ALGEBRA 2 and ... ANSWER KEY/PREPARING FOR THE REGENTS
EXAMINATION: ... No part of this book may be reproduced in any form without
written ... Contents. Chapter 1: The Integers. 1. Chapter 2: The Rational Numbers.
4. Chapter ...
ANSWER KEY Preparing for the
REGENTS EXAMINATION
ALGEBRA 2 and
TRIGONOMETRY Ann Davidian and Christine T. Healy
AMSCO
AMSCO SCHOOL PUBLICATIONS, INC. 315 Hudson Street, New York, N.Y. 10013
Composition and art by Progressive Information Technologies Please visit our Web site at: www.amscopub.com When ordering this book, please specify: N 238 K or ANSWER KEY/PREPARING FOR THE REGENTS EXAMINATION: ALGEBRA 2 AND TRIGONOMETRY Copyright © 2009 by Amsco School Publications, Inc. No part of this book may be reproduced in any form without written permission from the publisher. Printed in the United States of America 1 2 3 4 5 6 7 8 9 10
14 13 12 11 10 09
Contents Chapter 1: The Integers Chapter 2: The Rational Numbers Chapter 3: Real Numbers and Radicals Chapter 4: Relations and Functions Chapter 5: Quadratic Functions and Complex Numbers Chapter 6: Sequences and Series Chapter 7: Exponential Functions Chapter 8: Logarithmic Functions Chapter 9: Trigonometric Functions Chapter 10: More Trigonometric Functions Chapter 11: Graphs of Trigonometric Functions Chapter 12: Trigonometric Identities Chapter 13: Trigonometric Equations Chapter 14: Trigonometric Applications Chapter 15: Statistics Chapter 16: Probability Cumulative Reviews Practice Algebra 2 and Trigonometry Regents Examinations
1 4 8 11 18 26 30 35 38 42 45 53 57 59 62 69 74 116
CHAPTER
The Integers
1
1.1 Reviewing Real Numbers
1.2 Writing and Solving Equations and Inequalities
(page 3) In 1–15, explanations will vary. 1 Rational 2 Rational 3 Rational 4 Irrational 5 Rational 6 Irrational 7 Irrational 8 Irrational 9 True 10 False 11 True 12 False 13 False 14 True 15 True 16 3, 4 17 1, 2 18 0, 1 19 10, 11 20 0.444, 1.4, 4.4 兹8 , 5, 2 2 21 2 兹 兹 7 7 , , 兹7 22 3 3 1 1 1 , , 23 2 2 兹2 22 ,p 24 3.14, 7 1 3 , 0.3333, 25 10 3
(page 7) 1 x⫽2 2 x⫽9 3 x⫽4 4 x⫽8 5 x⫽7 6 x⫽6 7 x ⱖ ⫺4 8 x ⱕ ⫺2 3 9 x⬍ 5 10 x ⬎ ⫺7 11 2n ⫹ 8 ⫽ 3n; n ⫽ 8 12 3n ⫹ 8 ⫽ 2n ⫹ 17 ⫺ 5; n ⫽ 4 13 7 ⫹ 2n ⫺ 4 ⫽ n ⫹ 12; n ⫽ 9 14 5n ⫺ 11 ⱖ 3n ⫺ 1; n ⱖ 5 15 Feb. ⫽ Jan. ⫹ 3; Mar. ⫽ Jan. ⫹ 4; Apr. ⫽ 1 2(Jan. ⫹ Feb.); May ⫽ Mar.; June ⫽ 2 Mar. – 2; Jan. ⫹ Feb. ⫹ Mar. ⫹ Apr. ⫹ May ⫹ June ⫽ 50 Jan: 3; Feb: 6; March: 10; April: 18; May: 5; June: 8 16 313.50 ⫽ [2(5) ⫹ 2(6) ⫹ 8]8.90 ⫹ 3d; d ⫽ $15.50 17 c ⫽ 2s; t ⫽ s ⫹ 16; c ⫹ s ⫹ t ⫽ 452 a 109 swimsuits, 125 towels, 218 calendars b $6,671 18 Co ⫽ 2P ⫹ 8; Ca ⫽ 3P ⫹ 3; Co ⫹ Ca ⫹ P ⫽ 46 No. Colby: 20; Carlyn: 20; Paolo: 6
冪
冪
1.2 Writing and Solving Equations and Inequalities
1
1.3 Absolute Value Equations and Inequalities Absolute Value Equations
(page 10) 1 x ⫽ 6, x ⫽ ⫺2 2 y ⫽ 3, y ⫽ ⫺8 3 z ⫽ 4, z ⫽ 0 4 n ⫽ 5, n ⫽ ⫺1 5 c⫽3 5 6 r ⫽ , r ⫽ ⫺15 7 7 (3) {⫺6, 2} 8 (3) {⫺1, 5} 9 (1) x ⫽ 1, x ⫽ 9 10 (4) 兩4 ⫺ x兩 ⫹ 10 ⫽ 4 11 (2) 兩x ⫺ 7兩 ⫽ 6 3 x ⫺ 6 ⫽ 12 12 (1) 2 13 a 兩x ⫺ 10兩 ⫽ 7 b x ⫽ 3, x ⫽ 17 14 a 兩P ⫺ 0兩 ⫽ 2P ⫹ 5, 兩P兩 ⫽ 2P ⫹ 5 5 b P⫽⫺ 3 15 a 兩a ⫺ 16兩 ⫽ 0.5 b a ⫽ 15.5 or a ⫽ 16.5
兩
兩
Absolute Value Inequalities
(page 12) 1 ⫺10 ⱕ a ⱕ 10 2 x ⬍ ⫺5 or x ⬎ 9 3 ⫺4.8 ⱕ d ⱕ 4 4 x ⱕ ⫺19 or x ⱖ 1 5 n ⬍ 2 or n ⬎ 14 6 ⫺2 ⬍ y ⬍ 2 7 m ⱕ ⫺4 or m ⱖ 8 8 (4) –2 –1
9 10 11 12 13 14
15
2
0
1
2
3
Chapter 1: The Integers
a 兩t ⫺ 350兩 ⬎ 7 b The oven turns off at 357° and turns on again when the oven temperature reaches 343°.
1.4 Adding and Subtracting Polynomials (pages 14–15) 1 ⫺x 4 ⫺ x 3 ⫹ 5x 2 ⫹ 2x ⫹ 5; degree 4 2 ⫺c 5d 2 ⫺ 5c 4d 3 ⫹ 25c 3d 4; degree 7 3 ⫺c 4 ⫺ c 3 ⫹ 5c ⫹ 3; degree 4 4 ⫺9w 2 ⫹ 9w ⫹ 4; degree 2 5 3x 4 ⫹ 8x 3 ⫺ 8x 2 ⫺ 4x ⫹ 9; degree 4 6 ⫺xy ⫺ 4x 2; degree 2 7 7a2b 2 ⫺ 2a2b ⫹ 3ab 2; degree 4 8 (2) 14x2 ⫺ 4x ⫹ 21 9 (3) 3c 4 ⫺ 5c 2 ⫹ 10 ⫹ (⫺3c 4 ⫹ 3c 2 ⫺ 3) 10 (4) 3x2 ⫹ 7x ⫺ 22 11 (1) 27c 2 ⫺ 15c ⫹ 12 12 (3) 4a2 ⫺ 14ab ⫹ 8b2 13 (1) 14c 2 ⫺ 5c ⫹ 2 14 (1) 3x 2 ⫺ 5x ⫹ 13 15 (2) 12m ⫹ 5
1.5 Multiplying Polynomials
4
5
6
兩4x ⫺ 2兩 ⫺ 6 ⱕ 8 {x : x ⬍ ⫺2 or x ⬎ 3} {x : ⫺3 ⬍ x ⬍ 4} 兩a ⫺ 16兩 ⱕ 0.5 兩I ⫺ 100兩 ⱕ 15 兩t ⫺ 98.6兩 ⬍ 1.4 Temperatures greater than 100° or less than 97.6° would be considered unhealthy. a 兩a ⫺ .500兩 ⱕ .010 b Con’s batting average is between .490 and .510 inclusive.
(1) (1) (4) (4) (3) a b
16
(pages 17–18) 1 18x 5 ⫺ 12x 4 ⫹ 6x 3 ⫺ 6x 2 2 m4 ⫺ 16 3 12a5b4 ⫺ 9a4b5 ⫹ 15a3b6 4 10c3d 3 ⫹ 6c2d 2 ⫺ 5cd ⫺ 3 5 10c3 ⫹ 17c ⫺ 20 6 36z2 ⫹ 60z ⫹ 25 7 49y2 ⫺ 4 8 2p3 ⫹ p2 ⫹ p ⫺ 6 9 8x3 ⫹ 4x2 ⫺ 2x ⫺ 1 10 14 ⫺ 41y ⫹ 29y2 ⫺ 6y3 11 16c2a ⫹ 18ca ⫺ 9 12 ⫺6h4 ⫺ 5h3k ⫹ 14h2k 2 ⫺ hk 3 ⫺ 2k 4 13 7j 2 ⫹ 64j ⫹ 9 14 x 2 ⫺ 7x ⫺ 4 15 12m3n3 ⫺ 5m2n2 16 2m2b ⫹ 11mb ⫹ 7 17 2y3 ⫹ 7y ⫺ 15 18 4x2 ⫹ 82x 19 12x3 ⫺ 29x2 ⫺ 4x ⫺ 6
20 x3 ⫺ 9x2 ⫺ 8x ⫹ 30 21 (2) 2h2 ⫹ h ⫺ 10 22 (3) 7p3 ⫹ 30p2 ⫺ 27p ⫺ 10
12
(page 21) 1 (x ⫺ 5)(x ⫺ 2) 2 (x ⫹ 5)(x ⫺ 4) 3 (c ⫹ 6)(c ⫹ 2) 4 (y ⫺ 18)(y ⫹ 2) 5 (a ⫺ 6)(a ⫺ 5) 6 (x ⫹ 7)(x ⫺ 5) 7 (p ⫹ 10)(p ⫺ 3) 8 (m ⫹ 9)(m ⫺ 6) 9 (x ⫺ 7)(x ⫹ 2) 10 (2z ⫹ 9)(2z ⫺ 9) 11 2x3(1 ⫺ 2x)(1 ⫺ x) 12 3c2d (3c ⫺ d)(c ⫺ 2d) 13 (4x ⫺ 5)(x ⫹ 1) 5 5 6y ⫹ z 6y ⫺ z 14 7 7 15 (9 ⫹ a)(2 ⫺ a) 16 4p(p ⫹ 4)(p ⫺ 4) 17 2(x ⫺ 5)(x ⫺ 3) 18 (d ⫺ 8)(d ⫹ 6) 19 my(m ⫹ 5)(m ⫺ 5) 20 (3a ⫺ 1)(a ⫺ 3) 21 (ya ⫺ 3)(ya ⫹ 2) 22 (8c ⫺ 5)(c ⫺ 1) 23 (7a ⫺ 3)(a ⫹ 2) 24 4n2 ⫺ 12x ⫹ 9; cannot be factored
冣冢
10 11
1.6 Factoring
冢
9
冣
13 14 15 16 17 18 19 20 21 22 23 24
冦⫺ 34 , 32 冧 冦⫺ 52 , 4冧 {⫺2, 3} 1 3 , 3 2 2 ⫺ ,2 5 1 ,2 2 {⫺2, 2} 5 ⫺ ,3 2 2 ⫺ , 38 3 9 ⫺ ,4 4 6 ⫺ ,4 5 2 ⫺ ,5 3 {4} {2, 6} {13} {3, 7}
冦 冧 冦 冧 冦 冧 冦 冦 冦 冦 冦
冧 冧 冧 冧 冧
1.8 Quadratic Inequalities (pages 26–27) 1 x ⬍ ⫺7 or x ⬎ 8 –8 –6 –4 –2
1.7 Solving Quadratic Equations with Integral Roots (page 24) 1 {2} 2 {⫺6, 8} 3 {⫺3, 8} 4 {⫺2, 7} 5 {⫺3, 3} 6 {⫺7, 4} 7 ⫺ ,1 7 3 5 ⫺ ,3 8 2
冦 冦
冧 冧
2
4
0
2
6
8
–4
–2
4
6
⫺2 ⬍ x ⬍ 5 –6
4
2
x ⱕ ⫺3 or x ⱖ 3 –6
3
0
–4
–2
0
2
4
6
x ⱕ ⫺4 or x ⱖ ⫺1 –5
–4
–3
–2
5 x ⬍ ⫺3 or x ⬎ ⫺ –5
–4
–3
–1
0
1
–1
0
1
1 2
–2
–1 2
6
⫺6 ⱕ x ⱕ 2 –10 –8
–6
–4
–2
0
2
4
1.8 Quadratic Inequalities
3
7
–3
–2
–1
0 –
8
1 3
x ⱕ ⫺1 or x ⱖ ⫺
⫺
1
–2
–1
0
1
–3
2
3
3 2
2
15
x ⬍ 4 or x ⬎ 8 3
10
3
1 3
3 3 ⬍x⬍ 2 2 –3
9
2
6 7 8 9 10 11 12 13 14
4
5
6
7
8
9
0ⱕxⱕ5 –1
1
0
2
3
4
5
11 (2) {x : x ⬍ ⫺3 or x ⬎ 6} 12 (4) {x : ⫺6 ⱕ x ⱕ 6} 13 (4) x2 ⫹ 4x ⫺ 21 ⱖ 0 14 (3) 0
3
15
(1)
16
(4) {x : 5 ⬍ x ⬍ 6}
–4
–2
0
2
4
6
8
10
Chapter Review (pages 31–32) 1 x2 ⫹ x ⫺ 20 2 3a2 ⫺ 19a ⫹ 12 3 4c2 ⫺ 6c ⫹ 8 4 2w2 ⫹ 7wy ⫺ 15y2 5 2x3 ⫺ 5x2 ⫺ 8x ⫹ 6
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
9a2 ⫹ 6ab ⫹ b2 6a4b ⫺ 8a3b ⫹ 2a2b2 2x2 ⫹ 16xy ⫹ 5y2 2a2b ⫺ ab2 ⫺ 3b2 ⫹ 4b r 3 ⫹ 2r 2s ⫺ 4rs2 ⫹ s3 x ⫽ 2, 7 x ⫽ 1, ⫺1 ⫺8 ⱕ x ⱕ 2 x ⫽ ⫺2, 6 5 x ⫽ ⫺3, 2 x ⬍ ⫺8 or x ⬎ 7 ⫺3 ⱕ x ⱕ 8 x ⱕ 0 or x ⱖ 7 3(a ⫹ 3)(a ⫺ 3) (b ⫹ 1)(b ⫹ 12) 3(x ⫺ 3)(x ⫹ 1) (2x ⫹ 3)(x ⫺ 4) (3cd ⫹ 4)(3cd ⫺ 4) (2) 0.33333 . . . (2) 6 feet (2) ⫺3x2 ⫹ 11x ⫺ 7 (3) ⫺a2 ⫹ a ⫹ 4 (1) ⫺3.14 (4) {1, 6} (4) 4 (1) {x : x ⱕ ⫺6 or x ⱖ 8} (2) {x : x ⱕ ⫺3 or x ⱖ 11} (2) {x : ⫺2 ⬍ x ⬍ 0} (1) x2 ⫹ x ⱕ 6 (2) 兩x ⫺ 10兩 ⱕ 0.001
CHAPTER
2 2.1 Simplifying Rational Expressions (pages 37–38) 1 x⫽0 2 x ⫽2
4
Chapter 2: The Rational Numbers
The Rational Numbers
3 x ⫽ 0 or x ⫽ 5 4 x ⫽ ⫺9 5 x ⫽ ⫺2 or x ⫽ ⫺4 6 x ⫽ ⫺5 3a , (a ⫽ 0, b ⫽ 0, c ⫽ 0) 7 4b 3
8 9 10 11 12 13 14 15 16 17 18 19 20
2xyz10, (x 0, y 0) 1 , (x 0, 2) 2x 4 2, (x 3) 1 , (x 5, 5) x5 x6 , (x 4, 4) x4 x5 , (x 0, 3) x y3 2y2 4y 8, ( y 2) x , (x 3, 7) x3 7 (4) 2 x 3 2 (1) x5 x 2 4x (2) 4 (4) x 0, x 6 x4 (2) x8
2.2 Multiplying and Dividing Rational Expressions (page 41) 2a3 1 , (a 0, b 0, c 0) 9c 3 9xy 2 , (x 0, y 0, z 0) 2 5 (x 2)2 , (x 2, 0, 2) 3 2x 12 , (x 0, y 0, x y) 4 y 5 2, (a 3, 2, 3) 6 1, (x 1, 0, 1) 1 , (a b, b, 0) 7 2a 3 , (x 3, 2, 4) 8 2 3 9 , (a 0, 3, 6) a a(a 3) , (a 3, 2, 2, 5) 10 a3 1 1 11 2, w 2, , 2 2
冢
冣
12 13 14 15 16 17 18 19 20 21 22 23
d , (d 8, 0, 8) 2 2 2, x 1, , 1 3 1 , (y 6, 5, 3, 0, 4) y 1 , (z 7, 3, 0, 7) z 2, (x 3, 3) 15, (x 4, 4) 1 , (x 6, 6, 8) 3(x 8) (1) 4 (4) x 3 or x –1 or x 0 (2) 2 (1) 1 (3) 3
冢
冣
2.3 Adding and Subtracting Rational Expressions Expressions with the Same Denominator
(page 44) 3 1 , (a 0) a 2 5, (x 2) 3 x 4 , (x 0, 2) 4 x 3 , (x 3, 8) 5 x8 y5 , (y 7, 8) 6 y3 z2 , (z 0, 1) 7 z 1 , (a 2, 3) 8 a2 b1 , (x 5) 9 2 10 n 2, (n 0, 2) Expressions with Different Denominators
(pages 48–49) x 1 21 14 , (x 0) 2 5x 3 2x , (x 0) 3 x2 2.3 Adding and Subtracting Rational Expressions
5
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
10 3a , (a 0) 2a2 3, (c 5) 5x 16 , (x 4, 4) (x 4)(x 4) 21 , (x 3) 10(x 3) 5x 3 , (x 1, 0, 1) x(x 1)(x 1) 12 , (x 2, 0) x(x 2) 4 , ( y 1, 2) 3( y 2) 9 , (x 4, 3, 1) (x 1)(x 4) 1 , (x 7, 1, 2) (x 2)(x 1) x6 , (x 6, 3, 0) x(x 6) 4 , (x 4, 3) x4 2 , (x 1, 1) x1 4n 3 6a 10 5 r 2 2r 3 , (r 0) r (3) x2 3x 7 m (4) 12 x 2 4x 2 (1) x 4 p (2) 15
2.4 Ratio and Proportion (pages 52–54) 1 True, 24 24 2 False, 175 245 3 False, 15 16 4 True, 18 18 5 x 20 6 y 15 7 z 10 8 x2 9 y4
6
Chapter 2: The Rational Numbers
