. 10. f x x3 2x2 x 1

11. f x x3 2x2 5x 1

12. f x 2x3 3x2 4x 6

Use finite differences and a system of equations to find a polynomial function that fits the data. 13.

14.

x f(x)

1 5

2 19

3 49

4 101

5 181

6 295

15.

x f(x)

1 5

2 6

3 1

4 16

1 1

2 8

3 25

4 58

5 51

6 110

16.

x f(x)

1 4

2 4

3 2

4 2

5 8

6 16

x f(x)

5 6 113 196

17. Average Miles Traveled

The table shows the average miles traveled per vehicle (in thousands) from 1960 to 1996. Find a polynomial model for the data. Then predict the average number of miles traveled per vehicle in 2000. 1960 1965 9.7 9.8

1970 10.0

1975 9.6

1980 1985 9.5 10.0

1990 1995 11.1 11.8

1996 11.8

Lesson 6.9

t M

122



Answer Key Practice C 1. f x x 5x 2x 1 2. f x 2x 2x 1x 2 3. f x x 1x 22 3 4. f x 2x 1x 2x 3 3 1 5. f x 2x 2 x 1x 3 1 3 6. f x 12x 2 x 2 x 2 1 1 7. f x 12x 3 x 4 x 1 2 1 1 8. f x x 3 x 4 x 2 1 1 5 9. f x 108x 3 x 3 x 6

12.

f 1

f 2

f 3

f 4

f 5

f 6

3

22

213

966

3031

7638 16617

19

191 172

753 562

390

1312 750

360

10.

f 1

f 2

f 3

f 4

f 5

f 6

f 7

0

29

132

381

872

1725

3084

2065

2542

1230 480

120

4607

f 7

8979

4372

1830

600 120

13.

29

103 74

249 146

491 242

853

362

1359

f 1

f 2

f 3

f 4

f 5

f 6

f 7

7

0

57

224

585

1248

2345

506 7

72

96

120

57

24

110 60

f 1

f 2

f 3

f 4

f 5

f 6

f 7

2

9

76

265

666

1393

2584

67 56

189 122

66

401 212

90 24

727

326 114

24

663

1097

194

302

434

24

11.

11

361

144 50

24

167

464 138

24

1191

84 24

108 24

132 24

14. f x x3 10x2 8x 15 15. f x x3 8x2 12x 13 16. y 1.114t 3 45.50t 2 2829.5t 249,915;

about 274,774,000 people

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LESSON

6.9

NAME _________________________________________________________ DATE ___________



2.

y

3.

y

y

3 3

1 1

3

x

1

x

5 278

6 445

x

Write a cubic function whose graph passes through the given points. 5. 2, 0, 1, 0, 3, 0, 0, 4

4. 1, 0, 2, 0, 3, 0, 0, 9 6. 8.

1

12, 0, 32, 0, 2, 0, 0, 18 23, 0, 14, 0, 12, 0, 1, 25 24

9

7. 3, 0, 4, 0, 1, 0, 2, 49 1

1

9. 3, 0, 3, 0, 6, 0, 1, 16 1

1

5

Show that the nth order differences for the given function of degree n are nonzero and constant. 10. f x x 4 2x3 3

11. f x x 4 x3 3x2 2x 1

12. f x x5 4x2 6

13. f x x 4 8x

Use finite differences and a system of equations to find a polynomial function that fits the data. You may want to use a calculator. 14.

15.

x f(x)

1 16

2 31

3 54

4 5 6 79 100 111

x f(x)

1 10

2 29

3 76

4 157

16. The table shows the U.S. population (in thousands) from 1990 to 1997.

Find a polynomial model for the data. Then estimate the U.S. population in 2000. t y

1990 249,949

1991 252,636

1992 255,382

1993 258,089

t y

1994 260,602

1995 263,039

1996 265,453

1997 267,901 Lesson 6.9



123

Answer Key

18.

y

1 x

1

Domain: x ≥ 0 Range: y ≥ 0 19.

Domain: all real numbers Range: all real numbers Domain: x ≥ 0 Range: y ≥ 0

y

1 x

1

20. 1

21. 6

22. ± 32

23. mean 83.4

median 85 mode 84 range 30 standard deviation 9.25 24.

x

Exam Scores 60

70

65

80

75

90

100

85 91 95

Tally

Frequency

Exam Scores y 4 3 2 1 0 Interval

1 1

Interval 60–69 70–79 80–89 90–99

-6 9 70 -7 9 80 -8 9 90 -9 9

17.

y

25.

60

Test A 1 1. 2 2. 5 3. 9 4. 2 5. 6 6. 2xy 2z 7. x6y6 8. 72 9. 4x 5; Domain: all real numbers 10. 2x 5; Domain: all real numbers 11. 3x 2 15x; Domain: all real numbers 3x ; Domain: all real numbers except 5 12. x5 13. 3x 15; Domain: all real numbers 14. f x x 9 15. f x 2x 4 1 16. f x 3x 2

x

Frequency 1 2 3 4

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CHAPTER

NAME _________________________________________________________ DATE ____________

7


Evaluate the expression without using a calculator. 3 8 1.

2. 2512

Answers

4. 813

3. 2723

Simplify the expression. Assume all variables are positive. 5. 213 313

3

3 8x3y6z3 6.

7.

x3y3 xy3

8. 50 8

1. 2. 3. 4.

Perform the indicated operation and state the domain. Let fx 3x and gx x 5.

5.

gx

6.

9. f x gx 12.

10. f x gx

f x gx

11. f x

7.

13. f gx

8.

Find the inverse function. 14. f x x 9

9. 15. f x 2 x 2 1

10.

16. f x 3x 6

11.

Graph the function. Then state the domain and range. 17. f x x

18. f x

x13

y

13. y

14. 15.

1 1

1 1

x

19. gx x 3

12.

x

16. 17.

Use grid at left.

18.

Use grid at left.

19.

Use grid at left.

y

1 x

Review and Assess

1

106



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CHAPTER

7

CONTINUED

NAME _________________________________________________________ DATE ____________


Solve the equation. Check for extraneous solutions.

20.

20. x12 3 4

21.

21. 32x 4 12

3 2 x 93 22.

Exam Scores In Exercises 23–25, suppose your exam scores on the ten exams taken in Algebra 2 are: 65, 75, 84, 72, 90, 92, 86, 95, 84, and 91.

22. 23.

23. Find the mean, median, mode, and range of the exam scores. 24. Draw a box-and-whisker plot of the exam scores.

24.

Use space at left.

25.

Use space at left.

25. Make a frequency distribution using four intervals beginning with

60–69. Then draw a histogram of the data set.

Review and Assess



107

Answer Key

17.

18.

y

y

1 1

x

1 x

1

Domain: x ≥ 0 Range: y ≥ 1

19.

Domain: all real numbers Range: all real numbers Domain: x ≥ 7 Range: y ≥ 0

y

2 x

2

20. 32

21. 3

22. 4, 3

23. mean 976.4

median 971 mode 964 range 184 standard deviation 51.3 24.

Polar Bears 900

950

894 928

1000

971 1005

1050

1078

Interval 875–929 930–984 985–1039 1040–1094

Tally

92 9 098 984 510 103 40 9 -1 09 4

93

Frequency

Polar Bears y 4 3 2 1 0 5-

25.

87

Test B 1 1. 4 2. 9 3. 9 4. 5 5. 12 6. 2xyz 1 7. 8. 82 4y3 9. 3x 1; Domain: all real numbers 10. x 1; Domain: all real numbers 11. 2x 2 2x; Domain: all real numbers x1 12. ; Domain: all real numbers except 0 2x 13. 2x 1; Domain: all real numbers 1 3 14. f x 2 x 2 15. f x 2 x 6 16. f x x12

Interval

x

Frequency 3 3 3 1

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7



2. 8112

3. 2723

Answers

4. 12513

Simplify the expression. Assume all variables are positive.

4133

5. 313

3 8x3y3z3 6.

1

7.

4xy 16xy

8. 98 2

2

12.

fx gx

10. f x gx

2. 3. 4.

Perform the indicated operation and state the domain. Let f(x) x 1 and g(x) 2x. 9. f x gx

1.

11. f x

5. 6.

gx

7.

13. f gx

8. 9.

Find the inverse function. 14. f x 2x 4

15. f x

23 x

4

16. f x

x 2,

x ‡0

10.

Graph the function. Then state the domain and range.

11.

17. f x x 1

12.

18. fx 2x13 3

y

13.

y 1 1

x

14. 15.

1

16. 1

x

19. f x x 7

17.

Use grid at left.

18.

Use grid at left.

19.

Use grid at left.

y

2 x

Review and Assess

2

108



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CHAPTER

7

CONTINUED

NAME _________________________________________________________ DATE ____________



20.

3 2x 4 20.

21.

21. 3x x 6

22. x 2 x 3 3

Polar Bears In Exercises 23–25, suppose a scientific team gathered the weights (in pounds) of ten polar bears. The weights are 964, 1002, 1026, 978, 1078, 925, 928, 1005, 964, and 894.

22. 23.

23. Find the mean, median, mode, range, and standard deviation of the

weights. 24. Draw a box-and-whisker plot of the weights.

24.

Use space at left.

25.

Use space at left.

25. Make a frequency distribution using four intervals beginning with

875 929. Then draw a histogram of the data set.

Review and Assess



109

Answer Key 3. 3

4. 6

9x 4 4y8

7.

5. 3

3 8. 4 2

9. 2x; Domain: all real numbers 10. 2; Domain: all real numbers

Interval 55–59 60–64 65–69 70–74 75–79

11. x 2 1; Domain: all real numbers

13. 14. 16. 17.

18.

y

y

1 1

1

Domain: x ≥ 0 Range: y ≤ 1 19.

Domain: x ≥ 0 Range: y ≥ 2 Domain: all real numbers Range: all real numbers

y

1 1 x

20. 2, 3

21. no solution

23.

Boys 70.1 70 70 42 11.7

mean median mode range standard deviation 24. girls team 25.

Boys Basketball Points 50

48

x

x

1

60

70

80

90

65 70

80

90

22. 25

Girls 65.3 65 65 23 6.87

Girls Basketball Score Frequency

12.

x1 ; Domain: all real numbers except 1 x1 x; Domain: all real numbers f x 12 x 2 15. f x x 5; x ≥ 5 f x x3 7

Tally

y 4 3 2 1 0 -5 9 60 -6 4 65 -6 9 70 -7 4 75 -7 9

4 2x 6. 2xy

26. 56

55

Test C 1. 5 2. 9

Interval

x

Frequency 2 2 3 2 1

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2. 2723

4 81 3.

4.

Answers

1

216

13

1. 2.

Simplify the expression. Assume all variables are positive. 5. 312

313

4 32x5y4 6.

27x6 7. 8y12

23

3 3 54 2 8.

Perform the indicated operation and state the domain. Let f x x 1 and gx x 1. 9. f x g(x 12.

10. f x gx

f x gx

11. f x

gx

13. f gx

3. 4. 5. 6. 7. 8. 9.

Find the inverse function. 15. f x x 2 5; x ≥ 0

14. 4x 2y 8 16. f x x 713

Graph the function. Then state the domain and range. 18. f x x12 2

17. f(x) 1 x y

10. 11. 12. 13. 14.

y

15. 1 1

16. 1

x

1

x

17.

Use grid at left.

18.

Use grid at left.

19.

Use grid at left.

3 x 2 1 19. f x

y

Review and Assess

1 1 x

110



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7

Chapter Test C

CONTINUED


Solve the equation. Check for extraneous solutions. 20. 4 x 10 3x 22. 2x 2

13

21. 5 7y 3

6

21. 22.

Basketball In Exercises 23–26, use the tables below which give the points scored in each game played by the boys and girls basketball teams this season. Boys Team 56, 81, 80, 75, 48, 65, 90, 66, 70, 70

20.

23.

Girls Team 60, 72, 61, 58, 78, 65, 66, 55, 65, 73

23. Find the mean, median, mode, range, and standard deviation for

each data set. 24. Interpret the data as to which team is more consistent in their

scoring (use the standard deviation).

24. 25.

Use space at left.

26.

Use space at left.

25. Draw a box-and-whisker plot of the boys points.

26. Make a frequency distribution of the girls points using five intervals

beginning with 55–59. Then draw a histogram of this data.

Review and Assess



111

Answer Key Cumulative Review 1. inverse property of addition 2. commutative property of multiplication 3. associative property of multiplication 4. identity property of multiplication 5. distributive property 6. commutative property of addition 7. 4 8. 5 9. 5, 11 10. 1, 6 11. 6 1 12. 4 13. C 14. B 15. A 16. positive correlation 17. negative correlation 18. minimum: 0, maximum: 12 19. minimum: 1, maximum: 21 20. minimum: 5, maximum: 19 1 1 4 2 21. 22. 2 1 7 2 1 6 8 4 12 23. 24. 3 7 4 8 8 4 8 25. not defined; number of columns of first matrix is not equal to number of rows of second matrix 26. 1 27. 8 28. 19 29. 70 1 1 1 1 1 2 10 10 6 12

30.

3 2

31.

7 2

33. 2x2 12

1 5

45

32.

23

1 3

34. a 22a 22

35. x2 y2x yx y 36. 2a 54a2 10a 25 37. x2 y2x y 38. 3x 2y9x2 6xy 4y2 39.

40.

y

1

1 1

41.

y

x

42.

y

1

x

1

y

1 1

x

1 x

43.

44.

y

y 1

1

1 1

x

45. y 3x 32 1

46. y x 22

47. y 2x 4x 2 48. y x 2x 4 2

49. y 3x2 2x 1

1 1 52. x10 53. x9y12 9x4y6 5 54. x3y5 55. 4 56. 1254 F x 57. Drivers ≥ 35 years 50. x3

51.

x

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NAME _________________________________________________________ DATE ____________

7


Identify the property shown. (1.1)

997

1. 7 7 0

2. 7

4. 91 9

5. 25 3 2

3. 3 5 7 3 5 7

523

6. 4 7 7 4

Solve the equation. (1.3, 1.7) 7. 4x 5 11

8. 1.3x 3.5 10

10. 10 4x 14

11.

2 3x

5 6

1 6x

9. x 3 8 12. 42x 3 14

23 6

Match the equation with the graph. (2.3, 2.8) 1

13. y 3x 1 A.

14. y x B.

y

1 3

15. y x 3 C.

y

y

2 1 1

x

1

x

2

x

1

Draw a scatter plot of the data. Then state whether the data have a positive correlation, a negative correlation, or relatively no correlation. (2.5) 16.

x 1 y 1

1 2

2 3

4 5

5 5

6 7

x 2 1 y 5 4

17.

8 9

1 2

2 2

3 2

3 0

4 5 1 3

Find the minimum and maximum values of the objective function subject to the given constraints. (3.4) 18. Objective function:


20. Objective functions:

C x 3y

C 2x 3y

Constraints:

Constraints:

Constraints:

x ≥ 0 y ≥ 0 xy ≤ 6

x x y y

≤ ≥ ≤ ≥

6 1 5 0

x x y y

Perform the indicated matrix operation. If the operation is not defined, state the reason. (4.1, 4.2) 3 2 4 3 8 3 4 5 22. 21. 4 1 2 0 7 1 0 3

3 2 24. 4

4 1 0

1 1

2 3


25.

2 3

8 4

1 1

0 2

2 1

3 2

≥ ≤ ≤ ≥

23. 4

26.

4 2 5 1

21

12

1 2 1 3

12

3 2


6 0 4

117

Review and Assess

C 2x y

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Cumulative Review

CONTINUED


Evaluate the determinate of the matrix. (4.3) 27.

1 2

2 4

28.

31.

2

2 3 1

0 1 2

1 4 2

29.

32.

4

1 2 2

4 2 1

5 0 4

Find the inverse of the matrix. (4.4) 30.

4 2

3 7

8

1 1

8

1 2

Factor the expression. (5.2) 33. 4x4 4x2 1

34. a4 8a2 16

35. x 4 y 4

36. 8a3 125

37. x 3 x 2y y3 xy 2

38. 27x3 8y3

39. y ≥ x2 3

40. y ≥ x2 x 5

41. y < 3x2 12x 11

42. y < x 22

43. y ≤ x 32 1

44. y ≥ 4x2 3

Graph the inequality. (5.7)

Write a quadratic function in the specified form whose graph has the given characteristics. (5.8) 45. vertex form

vertex: 3, 1 point on graph: 4, 2

48. intercept form

46. vertex form

vertex: 2, 0 point on graph: 3, 1

47. intercept form

x-intercepts: 4, 2 point on graph: 3, 14

49. standard form

points on graph: 0, 1, 3, 11, 3, 1

x-intercepts: 2, 4 point on graph: 1, 3 Simplify the expression. (6.1) 50. x5

1

x2

53. 3x2y32

51. x3y43 54.

x1y2 x4y3

52.

x7 x3

55. 5x4y0

Review and Assess

56. Planet Temperatures Pluto’s surface temperature is believed to be

387F, the lowest temperature observed on a natural body in our solar system. Measurements by the Pioneer probe indicate that Venus’ surface temperature is 867F. What is the difference between the two temperatures? (1.1) 57. Driving For a driver aged x years, a study found that a driver’s reaction

time Vx (in milliseconds) to a visual stimulus such as a traffic light can be modeled by: Vx 0.005x2 0.23x 22, 16 < x < 70. At what age does a driver’s reaction time tend to be greater than 20 milliseconds? (5.7)

118



Answer Key Practice A 1. 1113 2. 514 3. 2315 4. 712 5. 1713 6. 216 7. 814 8. 1513 9. 1012 10. 317 3 4 2 14. 5 11. 615 12. 2118 13. 5 7 4 31 15. 11 16. 6 17. 23 18. 3 3 8 19. 103 20. 17 21. 4 22. 7 5 8 24. 1412 25. 2 26. 3 27. 2 23. 28. 4 29. 1 30. 5 31. 1 32. 2 33. 2 34. 1.71 35. 2.88 36. 1.78 37. 1.32 38. 1.74 39. 1.52 40. 1.43 41. 2.29 42. 1.63 43. 1.32 44. 3.07 45. 2.24 46. 6 in. 47. 8.08 cm.

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LESSON

7.1

NAME _________________________________________________________ DATE ___________


Rewrite the expression using rational exponent notation. 3 11 1.

4 5 2.

5 23 3.

4. 7

3 17 5.

6 2 6.

4 8 7.

3 15 8.

7 3 10.

5 6 11.

8 21 12.

9. 10

Rewrite the expression using radical notation. 14. 514

15. 1112

16. 615

17. 2317

18. 3114

19. 10312

20. 1713

21. 413

22. 718

23. 815

24. 12114

Lesson 7.1

13. 213


4 81 26.

5 32 27.

3 64 28.

4 1 29.

3 125 30.

31. 116

32. 1614

33. 813

Evaluate the expression using a calculator. Round the result to two decimal places. 3 5 34.

3 24 35.

4 10 36.

4 3 37.

5 16 38.

5 8 39.

40. 615

41. 1213

42. 714

43. 415

44. 2913

45. 12616

46.

Geometry Find the length of an edge of the cube shown below.

Volume 216 in.3 47.

Geometry Find the length of an edge of the cube shown below.

Volume 527 cm3



13

Answer Key Practice B 1. 713 2. 523 3. 1152 4. 1253 5. 1573 6. 953 7. 4227 8. 1083 4 3 3 5 43 3 62 12. 19 10. 9. 11. 9 4 3 3 62 15. 104 13. 83 14. 7 143 17. 16 18. 216 19. 8 16. 20. 729 21. 16 22. 4 23. 32 24. 4 25. 32 26. 2.65 27. 3.00 28. 2.61 29. 2.93 30. 2.47 31. 2.21 32. 25.92 33. 4148.54 34. 291,461.63 35. 4.50 cm 36. 12, 12 37. 9.00 38. 16 39. 511.48 cm3 40. 8 cm 41. 556.28 cm3 42. 8.22 cm

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LESSON

7.1

NAME _________________________________________________________ DATE ___________


Rewrite the expression using rational exponent notation. 3 7 1.

5.

2.

3 157

6.

3 52 3 95

3. 115

4.

7.

8.

7 422

6 1210 3 108

Rewrite the expression using radical notation. Lesson 7.1

9. 1913 13. 834

10. 4315

11. 623

12. 943

14. 623

15. 1043

16. 1437

Evaluate the expression without using a calculator. 17. 843

18. 3632

19. 1634

20. 8132

21. 6423

22. 3225

23. 452

24. 6413

25. 853

Evaluate the expression using a calculator. Round the result to two decimal places. 4 49 26.

9 19,422 27.

5 122 28.

29. 21515

30. 1513

31. 11616

32. 13223

33. 2852

34. 11283

35.

Geometry Find the radius of a sphere with a volume of 382 cubic centimeters.

Solve the equation. Round your answer to two decimal places when appropriate. 36. x2 5 139

Water and Ice

37. 5x3 3650

38. x 73 729


Water, in its liquid state, has a density of 0.9971 grams per cubic centimeter. Ice has a density of 0.9168 grams per cubic centimeter. You fill a cubical container with 510 grams of liquid water. A different cubical container is filled with 510 grams of solid water (ice). 39. Find the volume of the container filled with liquid water. 40. Find the length of the edges of the container in Exercise 39. 41. Find the volume of the container filled with ice. 42. Find the length of the edges of the container in Exercise 41.

14



Answer Key Practice C 1 1 1 1. 27 2. 729 3. 9 4. 2 5. 125 6. 100,000 1 1 7. 64 8. 256 9. 8 10. 3596.65 11. 106.17 12. 0.03 13. 0.15 14. 2002.65 15. 6.85 16. 13,593.93 17. 15.00 18. 0.10 19. 2.68 20. 1, 1 21. 1 22. 4.61, 2.39 23. 2.92 24. 0.99 25. 0.67, 1.67 26. 1.41, 1.41 27. 1.33 28. 29. > 30. < 31. n n 32. 1.4 in. 33. a a when a < 0 and n is even.

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LESSON

7.1

NAME _________________________________________________________ DATE ___________


Evaluate the expression without using a calculator. 3 272

1. 8134

2. 24365

3.

4. 1614

5. 2532

6. 10052

7. 25634

8.

9. 3235

3 644

10. 13653

11. 2875

12. 12434

13. 1723

14.

15.

16.

17.

3 3025

3 267 4 373

18.

Lesson 7.1

Evaluate the expression using a calculator. Round the result to two decimal places. 5 1232 5 433

Solve the equation. Round your answer to two decimal places when appropriate. 19. x5 137

20. 3x4 2 5

21. x 35 32

22. 2x 76 120

23. 2x3 50

24. x 17 125

25. 2x 14 10 20

26. 3 x2 1

27. 12 3x 23 20

Place the appropriate sign , or = between the two expressions. 15 23 3215 28. 32 29. 8 30. 1635 32.

31. 2723

1635

823 2723

Volume A cylindrical can of chicken broth holds 14.5 ounces of broth. One fluid ounce is approximately 1.8 cubic inches. What is the radius of a can that is 4.5 inches tall? (Hint: Use the formula V r2h for the volume of a cylinder.)

33.

Critical Thinking Use the following examples to determine when n an a.

3 23 a.


b. 22

3 23 c.

d. 22


15

Answer Key Practice A 2. 614 3. 523

1. 453 5. 1016 10.

1 5

3

7. 21 12. 32

4.

1 1354

6 5 8.

9. 2

13. x34

15. x110 16. 3 x12 17. 3x13

1 x72

19. x

23. 27 26. 3

218 318

10

11.

14. x83 18.

6.

323

1 x2

5 22 24. 15

3 x

27.

xy 2 29. yy y x

32. 6x2x

20.

33.

4 x 2

21.

10 x12

25. 5x 28. 7x x

30. yz2x3y y2xz x2

22. 63

3 yz 31. 3x

34. 22 in.

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LESSON

7.2

NAME _________________________________________________________ DATE ___________


Simplify the expression using the properties of rational exponents. 1. 413

443

4. 1354

2. 63413 5.

323

3. 5

1056 1046

6.

2 3

18

Simplify the expression using the properties of radicals. 7. 7 10.

3

251

8.

3 5

11.

5 3

20

9. 5 12.

6

30

10

Simplify the expression. Assume all variables are positive. 13. x14

x24

16. 3x12 19.

15. x3516

17. 27x13

18. x72

20.

x12 x52

21.

100x

12

Lesson 7.2

x53 x23

14. x234

Perform the indicated operation. Assume all variables are positive. 22. 23 43

23. 57 37

5 5 22 9 22 24. 6

25. 2x 7x

3 3 x 2 x 26. 5

4 x 6 4 x 27. 8

Write the expression in simplest form. Assume all variables are positive.

xy

2

28. 49x3

29.

3 x3yz 3 8x3yz 31.

32. x5 5xx3

34.

4 x3y5z8 30.

3

Geometry The area of an equilateral triangle is given by A

33.

x2y4z x5

3

s 2. 4 Find the length of the side s of an equilateral triangle with an area of 12 square inches.



25

Answer Key Practice B 1. 52 25 2. 312 3. 753 4. 1214 4 3 5. 8 2 6. 2 7. 323 8. 5 9. 1013 x12 3 2 x 12. x 13. 10. 3x 11. 14. 2x14 2 15. 3x

16. 2x

19. x3 5

20. x3

22. x43 3 3 25. 3

29.

3 5 3

4 x 18. 4y

17. x23

1 43

x

21. 42x2 23.

4 15 26.

30. 2

5

1 2

x

24. x6

27. 8213

28. 22

3 31. diameter 5.88 1017 miles; thickness 5.88 1016 miles 32. 12.5 in. 33. 2.5 in. 34. 0.2

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LESSON

7.2

NAME _________________________________________________________ DATE ___________


Simplify the expression using the properties of radicals and rational exponents. 312 3

1. 523

543

2.

4. 314

414

3 2 5.

7.

3 3

8.

3

3. 72352

64 125

3 4

6.

13

4 240 4 15

9. 101223

Simplify the expression. Assume all variables are positive. 10. 9x2 13.

Lesson 7.2

16.

4x

12

12x2 3

x13

3 2x3 11.

12. x23

14. 16x14

5 27x 15.

17.

1 x213

5 9x4

4 256xy4 18.

20. x3 3

21. 4x2

23. x2

24. 3x6 2x6

3 3 3 3 25. 2

4 15 2 4 15 26. 3

27. 3213 5213

28. 42 8

3 40 3 5 29.

5 5 3 96 4 30.

3 19. x

22.

x5

x3 x53

Perform the indicated operation.

31.

Milky Way The Milky Way is 105 light years in diameter and 104 light years in thickness. One light year is equivalent to 5.88 1012 miles. What is the diameter and thickness of the Milky Way in miles?

Archery Target In Exercises 32–34, use the following information. The figure at the right shows a National Field Archer’s Association official hunter’s target. The area of the entire hunter’s target is approximately 490.9 square inches. The area of the center white circle is approximately 19.6 square inches. 32. Find the radius of the target. 33. Find the radius of the center white circle. 34. Find the ratio of the radius of the white circle to the radius

of the target.

26



Answer Key Practice C 1. 5 334 2. 21324 5. 10. 14.

1 8

6.

x14 y13 4x4

x 18. 10

24

2

11.

7. 2

4. 6110

8. 4 9.

9y13z2 2z18 12. 13 54 32 4x x y

4 x 15.

19.

3. 5415

3 2 2 y y

4 6 16.

1 10 10 10 13. 3x12

3 17. 5x2 y2z

5 y 20.

21. 15x2yy 22. 2.7 1012 meters 23. 5.3 1012 meters

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LESSON

NAME _________________________________________________________ DATE ___________

7.2


Simplify the expressions using the properties of radicals and rational exponents.

52332

1. 312 4.

66

7.

2. 213

12 35

23412

123 12

5.

13

108

8.

27

3. 523152

3

6.

12

3 18 9.

2 2

7

2 3

4 3 2

2 4

3

5

Simplify the expression. Assume all the variables are positive. 10. 13.

x54 y23 xy

3xy12 27x2y12

11.

1

3x14 y23z 2xy12

2

12.

x43y5 16z12

3 4 x 15.

5 2x232x27 14.

14

3 x

Perform the indicated operations. Assume all variables are positive.

46 36

3 3 8x6y2z x 27x3y2z 17.

19.

5y y9

5 y 10 y2 3 15 y3 20.

3

2

18.

5 x

Lesson 7.2

16.

5 32x

2 3 2 21. xy3x y3xy 6x yx54y

Halley’s Comet In Exercises 22 and 23, use the following information. Halley’s Comet travels in an elliptical orbit around the sun, making one complete orbit every 76 years. When the comet was closest to the sun 8.9 1010 meters), it developed its tail. In the diagram at the right, a is the length of the semi-major axis, A is the comet’s closest distance to the sun, and B is the comet’s farthest distance from the sun.

Halley's Comet A Sun

a B Not drawn to scale.

22. The length of the semi-major axis a can be found by the

equation a

GMT 2 4 2

13

where

G gravitational constant 6.67 1011 N m2kg2 M mass of sun 1.99 1030 kg T period 2.4 109 seconds (76 years). Find the length of the semi-major axis. 23. The comet’s farthest distance from the sun can be calculated by

B 2a A. What’s the comet’s farthest distance from the sun?



27

Answer Key Practice A 1. 3x 1 2. x2 2x 2 3. 2x2 2x 2 4. 7x12 5. x 3 6. x 2 7. x2 x 2 8. x32 9. 6x 3 10. 3x2 x 2 3x 11. 2x3 2x2 2x 12. 6x 13. x2 x2 x2 1 x16 14. 15. 2 16. x2 x x4 2 2 17. 2x 10 18. 4x 9 19. x 2x 3 20. x320 21. All real numbers 22. All real numbers 23. All real numbers 24. All real numbers except x 3 25. All real numbers 26. All real numbers 27. All real numbers 28. All real numbers except x 0 29. All real numbers 30. Px 0.75x 20,000; $730,000

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LESSON

NAME _________________________________________________________ DATE ___________

7.3


Find f x g x. Simplify your answer. 1. f x 4x, g x 1 x

2. f x 2x 3, g x x2 1

3. f x x2 3, g x x2 2x 1

4. f x x12, g x 6x12

Find f x g x. Simplify your answer. 5. f x 2x, g x x 3

6. f x x2 x, g x x2 2

7. f x x 1, g x x2 2x 3

8. f x 3x32, g x 4x32

Find f x g x. Simplify your answer. 9. f x 2x 1, g x 3

10. f x x 1, g x 3x 2

11. f x x2 x 1, g x 2x

Find

12. f x 2x23, g x 3x13

f x . Simplify your answer. g x

13. f x 3x, g x x 2

14. f x x2 1, g x x 2

2 15. f x x 2, g x x x 4

12 13 16. f x 2x , g x 22 x

Find f g x. Simplify your answer. 17. f x 2x, g x x 5

18. f x x, g x 4x 9

19. f x x 2, g x x 1

20. f x x15, g x x34

2

Lesson 7.3

Let f x x2 and g x x 3. Find the domain of the following functions. 21. f x g x 24.

f x g x

27. g x f x

30.

40

22. f x g x

23. f x g x

25. f g x

26. g f x

28.

g x f x

29. f f x

Profit A company estimates that its cost and revenue can be modeled by the functions C x 0.75x 20,000 and R x 1.50x where x is the number of units produced. The company’s profit, P, is modeled by P x R x C x. Find the profit equation and determine the profit when 1,000,000 units are produced.



Answer Key Practice B 1. 3x3 x2 12x 2; 3x3 3x2 2x 2. 7x23; x23 3. 2x3 x2 2x 3; 5 3 2x3 x2 8x 5 4. 8 x34; 8 x34 5. x3 x2 4x 2 6. x6 3x4 3x3 2x2 9x 6 3x2 x 1 8 7. 4x712 8. 8x12 12 9. x3 x 3x 5 314 10. 11. 2x53 12. 2 2x 1 x 13. f g x 6x 3, g f x 6x 1 14. f g x x2 4x 5, g f x x2 1 15. f g x x 412, g f x x12 4 16. f g x 3x25, g f x 3 x25 17. 4x12 x 3; nonnegative real numbers 18. x 3 4x12; nonnegative real numbers 19. 4x32 12x12; nonnegative real numbers

x3 ; positive real numbers 4x12 21. 4x 312; real numbers greater than or equal to 3. 22. 4x12 3; nonnegative real numbers 23. f x x 100 24. g x 0.75x 25. g f x 0.75x 75 26. f gx 0.75x 100 27. Discount 20.

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LESSON

7.3

NAME _________________________________________________________ DATE ___________


Find f x g x and f x g x. Simplify your answers. 1. f x 3x3 2x2 5x 1, gx x2 7x 1 2. f x 4x23, gx 3x23 3. f x 2x3 3x 4, gx x2 5x 1

4. f x 2 x 34, gx 8 x 34 1

1

Find f x g x. Simplify your answer. 5. f x x2 2x 2, gx x 1

6. f x x 4 3x 2, gx x2 3

7. f x 2x14, gx 2x13

8. f x 4x1, gx 2x12

Find


9. f x 3x2 x 1, gx x 3 11. f x 6x73, gx 3x23

10. f x 3x 5, gx 2x2 1 12. f x 3x14, gx x54

Find f g x and g f x. Simplify your answers. 13. f x 3x, gx 2x 1

14. f x x2 1, gx x 2

15. f x x12, gx x 4

16. f x 3x 45, gx x12

Let f x 4x1/2 and g x x 3. Perform the given operation and state the domain. 17. f x gx 20.

gx f x

gx

18. gx f x

19. f x

21. f gx

22. g f x

Lesson 7.3

Furniture Sale In Exercises 23–27, use the following information. You have a coupon for $100 off the price of a sofa. When you arrive at the store, you find that the sofas are on sale for 25% off. Let x represent the original price of the sofa. 23. Use function notation to describe your cost, f x, using only the coupon. 24. Use function notation to describe your cost, gx, with only the 25% dis-

count. 25. Form the composition of the functions f and g that represents your cost, if

you use the coupon first, then take the 25% discount. 26. Form the composition of the functions f and g that represents your cost if

you use the discount first, then use the coupon. 27. Would you pay less for the sofa if you used the coupon first or took the

25% discount first?



41

Answer Key Practice C 1. x5 x3 3x2 6x 9; x5 3x3 3x2 2x 7 2. 10x25 2x1; 2x25 8x1 3. x2 10x 1; x2 4x 3 4. 5x16 7x3 1; x16 3x3 1 5. x5 4x4 x3 15x2 16x 30 6. 5x58 5x14 3x38 3 16 7. 3x x13 8. 73 x 2 9. f g x x 412; g f x x 4 1 3 10. f g x ; g f x 2 1 3x 12 x 11. f g x 2x34; g f x 2x34 3 23 12. f g x 12; g f x 12 2x x 1 13. f g x ; all real numbers less x2 2x than 2 or greater than 0 1 2x12 ; positive real numbers x 1 f x 15. ; positive real numbers g x x52 2x32 g x 16. x52 2x32; nonnegative real numbers f x 17. f f x x14; nonnegative real numbers 18. g g x x4 4x3 6x2 4x; all real numbers 19. True 20. False; Examples vary. 21. True 22. False; Examples vary. 23. False; Examples vary. 24. False; Examples vary. 25. Sample answer: f x x, g x 2x 1 1 26. Sample answer: f x , g x 3x 2 x 14. g f x

27. Let f x 0.6x, g x x 5, h x 0.9x

f g h x 0.54x 3 f h g x 0.54x 2.7 g f h x 0.54x 5 g h f x 0.54x 5 h f g x 0.54x 2.7 h g f x 0.54x 4.5; First the store will deduct the $5 coupon. Then it makes no difference in what order they take the 40% and 10% discount.

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LESSON

7.3

NAME _________________________________________________________ DATE ___________


Find f x g x and f x g x. Simplify your answers. 1. f x x3 3x2 2x 1, gx x5 2x3 4x 8 2. f x 6x25 3x1, gx 4x25 5x1 3. f x x2 3x 1, gx 7x 2 4. f x 3x16 2x3 1, gx 2x16 5x3

Find f x g x. Simplify your answer. 5. f x x3 2x2 x 5, gx x2 2x 6

Find

6. f x 5x14 3, gx x38 1


7. f x 3x23 1, gx x13

8. f x 16x13, gx x2

Find f g x and g f x. Simplify your answers. 9. f x 3 x12, gx x2 1

10. f x x2, gx 3x 1

11. f x x34, gx 2x

12. f x 3x1, gx 2x12

Let f x x 1/2 and g x x2 2x. Perform the operation and state the domain. 13. f gx

Lesson 7.3

16.

gx f x

f x gx

14. g f x

15.

17. f f x

18. ggx

Critical Thinking State whether or not the following statements are always true. If they are false, give an example. 19. f x gx gx f x 21. f x

gx gx f x

23. f gx g f x

Function Composition

20. f x gx gx f x 22.

f x gx gx f x

24. f f x f x 2

Find functions f and g such that hx) f g x.

25. hx 2x 1

26. hx

1 3x 2

27. Holiday Sale

A department store is holding its annual end-of-year sale. Feature items are marked 40% off. In addition, a flyer was sent to the newspapers which included a coupon for $5 off any purchase. Also, if you open a charge account with the store, you can receive an additional 10% discount. There are six different ways in which these price reductions can be composed. Find all six compositions. Which of the six compositions is the store most likely to use?

42




x 3 y 2

5 1

7 0

9 1

11 2

2.

x y

2 1

4 2

1 3

0 4

6. B

7. A

3. no

1 0

4. yes 5. yes

8. C

9–12. Show f g x x and g f x x. 13. i

C ; 16.54 in. 2.54

14. r

C ; 4.46 in. 2

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LESSON

NAME _________________________________________________________ DATE ___________

7.4


Find the inverse relation. 1.

x 2 y 3

1 5

0 7

1 9

2.

2 11

x y

0 1

1 2

2 4

3 1

4 0

Use the horizontal line test to determine whether the inverse of f is a function. 3.

4.

y

5.

y

y

1

1 2

1

x

1

x

x

1

Match the graph with the graph of its inverse. 6.

7.

y

8.

y

y

2 1

1 x

1

A.

1

B.

y

1

x

C.

y

y

1

1 2

x

x

1

1

x

1

x

Verify that f and g are inverse functions. 9. f x x 5, gx x 5 Lesson 7.4

5 x 11. f x x5, gx

13.

10. f x 6x, gx 6 x 1

12. f x 2x 1, gx 2 x 1

1 2

Metric Conversions The formula to convert inches to centimeters is C 2.54i. Write the inverse function, which converts centimeters to inches. How many inches is 42 centimeters? Round your answer to two decimal places.

14.

Geometry The formula C 2r gives the circumference of a circle of radius r. Write the inverse function, which gives the radius of a circle of circumference C. What is the radius of a circle with a circumference of 28 inches? Round your answer to two decimal places.

54




2.

x 6 y 1 x

3 2

0 3

3 4

6 5

2

4

6

0

0

1 2

3

1

23

y 1 3. yes

4. no

5. no

6. no

7. yes

8. yes

9–16. Show f g x x and g f x x. 1

1

17. y 4x 18. y x 5 19. y 3x 1 9 20. y 4x 4 21. y 2x 12

12x 23. y x 3, y x 3 24. y x 1, y x 1 25. y x2, x ≥ 0 22. y

3 2

26.

27.

y

1

x 1

28.

y

1 1

x

29. C K 273.15,

y

1 1

x

21.85 C S 30. R , $26.51 0.75

1 3

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LESSON

NAME _________________________________________________________ DATE ___________

7.4


Find the inverse of the relation. 1.

x 1 y 6

2 3

3 0

4 3

2.

5 6

x 1 y 1

23 2

0 4

1 2

3 0

6

Use the horizontal line test to determine whether the inverse of f is a function. 3. f x 3x 5

4. f x 2x2 3

5. f x 1 x2

6. f x x

7. f x 3 2x

8. f x 2 x 4

1

Verify that f and g are inverse functions. 9. f x 2x, gx

10. f x 1 x, gx 1 x

x 2

11. f x x 2, gx x 2

12. f x 3x 6, gx 3 x 2

13. f x 2 x 4, gx 2x 8

14. f x 4x 1, gx 4 x

15. f x x2, x ‡ 0, gx x

3 x 16. f x x3, gx

1

1

1

1 4

Find an equation for the inverse of the relation. 17. y 4x

18. y x 5

20. y 4x 9

21. y

1 2x

23. y

24. y

x2

x2

3

19. y 3x 1

6

22. y 3 2x

1

25. y x

Sketch the inverse of f on the coordinate system. 26.

27.

y

28.

y

y

1 1

1 1

x

1 1

x

x

29. Temperature Conversion

The formula to convert temperatures from degrees Celsius to Kelvins is K C 273.15. Write the inverse of the function, which converts temperatures from Kelvins to degrees Celsius. Then find the Celsius temperature that is equal to 295 Kelvins.

A gift shop is having a storewide 25% off sale. The sale price S of an item that has a regular price of R is S R 0.25R. Write the inverse of the function. Then find the regular price of an item that you got for $19.88.


Lesson 7.4

30. Sale Price


55

Answer Key Practice C 1–6. Show f g x x and g f x x. 1 1 1 8 1 1 7. f x 4 4x 8. f x 3x 3 9. f 1 x x2 1, x ≥ 0 1 3 10. f1x 2x2 2, x ≥ 0 11. f1x 4 x2, x ≥ 0 1 3 12. f1x 5x3 5 13. f1x x 7; x ≥ 7 14. f1x

x 2 5 3

15. f1x x , x ≥ 0 16. Restrictions on the domain must be made in

inverse functions of all functions where n is even. 1 1 1 1 1 17. No. f1x x and ⇒ x 3 f x 3x 3 3x 1 3 3 3 3 g1x x and ⇒ x 2 g x 2x 2 2x 18.

y

1 1

x

19. f f x f

1x 11 x x

20. yes; f gx g fx x 21. no 23. yes

22. yes

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LESSON

NAME _________________________________________________________ DATE ___________

7.4


Verify that f and g are inverse functions. 1. f x 2x 7, gx 2 x 1

2. f x 5x 3, gx

7 2

3. f x x 4, gx x2 4, x ‡ 0

1 3 2x 4. f x 2 x3, gx)

1 4 3x 6 5. f x 3 x 4 2 ‡ 0, gx

6. f x 4 x4 2, gx 3

3 5

15 x

4 108x 216

3

Find the inverse function. 7. f x 1 4x

8. f x 3x 8

9. f x x 1

10. f x 2x 3

11. f x 4 x

3 5x 3 12. f x

13. f x x2 7, x ‡ 0

14. f x 2x3 5

15. f x x , x † 0

Consider the basic power function f x x n for n 1, 2, 3, 4, and 5. Make a conclusion about the values of n for which a restriction on the function’s domain must be made to ensure that the inverse of f is a function.



Consider the following pairs of inverse functions:

f x 3x

and

f 1x 13 x

gx 23 x

and

g1x 32 x

Does f 1x

1 ? Explain. f x

1 Visual Thinking In Exercises 18–20, consider the function f(x) , x which is its own inverse. 18. Sketch the graph of f x to verify that it is its own inverse. 19. Verify that f x is its own inverse by showing f f x x. 20. If g(x) af(x) where a is a nonzero constant, is it true that g(x) is its own

inverse? Explain. Use the horizontal line test to determine whether the inverse of the function is a function. 21. f x

x 2, x 1,

x < 0 x ‡0

22. f x

x 2, x 3,

Lesson 7.4

y

x < 0 x ‡0

x2, x,

y

x < 0 x ‡0

y

1

1 1

x

1

1 1

56

23. f x


x

x


Answer Key Practice A 1. F 2. C 3. B 4. E 5. A 6. D 7. Shift the graph 3 units up. 8. Shift the graph 2 units down. 9. Reflect the graph across the xaxis. 10. Shift the graph 1 unit left. 11. Shift the graph 4 units right. 12. Stretch the graph vertically by a factor of 2. 13. Shift the graph 3 units down. 14. Shift the graph 2 units up. 15. Shift the graph 7 units left. 16. Shift the graph 5 units right. 17. Shrink the graph vertically by a factor of 12. 18. Reflect the graph across the x-axis. 19.

20.

y

y

1 x

1 1 x

1

x ≥ 0, y ≥ 4 21.

x ≥ 0, y ≥ 3 22.

y

y

2 1 1

x

x ≥ 2, y ≥ 0 23.

x

1

x ≥ 3, y ≥ 0 24.

y

y 2

1 1

x

1

x

x, y are all real x, y are all real numbers. numbers. 25. Domain: 0 ≤ h ≤ 100, Range: 0 ≤ t ≤ 2.5 t 26. 27. 36 ft

1 20

h

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Lesson 7.5

LESSON

NAME _________________________________________________________ DATE ___________

7.5


Match the function with its graph. 3 x 2 1. f x

3 x 2 2. f x

3 x 2 3. f x

4. f x x 1

5. f x x 1

6. f x x 1

A.

B.

C.

y

y

y 1

1 1

D.

x

1

1

x

E.

y

x

1

F.

y

y

2 2 1

1 1

x

x

1

x

Describe how to obtain the graph of g from the graph of f x x. 7. gx x 3

8. gx x 2

10. gx x 1

11. gx x 4

9. gx x 12. gx 2x

3 x. Describe how to obtain the graph of g from the graph of f x 3 x 3 13. gx

3 x 2 14. gx

3 x 7 15. gx

3 x 5 16. gx

17. gx

3 x 18. gx

1 3 2 x

Graph the function. Then state the domain and range. 19. f x x 4

20. f x x 3

21. f x x 2

22. f x x 3

3 x 1 23. f x

3 x 2 24. f x

Falling Object In Exercises 25–27, use the following information. A stone is dropped from a height of 100 feet. The time it takes for the stone to 1 reach a height of h feet is given by the function t 4100 h where t is time in seconds. 25. Identify the domain and range of the function. 26. Sketch the graph of the function. 27. What is the height of the stone after 2 seconds?

68



Answer Key Practice B 1. E 2. B 3. F 4. A 5. C 6. D 7. Shift the graph 4 units left and 3 units up. 8. Shift the graph 4 units left and 2 units down. 9. Shift the graph 4 units left and reflect it across the x-axis. 10. Shift the graph 4 units right and 3 units down. 11. Shift the graph 4 units right and 2 units up. 12. Shift the graph 4 units right, reflect across the x-axis, and shift 2 units up. 13. Reflect the graph across the x-axis and shift 1 unit down. 14. Reflect the graph across the x-axis and shift 1 unit up. 15. Shift the graph 1 unit right and 5 units up. 16. Shift the graph 1 unit left and 5 units up. 17. Shift the graph 1 unit left and 2 units down. 18. Shift the graph 1 unit right and 2 units down. 19.

20.

y

y 1 x

1 1 x

1

x ≥ 3, y ≥ 2 21.

x ≥ 1, y ≥ 3 22.

y

y

1 1

x 1 x

1

x ≥ 1, y ≤ 3 23.

x, y are all real numbers. 24.

y 1

x 1

x, y are all real numbers.

y 1 1


x

25. Domain: t ≥ 273, Range: v ≥ 0 26.

27. 11.58 C

y

250 75

x

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LESSON

NAME _________________________________________________________ DATE ___________

7.5

Lesson 7.5


Match the function with its graph. 1. f x x 1 1

2. f x x 1 1

3. f x x 1 1

3 x21 4. f x

3 x21 5. f x

3 x21 6. f x

A.

B.

C.

y

y

y

1

1 1

1

x

D.

E.

y

F.

y

1

1 x

1

x

1 x

1

y

1 2

x

1

x

Describe how to obtain the graph of g from the graph of f x x. 7. gx x 4 3

8. gx x 4 2

10. gx x 4 3

11. gx x 4 2

9. gx x 4 12. gx x 4 2

3 x. Describe how to obtain the graph of g from the graph of f x 3 x 1 13. gx

3 x 1 14. gx

3 x 1 5 15. gx

3 x 1 5 16. gx

3 x 1 2 17. gx

3 x 1 2 18. gx

Graph the function. Then state the domain and range. 19. f x x 3 2

20. f x x 1 3

21. f x x 1 3

3 x 1 3 22. f x

3 x 4 2 23. f x

3 x 1 3 24. f x

Speed of Sound


The speed of sound in feet per second through air of any temperature measured in Celsius is given by V

1087273 t , 16.52

where t is the temperature. 25. Identify the domain and range of the function. 26. Sketch the graph of the function. 27. What is the temperature of the air if the speed of sound is 1110 feet per

second? Copyright © McDougal Littell Inc. All rights reserved.


69

Answer Key Practice C 1. B 2. A 3. C 4.

14.

y

1

5.

y

15.

y

1

y

1

x

2

x

2 1

1

x


x

1

x ≥ 3, y ≥ 0 6.

x ≥ 4, y ≥ 0 7.

y

x, y are all real numbers. 1, 1 and 0, 0

16.

y

1 1

x

1 x

1

x ≥ 1, y ≥ 4 8.

x ≥ 1, y ≤ 3 9.

y

17. On the interval 0, 1, the larger the root, the

steeper the graph. On the interval 1, , the larger the root the less steep the graph.

18.

19.

y

y

1

1 1

1

x

x

1 x

1

x ≥ 1, y ≤ 2 10.

x ≥ 12, y ≥ 2 11.

y

y

1

1 1

x

1

1, 1, 0, 0, and 1, 1 20. On the interval 1, 1, the larger the root, the steeper the graph. On the intervals , 1 and 1, , the larger the root, the less steep the y graph. 21.

x 1 1



y

y

1

1 x

x 1

Life expectancy (years)

22.

1


1994

f(t) 80 75 70 65 (0.3, 62.7) 60 0


x

0 10 20 30 40 50 60 t Years since 1940

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Lesson 7.5

LESSON

NAME _________________________________________________________ DATE ___________

7.5


Match the function with its graph. 1. f x x 2 1

2. f x x 2 1

3. f x x 2 1

A.

B.

C.

y

1

y

y

1 x

1

1 1

x

x

2

Sketch the graph of the function. Then state the domain and range. 4. f x 2x 3

1 5. f x 2x 4

6. f x x 1 4

7. f x

8. f x x 1 2

9. f x

23x

13

x 12 2

3 x 1 10. f x 4

3 3 x2 11. f x 2

3 x 3 2 12. f x

3 x 3 1 13. f x

3 x 1 2 14. f x 5

3 x 1 15. f x

4

1 3

Visual Thinking In Exercises 16–18, use the following information. 4 x, hx 6 x, 8 x Graph the functions f x x, gx and jx on the same coordinate plane. Use the window xmin 1, xmax 2, xscl 1, ymin 1, ymax 2, and Yscl 1.