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
z5 x 4, 2 x 2, 6 (3) 9 : 25 x 4 (3) x1 3 (2) x 9 (2) 3.788 (2) 1,000 (4) 8 3a (3) b (3) 40 (4) 14 inches 22.047 inches 240 inches 20 feet 275 13.59375 gallons 400 a 24, 55, 89, 144, 233 b 1.618 c approximately the same
2.5 Complex Rational Expressions (pages 58–59) 1 2 5 2 6 3 3 49 9 4 49 a1 , (a 0) 5 3a2 2(6a 5) , (a 0) 6 7 7 ab, (a, b 0) w9 , (w 0) 8 18 yx , (x 0, y 0, x y) 9 yx 3 , (n 0) 10 2 2 b , a 0, , b 0 11 2a b 2 y , (y 0, 3) 12 3y
冢
冣
13 14 15 16 17 18
2s , (s 1) s1 x4 , (x 0) x x7 , (x 0, 2) x(x 2) 3( y 2) , ( y 0, 1, 3) y R 1R 2 (4) R1 R2 s • s (3) s s
2.6 Solving Rational Equations (pages 64–65) 1 a 11 2 r 7 or r 3 3 x 7 or x 7 4 b2 5 x 3 or x 1 6 { } 7 a 13 8 y 4 9 a5 10 x 3 11 y 3 or y 1 12 x 1 13 z 3 or z 6 14 n 5 1 15 b 2 2 3 or 16 3 2 2 17 3 1 18 a 8 5 b 8 1 5 , c n n 5 5 1 d 8 n 1 e 13 hr 3
19
a b c
20
a b c
21
(2)
22
(4)
23
(3)
24
(1)
t 10 t 5 1 3 minutes 3 300 20w 300 20w w 300 20w 50; 10 weeks w {0} 50x 60y xy {6} 1,000 10w w
2.7 Solving Rational Inequalities (page 71) 1 a9 2 7 n 4 3 0s2 4 1x5 9 5 y 4 2 6 g 7 or g 5 7 0r4 8 0v9 9 z 0 or z 7 10 d 4 or d 1 11 5 k 4 12 4 q 2 or q 6 13 y 3 or 0 y 4 14 0 w 1 or w 2 15 x 6 or 2 x 3 16 a 25 0.03n 25 0.03n b n c Joey must use more than 1,250 minutes. 17 The resistance of the other resistor must be greater than 6 ohms. 18 The most that Dante can drive is 500 miles. Chapter Review (pages 75–77) 1 (4) x 0, x 6 x1 2 (4) 2 x 1 2.7 Solving Rational Inequalities
7
3
(1)
4
(2)
5
(1)
6
(4)
7
(2)
8
(3)
9
(1)
10
(2)
11
(3)
12
(2)
13 14 15 16
(4) (3) (1) (3)
2 x5 x4 x8 4 x x5 1 6 a2 1 x2 x4 2 1 2c 1 x1 1x {4} {6} 12 by 21 33
17 18 19 20 21 22 23 24 25
冢
冢 冢
–4
26
冣
c(2c 1) 1 , c 3, 5 2 a(a 3) , (a 4, 1, 2) a1 1 9 2y 1 , y 2, , 2 2 4 5 1 1 5y, y , , 2 3 5 6 , (x 3) x2 9 x2 , (x 0, 1) x x2 x5 {x : x 0 or 1 x 6} –2
冣 冣
0
2
4
6
–4
–2
0
2
–3.5
4 3.5
CHAPTER
Real Numbers and Radicals
3.1 Real Numbers and Absolute Value
– 11 7
2
8
4 3
8
Chapter 3: Real Numbers and Radicals
3 2 –3 2
3 33 x 2 2 33 – ––– 2
20 x 8 –20
3 x 9 or x –9
(pages 80–81) 11 or x 3 1 x 7
10
冦x : x 2兹3 or 2 x 2 or x 2兹3冧 –6
3
8
3 – –– 2
6
5
c⬍⫺
9 15 or c ⬎ 4 4
3.2 Simplifying Radicals
9 – –– 4
6
⫺4 ⬍ x ⬍ 6 –4
7 ⫺
6
6 ⱕcⱕ6 5 6 – –– 5
6
5 3
8 x ⱕ ⫺7 or x ⱖ
5 3
–7
9
⫺11 ⬍ n ⬍ 25 –11
10 y ⬍ ⫺
25
11 or y ⬎ 3 2 11 – ––– 2
11
(1)
–2 –1
3
0
1
2
3
12 (3) 兩5x ⫹ 10兩 ⱕ 15 13 (1) –2
14
(4)
15 (1) 16 (2) 17 a b c 18 a b c 19
–1
0
冦
1
4
2
5
6
3
7
4
冧
17
a 兩M ⫺ 37兩 ⱕ 8 b 29 ⱕ M ⱕ 45 c a 兩14 ⫺ C兩 ⱕ 6 b 8 ⱕ C ⱕ 20
8
5
5 ⫺ ⬍a⬍7 2 兩S ⫺ 24兩 ⱕ 3 兩w ⫺ 16兩 ⱕ 0.4 兩D ⫺ 132兩 ⬍ 26 106 ⬍ D ⬍ 158 yes 兩H ⫺ 23兩 ⬍ 6 17 ⱕ H ⱕ 29
29
20
(page 84) 1 2兹3 2 3兹6 3 25x 3兹2 4 2兹5 5 2兹7 6 ⫺8a2b 4兹3b 7 4兹2 8 2 9 ⫺3n兹3n 10 3兹3 3 11 8兹2 12 1 13 (4) 4 9 14 (3) ⫺ 兹2 4 15 (2) 8y 4兹5 16 (2) 2兹6 17 (2) 2兹30 18 (4) 250x 6
15 ––– 4
29
45
3.3 Operations with Radicals Adding and Subtracting Radicals
(page 86) 1 ⫺5p兹7 38 6 2 3 兹 3 2兹5 4 2兹3 5 13兹2 6 ⫺a2b 3兹3c 7 ⫺2 兹6 8 8 兹2 9 (2) 52 兹3 10 (3) 17 兹3 11 12 13 14
3
(1) 12 兹3 ⫹ 3 (4) 兹6 (1) 兹3 1 (4) x ⫽ 兹3 2
3.3 Operations with Radicals
9
3.4 Multiplying Radicals (page 89) 1 12x兹10 ⫹ 5x 2 12兹3 ⫺ 72 3 ⫺9 ⫺ 兹6 4 40 5 4 6 ⫺13 ⫺ 兹5 7 20 ⫺ 8兹2 8 29 3 3 9 ⫺4 ⫺ 8兹2 ⫹ 30兹4 10 19 11 58 ⫺ 12兹6 12 (3) 45兹2 13 (4) 冢6 ⫹ 3兹2冣冢6 ⫺ 3兹2冣 14 (1) 66 ⫹ 36兹3
3.5 Dividing by Radicals (pages 93–94) 1 1 2 10, (a ⫽ 0) 3 3 4 4 5 15 6 6兹2 7 6兹2 8 17 9 4 8 ⫺ 2兹7 10 3 6 ⫺ 兹15 11 3 12 18 ⫹ 9兹3 8 ⫺ 2兹2 ⫺ 4兹5 ⫺ 兹10 13 14 ⫺5 ⫹ 9兹5 14 20 21 ⫹ 7兹6 ⫹ 3兹2 ⫹ 2兹3 15 3 16 (2) 兹6 17 (2) 5 18 (3) 5 7 ⫹ 3兹5 19 (2) 2 20 (4) 8 ⫹ 2兹10
10
Chapter 3: Real Numbers and Radicals
3.6 Solving Radical Equations (pages 96–97) 1 x ⫽ 40 2 x⫽6 3 x ⫽ 43 4 x ⫽ 13 5 x ⫽ 58 6 x⫽4 7 x⫽3 8 { } 9 x⫽4 10 x ⫽ 31 11 x ⫽ 6 12 x ⫽ 10 13 { } 14 x ⫽ 4 15 x ⫽ 15 16 (1) 3兹2x ⫺ 5 ⫽ ⫺6 17 (3) 3 18 (4) {4} 19 (3) 5 20 (3) Subtract 2 from each side. Chapter Review (pages 97–98) 1 (2) {1, 3} 2 (2) 5c兹3c 3 (3) 10n 4 (4) –6
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
0
(4) 4 (3) 6y2 (4) ⫺6 7 ⫹ 3兹5 (3) 2 (4) from 20 to 40 minutes (2) {8} 4兹3 ⫹ 9兹2 (4) 6 (1) 4兹6 ⫺ 8 (1) 3x 2y 4兹3y (2) ⫺3 1 (1) 兩 a ⫺ 12 兩 ⱕ 4 12兹6 ⫺25兹3 ⫺16 ⫺ 31兹3 8 ⫺ 4兹3 11兹5 ⫺ 26 6兹6 ⫺ 6兹2 3
14
7兹6 3 24 10x 3兹2 3兹2 ⫺ 2兹3 25 6 26 x ⫽ ⫺3, 7 23
68 5 x⫽8 3 ⫺ ⱕ x ⱕ 15 2 x ⬍ ⫺5 or x ⬎ 4
27 x ⫽ 28 29 30
CHAPTER
Relations and Functions 4.1 Relations and Functions
4
7
Relations
(pages 100–101) 1 Domain: {Albany, Bismarck, Juneau} Range: {New York, North Dakota, Alaska} Independent variable: capital; Dependent variable: state 2 Domain: {1, 4, 11, 25} Range: {January, July, November, December} Independent variable: date of U.S. holiday; Dependent variable: month of U.S. holiday 3 Domain: {The Dark Knight, Legally Blonde, Mr. and Mrs. Smith, Pirates of the Caribbean, The Bourne Identity, Titanic} Range: {Christian Bale, Reese Witherspoon, Angelina Jolie, Johnny Depp, Matt Damon, Leonardo DiCaprio} Independent variable: movie title; Dependent variable: movie star 4 Domain: {3, ⫺2, 4, ⫺1, ⫺3, 5}; Range: {9, 4, 16, 1, 9, 25}; Rule: y ⫽ x2 5 Domain: {⫺10, ⫺8, ⫺4, ⫺2, 0}; 1 Range: {4, 3, 1, 0, ⫺1}; Rule: y ⫽ ⫺ x ⫺ 1 2 6 Domain: {2, 4, 5, 7, 9}; Range: {1, 5, 7, 11, 15}; Rule: y ⫽ 2x ⫺ 3
8
9
10
11 12 13 14 15
冦(5, rational number), 冢兹7, irrational num1 ber冣, 冢 , rational number冣 (p, irrational 2 1 number)冧; Domain: 冦5, 兹7, , p冧; 2 Range: {rational number, irrational number} {(0, integer), (0, whole number), (⫺1, integer), (2, integer), (2, whole number), (⫺5, integer), (⫺3, integer)}; Domain: {0, ⫺1, 2, ⫺5, ⫺3}; Range: {integer, whole number} {(⫺6, ⫺5), (⫺1, 5), (4, 15), (9, 25)}; Domain: {⫺6, ⫺1, 4, 9}; Range: {⫺5, 5, 15, 25} {(⫺4, 4), (⫺2, 0), (⫺2, ⫺1), (0, 3), (1, ⫺4), (2, 1), (4, 2), (4, ⫺3)}; Domain: {⫺4, ⫺2, 0, 1, 2, 4}; Range: {4, 0, ⫺1, 3, ⫺4, 1, 2, ⫺3} Domain: {x : ⫺4 ⬍ x ⱕ 5 and x ⫽ 2}; Range: {y : ⫺2 ⱕ y ⱕ 5} Domain: All real numbers; Range: {y : y ⱖ ⫺4} Domain: {x : ⫺2 ⬍ x ⬍ 4}; Range: {y : ⫺4 ⬍ y ⬍ 5} Domain: {x : ⫺5 ⱕ x ⬍ 4}; Range: {y : ⫺3 ⬍ y ⬍ 4} Domain: {x : x ⱖ ⫺3}; Range: All real numbers 4.1 Relations and Functions
11
Functions
18
(pages 105–108) 1 Function; it passes the vertical line test. 2 Function; it passes the vertical line test. 3 Not a function; it does not pass the vertical line test. 4 Function; it passes the vertical line test. 5 Function; it passes the vertical line test. 6 Function; it passes the vertical line test. 7 Not a function; it does not pass the vertical line test. 8 Not a function; it does not pass the vertical line test. 9 Not a function; it does not pass the vertical line test. 10 Function; for each x-value in the domain, there is exactly one y-value. 11 Function; for each x-value in the domain, there is exactly one y-value. 12 Function; for each x-value in the domain, there is exactly one y-value. 13 Function; for each value in the domain, there is exactly one corresponding value in the range. 14 Function; for each value in the domain, there is exactly one corresponding value in the range. 15 Not a function; there are values in the domain that correspond to two distinct values in the range. 16 (3)
17
54
43
13
32
16
71
(4) Distance
Time
12
Chapter 4: Relations and Functions
(1) Height
Time 19
(4) Money
Time
4.2 The Algebra of Functions (pages 111–113) 1 a f(3) ⫽ 10 1 5 ⫽⫺ b f 2 4 c f(⫺1) ⫽ ⫺2 2 a g(⫺2) ⫽ 5 b g(1.4) ⫽ ⫺7.92 c g(5) ⫽ ⫺72 3 x⫽6 4 x ⫽ ⫺1, 4 5 a h(⫺2) ⫽ ⫺11 b h(2.5) ⫽ 10.375 1 ⫽ 0.625 c h ⫺ 2 d x ⫽ 0, 1 6 a 3t ⫹ 9 b 5t ⫹ 15 c t⫹3 d 2 e 2t 2 ⫹ 12t ⫹ 18 7 f (2) ⫽ ⫺2 8 f (⫺1) ⫽ ⫺5 9 f (4) ⫽ 0 10 f (3.5) ⫽ ⫺4 11 2f (0) ⫽ 8 12 f (2) • f (3) ⫽ 10 13 f (0) ⫹ f (1) ⫽ 7 14 3 15 {⫺1, 3} 16 a g(1) ⫽ 9 b g(⫺2) ⫽ 18
冢 冣
冢 冣
17
22 23 24 25 26 27
a b a b a b a b a b (3) (2) (4) (4) (2) (2)
28 29
(4) a
18 19 20 21
b
30
c a b c d
31
a b
c d e
g(1) ⫽ ⫺3 g(⫺2) ⫽ 0 g(1) ⫽ 3 g(⫺2) ⫽ 1.5 g(1) ⫽ 5 g(⫺2) ⫽ 5 g(1) ⫽ 6.25 g(⫺2) ⫽ 8.5 g(1) ⫽ 0 g(⫺2) ⫽ 21 3 (2, ⫺1) 22 ⫺13 7 The point (⫺1, 5) would appear on the graph of this function. f(x)x ⫹ 6 No, f(1 ⫹ 4) ⫽ f(5) ⫽ 52 ⫽ 25 while f(1) ⫹ f(4) ⫽ 12 ⫹ 42 = 17. No, f(a ⫹ b) ⫽ (a ⫹ b)2 ⫽ a2 ⫹ 2ab ⫹ b2 while f(a) ⫹ f(b) ⫽ a2 ⫹ b2. Yes, if a ⫽ 0 or b ⫽ 0. In 2007, Mrs. Santiago had 10 girls in her class. In 2008, Mrs. Santiago had 5 more girls than boys in her class. In year y, Mrs. Santiago had twice the number of boys in her class as girls. The function t(y) is the total number of students, boys and girls, that Mrs. Santiago has in her class in year y. In three weeks, the Longarzo family buys 7 gallons of milk. In three weeks, the Longarzo family buys the same amount of milk and ice cream. In two weeks, the Longarzo family buys a total of 8 gallons of milk and ice cream. In w weeks, the Longarzo family buys three times as much milk as ice cream. In w weeks, the Longarzo family buys 2 more gallons of milk than they buy of ice cream.