16. What two points do all of the graphs have in common? 17. Describe how the graphs are related. 18. Using what you have learned in Exercises 16 and 17, sketch the graph of 4 x 3 2. f x 3 x, gx 5 x, h x 7 x, Visual Thinking Graph the functions f x 9 and jx x on the same coordinate plane. Use the window xmin 2,

xmax 2, xscl 1, ymin 2, ymax 2, and Yscl 1.

19. What three points do all of the graphs have in common? 20. Describe how the graphs are related. 21. Using what you have learned in Exercises 19 and 20, sketch the graph of 5 x 2 1. f x

22.

Life Expectancy From 1940 through 1996 in the United States, the age to which a newborn can expect to live can be modeled by f t 1.78t 0.3 62.7, where t is the number of years since 1940. Graph the model. In what year was the life expectancy at birth 75.7 years?

70



Answer Key Practice A 1. yes 2. yes 3. no 4. yes 5. no 6. yes 7. 16 8. 64 9. 64 10. 8 11. 4 12. 125 1 13. 16 14. 8 15. 81 16. 1 17. 27 4 18. 27 19. 3 20. no solution 21. 3 22. 10 23. 3 24. 2 25. 0.81 ft 26. 0.20 ft 27. 3.24 ft 28. 100 ft 29. 36 ft 30. 225 ft

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LESSON

NAME _________________________________________________________ DATE ___________

7.6


Check whether the given x-value is a solution of the equation. 1. x 5 8; x 169

2. 2x 1 2 5; x 5

3. x 4 10; x 25

4. 1 x 3 5; x 3

5. 2x 5 12; x 20

3 x 1 3 2; x 0 6.

Lesson 7.6

Solve the equation. Check for extraneous solutions. 7. x14 2

8. x23 16

10. x13 2 0

11. x32 4 12

9. x12 8 12. 4x23 100

Solve the equation. Check for extraneous solutions. 13. x

1 4

5 x34 16.

3 x 2 14.

4 3 x 27 15.

3 x6 17. 2

3 x 15 18. 5

Solve the equation. Check for extraneous solutions. 19. x 3 6

20. 2x 1 x

21. 3x 2 2

3 x 5 3 5 22.

4 3x 1 4 2x 2 23.

3 5x 6 3 4 24.

Pendulums In Exercises 25–27, use the following information. The period of a pendulum is the time T (in seconds) it takes for a pendulum of length L (in feet) to go through one cycle. The period is given by T 2

32L .

Given the period of a pendulum, find its length. Round your answers to two decimal places. 25. T 1 second

26. T 0.5 second

27. T 2 seconds

Velocity of a Free-Falling Object In Exercises 28–30, use the following information. The velocity of a free-falling object is given by V 2gh, where V is velocity (in feet per second), g is acceleration due to gravity (in feet per second) and h is the distance (in feet) the object has fallen. On Earth g 32 fts2. How far did an object fall if it hits the ground with the given velocity? 28. 80 ft/s

82


29. 48 ft/s

30. 120 ft/s


Answer Key Practice B 1. 8 2. 16 3. 8 4. 9 5. 8 6. 8 7. 12 5 1 8. 1 9. 3 10. 3 11. 3 12. no solution 3 7 13. 2 14. 12 15. 3 16. 4 17. 0 18. 2 7 19. 2 20. no solution 21. 5 22. 2, 1 23. 6 24. 3 25. 3, 4 26. 8 27. 5 28. 2.25 29. 3.24 30. 6.5 31. 3.85 32. 6.75 33. 1.10 34. 9.77 ft 35. 10.78 ft 36. 34,722.22 ft

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LESSON

7.6

NAME _________________________________________________________ DATE ___________


Solve the equation. Check for extraneous solutions. 1. x 43 5 11

2. 2x34 7 23

3. 2x34 8

4. x 123 4

5. 2x 132 54

6. 2x53 64

7. 2x 313 5 2

8. 2x 115 2 3

9. 3x 412 3 0

Solve the equation. Check for extraneous solutions. 11. 3x 6 5 14

12. 5x 1 8 2

3 2x 1 2 4 13.

3 5x 4 1 3 14.

3 3x 1 5 3 15.

5 3x43 16.

3 1 3x 4 6 17. 2

18. 5 2x 1 3

Lesson 7.6

4 3x 5 6 10.

Solve the equation. Check for extraneous solutions. 3 3 2x 1 8 19.

20. 3x 1 x 5

4 4 2x 1 x6 21.

22. x 2 x 2

23. 2x 3 x 3

24. 12x 13 2x 1

25. 3x 13 x 5

26. 2x x 4

27. 2x 4 1 x

Use the Intersect feature on a graphing calculator to solve the equation. 28.

2 12 3x

1

31. 1.3x 11 4

29. 6x 335 18

30. 2x 513 2

3 43 5x 2.1 32.

33. 2x 323 3

Velocity of a Free-Falling Object In Exercises 34–36, use the following information. The velocity of a free-falling object is given by V 2gh where h is the distance (in feet) the object has fallen and g is acceleration due to gravity (in feet per second squared). The value of g depends on your altitude. If an object hits the ground with a velocity of 25 feet per second, from what height was it dropped in each of the following situations? 34. You are standing on the earth, so g 32 fts2. 35. You are on the space shuttle, so g 29 fts2. 36. You are on the moon, so g 0.009 fts2.



83

Answer Key Practice C 17 5 1. 65 2. 516 3. 2 4. 3 5. no solution 6751 6. 3 7. 18 8. no solution 9. 4 10. 78, 78 11. 10, 10 12. 27, 27 13. 6 14. no solution 15. 7, 8 16. 3, 5 17. 1 18. 3, 6, –6 169 19. 5 20. 1, 2, 2 21. 4 22. 64 16 23. no solution 24. no solution 25. 45 26. 0 27. 1, 3 28. 4 in. 29. 24 in.

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LESSON

7.6

NAME _________________________________________________________ DATE ___________


Solve the equation. Check for extraneous solutions. 1. 3x 123 4 52 4.

1 2 3x

134 3 1

2. 2x 413 7 9 5.

1 3 2x

332 2 7

3. 2x 323 5 1 6.

1 3 2x

9.

1 3 5

332 2 7

Solve the equation. Check for extraneous solutions. 7. 32 x 5 1 7 Lesson 7.6

1

10. x2 3 5 4

8. 4 3x 1 5 11. 2x2 1 4 10

2x 12 3 6

3 1 x2 1 8 12. 3


23 x 2x 13

5 5 3x 7 2x 1 13.

14.

15. x 7 x 7

16. 3x2 12x 10 2x 5

4 2x2 1 x 17.

3 9x 19 x 1 18.

3 2x2 14 x 1 19.

5 4x3 x2 4 x 20.

21. x 3 x 5

Solve the equation. Check for extraneous solutions. 22. x 3 4 x

23. x 5 2 x

24. x 5 2 x

25. 5x 1 3 5x

26. 2x 1 1 2x

27. 2x 3 1 x 1

28. Geometry

The lateral surface area of a cone is given by h2. The surface area of the base of the cone is S given by B r2. The total surface area of a cone of radius 3 inches is 24 square inches. What is the height of the cone?

rr 2

h

3 in.

29. Geometry

A container is to be made in the shape of a cylinder with a conical top. The lateral surface areas of the cylinder and cone are S1 2rh and S2 2rr 2 h2. The surface area of the base of the container is B r 2. The height of the cylinder and cone are equal. The radius of the container is 5 inches and its total surface area is 275 square inches. Find the total height of the container.

h

h 5 in.

84



Answer Key Practice A 1. 1, 1, 1, 2, 2, 3, 4, 5, 5, 6, 7, 7, 8, 8, 9; 4.6; 5; 1 2. 8, 10, 10, 10, 12, 12, 13, 15, 16, 19; 12.5; 12; 10 3. 24 4. 48 5. 3 6. 7 7. 149 8. 21 9. 7.5, 14 10. 136, 154.5 11. 1, 4.5 12. 35, 42 13. 20

30

50

40

28 32 35 40

60 54

14. 100

15.

120

140

160

120

140

160

Interval 1–2 3–4 5–6 7–8 9–10

16.

Interval 1–2 3–4 5–6

Tally

Tally

7–8 9–10 17. Exercise 16

180

200

185 200

Frequency 7 4 4 7 3 Frequency 8 3 3 0

3

18. Exercise 15

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Page 97

LESSON

NAME _________________________________________________________ DATE ___________

7.7


Write the numbers in the data set in ascending order. Then find the mean, median, and mode of the data set. 1.

2, 3, 7, 1, 8, 7, 4, 5, 1, 8, 2, 6, 5, 9, 1

2.

10, 15, 8, 19, 12, 13, 10, 16, 12, 10

Find the range of the data set.

18, 24, 37, 29, 13, 22, 25, 30 5. 3, 2, 1, 2, 3, 3, 1, 4 7. 2, 7, 150, 125, 3, 2, 1, 20 3.

123, 100, 132, 112, 148, 129, 138, 118 6. 105, 110, 104, 109, 110, 111, 108, 106 8. 88, 72, 84, 71, 73, 85, 90, 92 4.

Find the lower and upper quartiles of the data set.

5, 10, 7, 13, 12, 8, 15, 20, 10 11. 0, 3, 2, 4, 1, 6, 3, 5, 1 9.

153, 146, 128, 144, 156, 120, 148, 160 12. 38, 43, 32, 33, 37, 41, 44, 40, 38 10.

Use the given information to draw a box-and-whisker plot of the data set. 14. minimum 120

maximum 54 median 35 lower quartile 32 upper quartile 40

Lesson 7.7

13. minimum 28

maximum 200 median 160 lower quartile 140 upper quartile 185

Use the given intervals to make a frequency distribution of the data set. 15. Use five intervals beginning with 1–2.

1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 5, 5, 6, 7, 7, 8, 8, 8, 8, 8, 9, 9, 10 16. Use five intervals beginning with 1–2.

1, 1, 2, 2, 1, 3, 2, 10, 4, 1, 6, 5, 3, 1, 9, 10, 6


9– 10

5– 6 7– 8

9– 10

0 7– 8

0 5– 6

6 4 2

10 8 6 4 2 3– 4

18.

10 8

3– 4

17.

1– 2

Match the histograms with the data sets from Exercise 15 and Exercise 16.

1– 2

MCRB2-0707-PA.qxd


97

Answer Key Practice B 1. 11.4; 9; 8 2. 20.4; 22; 22 3. 49.5; 48; 44 4. 127.2;130; 100 5. 47, 16.7 6. 15.5, 5.01 7. 4.5 8. 25.2 oz 9. 25 oz 10. 28 oz 11. 6 12. 5; 7 13. 0 0

1

2

3

4

5

6

7

5

6

7

8

9

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LESSON

7.7

NAME _________________________________________________________ DATE ___________


Find the mean, median, and mode of the data set.

6, 22, 4, 15, 10, 8, 8, 7, 14, 20 3. 53, 51, 47, 44, 60, 48, 44, 55, 44

10, 15, 12, 20, 25, 22, 28, 24, 22, 26 4. 100, 150, 100, 120, 130, 125, 135, 140, 145

1.

2.

Find the range and standard deviation of the data set. 5.

47, 18, 65, 28, 43, 18

6.

35.8, 29.4, 32.1, 24.9, 30.5, 20.3

7. Reading Levels

The Pledge of Allegiance contains 31 words. The bar graph at the right shows the number of words of different lengths in the pledge. Find the mean word length of the set of 31 words.

Word Lengths in the Pledge of Allegiance

Frequency

12 10 8 6 4 2 0

1

2 3 4

5 6

7

8

9 10 11

Lesson 7.7

Word Length (number of letters)

Walking Shoes


An important feature of walking shoes is their weight. The graph below shows the weight of the top-10 rated men’s walking shoes. 8. Find the mean of the ten weights. 9. Find the median of the ten weights. 10. Find the mode of the ten weights.

Ranking 1 2 3 4 5

Weight 24 oz 22 oz 26 oz 28 oz 24 oz

Ranking 6 7 8 9 10

Weight 28 oz 22 oz 28 oz 22 oz 28 oz

World Series In Exercises 11–13, use the following information. The World Series is a best-of-seven playoff between the National League champion and the American League champion. The table shows the number of games played in each World Series for 1981 through 1998. Year 1981 1982 Games 6 7

1983 5

1984 1985 5 7

1986 1987 7 7

1988 5

1989 4

Year 1990 1991 Games 4 7

1992 6

1993 1994 6 0

1995 1996 6 6

1997 7

1998 4

11. Find the median of the number of games played. 12. Find the lower and upper quartiles of the number of games played. 13. Construct a box-and-whisker-plot of the number of games played.

98



Answer Key Practice C 1. 3; 1.05 2. 0.6; 0.208 3. 3; 0.957 4. 20; 6.54 5. b 6. a. 6.7, 2 b. 23.3, 25.5; The median is a more accurate measure of central tendency when a small number of data is much different than the majority of the data. 7. 30 8. 107.4 9. 8.052 10. Machine #1: 1.0008; Machine #2: 0.9993 11. Machine #1: 0.00098; Machine #2: 0.0009 12. Machine #2 13.

Children of U.S. Presidents Interval 0–2

Tally

3–5 6–8 9–11 12–14

Children of U.S. Presidents 20 16 12 8 4 4 –1

12

8

11 9–

5

6–

3–

2

0 0–

Number of Presidents

14.

Number of children

Frequency 16 16 7 1 1 15. median

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LESSON

7.7

NAME _________________________________________________________ DATE ___________


Find the range and the standard deviation of the data set.

1, 4, 3, 2, 1, 2, 1, 3, 1 3. 105, 106, 104, 105, 107, 106

6.5, 7.1, 6.8, 6.6, 6.8, 7.0 4. 20, 18, 36, 16, 16, 17, 21

1.

2.

5. Test Scores

The bar graphs below represent three collections of test scores. Which collection has the smallest standard deviation?

A.

B. 9 8 7 6 5 4 3 2 1 0

60 65 70 75 80 85 90 95

C. 9 8 7 6 5 4 3 2 1 0

9 8 7 6 5 4 3 2 1 0

60 65 70 75 80 85 90 95

60 65 70 75 80 85 90 95

6. Critical Thinking a.

1, 1, 2, 3, 3, 2, 1, 50, 1, 3

b.

Lesson 7.7

Find the mean and median of the following data sets. When is the median a more accurate measure of central tendency?

20, 25, 30, 24, 26, 1, 28, 25, 26, 28

Breakfast Cereals In Exercises 7–9, use the following information. The number of calories in a 1-ounce serving of ten popular breakfast cereals is 116, 113, 104, 110, 119, 101, 106, 110, 106, 89. 7. Find the range of this data.

8. Find the mean of this data.

9. Find the standard deviation of this data. Round to three decimal places.

Manufacturing Couplers In Exercises 10–12, use the following information. A company that manufactures hydraulic couplers takes ten samples from one machine and ten samples from another machine. The diameter of each sample is measured with a micrometer caliper. The company’s goal is to produce couplers that have a diameter of exactly 1 inch. The results of the measurements are shown below. Machine #1: 1.000, 1.002, 1.001, 1.000, 1.002, 0.999, 1.000, 1.002, 1.001, 1.001 Machine #2: 0.998, 0.999, 0.999, 1.000, 0.998, 0.999, 1.000, 1.000, 1.001, 0.999 10. Find the mean diameter for each machine.

11. Find the standard deviation for each machine.

12. Which machine produces the more consistent diameter?

History In Exercises 13–15, use the following information. The table at the right gives the number of Number of Children of U.S. Presidents children of the Presidents of the United States. 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 7, 8, 10, 14 13. Make a frequency distribution of the data set using five intervals beginning with 0–2. 14. Draw a histogram of the data set. 15. Based on the histogram, which is the better measure of central tendency, the mean or median? Copyright © McDougal Littell Inc. All rights reserved.


99

Answer Key Test A Graph 1–6 on graph paper 1.

2.

y

y

2 1 1

x

Domain: all real numbers Range: y > 0 3.

1

x


y

y

1

1

x


x

1

1

Domain: x > 0 Range: all real numbers 6.

y

y

1 1

x 1 1

x

Domain: x > 1 Domain: Range: all real numbers all real numbers Range: y > 0 3 7. e5 8. 9. 4 10. 3 11. 3 12. 1 e 13. 2 14. 0 15. 1 16. 4 17. 5 18. 3 19. 4 8 is extraneous 20. exponential growth 21. y 5x 22. 1.398 23. 1.079 24. log 7x3 25. ln 3 ln x ln y 26. 1.431 1 27. y 4 2x 28. y 2x1 2 29. y 20,000.90t; $18,000 30. $1127.50

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CHAPTER

NAME _________________________________________________________ DATE ____________

8


Graph the function. State the domain and range.

Answers

2. y 2x1 1

1. y 2x y

1.

Use grid at left.

2.

Use grid at left.

3.

Use grid at left.

4.

Use grid at left.

5.

Use grid at left.

6.

Use grid at left.

y

1

1 x

1

x

1

1

3. y 3 ex

4. y log x y

y

1 x

1

1 x

1

5. y ln x 1

7.

6. y 2ex

8.

y

y

9. 10.

1 x

1

11.

1 x

1

12. 13. 14.

Simplify the expression. 7. e3e2

8. 3ee2

10. log3 27

11.

e4 e3

9. log 10,000

3 e

16. 17.

13. log12 4

14. log3 1

15. ln

Review and Assess

Evaluate the expression without using a calculator. 12. log2 0.5

15.

18.

e1

19.

Solve the equation. Check for extraneous solutions. 16. 103x5 10x3

17. log32x 1 2

18. log54x 1 log52x 7

19. log2 y 4 log2 y 5



119

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CHAPTER

NAME _________________________________________________________ DATE ____________

8

Chapter Test A

CONTINUED


20. Tell whether the function f x 42 represents exponential 3 x

growth or exponential decay.

21.

21. Find the inverse of the function y log5 x.

Use log 5 ≈ 0.699 and log 12 ≈ 1.079 to approximate the value of the expression. 22. log 25

20.

22. 23. 24.

1

23. log 12

24. Condense the expression 3 log x log 7.

25. 26.

25. Expand the expression ln 3xy. 26. Use the change-of-base formula to evaluate the expression log5 10. 27. Find an exponential function of the form y abx whose graph

passes through the points 2, 1 and 3, 2.

28. Find a power function of the form y axb whose graph passes

through the points 4, 4 and 16, 8.

27. 28. 29. 30.

29. Car Depreciation

The value of a new car purchased for $20,000 decreases by 10% per year. Write an exponential decay model for the value of the car. Use the model to estimate the value after one year. You deposit $1000 in an account that pays 6% annual interest compounded continuously. Find the balance at the end of 2 years.

Review and Assess

30. Earning Interest

120



Answer Key Test B Graph 1–6 on graph paper 1.

2.

y

y

1 2

x

1


x

1


y

y

2 1 x

1


x

1


y

y

1 2

x

2

1

x

Domain: Domain: all real numbers all real numbers Range: y > 3 Range: 0 < y < 4 8 7. e 8. 9. 2 10. 5 11. e7 e 12. 4 13. 4 14. 0 15. 3 16. 16 17. 1 18. ln 5 19. 6 2 is extraneous 20. exponential growth 21. y 7x 7 22. 6.644 23. 0.903 24. log b 25. ln 5 ln x ln 2 26. 3.169 1 x 27. y 1002 28. y 2.583x0.6309 29. y 18,000.88t; $13,939 30. $1161.83

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Page 121

CHAPTER

NAME _________________________________________________________ DATE ____________

8


Graph the function. State the domain and range.

Answers

2. y 2 log x

1. y 3x y

2.

Use grid at left.

3.

Use grid at left.

4.

Use grid at left.

5.

Use grid at left.

6.

Use grid at left.

x

1

1 1

x

2

3. y 3x1 1

4. y 3 ex

y

y

1

1 1

5. y

Use grid at left.

y

1

3x

1.

x

x

1

7.

4 6. y 1 2ex

3 y

8. 9.

y

10. 1

11. 1

x 1

12. x

1

13. 14.

Simplify the expression. 7. e2e1

8. 2e4ee3

1

9. log 100

16.

e3 e2 e 11. e1

10. log5 3125

17.

13. log12 16

14. log12 1

15. ln e3

Review and Assess


15.

18. 19.

Solve the equation. Check for extraneous solutions. 16. log4 x 2

17. 104x1 1000

18. 2ex 1 9

19. 2 log5 x log52 log52x 6



121

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CHAPTER

NAME _________________________________________________________ DATE ____________

8

Chapter Test B

CONTINUED

For use after Chapter 8 1

20. Tell whether the function y 4 e2x represents exponential growth or

exponential decay.

21.

21. Find the inverse of the function y log7 x.

Use log210 ¯ 3.322 and log 8 ¯ 0.903 to approximate the value of the expression. 1

23. log 8

22. log2 100

24. Condense the expression log 7 log b. 25. Expand the expression ln

20.

5x . 2

26. Use the change-of-base formula to evaluate the expression log2 9. 27. Find an exponential function of the form y abx whose graph

passes through the points 1, 50 and 2, 25.

b 28. Find a power function of the form y ax whose graph passes

22. 23. 24. 25. 26. 27. 28. 29. 30.



The value of a new car purchased for $18,000 decreases by 12% per year. Write an exponential model for the value of the car. Use the model to estimate the value after two years. You deposit 1000 in an account that pays 5% annual interest compounded continuously. Find the balance at the end of 3 years.

Review and Assess

30. Earning interest

122



Answer Key Test C Graph 1–6 on graph paper 1.

2.

y

y

1 2

1

x


x

1


y

y

2 1 1

x


1

x


y

1

y

10 1

x

2

x

Domain: x > 0 Domain: Range: all real numbers all real numbers Range: y > 0 7. e 4 8. 9e 2 9. 3 10. 5 11. 2 12. 2 13. 3 14. 0 15. 2 16. 625 17. 2, 2 18. 8; 4 is extraneous 19. 2 20. exponential decay 21. y 8x 22. 4.428 23. 1.176 24. log424 25. ln 2 ln y ln x 26. 2.481 27. y 3x 28. y 3.227x0.631 29. y 28,000.92t; $18,454 30. $1053.22

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Page 123

CHAPTER

NAME _________________________________________________________ DATE ____________

8


Graph the function. State the domain and range. 1. y

3 x 2

Answers

2. y log4 x y

1.

Use grid at left.

2.

Use grid at left.

3.

Use grid at left.

4.

Use grid at left.

5.

Use grid at left.

6.

Use grid at left.

y

1 x

1

1 1

x

1 3. y 2

x1

4. y ex y

y

1

1 1

x

x

1

7. 5. y lnx 2

50 6. y 1 125ex

8.

y

9.

y

10. 11. 1

10 1

12.

x

x

2

13. 14.

Simplify the expression. 7. ee3

8. 3e2

10. log2 32

4e4 11. 5 e

15. 1

9. log 1000

e 2

16. 17. Review and Assess


13. log12 8

14. log2 1

15. ln e2

Solve the equation. Check for extraneous solutions. 16. log5 x 4


2

17. 10x

1

100,000


123

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CHAPTER

NAME _________________________________________________________ DATE ____________

8

Chapter Test C

CONTINUED


18. 2 log3 y log3 4 log3 y 8

18.

19. ln3x 1 lnx 5 0

19.

20. Tell whether the function f x 32 represents exponential 1 2

growth or exponential decay.

21. Find the inverse of the function y log8 x. 1 Use log8100 2.214 and log 15 ≈ 1.176 to approximate the value of the expression.

22. log810,000

23. log 15

24. Condense the expression log43 3 log42. 25. Expand the expression ln

2y . x

26. Use the change-of-base formula to evaluate the expression log7125. 27. Find the exponential function of the form y abx whose graph 1 passes through the points 3, 27 and 0, 1.

28. Find a power function of the form y axb whose graph passes

20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.



The value of a new car purchased for $28,000 decreases 8% per year. Write an exponential decay model for the value of the car. Use the model to estimate the value after 5 years. 1 You deposit $800 in an account that pays 52 % annual interest compounded continuously. Find the balance at the end of 5 years.

Review and Assess

30. Earning Interest

124



Answer Key Cumulative Review 13 7 1 1. 4 2. 4 3. 3 4. 20 7.

5.

13 14

6. 2.25

8. y

y

1

1

x

x

1

1

9.

10. y

y

1

x

1

1

11.

1

x

1

x

12. y

y

1 x

1

1

9

13. y 5 x 3

15. y 4x 18. y 5 21. 0, 5

23 5 5 4

2

14. y 3 x 60 16. y x 1

17. x 2

20. 2, 1

19. 1, 5

22. 2, 3

23. 4, 1

25. 2, 1

26. 5, 3

28. 6, 5

29. 12, 2

30.

31. y x2 7x 12

24. 2, 0

27. 2, 0

12, 13

32. y 2x2 13x 15 33. y 2x2 12x 19 34. y 9x2 18x 4 35. y 5x2 20x 22 36. y 2x2 6x 5 37. 4, 1 41. 3, 3

2

38. 3

39. 0, 2

42. 4, 1

40. 11, 8

43. 33

14 25 48. 5 8 49. 3 4i 50. 45 11i 51. 5 12i 16 8 3 52. 5 5 i 53. 2 2i 54. 1 i 5 ± i3 55. 2 ± 2 56. 57. 3 ± 17 2 5 1 2 ± 6 3 ± 15 58. 59. 60. , 2 3 2 2 61. 0; one real solution 62. 1; two real solutions 63. 23; two imaginary solutions 64. 160; two real solutions 65. 0; one real 1 solution 66. 81; two real solutions 67. 25 9 1 1 68. 216 69. 729 70. 27 71. 729 72. 4 5 73. x 2 74. x 2 75. 2x 1 3x 2 4 3x 1 76. x2 x 3 77. x 3 2 2x 1 x 3 5 78. x2 2x 2 79. 25 80. 32 x 5 17 81. 3 82. 125 83. 11 84. 8, 2 85. $109,556.16 86. 1.9 years 87. $1176.43

45. 52

44. 102

46.

1 3

47.

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CHAPTER

NAME _________________________________________________________ DATE ____________

8


Solve the equation. (1.3) 2. x 5 4 32 x

1. 3a 5 7a 8 4.

2 3 a

2 6 18

5.

1 4x

1 5

3 5x

3. 10b 5 5b

1 8

6. x 0.05 2.3

Use slope-intercept form to graph the equation. (2.3) 7. y 3x 1

8. y 3 x 2

1

9. y 4x

3 4

10. 4x 2y 8

11. 3x 6y 18

12. y 2 x

5

2 3

Write an equation of the line from the given information. (2.3, 2.4) 13. The line passes through 2, 1 and 7, 8. 14. The line has a slope of

2 3

and a y-intercept of 60.

15. The line passes through 1, 2 and is perpendicular to the line y 3 x 3. 4

16. The line passes through 3, 4 and is parallel to the line that passes through 3, 8 and 5, 10. 17. The line passes through 2, 6 and is parallel to x 8. 18. The line passes through 3, 5 and is perpendicular to x 10.

Graph the linear system and estimate the solution. Then check the solution algebraically. (3.1) 19. 4x 2y 14

20. x 3y 5

3x 5y 22 22. 3x 2y 12

2x 2y 6 23. 3x 5y 7

2x y 1

2x y 7

21. 5x 2y 10

4x 3y 15 24. 5x 3y 10

4x 8y 8

Use an inverse matrix to solve the linear system. (4.5) 25. 4x 2y 10

26. x y 8

3x y 7 28. 2x 3y 27

2x 8y 14 29. x 5y 2

3x y 23

2x 6y 12

27. 4x 2y 8

8x 4y 16 30. 4x 9y 5

6x 6y 5

Review and Assess

Write the quadratic function in standard form. (5.1) 31. y x 3x 4

32. y x 52x 3

33. y 2x 32 1

34. y 3x 22 6x

35. y 5x 22 2

36. y 2 2x 32 1

1 2

Solve the quadratic equation. (5.2) 37. x2 5x 4 0

38. 9a2 12a 4 0

40. a2 19a 88 0

41. 5a2 16 4a2 7

39. 30x2 60x 0

42 3x2 5x 7 2x2 2x 11

Simplify the expression. (5.3) 43. 27 130


44. 250

45. 5

10


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CHAPTER

NAME _________________________________________________________ DATE ____________

8

Cumulative Review

CONTINUED


Simplify the expression. (5.3) 46.

19

47.

2 5

48.

7 32

Write the expression as a complex number in standard form. (5.4) 49. i4 3i 52.

8 2i

50. 6 i7 3i 53.

4 3i 2i

51. 3 2i2 54.

7 3i 2 5i

Solve the equation by completing the square. (5.5) 55. x2 4x 2

56. x2 5x 7 0

57. u2 2u 4u 8

58. 2x2 4x 1

59. 3x2 6x 2 0

60. 4r 2 9r r 5

Find the discriminant of the quadratic equation and give the number and type of solutions of the equation. (5.6) 61. x2 4x 4 0

62. 2x2 5x 3 0

63. 3x2 x 2 0

64. 4x2 10 0

65. 64x2 16x 1 0

66. 4x2 9x 0

Evaluate the expression. (6.1) 67. 54 70.

13

52 3

68.

62 65

71. 93

69. 332

90

72.

23

2

Divide using polynomial long division. (6.5) 73. x2 6x 8 x 4

74. 2x2 3x 2 2x 1

75. 6x2 x 3 3x 2

76. 2x3 x2 7x 7 2x 1

77. x3 3x2 6x 8 x2 3

78. x 4 2x3 5x2 10x 5 x2 5

Solve the equation. Check for extraneous solutions. (7.6) 79. x32 125

80. x15 2 0

81. 23x 112 8

82. x 25 10

83. x 8 x 2

84. x 5 20x 9

85. Land Value

Review and Assess

You purchased land for $50,000 in 1980. The value of the land increased by approximately 4% per year. What is the approximate value of the land in the year 2000? (8.1)

86. Depreciation

You buy a new car for $21,000. It depreciates by 10.5% each year. Estimate when the car will have a value of $17,000. (8.2)

87. Continuous Compounding

You deposit $850 in an account that pays 6.5% annual interest compounded continuously. What is the balance after 5 years? (8.3)



131

Answer Key Practice A 1. F 2. B 3. A 4. E 5. C 6. D 7. Shift graph of f 2 units up. 8. Shift graph of f 5 units down. 9. Shift graph of f 1 unit left. 10. Shift graph of f 3 units right. 11. Reflect graph of f across x-axis. 12. Shift graph of f 2 units up. 13. 1; x-axis 14. 1; x-axis 1 15. 2; x-axis 16. 2; x-axis 17. 2; x-axis 1 18. 4; x-axis 19. a. $2100 b. $2101.89 c. $2102.32

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LESSON

NAME _________________________________________________________ DATE ___________

8.1


Match the function with its graph. 1. f x 3x

2. f x 3x

1 2

3. f x 23x

1 2

4. f x 3x

5. f x 3x

6. f x 23x

A.

B.

C.

y

y

(1, 6)

Lesson 8.1

y

1

1 x

1

x 1

(0, ) (1, ) 1 2

(0, 1)

3 2

(1, 3)

2

(0, 2) 1

D.

x

E.

y 1

F.

y

y

x

(0, 2)

2

(1, 3)

(0, ) 1 2

(1, ) 3 2

1

(1, 6)

1

1

(0, 1)

x

1

x

Explain how the graph of g can be obtained from the graph of f. 7. f x

4 3

x

g x

x

4 3

8. f x 2x

2

10. f x 5x

g x 5x3

9. f x

g x

53

g x 2x 5 11. f x 2x

5 3

x

x1

12. f x 32x

g x 2x

g x 32x 2

Identify the y-intercept and asymptote of the graph of the function. 13. y 3x

1 2

16. y 4x

14. y

65

x

17. y 24x

15. y 24x

1 4

18. y 4x

19. Account Balance

You deposit $2000 in an account that earns 5% annual interest. Find the balance after 1 year if the interest is compounded with the given frequency. a. annually

14


b. quarterly

c. monthly


Answer Key Practice B 1. B 2. A 3. C 4. E 5. F 6. D 7. Shift the graph of f 1 unit right and 2 units up. 8. Shift the graph of f 2 units left and reflect across the x-axis. 9. Shift the graph of f 2 units left and 4 units down. 10. 3; y 2 1 11. 27; x-axis 12. 1; y 2 13.

14.

y

1

y

1 1 x

15.

x

1

16.

y

y 1 1

x

1

x

1 x

1

17.

18.

y

y 1

1 1 x

19.

20.

y 1

y

2 x 2

1

22. 25.2; 1.15; 15%

y

23. 2

1

24. 88.7

x

Number of computers (per thousand people)

21.

x

C 45 40 35 30 25 0 0 1 2 3 4 t Years since 1991

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LESSON

NAME _________________________________________________________ DATE ___________

8.1


Match the function with its graph. 1. f x

4 3

x

2. f x

2

4. f x 3x1 A.

y

x1

1

6. f x 3x1 2

B.

C.

y

y

(1, 2 ) 3 4

(0, )

(0, 3)

1

(2, 0)

x

1

E.

y

(0, ) 7 9

1

1

D.

x2

5. f x 3x1

1 33

1

4 3

3. f x

2

x

1

x

F.

y

y 1

1

(1, 1)

1

(0, 1)

x

(0, ) 1 3

x

1

(0, ) 1 3

(2, 3)

(2, 3)

1 1

x

Explain how the graph of g can be obtained from the graph of f. 7. f x

g x

1 2

1 2

x

8. f x 10x

x1

2

9. f x 3

g x 10x2

x

g x 3x2 4

Identify the y-intercept and the asymptote of the graph of the function. 10. y 3x 2

11. y 3x3

12. y 3x1 2

13. y 4x2

14. y 2x3

15. y 3x 1

16. y 2x 3

17. y 3x1 2

18. y 2x1 3

19. y 2x1 4

20. y 3x2 1

21. y

Graph the function.

Computer Usage

3 2

x2

1


From 1991 through 1995, the number of computers C per 100 people worldwide can be modeled by C 25.21.15t where t is the number of years since 1991. 22. Identify the initial amount, the growth factor, and the annual percent increase. 23. Graph the function. 24. Estimate the number of computers per 1000 people worldwide in 2000. Copyright © McDougal Littell Inc. All rights reserved.


15

Lesson 8.1

(1, 3)

4 3

Answer Key Practice C 1 1. 1; y 2 2. 5; x-axis 3. 7; y 4 1 4. 75; x-axis 5. 6; y 7 6. 3.2 105; x-axis 7. domain: all real numbers; range: y > 3 8. domain: all real numbers; range: y > 2 9. domain: all real numbers; range: y > 4 10. domain: all real numbers; range: y > 2 11. domain: all real numbers; range: y > 4 12. domain: all real numbers; range: y < 3 13.

14.

y

22.

All three graphs have a y-intercept of 1. The larger a is, the steeper the graph.

y

y 3x y 2x 2

y

3 x 2

()

x

1

23. a.

b.

y

y 3x

y 2x

2

y

x

1

2

y

y 2x y 3x

3

1

x

2

c. x

1

1

y

x

y

4 x 3

()

y

4 x 3

()

2

15.

16.

y

y 1

x

Reflection across the y-axis 24. a. $1077.80 b. $1077.88 c. $1077.88

1 1

x

1

27.

18.

y

y

30,000 25,000 20,000 15,000 0

1

1

19.

20.

y

1

21.

x

1

x

y 1

x

y

1 1

0 1 2 3 4 5 6 7 8 9 t Years since 1990

1

x

1

26. C 15,0001.072t

C

x Cost of tuition (dollars)

1

17.

25. yes; $1077.88

x 1

28. 1994 29. $60,254.15

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LESSON

8.1

NAME _________________________________________________________ DATE ___________


Identify the y-intercept and asymptote of the graph of the function. 1. y 5x 2

1 3

4. y 5x2

2. y 5x1

3. y 35x 4

5. y 3x 7

6. y 5x6

1 2

Lesson 8.1

State the domain and range of the functions. 7. y 8x1 3

8. y 5x3 2

10. y 7x5 2

11. y 32x 4

9. y 6x1 4 12. y 23x1 3

Graph the function.

13. y 4x2 1

14. y

16. y 32x1 4

17. y 3

19. y 2x32

1 3

3 2

x3

15. y 22x1 4

2

3x2 1 2

18. y 3x12 1

1 2

21. y 32x13

20. y 2x1 5

1 2

3 x on 2 the same coordinate plane. Explain how the value of a in the equation y ax affects the graph. Assume that a > 0.

22. Visual Thinking

Sketch the graphs of y 2x, y 3x, and y

23. Visual Thinking

Sketch the following pairs of graphs in the same coordinate plane. Assuming a > 0, explain the difference between y ax and y ax. a. y 2x

b. y 3x

y 2x

c. y

y

y 3x

4 3

4 3

x

x

24. Account Balance

You deposit $1000 in an account that earns 2.5% annual interest. Find the balance after 3 years if this interest is compounded with the given frequency. a. monthly

b. daily

c. hourly

25. Use your results from Exercise 24 to determine if there is a limit to how

much you can earn. If there is a limit, what is the maximum amount? College Tuition


In 1990, the tuition at a private college was $15,000. During the next 9 years, tuition increased by about 7.2% each year. 26. Write a model giving the cost C of tuition at the college t years after 1990. 27. Graph the model. 28. Estimate the year when the tuition was $20,000. 29. Estimate the tuition in 2010. 16



Answer Key Practice A 1. exponential decay 2. exponential growth 3. exponential growth 4. exponential decay 5. exponential decay 6. exponential growth 7. A 8. E 9. D 10. F 11. C 12. B 13. 1; x-axis 14. 1; x-axis 15. 2; x-axis 1 2 16. 4; x-axis 17. 5; x-axis 18. 3; x-axis 19. 8.78 grams

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LESSON

NAME _________________________________________________________ DATE ___________

8.2


Tell whether the function represents exponential growth or exponential decay.

2 3

1. f x

x

4. f x 0.7x

2. f x

5 4

x

5. f x

1 3

x

8. y

x

3. f x 6x

1 2

6. f x 3x

Match the function with its graph. 7. y

10. y

1 1 2 3

1 3

x

x

A.

11. y 2

1 3

1 3

B.

y

9. y 2

1 3

x

12. y

1 1 2 3

C.

y 1 2

3 x

1 6

1 3

(1, ) 2 3

(0, 2)

(1, )

(1, ) x

1

D.

x

E.

y

F.

y

y

1

(0, 1) (0, 2)

(1, ) 1 3

(1, ) 2 3

1 1

x

2

2

(1, ) 1 6

(0, ) 1 2

x

1

x

Identify the y-intercept and asymptote of the graph of the function. 13. y

16. y

1 8 4 9

2 3

x

14. y 0.3x x

17. y 5

12

15. y 2 x

18. y

1 3

x

2 1 3 5

x

19. Radioactive Decay

Ten grams of Carbon 14 is stored in a container. The amount C (in grams) of Carbon 14 present after t years can be modeled by C 100.99987t. How much Carbon 14 is present after 1000 years?



29

Lesson 8.2

(0, 1)

1 2

(0, )

x

y

1 3

x

Answer Key Practice B 1. exponential decay 3. exponential decay 7. F 8. D 9. B 10.

2. exponential growth 4. E 5. A 11.

y

6. C y

1 x

1

1 x

1

12.

13.

y

y

1

1 x

1

14.

1 x

15.

y

1

1 x

1

17. Value of a dollar

16. $1.14

1

V 1.24 1.18 1.12 1.06 1.00 .94 0

18. 1995

y

0 1 2 3 4 5 6 7 8 9 t Years since 1990

x

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LESSON

NAME _________________________________________________________ DATE ___________

8.2


Tell whether the function represents exponential growth or exponential decay.

1 5 2 7

1. f x

x

1 7 3 5

2. f x

x

3. f x 34x


1 5

7. y 2

x

5. y

1 5

x

3

A.

1 5

8. y 3

x2

1 5

x2

B.

y

6. y

1

y

1 5

x3

9. y

C.

y

1 5

x

(0, 125)

1 2

x

(0, 1)

Lesson 8.2

(2, 1)

( )

1 0,

1

D.

1 25

25

E.

y

(3, 1)

(1, 5)

x

x

1

F. (1, 13)

y

y

(1, 5)

(2, 2) 1 1

(0,

22 25

x 1

)

(0, 5)

(0, 1) 1

2

x

1

x

Graph the function. 10. y 2

x

13. y 2

x3

1 2 1 5

3

Value of the Dollar

11. y 2 14. y

1 3

3 5

x

4

x1

3

12. y 3 15. y

1 4

1 2

x1

x2

1


From 1990 through 1998, the value of the dollar has been shrinking. That is, you cannot buy as much with a dollar today as you could in 1990. The shrinking value can be modeled by V 1.240.973t, where t is the number of years since 1990. 16. How much was a 1998 dollar worth in 1993? 17. Graph the model. 18. Estimate the year in which the 1998 dollar was worth $1.07.

30



Answer Key

16.

17.

y

24.

27.

y 250,000

x

1

x

1 x

18.

19.

y

y

1 1 x

1

20.

21.

y

y

2

x

1

1 x

1

22.

23.

y

1

x 1

y

1 1

x

x

26. $131,932.98

28. 10 years (0, 250,000)

200,000 150,000 100,000 (5, 131,932.98) 50,000 0

1

1

1

1

y

1

25. y 250,0000.88t

y

Value (dollars)

Practice C 1. exponential decay 2. exponential growth 3. exponential decay 4. exponential growth 5. exponential growth 6. exponential decay 8 3 7. 4; y 3 8. 27; x-axis 9. 16; x-axis 10. domain: all real numbers; range: y > 3 11. domain: all real numbers; range: y > 4 12. domain: all real numbers; range: y > 1 13. domain: all real numbers; range: y > 2 14. domain: all real numbers; range: y > 7 15. domain: all real numbers; range: y < 4

29. y

0 1 2 3 4 5 6 7 8 9 t Years since purchase

830, 0 ≤ t ≤ t14 8300.87 , t > 14

1 4

30. $747.68

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LESSON

NAME _________________________________________________________ DATE ___________

8.2


Tell whether the function represents exponential growth or exponential decay. 1. f x

x

4. f x

x

2 3 2 3

2. f x

3 2

5. f x

x

3. f x

3 2

x

3 2

6. f x

x

2 3

x

Identify the y-intercept and asymptote of the graph of the function. 7. y

1 2

x

3

8. y

2 3

x3

9. y

1 3 4 4

12. y

25

x1

State the domain and range of the function. 10. y

12

x1

13. y

35

x3

13

x2

3

11. y

2

14. y 3x 7

4

x4

1

15. y 23x 4 Lesson 8.2

Graph the function. 16. y

12

x1

19. y 3 22. y

12

23

3 x1

x32

17. y

13

x1

34

2

20. y 2

1 4

23. y 2

Equipment Depreciation

x1

13

18. y 2

2

21. y

2

x1

3

12

12

x3

x13

24. y 3

23

1 2

x12

4 3


A tool and die business purchases a piece of equipment for $250,000. The value of the equipment depreciates at a rate of 12% each year. 25. Write an exponential decay model for the value of the equipment. 26. What is the value of the equipment after 5 years? 27. Graph the model. 28. Use the model to estimate when the equipment will have a value of

$70,000. Stereo System


You purchase a stereo system for $830. After a 3 month trial period, the value of the stereo system decreases 13% each year. 29. Write an exponential decay model for the value of the stereo system in

terms of the number of years since the purchase. 30. What was the value of the system after 1 year?



31

Answer Key Practice A 1. 54.598 2. 0.368 3. 1096.633 4. 1 5. 0.135 6. 1.948 7. 0.607 8. 9.974 9. exponential growth 10. exponential decay 11. exponential growth 12. exponential growth 13. exponential decay 14. exponential decay 1 15. e8 16. e6 17. e10 18. e3 19. e5 5 e 15 20. 8e 21. A 22. C 23. B 24. $829.79 25. 273,544

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LESSON

NAME _________________________________________________________ DATE ___________

8.3


Use a calculator to evaluate the expression. Round the result to three decimal places. 1. e4

2. e1

3. e7

4. e0

5. e2

6. e23

7. e12

8. e2.3

Tell whether the function is an example of exponential growth or exponential decay. 9. f x e x 12. f x

1 2

ex

10. f x ex

11. f x 2e x

13. f x e2x

14. f x e13x

Simplify the expression. 15. e3 18.

e5

16. e2

e8 e5

19.

e8

17. e25

e3 e2

20. 2e53

Match the function with its graph. 21. f x 2e x 1 A.

22. f x 2e x1 B.

y

23. f x e2x C.

y

y

3 1

1 1

1

x

1

x

Lesson 8.3

24.

x

Continuous Compounding You deposit $725 in an account that pays 4.5% annual interest compounded continuously. What is the balance after 3 years?

25.

Population The population P of a city can be modeled by P 250,000e0.01t where t is the number of years since 1990. What was the population in 1999?



41

Answer Key Practice B 1. 148.413 2. 0.717 3. 0.247 4. 4.113 5. exponential growth 6. exponential decay 7. exponential decay 8. exponential growth 9. exponential decay 10. exponential growth 1 2 11. 8 12. 3e4 13. 14. 16e6 15. 12e3 e e 1 16. 2e2x3 17. 8e2x 18. e 19. x e 20.

21.

22.

23.

x f(x)

2 0.27

1.5 0.45

1 0.74

x f(x)

1 5.44

1.5 8.96

2 14.78

x f(x)

2 14.78

1.5 8.96

1 5.44

x f(x)

1 0.74

1.5 0.45

2 0.27

x f(x)

2 3.02

1.5 3.05

1 3.14

x f(x)

1 10.39

1.5 23.09

2 57.60

x 2 f(x) 401.43

1.5 88.02

1 18.09

x f(x)

1.5 1.99

2 2.00

1 1.95

24.

25.

y

0 2

0 2

0 4

0 1

y

2 1 1

x

y0

1

x

1

x

y0

26.

27.

y

y

2 1 1

y2

x

y1

28.

29.

y

y

1

1 1

x

y 1 y 3 30. $1972.34 31. $1978.47 32. Continuous compounding

2

x

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Page 42

LESSON

NAME _________________________________________________________ DATE ___________

8.3


Use a calculator to evaluate the expression. Round the result to three decimal places. 2. e13

1. e5

3. e1.4

4. e2

Tell whether the function is an example of exponential growth or exponential decay. 5. f x 2e3x

6. f x e3x

8. f x

9. f x

1 5

e5x

1 2

7. f x 2e3x 10. f x 4e5x

ex

Simplify the expression. 3e5 e

11. e42

12.

14. 4e32

15. 3e

17. 64e4x

18. e2x

13.

4e2

2e

16. 2ex

e12x

19.

1

e x3

e e x1

Complete the table of values. Round to two decimal places. 20. f x 2e x

x f(x)

2 1.5

21. f x 2ex

1

0

1

1.5

2

x f(x)

22. f x e2x 3

Lesson 8.3

x f(x)

2 1.5

2 1.5

1

0

1

1.5

2

1

0

1

1.5

2

23. f x e3x 2

1

0

1

1.5

2

x f(x)

2 1.5

Graph the function and identify the horizontal asymptote. 24. f x 2e x

25. f x 2ex

26. f x e x 2

27. f x e3x 1

28. f x

29. f x e2.5x 3

1 2

e2x 1

Interest In Exercises 30–32, use the following information. You deposit $1200 in an account that pays 5% annual interest. After 10 years, you withdraw the money. 30. Find the balance in the account if the interest was compounded quarterly. 31. Find the balance in the account if the interest was compounded

continuously. 32. Which type of compounding yielded the greatest balance?

42



Answer Key

6. 81e8

7.

e6x2 10. 2e 4x 4 13. y 0 9.

14.

3

1

15.

8.

y

1

x

1 4096e3x

11. y 1

12. y 4 20. exponential decay 21.

Domain: All real numbers; Range: y > 2

2

24.

y 1 1

x


18.


y

2

1

8 10 13 6 10 13 4 10 13 2 10 13 0

0

x

2000 4000 6000 8000 Years

t

23. exponential growth 25. 13 units 26. 26 days

y 30 25 20 15 10 5 0

x

1

R 1 10 12

22. 10,000 years


y

x

1

x

1

17.

8 e6

4. 15.154


y

16.

3. 0.493


y 1

Units produced

5. 8e14

19.

Ratio (Carbon 14 to Carbon 12)

Practice C 1. 5.652 2. 0.074

0

10

20 30 Days

40

t

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LESSON

NAME _________________________________________________________ DATE ___________

8.3


Use a calculator to evaluate the expression. Round the result to three decimal places. 1

2. e2.6

1. e3

3. e2

4. ee

Simplify the expression. 5. e 2e

6.

8. 4e 0.5x6

9.

e2e

2

4 3

1 2 e 3

4

2 3

7.

3x 2

e2

3 8e12x 10.

Identify the horizontal asymptote of the function. 11. f x 3e2x 1

12. f x

1 2

13. f x 245e0.023x

e3x1 4

Graph the function. State the domain and range. 14. f x 2e3x 1

15. f x

17. f x

18. f x

1 2

e2x1 5

1 4 2 3

ex 2

16. f x 2ex4 1

e3x 1

19. f x

5 4

e2x1 3

Carbon Dating In Exercises 20–22, use the following information. Carbon dating is a process to estimate the age of organic material. In carbon dating the formula used is R

1 t8233 e 1012

where R is the ratio of Carbon 14 to Carbon 12 and t is time in years. 20. Is the model an example of exponential growth or exponential decay? 21. Graph the function. Lesson 8.3

22. Use the graph to estimate the age of a fossil whose Carbon 14 to Carbon

12 ratio is 3 1013.

Learning Curve In Exercises 23–26, use the following information. The management at a factory has determined that a worker can produce a maximum of 30 units per day. The model y 30 30e0.07t indicates the number of units y that a new employee can produce per day after t days on the job. 23. Is the model an example of exponential growth or exponential decay? 24. Graph the function. 25. How many units can be produced per day by an employee who has been

on the job 8 days? 26. Use the graph to estimate how many days of employment are required for

a worker to produce 25 units per day.



43

Answer Key Practice A 1. 23 8 2. 52 25 3. 33 27 4. 72 49 5. 24 16 6. 61 6 7. 2 8. 5 9. 2 10. 2 11. 0 12. 1 13. 0.778 14. 0.398 15. 0.571 16. 2.079 17. 1.470 18. 1.812 19. x 20. x 21. x 22. x 23. x 24. x 25. A 26. C 27. B 28. C 29. A 30. B 31. 110 decibels

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LESSON

NAME _________________________________________________________ DATE ___________

8.4


Rewrite the equation in exponential form. 1. log2 8 3

2. log5 25 2

3. log3 27 3

4. log7 49 2

5. log2 16 4

6. log6 6 1

Evaluate the expression without using a calculator. 7. log2 4

9. log8 64

8. log2 32

10. log10 100

11. log7 1

12. log8 8

Use a calculator to evaluate the expression. Round the result to three decimal places. 13. log 6

14. log 0.4

15. log 3.72

16. ln 8

17. ln 0.23

18. ln 6.12

19. 7log7 x

20. 27log27 x

21. 13log13 x

22. log33x

23. log1515 x

24. log221221x

Simplify the expression.

Match the function with its graph. 25. f x log3 x A.

26. f x log5 x B.

y

1

27. f x log12 x C.

y

y

1

1 x

1

x

1

x

1

Match the function with the graph of its inverse. 28. f x log x A.

29. f x log13 x B.

y

30. f x ln x C.

y

y

2 1

Lesson 8.4

1 x

1

31. Sound

1

x

1

x

The level of sound V in decibels with an intensity I can be modeled by

V 10 log

10I , 16

where I is intensity in watts per centimeter. Loud music can have an intensity of 105 watts per centimeter. Find the level of sound of loud music. 56



Answer Key Practice B 1. 42 16 2. 34 81 3. 20 1 1 1 4. 912 3 5. 51 5 6. 23 8 7. 0.549 8. 1.061 9. 0.405 10. 3 2 1 11. 0 12. 1 13. 3 14. 3 15. undefined 16. f 1 x 3x 17. f 1 x ex 10x 1 x 1 19. f 1 x 18. f x 2 3 20. f 1 x 2x 1 21. f 1 x 4x2

22.

23.

y

y

1

1 x

1

24.

25.

y

x

1

x

y

1

1 1

26.

1

x

27.

y

y 1

1

1 1

28.

x

x

127 strides 29. 267.4 miles per hour

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LESSON

NAME _________________________________________________________ DATE ___________

8.4


Rewrite the equation in exponential form. 1. log4 16 2 4. log9 3

1 2

2. log3 81 4 5.

log5 15

3. log2 1 0 1

1

6. log2 8 3

Use a calculator to evaluate the expression. Round the result to three decimal places. 7. ln 3

9. ln 3 2

8. log 11.5

Evaluate the logarithm without using a calculator. 1

10. log3 27

11. log4 1

12. log2 2

13. log8 2

14. log5 523

15. log6 1

16. f x log3 x

17. f x ln x

18. f x log13 x

19. f x log 2x

20. f x log2 x 1

21. f x log4 16x

22. f x log6 x

23. f x 1 log6 x

24. f x log6 x 1

25. f x log6 x

26. f x log6 2x

27. f x 1 log6 x

Find the inverse of the function.

Graph the function.

28. Galloping Speed

Four-legged animals run with two different types of motion: trotting and galloping. An animal that is trotting has at least one foot on the ground at all times. An animal that is galloping has all four feet off the ground at times. The number S of strides per minute at which an animal breaks from a trot to a gallop is related to the animal’s weight w (in pounds) by the model S 256.2 47.9 log w.

Approximate the number of strides per minute for a 500 pound horse when it breaks from a trot to a gallop. 29. Tornadoes

The wind speed S (in miles per hour) near the center of a tornado is related to the distance d (in miles) the tornado travels by the model S 93 log d 65.

Approximate the wind speed of a tornado that traveled 150 miles. Lesson 8.4



57

Answer Key Practice C 1 1. 53 125 2. 813 2 3. 33 27 4. 2.099 5. 0.092 6. 1.199 7. 5 2 3 2 3 8. 3 9. 3 10. 4 11. 3 12. 2 2x 13. f 1 x 4x 14. f 1 x 7 x 10 2 15. f 1 x 16. f 1 x ex3 3 1 x1 2 17. f x e x4 1 1 or f 1 x x 2 18. f x 2 2 19.

20.

y

1

22.

y

1

23.

24.

y

1

x

1

12

1

1

1

1

0

1 2

1

2

27. no

y

28. no

x

2

30.

1

x

1

x

1

x

y

1

1

1

29. 41.9 seconds

y

x

2

1

1

1 1

1

26.

y

x

x y

1 1

21.

25.

31. 41.2 seconds

70 65 60 55 50 45 40 35 0

20

40

60

80

100

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LESSON

8.4

NAME _________________________________________________________ DATE ___________


Rewrite the equation in exponential form. 1. log5 125 3

2. log8 2

1 3

1

3. log3 27 3

Use a calculator to evaluate the expression. Round the result to three decimal places. 4. ln 3 1

5.

ln 2.5 10

6.

log 4 3 2

Evaluate the expression without using a calculator. 1

1

7. log2 32

9. log8 4

8. log 1000

10. log16 8

11.

log27 19

1

12. log100 1000

Find the inverse of the function. 13. f x log4 x

14. f x log2 7x

15. f x log 3x 2

16. f x ln x 3

17. f x ln x 2 1

18. f x log 100 x2

19. f x log3 x

20. f x log3 x 2

21. f x log3 x 1

22. f x log3 x 2 1

23. f x log3 x 2

24. f x log3 x 2 1

Graph the function.

Critical Thinking In Exercises 25–28, use the following information. By definition of a logarithm, the base b of a logarithmic function must be a positive number and b 1. 25. Assuming that b 1, the “logarithmic function’ would be written y log1 x.

Complete the table of values for this “logarithmic function.” x y

2 1

12

0

1 2

1

2

26. Use the data to sketch a graph. 27. Does the graph look like a typical logarithmic graph? 28. Is the relation a function?

Lesson 8.4

400-Meter Relay In Exercises 29–31, use the following information. The winning time (in seconds) in the women’s 400-meter relay at the Olympic Games from 1928 to 1996 can be modeled by the function f t 67.99 5.82 ln t, where t is the number of years since 1900. 29. In 1988 the United States team won the 400-meter relay. What was its

winning time? 30. Use a graphing calculator to graph the model. 31. Use the graph to approximate the winning time in the 2000 Olympic Games. 58



Answer Key Practice A 1. 2 log 2 0.602 2. log 7 log 2 1.146 3. log 7 log 2 0.544 4. log 2 log 7 0.544 5. 3 log 7 2.535 6. 2 log 7 1.69 7. log2 3 log2 x 8. 2 log3 x 9. log x log 5 10. 1 log6 x 11. 5 log3 x 1 1 1 12. 3 ln x 13. 3 log x 14. 2 2 log2 x 15. 6 2 log3 x 16. log 15 17. log2 7x 4 x 18. log3 14y 19. log 20. ln x 3 x1 2 21. log 22. ln 6 x2 23. log34x 20 24. log 8x2 log 5 ln 5 25. 2.322 log 2 ln 2 log 10 ln 10 26. 1.183 log 7 ln 7 log 17 ln 17 27. 2.579 log 3 ln 3 log 200 ln 200 28. 2.957 log 6 ln 6 1 1 log 2 ln 2 29. 0.431 log 5 ln 5 log 1235 ln 1235 30. 5.135 log 4 ln 4 ln I ln I0 31. t 32. I 2000 3000 4000 0.049 t 14.1 22.4 28.3

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Lesson 8.5

LESSON

NAME _________________________________________________________ DATE ___________

8.5


Use the properties of logarithms to rewrite the expression in terms of log 2 and log 7. Then use log 2 ≈ 0.301 and log 7 ≈ 0.845 to approximate the expression. 1. log 4

2. log 14

4. log 7

3. log 2

5. log 73

6. log 49

8. log39x

9. log 5

2

7

Expand the expression. 7. log23x

6x

x

11. log3 x5

12. ln x3

14. log22x

15. log327x2

16. log 3 log 5

17. log2 x log2 7

18. log3 14 log3 y

19. log 4 log x

20. ln x ln 3

21. log x 1 log 6

22. ln 2 ln x 2

23. log3 x 5 log3 4

24. 2 log x log 8

10. log6

3 x 13. log

Condense the expression.

Use the change-of-base formula to rewrite the expression. Then use a calculator to evaluate the expression. Round your result to three decimal places. 25. log2 5

26. log7 10

28. log6 200

29.

log5 12

27. log3 17 30. log4 1235

Investments In Exercises 31 and 32, use the following information. You want to invest in a stock whose value has been increasing by approximately 5% each year. The time required for an initial investment of I0 to grow to I can be modeled by ln t

II 0

0.049

,

where I0 and I are measured in dollars and t is measured in years. 31. Expand the expression for t. 32. Assume that you have $1000 to invest. Complete the table to show how

long your investment would take to double, triple, and quadruple. I t

70

2000

3000

4000



Answer Key Practice B 1. log 3 log 4 0.125 2. log 3 log 4 1.079 3. 2 log 3 0.954 4. 2 log 4 1.204 5. log 4 0.602 6. log 4 3 log 3 0.829 7. log6 3 log6 x 8. log2 x log2 5 9. log x 2 log y 10. log 4 x log 4 y log 4 3 1 11. 2 log3 x log3 y log3 z 1 12. log5 2 2 log5 x 13. 2 log x log 4 1 14. 1 2 log x 15. 2 log2 x log2 y log2 z 16. log3 7x 17. log5 3x2 18. log4 5xy 3 2 x x 4x2 19. log 20. log2 3 21. log3 4 y 5 log 12 ln 12 22. 2.262 log 3 ln 3 log 2 ln 2 23. 0.387 log 6 ln 6 log 0.5 ln 0.5 24. 0.5 log 4 ln 4 ln 12 log 12 25. 11.136 log 0.8 ln 0.8 log 2.8 ln 2.8 26. 2.539 log 1.5 ln 1.5 log 6 ln 6 27. 1 2.585 log 12 ln 2 28. pH 6.1 log B log C 29. 7.2 30. below normal 31. pH 7.48 log C 32. 1.2

pH

pH 9 8 7 6 5 4 3 0

0 1 2 3 4 5 6 7C Carbonic acid

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LESSON

Practice B

Lesson 8.5

8.5

NAME _________________________________________________________ DATE ___________


Use the properties of logarithms to rewrite the expression in terms of log 3 and log 4. Then use log 3 ≈ 0.477 and log 4 ≈ 0.602 to approximate the expression. 1. log 4

2. log 12

4. log 16

5. log 4

3

3. log 9 6. log 27

1

4

Expand the expression. 7. log6 3x 10. log4 13. log

xy 3

x2 4

8. log2

x 5

2 9. log xy

11. log3 x y z 14. log

12. log5 2x

10 x

15. log2

x2y z

Condense the expression. 16. log3 7 log3 x

17. 2 log5 x log5 3

18. log4 5 log4 x log4 y

19.

20.

2 3

log2 x 3 log2 y

1 2

log x log 4

21. log3 4 2 log3 x log3 5

Use the change-of-base formula to rewrite the expression. Then use a calculator to evaluate the expression. Round your result to three decimal places if necessary. 22. log3 12

23. log6 2

24. log4 0.5

25. log0.8 12

26. log1.5 2.8

27. log12 6

Henderson-Hasselbach Formula In Exercises 28–32, use the following information. The pH of a patient’s blood can be calculated using the Henderson-Hasselbach B Formula, pH 6.1 log C , where B is the concentration of bicarbonate and C is the concentration of carbonic acid. The normal pH of blood is approximately 7.4. 28. Expand the right side of the formula. 29. A patient has a bicarbonate concentration of 24 and a carbonic acid

concentration of 1.9. Find the pH of the patient’s blood. 30. Is the patient’s pH in Exercise 29 below normal or above normal? 31. A patient has a bicarbonate concentration of 24. Graph the model. 32. Use the graph to approximate the concentration of carbonic acid required

for the patient to have normal blood pH.



71

Answer Key Practice C 1. ln 2 ln 3 1.792 2. ln 2 ln 5 ln 3 1.203 3. ln 2 ln 3 ln 5 3.401 4. 2 ln 2 ln 3 2.485 5. ln 2 ln 5 0.916 6. ln 5 ln 3 ln 2 0.183 7. log 8 log x 9. 10. 11. 12. 14. 15. 16. 19. 20. 21. 22. 23. 24. 25. 26.

8. log3 x log3 y log3 z

1 2

log 4 x log 4 y log 4 z ln x ln y ln z 1 2 log 3 log x log y 1 1 2 log5 x log5 y 13. ln 3 ln y 4 ln x 3log 3 log x log y 2 log z 4log2 x log2 y 2 log2 z 3 3xz x3 17. ln 18. ln 2 4 log 28 y yz 5 x 4x 1 log2 x 13 x 5 log2 ln y x2 x 2x 12 3 ln x 2x 15 ln x log x or y y log 3 ln 3 lnx 3 logx 3 or y y log 6 ln 6 logx 1 lnx 1 y 3 or y 3 log 2 ln 2 lnSr Pn ln P ln n t nlnn r ln n 19.7 years

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Lesson 8.5

LESSON

8.5

NAME _________________________________________________________ DATE ___________


Use the properties of logarithms to rewrite the expression in terms of ln 2, ln 3, and ln 5. Then use ln 2 ≈ 0.693, ln 3 ≈ 1.099, and ln 5 ≈ 1.609 to approximate the expression. 2. ln 3

1. ln 6 4. ln 12

10

3. ln 30

5. ln5

6. ln6

8. log3 xyz

9. log4 z

2

5

Expand the expression. 7. log 8x

2xy

10. ln

x yz

11. log 3xy

12. log5

13. ln

3y x

14. log3xyz23

15. log2

4

x

y

xy4 z2

Condense the expression. 16. log 3 log 4 log 7

17. ln x ln y ln z ln 3

18. 3 ln x 2 ln y 4 ln z

19. log2x 4 5 log2x 1 3 log2x 1

20.

1 logx 5 2 log x ln y 2

21. 3 lnx 2 2 lnx 1 lnx 2 5 lnx 1

Use the change-of-base formula to rewrite the function in terms of common (base 10) or natural (base ln) logarithms. 22. y log3 x

23. y log6x 3

24. y log2x 1 3

Annuities In Exercises 25 and 26, use the following information. An ordinary annuity is an account in which you make a fixed deposit at the end of each compounding period. You want to use an annuity to help you save money for college. The formula Pn Sr Pn t nr n ln n ln

gives the time t (in years) required to have S dollars in the annuity if your periodic payments P (in dollars) are made n times a year and the annual interest rate is r (in decimal form). 25. Expand the right side of the formula. 26. How long will it take you to save $20,000 in annuity that earns an annual

interest rate of 5% if you make monthly payments of $50?

72



Answer Key Practice A 1. yes 2. no 3. no 4. no 5. yes 6. no 7. no 8. yes 9. no 10. yes 11. no 12. yes 13. 1 14. 5 15. 2 16. 7 7 4 17. 3 18. 4 19. log29 20. log310 21. ln 5 ln 6 log510 22. 23. log27 24. 25. 7 26. 7 2 3 3 3 27. 3 28. 6 29. 4 30. 2 31. 32 e2 3 32. 6562 33. 499,998.5 34. 35. 0 5 36. 25 37. 3.2 years 38. 13.5 years 39. 23.1 years

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LESSON

8.6

NAME _________________________________________________________ DATE ___________


Tell whether the x-value is a solution of the equation. 1. ln x 9, x e9

2. ln x 3, x 3e

4. ln 2x 8, x e8

5. ln 6x 4, x

e4 6

3. ln x 7, x 7e 6. ln 2x 14, x 2e14

Tell whether the x-value is a solution of the equation. 10. 2e x 8, x ln 4

8. e x 7, x ln 7

9. e x 3, x log 3

11. 3e x 1 11, x 4

12. 5e x 2 17, x ln 3

13. 4x 42x1

14. 32x 3x5

15. 2 4x1 22x3

16. e3x e2x7

17. e2x1 e3x

18. 10 x 1073x

Lesson 8.6

7. e x 5, x 5

Solve the equation.

Solve the equation by taking the appropriate log of each side. 19. 2x 9

20. 3x 10

21. e x 5

22. e2x 6

23. 2x 5 12

24. 53x 2 8

Use the following property to solve the equation. For positive numbers b, x, and y where b 1, logb x logb y if and only if x y. 25. log x log 7

26. logx 2 log 9

27. log24x log2 12

28. log3x 1 log32x 5

29. lnx 3 ln6 3x

30. log3x 2 logx 1

Solve the equation by exponentiating each side. 31. log2 x 5

32. log3x 1 8

33. log2x 3 6

34. ln5x 3 2

35. ln3x 1 0

36. log4x 1 3

Compound Interest You deposit $100 in an account that earns 3% annual interest compounded continuously. How long does it take the balance to reach the following amounts? 37. $110


38. $150

39. $200


83

Answer Key Practice B 1. 2.890 2. 2.544 3. 1.869 4. 1.609 5. 1.585 6. 0.646 7. 0.667 8. 0.805 9. 0.886 10. 0.462 11. 0.576 12. 2.322 13. 0.5 14. 0.973 15. 1.946 16. 1.609 17. 2 18. 1.792 19. 0.229 20. 0.308 21. 0 22. 0.347 23. 1.099 24. 25.850 25. 2.485 26. 1.445 27. 1.528 28. 148.413 29. 0.01 30. 2.828 31. 20.086 32. 100,000 33. 0.001 34. 2980.958 35. 20.086 36. 148.413 37. 10,000 38. 46.416 39. 3 40. 0.4 41. 0.002 42. 300,651.071 43. 21.333 44. 1 45. no solution 46. 1.5 47. no solution 48. 3 49. 11.185 years 50. 20.086

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LESSON

8.6

NAME _________________________________________________________ DATE ___________


Lesson 8.6

Solve the exponential equation. Round the result to three decimal places if necessary. 1. e x 18

2. 10 x 350

3. e2x 42

4. e x 3 8

5. 2x 7 10

6. 52x 8

7. 23x 4

8. e2x 5

9. 32x 3 4

10. e3x 6 10

11. e4x 3 7

12. 2x 1 6

13. 42x 3 1

14. e2x 5 12

15. ex 6 1

16. 2ex 10

17. 42x 16

18. 3ex 18

19. 2e4x 5

20. 3e5x 14

21. 223x 2

22. 4e2x 3 5

23. 3ex 4 13

24. 20.1x 6 12

1

25. 3e x 1 5

2

26. 3e2x 12

27. 823x 1 10 3

Solve the logarithmic equation. Round the result to three decimal places if necessary. 28. ln x 5

29. log10 x 2

30. log2 x 1.5

31. 7 ln x 21

32. 2 log10 x 10

33. 7 log10 x 4

34. 3 ln x 5

35. 4 ln x 1

36. 5 2 ln x 5

37. 3 log10 x 1 13

38. 9 log10 x 4 11

39. log3 3x 2

40. log2 5x 1

41. 2 log3 2x 3

42. ln 4x 6 8

43. 2 log2 3x 8

44. log2 x 2 log2 3x

45. log3 2x 1 log3 x 4

46. ln 5x 1 ln 3x 2

47. ln 2x 3 ln 2x 1

48. ln 4x 9 ln x


You deposit $2000 into an account that pays 2% annual interest compounded quarterly. How long will it take for the balance to reach $2500?

50. Rocket Velocity

Disregarding the force of gravity, the maximum velocity v of a rocket is given by v t ln M, where t is the velocity of the exhaust and M is the ratio of the mass of the rocket with fuel to its mass without fuel. A solid propellant rocket has an exhaust velocity of 2.5 kilometers per second. Its maximum velocity is 7.5 kilometers per second. Find its mass ratio M.

84



Answer Key Practice C 1. 2.197 2. 0.333 3. 3.386 4. 0.349 5. 1.436 6. 6.447 7. 0.376 8. 1.269 9. 0.258 10. 0 11. 1, 2 12. 1, 0.667 13. 4.5 14. 22,023.466 15. 11 16. 181.939 17. 2.414 18. 1 19. 3 20. 0.143 21. 3.333 22. no solution 23. 2, 3 24. 7 25. 4, 6 26. no solution 27. 0.461 28. 3.697 29. 8.266 30. no solution 31. 5.303 32. 7.193 33. 5.2 years 34. 30 years 35. $211,320 36. $131,320

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LESSON

8.6

NAME _________________________________________________________ DATE ___________


Solve the exponential equation. Round the result to three decimal places if necessary. 1. e x 9

2. 23x1 4

3. 32x5 7

4. e4x1 3 8

5. e53x 4 6

6. 30.4x 7 10

7.

2 3

e4x 5 8

8.

2

2

11. e x

1

25

ex3

9.

5 3

e1x 1 92

12. 23x1 22x

Lesson 8.6

10. e x 3 4

1 3x1 4 2

Solve the logarithmic equation. Round the result to three decimal places if necessary. 13. log2x 1 1

14. lnx 3 2 8

15. log3x 2 5 7

16. ln6x 5 7

17. lnx 2 ln x 0

18. log2 x log2x 1 1

19. log3 x log3x 2 1

20. log2x 1 log2 x 3

21. log4x 2 log4x 3 2 22. log3x 2 log2x 1 23. logx2 1 logx 5

24. logx 2 logx 3 logx 29

25. log2 x log2x 2 log2x 3 3 26. log2x 3 log2x 1 log2x 3 1

Solve the exponential equation. Round the result to three decimal places. 27. 2x1 32x

28. e x3 10 4x

29. 52x1 24x3

Solve the logarithmic equation. Round the result to three decimal places. 30. log2x 1 log42x 3

31. log3x 3 log9 x

32. logx 4 log100x 3


You deposit $2500 into an account that pays 3.5% annual interest compounded daily. How long will it take for the balance to reach $3000? In Exercises 34–36, use the following information.

Loan Repayment

1 1

The formula L P

r n

r n

nt

gives the amount of a loan L in terms

of the amount of each payment P, the interest rate r, the number of payments per year n, and the number of years t. 34. When purchasing a home, you need a loan for $80,000. The interest rate

of the loan is 8% and you are required to make monthly payments of $587. How long will it take you to pay off the loan? 35. When the loan is paid off, how much money will you have paid the bank? 36. How much did you pay in interest? Copyright © McDougal Littell Inc. All rights reserved.


85

Answer Key Practice A 1. y 3x 2. y 3 2x 3. y 2 5x 1 4. y 2 4x 5. y 2x 6. y 2 3x 7. yes 8. no 9. yes 10. yes 11. y 4.961.38t 12. y 1.523.33t 13. y 171.40186,278.85t 14. y 3.1024.70t 15. y 3459.922.81t 16. y 5.079.98t 17. y 2x3 18. y 3x2 19. y x1.5 20. no 21. yes 22. y x2.4 23. y x1.3 24. y x0.8

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LESSON

NAME _________________________________________________________ DATE ___________

8.7


Write an exponential function of the form y ab x whose graph passes through the given points. 1. 0, 1), 3, 27

2. 1, 6, 2, 12

3. 1, 10, 2, 50

4. 1, 2, 2, 8

5. 4, 16, 6, 64

6. 2, 18, 3, 54

Use the table of values to determine whether or not an exponential model is a good fit for the data t, y. 7. t

ln y 8. t

ln y 9. t

ln y

1 0.23

2 0.64

3 1.07

4 1.47

5 1.88

6 2.31

7 2.72

8 3.12

1 1.32

2 1.52

3 1.92

4 2.72

5 2.88

6 3.52

7 4.32

8 5.6

1 0.05

2 0.17

3 0.27

4 0.40

5 0.52

6 0.63

7 0.75

8 0.85

10. t

3 4 5 6 14.82 16.04 17.29 18.49

7 8 19.76 21.01

Lesson 8.7

1 2 ln y 12.31 13.56

Solve for y. 11. ln y 0.324t 1.601

12. ln y 1.203t 0.418

13. ln y 12.135t 5.144

14. ln y 3.207t 1.132

15. ln y 1.032t 8.149

16. ln y 2.301t 1.624

Write a power function of the form y ax b whose graph passes through the given points. 17. 1, 2, 3, 54

18. 1, 3, 2, 12

19. (1, 1, 4, 8

Use the table of values to determine whether or not a power function model is a good fit for the data x, y. 20. ln x

1.099 3.924

1.386 5.254

1.609 6.584

21. ln x

1.099 3.030

1.386 3.605

1.609 4.052

0 0.693 ln y 1.264 2.594 0 0.693 ln y 0.833 2.219

Solve for y. 22. ln y 2.4 ln x


23. ln y 1.3 ln x

24. ln y 0.8 ln x


97

Answer Key Practice B 1. y

2. y

3. y 32

4. 7.

5.

6. y

ln y 8 7 6 5 4 3 2 1 0

13 2x x y 623

251 4x y 12 5x

5 1 x 4 3

8.

0 1 2 3 4 5 6 7 8 x

y 42x 9.

1 x

ln y 8 7 6 5 4 3 2 1 0

0 1 2 3 4 5 6 7 8

ln y 8 7 6 5 4 3 2 1 0

0 1 2 3 4 5 6 7 8 x

y 1.52.4x 10. y 4x2 1 11. y 2x3 12. y 2x5 13. y 1.2x1.5 14. y 3x3.5 1.2 15. y 4.3x x

y 21.5x 16.

17.

ln y

ln y

4

4

3

3

2

2

1

1

0

0

1

2

3

4 ln x

0

0

1

2

y 1.5x2 y 2.4x1.6 18. y 29.623x0.809; 3.313 million

3

4 ln x

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Page 98

LESSON

NAME _________________________________________________________ DATE ___________

8.7


Write an exponential function of the form y ab x whose graph passes through the given points. 1.

1, 32, 2, 43 83

4. 1, 4, 2,

2.

64 , 3, 2, 16 25 25

3.

2, 34, 3, 38

5.

1, 52, 2, 252

6.

5 2, 365 , 3, 108

Lesson 8.7

Use the table of values to draw a scatter plot of ln y versus x. Then find an exponential model for the data. 7.

x y

1 8

2 16

3 32

4 64

5 128

6 256

7 512

8 1024

8.

x y

1 3.6

2 8.64

3 20.736

4 49.766

5 119.439

6 286.654

7 687.971

8 1651.13

9.

x y

1 3

2 4.5

3 6.75

4 10.125

5 15.188

6 22.781

7 34.172

8 51.258

Write a power function of the form y ax b whose graph passes through the given points. 10. 2, 16, 3, 36

11. 2, 4, 4, 32

12. 2, 64, 3, 486

13. 4, 9.6, 9, 32.4

14. 4, 384, 16, 49,152

15. 2, 9.879, 3, 16.070

Use the table of values to draw a scatter plot of ln y versus ln x. Then find a power model for the data. 16.

x y

1 1.5

2 6

3 13.5

4 24

5 37.5

6 54

7 73.5

8 96

17.

x y

1 2.4

2 7.275

3 13.919

4 22.055

5 31.518

6 42.194

7 53.997

8 66.858

18. Consumer Magazines

The table shows the circulation of the top 10 consumer magazines in 1997 where x represents the magazine’s ranking. Use a graphing calculator to find a power model for the data. Use the model to estimate the circulation of the 15th ranked magazine. Rank 1 2 3 4 5

98

Circulation (millions) 20.454 20.432 15.086 13.171 9.013

Rank 6 7 8 9 10


Circulation (millions) 7.615 5.054 4.643 4.514 4.256


Answer Key Practice C 1. y 2.31.6x 2. y 4.50.2x 3. y 5.32.8x 4. y 43.5x 5. y 1.52x 6. y abx ln y lnabx ln y ln a ln bx ln y ln a x ln b constant Thus, there is a linear relationship between x and ln y. 7. y 1.5x0.5 8. y 2.4x1.5 9. y 8.3x0.25 10. y 3x1.2 11. y 2.5x2.5 12. y axb ln y ln axb ln y ln a ln xb ln y ln a b ln x constant Thus, there is a linear relationship between ln x and ln y. 13. y 28.381.14x 14. y 30.84x0.33 15. The exponential model is better because the relationship between x and ln y is closer to linear than the relationship between ln x and ln y.

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Page 99

LESSON

NAME _________________________________________________________ DATE ___________

8.7


Write an exponential function of the form y ab x whose graph passes through the given points. 1. 2, 5.888, 3, 9.4208

2. 1, 0.9, 2, 0.18

3. 2, 41.552, 3, 116.3456

Find an exponential model for the data. 4.

x y

1 14

2 49

3 171.5

4 600.25

5 2100.9

6 7353.1

7 25,736

8 90,075

5.

x y

1 3

2 6

3 12

4 24

5 48

6 96

7 192

8 384


To determine whether an exponential model fits the data, you need to determine whether the data of the form x, ln y is linear. To see that this test works, start with y a b x, take the natural logarithm of both sides, and use the properties of logarithms to verify that there is a linear relationship between x and ln y.

7. 4, 3, 9, 4.5

8. 4, 19.2, 9, 64.8

Lesson 8.7

Write a power function of the form y ax b whose graph passes through the given points. 9. 16, 16.6, 81, 24.9

Find a power model for the data. 10.

x y

1 3

2 6.8922

3 11.212

4 15.834

5 20.696

6 25.757

7 30.991

11.

x y

1 2.5

2 14.142

3 38.971

4 80

5 139.75

6 220.45

7 324.1


To determine whether a power model fits the data, you need to determine whether the data of the form ln x, ln y is linear. To see that this test works, start with y ax b, take the natural logarithm of both sides, and use the properties of logarithms to verify that there is a linear relationship between ln x and ln y.

Volunteer Work In Exercises 13–15, use the following information. The table below shows the percent of the adult population P that participates in volunteer work as a function of household income where t 1 represents a household income under $10,000, t 2 represents a household income between $10,000 and $19,000, and so on. t P

1 34.7

2 34.3

3 41.2

4 46.0

5 52.7

6 64.1

13. Use your graphing calculator to find an exponential model for the data. 14. Use your graphing calculator to find a power model for the data. 15. Which model is the better fitting model? Explain your answer.



99

Answer Key Practice A 1. about 1.4621 2. about 0.5379 3. about 1.9951 4. 1 5. about 1.2449 6. about 1.9354 7. about 0.9003 8. about 1.5546 9. C 10. A 11. B 12. y 0, y 1 13. y 0, y 5 5 1 14. y 0, y 6 15. 3 16. 2 17. 2 18. 0, 2 19. 1.1, 0.5 20. 0.23, 1 21. 89,963 units 22. No more than 100,000 units will be sold each year.

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Page 110

LESSON

NAME _________________________________________________________ DATE ___________

8.8


Evaluate the function f x 1. f 1 5. f

1 2

2 for the given value of x. 1 ex

2. f 1

3. f 6

6. f 3.4

7. f 0.2

4. f 0 8. f 4 5


3 1 ex

A.

10. f x

3 1 e2x

B.

y

11. f x C.

y

y

1 2

1 1 2ex

2 1

x 1

1

x

x

Identify the horizontal asymptotes of the function. 12. f x

1 1 4e2x

13. f x

5 1 e2x

14. f x

6 1 2ex

Identify the y-intercept of the function.

Lesson 8.8

15. y

1 1 2ex

16. y

4 1 ex

17. y

5 1 e3x

Identify the point of maximum growth of the function. 18. f x

4 1 e2x

19. f x

1 1 3ex

20. f x

2 1 2e3x

Advertising In Exercises 21 and 22, use the following information. A company decides to stop advertising one of its products. The sales of the product S can be modeled by S

100,000 1 0.5e0.3t

where t is the number of years since advertising stopped. 21. What are the sales 5 years after advertising stopped? 22. What can the company expect in terms of sales in the future?

110



Answer Key Practice B 1. exponential decay 2. logarithmic 3. logistics growth 4. exponential decay 5. exponential growth 6. logarithmic 7. A 8. C 9. B 10. y 0, y 20 11. y 5, y 4 12. y 10, y 12 13.

14. y

y

2

2

x

1

15.

1

x

16. ln 2 0.693

y

2

x

1

5

17. ln 3 0.511 Population

19.

P 500 400 300 200 100 0

18.

0 2 4 6 8 10 t Year

1 2

ln 5 0.805 20. y 0, y 500 21. 500 22. 451

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Page 111

LESSON

NAME _________________________________________________________ DATE ___________

8.8


Tell whether the function is an example of exponential growth, exponential decay, logarithmic, or logistics growth. 1. f x

1 2

x

4. f x e2x

1 1 3ex

2. f x ln 3x

3. f x

5. f x 2.5x

6. f x log 6x


4 1 2ex

A.

8. f x

2 1 2ex

B.

y

9. f x C.

y

2

4 1 e2x y

2

1 x

1 1

x

1

x

Identify the horizontal asymptotes of the function. 10. f x

20 1 0.4ex

11. f x 5

1 1 ex

12. f x 10

2 1 ex

Sketch the graph of the function. 13. f x

3 1 ex

14. f x

1 1 5ex

15. f x 1

5 1 ex Lesson 8.8

Solve the equation. 16.

4 2 1 2ex

17.

8 5 1 ex

18.

12 6 1 5e2x

Wildlife Management In Exercises 19–22, use the following information. A wildlife organization releases 100 deer into a wilderness area. The deer population P can be modeled by P

500 1 4e0.36t

where t is the time in years. 19. Sketch the graph of the model. 20. Identify the horizontal asymptotes of the graph. 21. What is the maximum number of deer the wilderness area can support? 22. What is the deer population after 10 years?



111

Answer Key Practice C 1. 2.667 2. 7.276 3. 0.194 4. 8 5. 0.0000012 6. 4.993 7. 0.400 8. 7.985 9.

10.

y 3

y

(0, 1)

(0.231, 2)

2

x

1

(0, ) 4 3

x

1

11.

12.

y

y

(0.549, 3.5) 2

(0, ) 7 4

(0.277, 1.5) 1

x

1

(0, ) 3 5

1

13.

14.

y

y

(3.2, 5)

(1.24, 12.5) 5

(

0,

25 13

)

15. ln 2 0.693 1 3 5 4

2 x

1

18.

x

ln 6 0.597

( ) 0,

5 3

2

16. 0 19.

17. 2 5

1 2

x

ln 15 4 0.661

ln 81 4 0.702

ln 136 63 0.962 21. k 0.186 22. 6.5 years 23. 2000 c 24. y 25. rx → 1 aer 0 c y 1 ae0 c y 1a 20.

26. erx → 0 29.

27. aerx → 0

c →c 1 aerx

28. 1 aerx → 1

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Page 112

LESSON

8.8

NAME _________________________________________________________ DATE ___________


Evaluate the function f x 1. f 0

8 for the given value of x. 1 2e3x

2. f 1 6. f

5. f 5

2 5

3. f 1

7. f

34

4. f 5

8. f 3 7

Graph the function. Identify the asymptotes, y-intercept, and point of maximum growth. 9. y 12. y

2 1 ex

10. y

4 1 2e3x

11. y

7 1 3e2x

3 1 4e5x

13. y

25 1 12e2x

14. y

10 1 5e12x

Solve the equation. Round the answer to three decimal places. 15.

10 5 1 2ex

16.

12 3 1 3e4x

17.

13 5 1 6e2x

18.

28 14 1 6e3x

19.

32 18 1 4.5e2.5x

20.

40 8.5 1 8e0.8x

Conservation In Exercises 21–23, use the following information. A conservation organization believes that the growth of a population P of an endangered species at its preserve can be modeled by the curve P

2000 1 10e kt

Lesson 8.8

where t is time in years. 21. After 1 year, the preserve’s population of endangered species is 215. Find k. 22. When will the population reach 500? 23. What is the maximum population the preserve can maintain? 24. Analyzing Models

The graph of the logistic growth function c c has a y-intercept of y . Verify this formula by rx 1 ae 1a setting x equal to 0 and solving for y.

Analyzing Models

In Exercises 25–29, use the function y

c . 1 aerx

25. As x → , what is the behavior of rx? 26. As x → , what is the behavior of erx ? 27. As x → , what is the behavior of aerx ? 28. As x → , what is the behavior of 1 aerx ? 29. As x → , what is the behavior of 112


c 1 aerx Copyright © McDougal Littell Inc. All rights reserved.

Answer Key Test A 2 x

4 x

1. y ; 1

2. y ; 2

1 8

4. z xy; 1 6.

12 ;6 x

5. z –xy; 8 7.

y

1

y

1 1

8.

3. y

x

2

x

9.

y

1 x

1

1 x

1

10.

11.

y

y

1 2

1 1

x

x

x x2 5x 13. 14. 2 15. 2 x1 x2 4x 3 63 xx 12 16. 17. 18. 19. x3 4x 32 35x 1 3 x 3 3 20. 21. 2 22. 4 18x 4 xy 23. 4 (7 is extraneous) 24. z x2 25. 14,000 dozens of golf balls 12.

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CHAPTER

NAME _________________________________________________________ DATE ____________

9


The variables x and y vary inversely. Use the given values to write an equation relating x and y. Then find y when x 2. 1. x 1, y 2

2. x 4, y 1

3. x 6, y 2

Answers 1. 2.

The variable z varies jointly with x and y. Use the given values to write an equation relating x, y, and z. Then find z when x 2 and y 4. 4. x 2, y 4, z 1

5. x 2, y 1, z 2


1 x

7. y

2 x1

y

y

1

1 1

8. y

x

x x2

1

9. y

3. 4. 5. 6.

Use grid at left.

7.

Use grid at left.

8.

Use grid at left.

9.

Use grid at left.

10.

Use grid at left.

11.

Use grid at left.

x

1 x2 y

y

1 x

1

1

11. y

10. y x 2

1

x

1

x

x2 1 x

y

y

1 1


Review and Assess

1

x


93

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Page 94

CHAPTER

NAME _________________________________________________________ DATE ____________

9

Chapter Test A

CONTINUED


Perform the indicated operation. Simplify the result. 12.

x3 4

2

x2

13.

x3

x 12

3x 1 2x 1 15. x2 x2

x5 x5 14. x 2x 16.

x1 x

5x 2 8x 4x 9x 2 2 x2 9 x 9

17.

9x3 8x 32

2x 8 3x 4

Simplify the complex fraction. x3 3x 2 20. 6x 2 x 32

Solve the equation using any method. Check each solution. 21.

3x x 1 4 2

13. 14. 15. 16. 17. 18.

x 4 3 19. 1 5 x

5 14 18. 2 23

12.

22.

10 10 6 x3 3

2x 9 x 5 23. x7 2 x7

19. 20. 21. 22. 23. 24. 25.

24. Geometry Connection

The similar triangles below have congruent angles and proportional sides. Express z in terms of x and y. x2 x y

z

You start a business manufacturing golf balls, spending $42,000 for supplies and equipment. You figure it will cost $12 per dozen to manufacture the golf balls. How many dozens of golf balls must you produce before your average total cost per dozen is $15?

Review and Assess

25. Starting a Business

94



Answer Key Test B 8 4 4 2. y ; 1 3. y ; 2 x x x 1 1 4. z xy; 1 5. z xy; 2 4 2

1. y ; 1

6.

7.

y

y

1

1 x

1

8.

x

2

9.

y

1 1

x

2

x

1

10.

11.

y

1 2

1 x

2

12.

11x 2y

16. 2 20.

14. 2x 2

13. 1 17. 2x

18.

10xy 15x 15y x 2y

25 x2

yx 4 x

x1 x2

xy xy

21. 14

22. 1; (1 is extraneous) 24. z

19.

15.

23.

3 7 , 2 4

25. 160,000 hats

x

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Page 95

CHAPTER

NAME _________________________________________________________ DATE ____________

9



2. x 8, y

1 2

2

3. x 3, y 12

Answers 1. 2.


1

5. x 4, y 2, z 1

Graph the function. 6. xy 1

7. y

3 x1

y

y

1

1 1

8. y

x

x x4

9. y

1

x

1

x

1

x

3. 4. 5. 6.

Use grid at left.

7.

Use grid at left.

8.

Use grid at left.

9.

Use grid at left.

10.

Use grid at left.

11.

Use grid at left.

1 2x 2 y

y

1 1

x

1

10. y x 2 1

11. y

x 2 x1

y

y 1

Review and Assess

1 1


x


95

MCRB2-0911-TB.qxd

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Page 96

CHAPTER

NAME _________________________________________________________ DATE ____________

9

Chapter Test B

CONTINUED



x 5x 2y y

13.

x 12 2x 2 14. x 1 x 1 16.

y 3 y3 y3

x 2 5x 4 x 4 15. x3 x

6x 5 2x 7 2x 6 2x 6

17.

x3 3x 2 x3 8x 2 15x 2 3x 6 6x 18x 60

Simplify the complex fraction. xy 5 xy x 18. x 19. x 2 2xy y 2 5 x 2 2xy y 2

21.

3 2 x1 x4

22.

13. 14. 15. 16. 17. 18.

3 2 x xy 20. 3 1 2 x 5

Solve the equation using any method. Check each solution. x2

12.

2x 2 2x 3 x1 x1

7 2 13y 23. y 2 4 3 6

19. 20. 21. 22. 23. 24. 25.


The similar triangles below have congruent angles and proportional sides. Express z in terms of x and y. x4

x y

z

You start a business manufacturing hats, spending $8,000 for supplies and equipment. You figure it will cost $4.95 per hat to manufacture the hats. How many hats must you produce before your average total cost per hat is $5?

Review and Assess

25. Starting a Business

96



Answer Key Test C 9 1 1 2 2 2. y ; 3. y ; x x 3 x 3 1 3 4. z xy; 4 5. z 9xy; 54 8

1. y ; 3

6.

7.

y

y

1

1 x

1

8.

x

1

9.

y

y

2 6

x 1 x

1

10.

11.

y 6 1

12. 15. 18. 21. 25.

y 5

x

5yz 3xz 2xy xyz

1

x

3 7 14. x3 10 3x 1 17. 1 8xx 3x 2 16. 2x 3 2y 2 20x 2y 20 x 19. 20. 2 2 7xy 15x x7 x1 yx 3 4, 4 22. 5 23. 3, 2 24. z x about 130 boxes 13.

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Page 97

CHAPTER

NAME _________________________________________________________ DATE ____________

9



3

2. x 5, y

5 3

3. x 6, y

1 3

Answers 1. 2.


5. x

1 2,

1 3,

y

z

3 2

7. y

2 x2

y

6.

Use grid at left.

7.

Use grid at left.

8.

Use grid at left.

9.

Use grid at left.

10.

Use grid at left.

11.

Use grid at left.

y

1

1 1

8. y

4. 5.

Graph the function. 6. xy 2

3.

x

2x x4

x

1

9. y

4 x2

y

y

2 2

x 1 x

1

10. y

2x 2 x2

11. y

x 2 3x 5 x1 y

y

Review and Assess

5

6 1


x

1

x


97

MCRB2-0911-TC.qxd

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Page 98

CHAPTER

NAME _________________________________________________________ DATE ____________

9

Chapter Test C

CONTINUED



5 3 2 x y z

16.

3x 5 7x 1 x2 9 9 x2 x 3

13.

15x 2 3x 15. 30x 6x 2 4x 4x 24

15.

13.

3y 5 4y 2 14. 2y 6 5y 15

3

3x 3x 6 2 2x 3 2x x 6

12.

17.

2

6x 2 x 2 6x 2 7x 2

2x 2 9x 4

4 7x 2x 2

Simplify the complex fraction. 1 2 5x 2 y 18. 7 3 10x 2y 2

19.

14.

16. 17. 18.

x 10 1

19. 20. 2

2

2 2 x

20.

Solve the equation using any method. Check each solution.

22.

21.

1 3 2 x1

2 x 2 x x 8

22.

2

3 4 6 2 x 2 x 3 x 5x 6

3 x2 13 23. x1 3 3x 3

21.

23. 24. 25.


The similar triangles below have congruent angles and proportional sides. Express z in terms of x and y. x

x3 z

y

As a fund raiser, your junior class will make and sell holiday greeting cards. You spend $750 as an initial startup cost. It will cost you $4.25 per box to print, and you will sell the cards at $10 per box. How many boxes must you sell to show a profit?

Review and Assess

25. Fund Raiser

98



Answer Key

14 7 8 12 22. 2 16

Cumulative Review Chapter 9

21.

73

72

y

17 2

or 3. 56 or 8 4. n > 2 or n < 1 5. 8 ≥ n ≥ 4 6. x ≥ 7 or x ≤ 29 1. 3 or

2.

7.

1 1

x

8. y

y

23.

1 1

x

1

x

1

9.

25. 27.

10. y

29.

y

33. 36.

1 1

x

1 1

39.

x

42. 11.

12. y

43. y

1 1

45.

x

49. 1 1

x

9 15 13 3 24. x 1x 6 5 15 3x 53x 10 26. 33x 1x 1 7x 97x 9 28. 3y 83y 2 xx 32 30. 34 31. 3 32. 257 5 34. 32 35. 17 3 ± i7 2 ± i14 37. 38. 3, 1 2 5 ± 37 2 ± 10 40. 41. 4, 2 2 2 y 3x 42 6 y 2x 12 9 44. y x2 3 y5 y6 x 8y3 46. 47. 48. x3 y14 3x 16x 4 3 50. x 6 51. 17 52. 281 53. 18 x2

54. 25

55. f x 6x 13; first; linear; 6

56. f x 3 x3 2x 8; third; cubic; 1

13. C

14. A

15. B

16. infinite

17. none

18. one 19.

57. no

1 3

58. 2x2 6x 9; all real numbers

59. 6x 9; all real numbers 60. 2x2 12x; all real numbers

20. y

61. x 4 6x3 9x2 54x; all real numbers

y

62. x 4 12x2 27; all real numbers 63. x 4 18x2 72; all real numbers 1 1

x 1 1

x

64. 8 units left 65. 6 units left, reflected over xaxis 66. 1 unit left, 5 units up 67. 1 unit right 68. 1; y 0 69. 3; y 0 70. 4; y 3 1 3 71. 2; y 0 72. 2; y 0 73. 10; y 5 e2x 74. 75. 3e3x 76. 3e3x1 77. 2.398 3 78. 0.239 79. 1.375

MCRB2-0911-CR.qxd

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CHAPTER

NAME _________________________________________________________ DATE ____________

9


Solve the equation or inequality. (1.7)

4n 2 > 6

6 3n ≤ 18

x 6 8 1 4

1. 3x 1 8

2. 12 2x 5

3.

4.

5.

6. 11 x ≥ 18

Graph the equation. (2.3) 3

7. y 4 x 2

8. y 2x 6

10. x 0

9. 5x 3y 15

11. y 5

12. 3x 6y 18

Match the equation with its graph. (2.8)

13. y x 3 3 A.

14. y x 3

15. y x 3

B.

y

C.

y

y

1 1

1 x 1 x

1 1

x

Tell how many solutions the system has. (3.1) 16. x 2y 6

17. 2x y 5

3x 18 6y

18. 3x 4y 5

2y 4x 20

x 2y 9

Graph the system of inequalities. (3.3) 20. x y ≤ 0

21. 5x 3y ≤ 15

y < x

xy ≥ 8

3x y ≤ 3

x > 4

y ≤ 6

x ≤ 0

19. 2x y > 6

Perform the indicated operations. (4.1, 4.2) 22.

2 0 2

3 3 4 4 2 4 1

1 2 5

23.

3 1 5

3 3 0

4 1

3 2

Review and Assess

Factor the expression. (5.2) 24. x2 7x 6

25. 9x2 45x 50

26. 9x2 12x 3

27. 49x2 81

28. 9y2 30y 16

29. x2 32x

Find the absolute value of the complex number. (5.4) 30. 3 5i

31. 3i

32. 16 i

33. 2 i

34. 3 3i

35. 1 4i

104



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CHAPTER

NAME _________________________________________________________ DATE ____________

9

Cumulative Review

CONTINUED


Use the quadratic formula to solve the equation. (5.6) 36. x2 4x 18

37. x2 3x 4 0

38. x2 2x 3

39. 16x x2 11x 3

40. 2c 12 4 1

41. x2 6x 9 1

Write a quadratic function in vertex form whose graph has the given vertex and passes through the given point. (5.8) 42. vertex: 4, 6

43. vertex: 1, 9

point: 2, 18

44. vertex: 0, 3

point: 2, 27

point: 5, 22

Simplify the expression. (6.1) 45.

x5y3 x3y0

48.

xy8 3x2

46.

6x3y 4

2x3y2

2x2y 6x3y4

47. 4x2y32

49. 3x2y0

50.

x4 x2

Use synthetic substitution to evaluate. (6.2) 51. f x 2x3 x2 2x 1, x 2

52. f x 4x 4 3x2 5x 1, x 3

53. f x x 4 x2 5x 11, x 1

54. f x 3x3 x2 x 3, x 2

Decide whether the function is a polynomial function. If it is, write the function in standard form and state the degree, type, and leading coefficient. (6.2) 55. f x 13 6x

56. f x 2x

1 3 x 8 3

57. f x 6x2 x

2 x

Let f x x2 6x and g x x2 9. Perform the indicated operation and state the domain. (7.3) 58. f x gx 61. f x

gx

59. f x gx

60. f x f x

62. f gx

63. ggx

Describe how to obtain the graph of g from the graph of f. (7.5) 64. gx x 8, f x x

65. gx x 6, f x x

66. gx x 1 5, f x x

67. gx 5x 1, f x 5x

Identify the y-intercept and the asymptote of the graph of the function. (8.1) 69. y 3

71. y 2x1

72. y 3

5x

Review and Assess

68. y 6 x

70. y 2x 3

2x1

73. y 5x1 5

Simplify the expression. (8.3) 74. 3e2x1

3 27e9x 75.

76. ex

3e2x1

Use a calculator to evaluate the expression. Round the result to three decimal places. (8.4) 77. ln 11 Copyright © McDougal Littell Inc. All rights reserved.

78. log 3

79. log 23.724 Algebra 2 Chapter 9 Resource Book

105

Answer Key Practice A 1. direct variation 2. inverse variation 3. neither 4. inverse variation 8 9 9 5. y ; 2 6. y ; x x 4 12 36 7. y ; 9 8. y ; 3 x x 5 5 4 9. y ; 1 10. y ; 11. direct variation x x 4 12. neither

13. inverse variation

14. direct

2 variation 15. z 3xy; 36 16. z 3xy; 8 1 17. z 2xy; 24 18. z 3xy; 4 19. k 0.055 20. I 0.055Pt 21. $220.00

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LESSON

9.1

NAME _________________________________________________________ DATE ___________


Tell whether x and y show direct variation, inverse variation, or neither. 2. y

Lesson 9.1

1. y 3x

2 x

3. x y 7

4. xy 5

The variables x and y vary inversely. Use the given values to write an equation relating x and y. Then find y when x 4. 5. x 2, y 4 8. x 3, y 4

6. x 3, y 3 9. x 16, y

7. x 4, y 9

1 4

10. x 10, y

1 2

Determine whether x and y show direct variation, inverse variation, or neither. 11.

x 3 8 11 0.5

y 12 32 44 2

12.

x 1 2 4 5

y 6 5 3 2

13.

x 3 6 10 12

y 1 0.5 0.3 0.25

14.

x 8 10 24 2

y 4 5 12 1


16. x 2, y 3, z 4

17. x 4, y 3, z 24

18. x 8, y 54, z 144

Simple Interest


The simple interest I (in dollars) for a savings account is jointly proportional to the product of the time t (in years) and the principal P (in dollars). After six months, the interest on a principal of $2000 is $55. 19. Find the constant of variation k. 20. Write an equation that relates I, t, and P. 21. What will the interest be after two years?