4.3 Domain and Range (pages 120–121) 1 (3) ⫺16 2x ⫹ 5 2 (3) y ⫽ x⫺6
3 (2) 84 4 (2) {x: x ⫽ 0, 5} 5 (2) x ⱖ ⫺4 6 (2) {⫺2, 0, 2} 7 (2) ⫺2 4 8 (4) y ⫽ x⫺2 9 Domain: All real numbers; Range: All real numbers 10 Domain: All real numbers; Range: { y : y ⱖ ⫺5} 11 Domain: {x : ⫺5 ⱕ x ⬍ 5}; Range: {y : ⫺5 ⱕ y ⱕ 6} 12 Domain: {x : ⫺7 ⱕ x ⬍ 8}; Range: {⫺1, 1, 3, 5, 7} 13 {x : x ⫽ ⫺2, 2} 14 {x : x ⬎ ⫺3} 15 All real numbers 16 Domain: All real numbers; Range: { y : y ⱕ 16} 17 Domain: All real numbers; Range: All real numbers 18 Domain: All real numbers; Range: { y : y ⱖ 0} 19 Domain: {x : –3 ⱕ x ⱕ 3}; Range: {y : 0 ⱕ y ⱕ 3} 20 Domain: All real numbers; Range: { y : y ⱕ 3} 21 Domain: All real numbers; Range: All real numbers
4.4 Composition of Functions (pages 125–127) 1 2x2 ⫺ 3x ⫺ 20 2 ⫺5x3 ⫹ 2x2 ⫹ 80x ⫺ 32 3 ⫺3x ⫹ 12 x⫹4 , (x ⫽ ⫺1, 4) 4 x⫹1 5 4 6 48 7 62 8 ⫺68 9 234 10 ⫺7 11 兹82 12 x, (x ⱖ ⫺16) 13 兹x 2 ⫺ 3x ⫹ 12 14 82 ⫺ 5x2 15 (2) ⫺1 16 (4) ( f ⴰ g)(4) 17 (2) 0 4.4 Composition of Functions
13
(3) (4) (3) a b c d 22 a b c d e
18 19 20 21
{⫺0.5, 0.5, 3.5, 4.5} ( f ⴰ g)(2) ⫽ ⫺1 3 t(x) ⫽ 1.0825x c(x) ⫽ 0.8x t(c(x)) ⫽ 0.866x $259.80 f(d) ⫽ 90d b(d) ⫽ 95d c(x) ⫽ 1.3849x c( f(d )) ⫽ 124.641d At $95 per day, the cost for 5 days would be $475. At 90 euros per day, the cost for 5 days would be $623.21.
4.5 Inverse Functions (pages 133–134) 1 A⫺1: {(5, 8), (8, 6), (11, 4), (14, 2)} 2 B⫺1: {(3, ⫺2), (5, ⫺5), (7, ⫺8) (9, ⫺11)} 3 C⫺1: {(&, *), (%, $), (⫹, @), (!, #)} 4 D⫺1: {(9, 2), (⫺5, 4), (8, 13), (⫺10, ⫺1)} 5 Beauty and the Beast x⫺7 6 y ⫺1 ⫽ 4 ⫺x ⫺ 1 7 y ⫺1 ⫽ 3 5 8 f ⫺1(x) ⫽ x ⫹ 15 2 9 f ⫺1(x) ⫽ x3 ⫹ 4 x⫹6 10 g ⫺1(x) ⫽ 3 3 1 1 11 y ⫺1 ⫽ x ⫺ 4 6 1 12 (3) 13 3 13 (2) f(x) ⫽ x 3, g(x) ⫽ 兹x 14 (3) x2 ⫹ 4 15 (3) ⫺1 16 (4) 123 17 (2) 2 18 (4) {(x, y), ( y, z), (z, a)} 1 19 (4) f ⫺1(x) ⫽ ⫺ x ⫹ 5 2 5 20 (3) 2
4.6 Transformations of Linear, Absolute Value, and Polynomial Functions (pages 143–144) 1 (1) (0, 3) 2 (4) (⫺3, 0) 3 (3) 10 units to the right 4 (4) 4 5 (2) 0 6 (4) 5 units to the left and 3 units down 7 (3) 2 units to the right and 1 unit up 8 (1) (1, 5) 9 (1) y ⫽ ⫺x2 ⫹ 6 10 (3) y ⫽ 兩x ⫹ 1兩 11 a
12
zeros at x ⫽ ⫺2, 0, 1, 3 b 4th degree a
冪
14
Chapter 4: Relations and Functions
b 3 c between ⫺3 and ⫺2, between ⫺2 and ⫺1, between 1 and 2 d zeros at x ⫽ ⫺2.38, ⫺1.27, 1.65 13 a
b 2 c between 1 and 2 and between 3 and 4 d zeros at x ⫽ 1.49, 3.18
14
18
y3 y2
y2
y1
y1
y3
The graph of y2 is the graph of y1 shifted 3 units to the left. The graph of y3 is the graph of y1 shifted 3 units up. 15
The graph of y2 is the graph of y1 condensed by a factor of 4; that is, each y-value of y2 is 4 times the corresponding y-value of y1 . The graph of y3 is the graph of y1 expanded by a 1 factor of 2; that is, each y-value of y3 is 2 the corresponding y-value of y1 .
y3 y1
y2
19 y2
The graph of y2 is the graph of y1 shifted 3 units to the right. The graph of y3 is the graph of y2 shifted 2 units up. 16 y2
y1
y1
y3
The graph of y2 is the graph of y1 shifted 3 units to the right. The graph of y3 is the graph of y1 shifted 2 units down.
y3
20 y1
The graph of y2 is the graph of y1 condensed by a factor of 4; that is, each y-value of y2 is 4 times the corresponding y-value of y1 . The graph of y3 is the graph of y1 expanded by a 1 factor of 2; that is, each y-value of y3 is 2 the corresponding y-value of y1 . 17 y2 y1 y3
The graph of y2 is the graph of y1 shifted 3 units to the right. The graph of y3 is the graph of y1 shifted 2 units to the left and 3 units down.
y3
y2
The graph of y1 is a parabola. The graphs of y2 and y3 are both 3rd degree. The graph of y2 is y ⫽ x3 condensed by a factor of 3 and shifted 1 to the right. The graph of y3 is y ⫽ x3 expanded by a factor of 3 and shifted 2 down.
4.7 Circles (pages 146–148) 1 (4) (4, ⫺2) 2 (3) (x ⫺ 1) ⫹ y2 ⫽ 42 3 (4) (⫺4, ⫺3) 4 (1) center (2, ⫺3), radius 4 5 (4) (x ⫺ 2)2 ⫹ (y ⫹ 4)2 ⫽ 25
4.7 Circles
15
6
(4)
–8
7 8 9 10 11 12 13 14 15 16 17 18 19 20
9
y
–6
–4
y
8
8
6
6
4
4
2
2 O
–2
2
4
6
8
x
–8
–6
–4
O
–2
–2
–2
–4
–4
–6
–6
–8
–8
(1) (x ⫺ 4)2 ⫹ ( y ⫹ 6)2 ⫽ 5 (3) (x ⫹ 2)2 ⫹ ( y ⫺ 1)2 ⫽ 36 (4) (x ⫺ 6)2 ⫹ ( y ⫹ 5)2 ⫽ 20.25 (2) 5.6 center (⫺7, 0), r ⫽ 5.4 center (0, 3), r ⫽ 3.7 1 ,r⫽4 center ⫺2, ⫺ 2 center (5, ⫺2), r ⫽ 4.8 center (⫺1.5, 3.6), r ⫽ 兹10 (x ⫹ 1)2 ⫹ ( y ⫹ 5)2 ⫽ 64 x2 ⫹ ( y ⫺ 3)2 ⫽ 1 (x ⫺ 2)2 ⫹ y2 ⫽ 1.44 (x ⫺ 4)2 ⫹ ( y ⫺ 8)2 ⫽ 8 (x ⫹ 2)2 ⫹ ( y ⫺ 1)2 ⫽ 41
冢
(3)
冣
2
10 Yes, D ⫽ 1.2C 11 Not a direct variation 12 Yes, M ⫽ 3.6K 13 Not a direct variation 14 750 15 11.811 16 a C(g) ⫽ 4.074g b g 10 15 C(g)
c
$40.74
$61.11
4
6
8
x
20 $81.48
C(g) 85 80 75 70 65 60
4.8 Direct and Inverse Variation
55 50 45
Direct Variation
(pages 150–152) 1 (3) 15 2 (2) 0.8 x ⫽k 3 (4) y 4 (4) 263 5 (3) $97.75 6 (2) 28.8 7 (2) 0 y 8 (3) x
16
Chapter 4: Relations and Functions
40 O 5
17
18
10 15 20
g
a 19,610,620 b No, cities are more densely populated. The population density is an average and includes sparsely populated regions such as forests. 5 p a w⫽ 11 b 10 kg c 30 mg per day 15 p d d⫽ 11
Yes, all direct variations are a function of a variable times a constant. b No, some linear equations are translated so that they do not go through the origin. c Answers will vary. Example: y ⫽ 3x ⫹ 6 20 a p b C ⫽ pd c 40p in.
19
a
Inverse Variation
(pages 155–156) 1 (4) 8% 2 (2) 16 3 (3) 21 4 (1) 8 5 (4) 18 6 (2) 32 7 (2) 12 8 (4) 4 9 (4) 12 10 (1) 10 11
y 8 6 4 2
–8
–6
–4
O
–2
2
4
6
8
2
4
6
8
x
–2 –4 –6 –8
12
y 8 6 4 2 –8
–6
–4
O
–2 –2 –4 –6 –8
x
16 x
13
xy ⫽ 16 or y ⫽
14
xy ⫽ ⫺6 or y ⫽ ⫺
6 x
Chapter Review (pages 160–162) 1 Function; Domain: {⫺2, 3, 8, 9}; Range: {5, 7, 9, 11} 2 Function; Domain: {8, 5, 3, 0}; Range: {⫺3} 3 Not a function; two different elements of the range correspond to the element Drew Carey of the domain. 4 Not a function; two different elements of the range correspond to the element 9 of the domain. 5 Function; Domain: {x : ⫺7 ⬍ x ⬍ 7}; Range: { y : 1 ⬍ y ⬍ 7} 6 Not a function; the graph fails the vertical line test. 7 Function; Domain: {3, 17, 23}; Range: {25, 16} 8 Not a function; the graph fails the vertical line test. 9 Function; Domain: All real numbers; Range: All real numbers 10 Function; Domain: All real numbers; Range: All real numbers 11 (3) 0 12 (1) 1 13 (2) (8, 12) 14 (1) {2, 4} 8 ⫺ 3x 15 (4) g(x) ⫽ x⫹5 16 (4) ⫺16 17 (2) ⫺5 1 18 (4) f ⫺1(x) ⫽ x ⫺ 4 4 19 (1) {(3, ⫺2), (⫺2, 3), (3, ⫺1), (⫺2, 4)} 20 (4) 4 21 (1) ⫺3 22 (1) reflection in y ⫽ x 23 (3) 3x2 ⫹ 5 24 (2) 2 25 (2) divided by 3 26 (2) 2.5 27 (1) ⫺1 28 (3) (g ⴰ f )(⫺4) ⫽ 4
Chapter Review
17
CHAPTER
5
Quadratic Functions and Complex Numbers
5.1 Alternate Methods of Solving Quadratics Completing the Square
(page 165) 1 x ⫽ ⫺3 ⫾ 兹17 2 x ⫽ 4 ⫾ 兹7 3 x ⫽ 6 ⫾ 2兹6 4 x ⫽ 5 ⫾ 4兹3 5 x ⫽ 3 ⫾ 2兹3 6 x ⫽ 1 ⫾ 兹6 7 x ⫽ 2 ⫾ 兹2 8 x ⫽ ⫺1, 7 9 x ⫽ ⫺1, 2 10 x ⫽ ⫺6 ⫾ 兹31 The Quadratic Formula
(page 168) 1 x ⫽ 2 ⫾ 兹3 1 5 2 x⫽⫺ , 2 2 3 x ⫽ 4 ⫾ 兹6 4 x ⫽ 3 ⫾ 兹5 1 ⫾ 兹2 5 x⫽ 2 2 6 x⫽⫺ ,1 5 7 x ⫽ ⫺1 ⫾ 兹3 3 7 1 ⫾ 兹 8 x⫽ 2 2 2 9 x ⫽ ⫺ ⫾ 兹2 3 10 x ⫽ 12 11 x ⫽ 1 ⫾ 3兹2 4 12 x ⫽ ⫺ , 2 3 13 x ⫽ ⫺3
18
14 x ⫽ ⫺1, ⫺ 15
x⫽⫺
2 3
2 14 ⫾ 兹 5 5
5.2 The Complex Number System Imaginary Numbers
(pages 170–171) 1 7i 2 88i 3 4i 4 ⫺7i 5 4i兹5 6 ⫺8i兹5 7 12i兹11 8 8i兹2 i 9 2 10 2i兹3 11 5i 12 2i兹6 13 ⫺i 14 i 15 i 16 ⫺i 17 1 18 ⫺1 19 i 20 ⫺1 21 12i 22 3i 23 72 24 2i 25 (3) ⫺19
Chapter 5: Quadratic Functions and Complex Numbers
26 (1) ⫺70i 27 (3) ⫺100 28 (2) i 9 29 (4) 48 30 (2) 3i兹3 Complex Numbers
(page 173) 1 a ⫽ 2, b ⫽ 5 2 a ⫽ 0, b ⫽ 7 3 a ⫽ 12, b ⫽ 0 4 a ⫽ 11, b ⫽ 7 5 a ⫽ 6, b ⫽ 2 6 a ⫽ 5, b ⫽ ⫺2 7 a ⫽ ⫺2, b ⫽ 12 8 6 ⫹ 5i 9 ⫺2 ⫹ 7i 10 7 ⫺ 3i兹3 11 5 ⫹ 6i兹2 12 ⫺6 ⫹ 4i 13 True; real numbers are of the form a ⫹ bi with b ⫽ 0. 14 False; if b ⫽ 0, the number is not real. 15 True; integers are of the form a ⫹ bi with a an integer and b ⫽ 0. 16 True; if b ⫽ 0, the number is not real. 17 False; an imaginary number is a complex number when a ⫽ 0. 18 False; an imaginary number is a complex number when a ⫽ 0. 19 A: ⫺5 ⫹ I, B: 2, C: 4 ⫹ 2i, D: 4 ⫺ 3i, E: ⫺2 ⫺ 4i 20 A: ⫺4 ⫹ 5i, B: 2i, C: 5 ⫹ 3i, D: 5 ⫺ 4i, E: ⫺4 ⫺ 2i
5.3 Operations with Complex Numbers Addition and Subtraction of Complex Numbers
(pages 176–177) 1 4i 2 2 ⫹ 3i 3 1⫺i 4 4 ⫹ 10i 5 9 ⫹ 15i 6 ⫺7 ⫺ 40i 7 7 ⫹ 13i兹2 8 ⫺9 ⫺ 16i兹3 9 14 ⫹ 22i兹5
10 ⫺7 ⫹ 5i兹3 11 a ⫽ 5, b ⫽ 5 12 a ⫽ 4, b ⫽ ⫺2 13 a ⫽10, b ⫽ 8 14 a ⫽ 3, b ⫽ 9 15 (4) 5 ⫺ 10i 16 (1) I 17 (1) 20 18 (2) II 19 (1) 4i兹2 20 (4) 8 ⫺ 12i兹5 21 a
yi 6 5 4
Z1 ⫹ Z2
3
Z2
2 1
Z1
–4 –3 –2 –1 O 1 –1 –2
2
3
4
x
b ⫺1 ⫹ 5i yi 22 a 4 3 2
Z2 Z1 ⫹ Z2
1
–1 O 1 2 –1 Z1 –2
3
4
5
6
7
2
3
4
5
6
x
–3 –4
b 5⫹i 23 a
yi 4
Z2
3 2 1
–2 –1 O –1 –2 –3 –4
1
x
Z1 Z1 ⫺ Z2
–5
b 6 ⫺ 6i
5.3 Operations with Complex Numbers
19
24
a
5.4 Nature of the Roots and the Discriminant
yi 4
Z2
3 2
Z1
1
–4 –3 –2 –1 O 1 2 3 –1 Z ⫺ Z2 –2 1
4
–3 –4
b 4 ⫺ 2i Multiplication and Division of Complex Numbers
(pages 179–180) 1 47 ⫺ 16i 1 1 ⫺ i 2 2 2 3 ⫺12 ⫺ 24i 6 4i ⫺ ⫺ 4 5 5 5 33 ⫺ 31i 6 40 ⫹ 42i 7 18 ⫺ i 5i 2 ⫹ 8 29 29 9 8 ⫺ 32i i 8 ⫹ 10 13 13 11 8 ⫹ 4i 11 23i ⫹ 12 25 25 ⫺9 ⫺ 11i兹2 13 3i ⫺1 ⫺ 14 2 30 ⫺ i兹3 15 16 100 5 3i ⫺ 17 34 34 4i 7 ⫹ 18 (3) 65 65 19 (4) 85 20 (1) 3 ⫹ 4i 3 3 ⫺ i 21 (4) 4 4 3 i兹5 ⫹ 22 (1) 14 14 23 (2) 47 ⫺ 29i 24 (4) 2i
20
x
(pages 183–184) 1 ⫺80; two imaginary, unequal roots 2 49; two real, rational, unequal roots 3 88; two real, irrational, unequal roots 4 ⫺143; two imaginary, unequal roots 5 81; two real, rational, unequal roots 6 0; two real, rational, equal roots 7 (2) 3x2 ⫺ 2x ⫺ 5 ⫽ 0 8 (2) II and III 9 (2) 2 10 (4) The parabola intersects the x-axis at two distinct points. 11 (1) 9x2 ⫹ 6x ⫹ 1 ⫽ 0 12 (1) two real, unequal, irrational roots 13 (4) ⫺11 14 (4) 10 15 (2) 6 16 (2) x2 ⫺ 5x ⫹ 2 ⫽ 0 17 (2) a ⬎ 1 only 18 (4) 6 19 (3) x(x ⫹ 6) ⫽ ⫺10 1 20 x ⫽ , 3 2 21 x ⫽ ⫺2, 10 22 x ⫽ 1, 5 3 ⫾ 兹2 23 x ⫽ 2 24 x ⫽ 2, 12 3 19 ⫾ 兹 25 x ⫽ 2 2
5.5 Complex Roots of Quadratic Equations (page 187) 1 x ⫽ 2 ⫾ i兹3 2 x ⫽ 3 ⫾ 2i 1 1 ⫾ i兹19 3 x⫽ 4 4 4 x⫽2⫾i 3 i ⫾ 5 x⫽ 2 2 1 i ⫾ 6 x⫽ 3 3 1 7 x ⫽ 1 ⫾ i兹2 2
Chapter 5: Quadratic Functions and Complex Numbers
8 9 10 11 12 13 14 15 16 17 18 19 20
1 5 ⫾ i 2 2 x⫽1⫾i 2 x ⫽ 2 ⫾ i兹21 3 1 5 ⫾ i兹11 x⫽ 2 2 x ⫽ 1 ⫾ 4i 1 x⫽1⫾ i 2 4 1 x⫽ ⫾ i兹5 3 3 1 x⫽ ⫾ 3i 2 1 1 x⫽ ⫾ i 2 2 x ⫽ 1 ⫾ i兹3 x⫽7⫾i 3 x⫽ ⫾ 2i 2 1 x⫽ ⫾ 2i 2 x⫽
5.6 Sum and Product of the Roots (pages 189–190) 1 Sum ⫽ 3; Product ⫽ 5 2 x ⫽ 7; x2 ⫺ 12x ⫹ 35 ⫽ 0 3 3 Sum ⫽ ; Product ⫽ ⫺3 4 4 x ⫽ 6 ⫺ 2i; x2 ⫺ 12x ⫹ 40 ⫽ 0 5 x ⫽ 10; k ⫽ 48 6 r2 ⫽ 2 ⫹ 兹5; x 2 ⫺ 4x ⫺ 1 ⫽ 0 7 x ⫽ ⫺5; k ⫽ 9 8 3x2 ⫺ 11x ⫺ 4 ⫽ 0 9 x2 ⫺ 6x ⫹ 25 ⫽ 0 10 81x2 ⫺ 36x ⫹ 5 ⫽ 0 11 (4) 5 12 (3) x2 ⫹ 13 ⫽ 13x 13 (2) x2 ⫺ 14x ⫹ 50 ⫽ 0 14 (3) 7 15 3 15 (1) sum: ⫺ , product: ⫺ 4 4
5.7 Solving Higher Degree Polynomial Equations Factoring by Grouping
(page 190) 1 (3x ⫹ 5)(x ⫹ 2)(x ⫺ 2) 2 (3a ⫹ 2)(2a2 ⫺ 3) 3 (4x ⫹ 3)(2x2 ⫹ 1) 4 (3y ⫹ 4)(y2 ⫹ 2) 5 (2x ⫹ 7)(x ⫹ 2)(x ⫺ 2) 6 (d ⫺ 2)(d ⫹ 3)(d ⫺ 3) 7 (3x ⫹ 1)(x ⫹ 1)(x ⫺ 1) 8 (3c ⫺ 2)(c ⫺ 2)(c ⫹ 2) 9 (2x ⫺ 3)(x ⫹ 3)(x ⫺ 3) Factoring the Sum and Difference of Cubes
(page 192) 1 (a ⫹ 4b)(a2 ⫺ 4ab ⫹ 16b2) 1 1 5 ⫺ 5a ⫹ a ⫹ 25a2 2 2 4 2 3 (3x2 ⫹ 4)(9x4 ⫺ 12x2 ⫹ 16) x x2 7 ⫺7 ⫹ x ⫹ 49 4 10 100 10 5 (1 ⫹ y2)(1 ⫹ y)(1 ⫺ y)(1 ⫹ y4 ⫹ y8) 3 9 2 15 b⫺5 b ⫹ b ⫹ 25 6 2 4 2 7 (cd ⫹ 5)(c2d 2 ⫺ 5cd ⫹ 25) x3 3 x6 6⫹ 36 ⫺ x 3 ⫹ 8 4 2 16 9 (2x2 ⫹ 5)(3x ⫺ 4) 10 (3x ⫹ 2)(x ⫺ 3)(x ⫹ 3) 11 2(2a2 ⫺ b)(4a4 ⫹ 2a2b ⫹ b2) 12 (x2 ⫺ 5)(3x ⫹ 1) 13 (p ⫺ 5q)(p2 ⫹ 5pq ⫹ 25q2) 14 4x(x ⫹ 4)(x ⫺ 1) 15 (4z2 ⫹ 3)(16z4 ⫺ 12z2 ⫹ 9)
冢
冣冢
冢
冣冢
冣
冢
冣冢
冣
冢
冣冢
冣
冣
Algebraic Solutions
(page 194) 1 2 3 4 5 6 7
1 ,2 2 x ⫽ ⫺3, ⫺2, 2, 3 5 x ⫽ ⫺2, 0, 3 3 x ⫽ ⫺1, ⫺ , 1 4 x ⫽ ⫺1, 1, 9 x ⫽ ⫺3, 0, 3, ⫾3i 3 x ⫽ 0, , 2 2 x ⫽ 0,
5.7 Solving Higher Degree Polynomial Equations
21
1 1 x ⫽ ⫺1, ⫺ , , 1 2 2 2 9 x ⫽ ⫺ , 0, 2 3 10 x ⫽ 0, 1 11 x ⫽ ⫺4, ⫺1, 0, 1, 4 5 1 1 12 x ⫽ ⫺ , ⫺ , 3 3 3
Graphic Solutions to Systems
8
(page 202) 1 {(⫺5, ⫺3), (1, 3)}
Graphic Solutions
(page 197) 1 x ⫽ ⫺4, ⫺2, 2, 2 3 4 5 6 7 8 9 10
5 2
x ⫽ ⫺2, ⫺1, 1 1 1 5 x ⫽ ⫺1, ⫺ , , 2 2 2 4 x ⫽ ⫺3, ⫺2, 0, , 3 3 1 x ⫽ ⫺1, ⫺ , 1, 2 3 3 1 x ⫽ ⫺2, ⫺ , ⫺ , 1 4 2 3 x ⫽ ⫺3, , 3 2 4 4 x ⫽ ⫺2, , 5 3 2 x ⫽ ⫺3, ⫺2, , 2 3 x ⫽ ⫺4, ⫺3, 4
5.8 Systems of Equations
2
冦冢⫺ 23 , 203 冣, (1, 5)冧
3
{(0, 1), (2, ⫺1)}
4
冦冢 12 , ⫺ 74 冣, (4, 0)冧
5
{(0, ⫺3), (3, 3)}
Algebraic Solutions to Systems
(page 199) 1 {(2, ⫺11), (5, ⫺8)} 11 41 ⫺ , , (2, ⫺2) 2 4 8 3 {(1, 3), (2, 4)} 4 {(⫺4, 6), (2, 0)} 5 {(⫺3, ⫺10), (2, 0)} 6 {(1, ⫺2), (3, 0)} 7 {(1, 2), (7, 20)} 1 49 ⫺ , , (3, 1) 8 3 9 9 {(⫺1, 2), (2, 5)}
冦冢
冦冢
22
冣
冣
冧
冧
Chapter 5: Quadratic Functions and Complex Numbers
6
4
{(⫺1, 6), (3, 2)}
5 7 {(1, 4), (3, 8)} 8 {(⫺3, 0)} 9 {(⫺1, 3), (2, 0)} 10 {(⫺2, 5), (10, ⫺19)} 11 {(0, ⫺3), (4, 9)} 2 16 ⫺ ,⫺ , (⫺1, ⫺6) 12 3 3
冦冢
冣
冧
6
5.9 Quadratic Inequalities (pages 208–209) 1
7
2
8
3
9
5.9 Quadratic Inequalities
23
10
1 2
17 x ⱕ ⫺4 or x ⱖ
y 12 10
y ⫽ 2x2 + 7x
8
4 2
–2
14 15
6
y⫽4
11 (4) (4, 5) 12 (4) y ⱕ ⫺2x2 ⫺ 5x ⫹ 2 13 (3)
4 –– 3
–8
–6
–4
18
4
x ⬍ ⫺2 or x ⬎ 7 y
y ⫽ x2 – 4x
24
18
y ⫽ x2 + 5x
15 12
8
y ⫽ x + 14
y⫽6
6
9 6
4
3
2 O
2
4
6
8
x
–10 –8 –6 –4 –2
–2 –4 –6 –8
24
O 2
–8
10
–2
x
8
–6
12
–4
6
21
14
–6
4
–4
y
–8
2
–2
(2) (⫺2, 0) (1) The solution includes all values of (x, y) that lie within the parabola, including those values on the parabola.
In 16–20, graphs will vary. The graphs in 16–18 represent one type of acceptable solution; the graphs in 19–20 represent another. 16 ⫺6 ⱕ x ⱕ 1
O
–2
Chapter 5: Quadratic Functions and Complex Numbers
–3
6
8 10
x
19
1 ⬍x⬍4 3
4
6 7 8 9 10 11 12
y ⫽ 3x2 – 11x – 4 2
13
⫺
y 8 6
–8
–6
–4
O
–2
2
4
6
8
x
14
–2
15
–4
16 17 18 19 20 21 22 23
–6 –8 –10 –12 –14
20
⫺8 ⬍ x ⬍ 3 y
28
24
24 20
y ⫽ x2 – 5x + 24
16
25
12 8 4 –12 –9
O 3
–6 –3
6
9
x
–4 –8 –12
21 22 23
26 –6
1
–4
1 –– 2
–2
7
Chapter Review (pages 212–214) 1 (3) real, irrational, and unequal 2 (3) ⫺2 3 (1) 1 4 (4) 25 5 (1) ⫺12
27 28
(2) (3) (2) (3) (1) (4) (4)
II 1⫹i x⫺5 real, rational, and unequal ⫺10 4 {⫺5i, 5i} 6 ⫺ 9i (1) 13 (3) 2 ⫺ 兹5 12 ⫹ 3i (3) 153 (4) 0 (1) 5 (4) ⫺3 (1) x ⫹ 4 (1) ⫺36 (1) 18i兹2 (3) III a x ⫽ 1, 5 b x ⫽ 2 ⫾ 兹7 c x ⫽ ⫺1 ⫾ 2兹2 a x ⫽ 5 ⫾ 2兹5 1 15 ⫾ 兹 i b x⫽ 2 2 7 3 ⫾ 兹 i c x⫽ 4 4 a x ⫽ 1, 2 1 兹19 i b x⫽⫺ ⫾ 2 2 c x ⫽ ⫺2 ⫾ 兹3 d x ⫽ 2 ⫾ 兹6 e x ⫽ ⫺5, 8 f x ⫽ ⫺1 ⫾ i 10 1 g x⫽ ⫾ 兹 3 3 h x ⫽ 0, 3, 4 i x ⫽ ⫺4, ⫺2, 2 a 1,224 ft b 1,288 ft c approximately 2 seconds d approximately 10.972 seconds 2 ⫺ 20i a {(⫺3, 10), (2, 15)} b {(5, 6)} c {(⫺5, 23), (–2, ⫺1)}
Chapter Review
25
29
30 31 32 33 34 35
The coordinates of P are (0, 228). This tells us that the bridge is suspended 228 feet above the water at its center. b A(⫺2,130, 700), B(2,130, 700) c 100 feet from the center of the bridge, the cables are 229.04 feet above the water. d y ⫽ 332; at a distance of 1,000 feet from the center of the bridge, the cables are 332 feet above the water. e y ⫽ 332; at a distance of 1,000 feet from the center of the bridge in the opposite direction, the cables are 332 feet above the water. f x ⫽ 1,617.21508 or ⫺1,617.21508; the cables are suspended 500 feet above the water at 1,617.2151 feet from the center of the bridge, in either direction. 7 4 ⫺ ⫺ i 5 5 x ⫽ ⫺1 ⫾ i 兹2 (3, ⫺12) x2 ⫹ 2x ⫺ 48 ⫽ 0 a x ⫽ ⫺3.3, ⫺0.4, 1.7 b x ⫽ ⫺2.6, 1.4 x2 ⫺ 4x ⫹ 29 ⫽ 0 a
36 37 38
x ⫽ 7.4 ⫺2 a, b
yi 1
–4 –3 –2 –1 O 1 –1 –2 Z2 = –3 – 5i –3 –4 –5 –6 –7
39 ⫺2 ⱕ x ⱕ
3
4
5
6
x
Z1 = 5 – 2i
Z1 + Z2 = 2 – 7i
5 4
–2
40
2
5 –– 4
⫺2 ⬍ x ⬍ 3.5
CHAPTER
6
Sequences and Series
6.1 Sequences
4
(pages 217–218) 1 Each term is 5 times the number of the term; 35, 40, 45 2 Each term is 5 more than the preceding term, starting with 7; 37, 42, 47 3 Each term is twice the preceding term, starting with 11; 352, 704, 1,408
5
26
Chapter 6: Sequences and Series
6 7 8 9 10
Each term is the cube of the number of the term; 216, 353, 512 Each term is 1 more than the cube of the number of the term; 217, 354, 513 3, 7, 11 6, 9, 12 ⫺7, 0, 9 2, 5, 14 6, 10, 17
a1 ⫽ 4, an ⫽ an⫺1 ⫹ 2 a1 ⫽ 5, an ⫽ an⫺1 ⫹ 2 a1 ⫽ 5, an ⫽ 2an⫺1 a1 ⫽ 2, an ⫽ 2an⫺1 ⫹ 1 a1 ⫽ 1, an ⫽ 3an⫺1 ⫺ 1 an ⫽ 2n an ⫽ 2n ⫺ 1 an ⫽ n2 an ⫽ n2 ⫺ 1 an ⫽ n(n ⫹ 1) a 144 b a1 ⫽ 1, a2 ⫽ 1, an ⫽ an⫺1 ⫹ an⫺2 22 a 100 b an ⫽ n2
11 12 13 14 15 16 17 18 19 20 21
6
14.25 or
7 8 9
121 110 124
10
6
13
兺 (1 ⫹ 4k)
k⫽1 10
14
兺 ( j 2 ⫺ 1)
j⫽6 6
6.2 Arithmetic Sequences
兺 冢2 ⫺ n⫽2 兺冢 5
16
k⫽2
冣 冣
1 n 2 ⫺k 2k ⫹ 1
5
17
(3)
兺 ( 3j ⫺ 12)
j⫽2
18 19
(2) 20 (3) 91
20
(2)
(n 2 ⫺ 10) 兺 n⫽4
21
(4)
兺 (20,000 ⫹ 5,000(k ⫺ 1)) k⫽1
22
(1)
兺 (4 ⫹ 0.5i) i⫽0
23
(3) 300
7
4
51
6.4 Arithmetic Series (page 228) In 1–5, answers may vary. 8
1
兺 (8i ⫺ 1) ⫽ 280
i⫽1 7
2
兺 (1.5n) ⫽ 42
n⫽1 5
3
6.3 Sigma Notation (pages 224–225) 1 20 2 65 3 11 4 42 5 30
15 4
11 12 12 5 In 13–17, answers may vary.