16



Answer Key Practice B 1. direct variation 2. direct variation 3. inverse variation 4. neither 5. direct variation 6. neither 4 4 3 54 7. y ; 18 8. y ; 9. y ; 1 x 3 x x 3 9 xy; z 10. 4 2 11. z 32xy; 192 12. z 3xy; 18 13. k 0.035 14. I 0.035Pt 15. $612.50 16. k 12.84 17. PV 12.84 18. 10.7 cubic liters

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LESSON

NAME _________________________________________________________ DATE ___________

9.1


Tell whether x and y show direct variation, inverse variation, or neither. 1. x 5.

1 2

y 9

2. y x 6.

y 15 24 4.5 1.5

x 3 5 4.5 7

4. y x 5

y 5 21 16.25 45

Lesson 9.1

x 5 8 1.5 0.5

3. xy 0.1


8. x 72, y

1 18

9. x 6, y

1 2


Simple Interest

1

1

11. x 1, y 8, z 4

12. x 2, y 8, z 12


The simple interest I (in dollars) for a savings account is jointly proportional to the product of the time t (in years) and the principal P (in dollars). After nine months, the interest on a principal of $3500 is $91.88. 13. Find the constant of variation k. 14. Write an equation that relates I, t, and P. 15. What will the interest be after five years?

Boyle’s Law


Boyle’s Law states that for a constant temperature, the pressure P of a gas varies inversely with its volume V. A sample of hydrogen gas has a volume of 8.56 cubic liters at a pressure of 1.5 atmospheres. 16. Find the constant of variation k. 17. Write an equation that relates P and V. 18. Find the volume of the hydrogen gas if the pressure changes to 1.2

atmospheres.



17

Answer Key Practice C 1. direct variation 2. inverse variation 3. direct variation 4. direct variation 5. inverse variation 6. neither 0.3 1 18 1 1 ; 7. y ; 3 8. y 9. y ; x 20 x 5x 30 10. z 12xy; 144 11. z 16xy; 192 11 12. z 6 xy; 22 13. k 150,000 14. dp 150,000 15. 12,500 units 16. k 0.22 17. H 0.22mT 18. 3.9424 kilocalories

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LESSON

NAME _________________________________________________________ DATE ___________

9.1


Tell whether x and y show direct variation, inverse variation, or neither. 1. x 4y

x 1 2 0.5 0.25

5 y

3. x 6.

y 4 2 8 16

x 3 7 2.5 5.7

y 3

4.

2 7 x y

y 6 10 5.5 8.7

Lesson 9.1

Lesson 9.1

5.

2. x

The variables x and y vary inversely. Use the given values to write an equation relating x and y. Then find y when x 6. 3

7. x 2, y 12

8. x 3, y 0.1

1

2 5

2

3

9. x 2, y

The variable z varies jointly with x and y. Use the given values to write an equation relating x, y, and z. Then find z when x 3 and y 4. 1

10. x 2, y 8, z 3

Product Demand

3

5

11. x 4, y 6, z 10

12. x 3, y 4, z

11 12


A company has found that the monthly demand d for one of its products varies inversely with the price p of the product. When the price is $12.50, the demand is 12,000 units. 13. Find the constant of variation k. 14. Write an equation that relates d and p. 15. Find the demand if the price is reduced to $12.00.

Specific Heat


The amount of heat H (in kilocalories) necessary to change the temperature of an aluminum can is jointly proportional to the product mass m (in kilograms) and the temperature change desired T (in degrees Celsius). It takes 1.54 kilocalories of heat to change the temperature of a 0.028 kilogram aluminum can 250 C. 16. Find the constant of variation k. 17. Write an equation that relates H, m, and T. 18. How much heat is required to melt the can (at 660 C if its current

temperature is 20 C?

18



Answer Key Practice A 1. all real numbers except 5 2. all real numbers except 6 3. all real numbers except 0 1 4. x 1 5. x 2 6. x 3 7. y 2; all real numbers except 12 8. y 1; all real numbers except 1 9. y 6; all real numbers except 6 10. B 11. C 12. A 13.

14. y

y

2 1 x

1

15.

1

16. C 7x 250

y

17. A 1 1

18.

x

y

10 10

x

x

7x 250 x

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Page 29

LESSON

NAME _________________________________________________________ DATE ___________

9.2


Find the domain of the function. 1. f x

3 x5

2. f x

x4 x6

3. f x

2 5 x

6. f x

x7 x3

9. f x

4 6 x

12. f x

x1 x2

Find the vertical asymptote of the graph of the function. 4. f x

x x1

5. f x

6 7 x2

Find the horizontal asymptote of the graph of the function. Then state the range. 7. f x

x3 2x 1

8. f x

7 1 x2


2x 1 x1

B.

y

x1 x2

C.

y

Lesson 9.2

A.

11. f x

y

2 2 1 x

1 2

x

1

x

Graph the function. 13. f x

x1 x

Sports Banquet

14. f x

3 x2

15. f x

3 2 x2


You are organizing your high school’s sports banquet. The banquet hall rental is $250. In addition to this one-time charge, the meal will cost $7 per plate. Let x represent the number of people who attend. 16. Write an equation that represents the total cost C. 17. Write an equation that represents the average cost A per person. 18. Graph the model in Exercise 17.



29

Answer Key Practice B 1. y 5; x 4; domain: all real numbers except 4; range: all real numbers except 5 3 1 2. y 4; x 4; domain: all real numbers except 14; range: all real numbers except 34 2 3. y 2; x 3; domain: all real numbers except 2 3 ; range: all real numbers except 2 4. B 5. C 6. A 7. domain: all real 8. domain: all real numbers except 0; numbers except 2; range: all real numbers range: all real numbers except 0 except 3 y

y

1 x 1 1 x

1

9. domain: all real numbers except 3; range: all real numbers except 1


y

y

1 1 x

1 x

1

11. domain: all real 3 2;

numbers except range: all real numbers except 32 y

12. domain: all real

numbers except 21; range: all real numbers except 12 y

1 1

x

1 1

x

13.

14. y 2; the amount

r

4 4

t

of rain will be less than 2 inches 15. 0.8 inch

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Page 30

LESSON

NAME _________________________________________________________ DATE ___________

9.2


Identify the horizontal and vertical asymptotes of the graph of the function. Then state the domain and range. 1. y

2 5 x4

2. y

3x 4 4x 1

3. y

2x 1 2 3x 2

6. y

x3 x2


2 1 x3

A.

5. f x B.

y

2

2x 3 x3

C.

y

y

2 2 1 x

1

x

Lesson 9.2

1

x

Graph the function. State the domain and range. 2 x

8. y

x1 x3

11. y

7. y 10. y

Inches of Rain

4 3 x2 3x 2 2x 3

9. y 12. y

2 1 x3

x 2x 1


The total number of inches of rain during a storm in a certain geographic area 2t can be modeled by r where r is the amount of rain (in inches) and t is t8 the length of the storm (in hours). 13. Graph the model. 14. What is an equation of the horizontal asymptote and what does the

asymptote represent? 15. Use the graph to find the approximate number of inches of rain during a

storm that lasts 5 hours.

30



Answer Key

y

y

1 x 1

1 x

1



y

y

1 1

x

1 x

1


numbers except 23; range:all real numbers except 13

y

1 1

x


numbers except 3; range:all real numbers except 5

13. Child's dosage (milligrams)

Practice C 1 1. y 5; x 2; domain: all real numbers except 1 2 ; range: all real numbers except 5 3 1 2. y 4; x 8; domain: all real numbers except 18; range: all real numbers except 34 3. y 10; x 7; domain: all real numbers except 7; range: all real numbers except 10 4. C 5. A 6. B 7. domain: all real 8. domain: all real numbers except 0; numbers except 3; range: all real numbers range: all real numbers except 1 except 4

c 100 80 60 40 20 0

y

2

2

x

14. 40 milligrams

0 2 4 6 8 10 12 t Age (years)

15. y 100; the child’s dose will be less than 100 milligrams

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Page 31

LESSON

NAME _________________________________________________________ DATE ___________

9.2


Identify the horizontal and vertical asymptotes of the graph of the function. Then state the domain and range. 1. y

1 5 2x 1

2. y

6x 5 8x 1

3. y

12 10 x7

Match the function with its graph. 4. f x A.

1 3 x2

1 3 x2

6. f x

y

C.

5. f x B.

y

x2 x1

y

2

2

1

x 1

x

1

1

x

7. y

4 1 x

10. y

4x 1 2x 3

Young’s Rule

8. y 11. y

2 4 x3

9. y

x5 3x 2

12. y

Lesson 9.2

Graph the function. State the domain and range. 2 3 4x 1 5x x 3


Young’s Rule is a formula that physicians use to determine the dosage levels of medicine for young children based on adult dosage levels. The child’s dose can ta be modeled by c where c is the child’s dose (in milligrams), a is the t 12 adult’s dose (in milligrams), and t is the age of the child (in years). 13. Graph the model for t > 0 and a 100. 14. Use the graph to find the approximate dose for an eight-year-old child. 15. What is an equation of the horizontal asymptote and what does the

asymptote represent?



31

Answer Key Practice A 1. x-intercept: 0; vertical asymptotes: x 5, x 5 2. x-intercept: 1; vertical asymptotes: 1 x 3, x 2 3. x-intercept: 2; vertical asymptote: x 0 4. no x-intercepts; vertical asymptote: x 5 5. x-intercepts: 9, 1; no vertical asymptotes 6. x-intercepts: 6, 6; vertical asymptote: x 0 7. B 8. A 9. C 10.

11. y

y

1

1 x

1

12.

x

1

13. y

y

1

1 x

1

14.

2

x

15. y

y

1 1

x 2 2

16. 15 ft by 30 ft

x

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Page 41

LESSON

NAME _________________________________________________________ DATE ___________

9.3


Identify the x-intercepts and vertical asymptotes of the graph of the function. 1. y 4. y

x2

x 25

x1 x6

3. y

2x 1 x2

5. y

x2 8x 9 x2 2

6. y

x2 6 x

8. y

5x 2 x 4

9. y

x2 4x 5 x2

2. y

2 x5

x2


x2 x2 4

A.

B.

y

1

C.

y

y 3

1 1

x

2

x

x

1

Graph the function. 10. f x 13. f x

x 1

11. f x

x2 1 x2 4

12. f x

x1 x2 4

14. f x

x2 x1

15. f x

x2

x

x 1x 3 x x2 4x 12

16. Garden Fencing


Lesson 9.3

Suppose you want to make a rectangular garden with an area of 450 square feet. You want to use the side of your house for one side of the garden and use fencing for the other three sides. Find the dimensions of the garden that minimize the length of fencing needed.


41

Answer Key Practice B 1 1. x-intercepts: 2, 4; vertical asymptote: x 5 2. no x-intercepts; vertical asymptotes: x 1, x 1 3. x-intercepts: 0; vertical asymptotes: x 4 4. B 5. C 6. A 7.

8. y

y

2

x

1

1

2 x

9.

10. y

y

2

1 x

2

2

11.

x

12. y

y

2 1 1

x

1

x


Sample answer: y

Oxygen level

14.

L 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

x5 x2 3x 15. The oxygen level

dropped to 50% of normal, then slowly increased to 93% of normal.

0 2 4 6 8 10 12 14 t Weeks

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9:45 AM

Page 42

LESSON

NAME _________________________________________________________ DATE ___________

9.3


Identify the x-intercepts and vertical asymptotes of the graph of the function. 1. y

2x2 7x 4 x5

2. y

x2 1 x2 1

3. y

3x2 6x x2 6x 8

5. y

x2 2 x2 16

6. y

x3 x2


4 2 x 5x 4

A.

B.

y

1

C.

y

y

1

2 x

1

x

2

4

x


2x 6 x4

10. y

3x2 2x 6

8. y 11. y

5x 1 x2 1 2x2 x 9 3x2 12

9. y 12. y

3x2 4x 4 x2 5x 6 3x2 1 x3


Give an example of a rational function whose graph has two vertical asymptotes: x 3 and x 0, and one x-intercept: 5.

Lesson 9.3

Pollution


Suppose organic waste has fallen into a pond. Part of the decomposition process includes oxidation, whereby oxygen that is dissolved in the pond water is combined with decomposing material. Let L 1 represent the normal oxygen level in the pond and let t represent the number of weeks after the waste t2 t 1 is dumped. The oxygen level in the pond can be modeled by L . t2 1 14. Graph the model for 0 ≤ t ≤ 15. 15. Explain how oxygen level changed during the 15 weeks after the waste

was dumped.

42



Answer Key Practice C 2 1. no x-intercepts; vertical asymptotes: x 3, x 3 2. x-intercept: 2; vertical asymptote x 0 3. x-intercepts: 3, 5; no vertical asymptotes 4. A 5. C 6. B 7.

8. y

y

2 x

2

1 x

1

9.

10. y

y

1

2 1 x x

1

11.

12. y

y

1

1 x

1

1 x


Sample answer: y 14. h

300 πr2

Surface area(cm2)

15. S 2πr2 16.

2x2 x2 x 12

600 r

S 500 400 300 200 100 2 4 6 8 10 r Radius (cm)

17. r 3.67 cm and h 8.42 cm

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Page 43

LESSON

NAME _________________________________________________________ DATE ___________

9.3


Identify the x-intercepts and vertical asymptotes of the graph of the function. 1. y

5 2 3x 7x 6

2. y

x3 8 x2

3. y

x2 8x 15 x2 4

5. y

x3 1 x2 2

6. y

x2 2x 3 2x2 x 3


6 2 x 9

A.

B.

y

C.

y

1

y

1 1

x

1 2

x

1

x

Graph the function. 2x2 x2 9

8. y

x2 x 2 x1

11. y

7. y 10. y

x2 10x 24 3x

9. y

3x3 1 4x3 32

12. y

x2 6x 9 x3 27 3 4x 10


Give an example of a rational function whose graph has two vertical asymptotes: x 4 and x 3, one x-intercept: 0, and one horizontal asymptote: y 2. Lesson 9.3

Manufacturing In Exercises 14–17, use the following information. A manufacturer of canned soup wants the volume of its cylindrical cans to be 300 cubic centimeters. 14. Use the volume formula V πr2h to express the can’s height h as a

function only of the can’s radius r. 15. Use the surface area formula S 2πr2 2πrh and your answer to

Exercise 14 to express the can’s surface area as a function only of the can’s radius. 16. Graph the function from Exercise 15 on the domain 0 < r < 10. 17. Find the dimensions of the can that has a volume of 300 cubic centimeters

and uses the least amount of material possible.



43

Answer Key Practice A x6 4x x3 x4 1. 2. 3. 4. 2x 3 x1 x3 x5 8x x1 3 5. not possible 6. 7. 8. 5 x1 4x2 xx 3 x2 x3 20x 9. 10. 11. 12. x1 x1 2 3y3 x 1 9x5y x 11 13. 14. 15. 16. 3x 1 2x2 x8 2 2x 2 4x 1 17. x 4x 3 18. 19. x6 3x 2

MCRB2-0904-PA.qxd

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Page 55

LESSON

NAME _________________________________________________________ DATE ___________

9.4


If possible, simplify the rational expression. 1.

4x2 2x2 3x

2.

x2 2x 15 x2 4x 5

3.

x2 16 x2 x 12

4.

x2 8x 12 x2 3x 10

5.

x2 2x 8 x2 3x 4

6.

x2 2x 1 x2 1

Multiply the rational expressions. Simplify the result. 7.

12x2y 5y2

9.

x2 2x x2 2x 1

2xy

3x2

8.

x2 4x 3 x2 3x

10.

4y2 9x

27

16xy2

x2 2x 3 x2

x2 2x x2 1

Divide the rational expressions. Simplify the result. 11.

5x5 15x2 8 12

12.

48x2 36xy2 y 5

13.

3x x2 2 x 1 x1

14.

x2 9x 22 x 2 x2 5x 24 x 3

Perform the indicated operations. Simplify the result. 15.

5x2y 2xy

6x3y5 10y

3x y3

17. x2 x 30

x2 11x 30 x2 7x 12

x6

x6

16.

x 11 x2 8x 33 2x 10 x5

18.

x2 5x 14 x2 6x 7

x3 x2

x2 4x 5

x2 x 30 2

19. CDs and Cassettes

Use the diagrams below to find the ratio of the volume of the compact disc storage crate to the volume of the cassette storage crate. 4x

s

CD

s

3x

tte

e ss

Ca

x x2 x1

x

Lesson 9.4



55

Answer Key Practice B x9 1 1. 2. 3. not possible x1 x2 9x6 1 3 6x 2 8x 4. 5. 6. 7. 8. 5 2 5 5x y 4y x3 10 5 xx 2 x 2x 6 9. 10. 11. x 3 5 5x2 2x 4 8 12. x 5x 4 13. 14. 3x x 2 7 x y 15. 16. 17. 2 6xx 5 3x 3 18x

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Page 56

LESSON

NAME _________________________________________________________ DATE ___________

9.4



x2 8x 9 x2 1

2.

x3 2 x 5x 6

3.

x2 4 x2 4

Multiply the rational expressions. Simplify the result. 4.

4x2y3 x5y6

6.

x2 4x 12 x4 9x3 18x2

xy

20x3 6x2

5.

81x9 y4

x2

7.

3x2 12 5x 10

36x5y 1

2x 4

Divide the rational expressions. Simplify the result. 8. 10.

12x2y 3x2 5y2 2xy

9.

5x2 20 x2 6x 8 25x2 x2 10x 24

x2 3x 2 x 1 25x 5x2

11. x 7

x2 9x 14 x2 5x 6

Perform the indicated operations. Simplify the result. 12. x2 x 30 14.

x2 6x 7 3x2

x2 2x 15 x2 7x 12 6x

x7

x1 4

x5

x6

13.

x2 x 20 33x2 132x 8x 40 x1 16x 16 11x 44

15.

3xy3 x3 y

y

6x

9y2 xy

Geometry Find the ratio of the area of the shaded region to the total area. Write your result in simplified form. 16.

17. 7 x 2 5x

x1

x

x

x 3

x3

Lesson 9.4

6(x 1)

56



Answer Key Practice C x5 3x 1 1. 2. not possible 3. 2 x 5x 25 x2 3 x2 3x11y 4. x 6 5. 6. 7. 10 x1 25 2 xx 2x 4 7x 3 8. 3x 2 9. 10. 4x 1 xx 5 x6 6 5x 5 11. 12. 13. 4x 2 5 4x3

x 10x 2 x 8x 1 1

15. 2 16. x x 3x 3x 4 1 14.

17. about 60,769 gallons

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Page 57

LESSON

NAME _________________________________________________________ DATE ___________

9.4



3x2 5x 2 x2 4

2.

2x 6 2 x 6x 9

3.

x2 25 x3 125

Multiply the rational expressions. Simplify the result. 4. x 5 6.

x2 36

x2 11x 30

3x2 12 5x 10

1

2x 4

5.

x2 2x x2 2x 1

7.

21x10y5 5x2

x2 4x 3 x2 3x

x3

35y4

Divide the rational expressions. Simplify the result. 8. x2 10x 24 10.

x2 144 3x 36

9.

x3 8 x2 x 2 64x 16x2

11.

7x2 21x x2 x2 2x 35 x 7 2x3 12x2 8x3 24x2 x2 4x 12 x2 9x 18

Perform the indicated operations. Simplify the result. 12.

x2 3x 2 x2

3x

14. x2 7x 30

Swimming Pools

2x 4

x 2 5x2 5x x2 5x 24 x2

13.

x2

x2 3x 2

15.

x2 100 4x2 x3

x3 5x2 50x x 102 x4 10x3 5x

x2 9 1 2 10x x3

x 10

x2 7x 12


You are considering buying a swimming pool and have narrowed the choices to two—one that is circular and one that is rectangular. The width of the rectangular pool is three times its depth. Its length is 6 feet more than its width. The circular pool has a diameter that is twice the width of the rectangular pool, and it is 2 feet deeper. 16. Find the ratio of the volume of the circular pool to the volume of the

rectangular pool. 17. The volume of the rectangular pool is 2592 cubic feet. How many gallons

of water are needed to fill the circular pool if 1 gallon is approximately 0.134 cubic foot?

Lesson 9.4



57

Answer Key Practice A 2x 1 4 5x 1. 2. 3. x3 x x1 4. x 1x 1 5. x 4x 4 6. x 2x 1 7. 2xx 12 15x 2 x2 4x 6 8. 9. 2 3x 2x2 6x 13 x 17 10. 11. x3 x 5x 1 x6 x2 4x 12. 13. x 2x 2 x 1x 1 2x 1 x1 3x x2 14. 15. 16. x2 x1 x1 R1R2 4 17. R 18. ohms R1 R2 3

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LESSON

Practice A

Lesson 9.5

9.5

NAME _________________________________________________________ DATE ___________


Perform the indicated operation and simplify. 1.

7 9 4x 4x

2.

x 5 x1 x1

3.

2x 1 x3 x3

Find the least common denominator. 4.

6 5 , x 1 x2 1

5.

x 3 , x 4 x2 16

6.

x 1 , x x2 x2

7.

5 1 3 , , 2x 1 2x 2x 12

2

Perform the indicated operation(s) and simplify. 8. 10. 12.

5 2 2 x 3x

9.

3 4 x5 x1 x2

1 2 3 2 2 x x

11. 6

4x 3 4 x2

13.

5 x3

x 3x 2 x1 x 1

Simplify the complex fraction.

14.

1 1x 15. 1 1x

2 1x x

Electrical Resistors

x 6 1 3 16. 3x


When two resistors with resistances R1 and R2 are connected in parallel, the 1 . total resistance R is given by R 1 1 R1 R2

17. Simplify this complex fraction. 18. Find the total resistance (in ohms) of a 4 ohm resistor and a 2 ohm resistor

that are connected in parallel.



69

Answer Key Practice B 1. 2x 12x 1 3. xx 1x 1 5. 7. 9. 11. 13. 15.

2x 1 x 2x 1

2. 4x 4

7x x2 6 6. x 6x 5 4.

24x2 5x 3 x 8. 2 x x 3 x1 2 4x 7 x 3x 9 10. xx 3x 3 x3 2 x 9x 12 3x 1 12. 2 3x 4x2 11x 2 1 14. 3x2 x 2 6 1374t2 20,461t 1,627,410 I 85 t55 2t

16. about 554,000 MDs; about 24,000 DOs

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Lesson 9.5

LESSON

9.5

NAME _________________________________________________________ DATE ___________



5 6 , 2 2x 1 4x 1

2.

3 x2 , x4 4

3.

x2

4 5 , 1 xx 1

Perform the indicated operation(s) and simplify. 4.

7 x x2 x2

5.

x 1 x x2 x2

6.

x 1 x x 30 x 5

7.

4 2 4 2 x x x3

8.

x2 2 14 2 x 1 x 6 x 5x 6

9.

x 3 x 9 xx 3

2

10. 4

5 x3

11.

2

2

1 3 4 2 3 x x


1x 2x 1 1 12. 2x4x 1 Doctors

3x1 x 4 2 13. x x 2 1x

14.

4x 2 12 2x 4 6 x 1 3


Over a twenty-year period the number of doctors of medicine M (in thousands) 28,390 693t in the United States can be approximated by M where t 0 85 t represents 1980. The number of doctors of osteopathy B (in thousands) can be 776 12t approximated by B . 55 2t 15. Write an expression for the total number I of doctors of medicine (MD)

and doctors of osteopathy (DO). Simplify the result. 16. How many MDs and DOs did the United States have in 1995?

70



Answer Key Practice C 1. 4x 4x 4 2. xx 1x 12 1 3. xx 2x 6 4. x1 2x2 8x 21 5x2 2 5. 6. 3x 4x 3 x3 2x 1 42x 5 x 5 7. 8. 9. x2 x x 22 22x2 4x 3 2x 12x 1 10. 11. 2 x 3 x 3 2xx 12 x2 15x 14 x3 12. 13. 10 x x3x 4 14. 4x3 9x 36 R1R2R3R4 15. Rt R1R2R3 R1R2R4 R1R3R4 R2R3R4 16. Rt

R1R2R3R4R5 R1R2R3R4 R1R2R3R5 R1R2R4R5 R1R3R4R5 R2R3R4R5

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LESSON

Practice C

Lesson 9.5

9.5

NAME _________________________________________________________ DATE ___________



3 x x2 , 2 , x 4 x 16 4

3.

7 3 5x , 2 , x 6 xx 2 x 8x 12

2.

x2

13 4 5 , 2 , 2x 1 x 1 xx 1

Perform the indicated operation(s) and simplify. 4.

2x 1 1 x x2 x2

5.

5 3x 1 2 2 3x 12 x x 12 3

6.

2 3x 1 x 2 x x2 x3

7.

3x 5x 40 2 x2 x2 x 4

8.

2x 8 3 2 x 2 x 2x x

9.

2x 1 6x 3 2 x 4x 4 x 4 x 2

10.

2

2x 5 x 1 2 6x 9 x 9 x 3

x2

11.

2

5 3 1 2x 1 2x 2x 12


12.

x 1 9 15 x

2

2 10x 9

Electronics Pattern information.

x 4 25 x 2 5 13. x 1 5 x 1 5

x x 4 14 14. 4x9 x x 4

2

2

In Exercises 15 and 16, use the following

The total resistance Rt (in ohms) of three resistors in a parallel circuit is given 1 by the formula Rt , which can be simplified to 1 1 1 R1 R2 R3 Rt

R1R2R3 . R1R2 R1R3 R2R3

15. Simplify the similar formula for four resistors in a parallel circuit given by

the formula Rt

1 . 1 1 1 1 R1 R2 R3 R4

16. Following the pattern (without algebraically simplifying the complex

fraction), write the simplified formula for the total resistance Rt (in ohms) of five resistors in a parallel circuit.



71

Answer Key Practice A 1 1. no 2. yes 3. no 4. no 5. 3 9 12 6. no solution 7. 8. 4 9. 7 7 10. 4 11. no solution 12. 5, 1 13. 6 14. 2, 5 15. 7, 4 16. 5, 6 4 9 17. 11 18. 3, 3 19. 3 20. 3 21. no solution 22. no solution 23. 4.5 miles, 8 miles

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Page 82

LESSON

9.6

NAME _________________________________________________________ DATE ___________


Lesson 9.6

Determine whether the given x-value is a solution of the equation. 1.

2 3 , x 1 x3 x1

2.

7 x ,x4 x3 4

3.

x 1 4 ,x4 x5 x3

4.

3x 1 x 3 , x 2 x2 x2

Solve the equation by using the LCD. Check each solution. 5.

3 2 4 x x1 x

6.

4 x 1 x4 x4

8.

1 2 4 x x2 x

9.

3 2x 5 x3 x3

10.

7.

15 6 4 3 x x

1 4 1 x 2 x 2 x2 4

Solve the equation by cross multiplying. Check each solution. 11.

2x 3 3x x3 x4

14.

x8 2 x1 x1

12.

5 x 2x 1 4 x

13.

6 x x3 x3

15.

7 x x3 4

16.

x 3 x2 10 2x 1

19.

5x 7 8 x2 x2


3 5x 6 x1 x1

18.

5x 14 2 2 x1 x 1

20.

1 1 x3 2 x 5 x 5 x 25

21.

2x 4 4 x4 x4

22.

1 5 1 2 x2 x3 x x6

23. Population Density

The population density in a large city is related to

the distance from the center of the city. It can be modeled by 5000x D 2 x 36 where D is the population density (in people per square mile) and x is the distance (in miles) from the center of the city. Find the areas where the population density is 400 people per square mile.

82



Answer Key Practice B 1 1. yes 2. no 3. no solution 4. 3 5. 0 7 6. no solution 7. 3 8. 5 9. 5 10. 11 11. 7, 4 12. 3 13. 4, 4 14. 5 15. no solution 16. 0, 2 17. 5, 2 18. 2, 2 1 19. 6 20. 1 21. 12,000 dozen cards 22. 30 miles per hour

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LESSON

NAME _________________________________________________________ DATE ___________

9.6


Determine whether the given x-value is a solution of the equation. 1.

1 1 10 2 ,x5 x3 x3 x 9

2.

x 4 1 ,x4 x4 x4

Solve the equation by using the LCD. Check each solution. 3x 6 1 x2 x2

4.

3x 1 4 2 x2 x2 x 4

5.

2 3 5x 5 2x 5 2x 5 4x2 25

6.

5 4 14x 3 2 2x 3 2x 3 4x 9

8.

3x 1 x 3 x2 x2

7.

15 6 4 3 x x

Lesson 9.6

3.

Solve the equation by cross multiplying. Check each solution. 9.

x1 2 x3

10.

2 3 x3 x1

11.

7 x x3 4

12.

6 5x 7 3x x

13.

x 2 2 x 8 x

14.

2x x2 5x 5 5x


5x 10 7 x2 x2

16.

2x x2 4x x4

17.

3x 12 2 2 x1 x 1

18.

6 7x x x 5 10

19.

3 4 12 2 x 3x

20.

x2 2x 2 2x 3 x1 x1

21. Average Cost

A greeting card manufacturer can produce a dozen cards for $6.50. If the initial investment by the company was $60,000, how many dozen cards must be produced before the average cost per dozen falls to $11.50?

s2 20 where d is the distance (in feet) that the car travels before coming to a stop, and s is the speed at which the car is traveling (in miles per hour). Find the speed that results in a braking distance of 75 feet.

22. Brakes

The braking distance of a car can be modeled by d s



83

Answer Key Practice C 1. 20 2. no solution 3. 10 4. 2 5. 2 3 6. 1, 3 7. no solution 8. 17 9. 1, 3 7 10. 8 11. 2, 5 12. 3 13. 2 14. no solution 15. 12 16. 12 17. 85 18. 4 19. 1.17 (Jan.), 12 (Dec.) 20. 50,000 baskets

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LESSON

9.6

NAME _________________________________________________________ DATE ___________


Lesson 9.6

Solve the equation by using the LCD. Check each solution. 1.

2 3 6 x 10 x 2 x2 12x 20

2.

2 1 2 x2 6x 8 x 4 x 2

3.

100 4x 5x 6 6 3 4

4.

4 3 8 x 2 x 1 x2 x 2

5.

3 4 28 x 8 x 2 x2 10x 16

6.

2x 4x 1 17x 4 x 2 3x 2 3x2 4x 4

Solve the equation by cross multiplying. Check each solution. 7. 10.

3x 1 2x 5 6x 2 4x 13 2 2 2x 3 x 5

8. 11.

5x 2 x8 10x 3 2x 3

9.

x 2 3x 5 x 1

12.

x 3 2x 1 x 2 x1 2 2x 3


15.

x2 4 3x 5

14.

2x 5 18 4 2 x3 x x 3x

1x x 1 1 2 16. 1 x 1

x 4 3 3 1 17. 4x 1 4 x3

4 1 5x 6 2 x2 x2 x 4

x 7 1 x 3 1 3 18. 2 x 1 2

19. Temperature

The average monthly high temperature in Jackson, 191t 30 Mississippi can be modeled by T 2 where T is t 16.5t 114

measured in degrees Fahrenheit and t 1, 2, . . . 12 represents the months of the year. During which month is the average monthly high temperature equal to 57.3 F? 20. Average Cost

You invest $40,000 to start a nacho stand in a shopping mall. You can make each basket of nachos for $0.70. How many baskets must you sell before your average cost per basket is $1.50.

84



Answer Key Test A 1. 72 8.49; 3, 3 5 2. 72 8.49; 3, 3 3. 13; 4, 2 4. 10; 0, 0 5.

4

6.

y x2

4

y x2

x

4

(2, 16)

7.

4

(2, 16)

8.

y x2

4

y x2

x

4

x

4

(2, 16)

(2, 16)

9.

x

4

10.

y

4

y x2 x

4

2 x

2

(2, 16)

11. y 2 12x

12. x 2 24y

13. x 2 y 2 16 15. 17. 18. 19. 20. 21. 22. 23.

14. x 12 y 12 25

x2 y2 x2 y 2 1 16. 1 25 36 1 3 2 2 circle; x y 16 parabola; y 2 2x circle; x 2 y 2 16 y2 x2 hyperbola; 1 4 25 y2 x2 ellipse; 1 4 16 y2 x2 hyperbola; 1 4 16 x2 y2 hyperbola; 1 25 4

24. circle; x 62 y 62 36 26. 2, 0

27. about 0.5 cm

25. none

28. parabolic

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CHAPTER

NAME _________________________________________________________ DATE ____________

10


Find the distance between the two points. Then find the midpoint of the line segment connecting the two points. 1. 0, 0, 6, 6

2. 0, 0, 6, 6

3. 10, 5, 2, 0

4. 4, 3, 4, 3

Answers 1. 2. 3.

Graph the equation. 5. x 2 y 2 16

4.

6. y 2 4x

y

y

2

1 2

7.

x2

y2

x

16

8.

x2

Use grid at left.

6.

Use grid at left.

7.

Use grid at left.

8.

Use grid at left.

9.

Use grid at left.

10.

Use grid at left.

x

1

y2

5.

16

y

y

2

2 2

x

9. 4x 2 9y 2 100

x

2

10.

x 22 y 2 1 4 1

y

y

2

1 2

x

1 x

Review and Assess



105

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CHAPTER

NAME _________________________________________________________ DATE ____________

10

Chapter Test A

CONTINUED


Write an equation for the conic section.

11.

11. Parabola with vertex at 0, 0 and focus at 3, 0

12.

12. Parabola with vertex at 0, 0 and directrix y 6

13.

13. Circle with center at 0, 0 and radius 4

14.

14. Circle with center 1, 1 and radius 5 15. Ellipse with center 0, 0, vertex at 0, 6, and co-vertex at 5, 0 16. Hyperbola with center 0, 0, foci at 2, 0 and 2, 0, and vertices

at 1, 0 and 1, 0

17. x y 16 0 19.

3x 2

2

3y 2

48 0

21. 4x 2 y 2 16 0

16. 17.

Classify the conic section and write its equation in standard form. 2

15.

18.

y2

2x 0

20. 25x 2 4y 2 100 22. 4x 2 y 2 16

18. 19. 20. 21.

23. 4x 2 25y 2 100

22.

24. x 2 y 2 12x 12y 36 0

23. 24.

Find the points of intersection, if any, of the graphs in the system. 25. x 2 y 2 16

y5

26.

x2 y2 1 4 16 x2

27. Telescope

25. 26. 27. 28.

The equation of a mirror in a particular telescope is

2

x , where x is the radius (in centimeters) and y is the depth 520 (in centimeters). If the mirror has a diameter of 32 centimeters, y

what is the depth of the mirror? 28. Classify the mirror in Exercise 27 as parabolic, elliptical, or

Review and Assess

hyperbolic.

106



Answer Key y 22 x 32 1 9 36 24. parabola; x 2 12y 25. 0, 0 26. none y 27. 28. parabolic

Test B 7 3 1. 13; 2, 2 2. 13 3.61; 2, 1 9 1 3. 82 9.06; 2, 2 15 4. 41 6.40; 2 , 6 5.

6.

y

23. hyperbola;

y 0.1

2

2

7.

2

x

2

2

8.

y

x

y

1 x

1

2 x

2

9.

10.

y

y

2

2 2

11. y 2 12x

x

6 x

12. x 2 y 2 25

13. x 12 y 22 16

14.

x2 y2 x2 y 2 1 16. 1 16 9 4 5 2 2 y x 17. hyperbola; 1 9 5 2 2 y x 18. ellipse; 1 9 4 2 2 19. circle; x y 25 15.

20. hyperbola; y2 x2 1 21. ellipse; x 12 y 2 1 22. circle; x 42 y 12 25

x2 y 2 1 9 4

x

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CHAPTER

NAME _________________________________________________________ DATE ____________

10



2. 3, 2, 0, 0

3. 0, 0, 9, 1

4. 10, 8, 5, 4

Answers 1. 2. 3.


4.

6. y 2 25x

y

y

2

2 x

2

7.

x2 9

y2 16

2

8.

Use grid at left.

6.

Use grid at left.

7.

Use grid at left.

8.

Use grid at left.

9.

Use grid at left.

10.

Use grid at left.

x

x 8 y 5 1 16 4 2

1

5.

y

2

y

1 x

1

2 x

2

9. x 22 y 32 16

10. y 32 2 x 2 9

y

y

2

2 2

x

2

x

Review and Assess



107

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CHAPTER

NAME _________________________________________________________ DATE ____________

10

Chapter Test B

CONTINUED



11.

11. Parabola with vertex at 0, 0 and directrix x 3

12.

12. Circle with center at (0, 0 and passing through 3, 4

13.

13. Circle with center at 1, 2 and radius 4

14.

14. Ellipse with center at 0, 0, x-intercepts of 3, 0 and 3, 0, and

y-intercepts 0, 2 and 0, 2

15.

15. Ellipse with center at 0, 0, vertex 4, 0, and co-vertex 0, 3

16.

16. Hyperbola with foci at 3, 0 and 3, 0 and vertices at 2, 0 and

17.

2, 0

18.

Classify the conic section and write its equation in standard form.

19.

17. 5x 2 9y 2 45

18. 4x 2 9y 2 36

20.

19. x 2 y 2 25 0

20. y2 1 x2

21.

21. 4x 2 8x 4y 2 0

22. x 2 y 2 8x 2y 8 0

22.

23. x 2 4y 2 6x 16y 29 0

23.

24. x 2 y 32 y 32

24.

Find the points of intersection, if any, of the graphs in the system. 25. x 2 y 2 2x 2y

x 2 y 2 2x 2y 0

26. y x 2

yx2

25. 26. 27. 28.

27. Telescope The equation of a mirror in a particular telescope is

x2 , where x is the radius (in centimeters) and y is the depth 780 (in centimeters). Graph the equation of the mirror. y

28. Classify the mirror in Exercise 27 as parabolic, elliptical, or

Review and Assess

hyperbolic.

108



Answer Key Test C 1. 80 8.94; 4, 2 1 1 2. 162 12.7; 2, 2 3. 116 10.8; 8, 3 4. 8 2.83; 6, 2 5.

15

20. parabola; x 2 2 y

22.

6.

y

23.

y 6

24. 4 x

4

x

2

25. 26. 7.

8.

y

y

1 x

1

1 1

9.

10.

y

x

y 4

2

x

2 2 x

11. x 2 12y

12. x 32 y 22 25

x2 y2 x 32 y 12 1 14. 1 25 4 4 1 x2 y2 x 32 y 22 15. 1 16. 1 16 20 9 16 x2 y2 17. ellipse; 1 16 9 5 2 1 11 18. parabola; x 6 3y 12 x2 y2 19. hyperbola; 1 9 16 13.

y 22 x 62 1 1 9 circle; x 42 y 32 25 circle; x 22 y 32 16 x 82 y 52 ellipse; 1 16 4 23 23 11 11 , , , 0, 3, 2 4 2 4 1 2 0, 4 27. x 20y 28. 14 feet

21. hyperbola;

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CHAPTER

NAME _________________________________________________________ DATE ____________

10



2. 5, 4, 4, 5

3. 10, 8, 6, 2

4. 7, 3, 5, 1

Answers 1. 2. 3.


4.

6. y 2 9x

y

y

4

1 x

4

7. 3x 2 y 2 12

5.

Use grid at left.

6.

Use grid at left.

7.

Use grid at left.

8.

Use grid at left.

9.

Use grid at left.

10.

Use grid at left.

x

1

8. x 2 y 2 2x 4y 1

y

y

2 x

2

1 1

9.

x 42 y 52 1 9 4

10.

x

y 22 x 32 1 9 36

y

y 2 2

x

2 2 x

Review and Assess



109

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CHAPTER

NAME _________________________________________________________ DATE ____________

10

Chapter Test C

CONTINUED



11.

11. Parabola with vertex at 0, 0 and focus at 0, 3

12.

12. Circle with center at (3, 2 and radius 5

13.

13. Ellipse with vertices at 5, 0 and 5, 0, and co-vertices at 0, 2

14.

and 0, 2

14. Ellipse with center at 3, 1, vertices at 1, 1 and 5, 1, and

co-vertices at 3, 0 and 3, 2

16.

15. Hyperbola with vertices at 4, 0 and 4, 0 and foci at 6, 0

and 6, 0

16. Hyperbola with foci at 2, 2 and 8, 2 and asymptote with

slope

15.

4 3

17. 18. 19.

Classify the conic section and write its equation in standard form. 17. 9x 2 16y 2 144 0

18. y 3x 2 5x 3

19. 16x 2 9y 2 144 0

20. 2x 2 15y 0

20. 21. 22.

21. x 2 9y 2 12x 36y 9 0

23.

22. x 2 y 2 8x 6y 0

24.

23. x 2 y 2 4x 6y 3 0

25.

24. x 2 4y 2 16x 40y 148 0

26.

Find the points of intersection, if any, of the graphs in the system.

27.

25. x 2 4y 2 36

x2 y 3

2

26.

28.

y2

x 1 25 16 16x 2 y 2 2y 8 0

27. Communications The cross section of a television antenna dish is

a parabola. The receiver is located at the focus, 5 feet above the vertex. Find an equation for the cross section of the dish. (Assume the vertex is at the origin.) Review and Assess

28. If the television antenna dish in Exercise 27 is 10 feet wide, how

deep is it?

110



Answer Key Cumulative Review 7 1. 60 meters 2. 3744 hours 3. 712 liters 4. yes; 18 5. yes; 6 6. yes; 12 7. no; 89 8. yes; 0 9. no; 52

38. 4712 or 276

10.

55. 2

42. no

11. y

y 2

x

1

2

x

43. yes

44. no

40. 2715 45. no

47. decay

48. growth

49. decay

51. decay

52. growth

53. 5

60.

1

39. 1014

8 5

56. 4

61.

8 7

57. 3

62. 2.46

41. yes

46. no 50. growth

54. 6 9

58. 4 63. 1.03

59. 4 64. 0.981

65. y 1; x 0

66. y 2; x 3

67. y 1; x 2

68. y 1; x 4

3 5;

x 1 70. y 0; x 11 71. 3; $225 69. y

12. x 4; y 4

13. x 2; y 3

14. y x 3x 1; 3, 1 15. y x 5x 3; 5, 3 16. y xx 4; 0, 4 17. y 3x 2x 1; 2, 1 18. y x 3x 2; 3, 2 19. y 2x 5x 3; 2, 3 5

20. 9 18i

21. 18 i

23.

8 4 i 5 5

22.

24. y

y

2

1 2

x

x

1

25. 26. A

y 1 1

x

32. x 4

20 x4

34. x2 3x 9

27. C

28. B

37 x3 2 30. x 5 x1 7 31. 3x 7 x1 30 33. x 5 x5 29. 6x 13

35. 8

36. 5

37. 7

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CHAPTER

NAME _________________________________________________________ DATE ____________

10


Give the answer with the appropriate measure. (1.1) 1.

meters 16 minute 10 minutes

2.

hours 721 week 52 weeks

1

3

3. 153 liters 74 liters

Decide whether the function is linear. Then find the indicated value of f x. (2.1) 4. f x x 13; f 5

5. f x 6; f 7

7. f x 9x3 4x2 x 1; f 2

8. f x 2 5x; f

2 5

6. f x x 5; f 7

9. f x 4 x2 4; f 8 3

Graph the step function. (2.7)

10. f x

2, 3, 4, 5, 6,

if 0 if 1 if 2 if 3 if 4

< < < <
x2 4

Review and Assess

Use what you know about end behavior to match the polynomial function with its graph. (6.2) 26. f x 2x3 6x2 8x 9 A.

y

27. f x x 4 4x2 3

28. f x 3x3 x2 1

B.

C.

y

y

4 2

x 1

2 2

116


x

1

x


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CHAPTER

NAME _________________________________________________________ DATE ____________

10

Cumulative Review

CONTINUED


Divide using synthetic division. (6.5) 29. 6x2 5x 2 x 3

30. x2 4x 3 x 1

31. 3x2 4x x 1

32. x2 4 x 4

33. x2 5 x 5

34. x3 27 x 3

Simplify the expression. (7.2) 35. 16 3 4 38.

4 16

4 4

4 5 36.

39.

4 125

4 7 5 3 5 81

4 7 37.

10

40.

4 10

5 9

Graph the function f. Then use the graph to determine whether the inverse of f is a function. (7.4) 41. f x 3x 5

42. f x x2 8

43. f x 3x3

44. f x x 7

45. f x x 2x 3

46. f x 4x 4 2x 1

Tell whether the function represents exponential growth or exponential decay. (8.2) 47. f x 32

1 x 2 x

50. f x 53

48. f x 5

4x

49. f x 4

51. f x 45

2 x

3x

52. f x 40.25x

Use a property of logarithms to evaluate the expression. (8.5) 53. log39

27

56. log4 162

54. log2 43 57. log

1 1000

55. log5 58. ln

1 25

1 e4

Solve the exponential equation. (8.6) 59. 102x1 1003x4

60. 3x7 272x5

61. 83x 164x2

62. 3x 15

63. 102x 5 120

64. 3 3ex 5

Identify the horizontal and vertical asymptotes of the graph of the function. (9.2) 4 1 x

66. y

3 2 x3

67. y

x1 x2

68. y

x x4

69. y

3x 4 5x 5

70. y

4 x 11

Review and Assess

65. y

71. Breaking Even

You start a business selling wooden carvings. You spend $180 on supplies to get started; the wood for each carving costs $15. You sell the carvings for $75. How many carvings must you sell for your earnings to equal your expenses? What will your earnings and expenses equal when you break even? (3.2)



117

Answer Key Practice A 3 5 1. 5; 2, 2 2. 5; 3, 2 11 11 3. 37 6.08; 5, 2 4. 5; 2 , 4 1 1 5. 52 7.07; 2, 2 1 6. 13 3.61; 4, 2 5 1 1 7. 5; 2, 1 8. 82 9.06; 2, 2 5 1 9. 52 7.07; 2, 2 10. 253 14.56; 3, 1 11. 213 7.21; 1, 1 12. 217 8.25; 1, 4 13. isosceles 14. isosceles 15. scalene 16. y x 7 3 21 17. y x 3 18. y 2 x 4 2 31 3 11 19. y 3 x 6 20. y 4 x 2 21. y 2x 10 22. 4, 6 23. 0, 2 24. 1, 7 25. 0, 8 26. 16, 20 27. 0, 28 28. about 50.6 mi

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LESSON

NAME _________________________________________________________ DATE ___________

10.1


Find the distance between the two points. Then find the midpoint of the line segment joining the two points. 1. 0, 0, 4, 3

2. 5, 4, 1, 1

3. 2, 5, 8, 6

4. 7, 6, 4, 2

5. 4, 0, 3, 1

6. 5, 1, 3, 2

7. 1, 1, 4, 3

8. 5, 1, 4, 0

9. 0, 3, 5, 2

10. 1, 8, 5, 6

11. 3, 2, 1, 4

Lesson 10.1

12. 2, 0, 0, 8

The vertices of a triangle are given. Classify the triangle as scalene, isosceles, or equilateral. 13. 0, 4, 8, 3, 8, 11

14. 0, 0, 3, 4, 4, 3

15. 1, 2, 1, 6, 0, 4

Write an equation for the perpendicular bisector of the line segment joining the two points. 16. 0, 2, 5, 7

17. 2, 7, 4, 5

18. 0, 2, 3, 4

19. 2, 6, 0, 3

20. 1, 0, 5, 8

21. 2, 2, 6, 6

Use the given distance d between the two points to solve for x. 22. 3, 5, 0, x; d 10

23. 1, 4, x, 2; d 5

24. 6, x, 2, 3; d 42

25. x, 6, 4, 9; d 5

Rhode Island


A coordinate plane is placed over the map of Rhode Island shown at the right. Each unit represents four miles.

y Woonsocket

26. Approximate the coordinates of the point representing Westerly. 27. Approximate the coordinates of the point representing Woonsocket.

Providence

28. Use the distance formula to approximate the distance between

Westerly and Woonsocket.

RI x

Westerly



13

Answer Key Practice B 1 1. 29 5.39; 5, 2 1 2. 157 12.53; 1, 2 7 3. 13 3.61; 2, 1 4. 213 7.21; 2, 2 5. 81.64 9.04; 0.4, 3 1 6. 17 4.12; 6, 4 7. scalene 8. isosceles 9. scalene 10. y 3x 9 3 21 11. y 2 x 4 12. y x 4 13. 0, 4 14. 1, 15 15. about 166 miles 16. about 0.74 hours or 44 minutes 17. about 74 miles 18. about 14.8 hours

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LESSON

NAME _________________________________________________________ DATE ___________

10.1


Lesson 10.1


2. 2, 5, 4, 6

3. 5, 0, 2, 2

4. 6, 1, 2, 5

5. 2.5, 1, 1.7, 7

6.

23, 6, 13, 2


8. 9, 2, 3, 6, 3, 2

9. 8, 5, 1, 2, 3, 2


11. 2, 5, 1, 7

12. 0, 6, 2, 4

Use the given distance d between the two points to solve for x. 13. 3, 2, 10, x; d 53

Wisconsin

14. 3, x, 5, 7; d 217


A coordinate plane is placed over the map of Wisconsin shown at the right. Each unit represents 10.5 miles.

y

15. Approximate the distance in miles between LaCrosse and

Green Bay. 16. How long would a flight from LaCrosse to Green Bay take

WISCONSIN EauClaire

traveling at 225 miles per hour? 17. Approximate the distance in miles between EauClaire and

LaCrosse.

Green Bay LaCrosse

x

18. What is the minimum time necessary to walk from EauClaire

to LaCrosse walking at a rate of five miles per hour?

14



Answer Key Practice C 1. 2173 26.31; 8, 1 2. 61.61 7.85; 0.75, 1 3. 109.8 10.48; 3.9, 1.9 4. 5. 6.

17 493 9 7.40; 6 , 1 1 19 1813 64 5.32; 16 , 8 1 1 13,549 3600 1.94; 12 , 40

7. isosceles

8. scalene 37 10

9. isosceles

9 11. y 2 12. x 3 13. 8.8, 1.2 14. 2, 8 15. about 169 miles 16. about 3.1 hours 17. 21, 31.5 18. 12:03 P.M.

10. y

45 x

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LESSON

NAME _________________________________________________________ DATE ___________

10.1



2. 2.5, 3.2, 4, 1.2

4. 5, 2, 4 2 3,

5.

1 2,

5,

38,

14

3. 5.1, 7, 2.7, 3.2

6. 3, 5 , 2, 4 2

4

1 3

7. 4, 2, 3, 1, 1, 4

Lesson 10.1


9. 5, 8, 1, 6, 2, 1


11. 7, 3, 7, 12

12. 9, 2, 3, 2

Use the given distance d between the two points to solve for x. 13. 3.5, x, 6, 3.8; d 115.25

Wisconsin

14. 5, 8, x, 11; d 32


A coordinate plane is placed over the map of Wisconsin shown at the right. Each unit represents 10.5 miles.

y

15. Approximate the distance in miles between Green Bay and

EauClaire. 16. How long would a trip from Green Bay to EauClaire take

WISCONSIN EauClaire

traveling at 55 miles per hour? 17. At the halfway point of your trip from Green Bay to EauClaire,

you need to pick up your friend. Approximate the coordinates of the meeting point.

Green Bay LaCrosse

x

18. If you leave Green Bay at 10:30 A.M., at what time will you

meet your friend?



15

Answer Key Practice A 1. B 2. C 3. A 4. right 7. down 8.

5. up

6. left

9. y

y

1

1 1

x

1

2, 0; x 2

0, 3; y 3

10.

11. y

x

y

1

1 1

x

1

x

4, 0; x 4 0, 12 ; y 12 12. x2 24y 13. y2 8x 14. y2 12x 15. x2 4y 16. x2 12y 17. y2 4x 18. y2 16x 19. x2 24y 20. x2 80y

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Page 27

LESSON

NAME _________________________________________________________ DATE ___________

10.2


Match the equation with its graph. 1. y2 2x A.

2. x2 2y

3. x2 2y

B.

y 1

C.

y

y

2 2

x 1

1

x

1

x

Tell whether the parabola opens up, down, left or right. 4. 2y2 x

5. x2 8y

7. y 4x2

6. y2 10x

Graph the equation. Identify the focus and directrix of the parabola. 8. y2 8x

9. x2 12y

10. y2 16x

11. x2 2y Lesson 10.2

Write the standard form of the equation of the parabola with the given focus and vertex at (0, 0). 12. 0, 6

13. 2, 0

14. (3, 0

15. 0, 1

Write the standard form of the equation of the parabola with the given directrix and vertex at (0, 0). 16. y 3

17. x 1

18. x 4

The course for a sailboat race includes a turnaround point marked by a stationary buoy. The sailboats must pass between the buoy and the straight shoreline. The boats follow a parabolic path past the buoy, which is 40 yards from the shoreline. Find an equation to represent the parabolic path, so that the boats remain equidistant from the buoy and the straight shoreline.

19. y 6

20. Sailboat Race

40 yards

Shoreline



27

Answer Key Practice B 1. down 2. up

3. right

5.

4. left

6. y

y

1

1 x

1

0, 12 ;

x

1

4, 0; x 4

y 12

7.

8. y

y

2 1 x

1

14, 0;

x 14

x

1

0, 3; y 3

9.

10. y

y

1 2

x

1

1

0, 161 ;

1, 0; x 1

1 y 16

11.

12. y

y

1

1 1

32, 0;

x

x

2

x 32

18, 0;

13. y2 8x

14. x2 4y

16. x2 12y

17. y2 4x

19. y2 x

20. x2 2y

22. x2 5y

x

x 18 15. y2 2x

18. x2 12y 21. x2 14y

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LESSON

10.2

NAME _________________________________________________________ DATE ___________


Tell whether the parabola opens up, down, left or right. 1. x2 8y

2. x2 12y

3. y2 16x

4. y2 24x

Graph the equation. Identify the focus and directrix of the parabola. 5. x2 2y 9. 4x2 y 0

6. y2 16x 10. 4x y2 0

7. y2 x

8. x2 12y

11. y2 6x 0

12. x 2y2


14. 0, 1

15.

12, 0

16. (0, 3

Write the standard form of the equation of the parabola with the given directrix and vertex at (0, 0). 17. x 1

18. y 3

19. x

1 4

20. y

1 2

Lesson 10.2

21. Television Antenna Dish The cross section of a television antenna

dish is a parabola. For the dish at the right, the receiver is located at the focus, 3.5 feet above the vertex. Find an equation for the cross section of the dish. (Assume the vertex is at the origin.)

3.5 feet

22. Sailboat Race

The course for a sailboat race includes a turnaround point marked by a stationary buoy. The sailboats must pass between the buoy and the straight shoreline. Find an equation to represent the parabolic path, so that the boats remain equidistant from the buoy and the straight shoreline. 2.5 miles

Shoreline

28




2. y

y

5 1 x

1 1

x

1 0, 321 ; y 32

0, 169 ;

3.

9 y 16

4. y

y

1

2 x

1

163 , 0;

x

2

8, 0; x 8

3 x 16

5.

6. y

y

1

2 1

x

3, 0; x 3

x

2

0, 5; y 5

7.

8. y

y

1

1 1

x

1

x

0, 2; y 2 16, 0; x 16 9. y2 32x 10. x2 8y 11. y2 48x 12. x2 64y 13. x2 2y 14. y2 3x 5 15. x2 y 16. y2 2x 17. y2 4x 18. y2 20x 19. x2 24y 20. x2 8y 1 21. y2 2x 22. x2 y 23. x2 3 y 3 24. y2 2 x 25. 6.25 ft 26. about 13,741 ft3

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LESSON

10.2

NAME _________________________________________________________ DATE ___________


Graph the equation. Identify the focus and directrix of the parabola. 9

1. 8x2 y

2. x2 4 y

3. 3x 4y2

4. y2 32x 0

5. y2 12x 0

6. 20y x2 0

7. x2 8y 0

8. 2x 3y2 0


13. 0,

12

10. 0, 2 14.

34,

0

11. 12, 0

15. 0,

1 4

12. 0, 16

16.

58, 0

Write the standard form of the equation of the parabola with the given directrix and vertex at (0, 0). 17. x 1

18. x 5

1 2

22. y 4

21. x

1

19. y 6 23. y

20. y 2

1 12

3

24. x 8

25. Solar Energy

Cross sections of parabolic mirrors at a solar-thermal complex can be modeled by the equation

y

where x and y are measured in feet. The oil-filled heating tube is located at the focus of the parabola. How high above the vertex of the mirror is the heating tube?

5

26. Storage Building

A storage building for rock salt has the shape of a paraboloid which has vertical cross sections that are parabolas. The 1 2 equation of a vertical cross section is y 12 x . If the building is 27 feet high, how much rock salt will it hold? (Hint: The volume of a paraboloid is v 12 r 2h, where r is the radius of the base and h is the height.)

5

x

y x

27 ft

r



29

Lesson 10.2

heating tube

1 2 x y 25

Answer Key Practice A 1. B 2. C 3. A 4.

5. y

y

1

3 1

x

r2

x

3

r 10

6.

7. y

y

2

1 2

x

r5

1

x

r 6 2.45

8.

9. y

y

1

1 1

x

1

r 23 3.46 r 2 1.41 10. x2 y2 4 11. x2 y2 64 12. x2 y2 6 13. x2 y2 2 14. x2 y2 1 15. x2 y2 25 16. x2 y2 20 17. x2 y2 29 1 10 18. y 4x 17 19. y 3 x 3 20. x2 y2 400

x

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LESSON

NAME _________________________________________________________ DATE ___________

10.3


Match the equation with its graph. 1. x2 y2 16

2. x2 y2 36

3. x2 y2 3

A

B.

C.

y

1

y

2 1

x

y

2 2

x

2

x

Graph the equation. Give the radius of the circle. 4. x2 y2 4

5. x2 y2 100

6. x2 y2 25

7. x2 y2 6

8. x2 y2 12

9. x2 y2 2

Write the standard form of the equation of the circle with the given radius and whose center is the origin. 10. 2

11. 8

12. 6

13. 2

Write the standard form of the equation of the circle that passes through the given point and whose center is the origin. 14. 0, 1

15. 5, 0

16. 2, 4

17. 5, 2

The equation of a circle and a point on the circle is given. Write an equation of the line that is tangent to the circle at that point. 18. x2 y2 17; 4, 1

19. x2 y2 10; 1, 3 Lesson 10.3

20. Garden Irrigation

A circular garden has an area of about 1257 square feet. Write an equation that represents the boundary of the garden. Let 0, 0 represent the center of the garden.



41


9.

1.

10. x2 y2 22

y

2 2 11. x y 42 12. x2 y2 45

2. y

y 1 x

1 1

14. x2 y2 65

1 x

1

1

x

15. x2 y2 40 16. x2 y2 5

r4

2

18. y 3x

r 13 3.61

3.

y

2

3 x

2

r 210 6.32

3

x

1

x

1

x

r 11

5.

6. y

y

1

1 x

1

r 7 2.65

r3

7.

8. y

y

1 2

x

17. x2 y2 41 13 3

1

4

19. y 5x

20. x2 y2 160,000

4. y

13. x2 y2 490

41 5

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LESSON

10.3

NAME _________________________________________________________ DATE ___________


Graph the equation. Give the radius of the circle. 1. x2 y2 16

2. x2 y2 13

3. x2 y2 40

4. x2 y2 121

5. 3x2 3y2 21

6. 4x2 4y2 36

The equations of both circles and parabolas are given. Graph the equation. 7. x2 4y 0

8. 2x2 2y2 16

9. x2 12y 0


11. 42

12. 35

13. 710


15. 2, 6

16. 2, 1

17. 4, 5


19. x2 y2 41; 4, 5

Jacob’s Field is the home field of the Cleveland Indians major league baseball team. The stadium is approximately circular with a diameter of 800 feet. Suppose a coordinate plane was placed over the base of the stadium with the origin at the center of the stadium. Write an equation in standard form for the outside boundary of the stadium.

Lesson 10.3

20. Jacob’s Field

42




9.

1.

10. x2 y2 35

y

11. x2 y2 32

2. y

2

y

12. x2 y2 14 1

1

2

x

15. x2 y2 16. x2 y2 145

r 7 2.65

r6

3.

18. y 7x 50

4. y

20. 17.05 cm

y

1

2 x

1

r4

2

x

r 26 4.90

5.

6. y

y

2

2 x

2

r6

2

x

r 25 4.47

7.

8. y

y

1

1

1 1

x

13. x2 y2 54 14. x2 y2 244

2 x

1

x

x

17. x2 y2 10 19. y

2

2

x6

257 4

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Page 43

LESSON

10.3

NAME _________________________________________________________ DATE ___________


Graph the equation. Give the radius of the circle. 1 2 2x

3. 3x2 3y2 48

12 y 2 18

1. x2 y2 7

2.

4. 6x2 6y2 144

5. 7x2 7y2 252

6. 20x2 20y2 400

The equations of both circles and parabolas are given. Graph the equation. 7. 4x2 4y2 1

8. 4x2 y 0

9. 4x y2 0


11. 42

12. 14

13. 36


15.

12, 8

16. 12, 1

17. 1, 3


19. x2 y2 24; 22, 4

20. Water Lily


10 cm 21 cm x

Lesson 10.3

In his novel Kavenaugh, Henry Wadsworth Longfellow stated the following puzzle about the water lily. When the stem of the water lily is vertical, the blossom is 10 centimeters about the surface of the lake. If you pull the lily to one side, keeping the stem straight, the blossom touches the water at a spot 21 centimeters from where the stem formerly cut the surface. How deep is the water?


43

Answer Key Practice A 1. vertices: ± 9, 0; co-vertices: 0, ± 2; foci: ± 77, 0 2. vertices: 0, ± 5; co-vertices: ± 4, 0; foci: 0, ± 3 3. vertices: 0, ± 4; co-vertices: ± 23, 0; x2 y2 foci: 0, ± 2 4. 1; vertices: 0, ± 2; 1 4 co-vertices: ± 1, 0; foci: 0, ± 3 x2 y2 5. 1; vertices: ± 13, 0; 169 1 co-vertices: 0, ± 1; foci: ± 242, 0 x2 y2 6. 1; vertices: 0, ± 5; 4 25 co-vertices: ± 2, 0; foci: 0, ± 21 7.

8. y

y

2

2 x

2

4

vertices: ± 7, 0; co-vertices: 0, ± 4; foci: ± 33, 0 9.

vertices: 0, ± 8; co-vertices: ± 2, 0; foci: 0, ± 215 vertices: ± 6, 0; co-vertices: 0, ± 2; foci: ± 42, 0

y

4

2

10.

x

x

vertices: 0, ± 6; co-vertices: ± 3, 0; foci: 0, ± 33

y

2 2

x

11.

12. y

y

1 1

x

2

x

vertices: 0, ± 4; vertices: 0, ± 10; co-vertices: ± 2, 0; co-vertices: ± 1, 0; foci: 0, ± 23 foci: 0, ± 311 x2 y2 y2 x2 1 14. 1 13. 49 25 25 9 2 2 2 x y y2 x 1 16. 1 15. 1 4 16 100 x2 y2 x2 y2 1 18. 1 17. 64 36 9 8 2 2 2 x y y2 x 1 20. 1 19. 32 36 25 16 x2 y2 1 22. 3, 0, 3, 0 21. 4 9

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Page 54

LESSON

10.4

NAME _________________________________________________________ DATE ___________


Write the equation in standard form (if not already). Then identify the vertices, co-vertices, and foci of the ellipse. 1.

x2 y2 1 81 4

4. 144x2 36y2 144

2.

x2 y2 1 16 25

5. x2 169y2 169

3.

x2 y2 1 12 16

6. 25x2 4y2 100

Graph the equation. Then identify the vertices, co-vertices, and foci of the ellipse. 7.

x2 y2 1 49 16

8.

x2 y2 1 4 64

9.

x2 y2 1 36 4

10.

x2 y2 1 9 36

11.

x2 y2 1 4 16

12.

x2 y2 1 1 100

Write an equation of the ellipse with the given characteristics and center at (0, 0). 13. Vertex: 7, 0

Co-vertex: 0, 5 16. Vertex: 0, 10


Focus: 0, 2

14. Vertex: 5, 0


Co-vertex: 0, 6 20. Co-vertex: 0, 4

Focus: 3, 0

A semi-elliptical archway is to be formed over the entrance to an estate. The arch is to be set on pillars that are 10 feet apart. The arch has a height of 4 feet above the pillars. Where should the foci be placed in order to sketch the plans for the semi-elliptical archway?

15. Vertex: 0, 2


Focus: 1, 0 21. Co-vertex: 2, 0

Focus: 0, 5

22. Archway

y

4 ft (0, 0)

x

Lesson 10.4

10 ft

54




7.

1.

8. y

2.

y

y 8 5

2 5

2 x

2

2

x

3.

vertices: 0, ± 7; co-vertices: ± 1, 0; foci: 0, ± 43 vertices: ± 11, 0; co-vertices: 0, ± 10; foci: ± 21, 0

y

3

1 x

11.

x

12. y

5.

2

1 1

y

2 1

x2 y2 1; 16 9

x

y

1

4.

1

10. y

x

3

x

x

9.


2

x

x2 y2 1; 25 4 y

y

4

13.

1 1

x

2

x

15. 17.

vertices: ± 4, 0; co-vertices: 0, ± 3; foci: ± 7, 0

vertices: ± 5, 0; co-vertices: 0, ± 2; foci: ± 21, 0

x2 y2 6. 1; 1 169 vertices: 0, ± 13; co-vertices: ± 1, 0; foci: 0, ± 242

y

3

x

19. 20.

x2 y2 y2 x2 1 14. 1 9 36 4 16 x2 y2 x2 y2 1 16. 1 49 16 1 4 2 2 2 x y y2 x 1 18. 1 9 36 36 4 y2 x2 1 2 57.95 56.712 y2 x2 y 1; 16 36 2 2

x

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Page 55

LESSON

10.4

NAME _________________________________________________________ DATE ___________


Write the equation in standard form (if not already) and graph the equation. Then identify the vertices, co-vertices, and foci of the ellipse. 1.

x2 y2 1 64 81

4. 9x2 16y2 144

2.

x2 y2 1 1 49

3.

5. 4x2 25y2 100

x2 y2 1 121 100

6. 169x2 y2 169

In Exercises 7–12, the equation of parabolas, circles, and ellipses are given. Graph the equation. 7. x2 y2 152 10. 18y x2 0

8.

x2 y2 1 49 9

9. 6x2 3y2 24

11. 4y2 8x

12. x2 y2 82



Focus: 0, 3

14. Vertex: 0, 4

15. Vertex: 7, 0

Focus: 33, 0

Co-vertex: 2, 0 17. Co-vertex: 3, 0

18. Co-vertex: 0, 2

Focus: 0, 33

Focus: 42, 0

19. Astronomy

In its orbit, Mercury ranges between 46.04 million kilometers and 69.86 million kilometers from the sun. Use this information and the diagram shown at the right to write an equation for the orbit of Mercury.

a

y

a

20 c 20

20. Swimming Pool

An elliptical pool is 12 feet long and 8 feet wide. Write an equation for the swimming pool. Then graph the equation. (Assume that the major axis of the pool is vertical.) 69.86

x

46.04

Lesson 10.4



55

Answer Key Practice C x2 y2 1. 1; 4 16

5.

1 1

2.

x2

259

1 x

x

1

6.

499 1; y

co-vertices: ± 2, 0; foci: 0, ± 4

1

x

y2

1

vertices: 0, ± 25;


y

x2 y2 1; 4 20

x2 y2 1; 24 64 vertices: 0, ± 8;

vertices: 0, ± 73 ; co-vertices: ± 53, 0; 26 foci: 0, 3

2 2

x

7.

3.

x2 y2 1; 324 225

8.

1

vertices: ± 18, 0;

y

5 5

x

1 1

x

x

1

x

10.

2

x2 y2 1; 15 10

2

vertices: ± 15, 0;

y

co-vertices: 0, ± 10; 1 1

1

co-vertices: 0, ± 15; foci: ± 311, 0 9.

4.

co-vertices: ± 26, 0; foci: 0, ± 210

x

foci: ± 5, 0

8 x

Answer Key 11.

12.

2

2 2

x

2

13.

x2 x2 y2 y2 1 14. 1 64 81 16 9

15.

x2 y2 1 121 100

17.

x2 y2 1 25 4

19.

18.

x2 y2 1 or 42 15 2 4

20. 15 in.2

16.

x2 y2 1 1 49

x2 y2 1 1 169 x2

154 2

y2 1 42

x

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LESSON

10.4

NAME _________________________________________________________ DATE ___________


Write the equation in standard form (if not already) and graph the equation. Then identify the vertices, co-vertices, and foci of the ellipse. 1.

x2 y2 1 8 32 2

2.

4. 2x2 3y2 30

9x2 9y2 1 25 49

3.

5. 100x2 20y2 400

x2 y2 3 108 75

6. 64x2 24y2 1536

In Exercises 7–12, the equation of parabolas, circles, and ellipses are given. Graph the equation. 8. x2 y2 2

2

7. 12x2 24y 10. x2 15y2 15

11.

y2 16x 0 4

9.

x2 y2 1 49

12. x2 y2 42

2


14. Vertex: 4, 0


17. Co-vertex: 0, 2

Focus: 0, 43

Bicycle Chainwheel

Co-vertex: 0, 3 Focus: 21, 0

15. Vertex: 11, 0

Focus: 21, 0

18. Co-vertex: 1, 0

Focus: 0, 242


The pedals of a bicycle drive a chainwheel, which drives a smaller sprocket wheel on the rear axle. Many chainwheels are circular. However, some are slightly elliptical, which tends to make pedaling easier. The front chainwheel on the bicycle shown at 1 the right is 8 inches at its widest and 72 inches at its narrowest.

Front Chainwheel

8 in.

19. Find an equation for the outline of this elliptical chainwheel. 20. What is the area of the chainwheel?

7 1 in.

Lesson 10.4

2

56



Answer Key Practice A x2 y2 1. A 2. C 3. B 4. 1 1 2 36 2 2 2 y x y x 5. 6. 1 1 4 9 4 49 7. vertices: ± 12, 0; foci: ± 65, 0 8. vertices: 0, ± 3; foci: 0, ± 109 9. vertices: 0, ± 11; foci: 0, ± 221

15.

10.

16.

2 2

11. y

y

2

3 x

2

3

x

foci: ± 41, 0; asymptotes: y ± 45 x

foci: 0, ± 58; asymptotes: y ± 73 x

12.

13. y

y

2

3 x

2

3

14.

foci: ± 26, 0; asymptotes: y ± 5x

y 6

2

x

foci: foci: 0, ± 113; asymptotes: y ± 87 x

± 61, 0 asymptotes: y ± 56 x

x


y

x

x2 y2 y2 x2 1 17. 1 4 5 16 9 y2 x2 y2 x2 1 19. 1 18. 1 35 1 15 y2 x2 9 1 20. 9 4

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LESSON

NAME _________________________________________________________ DATE ___________

10.5

Lesson 10.5


Match the equation with its graph. 1.

x2 y2 1 36 4

A.

2.

y2 x2 1 4 36

B.

y

3. C.

y

3

1 3

x

x2 y2 1 4 36 y

1 1

x

1

x

Write the equation of the hyperbola in standard form. 4. 36x2 y2 36

5. 9y2 4x2 36

6. 49x2 4y2 196

Identify the vertices and foci of the hyperbola. 7.

x2 y2 1 144 36

8.

y2 x2 1 9 100

9.

x2 y2 1 121 100

Graph the equation. Identify the foci and asymptotes. 10.

x2 y2 1 25 16

11.

y2 x2 1 49 9

12.

x2 y2 1 36 25

13.

y2 x2 1 64 49

14.

x2 y2 1 1 25

15.

x2 y2 1 16 25

Write an equation of the hyperbola with the given foci and vertices. 16. Foci: 3, 0, 3, 0

Vertices: 2, 0, 2, 0 18. Foci: 0, 6, 0, 6

Vertices: 0, 1, 0, 1

17. Foci: 5, 0, 5, 0

Vertices: 4, 0, 4, 0 19. Foci: 4, 0, 4, 0


20. Write an equation for the hyperbola having vertices at 0, 3 and 0, 3

and with asymptotes y 2x and y 2x.



67

Answer Key Practice B x2 y2 x2 y2 1. 1 2. 1 25 4 64 4 2 2 x y 3. 1 16 9 4. vertices: ± 9, 0; foci: ± 313, 0 5. vertices: ± 6, 0; foci: ± 210, 0 6. vertices: 0, ± 3; foci: 0, ± 109 7.

11. y

4

x

x

2

foci: ± 37, 0; asymptotes: y ± 6x


13.

14. y

1 1

x

2

y

1

y

8

8. y

12.

y

x

1

1 x

1


foci: (0, ± 5; asymptotes: y ± 34 x 9.

1


y

1

15.

2 2

1

x

1

10.

foci: ± 3, 0; asymptotes:

y

1 1

x

y±

2

2

x

19.

y2 x2 1 9 72 y2 x2 1 17. 9 40 y2 x2 1 18. 9 16.

y

x

x

x2 R2 r2 y2 1 20. 16 8 1 64

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Lesson 10.5

LESSON

10.5

NAME _________________________________________________________ DATE ___________


Write the equation of the hyperbola in standard form. 1. 4x2 25y2 100

2. y2 16x2 64

3. 9x2 16y2 144 0


x2 y2 1 81 36

5.

x2 y2 1 36 4

6.

x2 y2 1 9 100

9.

y2 x2 1 25 9


y2 x2 1 9 16

10. x2 2y2 2

8.

x2 y2 1 4 9

11. 36x2 y2 36

12. 49y2 4x2 196

In Exercises 13–15, the equations of parabolas, circles, ellipses, and hyperbolas are given. Graph the equation. 13. 9x2 4y2 36

14. 9x2 4y 0

15. 9x2 4y2 36



17. Foci: 7, 0, 7, 0


18. Foci: 0, 10, 0, 10

19. Foci: 65, 0, 65, 0



20. Machine Shop

A machine shop needs to make a small engine part by drilling two holes of radius r from a flat circular piece of radius R. The area of the resulting part is 16 square inches. Write an equation that relates r and R.

68



Answer Key Practice C y2 x2 y2 x2 1. 1 2. 1 100 4 4 49 x2 y2 3. 1 4 9 4. vertices: ± 4, 0; foci: ± 25, 0 5. vertices: 0, ± 2; foci: 0, ± 6 6. vertices: ± 53, 0; foci: ± 57, 0 y 7. foci: (0, ± 9; asymptotes: 817 4 y± x x 4 17

8.

y

4 x

4

12.

foci:

y

4

6

x

13.

14. y

y

4

4 x

4

foci: ± 8, 0; asymptotes: 3 y± x 3

15.

4

16

foci: ± 7, 0;

2 x

2

10.

asymptotes: 310 y± x 20

x2 y2 1 2 y2 x2 1; about 15.8 m 20. 50 225 19.

11. y

y

1

1 1

x

foci: ± 6.1, 0; asymptotes: y ± 56 x

2

x

x

foci: ± 4, 0 asymptotes: y ± 15x

x

y2 x2 1 4 21 x2 1 17. y2 48 y2 x2 18. 1 15 1

16.

y

2

y

109

;

2 asymptotes: y ±

2

9.

0, ±

16

3 10 x

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LESSON

Practice C

Lesson 10.5

10.5

NAME _________________________________________________________ DATE ___________


Write the equation of the hyperbola in standard form. 1. y2 25x2 100

2. 49x2 4y2 196 0

3. 4y2 9x2 36 0


x2 y2 1 16 4

5.

y2 x2 1 4 2

6.

y2 x2 1 75 100

9.

y2 x2 1 9 40

12.

4y2 x2 1 9 25


y2 x2 1 64 17

8.

10. 2.5x2 3.6y2 9

x2 y2 1 48 16

11. 15x2 y2 15

In Exercises 13–15, the equations of parabolas, circles, ellipses, and hyperbolas are given. Graph the equation. 13.

y2 x2 1 169 225

14.

x2 y2 1 169 225

15. 4x2 4y2 100


17. Foci: 0, 7, 0, 7

Vertices: 2, 0, 2, 0 18. Foci: 0, 1, 0, 1

Vertices: 0,

14

, 0, 1 4


19. Foci: 3, 0, 3, 0


20. Modeling a Hyperbolic Lobby

The diagram at the right shows the hyperbolic overview of a building’s lobby. Write an equation that models the curved sides of the lobby. Then find the width of the lobby halfway between the main entrance and the front desk. (Note: x and y are measured in meters.)

y

Front desk

5 5

x

Main entrance



69

Answer Key Practice A 1. x 42 y 62 49 2. x 22 12y 2 y 42 x 12 1 3. 9 4 x2 y 52 1 5. B 6. E 7. A 25 24 8. D 9. C 10. F 11. x 62 y 92 121; circle; center: 6, 9; radius 11 12. x 52 4y 6; parabola; vertex: 5, 6; focus: 5, 5 x 32 y 12 13. 1; ellipse; 4 1 vertices: 5, 1, 1, 1; foci:3 3, 1, 3 3, 1 y2 x 12 14. 1; hyperbola; 4 3 vertices: 1, 2, 1, 2; foci: 1, 7, 1, 7 405 15. x 452 y 20 4 4.

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Page 81

LESSON

10.6

NAME _________________________________________________________ DATE ___________


Write an equation for the conic section. 1. Circle with center at 4, 6 and radius 7 2. Parabola with vertex at 2, 2 and focus at 2, 5 3. Ellipse with vertices at 1, 7 and 1, 1 and co-vertices at 3, 4 and

1, 4

Match the equation with its graph. 5.

x 52 y 42 1 25 16

6.

y 42 x 52 1 16 25

7.

y 52 x 42 1 16 25

8.

x 52 y 42 1 16 25

9.

y 42 x 52 1 25 16

10.

x 42 y 52 1 16 25

A.

B.

y

C.

y

2

y 2

x

8

4

x

2 2

D.

E.

y

x

F.

y

y

2 2 x

2

2

x 4

2

x

Classify the conic section and write its equation in standard form. For circles, identify the radius and center. For parabolas, identify the vertex and focus. For ellipses and hyperbolas, identify the vertices and foci. 11. x2 y2 12x 18y 4 0

12. x2 10x 4y 1 0

13. x2 4y2 6x 8y 9 0

14. 4x2 3y2 8x 16 0

15. Sprinkler System

A sprinkler system shoots a stream of water that follows a parabolic path. The nozzle is fastened at ground level and water reaches a maximum height of 20 feet at a horizontal distance of 45 feet from the nozzle. Find the equation that describes the path of the water.



81

Lesson 10.6

4. Hyperbola with vertices at 5, 5 and 5, 5 and foci at 7, 5 and 7, 5

Answer Key 16. y 12 16x 2;

Practice B 1. x 32 y 12 4 2. y 22 8x 3 x 32 y 12 1 3. 4 1 y2 x 82 1 4. 16 20 5.

y

2 4

x

6. y

y

17.

1 1

x 12 y 22 1; 4 16

x

2

y 2

x

2

x

1

vertex: 1, 4; focus: 3, 4 7.

y

2 x

2

center: 1, 2; radius 1 center: 10, 2; vertices: 7, 2, 13, 2; foci: 10 5, 2, 10 5, 2

18.

y 32 x 12 1; 2 18 y

2 2

8.

x

y

2 x

2

19.

center: 1, 1; vertices: 1, 1 30, 1, 1 30; foci: 1, 1 55, 1, 1 55 9. hyperbola

10. parabola

11. parabola

12. ellipse 13. circle 14. hyperbola 15. x 32 y 22 1; y

1 1

x

y 202 x2 1; 200 32

y

10

5

x

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Page 82

LESSON

10.6

NAME _________________________________________________________ DATE ___________


Write an equation for the conic section. 1. Circle with center at 3, 1 and radius 2 2. Parabola with vertex at 3, 2 and focus at 5, 2 3. Ellipse with vertices at 5, 1 and 1, 1 and co-vertices at 3, 2 and

Lesson 10.6

3, 0

4. Hyperbola with vertices at 8, 4 and 8, 4 and foci at 8, 6 and

8, 6

Graph the equation. Identify the important characteristics of the graph, such as the center, vertices, and foci. 5. y 42 8x 1

6. x 12 y 22 1

x 102 y 22 1 7. 9 4

8.

y 12 x 12 1 30 25

Classify the conic section. 9. 4x2 4y2 2x 4y 5 0

10. 3y2 2x 3y 1 0

11. x2 2x 3y 5 0

12. x2 3y2 x 2y 4 0

13. 3x2 3y2 3x 3y 1 0

14. 5x2 3y2 2x 3y 4 0

Write the equation of the conic section in standard form. Then graph the equation. 15. x2 y2 6x 4y 12 0

16. y2 2y 16x 31 0

17. 4x2 y2 8x 4y 8 0

18. 9y2 x2 2x 54y 62 0

19. Designing a Menu

You are opening a restaurant called the Treetop Restaurant. You are using a computer program to design the menu cover as shown at the right. The equation for the tree trunk is 25x2 4y2 160y 800 0. Write this equation in standard form and then sketch its graph.

The TREETOP

Restaurant and lounge

82



Answer Key Practice C 1. x 52 y 32 2 2. x 42 8y 1 x 22 y 22 3. 1 4 9 x 32 y 32 4. 1 1 9 5.

y 22 x 32 1; 9 16

16.

y 1 1

x

6. y

y

17. x 12 y 32 4; 4

y 1

1 x

4

1 x

vertex: 4, 1; focus: 4, 3 7. 4

8.

x

y

4

2

9. hyperbola

x

1

x

center: 5, 2; radius: 3

y

2

1

x

center: 0, 3; vertices: 0, 1, 0, 7; foci: 0, 3 32 ,

y 22 1; 4

0, 3 32

center: 3, 4; vertices: 8, 4, 2, 4; foci: 3 7, 4, 3 7, 4

10. ellipse

12. circle 13. hyperbola 15. y 12 4x 2

18. x 12

11. ellipse 14. parabola y

1 1

x

y

1

19.

x2 y 852 1; 8500 yd2 2 100 852

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Page 83

LESSON

10.6

NAME _________________________________________________________ DATE ___________


Write an equation for the conic section. 1. Circle with center at 5, 3 and radius 2 2. Parabola with vertex at 4, 1 and directrix y 1

3. Ellipse with vertices at 2, 1 and 2, 5 and foci at 2, 2 5 and

2, 2 5

4. Hyperbola with vertices at 4, 3 and 2, 3 and foci at 3 10, 3 and

Lesson 10.6

3 10, 3

Graph the equation. Identify the important characteristics of the graph, such as the center, vertices, and foci. 5. x 42 16y 1

6. x 52 y 22 3

y 32 x2 1 7. 16 2

8.

x 32 y 42 1 25 18

Classify the conic section. 9. x2 25y2 14x 100y 76 0

10. 25x2 y2 100x 2y 76 0

11. x2 36y2 16x 72y 64 0

12. x2 y2 2x 6y 9 0

13. x2 y2 2x 12y 31 0

14. x2 6x 2y 13 0

Write the equation of the conic section in standard form. Then graph the equation. 15. y2 2y 4x 7 0

16. 9x2 16y2 54x 64y 161 0

17. x2 y2 2x 6y 6 0

18. 4x2 y2 8x 4y 4 0

19. Aussie Football

In Australia, football (or rugby) is played on elliptical fields. The field can be a maximum of 170 yards wide and a maximum of 200 yards long. Let the center of the field of maximum size be represented by the point 0, 85. Write an equation of the ellipse that represents this field. Find the area of the field.



83

Answer Key Practice A 1. no 2. yes 3. yes 4. yes 5. no 6. no 7. 5, 4, 4, 5 8. 4, 3, 4, 3 9. 3, 6, 3, 6 10. (4, 3, 3, 4 11. none 12. 2, 1, 2, 1 13. 3, 3, 3, 5 14. 2, 0 15. 2, 3 16. none 17. 220 ft by 1980 ft or 990 ft by 440 ft

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Page 96

LESSON

NAME _________________________________________________________ DATE ___________

10.7


Determine whether the given point is a point of intersection of the graphs in the system. 1. x2 y2 5

2. x2 y2 13

3. x2 y2 18

y4

yx1

xy0

Point: 1, 2

Point: 2, 3

Point: 3, 3

5. 4x2 5y2 16

6. 6x2 3y2 12

y x

3x y 6

y x 2

Point: 6, 6

Point: 0, 2

Point: 0, 2

4. x2 6y

Find the points of intersection, if any, of the graphs in the system. 7. x2 y2 41

8. x2 y2 25

y x 1 10.

x2

y2

25

Lesson 10.7

yx1 13. x2 y2 x 2y 21 0

x2 y2 5x 2y 9 0

y 3 11.

x2

y2

9. x2 y2 45

y 2x 36

12. x2 y2 3

2y x2

x y 12 14. x2 8y2 4x 16y 4 0

x2 4x 16y 4 0

Find the points, if any, that the graphs of all three equations have in common. 15. y x 1

16. x2 y2 4

4x y 11

x2 y2 14

x2 4x y2 5

x 2y 5

17. Farming

A farmer has 2420 feet of fence to enclose a rectangular area that borders a river as shown in the figure at the right. Notice that no fence is needed along the river. Find the possible dimensions to enclose 10 acres. 1 acre 43,560 ft2

96



Answer Key Practice B 1. yes 4.

2. yes

14 214

7

,

7

3. no

,

14

7

,

214 7

5. 22, 1, 22, 1, 22, 1,

22, 1 6. 2, 2, 2, 2 12 16 7. 5 , 5 , 4, 0 8. 2, 2, 4, 8 1 3 9. 1, 2 , 9, 2 10. 0, 1, 0, 1, 4, 1, 4, 1 11. 0, 0, 0, 4,

103, 6 32 14, 103, 6 32 14

12. 1, 1, 3, 7 13.

23, 23, 23, 23

14. 3, 3

15. none 16. 563.5 ft by 773 ft or 386.5 ft by 1127 ft

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LESSON

10.7

NAME _________________________________________________________ DATE ___________


Determine whether the given point is a point of intersection of the graphs in the system. 1. x2 y 5

2. x2 y2 5

3. 3x2 y2 6

3x y 7

y 3x 5

y 2x 1

Point: 2, 1

Point: 2, 1

Point: 1, 3

Find the points, if any, that the graphs of all the equations in the system have in common. 4. 2x2 3y2 4

y 2x 7. x2 y2 16

x 2y 4 10. 2x2 2y2 8x 2 0

x2 5y2 4x 5 0 2x2 y 6x 7 0 14. x2 y2 x 2y 21 0

6. x2 y2 4

xy

x2 y2 9 8. x2 2y

9. 4y2 x

xy4

x 4y 3

11. x2 y2 4x 4y 0

2x2 y2 6x 4y 0 13. x2 y2 3 0

Lesson 10.7

12. x2 y 2 0

5. 2x2 3y2 19

2x2 y 0 15. x2 y2 16

x2 y2 5x 2y 9 0

x2 5y 5

9y 4x 15

x2 y2 27 0

16. Farming

A farmer has 1900 feet of fence to enclose a rectangular area that borders a river as shown in the figure at the right. No fence is needed along the river. Is it possible for the farmer to enclose 10 acres? 1 acre 43,560 ft2 If possible, find the dimensions of the enclosure. If not possible, justify your answer.



97

Answer Key Practice C 1. no 2. yes 3. yes 4. 2 21, 7 221, 2 21, 7 221 5. 5, 252 , 1, 12 6. 0, 0, 2, 1 7. 3, 4, 5, 0 8. 4, 3 , 2, 3 4

1

9. 5, 5 , 0, 1 8

3

10. 0, 2 11. 0, 1, 5, 6, 5, 6 12. 2, 4, 2, 2

13. 10, 12

14. none

15. 2, 0 16. yes; about 3.12 ft by 3.12 ft by 1.64 ft or 2 ft

by 2 ft by 4 ft

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Page 98

LESSON

10.7

NAME _________________________________________________________ DATE ___________


Determine whether the given point is a point of intersection of the graphs in the system. 1. x 2y2 6

x 8y 0 Point: 8, 1

2. y2 2x2 6

3. 5x2 3y2 17

y 2x

x y 1

Point: 3, 23

Point:

74, 34

Find the points, if any, that the graphs of all the equations in the system have in common. 4. 2x2 4y 22

y 2x 3

8. y

y 10 2x

y

Lesson 10.7

x2 y2 4x 4y 4 0 12. x2 y2 4x 6y 4 0

x2 y2 4x 6y 12 0 14. x2 2x 4 y2 10 0

6. y2 2 x

4x 2y 5

7. x2 y2 25 10. x2 y2 4x 4y 4 0

1

5. x2 2y 1 2 12 x 16 x

x 2y 0 9. x2 4y2 4

23

yx1

11. x2 y2 8y 7 0

x2 y 1 0 13. 4x2 y2 32x 24y 64 0

4x2 y2 56x 24y 304 0 15. x2 4y2 4x 8y 4 0

y 3x 5

x2 4y 4 0

2y2 x 3 0

7x 5y 14

16. Aquarium

You want to construct an aquarium with a glass top and two square ends. The aquarium must hold 16 cubic feet of water and you only have 40 square feet of glass to work with. Is it possible to construct such an aquarium? If possible, find the approximate dimensions of the aquarium. If not possible, justify your answer.

x

x

y

98



Answer Key Test A 1. arithmetic; d 2 2. neither; no constant d or r 3. geometric; r 2 4. arithmetic; d 5 5. 3, 4, 5, 6, 7, 8 6. 2, 6, 12, 20, 30, 42 7. 3, 6, 9, 12, 15, 18 8. 25; 5n 9. 19; 4n 1 10. 21; 3n 2 11. 243; 3n 12. 1; 6 n 1 1 13. 14. a1 5; an 0.4an1 ; 32 2n 15. a1 1; an an1 5 16. a1 36; an 19. 5

20. 62

12 an1

21. 2800

4

24. 16

25.

6n

n1

28. about $87,900

26.

5 9

17. 1275 22.

121 243

27. 610

18. 1000 23. 650

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Page 81

CHAPTER

11

NAME _________________________________________________________ DATE ____________


Tell whether the sequence is arithmetic, geometric, or neither. Explain your answer. 1. 1, 1, 3, 5, . . .

2. 3, 8, 9, 12, . . .

3. 2, 4, 8, 16, . . .

4. 6, 1, 4, 9, . . .

Answers 1. 2. 3.

Write the first six terms of the sequence. 5. an n 2

6. an nn 1

4.

7. an 3n

5.

Write the next term of the sequence, and then write the rule for the nth term. 8. 5, 10, 15, 20, . . . 11. 3, 9, 27, 81, . . .

9. 3, 7, 11, 15, . . . 12. 5, 4, 3, 2, . . .

10. 9, 12, 15, 18, . . . 13.

1 1 1 1 2 , 4 , 8 , 16 ,

. . .

Write a recursive rule for the sequence. (Recall that d is the common difference of an arithmetic sequence and r is the common ratio of a geometric sequence.) 14. r 0.4, a1 5

15. d 5, a1 1

16. 36, 18, 9, . . .

6. 7. 8. 9. 10. 11. 12.

Find the sum of the series. 50

17.

25

i

18.

i1 5

20.

3n 1

10

19.

n1

2i

i1

4

21.

7n

n1

13.

5 n

n1 5

22.

i1

1 i 3

14. 15. 16. 17. 18. 19. 20. 21. 22.

Review and Assess



81

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Page 82

CHAPTER

11

NAME _________________________________________________________ DATE ____________

Chapter Test A

CONTINUED


23. Find the sum of the first 25 terms of the arithmetic sequence

2, 4, 6, 8, . . . .

23. 24.

24. Find the sum of the infinite geometric series

8421. . . .

25.

25. Write the series 6 12 18 24 with summation notation.

26.

26. Write the repeating decimal 0.5 as a fraction.

27.

27. Stacking Containers Containers are stacked in 20 rows, with 2 in

28.

the top row, 5 in the second row, 8 in the third row, and so on. How many containers are in the stack? 28. Land If a parcel of land originally worth $25,000 increases in value

Review and Assess

15% per year, what will the land be worth in the tenth year?

82



Answer Key Test B 1. arithmetic; d 3 2. geometric; r 2 3. neither; no common d or r 4. geometric; r 3 5. 0, 3, 8, 15, 24, 35 6. 2, 5, 8, 11, 14, 17 7. 10, 17, 26, 37, 50, 65 8. 3125; 5n 9. 28; 6n 2 10. 5; n 1 1 11. 30; nn 1 12. ; 25 n2 14 n 9 13. 14. a1 10; an 0.5an1 ; 3 3 15. a1 1; an an1 10 a 16. a1 22; an n1 17. 15 18. 1890 2 61 19. 420 20. 1364 21. 31 22. 27 23. 620 24.

2 3

5

25.

3n 7

n1

28. about $412,000

26.

5 11

27. $46.50

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11



2. 3, 6, 12, 24, . . .

3. 0, 1, 3, 8, 15, . . .

4. 4, 12, 36, 108, . . .

Answers 1. 2. 3.

Write the first six terms of the sequence. 5. an n2 1

4.

6. an 3n 1

5.

7. a1 10 , an an1 2n 3

6.

Write the next term of the sequence, and then write the rule for the nth term. 8. 5, 25, 125, 625, . . .

9. 4, 10, 16, 22, . . .

10. 1, 2, 3, 4, . . . 12. 1,

1 1 1 4 , 9 , 16 ,

8.

11. 2, 6, 12, 20, . . .

. . .

13.

10 11 3, 3,

4,

13 3,

9.

. . .

10.

Write a recursive rule for the sequence. (Recall that d is the common difference of an arithmetic sequence and r is the common ratio of a geometric sequence.) 14. r 0.5, a1 10

15. d 10, a1 1

16. 22, 11,

11 11 2, 4,

5

20.

a

35

18.

90 2n

15

19.

a1

n1

i1

5

5

5

4

n

n1

21.

2

i1

11. 12.

. . .

13. 14.

Find the sum of the series. 17.

7.

i1

22.

3i 4

3

1 n1 3

15. 16. 17.

n1

18. 19. 20. 21. 22.

Review and Assess



83

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CHAPTER

11

NAME _________________________________________________________ DATE ____________

Chapter Test B

CONTINUED



8, 20, 32, 44, . . . .

23. 24.


1 12 14 18 . . . .

25. Write the series 10 13 16 19 22 with summation nota-

tion.

25. 26. 27.

26. Write the repeating decimal 0.45 as a fraction.

28.

27. Saving Dimes Your little sister decides to save dimes. She saved

one dime the first day, two dimes the second day, and so on. How much money did she save in 30 days? 28. Value of a Home Suppose the average value of a home increases

Review and Assess

5% per year. How much would a house costing $100,000 be worth in the 30th year?

84



Answer Key Test C 1. arithmetic; d 6 2. neither; no common d or r 2 3. arithmetic; d 1.5 4. geometric; r 3 5. 0, 2, 4, 6, 8, 10 3 8 15 24 35 6. 0, 2, 3, 4 , 5 , 6 7. 6, 0, 6, 12, 18, 24 7 n2 8. 4; n 1 9. ; 10. 1024; 4n 8 n3 6 n1 11. 125; n3 12. ; 13. 120; n! 5 n 14. a1 10; an 2an1 4 15. a1 1; an an1 3 a 16. a1 55; an n1 17. 63 18. 134 10 1360 31 22. about 6.67 19. 195 20. 32 21. 81 13

23. 500

24. 4

25.

4 3n

n1

26.

11 90

27. 0.25 meters

28. 120

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11



2. 0, 4, 9, 15, 22, . . .

3. 2, 0.5, 1, 2.5, . . .

4. 6, 4, 3,

8 16 9,

Answers 1. 2.

. . .

3.

Write the first six terms of the sequence. 4.

1 6. an n n

5. an 2 2n

7. a1 6

5.

an an1 6

6.

Write the next term of the sequence, and then write the rule for the nth term. 3 4 5 6

8. 0, 1, 2, 3, . . .

9. 4, 5, 6, 7, . . .

10. 4, 16, 64, 256, . . . 3

4

5

10.

13. 1, 2, 6, 24, . . .

Write a recursive rule for the sequence. (Recall that d is the common difference of an arithmetic sequence and r is the common ratio of a geometric sequence.) 4

14. a1 10, r 2

8. 9.

11. 1, 8, 27, 64, . . .

12. 2, 2, 3, 4, . . .

7.

15. a1 1, d 3

11. 12. 13. 14.

16. 55, 5.5, 0.55, 0.055, . . .

15.

Find the sum of the series. 6

17.

3n

43

18.

n1 5

20.

1 n 2

n1

8 n

n40

n4 19. 3 n1 30

4

21.

5 i 3

i1

10

22.

10

i0

1 i 2

16. 17. 18. 19. 20. 21. 22.

Review and Assess



85

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11

Chapter Test C

CONTINUED



4, 2, 0, 2, . . . .

23. 24.


2 1 12 14 . . . .

25. Write the series 7, 10, 13, . . . , 43 with summation

notation. 26. Write the repeating decimal 0.12 as a fraction.

25. 26. 27. 28.

27. Ball Bounce You drop a ball from a height of 128 meters. Each

time it hits the ground, it bounces 50% of its previous height. How high does the ball go after the ninth time it hits the ground? 28. Invitations You ask your friends to help you spread the word about

Review and Assess

your upcoming picnic that you are hosting at your house. You give an invitation to each of your three best friends. They in turn each give invitations to three more friends. How many friends will have received an invitation after the fourth level of friends has been invited?

86



Answer Key Cumulative Review

3, if 2 5, if 1 1. f x 7, if 3 9, if 7

≤ ≤ ≤ ≤

32.

x x x x

< < <
1 7. no; r 4, 4 > 1 8. no; r 2, 2 > 1 20 9. 6 10. 16 11. 9 12. no sum 13. 3 1 50 3 1 14. 9 15. no sum 16. 2 17. 4 18. 10 5 1 2 2 1 19. 2 20. 2 21. 9 22. 5 23. 9

24.

1 9

25.

8 9

26.

29. 8 revolutions

4 33

27.

3 11

28.

31 99

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LESSON

NAME _________________________________________________________ DATE ___________

11.4


Decide whether the infinite geometric series has a sum. Explain why or why not. 1.

n1

n

3

n1

5.

7 2

3

n0

4 5

2.

4

8 7

n1

6.

1 6

n1

n1

3.

101n1

n1

n1

7.

4.

2 5

5

n1

24

n

1

n1

1

n1

4 2

8.

n0

n1

n1

Find the sum of the infinite geometric series if it has one. 9.

n1

2

2 3

n1

1 10 13. 2 n1

10.

n0 n1

14.

12

1 4

n

50.1 n

n0

11.

n1

n

n1

5

4 9

3 4 15. 2 n0

8 8

12.

n1

16.

n0

1 3

n

Find the common ratio of the infinite geometric series with the given sum and first term. 17. S 4, a1 3

18. S 20, a1 18

19. S 6, a1 3

20. S 10, a1 15

21. S 9, a1 11

22. S 7, a1 1

5

Write the repeating decimal as a fraction. 23. 0.555. . .

24. 0.111. . .

25. 0.888. . .

26. 0.1212. . .

27. 0.2727. . .

28. 0.3131. . .

29. Compact Disc

In coming to a rest, suppose that a compact disc makes one half as many revolutions in a second as in the previous second. How many revolutions does the compact disc make in coming to a rest if it makes 4 revolutions in the first second after the stop function is activated?

Lesson 11.4



57

Answer Key Practice B 1 1 4 4 1. yes; r 4, 4 < 1 2. no; r 3, 3 > 1 2 2 3. no; r 2, 2 > 1 4. yes; r 9, 9 < 1 50 4 5. 6 6. no sum 7. 3 8. 9 9. 4 10. 10 1 2 5 20 4 1 11. 7 12. 9 13. 5 14. 4 15. 3 16. 5 74 4 7 2 2 35 17. 9 18. 8 19. 3 20. 9 21. 99 22. 99 400 9 9 181 23. 333 24. 11 25. finite; r 10, 10 < 1; 200 in. 26. 300 ft

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LESSON

NAME _________________________________________________________ DATE ___________

11.4


Decide whether the infinite geometric series has a sum. Explain why or why not. 1.

4

n0

1 4

n

2.

2 4 n1 3 3

n1

3.

6 2n

4.

n0

n1

2 9

n1


3

n0

9.

1 2

n

7

n1

3 4

6.

n0 n1

10.

3

4 3

0.9

n

n

7.

n0

11.

n0

2

1 2

0.4

n

8.

5

n1

n

12.

n0

1 10

2 0.1

n1

n

n0

Find the common ratio of the infinite geometric series with the given sum and first term. 13. S 15, a1 3 25 7,

16. S

a1 5

3

14. S 16, a1 12

15. S 4, a1 1

17. S 9, a1 5

18. S 2, a1

1 4


20. 0.222. . .

21. 0.3535. . .

22. 0.7474. . .

23. 0.543543. . .

24. 36.3636. . .

25. Length of a Spring

The length of the first loop of a spring is 20 inches. 9 The length of the second loop is 10 the length of the first. The length of 9 the third loop is 10 the length of the second, and so on. Suppose the spring had infinitely many loops. Does it have a finite or infinite length? Explain. If it has a finite length, find the length.