15 (pages 221–222) 1 Arithmetic; d ⫽ 6; 30, 36 2 Not arithmetic 3 Arithmetic; d ⫽ ⫺4; ⫺9, ⫺13 4 Arithmetic; d ⫽ 4n; 17n, 21n 5 Not arithmetic 6 Arithmetic; d ⫽ 16; 56, 72 7 Arithmetic; d ⫽ b; 3b, 4b 8 Arithmetic; d ⫽ ⫺6; ⫺36, ⫺42 9 Arithmetic; d ⫽ ⫺4; 15, 11 10 Not arithmetic 11 12, 9, 6, 3 12 d ⫽ 6 13 a20 ⫽ 83 14 18, 22, 26, 30; an ⫽ 18 ⫹ 4(n ⫺ 1) 15 13 16 an ⫽ 102 ⫺ 4(n ⫺ 1) 17 $900 18 61 19 8 20 No, the nth term’s value would be increased by a1 .
3.75 or
57 4
兺 (150 ⫺ 10k) ⫽ 750
k⫽0 10
4
兺 (2n ⫺ 1)k ⫽ 100k
n⫽1 9
5
兺 冢⫺ 2 i冣 ⫽ ⫺22.5 i⫽1
6 7 8 9
5, 6, 7; 126 1, 5, 9; 435 28, 32, 36; 228 32, 38, 44; 1,232
1
6.4 Arithmetic Series
27
10 11 12 13 14 15 16 17 18 19 20
3 5 , 2, ; 51 2 2 3, 10, 17; 498 3, 6, 9; 15,150 Sodds ⫽ 2,500; Sevens ⫽ 2,550; S ⫽ 5,050; this is the sum of the first 100 integers. 405 15; starting at the bottom: 15, 13, 11, 9, 7, 5 $825 day 1: 120 miles, day 2: 180 miles, day 3: 240 miles, day 4: 300 miles, day 5: 360 miles 500,500 615 No, they will have only 1,500 calculators.
6.5 Geometric Sequences (page 232) 1 Not geometric 2 Geometric; r ⫽ 3; 81, 243 1 11 11 , 3 Geometric; r ⫽ ; 3 3 9 4 Geometric; r ⫽ n; n4, n5 1 1 1 5 Geometric; r ⫽ ⫺ ; , ⫺ 2 4 8 16 32 , 6 Geometric; r ⫽ 2; 3 3 7 No 8 Geometric; r ⫽ 0.4; ⫺0.0512, ⫺0.02048 3 243 729 , 9 Geometric; r ⫽ , 4 1,024 4,096 10 Geometric; r ⫽ 兹2; 4兹2, 8 11 12, ⫺36, 108 12 r ⫽ 0.2 1 13 512 14 13, 91, 637, 4,459 15 12 16 an ⫽ 10(⫺4)n⫺1 17 $1,310,720 18 Yes 19 $51,201; $46,080.90; $41,472.81; $37,325.53; $33,592.98 20 1st bounce: 9 ft, 5th bounce: 2.8 ft 21 $39,871.04
28
Chapter 6: Sequences and Series
6.6 Geometric Series (page 235) 1 510 2 ⫺510 364 3 81 4 ⫺6,825 5,461 5 64 6 5, 25, 125; 12,207,030 7 21, 63, 189; 5,580,120 8 4,096, ⫺16,384, 65,536; 839,680 45 135 405 35,145 , , ; 9 4 16 64 1,024 5 25 125 278,835 , ; 10 ⫺ , 2 16 8 256 1,093 11 9, 3, 1; 81 12 5, 30, 180; 7,775 7
13
兺 7(3i) ⫽ 22,960
i⫽0
14 15 16 17 18 19 20
19,682 243 31 0.333333 172.479963089 in. 9 $37,044,180.04 $4,682.89
6.7 Infinite Series (page 240) 320 1 3 2 No finite sum 3 No finite sum 2 4 3 5 10 25 6 3 7 No finite sum 8 No finite sum
9 10 11 12 13 14 15 16 17 18 19 20 21
8 17 9 340 2 3 7 9 31 99 52 99 41 333 634 999 17 3 17 11 11 45 86 165 325 in.
Chapter Review (pages 244–245) 1 Arithmetic; d ⫽ 7; 25, 32, 39 2 Geometric; r ⫽ ⫺0.5; 9, ⫺4.5, 2.25 64 2 3 Geometric; r ⫽ ; 48, 32, 3 3 4 Neither 5 Arithmetic; d ⫽ ⫺9; 38, 29, 20 1 1 1 1 , 6 Geometric; r ⫽ ; , 2 9 18 36 7 Arithmetic; d ⫽ 11; 57, 68, 79 3 162 243 729 ,⫺ , 8 Geometric; r ⫽ ⫺ ; 2 7 7 14
9 10 11 12 13 14
Geometric; r ⫽ 兹2; 4兹3, 4兹6, 8兹3 Neither (4) an ⫽ 11 ⫹ 6n (2) a1 ⫽ 9 d ⫽ 4 (3) 616 (2) an ⫽ 25 ⫹ 6(n ⫺ 1)
15
(1)
16
(2) 30
17
(2) ⫺
5
18 19 20 21 22 23 24 25 26 27 28
29
30
兺 (11 ⫹ 8n) n⫽1
1 3 (4) 3兹3 (3) 0.0512 1 (2) ⫺ 2 (3) 4 ⫹ 5 ⫹
25 125 ⫹ ⫹ ⭈⭈⭈ 4 16
2,401 512 14 (4) 37 9 12 3 ,⭈ ⭈ ⭈ (2) , 2, , 2 4 5 a an ⫽ 150 ⫹ 10n; a52 ⫽ $660 b Yes, she will have $550 in the bank. 4,960 grains 92 a 11 ⫹ 17 ⫹ 23 ⫹ 29 ⫹ 35 ⫹ 41 ⫹ 47 ⫹ 53 ⫹ 59 ⫹ 65 b 380 c 20% a an ⫽ 48(0.6)n⫺1 b a3 ⫽ 17.28 ft c 116.64 ft 156 (2)
Chapter Review
29
CHAPTER
7
Exponential Functions
7.1 Review of Exponents (pages 250–251) 1 5 2 1 3 7.5a⫺2b5c2 8 4 ⫺ 9 5 8 2s 13 6 3t 4 7 ⫺3 8 1 1 9 4 1 10 ⫺ 4 10 11 c 7d 8 1 12 ⫺ 27 1 13 27 14 4.5r 7 15 ⫺0.07x5z14 2 16 3 17 12 18 4 1 19 ⫺ 9 2 20 27 3 21 4兹x 3 or 4冢兹x冣 5 5 4 22 兹(3xy)4 or 冢兹3xy冣 3 3 2 23 ⫺兹642 or ⫺冢兹64冣
30
Chapter 7: Exponential Functions
24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
1
兹93
or
1
3
兹(⫺8)4 5
1
冢兹9冣3 or
1
冢兹⫺8冣4 3
x6 1 2 ⫺1 ⫺2 2 a 5 b 5 or (ab 2)⫺5 3 2 3 12533 (de)2 2 ⫺(cd)⫺3 1 (3) 16 (3) ⫺1 (4) 4 1 (4) 10 9 (1) 46 1 (4) 4 81 (2) 2 4 (4) 3 (2) 5 (1) 3x⫹1
7.2 Exponential Functions and Their Graphs (pages 256–258) 1 (1) I and II 2 (4) They will never intersect. 3 (3) 1 and 2 4 (2) II only 5 (4) x2 ⫹ y2 ⫽ 5 6 (4) They are all in the domain.
7 8 9 10 11
13
(2) ⫺2 1 (3) 25 (3) 3 (2) y-axis a x ⫺2 y
0
1
2
1 4
1
4
16
1 16
b, c
y 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
y
B
–4
12
⫺1
a, b
16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
A
–4
14
–2 –1
2
d y ⫽ 4⫺x a
–2 –1
c x ⫽ ⫺1 a, b 8 7 6 5 4 3 2 1
–4
–2 –1
y 9 8 6
15
5
2
x
y
4
y⫽
2
4
B
c y ⫽ ⫺2x a, b
3
x
A
–2 –3 –4 –5 –6 –7 –8
7
4
y
x
4
2
x
( 12 )
8
1
–4 –3 –2 –1 –1
1
2
3
4
6
x
4
b f(⫺1.6) ⫽ 0.172427286 c
2
–4 –3 –2 –1
y⫽
( 12 )
x
1
2
3
4
x
– 3 –2 –4
f(x) ⫽ 1.6; x ⫽ 0.42781574
c x-intercept: none; y-intercept: 1 d x-intercept: ⫺1.584963; y-intercept: ⫺2
7.2 Exponential Functions and Their Graphs
31
16
f(x) ⫽
冢 23 冣 ; g(x) ⫽ 冢 32 冣 ; in functions of x
x
19
a, b 6
the form y ⫽ bx, the graph will increase from left to right if b ⬎ 1 and decrease from left to right if 0 ⬍ b ⬍ 1. 17
y
y⫽
3 x 4
()
2 1
y⫽
16 1 x 3
()
y⫽
1 x 4
()
–4 –3 –2 –1 –1
14
c
12
20
10
y⫽
y⫽
()
a, b
4
x
B
9 8
6
7 6 5
y ⫽ 2x
4
–2
2
4
x
3 2
All three graphs decrease from left to right. 1 x For x ⬍ 0, the graph of y ⫽ decreases 4 1 x , which decreases faster faster than y ⫽ 3 1 x . than 2
冢 冣
冢 冣
冢 冣
a, b
y 8 6 4
y ⫽ 3x
2
–4
3
y
2
18
2
x
4
–4
1
冢 43 冣
8 1 x 2
B
4 3
y 18
5
–2
2
4
x
–2 –4 –6
B
1
–4 –3 –2 –1
1
2
3
4
x
c y ⫽ 2x ⫹ 3 21 f(x) ⫽ 2⫺x, g(x) ⫽ 3⫺x, h(x) ⫽ 3x, j(x) ⫽ 2x, k(x) ⫽ ⫺3x 22 a x ⫽ 0 b g(x); 4 raised to any positive power is greater than 2 raised to the same power. c f(x); 2 raised to any negative power is greater than 4 raised to the same power. 23 a x ⫽ 0 b h(x); 2 raised to any positive power is 1 greater than raised to the same 2 power. c j(x); a negative exponent raises the reciprocal of the base to the positive power. d j(x) ⫽ 2⫺x e ry-axis
–8
c
y ⫽ ⫺3x
7.3 Solving Equations Involving Exponents (pages 262–263) 5 1 x ⫽ 5兹5 2 y ⫽ 625 1 3 z⫽ 27
32
Chapter 7: Exponential Functions
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 1,296 a ⫽ 243 1 b⫽ 125 1 r⫽ 81 1 s⫽⫺ 32 g ⫽ 26 7 w⫽⫺ 8 145 r⫽ 36 v ⫽ 17 x ⫽ 1,042 z⫽2 1 m⫽ 2 2 y⫽ 3 2 (1) y 3 ⫽ ⫺4 33 (4) 16 3 (4) (x ⫹ 1)2 ⫽ 4 (2) irrational a⫽
7.4 Solving Exponential Equations (pages 267–268) 1 x⫽3 2 z⫽5 3 r⫽1 4 x ⫽ ⫺3 5 x ⫽ ⫺2 6 x ⫽ ⫺1 7 s⫽6 8 x ⫽ ⫺2 3 9 x⫽⫺ 4 10 x ⫽ ⫺5 3 11 n ⫽ 2 12 p ⫽ ⫺3 13 x ⫽ 5 14 x ⫽ ⫺1, 2 2 15 x ⫽ ⫺ 5
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
(3) 3 (4) {0, 3} 8 (4) 7 3 (2) 2 1 (1) ⫺ 4 1 r⫽ 2 w ⫽ 2.16 p⫽3 x ⫽ 9.17 v ⫽ ⫺2 w ⫽ ⫺2.37 x ⫽ ⫺2 r ⫽ 1.89 x ⫽ ⫺1.26 m ⫽ 5.17
7.5 Applications of Exponential Equations (pages 273–275) 1 (4) P(t) ⫽ 15,000(1.02)t 2 (4) P(t) ⫽ 15,000(0.98)t 3 (1) P(t) ⫽ 15,000e0.02t 4 (2) P(t) ⫽ 15,000e⫺0.02t 5 (3) y ⫽ 2x 6 (2) 2012 7 (3) P(t) ⫽ 3t 8 (4) $168.07 9 a $6,381.41 b approximately 14.2 years 10 a $6,420.13 b approximately 13.86 years 11 a $25,809.23 b approximately 16.31 years 12 a 2 weeks b 25 13 a 20,000 people b 15% per year 14 $11,034.39 15 a 20 b Each day, there are 90% of the fish left in the tank. The fish are dying at a rate of 10% per day.
7.5 Applications of Exponential Equations
33
P(x) ⫽ 31,443,790(1.346)x 251,681,000 people In approximately 8.6 decades or 1946 The population was 452,190,734 fewer than predicted by the formula. 17 a 88.6025 micrograms b approximately 2,377 years c 5,728 years 18 i ⫽ C, ii ⫽ A, iii ⫽ B, iv ⫽ D, v ⫽ E 19 Answers will vary. y 20 16
a b c d
y ⫽ ex
24 22
y⫽
3x
10 11 12 13 14 15 16 17 18 19 20
20
21 22 23 24 25
18 16 14 12
y ⫽ 2x
10 8 6 4
26
2
–8 –6 –4 –2
21
2
4
6
8
x
a 9 days b At 9.96 days, the two options are worth the same amount. So, on the 10th day the second option is worth more. The second option then continues to be the better choice.