26. Ball Bounce

A ball is dropped from a height of 60 feet. Each time it hits the ground, it bounces two-thirds of its previous height. Find the total distance the ball has traveled before coming to rest. 40 ft 40 ft

26.7 ft 26.7 ft 17.8 ft 17.8 ft

Lesson 11.4

60 ft

58



Answer Key Practice C 1 21 2 1. 3 2. no sum 3. 2 4. 3 5. 3 6. 10 24 7. no sum 8. 5 9. no sum 10. 2 1 3 1 1 5 11. 6 12. 9 13. 4 14. 4 15. 2 40 8 300 1 1 4 16. 10 17. 11 18. 2 19. 9 20. 99 21. 11 22.

109 333

23.

653 999

24.

50,000 333

25. 120 cm 1.2 m;

after 6 swings 26. $3,000,000

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LESSON

NAME _________________________________________________________ DATE ___________

11.4



4

n1

5.

n1

0.3

n1

9.

1 3

1 10

2.

6.

n1

1 1 6 2

2

5 4

n

3.

10.

7

n0

40.6n

7.

n0

24 n

n0 n1

n0

12.

3

n1

1 7 n1 4 8

n1

11.

1 3

n0

7 2

n

4.

n

n0

n1

1 2 2 5

0.6 0.1

8.

4

n1

1 6

n1

n

n1

Find the common ratio of the infinite geometric series with the given sum and first term. 13. S 4, a1 7 16. S

10 9,

a1 1

8

14. S 9, a1 17. S

44 15 ,

2 3

a1 4

15. S

16 3,

a1 8

18. S 200, a1 100


20. 0.4040. . .

21. 27.2727. . .

22. 0.327327. . .

23. 0.653653. . .

24. 150.150150. . .

25. Pendulum

A pendulum is released to swing freely. On the first swing, the pendulum travels a distance of 24 centimeters. On each successive swing, the pendulum travels four fifths of the distance of the previous swing. What is the total distance the pendulum swings? After how many swings has the pendulum traveled 70% of its total distance?

26. Economy

A manufacturing company has opened in a small community. The company will pay two million dollars per year in employees’ salaries. It has been estimated that 60% of these salaries will be spent in the community, and 60% of this money will again be spent in the community. This process will continue indefinitely. Find the total amount of spending that will be generated by the company’s salaries.

Lesson 11.4



59

Answer Key Practice A 1. 4, 2, 0, 2, 4 2. 3, 15, 75, 375, 1875 5 5 5 3. 1, 7, 13, 19, 25 4. 10, 5, 2, 4, 8 5. 2, 5, 14, 41, 122 6. 7, 9, 12, 16, 21 7. 2, 4, 16, 256, 65,536 8. 5, 4, 8, 1, 15 9. 1, 4, 19, 364, 132,499 10. an 65n1; a1 6, an 5an1 11. an 23n1; a1 2, an 3an1 12. an 82

; a1 8, an 12 an1 an 6 10n; a1 4, an an1 10 an 2 2n; a1 0, an an1 2 an 4 3n; a1 7, an an1 3 a1 3, an an1 5 a1 2, an 4an1 a1 12, an an1 9 a1 48, an 12an1 a1 1, a2 3, an an1 an2 a1 1, an an12 1 a1 50, an 1.01an1 56; $766.56 a1 60, an an1 16; $112 1 n1

13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

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Lesson 11.5

LESSON

11.5

NAME _________________________________________________________ DATE ___________


Write the first five terms of the sequence. 1. a0 4

an an1 2 4. a1 10

an 12 an1 7. a0 2

an an12

2. a0 3

3. a0 1

an 5an1

an an1 6

5. a1 2

6. a1 7

an 3an1 1

an n an1

8. a0 5

9. a1 1

an an12 3

an n2 an1

Write an explicit rule and a recursive rule for the sequence. (Recall that d is the common difference of an arithmetic sequence and r is the common ratio of a geometric sequence.) 10. a1 6

r5 13. a1 4

d 10

11. a1 2

12. a1 8

r 12

r3 14. a1 0

15. a1 7

d 2

d3

Write a recursive rule for the sequence. The sequence may be arithmetic, geometric, or neither. 16. 3, 8, 13, 18, . . .

17. 2, 8, 32, 128, . . .

18. 12, 3, 6, 15, . . .

19. 48, 24, 12, 6, . . .

20. 1, 3, 4, 7, 11, . . .

21. 1, 2, 5, 26, . . .

22. Savings Account

On January 1, 2000, you have $50 in a savings account that earns interest at a rate of 1% per month. On the last day of every month you deposit $56 in the account. Write a recursive rule for the account balance at the beginning of the nth month. Assuming you do not withdraw any money from the account, what will your balance be on January 1, 2001? 23. Layaway Suppose you buy a $300 television set on layaway by making a down payment of $60 and then paying $16 per month. Write a recursive rule for the total amount of money paid on the television set in the nth month. How much will you have left to pay on the television set in the ninth month?

70



Answer Key Practice B 1. 3, 8, 13, 18, 23 2. 2, 8, 32, 128, 512 19 3. 32, 20, 14, 11, 2 4. 2, 2, 5, 7, 12 5. 1, 5, 29, 845, 714,029 6. 5, 9, 14, 23, 37 7. an 42n1; a1 4, an 2an1 8. an 105n1; a1 10, an 5an1 1 n1 ; a1 16, an 14 an1 9. an 16 4 10. an 2 3n; a1 1, an an1 3 14 1 11. an 3 5n; a1 3, an an1 5 11 1 1 12. an 2 2n; a1 6, an an1 2 13. a1 1, an an1 6 1 14. a1 36, an 3an1 15. a1 4, an an1 2 16. a1 1, an an12 3 17. a1 7500, an 0.88an1 600; 6022 trees 18. a1 200, an 1.005an1 70; $704.52 19. a1 50, an 0.4an1 50; about 83.3 mg

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LESSON

Practice B

Lesson 11.5

11.5

NAME _________________________________________________________ DATE ___________



an an1 5 4. a0 2

an n2 an1 3

2. a1 2

3. a1 32

an 12 an1 4

an 4an1 5. a0 1

6. a1 5, a2 9

an an12 4

an an1 an2

Write an explicit rule and a recursive rule for the sequence. (Recall that d is the common difference of an arithmetic sequence and r is the common ratio of a geometric sequence.) 7. a1 4

r2 10. a1 1

d 3

8. a1 10

9. a1 16

r 5

r 14

11. a1

1 3

12. a1 6 1

d5

d2

Write a recursive rule for the sequence. The sequence may be arithmetic, geometric, or neither. 4

13. 1, 7, 13, 19, . . .

14. 36, 12, 4, 3, . . .

15. 4, 2, 0, 2, . . .

16. 1, 4, 19, 364, . . .

17. Tree Farm

Suppose a tree farm initially has 7500 trees. Each year 12% of the trees are harvested and 600 seedlings are planted. Write a recursive rule for the number of trees on the tree farm at the beginning of the nth year. How many trees remain at the beginning of the eighth year?

18. Savings Account

On January 1, 2000, you have $200 in a savings account which earns interest at a rate of 12% per month. On the last day of every month you deposit $70 in the account. Write a recursive rule for the account balance at the beginning of the nth month. Assuming you do not withdraw any money from the account, what will your balance be on August 1, 2001?

19. Dosage

A person takes 50 milligrams of a prescribed drug every day. Suppose that 60% of the drug is removed from the bloodstream every day. Write a recursive rule for the amount of the drug in the bloodstream after n doses. What value does the drug level in the person’s body approach after an extended period of time?



71

Answer Key Practice C 35 89 1. 8, 10, 14, 22, 38 2. 81, 33, 17, 3 , 9 3. 2, 2, 4, 10, 22 4. 1, 2, 6, 15, 31 5. 1, 4, 4, 16, 64 6. 2, 6, 4, 2, 6 2 2 7. an 32n1; a1 3, an 2an1 8. an 3 4n; a1 7, an an1 4 5 1 1 1 9. an 6 3n; a1 2, an an1 3 10. an 60.2n1; a1 6, an 0.2an1 3 3 11. an 104 n1; a1 10, an 4 an1 12. an 3.5 1.5n; a1 2, an an1 1.5 13. a1 3, an 3an1

14. a1 4.3, an an1 0.6 15. a1 2, a2 3, an an1

an2

16. a1 24, a2 13, an an2 an1 17. a1 1, an 2an1 1 18. an 2n 1

19. 4.25 minutes;

18.20 hours; 136.19 years 20. a1 1, an an1 3n1

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Lesson 11.5

LESSON

11.5

NAME _________________________________________________________ DATE ___________



2. a1 81

an 2an1 6 4. a0 1

3. a0 2

an 13 an1 6

an n2 n an1

5. a1 1, a2 4

an an1 n2

an an1

6. a0 2, a1 6

an2

an an1 an2

Write an explicit rule and a recursive rule for the sequence. (Recall that d is the common difference of an arithmetic sequence and r is the common ratio of a geometric sequence.) 7. a1

2 3

8. a1 7

r 2

9. a1

d 13

d 4

10. a1 6

11. a1 10

r 0.2

r

1 2

12. a1 2

3 4

d 1.5

Write a recursive rule for the sequence. The sequence may be arithmetic, geometric, or neither. 13. 3, 33, 9, 93, . . .

14. 4.3, 4.9, 5.5, 6.1, . . .

15. 2, 3, 6, 18, . . .

16. 24, 13, 11, 2, . . .

Tower of Hanoi


This popular puzzle has three pegs and a number of discs of different diameters, each with a hole in the center. The initial position of the discs is shown in the figure. The objective is to move the tower to one of the other pegs by moving the discs to any peg one at a time in such a way that no disc is ever placed upon a smaller one. 17. Write a recursive rule for an, the number of moves required to transfer

n discs from one peg to another. 18. Find an explicit rule for an. 19. Suppose you can move one disc per second. Estimate the time required

to transfer the discs if n 8, n 16, and n 32. 20. Suppose the traditional rules for the Tower of Hanoi are modified. Now

you are required to move discs only to an adjacent peg. Write a recursive rule for an.

72



Answer Key Test A 1. 120 2. 210 3. 5 4. 35 5. 12 6. x3 3x 2y 3xy 2 y3 7. x 4 12x3 54x 2 108x 81 8. x5 5x 4y 10x3y 2 10x 2y3 5xy 4 y 5 13 9. 8x3 12x 2y 6xy 2 y3 10. 52 0.25 4 4 11. 52 0.0769 12. 52 0.0769 1 1 2 13. 52 0.0192 14. 52 0.0385 15. 2 1 16. 0% 17. 0.5 18. 0.30 19. 0.10 20. 1000 21. 68%; 68 students 22. 0.95 23. 0.475 24. 8; 2.19 25. 455

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12


Find the number of permutations or combinations. 1. 5P4

2. 7P3

3. 5C4

4. 7C3

5. Find the number of distinguishable permutations of the letters in

ERIE.

1. 2. 3. 4.

Expand the power of the binomial. 6. x y3

7. x 34

8. x y5

9. 2x y3

A card is drawn randomly from a standard 52-card deck. Find the probability of drawing the given card. 10. a diamond

11. a queen

13. the ten of spades

14. any black ace

12. an ace

Find the indicated probability. 15. PA

Answers

1 2

PA ?

16. PA 60%

17. PA ?

PB 40%

PB 0.8

PA or B 100%

PA or B 0.7

PA and B ?

PA and B 0.6

5. 6.

7.

8.

9.

10. 11. 12. 13. 14. 15. 16.

Review and Assess

17.

108



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NAME _________________________________________________________ DATE ____________


Find the indicated probability.

18.

18. A and B are independent events.

19.

PA 0.5

20.

PB 0.6

21.

PA and B ?

22.

19. A and B are dependent events. 23.

PA ?

PB A 0.6

24.

PA and B 0.06

25.

20. Suppose you play the three digit number 917 in your state’s lottery.

If you assume that digits can be repeated what is the probability you will win? (You must “hit” the number in the exact order.) 21. ACT Test One hundred students in your school took the ACT test.

Assuming that a normal distribution existed after the results, how many of the students scored within one standard deviation of the mean? (Give the percent and the number.) 22. A normal distribution has a mean of 8 and a standard deviation of 1.

Find the probability that a randomly selected x-value is in the interval between 6 and 10. 23. In Exercise 22, what is the probability that the randomly selected

x-value is between 8 and 10? 24. Find the mean and standard deviation of a normal distribution that

approximates a binomial distribution of 20 trials with a probability of success of 0.40. 25. Find the number of possible twelve member juries that can be

selected from fifteen qualified people.

Review and Assess



109

Answer Key Test B 1. 12 2. 181,440 3. 6 4. 36 5. 180 6. x5 5x 4y 10x3y 2 10x 2y3 5xy 4 y5 7. x6 12x5 60x 4 160x3 240x 2 192x 64 8. x8 4x6 6x 4 4x 2 1 9. 8x3 12x 2y 6xy 2 y 3 26 2 13 10. 52 0.5 11. 52 0.0385 12. 52 0.25 1 13. 52 0.0192 4 1 14. 52 0.0769 15. 4 1 16. 0% 17. 0.6 18. 0.24 19. 0.8 20. 1000 21. 95%; 475 students 22. 0.68 23. 0.475 24. 13.5; 2.72 25. 362,880

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12



2. 9P7

3. 4C2

4. 9C7


DALLAS.

2.

4.

7. x 26

8. x2 1 4

9. 2x y3

A card is drawn randomly from a standard 52-card deck. Find the probability of drawing the given card. 10. a red card

11. a red ace

13. the queen of diamonds

12. a spade

3 4

PA ?

5. 6.

7.

14. a jack


1.

3.


Answers

16. PA 30%

17. PA 0.5

PB 70%

PB 0.3

PA or B 100%

PA or B ?

PA and B ?

PA and B 0.2

8.

9.

10. 11. 12. 13. 14. 15. 16.

Review and Assess

17.

110



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NAME _________________________________________________________ DATE ____________



18.


19.

PA 0.40

20.

PB 0.60

21.

PA and B ?

22.

19. A and B are dependent events. 23.

PA ?

PB A 0.7

24.

PA and B 0.56

25.

20. Suppose you play a three digit number of your choice in the lottery.

If you assume that digits can be repeated calculate the probability of winning. (You must “hit” the number in the exact order.) 21. SAT Test Five hundred students in your school took the SAT test.

Assuming that a normal curve existed for your school, how many of those students scored within 2 standard deviations of the mean? (Give the percent and the number.) 22. A normal distribution has a mean of 10 and a standard deviation of

2. Find the probability that a randomly selected x-value is in the interval between 8 and 12. 23. In Exercise 22, what is the probability that the randomly selected


approximates a binomial distribution of 30 trials with a probability of success of 0.45. 25. Batting Orders Find the number of possible batting orders for the

nine starting players on a girls high school softball team.

Review and Assess



111

Answer Key Test C 1. 2520 2. 3,991,680 3. 21 4. 792 5. 90,720 6. x5 5x 4y 10x3y 2 10x 2y3 5xy 4 y 5 7. 8x3 12x 2y 6xy 2 y3 8. x6 12x5 60x 4 160x3 240x 2 192x 64 9. 1 5x 2 10x 4 10x6 5x8 x10 8 4 10. 48 0.167 11. 48 0.0833 12. 14. 15. 20. 24.

12 48 2 48 0 48 2 9

0.25 13. 16 48 0.333 0.0417 3 0 16. 52 17. 100% 18. 4 19. 0.65 21. 84%; 16,800 22. 0.95 23. 0.135 59 25.5; 4.10 25. 143 0.413

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12



2.

12P7

3. 7C5

4.

Answers 12C7


CLEVELAND.

2. 3. 4.


1.

7. 2x y3

8. x 2 6

9. 1 x2 5

5.

A card is drawn randomly from a standard 48-card pinochle deck. Find the probability of drawing the given card. (Note that a pinochle deck consists of all four suits. The cards 9, 10, jack, queen, king, ace appear twice in each suit. There are no 2, 3, 4, 5, 6, 7, or 8s.)

6.

10. any ace

11. any black queen

12. any heart

13. any 9 or 10

14. any ace of hearts

15. any 7


3 5

PA ?

17. PA 50%

18. PA ?

7.

8.

9.

PB 50%

PB 13

PA or B ?

PA or B 56

10.

PA and B 0%

PA and B 14

11. 12. 13. 14. 15. 16. 17.

Review and Assess

18.

112



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CONTINUED

NAME _________________________________________________________ DATE ____________



19.


20.

PA 0.35

21.

PB ?

22.

PA and B 0.2275

23.

20. A and B are dependent events.

PA 13

24.

25.

PB A 23 PA and B ? 21. ACT Test Twenty thousand students in your state took the ACT

test. On the math portion the mean was 21 and the standard deviation was 5. If the scores resulted in a normal distribution, how many students scored at least 16? (Give the percent and the number.) 22. A normal distribution has a mean of 200 and a standard deviation of

25. Find the probability that a randomly selected x-value is in the interval between 150 and 250. 23. In Exercise 22, what is the probability that the randomly selected


approximates a binomial distribution of 75 trials with a probability of success of 0.34. 25. Committee Selection A committee of 5 people is to be selected

from student council. Council has 6 boys and 7 girls. What is the probability that the committee will have at least 3 boys?

Review and Assess



113

Answer Key Cumulative Review 2 3 1. y 3x 4 2. y 3 x 5 3. y 2x 9 5 4. y 4 x 4 5. y 3x 4 6. y 4x 11 7.

8.

13. (0, 3

16. 1, 6

z

21. 25. 28. 32.

y

y

x

36.

x

40. 44. 9.

10.

47.

z

z

51. 55. y

y

x

58. 62.

x

66. 70. 11.

74.

12. z

77.

z

79. 81. y x

y x

17.

12, 13

19. x ≥ 3 or x ≤ 1 23.

z

14. 4, 1

85. 89. 93.

15. 5, 12 18. no solution

20. 1 ≤ x ≤

5 2

23 < x < 32 22. 34 < x < 34 x < 0 or x > 2 24. 52 ≤ x ≤ 3 1, 2 26. 1, 2 27. 2 2, 1 29. 2, 2 30. 1 31. 2 ± 3 33. 2 34. 5.62 35. 0, – 4 4 2 1.97 37. 53 38. 22 39. 12 3 3 5 2 3 5 42. 43. 8; 3.02 2 2 41. 3 2 13; 4.06 45. 8; 2.77 46. 23; 6.82 22; 6.54 48. 1.1; 0.368 49. B 50. C A 52. 54.598 53. 0.513 54. 9.974 0.149 56. 0.025 57. 0.034 0.845 59. 2.398 60. 0.349 61. 0.766 1.668 63. 2.303 64. 34 65. 1 no solution 67. 3.81 68. 0.42 69. 3.07 3.457 71. 5.457 72. 0.380 73. 4.820 2.860 75. 1.101 76. 25 4.47 42 5.66 78. 213 7.21 229 10.77 80. 31.33 5.60 10 0.79 82. down 83. up 84. right 4 down 86. left 87. right 88. 165 276 90. 2500 91. 143 92. 625 120

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Write an equation of the line using the given information. (2.4) 2

3 5

1. m 3, b 4

2. m 3, b

3. m 2, passes through (4, 1

4. passes through (0, 4 and 4, 1

5. passes through (1, 1 and is perpendicular to y 3 x 8 1

6. passes through (2, 3 and is parallel to y 4x 1

Plot the ordered triple in a three-dimensional coordinate system. (3.5) 7. 3, 2, 1 10. 2, 0, 0

8 2, 1, 0

9. 4, 1, 2

11. 1, 2, 3

12. 4, 0, 3

Use Cramer’s rule to solve the linear system. (4.3) 13. 2x y 3

14. x y 5

5x 2y 6 16. 5x y 1

15. x 4y 7

2x 3y 11

2x 6y 7

17. 4x 3y 1

18. 4x 3y 8

6x 3y 4

8x 6y 4

3x y 3

Solve the inequality algebraically. (5.7) 19. x2 2x 3 ≥ 0

20. 2x2 7x 5 ≤ 0

21. 6x2 5x 6 < 0

22. 16x2 9 < 0

23. 2x2 4x > 0

24. 2x2 x 15 ≤ 0

Use synthetic division to decide which of the following are zeros of the function: 1, 1, 2, 2. (6.6) 25. f x x3 2x2 5x 6

26. f x x3 7x2 14x 8

27. f x x 4 8x3 21x2 18x

28. f x x 4 3x3 7x2 15x 18

29. f x x5 4x3 8x2 32

30. f x x 4 5x3 x 5

Solve the equation. Round your answer to two decimal places when appropriate. (7.1) 31. x5 32

32. 3x 4 243

33. 4x3 32

34. x 33 18

35. x 2 4 1 15

36. x 4 3 12

Write the expression in simplest form. (7.2) 4 4 38.

5 64 40.

41.

4 16

19

4 8 39. 3

3

42.

Review and Assess

37. 75

4 4 2

3 5 16

Find the range and standard deviation of the data set. (7.7) 43. 1, 1, 3, 5, 6, 7, 9, 9

44. 12, 18, 15, 15, 16, 10, 19, 22, 23

45. 2, 3, 5, 9, 7, 7, 8, 8, 2, 10

46. 75, 77, 78, 84, 80, 80, 61

47. 19, 19, 18, 1, 14, 15, 23

48. 0.1, 0.8, 0.7, 1.2, 0.5, 1.1



119

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Cumulative Review

CONTINUED


Match the function with the graph. (8.1) 49. y 4x

50. y 3

A.

4x3

B.

y

51. y 3x 3 C.

y

y

1 2

x 1

1 1

x

1

x

Use a calculator to evaluate the expression. Round the result to three decimal places. (8.3 and 8.4) 52. e4

53. e23

54. e2.3

55. 3e3

56. 0.03e0.2

57. 5e5

58. log 7

59. ln 11

60. log5

61. log 5.83

62. ln 5.3

63. ln 10

Solve the equation. Round your answer to two decimal places when appropriate. (8.6) 64. 102x1 1003x1

65. 253x 125x1

66. 81x8 92x

67. 2x 14

68. 103x 2 20

69. 3x 2 27

6 for the given value of x. 1 2ex Round the result to three decimal places. (8.8) Evaluate the function f x

70. f 1

71. f 3

73. f 2.1

72. f 2

74. f 5

75. f 0.8

3

Find the distance between the two points. (10.1) 76. 5, 4, 7, 8

77. 6, 1, 2, 3

78. 3, 4, 3, 0

79. 8, 3, 2,1

80. 6.3, 9.2, 2.1, 5.5

81.

12, 14 , 34, 12

Review and Assess

Tell whether the parabola opens up, down, left, or right. (10.2) 82. y 4x2

83. 3y 8x2

84. 2x 3y2

85. 5y 2x2

86. x 7y2

87. x 3 y2

2

Find the sum of the first n terms of the arithmetic series. (11.2) 88. 3 6 9 12 15 . . . ; n 10

89. 1 5 9 13 17 . . . ; n 12

90. 1 3 5 7 9 . . . ; n 50

91. 7 4 1 2 5 . . . ; n 13

92. 40 45 50 55 60 . . . ; n 10

93. 22 20 18 16 14 . . . ; n 15

120



Answer Key Practice A 1. 8 2. 6 3. 21 4. 30 5. a. 6,760,000 b. 3,276,000 6. a. 2,600,000 b. 786,240 7. a. 45,697,600 b. 32,292,000 8. a. 118,813,760 b. 78,936,000 9. 720 10. 24 11. 6 12. 39,916,800 13. 24 14. 5 15. 20,160 16. 42 17. 6 18. 120 19. 24 20. 720 21. 3 22. 12 23. 2520 24. 180 25. 1920 26. 12

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LESSON

12.1

NAME _________________________________________________________ DATE ___________


Each event can occur in the given number of ways. Find the number of ways all of the events can occur. 1. Event 1: 2 ways, Event 2: 4 ways

2. Event 1: 6 ways, Event 2: 1 way

3. Event 1: 7 ways, Event 2: 3 ways

4. Event 1: 2 ways, Event 2: 5 ways,

For the given configuration, determine how many different computer passwords are possible if (a) digits and letters can be repeated, and (b) digits and letters cannot be repeated. 5. 4 digits followed by 2 letters

6. 5 digits followed by 1 letter

7. 4 letters followed by 2 digits

8. 5 letters followed by 1 digit

Evaluate the factorial. 9. 6!

10. 4!

Find the number of permutations. 13. 4P4 14. 5 P1

11. 3!

12. 11!

15. 8P6

16. 7P2

Find the number of distinguishable permutations of the letters in the word. 17. CAT

18. MONEY

19. UTAH

20. FAMILY

21. MOM

22. TENT

23. PHYSICS

24. FOLLOW

25.

Home Decor You are choosing curtains, paint, and carpet for your room. You have 12 choices of curtains, 8 choices of paint, and 20 choices of carpeting. How many different ways can you choose curtains, paint, and carpeting for your room?

26.

16

Naming a Dog You are choosing a name for your registered beagle. Your dog’s grandparent’s names were Willow-Sutton, Carolina-Downing, Hollybrook-Loner, and Starfire-Wolf. You want your dog’s first name to be the same as one of its grandparents’ first names, and its second name to be the same as one of its grandparents’ second names. However, your dog cannot have exactly the same name as one of its grandparents. How many names are possible?



Lesson 11.3

Lesson 12.1

Event 3: 3 ways

Answer Key Practice B 1. 12 2. 5 3. 48 4. 90 5. a. 6,760,000 b. 3,276,000 6. a. 2,600,000 b. 786,240 7. a. 17,576,000 b. 11,232,000 8. a. 118,813,760 b. 78,936,000 9. 720 10. 6,227,020,800 11. 1 12. 1.31 1012 13. 1,814,400 14. 1 15. 6 16. 720 17. 5040 18. 120 19. 24 20. 12 21. 3 22. 20,160 23. 10,080 24. 907,200 25. 72 26. 120 27. 201,600 28. 5040

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2. Event 1: 1 way, Event 2: 5 ways

3. Event 1: 4 ways, Event 2: 6 ways


Event 3: 2 ways

Event 3: 5 ways

5. 2 letters followed by 4 digits

6. 1 letter followed by 5 digits

7. 3 digits followed by 3 letters

8. 1 digit followed by 5 letters

Lesson 12.1

For the given configuration, determine how many different computer passwords are possible if (a) digits and letters can be repeated, and (b) digits and letters cannot be repeated.


10. 13!

11. 0!

12. 15!

15. 6P1

16. 6P6

Find the number of permutations. 13.

10P8

14. 5P0

Find the number of distinguishable permutations of the letters in the word. 17. ENGLISH

18. NORTH

19. MATH

20. BELL

21. EYE

22. ALPHABET

23. OKLAHOMA

24. CALIFORNIA

25.

School Lunch Your school cafeteria offers three salads, four main courses, two vegetables, and three desserts. How many different lunches consisting of a salad, main course, a vegetable, and dessert are possible?

26.

Stacking Books Five books are taken from a shelf and laid in a stack on a table. In how many different orders can the books be stacked?

27.

Batting Order A baseball coach is determining the batting order for the team. The team has nine members, but the coach does not want the pitcher to be one of the first four to bat. How many batting orders are possible?

28.

Scheduling Classes Next year you are taking math, English, history, keyboarding, chemistry, physics, and physical education. Each class is offered during each of the seven periods in the day. In how many different orders can you schedule your classes?



17

Answer Key Practice C 1. 360 2. 120 3. a. 10,000 b. 3024 4. a. 3125 b. 120 5. a. 50,000 b. 15,120 6. a. 20,000 b. 6048 7. 24 8. 1 9. about 8.72 1010 10. about 2.43 1018 11. 1 12. 40,320 13. 120 14. 10 15. 362,880 16. 60 17. 3 18. 120 19. 420 20. 19,958,400 21. 4,989,600 22. 39,916,800 23. 85,765,680; 4080 24. 64,000 25. 40,320

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LESSON

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Lesson 12.1

Event 3: 12 ways


Event 3: 5 ways, Event 4: 3 ways

For the given configuration, determine how many different 5-digit postal zip codes are possible if (a) digits can be repeated, and (b) digits cannot be repeated. 3. Begins with a 4.

4. Has all even digits.

5. Is divisible by 2.

6. Begins with a 3 or a 1.


8. 0!

9. 14!

10. 20!

Find the number of permutations. 11.

12P0

12. 8P8

13. 6P3

14.

10P1

Find the number of distinguishable permutations of the letters in the word. 15. CHEMISTRY

16. PAPER

17. EEL

18. ALASKA

19. SUCCESS

20. PERMUTATION

21. MATHEMATICS

22. BILLIONAIRES 23.

Dog Show In a dog show, how many ways can four Pomeranians, five golden retrievers, two Great Pyrenees, and six English terriers line up in front of the judges if the dogs of the same breed are considered identical? In how many different ways can three dogs win first, second, and third place?

24.

Combination Lock You have forgotten the combination of the lock on your school locker. There are 40 numbers on the lock, and the correct combination is “R -L -R .” How many possible combinations are there?

25.

Circular Permutations In how many different ways can nine

A

people be seated around a circular table? E

B

D

C

C B

D

E

A This is the same permutation.

18



Answer Key Practice A 1. 20 2. 21 3. 3 4. 5 5. 1 6. 6 7. 1 8. 7 9. 2600 10. 4 11. 220 12. 286 13. 52 14. x4 12x3 54x2 108x 81 15. x3 12x2 48x 64 16. x3 6x2 12x 8 17. x4 20x3 150x2 500x 625 18. x5 5x4y 10x3y2 10x2y3 5xy4 y5 19. x4 8x3y 24x2y2 32xy3 16y4 20. x4 12x3y 54x2y2 108xy3 81y4 21. 8x3 12x2y 6xy2 y3 22. 560 23. 61,236 24. 220 25. 126

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LESSON

12.2

NAME _________________________________________________________ DATE ___________


Find the number of combinations. 1. 6C3

2. 7C5

3. 3C1

4. 5C4

5. 2C2

6. 4C2

7. 5C5

8. 7C1

In Exercises 9–13, find the number of possible 3-card hands that contain the cards specified. 9. 3 red cards 10. 3 aces 11. 3 face cards 12. 3 hearts 13. 3 of one kind (kings, queens, and so on)

Expand the power of the binomial. 15. x 43

16. x 23

17. x 54

18. x y5

19. x 2y4

20. x 3y4

21. 2x y3

Lesson 12.2

14. x 34

22. Find the coefficient of x4 in the expansion of 2x 17. 23. Find the coefficient of x5 in the expansion of x 3y10. 24.

Pizza Toppings A pizza shop offers twelve different toppings. How many different three-topping pizzas can be formed with the twelve toppings? (Assume no topping is used twice.)

25.

Bowling Team Nine people in your class want to be on a 5-person bowling team to represent the class. How many different teams can be chosen?



31

Answer Key Practice B 1. 56 2. 120 3. 1 4. 792 5. 65,780 6. 3744 7. 658,008 8. 5148 9. x7 7x6 21x5 35x4 35x3 21x2 7x 1 10.

x6 12x5 60x4 160x3 240x2 192x 64 11. 8x3 36x2 54x 27 12. x4 16x3y 96x2y2 256xy3 256y4 13. 60,555,264 14. 2,449,440 15. 1287 16. 300 17. 90

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LESSON

12.2

NAME _________________________________________________________ DATE ___________


Find the number of combinations. 1. 8C5

2.

10C3

3.

10C10

4.

12C5

In Exercises 5–8, find the number of possible 5-card hands that contain the cards specified. 5. 5 black cards 6. 3 of one kind (kings, queens, and so on) and 2 of a different kind 7. 5 cards, none of which are face cards (either kings, queens, or jacks) 8. 5 cards of the same suit

Expand the power of the binomial. 9. x 17

10. x 26

11. 2x 33

12. x 4y4

13. Find the coefficient of x5 in the expansion of 2x 411. 15.

Basketball Starters A basketball team has five starting players. There are 13 girls on the team. In how many ways can the coach select players to start the game? (Assume each player can play each position.)

16.

School Faculty A high school needs four additional faculty members:

Lesson 12.1

Lesson 12.2

14. Find the coefficient of x6 in the expansion of 3x 210.

two math teachers, a chemistry teacher, and a Spanish teacher. In how many ways can these positions be filled if there are six applicants for mathematicians, two for chemistry, and ten applicants for Spanish? 17.

Geometry How many different rectangles occur in the grid shown below? (Hint: A rectangle is formed by choosing two of the vertical lines in the grid and two of the horizontal lines in the grid.)

32



Answer Key Practice C 1. 210 2. 462 3. 1 4. 495 5. 4 6. 3744 7. 123,552 8. 3120 9. 32x5 80x4 80x3 40x2 10x 1 10. 64x6 192x5y2 240x4y4 160x3y6 60x2y8 12xy10 y12 11. 256x4 256x3y3 96x2y6 16xy9 y12 12. x21 7x18y 21x15y2 35x12y3 35x9y4 21x6y5 7x3y6 y7 13. 69,672,960 14. 316,800,000 15. 302,400 16. 386 17. 126

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LESSON

12.2

NAME _________________________________________________________ DATE ___________


Find the number of combinations. 1.

10C6

2.

11C5

3.

14C14

4.

12C8

In Exercises 5–8, find the number of possible 5-card hands that contain the cards specified. 5. 4 aces and 1 king 6. 3 of one kind (kings, queens, and so on) and 2 of a different kind 7. 2 of one kind, 2 of a second kind, and 1 other card 8. 3 face cards (kings, queens, or jacks) of the same suit and 2 other cards

(none of which are face cards) Expand the power of the binomial. 9. 2x 15

10. 2x y26

11. 4x y34

12. x3 y7

13. Find the coefficient of x6 in the expansion of 4x 310. 15.

Lesson 12.2

Lesson 12.1

14. Find the coefficient of x7 in the expansion of 2x 512.

Football Starters A high school football team has 2 centers, 9 linemen (who can play either guard or tackle), 2 quarterbacks, 5 halfbacks, 5 ends, and 6 fullbacks. The coach uses 1 center, 4 linemen, 2 ends, 2 halfbacks, 1 quarterback, and 1 fullback to form an offensive unit. In how many ways can the offensive unit be selected?

16.

Ice Cream Sundaes An ice cream shop has a choice of ten toppings. Suppose you can afford at most four toppings. How many different types of ice cream sundaes can you order?

17.

Geometry How many different rectangles occur in the grid shown below? (Hint: A rectangle is formed by choosing two of the vertical lines in the grid and two of the horizontal lines in the grid.)



33

Answer Key Practice A 1. 0.5 2. 0.25 3. 1 4. 0.75 5. 0.5 6. 0 7. 0.182 8. 0.091 9. 0.455 10. 0.636 11. 0.306 12. 0.660 13. 0.665 14. 0.125 15. 0.0425 16. 0.01

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LESSON

12.3

NAME _________________________________________________________ DATE ___________


Spinning a Spinner You have an equally likely chance of spinning any value on the spinner. Find the probability of spinning the given event. 1. a shaded region 2. a factor of 27

9 4

3. a number less than 6 or a shaded region

2

4. an even number or perfect square 5. a prime number

5

6. a two-digit number

Choosing Marbles A jar contains 5 red marbles, 3 green marbles, 2 yellow marbles, and 1 blue marble. Find the probability of randomly drawing the given type of marble. 7. a yellow marble 8. a blue marble 9. a green or yellow marble

10. a red or yellow marble

School Mascot In order to choose a mascot for a new school, 2755 students were surveyed: 896 chose a falcon, 937 chose a ram, and 842 chose a panther. The remaining students did not vote. A student is chosen at random. 11. What is the probability that the student’s choice was a panther? 12. What is the probability that the student’s choice was not a ram? 13. What is the probability that the student’s choice was either a falcon or a

ram?

Hitting a Star


You are throwing a dart at the square shown at the right. Assume that the dart is equally likely to land at any point in the square. The square is 2 inches by 2 inches. Each star has an area of 0.01 square inch. Lesson 12.3

14. The dart has landed inside the square. What is the probability that

it hit a star? 15. The dart has landed inside the square. What is the probability that

it hit a star in the top three rows? 16. The dart has landed inside the square. What is the probability that

it hit one of the four corner stars?



45

Answer Key Practice B 1. 0.5 2. 0.417 3. 0.333 4. 0.25 5. 0.222 6. 0.667 7. 0.444 8. 0.518 9. 0.852 10. 0.019 11. about 2.48 105 12. 0.624

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LESSON

12.3

NAME _________________________________________________________ DATE ___________


Choosing Numbers You have an equally likely chance of choosing any integer from the set 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. Find the probability of the given event. 1. An even number is chosen.

2. A prime number is chosen.

3. A multiple of 3 is chosen.

4. A two-digit number is chosen.

Farm Animals Your cousin lives on a small farm. She is a member of the 4-H Club and is showing nine animals at the county fair. Two of her animals won a blue ribbon (1st place), one won a red ribbon (2nd place), and three won white ribbons (3rd place). You do not know which animals won which prizes. You choose one of your cousin’s animals at random. 5. What is the probability that the animal won a 1st place ribbon? 6. What is the probability that the animal won a ribbon? 7. What is the probability that the animal won a red or white ribbon?

Live Births


Of all live births in the United States in 1996, 12.9% of the mothers were teenagers, 51.8% were in their twenties, 33.4% were in their thirties, and the rest were in their forties. Suppose a mother is chosen at random. 8. What is the probability that the mother gave birth in her twenties? 9. What is the probability that the mother gave birth in her twenties or

thirties? 10. What is the probability that the mother gave birth in her forties?

Lesson 12.3

11.

Choosing Coins You have 8 pennies in your pocket dated 1972, 1978, 1979, 1985, 1989, 1991, 1993, and 1999. You take the coins out of your pocket one at a time. What is the probability that they are taken out in order by date?

12.

Geometry Find the probability that a dart thrown at the given target will hit the shaded region. Assume the dart is equally likely to hit any point inside the target.

5 7

46



Answer Key Practice C 1. 0.278 2. 0.722 3. 0.278 4. 0.167 5. 0.563 6. 0.813 7. 0.25 8. 0.188 9. 0.0218 10. 0.0654 11. 0.153 12. 0.455 13. 0.771 14. 0.033

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LESSON

12.3

NAME _________________________________________________________ DATE ___________


Rolling Dice You have an equally likely chance of rolling any value on each of two dice. Find the probability of the given event. 1. rolling a sum of either 7 or 9

2. rolling a sum greater than 5

3. rolling a 6 on exactly one die

4. rolling doubles

Genetics Common parakeets have genes that can produce four feather colors: green (BBCC, BBCc, BbCC, or BbCc), blue (BBcc, or Bbcc), yellow (bbCC or bbCc), or white (bbcc). BC Complete the Punnett square to the right to find the possible feather colors of the offspring of two green parents (both with Bc BcCc feather genes). Then find the probability of the given event. bC 5. green feathers 6. not blue feathers 7. yellow or white feathers

8. yellow feathers

BC

Bc

bC

bc

bc

Geometry A marble is dropped into a large box whose base is painted different colors, as shown at the right. The marble has an equal likelihood of coming to a rest at any point on the base. Find the probability of the given event. 9. the center circle 11. the third ring

10. the first ring 2 2 2 2 2

12. the border

24 in.

Ring 1 Ring 2 Ring 3 Ring 4

Lesson 12.3

24 in.

Test Scores Thirty-five students in an Algebra 2 class took a test: 9 received A’s, 18 received B’s, and 8 received C’s. Find the probability of the given event. 13. If a student from the class is chosen at random, what is the probability that

the student did not receive a C? 14. If the teacher randomly chooses 3 test papers, what is the probability that

the teacher chose tests with grades A, B, and C in that order?



47

Answer Key Practice A 1. 0.15; no 2. 0.35; yes 3. 0.45; no 4. 0.80; no 5. 0.70; no 6. 0; yes 7. 0.75 1 8. 3 9. 0.36 10. 1 11. 0.0833 12. 0.0556 13. 0.417 14. 0.923 15. 0.769 16. 0.692 17. 0.692 18. 0.60

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LESSON

12.4

NAME _________________________________________________________ DATE ___________


Find the indicated probability. State whether A and B are mutually exclusive. 1. PA 0.2

2. PA 0.45

3. PA 0.25

PB 0.55

PB ?

PB 0.32

PA or B 0.6

PA or B 0.80

PA or B ?

PA and B ?

PA and B 0

PA and B 0.12

4. PA 0.50

5. PA ?

6. PA 0.45

PB 0.40

PB 0.40

PB 0.15

PA or B ?

PA or B 0.80

PA or B 0.60

PA and B 0.10

PA and B 0.30

PA and B ?

Find P A . 7. PA 0.25

8. PA

9. PA 0.64

2 3

Rolling Dice Two six-sided dice are rolled. Find the probability of the given event. 10. The sum is even or odd.

11. The sum is 3 or 12.

12. The sum is greater than 8 and prime.

13. The sum is 10 or a multiple of 3.

Using Complements A card is randomly drawn from a standard 52-card deck. Find the probability of the given event. 14. not an ace

15. not a face card

16. less than 10 (an ace is one)

17. not a diamond or a five

18.

Snow The probability that it will snow today is 0.30, and the probability

Lesson 12.4

that it will snow tomorrow is 0.50. The probability that it will snow both days is 0.20. What is the probability that it will snow today or tomorrow?

58



Answer Key Practice B 6 13 1. 11; no 2. 0%; yes 3. 24; no 4. 0.3 5. 6. 37% 7. 0.308 8. 0.0577 9. 0.923 10. 0.308 11. 0.743 12. 0.917; 0.083 13. 50%

1 4

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LESSON

NAME _________________________________________________________ DATE ___________

12.4


Find the indicated probability. State whether A and B are mutually exclusive. 1. PA

4 11

PB ? PA or B

8 11

PA and B

2 11

2. PA 28%

3. PA

PB 14%

PB

PA or B 42%

PA or B ?

PA and B ?

PA and B 8

2 3 1 2

5

Find P A . 4. PA 0.7

5. PA

6. PA 63%

3 4

Choosing Cards A card is randomly drawn from a standard 52-card deck. Find the probability of the given event. 7. an ace or a club 9. not an ace 11.

8. a face card and a diamond 10. less than or equal to four (an ace is one)

Honors Banquet Of the 148 students honored at an academic banquet, 40 won awards for mathematics and 82 for English. Twelve of these students won awards for both mathematics and English. One of the 148 students is chosen at random to be interviewed for a newspaper article. What is the probability that the student won an award in mathematics or English?

12.

Hockey or Swimming The probability that you will make the hockey 2

3

team is 3. The probability that you will make the swimming team is 4. If 1 the probability that you make both teams is 2, what is the probability that you at least make one of the teams? that you make neither team?

13.

Weather Forecast The probability that it will snow today is 40%, and the probability that it will snow tomorrow is 20%. The probability that it will snow both days is 10%. What is the probability that it will snow today or tomorrow?

Lesson 12.4



59

Answer Key Practice C 1 1. 10%; no 2. 92%; yes 3. 3; no 4. 0.98 7 5. 63% 6. 12 7. 0.167 8. 0.944 9. 0.75 10. 0.0278 11. 5 to 3 12. 9 13. 0.31 14.

3 5

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LESSON

12.4

NAME _________________________________________________________ DATE ___________


Find the indicated probability. State whether A and B are mutually exclusive. 1. PA 12%

2. PA 85%

3. PA

3 4

PB 48%

PB 7%

PB ?

PA or B 50%

PA or B ?

5 PA or B 12

PA and B ?

PA and B 0%

PA and B 3 2

Find P A . 4. PA 0.02

5. PA 37%

6. PA

5 12

Rolling Dice Two six-sided dice are rolled. Find the probability of the given event. 7. The sum is even and a multiple of 3. 9. The sum is greater than 7 or odd.

Odds

8. The sum is not 2 or 12. 10. The sum is prime and even.


The odds in favor of an event occurring are the ratio of the probability that the event will occur to the probability that the event will not occur. 11. A jar contains three red marbles and five green marbles. What are the

odds that a randomly chosen marble is green? 12. A jar contains three red marbles and some green marbles. The odds are 3

to 1 that a randomly chosen marble is green. How many green marbles are in the jar? 13.

Science Class Students at Northwestern High School have three choices for a required science in their junior year: physics, chemistry, or biology. Experience has shown that the probability of a student selecting physics is 0.12 and the probability of a student selecting chemistry is 0.57. If each student can select only one science course, what is the probability that a randomly chosen student will select biology?

14.

Cars A parking lot has 25 cars. Eight are red and 13 have four doors. Six are both red and have four doors. Find the probability that a randomly chosen car will be red or have four doors.

Lesson 12.4

Lesson 12.3

60



Answer Key Practice A 1. independent 2. dependent 3. independent 1 4. independent 5. 3 6. 0.08 7. 0.80 8. 0.80 9. 0.06 10. 0.50 11. 0.216 12. 0.0234 13. 0.784

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Lesson 12.5

LESSON

NAME _________________________________________________________ DATE ___________

12.5


State whether A and B are independent or dependent. 1. A single coin is tossed twice. Event A is having the coin land heads up on

the first toss. Event B is having the coin land tails up on the second toss. 2. Two cards are drawn from a standard 52-card deck. The first card is not

placed back in the deck before the second card is drawn. Event A is drawing a queen for the first card. Event B is drawing a king for the second card. 3. Two cards are drawn from a standard 52-card deck. The first card is

placed back in the deck before the second card is drawn. Event A is drawing a queen for the first card. Event B is drawing a king for the second card. 4. You buy one state lottery ticket this week and one next week. Event A is

winning the lottery this week. Event B is winning the lottery next week. Events A and B are independent. Find the indicated probability. 5. PA

PB

6. PA 0.40

1 2 2 3

PA and B ?

7. PA 0.80

PB 0.20

PB ?

PA and B ?

PA and B 0.64

Events A and B are dependent. Find the indicated probability. 8. PA 0.50

9. PA 0.60

10. PA ?

PB A ?

PB A 0.10

PB A 0.70

PA and B 0.40

PA and B ?

PA and B 0.35

Marbles in a Jar


A jar contains 12 red marbles, 16 blue marbles, and 18 white marbles. 11. Three marbles are chosen from the jar without replacement. What is the

probability that none are white? 12. Four marbles are chosen from the jar with replacement. What is the

probability that all are white? 13. Three marbles are chosen from the jar without replacement. What is the

probability that at least one is white?

70



Answer Key Practice B 3 1. 4 2. 0.2 3. 1 4. 0.06 5. 0.75 6. 0.9 7. 0.25 8. 0.998 9. 0.985 10. 0.681 11. a. 0.0178 b. 0.0181 12. a. 0.00592 b. 0.00603 13. a. 0.00137 b. 0.00145 14. a. 0.000455 b. 0.000181

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LESSON

NAME _________________________________________________________ DATE ___________

12.5

Lesson 12.5


Events A and B are independent. Find the indicated probability. 1. PA ?

PB

2. PA ?

2 3

PA and B 12

3. PA 0.6

PB 0.4

PB ?

PA and B 0.08

PA and B 0.6


5. PA ?

6. PA 0.3

PB A 0.3

PB A 0.2

PB A ?

PA and B ?

PA and B 0.15

PA and B 0.27

File Cabinet


Each drawer in a file cabinet that has 4 drawers has 100 folders. You are searching for some information that is in one of the folders, but you do not know which folder has the information. 7. What is the probability that the information is in the first drawer you

choose? 8. What is the probability that the information is not in the first folder you

choose? 9. What is the probability that the information is not in the first six folders

you choose? 10.

Apples The probability of selecting a rotten apple from a basket is 12%. What is the probability of selecting three good apples when selecting one from each of three different baskets?

Drawing Cards Find the probability of drawing the given cards from a standard 52-card deck (a) with replacement and (b) without replacement. 11. a face card, then an ace

12. a 2, then a 10

13. an ace, then a face card, then a 7

14. a king, then another king, then a third king



71


7 12

2.

3 4

7. a. 0.097 9. a. 0.0355 b. 0.0261

3. 0.29

4. 0.01

b. 0.0928 b. 0.0379

11. 0.509

13. 503,159 tickets

5.

7 12

8. a. 0.153

6.

b. 0.148

10. a. 0.0266

12. 458 sets

5 6

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Lesson 12.5

LESSON

NAME _________________________________________________________ DATE ___________

12.5


Events A and B are independent. Find the indicated probability. 1. PA

PB

2. PA

2 3 7 8

PA and B ?

3. PA ?

1 2

PB ?

PB 0.80

PA and B 38

PA and B 0.232


5. PA

6. PA ?

2 3

PB A 0.1

PB A ?

PB A 34

PA and B ?

7 PA and B 18

PA and B 58

Marbles in a Jar


A jar contains 12 red marbles, 16 blue marbles, and 18 white marbles. Find the probability of choosing the given marbles from the jar (a) with replacement and (b) without replacement. 7. red, then blue 9. red, then white, then blue 11.

8. white, then white 10. red, then red, then white

Table Tennis Tim has 4 table tennis balls with small cracks. His friend accidentally mixed them in with 16 good balls. If Tim randomly picks 3 table tennis balls, what is the probability that at least 1 is cracked?

12.

Television Sets An electronics manufacturer has found that only 1 out of 500 of its television sets is defective. You are ordering a shipment of television sets for the electronics store where you work. How many television sets can you order before the probability that at least one defective set reaches 60%?

13.

Lottery To win a state lottery, a player must correctly match six different numbers from 1 to 60. If a computer randomly assigns six numbers per ticket, how many tickets would a person have to buy to have a 1% chance of winning?

Lesson 12.3

72



Answer Key Practice A 1. 0.0417 2. 0.196 3. 0.0916 4. 0.00320 5. 0.202 6. 0.00992 7. 0.00000343 8. 9.09 1013 9. 0.0705 10. 0.913 11. 0.472 12. 0.633 13.

0.5 0.4 0.3 0.2 0.1 0

14.

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0 1 2 3

k2

0 1 2

k2 15.

16.

0.5 0.4 0.3 0.2 0.1 0

0.4 0.3 0.2 0.1 0 0 1 2 3 4 5

k1 17.

0.6 0.5 0.4 0.3 0.2 0.1 0

0 1 2 3 4 5 6

k4 18.

0.5 0.4 0.3 0.2 0.1 0

0 1 2 3 4 5

0 1 2 3 4

k0 k1 19. 0.00856 20. Do not reject the claim because the probability of these findings is 0.158 which is greater than 0.1.

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LESSON

12.6

NAME _________________________________________________________ DATE ___________


Calculate the probability of tossing a coin 15 times and getting the given number of tails. 1. 4

2. 7

3. 10

4. 2

Lesson 12.6

Calculate the probability of randomly guessing the given number of correct answers on a 20-question multiple-choice exam that has choices A, B, C, and D for each question. 5. 5

6. 10

7. 15

8. 20

Calculate the probability of k successes for a binomial experiment consisting of n trials with probability p of success on each trial. 9. k ≥ 4, n 6, p 0.3

10. k ≥ 2, n 5, p 0.6

11. k ≤ 3, n 5, p 0.7

12. k ≤ 4, n 10, p 0.4

A binomial experiment consists of n trials with probability p of success on each trial. Draw a histogram of the binomial distribution that shows the probability of exactly k successes. Then find the most likely number of successes. 13. n 3, p 0.6

14. n 2, p 0.8

15. n 5, p 0.3

16. n 6, p 0.7

17. n 4, p 0.15

18. n 5, p 0.25

19.

Automobile Accidents An automobile-safety researcher claims that 1 in 10 automobile accidents is caused by driver fatigue. What is the probability that at least three of five automobile accidents are caused by driver fatigue?

20.

86

College Enrollment A guidance counselor claims that only 60% of high school seniors capable of doing college-level work actually go to college. The recruitment office polls a random sample of 12 high school seniors capable of doing college-level work. Five of the seniors said they had plans to attend collge. Would you reject the guidance counselor’s claim? Explain.



Answer Key Practice B 1. 0.00000894 2. 0.0974 3. 0.0143 4. 0.00000003 5. 0.00992 6. 0.0609 7. 0.000000002 8. 0.202 9. 0.294 10. 0.496 11.

0.5 0.4 0.3 0.2 0.1 0

12.

0 1 2 3 4

k2 13.

0 1 2 3 4 5

k4 14. 5 or 6

0.5 0.4 0.3 0.2 0.1 0

0.5 0.4 0.3 0.2 0.1 0

15. 0.274 16. 0.00301 0 1 2 3 4 5 6

k5 17. Reject the claim because the probability of these findings is 0.02, which is less than 0.1.

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LESSON

12.6

NAME _________________________________________________________ DATE ___________


Calculate the probability of tossing a coin 25 times and getting the given number of heads. 1. 2

2. 10

3. 18

4. 25

Calculate the probability of randomly guessing the given number of correct answers on a 20-question multiple choice exam that has choices A, B, C, and D for each question. 6. 8

7. 18

Lesson 12.6

5. 10

8. 5

Calculate the probability of k successes for a binomial experiment consisting of n trials with probability p of success on each trial. 9. k ≥ 4, n 8, p 0.35

10. k ≤ 5, n 10, p 0.55


Puppies

12. n 5, p 0.75

13. n 6, p 0.83


A registered golden retriever gives birth to a litter of 11 puppies. Assume that the probability of a puppy being male is 0.5. 14. Because the owner of the dog can expect to get more money for a male

puppy, what is the most likely number of males in the litter? 15. What is the probability at least 7 of the puppies will be male?

Automobile Theft

In Exercises 16 and 17, use the following infor-

mation. The probability is 0.58 that a car stolen in a city in the United States will be returned to its lawful owner. Suppose that in one day 30 cars were stolen. 16. What is the probability that at least 25 of these stolen cars will be returned

to their lawful owners? 17. The police department claims that 75% of stolen cars are returned to their

lawful owners. You decide to test this claim by polling a random sample of 10 stolen cars. Four of the stolen cars were returned. Would you reject the police’s claim? Explain.



87

Answer Key Practice C 1. 0.00545 2. 0.144 3. 0.0280 4. 0.0000255 5. 0.0355 6. 0.00000003 7. 1.57 1013 8. 1.07 1021 9. 0.937 10. 0.115 11.

12.

0.5 0.4 0.3 0.2 0.1 0

0.5 0.4 0.3 0.2 0.1 0

0 1 2 3 4 5 6

k5 13.

14. 6

0.6 0.5 0.4 0.3 0.2 0.1 0

0 1 2 3 4 5 6 7 8

k2 k0

0 1 2 3 4 5 6 7 8 9 10

15. no

16. Do not reject the claim

because the probability of this finding is 0.608, which is much greater than 0.1.

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LESSON

12.6

NAME _________________________________________________________ DATE ___________


Calculate the probability of tossing a coin 30 times and getting the given number of tails. 1. 8

2. 15

3. 20

4. 26

Lesson 12.6

Calculate the probability of randomly guessing the given number of correct answers on a 30-question multiple-choice exam that has choices A, B, C, D, and “none of these” for each question. 5. 10

6. 20

7. 25

8. 30

Calculate the probability of k successes for a binomial experiment consisting of n trials with probability p of failure on each trial. 9. k ≥ 3, n 8, p 0.42

10. k ≤ 4, n 7, p 0.18


Side Effects

12. n 8, p 0.245

13. n 10, p 0.066


According to a medical study, 40% of the people will experience an adverse side effect within one hour after taking an experimental drug to reduce cholesterol. Fifteen people participated in the study. 14. What is the most likely number of people experiencing an adverse effect

in the study? 15. If seven of the people experience an adverse effect, would you reject the

study’s claim? 16.

Class President You read an article in your school newspaper in which a candidate claims that 30% of the class will vote for her. To test this claim, you survey 20 randomly selected students in the class and find that 6 are planning on voting for her. Would you reject the claim? Explain.

Lesson 12.3

88



Answer Key Practice A 1. 68% 2. 97.5% 3. 47.5% 4. 4.7% 5. 0.815 6. 0.0235 7. 0.34 8. 0.5 9. 0.025 10. 0.84 11. 0.593 12. 0.951 13. 16, 1.79 14. 10.5, 2.71 15. 8.4, 2.59 16. 0.025 17. 0.8385

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LESSON

12.7

NAME _________________________________________________________ DATE ___________


Give the percent of the area under a normal curve represented by the shaded region. 2.

x

2σ

σ

σ

1.

x

x

3σ

2σ x

x

2σ x

x

2σ

x

3σ

4.

x

Lesson 12.7

3.

A normal distribution has a mean of 56 and a standard deviation of 8. Find the probability that a randomly selected x-value is in the given interval. 5. between 40 and 64

6. between 32 and 40

8. at most 56

9. at least 72

7. between 56 and 64 10. at most 64

A normal distribution has a mean of 100 and a standard deviation of 16. Find the given probability. 11. three randomly selected x-values are all 84 or greater 12. two randomly selected x-values are both 132 or less

Find the mean and standard deviation of a normal distribution that approximates a binomial distribution consisting of n trials with probability p of success on each trial. 13. n 20, p 0.8

Photography

14. n 35, p 0.3

15. n 42, p 0.2


The developing times of photographic prints are normally distributed with a mean of 15.4 seconds and a standard deviation of 0.48 second. 16. What is the probability that the developing time of a print will be at least

16.36 seconds? 17. What is the probability that the developing time of a print will be between

13.96 seconds and 15.88 seconds? 98



Answer Key Practice B 1. 0.3% 2. 49.85% 3. 0.815 4. 0.4985 5. 0.68 6. 0.975 7. 0.84 8. 0.025 9. 0.000625 10. 0.000332 11. 35, 3.24 12. 31.2, 4.805 13. 15.48, 3.75 14. 0.16 15. 0.025 16. 0.000000391

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LESSON

12.7

NAME _________________________________________________________ DATE ___________


Give the percent of the area under a normal curve represented by the shaded region.

3σ

2σ x

1σ x

x

σ

2σ

x

x

x

x

3σ

2.

3σ

1.

x

MCRB2-1207-PA.qxd

A normal distribution has a mean of 31 and a standard deviation of 3. Find the probability that a randomly selected x-value is in the given interval. 4. between 22 and 31


6. at least 25

7. at most 34

8. at least 37

Lesson 12.7


A normal distribution has a mean of 85 and a standard deviation of 15. Find the given probability. 9. two randomly selected x-values are both 55 or less 10. four randomly selected x-values are all between 55 and 70


Bank Loans

12. n 120, p 0.26

13. n 172, p 0.09


A loan officer at a bank may reject a loan application if the borrower does not have enough assets or has too many debts based on their income. At a certain bank, 20% of the loan applications are rejected. Assume there were 225 applications. 14. What is the probability that at most 39 will be rejected? 15. What is the probability at least 57 will be rejected? 16. Great

Danes The heights of adult great danes are normally distributed with a mean of 31 inches and a standard deviation of 1 inch. If you randomly choose 4 adult great danes, what is the probability that all four are 33 inches or taller?



99

Answer Key Practice C 1. 50% 2. 5% 3. 0.16 4. 0.84 5. 0.1585 6. 0.975 7. 0.4985 8. 0.0015 9. 0.0000156 10. 0.494 11. 38, 4.85 12. 2.1, 1.44 13. 115.5, 5.15 14. 0.0256 15. 0.16

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LESSON

NAME _________________________________________________________ DATE ___________

12.7


Give the percent of the area under a normal curve represented by the shaded region.

3σ

2σ x

x

2σ

3σ x

x

3σ

2σ x

x

1σ

x x

σ x

x

x

2σ

2.

3σ

1.

Lesson 12.7

A normal distribution has a mean of 47.3 and a standard deviation of 2.7. Find the probability that a randomly selected x-value is in the given interval. 3. at most 44.6

4. at most 50

5. between 39.2 and 44.6

6. at most 52.7

7. between 47.3 and 55.4

8. at least 55.4

A normal distribution has a mean of 24.5 and a standard deviation of 3.5. Find the given probability. 9. three randomly selected x-values are all 31.5 or greater 10. four randomly selected x-values are all between 14 and 28


12. n 210, p 0.01

13. n 150, p 0.77

Saint Bernards The weights of adult Saint Bernards are normally

14.

distributed with a mean of 70.5 kilograms and a standard deviation of 20.5 kilograms. If you randomly choose 2 adult Saint Bernards, what is the probability that both are at least 91 kilograms? 15. Medicine

According to a medical study, 23% of all patients with high blood pressure have adverse side effects from a certain kind of medicine. What is the probability that out of the 120 patients with high blood pressure treated with this medicine, more than 33 will have adverse side effects? Lesson 12.3

100



Answer Key Test A 4 3 4 1. sin 5, cos 5, tan 3, cot 34, sec 53, csc 54 12 13 , cos 5 12 , sec

5 , tan 12 13 5, 13 13 cot 5 , csc 12 2 2 sin , cos , tan 1, 2 2 cot 1, sec 2, csc 2 5. 6. 180 7. 720 8. 270 2 4 in.; in.2 10. 6 ft; 27 ft2 11. 0 2 2 2 14. 0 15. 0, 0 1 13. 2 , 45 17. , 45 18. , 180 4 4 B 60, a 5.77, c 11.5 C 145, a 4.51, b 5.96 A 73.9, B 46.1, c 7.21 A 20, C 40, b 24.6 23. 30 34.6 25. 159

2. sin

3.

4. 9. 12. 16. 19. 20. 21. 22. 24. 26.

27.

y

y 1 1

1 1

x

x 2 2 Domain: 2 ≤ 0 ≤ 4 y

28. 25 m

29. 200 m

x

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CHAPTER

NAME _________________________________________________________ DATE ____________

13


Evaluate the six trigonometric functions of . 1.

2.

θ

Answers

3.

5

3

1.

7

θ 12 7

θ

2.

5

Rewrite each degree measure in radians and each radian measure in degrees. 4. 90

5. 45

7. 4

8.

6.

3.

3 2

Find the arc length and area of a sector with the given radius r and central angle . 9. r 2 in., 45

5.

10. r 9 ft, 120

6.

Evaluate the function without using a calculator. 11. sin 180 13. sin

7.

12. tan 135

3 4

8.

14. tan

9.

Evaluate the expression without using a calculator. Give your answer in both radians and degrees. 15. sin1 0 17.

sin1

4.

16. tan1 1

10. 11. 12.

2 2

1

1

18. cos

13. 14.

Solve ABC. 19. B

20. A

15. 15

16.

10 20

Review and Assess

17. C

30 C

10

A

21. C 60, a 8, b 6 22. B 120, a 10, c 18

B

18. 19. 20. 21. 22.

104



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CHAPTER

NAME _________________________________________________________ DATE ____________

13

Chapter Test A

CONTINUED


Find the area of ABC. 23.

23. 24.

B 13 A

24.

C

25.

5

23 12

10

C

26.

Use grid at left.

27.

Use grid at left.

60 B

25.

8

A

B

40

12 A

32

28.

C

29.

Graph the parametric equations. Then write an xy-equation and state the domain. 26. x t 2 and y t for 0 ≤ t ≤ 4

27. x 2t 2 and y t 3 for 0 ≤ t ≤ 3

y

y 1 1

x

1 1

x

28. Measuring Lake Width You want to

measure the width across a lake before you swim across it. To measure the width, you plant a stake on one side of the lake, directly across from the dock. You then walk 25 meters to the right of the dock and measure a 45 angle between the stake and the dock. What is the width w of the lake?

w 45 25 m

29. Ski Lift From the base of a ski lift, the angle of elevation of the


Review and Assess

summit is 30. If the ride on the ski lift is 400 meters to the summit, what is the vertical distance between the base of the lift and the summit?


105

Answer Key Test B 5 12 5 1. sin 13, cos 13, tan 12, 13 13 cot 12 5 , sec 12 , csc 5

529 229 5 , cos , tan , 29 29 2 2 29 29 cot , sec , csc 5 2 5 3 3 1 sin , cos , tan , 2 2 3 23 cot 3, sec , csc 2 3 5. 6. 360 7. 180 8. 150 4 12 cm; 72 cm 2 10. 12 ft; 96 ft2 2 0 12. 1 13. 14. 0 15. , 45 2 4 0, 0 17. , 180 18. , 60 3 B 20, a 13.2, b 4.79 C 80, a 13.1, b 17.6 A 27.5, B 112.5, c 13.9 A 15.5, B 14.5, c 28 72.4 24. 56.9 25. 20.8

2. sin

3.

4. 9. 11. 16. 19. 20. 21. 22. 23. 26.

27.

y

y 2

1 1 1

x

x 1 y 2 2 Domain: 1 ≤ x ≤ 5 28.

50,000 52,360 mi2 3

29. 102 km

x

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13



2.

13

Answers

3.

1. θ

θ 12

5

6

12

θ

2. 2


5. 180

7.

8.

6. 2

3.

5 6

Find the arc length and area of a sector with the given radius r and central angle . 9. r 12 cm, 180

10. r 16 ft, 135

4. 5. 6.

Evaluate the function without using a calculator. 11. cos 90°

12. tan 225°

4

13. sin

14. cot

8.

3 2

9.

Evaluate the expression without using a calculator. Give your answer in both radians and degrees. 15. sin1

2 2

17. cos11

10. 11.

16. tan1 0

12.

18. tan1 3

13. 14.

Solve ABC.

15.

19. A

20. C 70

7.

14

60

B

16.

Review and Assess

17. C

B 40

21. C 40, a 10, b 20 22. C 150, a 15, b 14

A

20

18. 19. 20. 21. 22.

106



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CHAPTER

NAME _________________________________________________________ DATE ____________

13

Chapter Test B

CONTINUED



B

23. 24.

24.

B

25. 10

14

105 A

13

C

15

26.

Use grid at left.

27.

Use grid at left.

A 9

25. A

8

C

B

60

28.

6

29. C

Graph the parametric equations. Then write an xy-equation and state the domain. 26. x t and y t 1 for 0 ≤ t ≤ 3

27. x 2t 1 and y t for 1 ≤ t ≤ 2

y

y 1 x

1 1 1

x

28. Radar A radar system has been set up to track approaching weather

storms. The radar is set to reach 200 miles and cover an arc of 150. Find the area of the sector that the radar covers. 29. Ships Two ships leave Boston Harbor A d C 80 km

85

Review and Assess

at the same time. What is the distance between ships A and C after they have traveled 80 kilometers and 70 kilometers respectively?

70 km

Boston



107

Answer Key Test C 5 12 5 1. sin 13, cos 13, tan 12, 13 13 cot 12 5 , sec 12 , csc 5

526 1 , tan , 26 26 5 26 cot 5, sec , csc 26 5 7 3 37 sin , cos , tan , 4 4 7 7 47 4 cot , sec , csc 3 7 3 5. 4 6. 150 7. 270 8. 110 10 27 cm; 243 cm2 10. 0.2 ft; 0.08 ft2 0 12. 1 13. 1 14. 2 5 , 45 16. , 30 17. , 150 4 6 6 0, 0 19. B 70, b 1.92, c 2.05 C 74, b 8.89, c 10.2 B 19.9, C 25.1, a 25 A 81, B 36, C 63 23. 30.8 53.4 25. 45

2. sin

3.

4. 9. 11. 15. 18. 20. 21. 22. 24. 26.

26

, cos

27.

y

y

2 2

x 1 1

x

x y 1 3 Domain: 6 ≤ x ≤ 6 28. 36.8 m 29. x 95 cos 32t 80.6t y 16t 2 95 sin 32t 3 16t2 50.3t 3

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13



2.

13

5

Answers

3.

1.

θ

θ

9 10

θ

12

2.

2

Rewrite each degree measure in radians and each radian measure in degrees. 4. 18 7.

5. 720

3 2

8.

6.

5 6

3.

11 18 4.

Find the arc length and area of a sector with the given radius r and central angle . 9. r 18 cm, 270

10. r 0.8 ft, 45

13. tan

12. cos180

7 4

14. sec

8.

5 4

9. 10.

Evaluate the expression without using a calculator. Give your answer in both radians and degrees.

17. cos1

3

2

1

13.

18. sin10

14. 15.

Solve ABC.

Review and Assess

19.

11. 12.

16. sin12

15. tan11

6. 7.

Evaluate the function without using a calculator. 11. sin 720

5.

20.

B

16.

A

17.

0.7

49

20 C

18.

A

19. 57 C

8

B

20.

21. A 135, b 12, c 15

21.

22. a 10, b 6, c 9

22.

108



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CHAPTER

NAME _________________________________________________________ DATE ____________

13

Chapter Test C

CONTINUED


Find the area of ABC. 23. A

23. 24.

24.

C

25.

10

11

12 140 8 C

25.

B

A

26.

Use grid at left.

27.

Use grid at left.

B

13

C 10

28.

60

58

A

29. B

Graph the parametric equations. Then write an xy-equation and state the domain. 26. x 3t and y t 1 for 2 ≤ x ≤ 2

27. x 2t 4 and y 2t for 0 ≤ x ≤ 3 y

y

2 2

x 1 x

1

28. Ravine Width Use the diagram to

A

find the distance across the ravine. 31 45 m 110 C

a?

B

Review and Assess

29. Baseball A baseball is hit at a speed of 95 feet per second during a

high school baseball game. The baseball was hit from a height of 3 feet and at an angle of 32. Write a set of parametric equations for the path of the baseball.



109

Answer Key Cumulative Review 3 1. x ≥ 3 2. x ≥ 4 3. x > 4 4. 12 < x < 6 5. x ≥ 6 or x ≤ 1 1 2 3 6. 3 ≤ x ≤ 1 7. y 5 x; 15 8. y 2 x; 4 9. y 14 x; 24 10. y 18 x; 48 36 12 5 23 11. y 6 x; 5 12. y 2 x; 23 13. none 14. infinite 15. 2, 7 16. 4, 2 17. none 18. infinite 19. 2, 1 20. 2, 1 21. 0, 4 1 22. 1, 5 23. 3, 3 24. 1, 3 25. y x2 8x 15 26. y x2 3x 10 1 10 27. y 3x2 15x 18 28. y 3x2 x 3 3 3 29. y 4x2 4x 15 30. y 4x2 12x 1 31. f x 4x4 3x2 7; 4; quartic; 4 32. not a polynomial function 33. not a polynomial function 34. f x 5; 0; constant; 5 35. f x 3x 2; 1; linear; 3 36. f x x2 0.6x 3; 2; quadratic; 1 37. x 2x 3x 1 38. x 3x 5x 4 39. x 1x 5x 6 40. x 3x 5x 1 41. 2x 32x 3x 1 1 4 42. 2x 1x 5x 6 43. ;x 3x 1 3 12 12x 4 44. 1; x 0 45. ;x 0 x x 46. x ≥ 14; y ≥ 0 47. all reals; all reals 48. x ≥ 3; y ≥ 0 49. x ≥ 5; y ≥ 4 50. x ≥ 4; y ≤ 5 51. x ≥ 3; y ≥ 0 x 1 x 52. y 8x 53. y 54. y 3 2 x e 55. y 56. y ex 2 57. y ex 5 5

except 1

58.

78. a.

3 169

b.

4 221

79. a.

1 2197

b.

8 16,575

81. a.

3 169

59. y

y

1 1

x

60.

61. y

y

4 1 2

x

domain: all real numbers except 2; range: all real numbers except 1

62.

63. y

y

1 x

2

1 x

1

domain: all reals numbers domain: all real except 2; range: all real numbers except 0; 3 numbers except 4 range: all real numbers except 2 64. ± 4, 0; 0, ± 3; ± 5, 0 65. ± 6, 0; 0, ± 5; ± 11, 0 66. 0, ± 25; ± 23, 0; 0, ± 22 y2 x2 1; 0, ± 10; ± 2, 0; 0, ± 46 67. 4 100 x2 y2 1; 0, ± 4; ± 3, 0; 0, ± 7 9 16 x2 y2 1; 0, ± 5; ± 10, 0; 0, ± 15 69. 10 25 68.

70.

3 2

71.

75. none

2 3

72.

76. a.

b.

9 2 1 16

4 221

73. none b.

87. 0.117

88.

2

74.

77. a. 1 64

b.

3 10 1 169

b.

4 663

13 850

82. 0.00977

84. 0.0439

3

13 204

80. a.

83. 0.205

91.

domain: all real numbers except 4; range: all real numbers

1

domain: all real numbers except 8; range: all real numbers except 2

1 1 x

domain: all real numbers except 0; range all real numbers except 0

x

1 2

85. 0.117

89. 1

92. 2

2

2

93. 3

86. 0.00097

90.

3

3

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CHAPTER

NAME _________________________________________________________ DATE ____________

13


Solve the inequality. (1.6, 1.7) 1. 3x 5 ≥ 14

4. x 3 < 9

2. 2x 1 ≤ 7

3. 42x 2 < 14

5. 2x 5 ≥ 7

6. 4 3x 1 2 ≤ 10

The variables x and y vary directly. Write an equation that relates the variables. Then find x when y 6. (2.4) 7. x 5, y 2 10. x 16, y 2

8. x 8, y 12

9. x 4, y 1

3

12. x 0.2, y 2.3

11. x 2, y

5 4

Graph the linear system and tell how many solutions it has. If there is exactly one solution, estimate the solution and check it algebraically. (3.1) 13. 4x 2y 8

14. x y 10

6x 3y 9

2x 20 2y

16.

1 2x

2y 6

x

1 2y

15. y 3x 1

4x 2y 6

17. x 4y 0

18. 1.2x 4y 4.8

0.5x 2y 8

3

10y 3x 12

Use an inverse matrix to solve the linear system. (4.5) 19. 4x 3y 11

20. 3x 5y 1

5x 2y 12 22. 3x y 8

21. 5x 2y 8

4x 5y 13

9x 3y 12

23. 2x 6y 12

4x 2x 6

24. 7x 3y 6

7x 2y 15

4x 6y 2

Write the quadratic function in standard form. (5.1) 25. y x 3x 5 28. y

1 3 x

2x 5

26. y x 5x 2 29. y

34x

5x 4

27. y 3x 1x 6 30. y 4xx 3

Decide whether the function is a polynomial function. If it is, write the function in standard form and state the degree, type, and leading coefficient. (6.2) 31. f x 3x2 4x4 7 1

32. f x x3 3x

33. f x 5x2 4x2 x

34. f x 5

35. f x 3x 2

36. f x 0.6x x2 3

3 2 37. f x x 6x 11x 6; k 2

38. f x x3 4x2 17x 60; k 3

39. f x x3 2x2 29x 30; k 1

40. f x x3 x2 17x 15; k 5

3 2 41. f x 4x 4x 9x 9; k 1

42. f x 2x3 3x2 59x 30; k 6

Let f x 4x1 and g x 3x 1. Perform the indicated operation and state the domain. (7.3) 43. f gx Copyright © McDougal Littell Inc. All rights reserved.

44. g f x

45. f x g x Algebra 2 Chapter 13 Resource Book

115

Review and Assess

Factor the polynomial given that f k 0. (6.5)

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CHAPTER

NAME _________________________________________________________ DATE ____________

13

Cumulative Review

CONTINUED


Find the domain and range of the function without graphing. (7.5) 46. y x 14

3 x 2 47. y 2

48. y x 3

49. y x 5 4

50. y x 4 5

51. y 3x 3

Find the inverse of the function. (8.4) 52. y log 8 x

53. y log 1 3 x

x 54. y log 8 64

55. y ln 5x

56. y ln x 2

57. y ln x 5

Graph the function. State the domain and range. (9.2) 58. y

5 x

59. y

4 1 x4

60. y

1 2 x8

61. y

x3 x2

62. y

3x 5 4x 8

63. y

4x 3 2x

Write the equation in standard form (if not already). Then identify the vertices, co-vertices, and foci for the ellipse. (10.4) 64.

y2 x2 1 16 9

65.

67. 25x2 y2 100

x2 y2 1 36 25

66.

68. 16x2 9y2 144

x2 y2 1 12 20

69. 25x2 10y2 250

Find the sum of the infinite geometric series if it has one. (11.4) 70.

1 3

n0

73.

n

71.

n0

n0

2

5 3

n

74.

1 2

n

1 2 3 n1 2

72.

3

n0 n1

75.

1 3

1

n

23

n

n0

Find the probability of drawing the given cards from a standard 52-card deck (a) with replacement and (b) without replacement. (12.5) 76. a diamond, then a spade

77. a king, then a queen

78. a 3, then a face card (K, Q, or J)

79. an ace, then a 3, then a 5

80. a diamond, then a spade, then another diamond

81. a face card (K, Q, or J), then a 10

Review and Assess

Find the probability of getting the given number of heads in 10 tosses of a coin. (12.6) 82. 1

83. 4

84. 8

85. 3

86. 10

87. 7

Evaluate the function without using a calculator. (13.3) 88. sin 750

91. cos

116

5 6

89. cos 225


92. sin

11 6

90. tan 210

93. tan

5 3


Answer Key Practice A 685 785 ; cos ; 1. sin 85 85 85 85 6 7 tan ; cot ; sec ; csc 7 6 7 6 313 213 ; cos ; 2. sin 13 13 13 13 3 2 tan ; cot ; sec ; csc 2 3 2 3 3 4 3 3. sin ; cos ; tan ; 5 5 4 4 5 5 cot ; sec ; csc 3 4 3 26 526 ; cos ; 4. sin 26 26 26 1 tan ; cot 5; sec ; csc 26 5 5 35 35 1 ; tan ; 5. sin ; cos 6 6 35 635 cot 35; sec ; csc 6 35 534 334 5 ; cos ; tan ; 6. sin 34 34 3 34 34 3 cot ; sec ; csc 5 3 5 7. x 8; y 43 8. x 43; y 4 9. x 22; y 22 10. 0.2588 11. 0.6820 12. 2.1445 13. 3.2361 14. 1.1034 15. 0.5317 16. 0.9848 17. 0.9848 18. A 78; b 0.9; c 4.1 19. B 16; a 19.2; b 5.5 20. B 40; a 9.5; c 12.4 21. A 52; a 5.5; b 4.3 22. B 18; a 55.4; c 58.2 23. A 68; b 2.0; c 5.4 24. about 346 ft

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LESSON

NAME _________________________________________________________ DATE ___________

13.1


Evaluate the six trigonometric functions of the angle . 1.

2.

3. θ

6

5

4

12

θ

Lesson 13.1

7

θ 8

4.

5.

θ

2

6. θ

10 18

10

θ 6

3

Find the missing side lengths x and y. 7.

8. x

9.

4 8

30

4

x

x

y 45

60 y

y

Use a calculator to evaluate the trigonometric function. Round the result to four decimal places. 10. sin 15

11. cos 47

12. tan 65

13. csc 18

14. sec 25

15. cot 62

16. sin 80

17. cos 10

Solve ABC using the diagram and the given measurements. 18. B 12, a 4

19. A 74, c 20

20. A 50, b 8

21. B 38, c 7

22. A 72, b 18

23. B 22, a 5

A

C

24.

c

b

a

B

Redwood Trees You are standing 200 feet from the base of a redwood tree. You estimate the angle of elevation to the top of the tree is 60. What is the approximate height of the tree? 60

Not drawn to scale.

200 ft Copyright © McDougal Littell Inc. All rights reserved.


15

Answer Key Practice B 8 15 8 1. sin ; cos ; tan ; 17 17 15 15 17 17 cot ; sec ; csc 8 15 8 21 2 221 ; tan ; 2. sin ; cos 5 5 21 21 521 5 ; sec ; csc cot 2 21 2 22 1 ; cos ; tan 22; 3. sin 3 3 2 32 ; sec 3; csc cot 4 4 4. x 72; y 7 5. x 5; y 53 6. x 23; y 3 7. 0.8910 8. 0.0875 9. 0.7431 10. 0.1584 11. 2.5593 12. 2.4586 13. 4.3315 14. 0.5299 15. B 44; a 8.3; c 11.5 16. A 66; a 11.9; b 5.3 17. A 72; a 9.5; b 3.1 18. B 35; b 14.0; c 24.4 19. A 20; b 16.5; c 17.5 20. B 83; a 2.2; c 18.1 21. about 14.4 ft 22. about 21,477 ft or 4.1 mi

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LESSON

NAME _________________________________________________________ DATE ___________

13.1



2.

3. 5

8

θ

3

2

1

θ

θ

Lesson 13.1

15


5.

6. 10

x

x x

30

7

3

y 45

60 y

y

Use a calculator to evaluate the trigonometric function. Round the result to four decimal places. 7. cos 27 11. csc 23

8. tan 5

9. sin 48

10. cot 81

12. sec 66

13. cot 13

14. sin 32

Solve ABC using the diagram and the given measurements. 15. A 46, b 8

16. B 24, c 13

17. B 18, c 10

18. A 55, a 20

19. B 70, a 6

20. A 7, b 18

A

b

C

21.

c

a

B

Flagpole You are standing 25 feet from the base of a flagpole. The angle of elevation to the top of the flagpole is 30. What is the height of the flagpole to the nearest tenth?

22.

Mount Fuji Mt. Fuji in Japan is approximately 12,400 feet high. Standing several miles away, you estimate the angle of elevation to the top of the mountain is 30. Approximately how far way are you from the base of the mountain?

16



Answer Key Practice C 69 10 1069 1. sin ; cos ; tan ; 13 13 69 69 1369 13 cot ; sec ; csc 10 69 10 3 4 3 2. sin ; cos ; tan ; 5 5 4 4 5 5 cot ; sec ; csc 3 4 3 3 3 1 3. sin ; cos ; tan ; 2 2 3 23 ; csc 2 cot 3; sec 3 4. x 10; y 53 5. x 42; y 42 6. x 163; y 32 7. 0.1763 8. 1.5557 9. 0.7771 10. 0.0175 11. 1.1434 12. 0.9986 13. 1.2799 14. 2.5593 15. B 76; b 24.1; c 24.8 16. B 33; a 18.5; c 22.0 17. A 58; a 17.3; b 10.8 18. B 26; a 11.5; b 5.6 19. A 17; b 55.6; c 58.1 20. A 80; a 79.4; c 80.6 21. about 66.78 ft or 66 ft 9 in.

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LESSON

NAME _________________________________________________________ DATE ___________

13.1



2.

3. θ

10

9

13

6 3

9

θ

Lesson 13.1

12

θ


5. 5

6.

x 8

x

30

y

x

y 45

60 16

y

Use a calculator to evaluate the trigonometric function. Round the result to four decimal places. 7. tan 10 11. sec 29

8. csc 40 12. cos 3

9. sin 51

10. cos 89

13. cot 38

14. sec 67

Solve ABC using the diagram and the given measurements. 15. A 14, a 6

16. A 57, b 12

17. B 32, c 20.4

18. A 64, c 12.8

19. B 73, a 17

20. B 10, b 14

B c

A

21.

a

C

b

Baseball Diamond A baseball diamond is laid out so that the bases are 90 feet apart and at right angles as shown at the right. The distance from home plate to the pitcher’s mound is 60 feet 6 inches. Find the distance from the pitcher’s mound to second base. (Hint: The pitcher’s mound is not exactly halfway between home plate and second base.)

90 ft

Pitcher Home Plate



17

Answer Key Practice A 1. B 2. A 3. C 4.

5.

y

y

100 x

6.

45 x

7.

y

y

8π 9

x

x 12π 5

8–11. Sample angles are given. 8. 585; 135

9. 420; 300

3 ; 2 2 13 14. 9 10.

2 6 4 3 ; 12. 13. 9 5 5 4 43 15. 16. 105 17. 150 18. 120 36 2 4 2 5 25 2 in.; in. 21. m; m 19. 30 20. 3 3 12 24 1 22. 14 cm; 84 cm2 23. 2 24. 1 25. 0.9511 26. 0.9239 27. about 4.19 ft 11.

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LESSON

13.2

NAME _________________________________________________________ DATE ___________


Match the angle measure with the angle. 1. 320 A.

2. B.

y

6 5

3.

7 4

C.

y

x

y

x

x

Draw an angle with the given measure in standard position. 5. 45

4. 100

6.

8 9

7.

12 5

8. 225

9. 60

10.

15 2

11.

Lesson 13.2

Find one positive angle and one negative angle coterminal with the given angle. 16 5

Rewrite each degree measure in radians and each radian measure in degrees. 12. 135 16.

13. 40

7 12

17.

14. 260

5 6

18.

15. 215

2 3

19.

6

Find the arc length and area of a sector with the given radius r and central angle . 20. r 4 in.,

6

21. r 5 m,

12

22. r 12 cm, 210

Evaluate the trigonometric function using a calculator if necessary. If possible, give an exact answer. 23. cos

27.

3

24. tan

4

25. sin

2 5

26. cos

8

Pendulum The pendulum of a grandfather clock is 4 feet long and swings back and forth creating a 60 angle. Find the length of the arc of the pendulum, after one swing.



29


2.

y

y

215 x

x

135

3.

4.

y

y

7π 12

x

x 5π 6


6. 180; 180

7 2 4 2 8 8. 9. ; ; 6 3 3 5 5 17 5 11 10. 11. 12. 13. 585 9 9 12 7.

14. 480

15. 120

17. 18 ft; 108 ft2 19. 20 m; 200 m2

16. 15

3 3 2 in.; in. 2 2 3 20. 21. 0.4142 2

18.

23 25. 2 3 26. 0.1045 27. 0.4142 28. about 2.75 in. 29. 1080; 6 22. 1

23. 0.9010 24.

MCRB2-1302-PA.qxd

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LESSON

13.2

NAME _________________________________________________________ DATE ___________


Draw an angle with the given measure in standard position. 2. 135

1. 215

3.

7 12

4.

5 6

Find one positive angle and one negative angle coterminal with the given angle. 5. 340

6. 540

7.

20 3

8.

12 5


9. 210

Lesson 13.2

13.

13 4

14.

8 3

11. 165 15.

12. 100

2 3

16.

12

Find the arc length and area of a sector with the given radius r and central angle . 17. r 12 ft,

3 2

18. r 2 in.,

3 4

19. r 20 m, 180

Evaluate the trigonometric function using a calculator if necessary. If possible, give an exact answer. 20. sin

3

21. tan

8

22. sin

2

23. cos

7

24. sec

6

25. csc

4

26. cos

7 15

27. cot

3 8

28.

Fire Truck Ladder For the ladder on a fire truck to operate properly, the base of the ladder must be almost level. The diagram at the right shows part of a leveling device that is used to determine whether the level of the ladder’s base is within the allowable range. Find the length of the arc that describes the allowable range.

29.

30

7 in. Allowable range π 8

Snowboarding During a competition, a snowboarder performs a trick involving three revolutions. Find the measure of the angle generated as the snowboarder performs the trick. Give the answer in both degrees and radians.




2.

y

y

7π 15

15π 4

x

3.

x

4.

y

y

500

290

x

x


6. 285; 435

10 2 7 9 7 ; ; 8. 9. 3 3 4 4 4 37 5 10. 11. 12. 13. 150 90 180 12 7.

14. 135

15. 40

16. about 172

26 169 2 35 735 cm; cm 18. in.; in.2 3 3 4 16 3 28 112 2 m; m 20. 19. 21. 3 3 3 3 17.

22. 2.6131

23. 0.3090

25. 2.6131

26. 3

29. about 4190 mi

24. 0.9945

27. 2

28. 540; 3

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Page 31

LESSON

13.2

NAME _________________________________________________________ DATE ___________


Draw an angle with the given measure in standard position. 1.

7 15

2.

15 4

3. 290

4. 500

Find one positive angle and one negative angle coterminal with the given angle. 6. 75

5. 105

7.

4 3

8.

4


9. 315 13.

5 6

14.

3 4

11. 2 15.

12. 37

2 9

16. 3

17. r 13 cm,

2 3

18. r 10.5 in., 150

Lesson 13.2

Find the arc length and area of a sector with the given radius r and central angle . 19. r 8 m, 210

Evaluate the trigonometric function using a calculator if necessary. If possible, give an exact answer. 20. cot

3

21. tan

3

22. csc

8

23. cos

2 5

24. sin

7 15

25. sec

3 8

26. cot

6

27. csc

4

28.

Bicycles A bicycle’s gear ratio is the number of times the freewheel turns for every one turn of the chainwheel. The table shows the number of teeth in the freewheel and chainwheel for the first 5 gears on an 18-speed bicycle. In first gear, if the chainwheel completes 2 rotations, through what angle does the freewheel turn? Give your answer in both degrees and radians.

29.

Gear Number 1 2 3 4 5

Number of teeth in freewheel 32 26 22 32 19

Number of teeth in chainwheel 24 24 24 40 24

Earth Assuming that Earth is a sphere of diameter 8000 miles, what is the distance between city A and city B in the figure shown if the central angle is 60? 60

A

B



31

Answer Key Practice A 3 5

4 5

3 4

12.

y

11.

y

θ 315

1. sin ; cos ; tan ;

4 5 5 cot ; sec ; csc 3 4 3 5 25 1 ; cos ; tan ; 2. sin 5 5 2 5 ; csc 5 cot 2; sec 2 5106 9106 ; cos ; 3. sin 106 106 5 106 9 tan ; cot ; sec ; 5 9 9 csc

5

889 589 8 ; cos ; tan ; 89 89 5 5 89 89 ; csc cot ; sec 8 5 8 4. sin

5. sin

2

; cos ; tan 1; 2 2 cot 1; sec 2; csc 2 758 358 7 6. sin ; cos ; tan ; 58 58 3 58 58 3 ; csc cot ; sec 7 3 7 26 526 1 ; cos ; tan ; 7. sin 26 26 5 26 cot 5; sec ; csc 26 8. 5 sin 0; cos 1; tan 0;

cot undefined; sec 1; sec undefined 9. sin 1; cos 0; tan undefined; cot 0; sec undefined; csc 1 10. sin 0; cos 1; tan 0; cot undefined; sec 1; csc undefined

x

x

θ 45

θ 150

13.

14.

y

θ

y

7π 4

θ x

θ

15.

106

2

θ 30

3

2 3 19. 2

16. 2 20. 2

23. 0.9659

17.

27. 0.8391 30.

1 2

π 4

3

2

21. 2

24. 0.1736

26. undefined 29. 0.5774

θ

π 4

13π 4

x

18. 1 22. 3

25. 2.4751 28. 1.0101

31. no

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LESSON

NAME _________________________________________________________ DATE ___________

13.3


Use the given point on the terminal side of an angle in standard position. Evaluate the six trigonometric functions of . 1.

2.

y

3.

y

y

(9, 5) 3

θ

θ 3

3

x

x

6

3

x

(10, 5)

(8, 6)

4. 5, 8

θ

3

5. 4, 4

6. 3, 7

7. 10, 2

Evaluate the six trigonometric functions of the quadrantal angle . 8. 180

9. 90

10. 360

Sketch the angle. Then find its reference angle. 11. 150

12. 315

13.

7 4

14.

13 4

Evaluate the function without using a calculator. 15. sin 300

2 3

20. csc

11 4

17. cos 750 21. sec

5 4

18. tan 405 22. tan

17 3

Use a calculator to evaluate the function. Round the result to four decimal places. 23. sin 435 27. tan 31.

42

2 9

24. cos 100

25. tan 112

28. sec 3

29. cot

11 3

26. sec 450 30. cos

7 3

Baseball You are at bat and hit a baseball so that it has an initial velocity of 80 feet per second and an angle of elevation of 40. Assuming the ball is not caught and the fence is 305 feet away, did you hit a homerun?



Lesson 13.2

Lesson 13.3

19. sin

16. csc 225


10.

1. sin

997 497 ; cos ; 97 97

y

θ 220 θ 25

x

θ 40

9 97 4 tan ; cot ; sec ; 9 4 4 97 csc 9 541 441 2. sin ; cos ; 41 41 5 41 4 tan ; cot ; sec ; 5 4 4 csc

11.

y

x

θ 155

12.

13.

y

y

θ θ

x

π 3

θ

2π

θ 3

π 3

x

7π 3

41

5

14.

3 4 3 3. sin ; cos ; tan ; 5 5 4 4 5 5 cot ; sec ; csc 3 4 3 4. sin

θ

x

π 3

8π

16. 1 20. 3

89

cot undefined; sec 1; csc undefined 9.

y

y

θ θ 65

θ 45 x

θ 225

x

θ

θ 3

889 589 ; cos ; 89 89

8 5. sin 1; cos 0; tan undefined; cot 0; sec undefined; csc 1 6. sin 1; cos 0; tan undefined; cot 0; sec undefined; csc 1 7. sin 0; cos 1; tan 0;

8.

y

θ

89 8 5 tan ; cot ; sec ; 8 5 5

csc

15.

y

17.

3

2 23 21. 3

18. 22. 3

24. 0.3090 25. 1.1434 27. 1.0515

x

12π 5

3

2

2π 5

19. 2

23.

2

2

26. 0.5

28. The terminal side of a 10 angle would be in the first quadrant where the sine function is positive. Your friend’s calculator was in radian mode. 29. 307.75 ft; 312.5 ft; 307.75 ft

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Page 43

LESSON

13.3

NAME _________________________________________________________ DATE ___________


Use the given point on the terminal side of an angle in standard position. Evaluate the six trigonometric functions of . 1. 4, 9

2. 4, 5

3. 8, 6

4. 5, 8


6. 90

7. 180

Sketch the angle. Then find its reference angle. 8. 225 12.

2 3

9. 65 13.

11. 155

10. 220

7 3

14.

8 3

15.

12 5

Evaluate the function without using a calculator. 16. tan 135 20. cot

7 6

17. sin 60 21. csc

2 3

19. sec 315

18. cos 210 22. tan

7 3

23. sin

3 4

Use a calculator to evaluate the function. Round the result to four decimal places. 24. sin 18 28.

25. sec 29

26. cos

10 3

27. csc

18 5

Critical Thinking Your friend used a calculator to evaluate sin 10 and

29.

Baseball You are at bat and hit the baseball so that it has an initial velocity of 100 feet per second. Approximately how far will the ball travel horizontally if the angle of elevation is 40? 45? 50?


Lesson 13.3

Lesson 13.2

obtained 0.544. How can you tell this is incorrect? What did your friend do wrong?


43

Answer Key Practice C 1. sin

12.

213 13

; cos

313 13

θ

; θ

2 13 3 tan ; cot ; sec ; 2 3 3 13 csc 2 3 3 1 ; tan ; 2. sin ; cos 2 2 3 23 cot 3; sec ; csc 2 3 2 2 ; cos ; tan 1; 3. sin 2 2 cot 1; sec 2; csc 2 15 8 15 4. sin ; cos ; tan ; 17 17 8 8 17 17 cot ; sec ; csc 15 8 15

9.

10.

θ 200 θ 20

θ 15

11.

y

x

y

θ

π 3

2

2

21. 2

y

θ 60 x

θ 240

27. 0.9511

θ

2π 3

x

2π 5

17.

24. 0.0402

x

x

15.

12π 5

16. 2 20.

20π 3

x

x

y

θ 510

θ

π 3

x

θ

θ 345 θ 30

y

θ

y

θ

cot 0; sec undefined; csc 1 6. sin 0; cos 1; tan 0; cot undefined; sec 1; csc undefined 7. sin 0; cos 1; tan 0; cot undefined; sec 1; csc undefined y

21π 4

π 4

14.

5. sin 1; cos 0; tan undefined;

8.

13.

y

2

2

18.

22. 3

25. 0.5774

23 19. 3 3

23. 2

26. undefined

28. about 152 ft/sec; about 722 ft

29. about 6.70 ft

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Page 44

LESSON

13.3

NAME _________________________________________________________ DATE ___________


Use the given point on the terminal side of an angle in standard position. Evaluate the six trigonometric functions of . 1. 3, 2

2. 3, 1

3. 2, 2

4. 8, 15


6. 180

7. 360

Sketch the angle. Then find its reference angle. 8. 510 12.

21 4

9. 345 13.

20 3

11. 240

10. 200 14.

12 5

15.

2 3

Evaluate the function without using a calculator. 16. sec 225 20. cos

15 4

17. cos 225 21. csc

5 6

18. csc 120 22. cot

11 6

19. tan 240

23. sec

4 3

Use a calculator to evaluate the function. Round the result to four decimal places.

Lesson 13.3

24. tan 2.3

25. cot 420

26. sec

9 2

28.

Driving Golf Balls You and a friend are driving golf balls at a driving range. If the angle of elevation is 30 and the ball travels 625 feet horizontally, what is the initial velocity of the ball? Suppose you use the same initial velocity and hit the ball at an angle of 45. How far would the ball travel?

29.

Fishing You and a friend are fishing. Each of you casts with an initial

27. sin

18 5

velocity of 40 feet per second. Your cast was projected at an angle of 45 and your friend’s at an angle of 60. About how much further will your fishing tackle go than your friend’s?

44



Answer Key Practice A

; 30 3. ; 30 4. ; 30 6 6 6 0; 0 6. ; 45 7. ; 60 8. ; 45 3 4 4 33.7 10. 36.9 11. 39.8 12. 36.9 18.4 14. 71.6 15. 0.644; 36.9 2.42; 139 17. 1.35; 77.5 18. 1.82; 104 1.47; 84.3 20. 1.19; 68.2 1.12; 64.2 22. 0.412; 23.6 23. 166 214 25. 68.2 26. 320 35.8; 0.625

1. ; 180 5. 9. 13. 16. 19. 21. 24. 27.

2.

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Page 55

LESSON

NAME _________________________________________________________ DATE ___________

13.4


Evaluate the expression without using a calculator. Give your answer in both radians and degrees. 1. cos11

2. sin1

5. tan1 0

6. sin1

1 2

3. tan1

2

7. cos1

2

3

4. cos1

3 1 2

3

2

8. tan1 1

Find the measure of the angle . Round to three significant digits. 9.

10.

11. θ

θ 5

3

3

6

θ 5 2

12.

13.

14. 2

10

6

3

θ

θ 9

θ 8

Use a calculator to evaluate the expression in both radians and degrees. Round to three significant digits. 15. sin1 0.6

16. cos1 0.75

17. tan1 4.5

18. cos1 0.25

19. cos1 0.1

20. tan1 2.5

21. sin1 0.9

22. sin10.4

Solve the equation for . Round to three significant digits. 23. sin 0.25; 90 < < 180

24. cos 0.83; 180 < < 270

25. tan 2.5; 0 < < 90

26. sin 0.64; 270 < < 360 1

The height of an outdoor basketball backboard is 122 feet, and the backboard casts a shadow 1713 feet long, as shown below. Find the angle of elevation of the sun. Give your answer in both radians and degrees.

27. Basketball

Lesson 13.4

1

12 2 ft

θ 1 17 3


ft Algebra 2 Chapter 13 Resource Book

55

Answer Key Practice B 2 ; 120 2. ; 45 3. ; 30 1. 6 4 3 4. ; 60 5. 36.9 6. 25.4 7. 18.4 3 8. 1.43; 82.0

9. 0.222; 12.7

10. 0.644; 36.9 11. 0.381; 21.8 12. 259

13. 297 14. 127

16. about 127 17. about 11.5

15. 56.8

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LESSON

NAME _________________________________________________________ DATE ___________

13.4



12

1. cos1

2. sin1

2

2

3. tan1

3

3

4. tan1 3


6. 5

7. 3

7

6 2

θ 6

θ

θ

2

4

Use a calculator to evaluate the expression in both radians and degrees. Round to three significant digits. 8. cos1 0.14

9. sin1 0.22

10. sin1 0.6

11. tan1 0.4

Solve the equation for . Round to three significant digits. 12. tan 5.3; 180 < < 270

13. sin 0.89; 270 < < 360

14. cos 0.6; 90 < < 180

15. tan 1.53; 0 < < 90

Find the measure of angle in the diagram below. Round the result to three significant digits.

16. Geometry

3

θ 1 3

1

In a video game, a target appears on the left side of the television screen and moves at the rate of 2 inches per second across the screen. You fire a laser beam that travels 10 inches per second. If the player tries to hit the target as soon as it appears, at what angle should the laser beam be aimed?

Lesson 13.4

17. Video Games

θ

56



Answer Key Practice C 5 1. ; 150 2. ; 60 3. ; 45 3 4 6 4. ; 90 5. 17.4 6. 45 7. 44.4 2 8. 0.694; 39.8 10. 0.955; 54.7 12. 281

9. undefined 11. 1.48; 84.6

13. 164 14. 215

16. about 12

17. about 51.3

15. 221

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LESSON

NAME _________________________________________________________ DATE ___________

13.4



1. cos1

3

2

2. tan1 3

3. cos1

2

4. sin1 1

2


6.

7. θ

2.5

θ 8

2

5

θ

5

2 7

Use a calculator to evaluate the expression in both radians and degrees. Round to three significant digits. 8. sin1 0.64

9. cos1 1.3

10. tan1 2

11. tan1 10.5

Solve the equation for . Round to three significant digits. 12. tan1 5.3; 270 < < 360

13. sin1 0.28; 90 < < 180

14. cos1 0.82; 180 < < 270

15. tan1 0.88; 180 < < 270

16. Ramp Construction

A builder needs to construct a wheelchair ramp 24 feet long that rises to a height of 5 feet above level ground. Approximate the angle that the ramp should make with the ground.

17. Casting Shadows

At a certain time of the day a child five feet tall casts a four foot long shadow as shown below. Approximate the angle of elevation of the sun.