Chapter Review (pages 276–278) 1 (2) ⫺2 2 (2) ⫺12x7 3 (4) 48 4 (3) 2ab13 5 (3) 15 6 (1) 4x⫺2 7 (3) 28 3 8 (2) 4 4 9 (3) (x 2 ⫹ 3x)3 ⫽ 16
34
Chapter 7: Exponential Functions
27
28 29 30
31
(4) (3) (3) (4) (2) (4) (3)
6 9 3 {3, ⫺1} 3兹x 8b {⫺3, ⫺1} 5 (4) 4 (3) the amount of a person’s salary if she gets a $1,500 raise every year (2) $952 8 (4) 7 (1) N(t) ⫽ 5(1.08)t (2) 41 (2) y ⫽ c(1.05)x (2) $7.50 a C(t) ⫽ 10(1.079)t b 521 c week 61 a 720,500 is the population of Halycon in January 2007; 1.022 indicates the population is increasing by 2.2% per year. b 2022 a y ⫽ 39,389(0.82)x b $32,299 c in year 10: 2019 a $4,073.58 b 9.4 years a $11,472 b 25.8 years a 47,827 is the initial population in 2008; e⫺0.1779 indicates the population is decreasing at an annual rate of 17.79% compounded continuously. b 6.5 years a 1.55 billion b 8.006% c 7.8155 billion d 2041
CHAPTER
Logarithmic Functions
8.1 Inverse of an Exponential Function
18
4 g(x)
2 1
c
g(x) ⫽ log3 x
3
4
5
6
7
8
5
10
15
20
25
x
8.2 Logarithmic Form of an Exponential Equation
f(x)
2
O
b (1, 0)
7
–2 –1 O 1 –1 –2 –3
y = log5 x
–4
8
3
y
–2
9
5
a 4 2
(pages 282–283) 1 (4) y ⫽ log3 x 2 (2) I and IV 3 (3) f(x) ⫽ logb x 4 (4) 0 5 (2) y ⫽ 2x 6 (1) reflection in the y-axis 7 (4) 10 ⬍ x ⬍ 100 8 (4) g(x) ⫽ log3 x 9 (4) It will not intersect the y-axis. 10 (1) y ⫽ 2x 11 y ⫽ log6 x 12 f ⫺1(x) ⫽ 4x 13 y ⫽ log3 x 14 f ⫺1(x) ⫽ 10x 15 y ⫽ 2x 16 f ⫺1(x) ⫽ log10 x y 17 a, b
6
8
9
x
(pages 285–286) 1 5 ⫽ log3 243 1 ⫽ log 16 4 2 2 3 2 ⫽ log6 36 1 4 ⫺2 ⫽ log 2 4 25 5 2 ⫽ log 56 36 6 ⫺2 ⫽ log10 0.01 1 ⫽ log 49 7 7 2 8 a ⫽ logb c 9 26 ⫽ 64 10 53 ⫽ 125 11 10⫺3 ⫽ 0.001 1 12 42 ⫽ 2 13 112 ⫽ 121 1 14 2⫺2 ⫽ 4 15 x ⫽ 4 1 16 x ⫽ ⫺ 2 3 17 x ⫽ 2 8.2 Logarithmic Form of an Exponential Equation
35
18 x ⫽ 4 19 x ⫽ ⫺2 20 x ⫽ 3 21 x ⫽ 64 1 22 x ⫽ 81 23 x ⫽ 5 24
x ⫽ 0.001 or
25
x ⫽ 27 1 x⫽ 7 x⫽2 x⫽9 x ⫽ 10 x⫽2 x⫽6 x⫽2 (4) 4 1 (3) 4 (4) ⫺4 (4) 7 (3) 36 b (3) 2 (4) b y 1 (3) 125
26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
11 12 13 14 15 16 1 1,000
18 19 20
8.4 Common and Natural Logarithms
8.3 Logarithmic Relationships (pages 290–291) 1 (2) m ⫹ 2n 2
(4) log 10 a ⫹
3 4 5
7
(2) 5.6 (2) 8.48 (1) 1.935 1 (2) ⫺ 2 (2) 2
8
(4) 2log c 8 ⫹
9
(3)
6
10
36
17
1 (p ⫺ q) 2 (2) product and power rules (1) 216 (4) 5 (3) {6} (4) {12} p3 r x⫽ 兹 q 3 兹qr x⫽ 2 p logb x ⫽ 2logb p ⫹ 3logb q ⫹ logb r 1 1 logb x ⫽ logb p ⫹ logb r ⫺ 3logb q 2 2 (4)
1 log 10 b ⫺ log 10 c 2
1 log c 5 ⫺ log c 21 3
a2
b 3兹d (2) 2
Chapter 8: Logarithmic Functions
(pages 294–295) 1 (1) 3,781.8126 2 (3) 8.6365 3 (2) 3 ⫹ a 4 (3) 5 5 (3) 6.752270376 6 (4) 2w ⫹ 1 7 (3) b ⫹ 1 log n 8 (1) 1 ⫺ 2 ln n 9 (1) 1 ⫺ 2 10 (2) 5.3804 11 a 100.5171959 ⫽ 3.29 b e2.0347056 ⫽ 7.65 c 10b ⫽ a d ed ⫽ c 12 a log 13,182.56739 ⫽ 4.12 b ln 906.870869 ⫽ 6.81 c log y ⫽ x d ln z ⫽ x 13 a True; product rule b False; log A ⫹ log B ⫽ log AB c True; power rule 1 3 d False; ln 兹e ⫽ 3 e True; log 1 ⫽ ln 1 ⫽ 0 14 a 6 b ⫺4 c 5 d ⫺3
8.5 Exponential Equations (pages 299–301) 1 2.26 2 2.13 3 5.56 4 2.14 5 1.29 6 ⫺25.04 7 3.28 8 5.31 9 0.585 10 ⫺12.9801 11 2.8966 12 33.9709 13 a Yes b 2012 c 5.0 ounces 14 9.64 hours 15 9.64 hours; yes 16 a V(t) ⫽ 32,640(0.82)t b $26,765 c 2017 17 2013 18 2011 19 a $14.91 b N(t) ⫽ 14.91e 0.054t c 2010 20 10.9%; this is not a plausible interest rate. Plausible interest rates will vary. 21 2012 22 a W(t) ⫽ 17,420(0.97)t b 22.8 hours c 5,644.2 gallons 23 a 0.995 ⫽ e ⫺0.00012101x b x ⬇ 41.4; the painting is a forgery since it is only about 41 years old. 24 9.0 years 25 a 5.83 years b $73,637.65 c Deidre: 9.7 years or early 2018; Alan: 10.7 years or early 2019
8.6 Logarithmic Equations (page 305) 1 x ⫽ 17.78 2 x ⫽ 7.39 3 x ⫽ 4.57 4 x⫽2 5 x ⫽ 134.48
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
x ⫽ 3.21 x ⫽ 547.72 x ⫽ 148.41 x⫽6 x⫽3 1 x⫽ 2 x⫽5 x⫽2 x⫽3 x⫽1 x⫽2 x ⫽ ⫺4, 5 x ⫽ 10 x⫽e x ⫽ 3, 6
Chapter Review (pages 305–307) 1 1 (4) log a ⫹ log b ⫺ log c ⫺ log d 2 2 (1) y ⫽ x 3 (4) IV and I 1 1 x⫹ y 4 (3) 2 2 5 (1) 81 6 (2) x ⫽ log a y 7 (2) a ⫹ 2 8 (1) 64 9 (4) The range is { y : y 僆 ⺢} 10 (2) y ⫽ 4x 11 (1) 6logb 6 12 (4) 2e4x⫺1 13 (3) 10 14 (3) 2a ⫹ b 15 (2) logb 7 16 (4) ⫺4 17 (2) 2a ⫹ 1 1 18 (2) 512 1 19 (1) 5 20 (4) ⫺4 21 (2) 0 22 (3) 100 ⬍ x ⬍ 1,000 23 (2) log10 0 ⫽ 1 24 (2) 1.271 25 (2) h ⫹ 1 26 x ⫽ 3.25 27 x ⫽ 0.456 28 x ⫽ 2
Chapter Review
37
29 30 31 32 33 34 35
1 2x 3 log 兹b x 2.585 Domain: x 0; Range: All real numbers x2 x7 x6
36 37 38 39 40
0 x 2.74 9.6 years a 659 b 6.2 minutes 3.7 hours
CHAPTER
9 9.1 Trigonometry of the Right Triangle (pages 312–313) 1 6.4 cm 2 52.6° 3 14.7 cm 4 26.1 cm 5 16.2° 6 33.7 cm 7 64.6° 8 8.8 cm 9 28.5° 10 11.4 cm 11 32.9° 12 29.5 cm 13 77 in.
Trigonometric Functions 14
68.2°
150 ft
x 60 ft
15
7.3 ft
x
36° x
96 in.
53°
38
Chapter 9: Trigonometric Functions
10 ft
16
9.3 The Unit Circle
11.8 ft
(pages 320–322) 1 Quadrant Sin u 10 ft
x
58°
17
2
70°
15 ft
x
18
a b
16 ft
23°
17.5 ft x
4
a b
5 6 7 8 9 10
(2) (4) (1) (4) (2) (3)
11
(3)
12
a
41 ft
9.2 Angles as Rotations (page 316) 1 III 2 III 3 IV 4 I 5 II 6 IV 7 III 8 II 9 I 10 II 11 IV 12 I 13 II 14 I 15 IV 16 (2) 68° 17 (2) 34° 18 (1) 104°
b 13 14 15
I
II
III
IV
兹3
a b
3
Cos u
a b 1 2
2 1 2 兹2 2 兹2 2 III Negative; y-values are negative in Quadrant III. II 303° I IV II 210° 3 sin 120° 兹 2 兹5 3 2 3 AB OB
兹2 2
9.4 The Tangent Function (pages 325–326) 1 Quadrant sin u
2
cos u
tan u
I
II
III
IV
(2) II 9.4 The Tangent Function
39
3 4 5 6 7 8 9 10 11
(3) (1) (3) (4) (1) (4) (2) (3) (4)
12
(1)
13 14 15
(3) (3) (4)
III I I and III 313° sin u 0 IV 1 tan 240° 兹3 Sin u may be positive or negative. 12 5 PR III Sin u can equal cos u only in Quadrant I of the unit circle.
5 6 7 8 9 10 11 12 13 14 15
9.5 Special Angles and Reference Angles
16
(pages 331–332) 1 In the following table, U = undefined. u
0°
30°
45°
60°
sin u
0
1 2
兹2
兹3
2
2
cos u
1
兹3
兹2
2
0
兹3
tan u
3
1
0
1
0
2
1 2
0
1
0
1
1
兹3
U
2
3 4
40
90° 180° 270° 360°
0
U
Angle
Quadrant
Reference Angle Formula
210°
III
210° 180°
30°
330°
IV
360° 330°
30°
135°
II
180° 135°
45°
300°
IV
360° 300°
60°
120°
II
180° 120°
60°
240°
III
240° 180°
60°
225°
III
225° 180°
45°
0
Reference Angle
320° 30°
Chapter 9: Trigonometric Functions
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
80° 320° 240° 兹3 2 1 2 2 兹 2 兹3 兹2 2 1 13 12 a 1 b 兹2 c 2 a sin 40° b tan 50° c cos 20° d tan 40° e sin 40° (2) 90° (3) 1 1 (2) 2 (3) 240° (1) cos 70° (3) reflection in the x-axis (3) tan 240° (3) sin 30° 5 (4) 4 (4) 4 (3) tan 135° (2) 48° (3) 150° (2) sin x (4) sin 180° (4) Sin u cos u only in Quadrant I.
9.6 Reciprocal Trigonometric Functions (pages 334–335) 1 In the following table, U = undefined. u
0°
30°
45°
60°
90°
180°
270°
360°
sec u
1
2兹3 3
兹2
2
U
1
U
1
csc u
U
2
兹2
2兹3 3
1
U
1
U
cot u
U
兹3
1
兹3
0
U
0
U
2
3
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
3
a 90°, 270° b 0°, 180°, 360° c 0°, 90°, 180°, 270°, 360° a II b IV c IV d II e I 2 兹2 0 兹3 3 2 1 1 1.179 1.556 1.428 0.700 1.743 (4) varies depending upon the value of u 1 (1) sin 45° 4兹3 (4) 3 (2) 2 (3) cot 135°
Chapter Review (pages 336–338) 1 (2) II 2 (2) 32° 3 (1) I 4 (4) tan 30° 5 (3) 0 6 (4) 兹3 7 7 (1) 25 8 (3) 330° 9 (2) decreases from 1 to 0 10 (4) 225° 11 (4) 315 3 12 (4) 4 13 (2) cos 90° 14 (4) 45° and 225° 兹3 15 (2) 3 16 (4) II or IV 17 (2) 300° 18 (1) 150° 19 (4) 0 20 (2) 450° 21 (2) 23 22 (3) 28° 23 (1) 1 24 (1) 1 12 25 (4) 13 26 (2) 150° 27 (4) 9 28 (3) 680° 29 (3) csc 兹2 30 (2) 120° and 240° 31 (3) 210° 32 (2) 34° 33 (1) 2兹2
Chapter Review
41
CHAPTER
10
More Trigonometric Functions
10.1 Radian Measure (pages 343–344) 2p 1 3 3p 2 2 5p 3 ⫺ 18 7p 4 4 3p 5 ⫺ 4 4p 6 9 11p 7 6 8 ⫺p p 9 ⫺ 4 4p 10 3 11 270° 12 120° 13 225° 14 ⫺90° 15 150° 16 ⫺315° 17 36° 18 300° 19 ⫺30° 20 180° 21 2 m 22 6 ft 23 1 degree, 1 radian, 3 radians, 1 revolution
42
Chapter 10: More Trigonometric Functions
24
25 26 27 28 29 30
105° 7p b 12 6p ⬇ 18.85 in. (2) II (3) 25 (3) III (4) 300° (4) 2p a
10.2 Trigonometric Functions with Radian Measure (pages 348–349) p 1 ⫺sin 3 p 2 ⫺cos 4 p 3 ⫺cot 4 p 4 ⫺sec 3 p 5 ⫺csc 6 p 6 ⫺cos 6 p 7 ⫺tan 4 p 8 ⫺sin 6 9 1 10 ⫺1 11 ⫺兹3 12 2
13 14
15 16 17 18 19 20 21 22 23 24 25
2
兹7 , cos u ⫽ 3 , tan u ⫽ 兹7 , 13 sin u ⫽ 4 4 3 3 7 4兹7 , cot u ⫽ 兹 csc u ⫽ 7 7 2 14 (3) sec u 3 15 (2) ⫺ 5 16 (1) sec2 u 17 (2) ⫺0.75 18 (3) k2 3 19 (3) 4 1 20 (4) 2 sin u cos2 u
2 a ⫺兹 2 兹3 b ⫺ 2 c 0 ⫺3 (3) 0 (3) 3 1 (3) 2 3p (2) 4 3p (3) sin 4 (3) 0 11p (4) 6 5p (2) 6 (1) 1 (4) sin p
10.4 Range and Domain of Trigonometric Functions
10.3 Basic and Pythagorean Trigonometric Identities (pages 354–355) 1 sin2 u 2 1, (sin u ⫽ 0) 3 1, (sin u ⫽ 0, cos u ⫽ 0) 4 tan2 u 5 sec u, (sin u ⫽ 0) 6 cot u 7 sec u, (sin u ⫽ 0) 8 csc2 u 9 cos2 u, (cos u ⫽ 0) 10 sin2 u, (sin u ⫽ 0, cos u ⫽ 0) 12 12 13 11 sin u ⫽ , tan u ⫽ ⫺ , csc u ⫽ , 13 5 12 13 5 sec u ⫽ ⫺ , cot u ⫽ ⫺ 5 12 24 7 25 12 cos u ⫽ ⫺ , tan u ⫽ , csc u ⫽ ⫺ , 25 24 7 25 24 sec u ⫽ ⫺ , cot u ⫽ 24 7
(page 357) 3p 1 3p ⫽ 0 so sec ⫽ , which is 1 cos 2 2 0 undefined. 1 2 sin 0° ⫽ 0 so csc 90° ⫽ , which is 0 undefined. 1 3 sin p ⫽ 0 so cos p ⫽ , which is undefined. 0 1 4 cos 90° ⫽ 0 so sec 90° ⫽ , which is 0 undefined. 兹3 5 3 6 ⫺1 7 ⫺2 8 Undefined; sin p ⫽ 0 兹3 9 ⫺ 2 10 Undefined; tan 2p ⫽ 0 兹3 11 ⫺ 2 兹3 12 ⫺ 3 13 ⫺2 14 ⫺兹2 15 ⫺兹3 16 1 1 17 2
10.4 Range and Domain of Trigonometric Functions
43
18 19 20
2 3 ⫺ 兹 3 p (2) 2 (3) The domain of the tangent function is all real numbers.
10.5 Inverse Trigonometric Functions (pages 360–361) 1 ⫺45° 2 90° 3 ⫺45° 4 41.2° 5 ⫺60° 6 41.4° 7 22.6° 8 ⫺60° 9 0 p 10 ⫺ 6 5p 11 6 p 12 ⫺ 6 p 13 ⫺ 3 1 14 (1) ⫺ p 4 15 (3) 300° 16 (1) 0° p 17 (2) ⫺ 3
(3) arccos ⫺
19
(4)
20
兹2 (1) arcsin ⫺ 2
5p 3
冢
9 10 11 12 13 14 15 16 17 18 19 20 21
23 24 25 26
冣
10.6 Cofunctions (page 363) 1 cos 25° p 2 ⫺tan 6 3 ⫺sec 15°
44
8
22
冢 12 冣 ⫽ u
18
4 5 6 7
Chapter 10: More Trigonometric Functions
⫺cot 20° csc 43° ⫺cos 16° tan 27° p sin 15 ⫺cot 19° ⫺sec 33° u ⫽ 22 u ⫽ 11 u ⫽ 35 u⫽6 u ⫽ 10 u ⫽ 17 u⫽3 u ⫽ 18 u ⫽ 24 u ⫽ 12 (3) 35 7p (1) csc 30 兹3 (2) 3 (3) 14 (1) 34° (2) ⫺csc 12°
Chapter Review (pages 364–366) 2p 1 (2) 3 3 2 (3) 2 兹3 3 (2) 2 4 (2) 2 5 (1) 1 5p 6 (1) csc 3 7p 7 (3) 6 8 (3) sec2 x p 9 (2) 4 5p 10 (4) 6 11 (4) 13.5 12 (4) 315 1 13 (2) 4
14
(4)
15 16
(1) (4)
17
(3)
18
(1)
19
(1)
20
(4)
21 22 23
(1) (1) (4)
24
(3)
12 5 10 csc2 a 兹3 3 sin u p ⫺ 6 4p 3 ⫺150° 1 20 centimeters 5p 6
25
(1) ⫺
26 27 28 29 30 31
(4) (1) (1) (2) (4) (1)
32
(1)
33 34
(1) (1)
35
(2)
4 5
0 sin x and cos x 1 11 11 1 radian 5p 6 57° 0 p ⫺ 4
CHAPTER
Graphs of Trigonometric Functions 11.1 Graphs of the Sine and Cosine Functions (pages 371–372) 1
11 2
2 1
–2p 3p –
2
2
–p
–p
y = cos x
p 2
2
p
3p 2
2p
–1 y = sin x
1
–2 –2p 3p – 2
–p
–p
p 2
2
–1 –2
p
3p 2
2p
a 1 b ⫺1 c
冦⫺ 3p2 , ⫺ p2 , p2 , 3p2 冧
a 1 b ⫺1 c {⫺2p, ⫺p, 0, p, 2p}
11.1 Graphs of the Sine and Cosine Functions
45
3
2 y = sin x
1
p
p 2
2p
3p 2
6
b
1 2
c
x⫽⫺
2
–1
y = sin x
1
y = cos x
p ,0 2
–2
冦 冤 冤 冤 冤
冧 冥
b c d e
–1
a b c d e f
冥
冥
–p
y = sin x 1
–p
p
p 2
2
–1 –2
p 2 b The two graphs are identical. x⫽
a
2 1
–p
–p
p
p 2
2
–1 –2
5
2 y = cos x
4p
y = cos x
2 2 4 Answers will vary. Example: [0, 2p] Answers will vary. Example: [2p, 4p] The length of the interval is 2p.