5 ft

θ 4 ft

Lesson 13.4



57

Answer Key Practice A 1. one triangle 2. one triangle 3. no triangle 4. one triangle 5. two triangles 6. one triangle 7. C 105, b 14.1, c 19.3 8. C 78, b 5.82, c 6.58 9. B 55.2, C 87.8, c 18.3; or B 124.8, C 18.2, c 5.71 10. B 21.6, C 122.4, c 11.5 11. no solution 12. B 10, b 69.5, c 137 13. B 70.4, C 51.6, c 4.16; or

B 109.6, C 12.4, c 1.14 14. 408 units2 15. 120 units2 16. 12.0 units2 17. 2.6 units2 18. 23.8 units2 19. 361 units2 20. 24.3 units2 21. about 9.58 ft

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Lesson 13.5

LESSON

NAME _________________________________________________________ DATE ___________

13.5


Decide whether the given measurements can form exactly one triangle, exactly two triangles, or no triangle. 1. B 110, C 30, a 15

2. B 35, a 12, b 26

3. B 130, a 10, b 8

4. B 60, b 30, c 20

5. C 16, b 92, c 32

6. A 10, C 130, b 5

Solve ABC. (Hint: Some of the “triangles” have no solution and some have two solutions.) 7.

8.

C

9.

C

C

10 30 A

15

4.5

45 B 42

37

60

A

11

A

B

10. A 36, a 8, b 5

11. C 160, c 12, b 15

12. A 150, C 20, a 200

13. A 58, a 4.5, b 5

B

Find the area of the triangle with the given side lengths and included angle. 14. A 70, b 28, c 31

15. B 35, a 12, c 35

16. C 95, a 8, b 3

17. A 10, b 5, c 6


19.

C 82 6

A

30

C

45

20. A 5

8 34

A

C

76

10

B

B B

21. Surveying

A surveyor wants to find the width of a narrow, deep gorge from a point on the edge. To do this, the surveyor takes measurements as shown in the figure. How wide is the gorge? 105 10 50 ft

70



Answer Key Practice B 1. no triangle 2. one triangle 3. two triangles 4. one triangle 5. C 110, b 22.4, c 24.4 6. B 21.4, C 116.6, c 29.4 7. C 35, b 18.5, c 10.8 8. A 38, a 22.0, c 34.0 9. no solution 10. A 40.9, C 84.1, c 30.4 11. B 71.8, C 78.2, c 39.2; or B 108.2, C 41.8, c 26.7 12. 2290 units2 13. 10.4 units2 14. 24.3 units2 15. 23.8 units2 16. 361 units2 17. 24.3 units2 18. 1680 units2 19. about $5,680 20. about 550 feet

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Page 71

LESSON

NAME _________________________________________________________ DATE ___________

13.5

Lesson 13.5


Decide whether the given measurements can form exactly one triangle, exactly two triangles, or no triangle. 1. A 63, a 42, b 120

2. B 47, A 60, a 45

3. B 30, b 40, a 60

4. A 60, B 40, c 6


6.

C

10

60 A

7. B

C

4.5

22

12 42

B

80

A

17 C

B 65 A

8. B 34, C 108, b 20

9. A 42, a 10, b 21

10. B 55, a 20, b 25

11. A 30, a 20, b 38

Find the area of ABC. 13. A

12. A

14. A 5

52

C

75 10

6

102 C

B

120

B

90

C

4

B

15. C 82, a 8, b 6

16. A 45, b 30, c 34

17. B 76, a 10, c 5

18. A 43.75, b 57, c 85

19. Real Estate

You are buying the triangular piece of land shown. The price of the land is $2500 per acre (1 acre 4840 square yards). How much does the land cost? C 100 yd

95

A

B

250 yd

20. Measuring an Island

What is the width w of the island in the figure

shown below? 27 39 1200 ft w



71

Answer Key Practice C 1. no triangle 2. two triangles 3. two triangles 4. one triangle 5. B 51.4, C 53.6, b 4.85 6. A 29.3, C 132.7, c 28.5; or A 150.7, C 11.3, c 7.61 5 , a 32.2, b 39.4 7. C 12 8. C 100, a 4.76, b 10.2 9. C 100, b 25.8, c 30.2 10. B 55.2, C 87.8, c 18.3; or

B 124.8, C 18.2, c 5.7 11. no solution 12. 1680 units2 13. 366 units2 14. 110 units2 15. 7 units2 16. 1.41 units2 17. 46.8 units2 18. 98.3 units2 19. about 2.67 miles

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Lesson 13.5

LESSON

NAME _________________________________________________________ DATE ___________

13.5


Decide whether the given measurements can form exactly one triangle, exactly two triangles, or no triangle. 1. A 76.4, a 176, b 189

2. A 48.2, a 15, b 20

3. A 20, a 10, c 11

4. C 95, a 8, c 9


6.

C

7.

C 12 A

6

C

19 18

B A

π 4

44

75 A

π 3

B

B

5

8. A 23, B 57, c 12

9. A 23, B 57, a 12

10. A 37, a 11, b 15

11. B 130, a 10, b 8


13. A

B

C

45

14.

C 10 95

85 34 A

43.75 57

A

60

25

B

B

C

, a 4, c 1 4

15. B 150, a 7, c 4

16. B

17. A 60, b 9, c 12

18. B 25, a 15, c 31

19. Hot Air Balloon

You and a friend live 8.4 miles apart. A hot air balloon is floating between your houses as shown in the figure. Given the angles of elevation, approximate the height of the balloon. (Hint: The height of the balloon is the altitude of the triangle.)

24

48

8.4 mi Your house

72

Friend's house



Answer Key Practice A 1. A 26.7, C 33.3, b 141 2. A 95.3, B 24.7, c 27.0 3. A 76.7, B 38.7, C 64.6 4. A 54.3, B 79.7, c 100 5. A 57.5, B 71.5, c 283 6. A 44.4, B 44.4, C 91.2 7. B 74.5, C 43.5, a 51.3 8. A 142.0, B 12.8, C 25.2 9. A 62.9, B 79.6, C 37.5 10. A 48.8, B 65.6, C 65.6 11. A 122.2, B 19.8, c 29.1 12. B 104.9, C 15.1, b 33.5 13. 16.2 units2 14. 41.2 units2 15. 96.8 units2 16. 54 units2 17. 1350 units2 18. 713 units2 19. 56.9 units2 20. 10.4 units2 21. 6 units2 22. B 52.6 E of S; C 25.3 W of S

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Page 82

LESSON

NAME _________________________________________________________ DATE ___________

13.6


Solve ABC. 1. A

2.

3.

A

B

13 C

Lesson 13.6

89

60 31

B

13

14

120 B

73

C A

C

9

4. C 46, a 113, b 137

5. C 51, a 307, b 345

6. a 7, b 7, c 10

7. A 62, b 56, c 40

8. a 39, b 14, c 27

9. a 19, b 21, c 13

Use the Law of Sines, Law of Cosines, or the Pythagorean theorem to solve ABC. 10.

11.

C 40

C

40 38 16

33

B

12.

C

A 30

A

B

40


B

14.

B

10

5

15.

B

15 12

7

A

60 9 B

A

C

A

7

20

20

C A

10

C

16. a 9, b 12, c 15

17. a 75.4, b 52, c 52

18. a 47, b 36, c 41

19. a 13, b 14, c 9

20. a 2.5, b 10.2, c 9

21. a 3, b 4, c 5

22.

Boat Race A boat race occurs along a triangular course marked by buoys A, B, and C. The race starts with the boats going 8000 feet due north. The other two sides of the course lie to the east of the first side, and their lengths are 3500 feet and 6500 feet as shown at the right. Find the bearings for the last two legs of the course.

N

B 3500

W

E S

C 8000 6500

A 82



Answer Key Practice B 1. B 37.5, C 84.5, a 15.3 2. A 82.1, B 58.8, C 39.1 3. A 51.6, B 27.3, C 101.1 4. A 38.0, C 42, b 19.2 5. A 38.3, B 99.7, c 23.8 6. A 102.2, B 38.4, c 81.8 7. A 153.5, B 15.5, C 11.0 8. B 12, a 17.9, c 20.8 9. A 30, a 29.0, c 41.0 10. C 105, b 18.4, c 35.5 11. A 48.5, C 69.5, b 4.71 12. B 79, a 102, c 17.7 13. A 55.8, B 8.6, C 115.6 14. 2813 units2 15. 10.4 units2 16. 56.9 units2 17. 1.62 units2 18. 0.468 units2 19. 43.3 units2 20. 9.92 units2 21. about 110 ft 22. about 4 ft

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Page 83

LESSON

NAME _________________________________________________________ DATE ___________

13.6


Solve ABC. 1.

B

2.

3.

B

B

18 A

58

14

22

15 12

A

C

C

19

A

Lesson 13.6

11 C

7

4. B 100, a 12, c 13

5. C 42, a 22, b 35

6. C 39.4, a 126, b 80.1

7. a 21.46, b 12.85, c 9.179

Use the Law of Sines, Law of Cosines, or the Pythagorean theorem to solve ABC. 8. A 48, C 120, b 5

9. B 15, C 135, b 15

10. A 45, B 30, a 26

11. B 62, a 4, c 5

12. A 91, C 10, b 100

13. a 11, b 2, c 12

Find the area of ABC. 14. B

15.

16.

B 2.5 A

B

10.2 9

C 13

89 120 A

73

14

C A

9

17. a 4.25, b 1.55, c 3

18. a 1.42, b 0.75, c 1.25

19. a 10, b 10, c 10

20. a 11, b 2, c 12

21.

C

Measuring a Pond How wide is the pond shown in the figure below? 152 ft 45 131 ft

22.

60 ft

Softball The pitcher’s mound on a softball field is 46 feet from home plate. The distance between the bases is 60 feet. How much closer is the pitcher’s mound to second base than it is to first base?

46 ft 60 ft


60 ft

45

60 ft


83

Answer Key Practice C 1. A 48.0, B 82.0, c 15.5 2. A 117.9, B 29.4, C 32.7 3. A 62.5, C 77.5, b 72.4 4. A 52.4, B 82.6, c 16.4 5. B 42.1, C 20.4, a 9.92 6. A 27.3, B 33.7, C 119 7. A 40.9, B 82.2, C 56.9 8. B 17.4, C 109.6, c 9.4 9. A 50.5, C 89.5, c 15.6; or 10. 11. 12. 13. 14. 17. 20. 22.

A 129.5, C 10.5, c 2.83 A 34.0, B 122.9, C 23.1 A 15, a 3.7, c 12.2 C 98, a 9.68, c 18.1 A 70.5, B 86.6, C 22.9 0.496 units2 15. 159 units2 16. 116 units2 43.2 units2 18. 20.4 units2 19. 62.0 units2 0.959 units2 21. about 2.01 acres about 92 ft

MCRB2-1306-PA.qxd

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12:27 PM

Page 84

LESSON

NAME _________________________________________________________ DATE ___________

13.6


Solve ABC. 1.

2. B

B

3.

15

90

Lesson 13.6

20

110

55

50 A

B 40 100

C A

50

C A C

4. C 45, a 132, b 23

5. A 117.5, b 7.5, c 3.9

6. a 4.3, b 5.2, c 8.2

7. a 20.1, b 30.4, c 25.7

Use the Law of Sines, Law of Cosines, or the Pythagorean theorem to solve ABC. 8. A 53, a 8, b 3

9. B 40, a 12, b 10

10. a 10, b 15, c 7

11. B 45, C 120, b 10

12. A 32, B 50, b 14

13. a 17, b 18, c 7


15.

B

A 2

16.

B 12

B

32

34 40

26

C

2 A

A

1 2

34 16

C

17. a 4, b 24, c 26

18. a 12, b 9, c 5

19. a 21.5, b 14.3, c 10.2

20. a 2.32, b 5.76, c 3.48

21.

C

Farming A farmer has a triangular field with sides of lengths 125 yards, 160 yards, and 225 yards. Find the number of acres in 125 yd the field. (1 acre 4840 square yards)

160 yd

225 yd

22.

84

Guy Wire A vertical telephone pole 40 feet tall stands on the side of a hill as shown in the figure to the right. Find the length of the wire that will reach from the top of the pole to a point 72 feet downhill from the pole.


40 ft

t 72 f 17



2. y

y

4 2

(1, 2)

(11, 0)

x

2

x

2

(11, 2) (3, 4)

3.

4. y

y

(6, 4) (9, 1)

4

2 4 x

2

(10, 0) x

(29, 3)

5.

y

2

(8, 14)

(0, 2) 2

x

6. y 2x; 0 ≤ x ≤ 12 1

7. y 2x; 0 ≤ x ≤ 8 1

8. y 3x 4; 6 ≤ x ≤ 18 1

9. y 2x 1; 4 ≤ x ≤ 4 1 10. y 2x 1; 2 ≤ x ≤ 6 11. x 8.06 cos 65.6t or x 3.33t;

y 8.06 sin 65.6t or y 7.33t 12. x 9.13 cos 61.2t 2 or x 4.40t 2;

y 9.13 sin 61.2t 5 or y 8.00t 5 13. x 8.72 cos 83.4t 9 or x 1.00t 9;

y 8.72 sin 83.4t 24 or y 8.67t 24 14. x 18 cos 15t or x 17.4t;

y 4.9t2 18 sin 15t 2 or y 4.9t2 4.66t 2 15. about 22.1 m

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Page 95

LESSON

13.7

NAME _________________________________________________________ DATE ___________


Graph the parametric equations. 1. x 3t 1 and y t 2 for 0 ≤ t ≤ 4 2. x 2t 3 and y t 4 for 0 ≤ t ≤ 4 3. x 1 5t and y t 3 for 2 ≤ t ≤ 6 4. x t 5 and y 5 t for 1 ≤ t ≤ 5 5. x 2t and y 3t 2 for 0 ≤ t ≤ 4

Write an xy-equation for the parametric equations. State the domain. 6. x 3t and y 6t for 0 ≤ t ≤ 4 7. x 2t and y t for 0 ≤ t ≤ 4 8. x 3t 3 and y t 3 for 1 ≤ t ≤ 5 9. x 2t 8 and y t 3 for 2 ≤ t ≤ 6 10. x 2t 2 and y t 2 for 0 ≤ t ≤ 4 Lesson 13.7

Use the given information to write parametric equations describing the linear motion. 11. An object is at 0, 0 at time t 0 and then at 10, 22 at time t 3. 12. An object is at 2, 5 at time t 0 and then at 24, 45 at time t 5. 13. An object is at 12, 2 at time t 3 and then at 15, 28 at time t 6.

Snowboarding


A snowboarder jumps off a ramp at a speed of 18 meters per second. The ramp’s angle of elevation is 15, and the height of the end of the ramp above level ground is 2 meters.

15

2m

14. Write a set of parametric equations for the snowboarder’s jump. 15. Use the equation to determine how far from the ramp the snowboarder

landed.



95


2. y

y

(4, 10) 2

(0, 0) x

4

2 x

2

(0, 2)

(12, 8)

3.

4. y

y

(85, 190) (10, 1)

2 2

(2, 3)

x

25

(5, 10) 100

x

3

5. y 2x 1; 0 ≤ x ≤ 8 6. y x 10; 7 ≤ x ≤ 11 1

7. y 5x 2; 20 ≤ x ≤ 0 8. y 2x 13; 6 ≤ x ≤ 46 9. x 19.5 cos 62.6t or x 9.00t;

y 19.5 sin 62.6t or y 17.3t 10. x 21.4 cos 46.1t 6 or x 14.8t 6; y 21.4 sin 46.1t 15 or y 15.4t 15 11. x 7.35 cos 54.7t 1 or x 4.25t 1; y 7.35 sin 54.7t 7 or y 6.00t 7 12. x 6.43 cos 66.2t 5 or x 2.60t 5; y 6.43 sin 66.2t 6 or y 5.88t 6 13. x 140 cos 22.5t or x 129t; y 16t2 140 cos 22.5t 10 or y 16t2 53.6t 10 14. about 456 ft 15. about 3.53 seconds

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Page 96

LESSON

13.7

NAME _________________________________________________________ DATE ___________


Graph the parametric equations. 1. x 3t and y 2t for 0 ≤ t ≤ 4 2. x t and y 3t 2 for 0 ≤ t ≤ 4 3. x 3t 1 and y t 2 for 1 ≤ t ≤ 3 4. x 20t 5 and y 50t 10 for 0 ≤ t ≤ 4

Write an xy-equation for the parametric equations. State the domain. 5. x 2t and y 3t 1 for 0 ≤ t ≤ 4 6. x t 5 and y 5 t for 2 ≤ t ≤ 6 7. x 5 5t and y t 3 for 1 ≤ t ≤ 5 8. x 2t 6 and y 4t 1 for 0 ≤ t ≤ 20

Lesson 13.7

Use the given information to write parametric equations describing the linear motion. 9. An object is at 0, 0 at time t 0 and then at 27, 52 at time t 3. 10. An object is at 6, 15 at time t 0 and then at 80, 92 at time t 5. 11. An object is at 1, 7 at time t 2 and then at 18, 31 at time t 6. 12. An object is at 5, 6 at time t 4 and then at 20.6, 41.3 at time t 10.

Snow Skiing


A snow skier jumps off a ramp at a speed of 140 feet per second. The ramp’s angle of elevation is 22.5, and the height of the end of the ramp above level ground is 10 feet. 13. Write a set of parametric equations for the snow skier’s jump.

10 ft 22.5

14. Use the equation to determine how far from the ramp the skier landed. 15. Determine how many seconds the snow skier is in the air.

96




2. y

y

(8, 13)

2

(18, 23)

x

2

(10, 7)

4

(0, 1)

3.

4

x

4. y

y

(110, 180) (49, 28)

6 25

(0, 0)

(30, 20) 25

18

x

x

5. y x 19; 7 ≤ x ≤ 12 6. y 3x 10; 4 ≤ x ≤ 20 7. y 1.04x; 0 ≤ x ≤ 148 8. y 2.20x; 0 ≤ x ≤ 50 9. x 29.2 cos 69.3t or x 10.3t;

y 29.2 sin 69.3t or y 27.3t 10. x 4.75 cos 14.6t 4 or x 4.60t 4; y 4.75 sin 14.6t 1 or y 1.20t 1 11. x 2.49 cos 23.6t 2 or x 2.29t 2; y 2.49 sin 23.6t or y 1.00t 12. x 10.0 cos 39.8t 2 or x 7.71t 2; y 10.0 sin 39.8t 3 or y 6.43t 3 13. about 29.7 ft 14. x 6.1 cos 125t 3.5 or x 3.5t 3.5; y 6.1 sin 125t or y 5.0t

4-6-2001

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Page 97

LESSON

13.7

NAME _________________________________________________________ DATE ___________


Graph the parametric equations. 1. x 2t and y 3t 1 for 0 ≤ t ≤ 4 2. x 2t 6 and y 4t 1 for 2 ≤ t ≤ 6 3. x 20t 10 and y 40t 20 for 1 ≤ t ≤ 5 4. x 14.1 cos 30t and y 14.1 sin 30t for 0 ≤ t ≤ 4

Write an xy-equation for the parametric equations. State the domain. 5. x t 7 and y 12 t for 0 ≤ t ≤ 5 6. x 2t 4 and y 6t 2 for 0 ≤ t ≤ 8 7. x 21.4 cos 46.1t and y 21.4 sin 46.1t for 0 ≤ t ≤ 10 8. x 8.1 cos 65.6t and y 8.1 sin 65.6t for 0 ≤ t ≤ 15

Use the given information to write parametric equations describing the linear motion. Lesson 13.7

9. An object is at 0, 0 at time t 0 and then at 31, 82 at time t 3. 10. An object is at 4, 1 at time t 0 and then at 27, 7 at time t 5. 11. An object is at 2, 0 at time t 1 and then at 14, 7 at time t 8. 12. An object is at 2, 3 at time t 5 and then at 56, 42 at time t 12. 13.

Soccer You are a goalie in a soccer game. You save the ball and then drop kick it as far as you can down the field. Your kick has an initial speed of 30 feet per second and starts at a height of 2.5 feet. If you kick the ball at an angle of 50, how far down the field does the ball hit the ground?

14.

Bike Path A bike trail connects State and Peach Streets as shown. You enter the trail 3.5 miles from the intersection of the streets and pedal at a speed of 12 miles per hour. You reach Peach Street 5 miles from the intersection. Write a set of parametric equations to describe your path.

y

Peach Street

MCRB2-1307-PA.qxd

5 mi 3.5 mi x

State Street



97


2.

y

1

1 π 2

x

3.

4.

y

1

π

x

2π

x

y

1

π

5.

y

x

6.

y

y 2

2 π 2

7. cos x 10. sin x

π 2

x

8. cos x

1

sin x 1

9. 2 sin x 11. sin x

16. 19. 21. 22. 23. 25.

11 , 6 6

1

sin x 1

12. 2 tan2 x 1 1 tan2 x 14.

x

13.

2 4 , 3 3

5 2n, 2n 6 6 6 2 2n 17. 2 3 18. 4 2 6 2 3 20. 4 a: 2, P: 2, y 2 sin x a: 3, P: , y 3 cos 2x 1 a: 12, P: 2, y 12 cos x 24. a: 5, P: 60 5 15.

CHAPTER

NAME _________________________________________________________ DATE ____________

14


Draw one cycle of the function’s graph. 1. y sin x

Answers

2. y cos x

y

y

1

1 π 2

π

x

x

1

3. y tan x

4. y 2 cos 2 x y

1.

Use grid at left.

2.

Use grid at left.

3.

Use grid at left.

4.

Use grid at left.

5.

Use grid at left.

6.

Use grid at left.

7. 8.

y

9. 1

1

π

x

x

2π

10. 11. 12.

1

5. y 4 sin x

6. y 3 tan 2 x

y

y 2

2 π 2

π 2

x

x

Simplify the expression. 7. sin

2 x

8.

cotx cscx

9. sin x cos x tan x

10. sin x csc x 1

11. cos

Review and Assess

Verify the identity.

2 xcsc x 1

12. 2 sec2 x 1 tan2 x



103

CHAPTER

NAME _________________________________________________________ DATE ____________

14

Chapter Test A

CONTINUED


13.

Solve the equation in the interval 0 ≤ x < 2. Check your solutions. 13. 2 cos 1 0

14.

14. 5 cos x 3 3 cos x

15.

Find the general solution of the equation.

16.

15. 2 sin x 1 0

17.

16. 5 sec x 5 0

Find the exact value of the expression.

18.

17. tan 15

18. sin 15

19.

7 12

20.

19. tan

12

20. sin

21.

Find the amplitude and period of the graph. Then write a trigonometric function for the graph.

22.

21.

23.

22.

23.

y

y

y

24. 25.

1

1 π 2

x

1 π 2

x

2

x

24. The voltage E in an electrical circuit is given by E 5 cos 120 t.

Find the amplitude and the period.

Review and Assess

25. In Exercise 24, find E when t 0.

104




2.

y

1

1 2π

3.

x

4.

y

1

π

x

π 2

x

y

1 π 2

5.

y

x

6.

y

1

y

3

π

x

π

7. cot x

8. 1

9. sin2 x

10. tan y

x

1 1 tan y

11. sec2 x 1 4 sec2 x 3 12.

13. 15. 17. 20. 22. 23. 25.

cos 2x 1 cos 2x 1 sin 2x sin 2x sin 2x 1 cos2 x sin2 x sin 2x 1 cos2 x sin2 x sin 2x 2 x sin2 x sin sin 2x 2 sin2 x 2 sin x cos x sin x tan x cos x 7 11 7 11 14. , , , 6 6 2 6 6 5 2n, 2n 16. n 6 6 4 2 6 2 2 6 18. 2 3 19. 4 4 1 2 1 21. a: 1, P: 4, y cos 2 x a: 12, P: , y 12 sin 2x 1 a: 13, P: 2, y 13 sin x 24. a: 3.8, P: 25 3.8

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3:02 PM

Page 105

CHAPTER

NAME _________________________________________________________ DATE ____________

14


Draw one cycle of the function’s graph. 1. y sin 2x

Answers

2. y 2 cos x

y

y

1

1

π

π

x

1

3. y tan 2x

x

1.

Use grid at left.

2.

Use grid at left.

3.

Use grid at left.

4.

Use grid at left.

5.

Use grid at left.

6.

Use grid at left.

4. y 1 sin 2x

7.

y

8.

y

9. 1

1 π 2

π 2

x

x

10. 11. 12.

5. y sin x

2

6. y 2 tan x

y

y

1

π

1

x

π

x

Simplify the expression. 7. tan

8.

1 sin x

2

1 tan x

2

cos2x cot2 x

Review and Assess

9.

x 2

Verify the identity. 10. tan y cot y 1

11. tan2 x 4 sec 2 x 3

12. csc 2x cot 2x tan x



105

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Page 106

CHAPTER

NAME _________________________________________________________ DATE ____________

14

Chapter Test B

CONTINUED


13.

Solve the equation in the interval 0 ≤ x < 2. Check your solutions.

14.

14. 4 csc 6 csc

13. 2 sin2 x sin x 1

15.

Find the general solution of the equation.

16.

15. 3 sin x sin x 1

17.

16. tan2 x 1 0

Find the exact value of the expression.

18.

17. sin 75

19.

19. cos

18. tan 105

7 12

20. tan

8

20. 21.

Find the amplitude and period of the graph. Then write a trigonometric function for the graph.

22.

21.

23.

22.

23.

y

y

24.

y 1

2

25.

1 x 2π

x

π 2

x

1

24. The voltage E in an electrical circuit is given by E 3.8 cos 50 t.

Find the amplitude and the period.

Review and Assess

25. In Exercise 24, find E when t 0.

106




2.

y

1

2

π

3.

y

π 2

x

4.

y

x

y

1

1 π 2

5.

π 2

x

6.

y

x

y

1 π 2

x 1

π

7. sec x

8. sec x

x

9. 1

23.

cos x cot x sin x 1 1 cos x cos x 1 sec x sin x tan x sin x cos x sin x cos x cos x 1 sin x cos x sin x cos x 1 sin xcos x 1 1 csc x sin x 3 7 2 4 5 14. , 15. n , , , 4 4 3 3 3 3 5 2n, 2n, 2n 3 3 6 2 6 2 18. 19. 2 3 4 4 6 2 1 21. a: 3, P: 4, y 3 sin 2 x 4 2 2 a: 3, P: , y 3 cos 2x 1 1 a: 2, P: 2, y 2 sin x 24. P 3 sin 660 πt

25.

1 330

10. sin2 x cos2 x 1

12.

13. 16. 17. 20. 22.

11.

MCRB2-1411-TC.qxd

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Page 107

CHAPTER

NAME _________________________________________________________ DATE ____________

14


Draw one cycle of the function’s graph. 1 2

1. y 2 sin x

Answers

2. y 4 cos 2x

y

y

1

2

π

π 2

x

1.

Use grid at left.

2.

Use grid at left.

3.

Use grid at left.

4.

Use grid at left.

5.

Use grid at left.

6.

Use grid at left.

x

7.

3. y 2 cos x

2

8.

4. y 2 tan x

y

9.

y

1

1 π 2

π 2

x

5. y tan 2x

2

6. y 4 sin

y

x

1 x 2

y

1 π 2

x 1

π

x


9.

2 x

8.

cos x tan x 1 sin x

Review and Assess

7. csc

sin x cos x tan x tanx sin2 x



107

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Page 108

CHAPTER

NAME _________________________________________________________ DATE ____________

14

Chapter Test C

CONTINUED


Verify the identity.

10.

sin

1 1 10. sin2 x sec2 x 12.

11.

2 x sin x

11.

cot x

12.

1 secx csc x sinx tanx

Solve the equation in the interval 0 ≤ x < 2. Check your solutions. 13.

sin 1 cos

14. 4sin 2 3

13. 14.

Find the general solution of the equation. 15. sin x sin x cos x 0

15.

16. 2 sin2 x cos x 1 0

16.

Find the exact value of the expression. 17. cos 75 19. tan

17.

18. sin 255

12

20. sin

5 12

18. 19. 20.

Find the amplitude and period of the graph. Then write a trigonometric function for the graph. 21.

22.

21.

23.

y

y

22.

y

3 1 x

π

23.

1 x π 2

x

1

24. 25.

Review and Assess

24. Music A tuning fork vibrates with a frequency of 330 hertz (cycles

per second). You strike the tuning fork with a force that produces a maximum pressure of 3 Pascals. Write the sine model that gives the pressure P as a function of time (in seconds). 25. In Exercise 24, what is the period of the sound wave?

108



Answer Key Cumulative Review 3 1 1. y 2x 2. y 2x 3 3. y 5x 19 3 4. 1, 3, 3 5. 5, 5 6. 2, 0 7. 6 8. 28 9. 0 10. 5 3 11. 10 12. 36 14 1 13. 14. 3 11 15. 4 4 2 16. y x 3 2; 3, 2 17. y x 42 2; 4, 2 7 2 53 7 53 18. y x 2 4 ; 2, 4 19. y x 32 1; 3, 1 20. y 2x 12 1; 1, 1 2 9 2 29 9 29 21. y 3x 2 2 ; 2, 2 22. 1.1, 3.1 1.1, 3.1 maximums, 0, 1 minimum; 1.8, 1.8; 4th degree 23. 2.3, 11 maximum, 0, 2 minimum; 4.1; 3rd degree 24. 0, 3 maximum, 1.1, 4.1 minimum; 2.5; 3rd degree 25. 7.67, 7, 6 26. 85.4, 85, 85 27. 0.267, 0.25, 0 1 28. 242, 242.5, 230 29. y 24x 30. y 2x 1 31. y 31.5x 32. y 253x 33. y 54x x2 8x x3 34. y 3x 35. 36. 2 6x 6 9x 15x 18 x2 37. 38. x 22 y 32 25 2x 2 8x 24 39. x 22 16 y 3 x 62 40. y 62 1 4 x 52 y 2 41. 1 64 36 42. geometric 43. arithmetic 44. geometric 45. arithmetic 46. neither 47. arithmetic 48. 12 49. 2520 50. 20,160 51. 15,120 52. 1 53. 2 4 54. sin 5 cos 35 4 tan 3

3 cot 4

5 sec 3

5 csc 4

55. sin

2

2 tan 1 sec 2

cos

2

2 cot 1 csc 2

9 117 117 3 tan 2 117 sec 6 3 57. sin 2 56. sin

tan 3 sec 2 4

58. sin 5 4 tan 3

6 117 117 2 cot 3 117 csc 9 1 cos 2 3 cot 3 2 3 csc 3 cos

3 cos 5 3 cot 4

5 5 sec 3 csc 4 10 149 7 149 59. sin cos 149 149 10 7 tan cot 7 10 149 149 sec csc 7 10 3 58 6 37 4 60. 61. 2 62. 64. 58 37 3

MCRB2-1411-CR.qxd

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Page 114

CHAPTER

NAME _________________________________________________________ DATE ____________

14


Write an equation of the line described. (2.4) 1. passes through 2, 3 and is perpendicular to the line y 3x 5 2

2. passes through 4, 5 and is parallel to the line y 2x 6 1

3. passes through 3, 4 and 4, 1

Solve the system using the linear combination method or the substitution method. (3.6) 4. a 3b 3c 1

5. 4x 5y 45

2a 3b 4c 1

6. 2x 3y 3

x y

4x 6y 6

bc0 Evaluate the determinant of the matrix. (4.3) 7.

2 0 5 3

8.

1 2 1

4 1 3

5 3 2

9.

1 3 3

4 1 1

0 2 2

Simplify the expression. (5.3) 10. 75

11. 25

5

12. 43

161

14. 11

9

15.

13.

27

78

Write the quadratic function in vertex form and identify the vertex. (5.5) 16. y x2 6x 11

17. y x2 8x 14

18. y x2 7x 1

19. y x2 6x 10

20. y 2x2 4x 3

21. y 3x2 6x 1

2

Estimate the coordinates of each turning point and state whether each corresponds to a local maximum or a local minimum. Then list all the real zeros and determine the least degree that the function can have. (6.8) 22.

23.

y

24.

y

y 1 1

Review and Assess

1

x

2 1

x

2

x

Find the mean, median, and mode of the data set. (7.7) 25. 5, 6, 6, 8, 10, 11

26. 87, 85, 85, 86, 87, 85, 83, 81, 90

27. 0, 0, 0.2, 0.3, 0.4, 0.7

28. 230, 230, 240, 245, 247, 260

114



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Page 115

CHAPTER

NAME _________________________________________________________ DATE ____________

14

Cumulative Review

CONTINUED


Write an exponential function of the form y abx whose graph passes through the given points. (8.7) 29. 1, 2, 3, 32

30. 2, 4, 1, 0.5

31. 1, 4.5, 2, 6.75

32. 2, 225, 3, 675

33. 1, 5 , 3,

34. 3, 27, 5, 243

4

64 5

Simplify the complex fraction. (9.5) x 4 2 35. 3 3 x

36.

1 3x2 3

37.

2 x x 1 x2 2x 3

1 x3 8 4 2 x2 4 x2 2x 4

Write an equation for the conic section. (10.6) 38. Circle the center at 2, 3 and radius 5 39. Parabola with vertex at 2, 3 and focus at 2, 7 40. Ellipse with vertices at 4, 6 and 8, 6 and co-vertices at 6, 7 and 6, 5 41. Hyperbola with vertices at 5, 8 and 5, 8 and foci at 5, 10 and

5, 10

Decide whether the sequence is arithmetic, geometric, or neither. (11.3) 42. 1, 3, 9, 27, . . . 1

3

45. 2, 1, 2, 2, . . .

1

43. 1, 3, 5, 7, 9, . . .

44. 3, 1, 3, 9, . . .

46. 1, 4, 9, 16, 25, . . .

47. 9, 5, 1, 3, . . .

Find the number of permutations. (12.1) 48. 4P2

49. 7P5

50. 8P6

51. 9P5

52. 8P0

53. 2P1

Use the given point on the terminal side of an angle in standard position. Evaluate the six trigonometric functions of .(13.3) 54. 3, 4

55. 1, 1

56. 6, 9

57. 1, 3

58. 6, 8

59. 7, 10

3 7

60. tan , 0 < < 62. cot 6,

; find sin 2

< 0 < ; find cos 2


61. sec 5, 0 <
0, with the given amplitude and period. 14. Amplitude: 2

Period: 4

15. Amplitude:

1 8

16. Amplitude: 4

Period: 8

Period:

2

17. Sound Waves

Plucking or striking a stretched string, such as a guitar string, causes sound waves. Sound waves can be modeled by sine functions of the form y a sin bx, where x is measured in seconds. Write an equation of a sound wave whose amplitude is 2 and whose period is 1 264 second.

14



Answer Key Practice B 1. A 2. C 3. B 4. amplitude: 5, period: 2 1 5. amplitude: 1, period: 6. amplitude: 3, period: 7. amplitude: 4, period: 8 3 8. amplitude: 2, period: 2 1 9. amplitude: 4, period: 4 10.

11.

y

y 1

1 π 4

2π

x

12.

13.

y

1

y

2 π 16

14.

x

π 2

x

15.

y 1

x

y

1

π

16.

π 3

x

17.

y

1

x

y

1 π 4

x

1 8

1 1 x 19. y sin 6x 3 2 2 x 20. y 12 sin 21. 4 sec 22. 15 7 18. y 3 sin

23.

y

1 1

x

x

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LESSON

NAME _________________________________________________________ DATE ___________

14.1


Match the function with its graph. 1

1. y 2 sin 2x

2. y 2 cos 2 x

3. y 2 tan 2x

A.

B.

C.

y

y

1

y

1 π 8

x

x

x

2π

Find the amplitude and period of the graph of the function. 4. y 5 cos x 7. y 4 cos

1 4x

5. y sin 2x

6. y

8. y 2 cos x

9. y

1 3 1 4

sin 6x sin 12 x

Graph the function. 10. y sin 2x 14. y

1 3

cos x

1

11. y cos 2 x

12. y tan 4x

15. y 2 cos 3x

16. y

3 4

13. y 5 sin x 17. y tan 2x

sin 2x

Write an equation of the form y a sin bx, where a > 0 and b > 0, with the given amplitude and period. 18. Amplitude: 3

19. Amplitude:

Period: 4

Period:

3

1 3

20. Amplitude: 12

Period: 7

Respiration Cycle After exercising for a few minutes, a person has a respiratory cycle for which the velocity, v (in liters per second), of air flow is approximated by v 1.75 sin t, where t is time in seconds. Inhalation occurs 2 when v > 0 and exhalation occurs when v < 0. 21. Find the time for one full respiratory cycle. 22. Find the number of cycles per minute. 23. Sketch the graph of the velocity function.



15

Lesson 14.1

π 4

1

Answer Key Practice C 1. A 2. C 3. B 4. amplitude: 8, period: 2 1 5. amplitude: 1, period: 6. amplitude: , 2 2 period: 7. amplitude: 3, period: 6 8. amplitude: 4, period: 1 1 9. amplitude: 10, period: 8 10.

11.

y

y 1

1 3π 2

12.

π

x

13.

y

y

1

1 2π

14.

x

π 32

x

15.

y 1

x

y

1 x

1 2

16.

1 4

17.

y

1

x 1 24

y

1

x 2

2 x 1 19. y sin x 5 3 6 20. y 5 sin 16x 21. about 2.22 sec 18. y 6 sin

x

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LESSON

NAME _________________________________________________________ DATE ___________

14.1


Match the function with its graph. 1

1. y 4 tan 2x

2. y 4 tan 2 x

3. y 4 tan 2 x

A.

B.

C.

y

y

2

y

2

Lesson 14.1

π 8

2

x

1 8

π

x

x

Find the amplitude and period of the graph of the function. 4. y 8 sin x 1

7. y 3 sin 3x

5. y cos 4x

6. y

8. y 4 sin 2 x

9. y

1 2 cos 2x 1 1 10 cos 4 x

Graph the function. 1

10. y sin 3 x 14. y

1 3

11. y

sin x

1 2

1

cos x

15. y 3 sin 2 x

12. y 3 cos 2 x

13. y 2 tan 8x

16. y 2 tan 6 x

17. y

1 2

cos 14 x

Write an equation of the form y a sin bx, where a > 0 and b > 0, with the given amplitude and period. 18. Amplitude: 6

19. Amplitude:

Period: 10

2 3

20. Amplitude: 5

Period: 12 21. Pendulum Motion

y A cos

32t

L

Period:

8

The motion of a pendulum can be modeled by

, where y is the directed length (in feet) of the arc, A is

the length (in feet) of the arc from which the pendulum is released, L is the length (in feet) of the pendulum, and t is the time in seconds. How many seconds does it take a 2 foot pendulum that is released with an initial arc of 4 inches to swing through one complete cycle?

L

A

16



Answer Key Practice A 1. shift up 5 units

2. reflect in x-axis, shift left

units 3. shift right 4. B

5. A

7.

units, shift down 4 units 4

6. C 8.

y

y

1

π

x 1

9.

10.

y

1

π 2

x

π 2

x

y

1 π 2

x

11.

12.

y

y

1 π x 2

1 π 4

x

1

13. y 5 3 sin 2x

14. y 2 cos x

15. y 1 3 tan 4 x

Temperature

16. July, January T 80 70 60 50 40 30 20 10 0

0 1 2 3 4 5 6 7 8 9 10 11 t Month

2

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LESSON

NAME _________________________________________________________ DATE ___________

14.2


Describe how the graph of y sin x or y cos x can be transformed to produce the graph of the given function.

2. y cosx

3. y 4 cos x

4. y 2 sin2x

5. y 2 sinx

6. y 2 sinx

A.

B.

C.

1. y 5 sin x

4

Match the function with its graph.

y

y

y 2

1

1

Lesson 14.2

π 2

π 4

x

π 2

x

x

Graph the function. 7. y cosx

8. y 4

10. y 2 2 cosx

1 2

9. y 3 sin x

cos x

2

12. y 3 tanx

11. y 3 tan x

Write an equation of the graph described. 1

13. The graph of y 3 sin 2x translated down 5 units 14. The graph of y cos x translated up 2 units and left units 15. The graph of y 3 tan 4 x translated down 1 unit and right

units, and 2

then reflected in the line y 1 16. Average Temperature

A model for the average daily temperature, T (degrees Fahrenheit), in Kansas City, Missouri, is given by T 54 25.2 sin

212 t 4.3 ,

where t 0 represents January 1, t 1 represents February 1, and so on. Sketch the graph of this function. Which month has the highest average temperature? The lowest average temperature?

26




4.

5.

y

1

y

1

x

π 4

π 2

6.

7.

y

2π

8.

y

π 2

x

9.

y 1 1

x

y 1

x

1

10.

11.

y

1 x

x

y

1

π

12.

x

1

1

π 2

x

13. y 8 3 tan 2x

y

1 π 2

x

1 cos 6 x 2 6 1 15. y 6 sin x 2 16. Minimum of 1 at x 0, 3 Maximum of 5 at x , 2 2 1 17. y 3 sin x 3 14. y 5

1. reflect in x-axis, shift left units 4 2. shift left units, shift up 2 units 2 3. shift right units, shift down 2 units

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LESSON

NAME _________________________________________________________ DATE ___________

14.2



1. y cos x

4

2

2. y 2 sin x

5. y tan 2x

6. y 1 cos x

8. y 3 sin x

9. y 1 3 tan x

1 2

12. y 2 cosx

3. y 2 cosx


1 sin x 2 2

2

7. y 2 cos x 10. y 1

1 sin x 4

1 4

11. y cosx

Write an equation of the graph described. 13. The graph of y 3 tan 2x translated up 8 units and then reflected in the 14. The graph of y

Lesson 14.2

line y 8

1 cos 6x translated down 5 units and right unit 2 6 1

15. The graph of y 6 sin 2 x translated left units and reflected in the x-axis 16. Minimum and Maximum Values

What are the minimum and maximum values of y 3 2 cos 2x? Write two x-values at which the minimum occurs. Write two x-values at which the maximum occurs.

17. Write an equation of the graph below. y 1

π


x


27

Answer Key

5.

y

1 π 2

π

6.

x

x

7.

y

y

1 π 8

x 1 π 4

8.

9.

y

2 π 2

10.

4π

x

11.

y

x

y

1

1 π 2

12.

x

y

1

x

1

1 3 14. y 10 3 sin 2x

1 π 2

15. y 1

x

13. y 2 tan x

y x

1 cos 3 x 4 3

0 10 20 30 40 50 60 70 t Time (days)

17. d 375 tan 230;

y

1

y 16 12 8 4 0

Distance from top (feet)

4.

16. Brightest: 25th, 65th; Dimmest: 5th, 45th Brightness

Practice C 1. shift left units, shift up 4 units 2. reflect in x-axis, shift right units 3. reflect in x-axis, 2 shift left units, shift down 1 unit 4

d 250 200 150 100 50 0

0 5 10 15 20 25 30 35 θ Angle (degrees)

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LESSON

NAME _________________________________________________________ DATE ___________

14.2



1. y 4 sinx

2. y cos x

2

3. y 1 sin x

4

Graph the function.

1 2

2

4. y 4 sin x

5. y cos x

7. y 3 sin 2x

8. y 1 sinx

10. y tan x

2

1 2

11. y tan x

6. y 2 cos 4x

1 4

9. y 4 cos x 12. y 2 tan x

Write an equation of the graph described. 1

Lesson 14.2

13. The graph of y 2 tan 3 x translated right units and reflected in

the x-axis 14. The graph of y 3 sin 2 x translated down 10 units and reflected in the

line y 10

1 cos 3x translated up 1 unit, left units, and then 4 3 reflected in the line y 1

15. The graph of y

16. Stars

Suppose that the brightness of a distant star is given by t y 10.5 5.2 cos 40 , 20

where t is given in days. Sketch the graph for 0 ≤ t ≤ 80. Which day(s) is the brightness the greatest? Which day(s) is the brightness the least? 17. Mountain Climbing

You are standing 375 feet from the base of a 230 foot cliff. Your friend is rappelling down the cliff. Write and graph a model for your friend’s distance d from the top as a function of her angle of elevation .

230 ft You

θ 375 ft

28



Answer Key Practice A 4 4 5 1. sin 5, tan 3, sec 3, csc 54, cot 34 1 , tan 1, sec 2, 2. cos 2 csc 2, cot 1 3 4 5 3. sin 5, cos 5, sec 4, csc 53, cot 43 3 4 4 4. cos 5, sin 5, tan 3, csc 54, cot 34 15 8 15 5. sin 17, cos 17, tan 8 , 17 sec 17 8 , csc 15 8 15 8 6. sin 17, cos 17, tan 15, 15 sec 17 15 , cot 8 7. cos x 8. sec x 9. cot x 10. tan x 11. sec2 x 12. sin x 1 1 cot x tan x 13. tan x cos x tan x cot x 1 sin2 x cos2 x 1 sin x 14. sin x sin x sin x cos x cos x cot x cos x sin x 15. 1 sin x1 sin x 1 sin x1 sin x 1 sin2 x cos2 x 16. sin2 x sin4 x sin2 x 1 sin2 x 1 cos2 xcos2 x cos2 x cos4 x 1 tan2 x 1 tan2 x tan2 x 1 17. 1 tan2 x sec2 x sec2 x sec2 x 2 2 2 cos x sin x 1 sin x sin2 x 1 2 sin2 x 18. cos

2 x cos x tan 2 x

sin x cos x cot x sin x cos x

sin2 x cos2 x 1 csc x sin x sin x

cos x sin x

19.

t

0

4

2

3 4

x

3

2.1

0

2.1

y

0

2.8

4

2.8

5 4

3 2

7 4

x 3

2.1

0

2.1

y

2.8

t

0

4 2.8

y

1 1

sin2 t cos2 t

x

y2 x2 1, ellipse 9 16

MCRB2-1403-PA.qxd

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Page 41

LESSON

NAME _________________________________________________________ DATE ___________

14.3


Find the values of the other five trigonometric functions of . 3 3 < < 2 5 2

2. sin

1. cos ,

3 4

3. tan , 0 < < 5. cot

1 3 , < < 2 2 5 < < 3 2

2

4. sec ,

8 3 , < < 2 15 2

6. csc

17 , < < 8 2


2 x 9. cos x 2 sin

8. cos x sin x tan x

7. cot x sin x

10.

sinx cosx

11. sin2 x tan2 x cos2 x

12. cot

2 x cos x

Verify the identity. 13.

1 1 tan x cot x tan x cot x

14.

15. 1 sin x1 sinx cos2 x 17.

1 sin x cot x cos x sin x

16. sin2 x sin4 x cos2 x cos4 x

1 tan2 x 1 2 sin2 x 1 tan2 x

18. cos

2 x cos x tan2 x csc x

19. Conic Sections

t

0

4

2

3 4

5 4

3 2

Lesson 14.3

Complete the table of values for the parametric equations x 3 cos t and y 4 sin t. Then sketch the graph in the xy-plane. Then use a trigonometric identity to verify that the graph is a circle, an ellipse, or a hyperbola. 7 4

x y



41

Answer Key Practice B 8 15 17 , tan , sec , 17 8 8 17 8 csc , cot 15 15 1. cos

3 5

4 5

5 4

2. sin , cos , sec ,

5 4 csc , cot 3 3 3 , tan 3, sec 2, 3. sin 2 2 1 csc , cot 3 3 1 2 , sin , tan 2, 5 5 1 5 , cot csc 2 2 5. 1 6. csc x 7. tan2 x 8. sin x 9. sin x cos x 1 10. csc x 11. sec x cot x cos x sin x 1 csc x sin x tan2 x sec2 x 1 12. sec x sec x 2 1 sec x sec x cos x sec x sec x 13. tan x sin x cot x sin x 2 cos x sin x cos x sin x 4. cos

14.

cos x csc x tan x cos x csc x cos x tan x sin x cos x cos x cot x sin x sin x cos x y2 x2 1, circle 15. sin2 t cos2 t 16 16 x2 y2 , hyperbola 16. 1 sec2 t tan2 t 9 1 2 17. r cos x R rR r r2 cos2 x R2 r2 R2 r2 1 cos2 x R2 r2 sin2 x

MCRB2-1403-PA.qxd

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Page 42

LESSON

NAME _________________________________________________________ DATE ___________

14.3


Find the values of the other five trigonometric functions of . 1. sin

3 15 , < < 17 2

2. tan

1 < < 2 2

4. sec 5,

3. cos ,

3 ,0 < < 4 2 3 < < 2 2

Simplify the expression. 5. sec x cot x sin x 8. sin3 x cos

2 x cos

2

6.

x

cos2 x sin x sin x

7.

9. csc x csc x cos2 x

10.

1 cos2 x cos2 x 1 sec x sin x tan x

Verify the identity. 11. sec x cot x csc x 13. tan

2 x sin x cos x

12.

tan2 x sec x cos x sec x

14. cos x csc x tan x cot x sin x

Use a graphing calculator set in parametric mode to graph the parametric equations. Use a trigonometric identity to determine whether the graph is a circle, an ellipse, or a hyperbola. (Use a square viewing window.) 15. x 4 cos t, y 4 sin t

16. x 3 sec t, y tan t

While drawing the plans for the plumbing of a new house, the contractor finds it necessary for two water pipes to be joined at right angles. The expression r cos x2 R rR r is used. Show that this expression can be written as R2 r2 sin2 x.

Lesson 14.3

17. Plumbing

42



Answer Key Practice C 8 15 17 1. sin , cos , sec , 17 17 15 17 15 csc , cot 8 8 4 3 4 2. sin , cos , tan , 5 5 3 3 5 sec , cot 4 3 6 , tan 6, sec 37, 3. sin 37 37 1 csc , cot 6 6 3 1 2 , sec , , tan 4. cos 2 3 3 csc 2, cot 3 5. 1 6. cos x 7. cot2 x 8. sin2 x 9. sec x csc x 10. cos2 x 11. 2 sec2 x 2 sec2 x sin2 x sin2 x cos2 x 2 sec2 x 1 sin2 x sin2 x cos2 x 2 cos2 x 1 2 sec2 x cos2 x 1 cos2 x 211 1 sec x 1 sec x sin x tan x sin x tan x 1 1 cos x 1 cos x sin x cos x sin x sin x sin x cos x 1 cos x 1 csc x sin x cos x 1 sin x 13. 2 cos2 x 3 cos4 x 2 3 cos2 x1 cos2 x 2 3 cos2 x sin2 x sin2 x 2 3 cos2 x tan3 x 1 tan x 1tan2 x tan x 1 14. tan x 1 tan x 1 2 tan x tan x 1 x2 y2 1, ellipse 15. sin2 t cos2 t 25 1 12.

16. 1 sec2 t tan2 t

y2 x2 , hyperbola 16 1

3 , 1 1 cos2 x 2 3 18. x , tan2 x 1 4 17. x

MCRB2-1403-PA.qxd

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Page 43

LESSON

NAME _________________________________________________________ DATE ___________

14.3


Find the values of the other five trigonometric functions of . 1. tan 3. cos

8 , < < 15 2 1

37

,0 <