11.2 Amplitude, Period, and Phase Shift
2 y = cos x
3p
–2
冥
4
2p
p
p 5p , or {45°, 225°} 4 4 3p p, 2 p 0, 2 3p , 2p 2 p ,p 2
a
(pages 381–383) 1 (3) Cos x increases from 0 to 1. 2 (3) 3 3 (3) line y ⫽ x 4 (3) 3 5 (4) p p 6 (3) units to the left 3 7 (2) 2 8 (3) coincide 9 (1) y ⫽ sin (⫺x) 10 (4) y ⫽ ⫺2cos 2x 11 (3) ⫺2 ⱕ y ⱕ 4 12 (1) p 13 (4) 4 14 (2) y ⫽ sin (⫺x) y 15 a y = –3cos 2x
1
3 2
–p 2
p 2
y = sin x
–1 –2
a
1 2
1 O –1
p 2
Chapter 11: Graphs of Trigonometric Functions
3p 2
–2 –3
b {0, p, 2p}
46
p
y = 3 sin x 2
2p
x
16
a, b
The two graphs have the same amplitude (1), frequency (1), and period (2p). However, y ⫽ cos (x ⫹ p) is y ⫽ cos x shifted p units to the left, while y ⫽ cos x ⫹ p is y ⫽ cos x shifted p units up. a, b 2
c
y y = 3 sin 1 x 2
3 2 1 O
p 2
–1
3p 2
p
–2
20
x
2p
y = sin (x – p)
1 –2p 3p –
y = 2 cos x
–p
2
–p 2
–3
–1
p 2
p
3p 2
2p
–2 –3
17
c a
p
–4 –5
y
2 1 O p 2
–
p 3
–
p 6
p 6
p 3
x
p 2
21
–1 y = 3 sin 2x
The two graphs have the same amplitude (1), frequency (1), and period (2p). However, y ⫽ sin (x ⫺ p) is y ⫽ sin x shifted p units to the right, while y ⫽ sin x ⫺ p is y ⫽ sin x shifted p units down. a, b 4
c
y = –2cos 2x
–
y = sin x + 2
3
–2
2
y = sin x – p
2 1
b {⫺0.46, 1.11} 18 a, b y = 2sin 2x
–2p y
–2 2
O p 2
p 2
p
x
–1
The two graphs have the same amplitude (1), frequency (1), and period (2p). However, y ⫽ sin (x ⫹ 2) is y ⫽ sin x shifted 2 units to the left, while y ⫽ sin x ⫹ 2 is y ⫽ sin x shifted 2 units up. a, b 2
22
y = cos (x – 1)
1
–2 –2p 3p –
c d 19
2
4
冦
p p ⫺ , 2 2
冧
a, b
–p 2
–1
p 2
p
3p 2
2p
y = cos x – 1
–3 4
c
y = cos x + p
3 2 1 –p y = cos (x + p)
–p
–2
5
–2p
y = sin (x + 2)
c
y = 3 cos (x + p)
1
–
2p
p –1
2
–p
–p
p
2p
The two graphs have the same amplitude (1), frequency (1), and period (2p). However, y ⫽ cos (x ⫺ 1) is y ⫽ cos x shifted 1 unit to the right, while y ⫽ cos x ⫺ 1 is y ⫽ cos x shifted 1 unit down.
–1 –2
11.2 Amplitude, Period, and Phase Shift
47
11.3 Writing the Equation of a Sine or Cosine Graph
5
4 2
5 4 3 2 1
1
p
p 2
冢冢
冣冣
6
(2) y ⫽ 3sin
7
(3) y ⫽
8 9
1
10 p
p 2
2p
3p 2
–1
11 12
–2
13
y ⫽ 4sin
冢 冢
p 5 x⫹ 5 2
冣冣
5
6
1
y ⫽ 2sin
2
3
4
5
6
7
8
9 10
冢 冣
j(x) ⫽ ⫺1.5cos (x ⫹ p) p 15 g(x) ⫽ 1.5sin x ⫺ 2 p 16 h(x) ⫽ 1.5cos x ⫺ 2 17 f(x) ⫽ 1.5sin (x ⫹ p)
冣 冣
In 18–22, answers may vary. 18 y ⫽ ⫺6sin 2x p x 19 y ⫽ 5cos 2 p 20 y ⫽ 2sin 4 x ⫹ 6
冢冢
3 2
21
1 2p
1 sin 2x 2 amplitude ⫽ 3.5, period ⫽ p, frequency ⫽ 2 2p , frequency ⫽ 3 amplitude ⫽ 2, period ⫽ 3 p amplitude ⫽ 6, period ⫽ 8, frequency ⫽ 4 y ⫽ ⫺3.5sin 2x y ⫽ 2sin 3x p y ⫽ 6sin x 4
冢 冢
冢 12 (x ⫺ 3)冣 p
1 x 2
14
5 4 3 2 1
3p
4p
–1 –2 –3
48
4
–6
2
4
3
–4
p y ⫽ sin 3 x ⫺ 3
–1 –2 –3 –4 –5
2
–2
–1 –2 –3 –4 –5
3
冢 2p3 (x ⫺ 1)冣
6
(pages 386–389) 1 y ⫽ 4sin 6x
2
y ⫽ 5sin
Chapter 11: Graphs of Trigonometric Functions
22
冣冣
冢 冢 p6 冣冣 p y ⫽ 1.5cos 3x, y ⫽ ⫺1.5sin 冢3冢x ⫺ 冣冣 6 y ⫽ ⫺1.5sin 3x, y ⫽ 1.5cos 3 x ⫹
11.4 Graph of the Tangent Function
5
y = tan x
(pages 391–392) 1 No, the tangent function does not have an amplitude because its range is [⫺⬁, ⬁]. 5 2 y = tan x
p 2
y = 2 tan x
a p b Both curves have the same basic shape and have the same domain and range. However, the graph of y ⫽ 2tan x is stretched vertically, so all of the y-values are double those of y ⫽ tan x.
p 2
2
–5
6
a
1
b
冢
p
–5
y = cos x –p
5
5 y = tan 1 x
冥
p ⫺ ,0 2
2
3
5
p
y = tan x
2p
y = 1 tan x 2
y = sin x –5 –2p
–p
p
2p
Both curves are the same basic shape and have the same range. However, the period 1 of y ⫽ tan x is p and its domain is 2 p x:x⫽ ⫹ np for n an integer , while the 2 1 period of y ⫽ tan x is 2p and its domain 2 is {x : x ⫽ p ⫹ 2np for n an integer}.
–5
a 2 b 4 c 5 d 4
冦
冤⫺2p, ⫺ 3p2 冣, 冢⫺ p2 , p2 冣, 冢 3p2 , 2p冥
5
7
y = tan 2x
a p b {x : x ⫽ np for n an integer}
y = tan x p 2
c
p
8
–5
p 2 b Both curves are the same basic shape and have the same range. However, the p period of y ⫽ tan 2x is and its domain 2 p np ⫹ for n an integer , is x : x ⫽ 4 2 while the period of y ⫽ tan x is p and its domain is p x:x⫽ ⫹ np for n an integer . 2 a
冦
冦
冧
冧
冧
The function y ⫽ tan x was shifted
units to the left. 3p p p ,x⫽⫺ ,x⫽ ,x⫽ a x⫽⫺ 2 2 2 7p 5p 3p ,x⫽⫺ ,x⫽⫺ , b x⫽⫺ 4 4 4 p p 3p x⫽⫺ ,x⫽ ,x⫽ , 4 4 4 5p 7p x⫽ ,x⫽ 4 4 p p 3p ,x⫽⫺ ,x⫽ ,x⫽ c x⫽⫺ 2 2 2 d x ⫽ ⫺p, x ⫽ p p p 3p ,x⫽⫺ ,x⫽ ,x⫽ e x⫽⫺ 2 2 2
p 2
3p 2
3p 2 3p 2
11.4 Graph of the Tangent Function
49
9
3p p p 3p ,x⫽⫺ ,x⫽ ,x⫽ 2 2 2 2 11p 3p 7p ,x⫽⫺ ,x⫽⫺ , b x⫽⫺ 6 2 6 5p p p p x⫽⫺ ,x⫽⫺ ,x⫽⫺ ,x⫽ , 6 2 6 6 p 5p 7p 3p x⫽ ,x⫽ ,x⫽ ,x⫽ , 2 6 6 2 11p x⫽ 6 3p p p 3p ,x⫽⫺ ,x⫽ ,x⫽ c x⫽⫺ 2 2 2 2 3p 3p , x⫽ d x⫽⫺ 2 2 a
x⫽⫺
4
Graph each function using information from its reciprocal. Find the asymptotes where the reciprocal function equals zero. Plot points based on known values of the function’s reciprocal. Pay special attention to maxima, minima, and whether the function is increasing or decreasing. a The reciprocal of y ⫽ 2sec x is 1 y⫽ cos x. Asymptotes are at 2 p ⫹ pn for n an integer. When x⫽ 2 y ⫽ 2sec x is decreasing, the maximum value is ⫺0.5, which occurs at x ⫽ p ⫹ 2pn for n an integer. When y ⫽ 2sec x is increasing, the minimum value is 0.5, which occurs at x ⫽ 2pn for n an integer. b The reciprocal of y ⫽ ⫺csc x is y ⫽ ⫺sin x. Asymptotes are at x ⫽ pn for n an integer. When y ⫽ ⫺csc x is decreasing, the maximum value is ⫺1, which occurs p at x ⫽ ⫹ 2pn for n an integer. When 2 y ⫽ ⫺csc x is increasing, the minimum 3p value is 1, which occurs at x ⫽ ⫹ 2 2pn for n an integer. 1 c The reciprocal of y ⫽ cot x is y ⫽ 2 2tan x. Asymptotes are at x ⫽ pn for n p 1 an integer. cot x ⫽ 0 at x ⫽ ⫹ pn 2 2 1 for n an integer. The graph of y ⫽ cot x 2 decreases from ⬁ to ⫺⬁ for each interval [2pn, (2p ⫹ 1)n] for n an integer.
5
Answers may vary. a y ⫽ tan x, y ⫽ sec x b y ⫽ cot x, y ⫽ csc x c y ⫽ tan x, y ⫽ sec x d y ⫽ cot x, y ⫽ csc x
11.5 Graphs of the Reciprocal Functions (pages 398–399) 1 a y ⫽ tan x b y ⫽ cot x c y ⫽ sec x d y ⫽ csc x p p 3p 3p ,⫺ , , 2 a x⫽⫺ 2 2 2 2 b x ⫽ ⫺2p, ⫺p, 0, p, 2p 7p 5p 3p p p 3p ,⫺ ,⫺ ,⫺ , , , c x⫽⫺ 4 4 4 4 4 4 5p 7p , 4 4 3 a They are undefined when their reciprocal functions, cosine, sine, and tangent, equal zero. b The contangent function equals zero when the tangent function is undefined. The secant and cosecant functions never equal zero because their reciprocal functions are always defined. c Asymptotes occur when the functions are undefined, that is, when their reciprocal functions equal zero.
50
Chapter 11: Graphs of Trigonometric Functions
6
a
16
Function
Interval
冢0, p2 冣
冢 p2 , p冣
y ⫽ sin x
Increasing
Decreasing
Decreasing
Increasing
y ⫽ cos x
Decreasing
Decreasing
Increasing
Increasing
y ⫽ tan x
Increasing
Increasing
Increasing
Increasing
y ⫽ csc x
Decreasing
Increasing
Increasing
Decreasing
y ⫽ sec x
Increasing
Increasing
Decreasing
Decreasing
y ⫽ cot x
Decreasing
Decreasing
Decreasing
Decreasing
7
冢p, 3p2 冣 冢 3p2 , 2p冣
b When a function increases, its reciprocal function decreases. When a function decreases, its reciprocal function increases. p ⫹ np a Period: 2p; Domain: x : x ⫽ 2
冧
冦
for n an integer ; Range: {y : 兩y兩 ⱖ 1} b Period: 2p; Domain: {x : x ⫽ np for n an integer}; Range: {y : 兩y兩 ⱖ 1} 8 Period: p; Domain: {x : x ⫽ np for n an integer}; Range: All real numbers
11.6 Graphs of the Inverse Trigonometric Functions (pages 403–405) 1 (3) y ⫽ arccos x 2 (3) III 3 (3) ⫺75° 4 (1) Unless the domain of y ⫽ sin x is restricted, the reflection is not a function. 2 兹5 5 (2) ⫺ 5 6 (4) I and III 7 (1) y ⫽ arcsin x p 8 (1) ⫺ 6 2 兹 9 (2) 2 1 10 (3) ⫺ ⫽ arccos x 2 11 (4) sin 0 12 (1) ⫺60° 13 (3) 210° 14 (4) y ⫽ arccos x p 15 (1) 2
(1) An inverse will exist if the original p p function has a domain ⫺ ⱕ x ⱕ . 2 2
11.7 Trigonometric Graphs and Real-World Applications (pages 407–409) 1 (4) 4 seconds 2 (2) 15 seconds 3 (2) 262 4 (2) 2 seconds 5 a Lowest in July; highest in January b $58,934 on 746 snow blowers c Highest in June; lowest in December d during March and September e $31,752 (highest monthly profit) p t ⫹ 300; we ex6 a (2) S(t) ⫽ ⫺200cos 6 pect swimming pool sales to be lowest in the winter and highest in the summer. b 126 c June; 500 d March and October 7 a 10.8 b March 20 and September 23 (assuming that it’s not a leap year) c December 21, the first day of winter; 9.4 hours of daylight 8 a 0.75; the tide varies 0.75 foot above and below the average depth of 1.5 feet. b 12; every 12 hours, the tides complete one full cycle. c 6 hours d 1.875 feet e at approximately 1:47 A.M., 10:47 A.M., 1:37 P.M., and 10:47 P.M. 9 a January: 23°; February: 26° b July; 71° c 47° d April and October e 24; the average monthly temperature varies 24 degrees above and below the average temperature for a year in Syracuse. 10 a ⫺503.90 ⱕ P(t) ⱕ 933.94; Captain Freeze can lose at most $503.90 in a week and have a maximum profit of $933.94 in a week.
冢 冣
11.7 Trigonometric Graphs and Real-World Applications
51
b 52; these are the number of weeks in the year from January through December. c Week 29 (t ⫽ 28); $933.94 d Weeks 1 through 13 (t ⫽ 0 through 12) and weeks 45 through 52 (t ⫽ 44 through 51) e Yes, it is profitable. Explanations will vary f Answers will vary. Chapter Review (pages 413–416) 兹3 1 (3) 2 2 (4) y ⫽ ⫺3cos 2x 3 (2) y ⫽ cos x 4 (4) There is no minimum value. 5 (1) p 6 (2) 2 7 (3) 60° 8 (4) ⫺5 ⱕ y ⱕ 5 p p 9 (3) ⫺ ⱕ x ⱕ 2 2 10 (3) y ⫽ ⫺cos x 11 (2) y ⫽ ⫺cos x 12 (1) 1 1 13 (2) cos x 14 (2) decreases only p 15 (2) 2 16 (3) y ⫽ tan x 17 (1) 1 p 18 (3) 4 19 (1) 1.4 inches 20 (3) the range increases 21 (2) y ⫽ sec x p 22 (2) It is undefined at . 2 23 (3) 15 3 24 a
25
c 26
p
3p 2
y = 3cos 2x
p 3p , 4 4 y ⫽ ⫺3cos 2x 5 4 3 2 1
y = 2cos x y = tan x p
p 2
2 1 y = sin 1 x 2
–p
–
p 2
p 2
–1
p
b
–2
c 28
29
2p
–3
b (p, 0) c x⫽p
Chapter 11: Graphs of Trigonometric Functions
y ⫽ sin
冢 12 (x ⫺ p)冣 or y ⫽ ⫺cos 12 x
June 80 March 1: 50; May 1: 76 trigonometric (sinusoidal) function 800 p c 400 d high point: 200 feet; midline: 100 feet e 100 p x ⫹ 100 f y ⫽ 100cos 400 p px ⫹5 a ⫽ 4, b ⫽ , d ⫽ 5; y ⫽ 4sin 6 6 a b c a b
冢
y = –sin x
–2
52
e a
p
p 2
b x⫽0 c x⫽p p d x⫽ 2 27 a, b
2
p 2
c
–1 –2 –3 –4 –5
1
–1
4 3 2 1 –1 –2 –3 –4
y = 2cos 1 x
2
a p b, d
30
冣
CHAPTER
Trigonometric Identities
12
12.1 Proving Trigonometric Identities
6
(1 ⫹ cos u)(1 ⫺ cos u) ⱨ sin2 u 1 ⫺ cos2 u ⱨ sin2 u sin2 u ⫽ sin2 u ✔
(page 420) Answers may vary. 1 sin2 u ⫹ cot2 u ⫹ cos2 u ⱨ csc2 u 1 ⫹ cot2 ⱨ csc2 u csc2 u ⫽ cos2 u ✔
7
cos2 u (tan2 u ⫹ 1) ⫹ cot2 u ⱨ csc2 u cos2 u (sec2 u) ⫹ cot2 u ⱨ csc2 u 1 ⫹ cot2 u ⱨ csc2 u csc2 u ⫽ csc2 u ✔
sec2 u (1 ⫺ cos2 u) ⱨ tan2 u sec2 u (sin2 u) ⱨ tan2 u sin2 u ⱨ tan2 u cos2 u tan2 u ⫽ tan2 u ✔
8
2
3
4
5
1 ⫺ sin2 u • sec u ⱨ cot u sin u cos2 u 1 • ⱨ cot u sin u cos u cos u ⱨ cot u sin u cot u ⫽ cot u ✔ sin2 u ⫺ cos2 u ⫹ 1 ⱨ tan u 2sin u cos u 2sin2 u ⱨ tan u 2sin u cos u sin u ⱨ tan u cos u tan u ⫽ tan u ✔ sec u sin u ⫺ ⱨ cot u sin u cos u 1 ⫺ sin2 u ⱨ cot u sin u cos u cos2 u ⱨ cot u sin u cos u cos u ⱨ cot u sin u cot u ⫽ cot u ✔
9
10
1 1 ⫹ ⱨ sec 2 csc 2 2 sin cos2 cos2 ⫹ sin2 ⱨ sec 2 csc 2 sin2 cos2 1 ⱨ sec 2 csc 2 2 sin cos2 sec2 b csc2 b ⫽ sec2 b csc2 b ✔ csc u ⫺ (cos u)(cot u) ⱨ sin u 1 cos2 u ⫺ ⱨ sin u sin u sin2 u sin2 u ⱨ sin u sin u sin u ⫽ sin u ✔ 1 ⫺ cos2 u ⱨ sin2 u ⫹ 1 tan2 u sin2 u 2⫺ ⱨ sin2 u ⫹ 1 tan2 u sin2 u cos2 u 2⫺ ⱨ sin2 u ⫹ 1 sin2 u 2 ⫺ cos2 u ⱨ sin2 u ⫹ 1 1 ⫺ cos2 u ⫹ 1 ⱨ sin2 u ⫹ 1 sin2 u ⫹ 1 ⫽ sin2 u ⫹ 1 ✔ 2⫺
12.1 Proving Trigonometric Identities
53
12.2 Sum and Difference of Angles (pages 424–426) 117 1 a 125 4 b 5 117 c 44 2 0.96 兹6 ⫹ 兹2 3 4 4 a Answers may vary. Example: 60° and 45° 兹6 ⫹ 兹2 b 4 5 sin (p ⫺ u) ⱨ sin u sin p cos u ⫺ cos p sin u ⱨ sin u sin u ⫽ sin u ✔ 6
7
cos (360° ⫺ A) ⱨ cos A cos 360° cos A ⫹ sin 360° sin A ⱨ cos A cos A ⫽ cos A ✔ sin (180° ⫹ x) ⱨ ⫺sin x sin 180° cos x ⫹ cos 180° sin x ⱨ ⫺sin x ⫺sin x ⫽ ⫺sin x ✔ cos
8 cos
9
(3)
10 11 12 13 14 15 16 17
(2) (1) (4) (4) (1) (3) (2) (3)
18
(2)
19 (3) 20 (1) 21 (1) 22
(3)
23
(2)
54
24
(1)
25
(4)
26
(3)
27
(2)
28
(4)
12.3 Double-Angle Formulas (pages 429–431) 24 1 25 1 2 ⫺ 2 3 0.28 3 4 5 5 6 7
冢 3p2 ⫺ 冣 ⱨ ⫺sin
3p 3p cos ⫹ sin sin ⱨ ⫺sin 2 2 ⫺sin b ⫽ ⫺sin b ✔ 16 65 ⫺tan y sin 300° cos 270° sin 120° ⫺1 tan 40° ⫺cos y 0 1 ⫺ 2 k sin x cos y 兹2 ⫹ 兹6 4 ⫺sin u
Chapter 12: Trigonometric Identities
84 85 ⫺cos x 16 65 兹3 2 2sin A sin B
8
9
2sin 60° cos 60° ⫽
兹3
2 2cos2 90 ⫺ 1 ⫽ ⫺1 1 ⫺ tan u • sin 2u ⱨ cos 2u sin u 1⫺ • 2sin u cos u ⱨ cos 2u cos u 1 ⫺ 2sin2 u ⱨ cos 2u cos 2u ⫽ cos 2u ✔ cos 2 ⫹ 2sin a ⱨ csc a sin 1 ⫺ 2sin2 2sin2 ⫹ ⱨ csc a sin sin 1 ⱨ csc a sin csc a ⫽ csc a ✔ 2 tan u 1 ⫹ tan2 u 2 tan u sin 2u ⱨ sec 2 u 2 sin u sin 2u ⱨ • cos2 u cos u sin 2u ⱨ 2sin u cos u sin 2u ⫽ sin 2u ✔ sin 2u ⱨ
10
11
12
1 cos2 u 1 (2cos2 u ⫺ 1)(sec2 u) ⱨ 2 ⫺ cos2 u 1 2 ⫺ sec2 u ⱨ 2 ⫺ cos2 u 1 1 ✔ 2⫺ ⫽2⫺ 2 cos u cos2 u cos 2u (1 ⫹ tan2 u) ⱨ 2 ⫺
sin 2u sec u ⱨ 2sin u 1 2sin u cos u • ⱨ 2sin u cos u 2sin u ⫽ 2sin u ✔ a
b
c
13 14 15 16 17
(1) (1) (2) (1) (4)
18 (4) 19 (2) 20 (3) 21 (2) 22 (3) 23 (2) 24 (3) 25
(4)
26
(1)
27
(2)
cos 2A ⫽ cos (A ⫹ A) ⫽ cos A cos A ⫺ sin A sin A ⫽ cos2 A ⫺ sin2 A cos 2A ⫽ cos2 A ⫺ sin2 A ⫽ cos2 A ⫺ (1 ⫺ cos2 A) ⫽ 2cos2 A ⫺ 1 cos 2A ⫽ cos2 A ⫺ sin2 A ⫽ (1 ⫺ sin2 A) ⫺ sin2 A ⫽ 1 ⫺ 2sin2 A sin u ⫺1 ⱕ y ⱕ 1 sin 2x 1 cos 30° 527 625 1 ⫺ 9 tan (2u) ⫺0.96 3 28 ⫺ 100 1 ⫺ cos 2x sin2 x ⫽ 2 兹30 6 ⫺1 5 ⫺ 8
12.4 Half-Angle Formulas (pages 435–436) 2兹5 1 5 1 2 2 兹10 3 10 兹3 4 3
5
cos 30° ⫽
6
a
冪
1 ⫹ cos 60⬚ ⫽ 2
sin 15° ⫽
⫽ b 0.259 c 0.259; yes 7
a
9 10 11 12 13 14 15
1 2
2
冪
1 ⫺ cos 30⬚ 2
冪
3 1⫺ 兹 2 2 ⫺ 兹3 ⫽ 兹 2 2
cos 22.5° ⫽
⫽
8
冪
1⫹
b 0.924 c 0.924; yes (2) II 2 兹13 (2) ⫺ 13 兹2 ⫹ 兹3 (1) 2 (2) ⫺1 兹5 (3) 3 兹5 (3) 5 兹7 (1) 4 (3) 60°
3 ⫽ 兹 2
冪
1 ⫹ cos 45⬚ 2
冪
2 1⫹ 兹 2 兹2 ⫹ 兹 2 ⫽ 2 2
12.4 Half-Angle Formulas
55
Chapter Review (pages 438–439) 1 (4) 0 2 (4) ⫺cos x 3 (4) csc u 1 4 (3) ⫺ 2 5 (2) sin 120° 7 6 (1) 6 7 (2) 5 ⫺ sin2 A 8 (1) sin2 4x ⫹ cos2 4x ⫽ 1 1⫺c 9 (1) 2 10 (3) 3 11 (1) 1 13 12 (3) ⫺ 85 13 (2) ⫺sin u 14 (3) cos 90° 7 15 (3) 25 sin 2u ⱨ tan u 16 1 ⫹ cos 2u 2sin u cos u ⱨ tan u 1 ⫹ 2cos2 u ⫺ 1 sin u ⱨ tan u cos u tan u ⫽ tan u ✔ 17
56
sin 2u 2sin2 u 2 sin u cos u cot u ⱨ 2sin2 u cos u cot u ⱨ sin u cot u ⫽ cot u ✔ cot u ⱨ
Chapter 12: Trigonometric Identities
18
19
20
1 ⫹ cos u cos 2u ⱨ cot u sin u ⫹ sin 2u 1 ⫹ cos u ⫹ 2cos2 u ⫺ 1 ⱨ cot u sin u ⫹ 2sin u cos u cos u(1 ⫹ 2cos u) ⱨ cot u sin u(1 ⫹ 2cos u) cos u ⱨ cot u sin u cot u ⫽ cot u ✔ 2sin ⱨ sec 2 sin 2 cos 2sin ⱨ sec 2 2sin cos cos 1 ⱨ sec 2 cos2 sec2 b ⫽ sec2 b ✔ sec x ⫺ sin x tan x ⱨ cos x sin x 1 ⫺ sin x ⱨ cos x cos x cos x 1 ⫺ sin2 x ⱨ cos x cos x cos2 x ⱨ cos x cos x cos x ⫽ cos x ✔
CHAPTER
Trigonometric Equations 13.1 First-Degree Trigonometric Equations (pages 443–444) 1 u 35°, 315° 2 u 30°, 150° 3 u 0°, 180°, 360° 4 u 150°, 210° 5 u 30°, 210° 6 u 90° p 5p 7 u , 4 4 2p 4p , 8 u 3 3 5p 7p , 9 u 4 4 5p 7p , 10 u 6 6 3p 11 u 2 p 3p 12 u , 2 2 13 b 11.5°, 168.5° 14 b 76.0°, 256.0° 15 b 78.5°, 281.5° 16 b 14.5°, 165.5° 17 b 60°, 240° 18 m⬔B 225 19 x 180° 7p 20 u 4 21 x {70.5°, 180°, 289.5°} 22 u 90° 23 (4) 74° 7p 24 (3) 6 25 (3) III and IV
13
26
(3)
冦 3p4 , 5p4 冧
7p 6 28 (3) 5csc u 2 1 29 (3) 150° p 2p , 30 (2) 3 3 27
(3)
冦
冧
13.2 Second-Degree Trigonometric Equations (pages 448–449) 1 u 45°, 135°, 225°, 315° 2 u 45°, 135°, 225°, 315° 3 u 270° 4 u 30°, 150°, 210°, 330° 5 u 60°, 90°, 270°, 300° 2p 4p , , 2p 6 u 0, 3 3 p 2p 4p 5p , , 7 u , 3 3 3 3 p 5p 8 u , 3 3 9 u 0, , 2 p 10 u 0, , p, 2p 2 11 b 21.8°, 135°, 201.8°, 315° 12 b 81.8°, 180°, 278.2° 13 b 193.4°, 346.6° 14 b 109.5°, 120°, 240°, 250.5° 15 b 57.9°, 122.1° 16 b 63.4°, 161.6°, 243.4°, 341.6° 17 m⬔B 180 18 x 135° 13.2 Second-Degree Trigonometric Equations
57
19 20 21 22 23 24 25 26 27
u
11p 6
17
a
2
–1
18
p
2p
b p c p a 2 y = –cos x
–1
b c 19
a
p , 2 p , 2
7p , 6 7p , 6
3p 11p , 2 6 3p 11p , 2 6
2 y = sin 1 x
1
2
2p
p –1
y = –cos x
–2
20 21 22 23
1
p
y = sin 2x
–2
y = 2sin x
2p
y = cos 2x
–2
58
2p
y = –sin x
1
(pages 452–453) 1 u 0°, 90°, 180°, 270°, 360° 2 u 0°, 120°, 240°, 360° 3 u 90°, 210°, 330° 4 u 180° 5 u 90°, 210°, 270°, 330° 2p 4p , p, , 2p 6 a 0, 3 3 p 5p 7 a , p, 3 3 8 0, 2 4p 9 a 0, 3 p 5p , 2p 10 a 0, , p, 3 3 11 u 25.7°, 154.3°, 230.1°, 309.9° 12 u 38.2°, 141.8° 13 u 56.0°, 153.3°, 206.7°, 304.0° 14 u 90°, 221.8°, 318.2° 15 u 102.7°, 257.3° 16 a 2
c
p
–2
13.3 Trigonometric Equations That Use Identities
b
y = 2cos 1 x
1
(3) 3 (3) 120 5p (3) 6 (2) cos2 x 1 0 4p (4) 3 (2) 2 (3) 3 (4) 180°
–1
2
p 2 p 2 Chapter 13: Trigonometric Equations
b c (2) (4) (3) (4)
p p 2 p 90° 4
Chapter Review (pages 455–456) 1 (4) 180° p 2 (4) 6 3 (1) 1 4 (1) 1 5 (3) 180° 6 (3) Quadrant I or Quadrant III 7 (2) 90° 4p 8 (4) 3 9 (3) 3
10 11 12 13 14
p 2 66.4°, 180°, 293.6° 19.5°, 160.5° 60°, 300° 33.7°, 45°, 213.7°, 225° (1)
15 16 17 18 19 20
55.5°, 155.4°, 235.5°, 335.4° 221.8°, 270°, 318.2° 180° 48.6°, 131.4°, 270° 30°, 150°, 199.5°, 340.5° 228.6°, 311.4°
CHAPTER
Trigonometric Applications 14.1 Law of Cosines (pages 460–462) 1 (2) 18.0 2 (1) 11.87 3 (3) 12.3 centimeters 4 (4) obtuse 5 (1) 135.6° 6 (4) 0.8647 7 (2) 90° 8 (1) 17.2 inches 9 (4) 9.63 inches 10 (3) 22.2 inches 11 73.61°, 62.95°, 43.43° 12 6.87 miles 13 a 66.4° or 66°25 b 47.2° or 47°9 14 6.3 miles 15 6.1 miles
14.2 Area of a Triangle (pages 466–467) 1 60.79 in.2 2 25.95 cm2 3 81.27 cm2 4 131.06 cm2 5 11.66 in.2
14 6 7 8 9 10 11 12 13 14 15 16 17
(1) (3) (4) (3) (3) (2) (3) (4) (2) (2) a b a b
8 inches 9 兹3 228.8 0.6697 170.9 square centimeters 29.53 52.94 12.2 inches 88.16 385.28 47.4° 257.1 cm2 13.2 in. 361.1 in.2
14.3 Law of Sines and the Ambiguous Case (pages 472–474) 1 a 28.5, m⬔A 135.2 2 a 30.3, m⬔A 45.4 3 a 38.2 4 m⬔A 57.8 5 m⬔A 37.7 6 12.1 cm 7 13.8 in. 8 5.5 ft
14.3 Law of Sines and the Ambiguous Case
59
9 10 11
9.40 in. m⬔AMH ⫽ 77.9 a 2 b 49°, 131° c 22, 38 d
16 17 18 T 114°
31 C
H
38 T 32°
31 C
12 13 14 15 16
17° 22
131°
51.3° 77.4° 607 ft 3,065 ft 107.2° and 72.8° 252.2 ft2
12 49°
17°
a b a b a b
14.5 Forces and Vectors (pages 482–483) 1 50.5 lb
12
(2) 49.8° (1) 10.4 sin 61⬚ sin 58⬚ ⫽ (2) 12.5 x (1) cannot be determined (4) 0
x
21.8 lb
H
52.6° 34.2 lb
2
62.5 lb
14.4 Mixed Trigonometric Applications (pages 477–478) 1 (1) ⫺0.1141 2 (2) 5.4 3 (3) 99.6° 4 (1) 21 5 (1) 7.2 6 (3) 40 centimeters 7 (1) 䉭MAT is a right triangle. 8 (2) 2 9 (3) 0.32 10 (1) 1 11 a 49° b 218 square inches 12 No, the Law of Sines does not hold for these measurements: sin 69.98⬚ sin 43.74⬚ ⫽ 120 96 13 228 ft and 386 ft 14 a 57.5° b 14.1 cm c 66 cm2 15 a 16 cm b 228 cm2
60
Chapter 14: Trigonometric Applications
73.4 lb
56° 47° x
3
128.7°
37 N
48.4 N
x
62 N
4
34.2 lb
x 21.8 lb 110.6° 35.1 lb
5
81.0 lb 37 lb
29°15’ 52 lb
x
6
79°2⬘
41.6 N
83.4 N x 64.8 N
7 x
48.3 N 18°25’ 29°40’ y
a 72.6 N b 30.8 N 8
124.7 lb 78 lb
31.5° y x
a 71.0 lb b 35.0° 9 10.2 mph
8 mph y
x 12 mph
Chapter Review (pages 483–485) 1 36 sq units 2 0.306 3 90° 4 8.7 5 16兹3 sq units 6 19.9 7 48 8 84兹3 in.2 9 13.16 10 (1) 24 11 11 (2) 18 12 (2) 2兹19 13 (3) 84兹2 29 14 (3) 36 s2 3 15 (2) 4 兹 16 (4) 0 17 (3) must be an obtuse triangle n2 18 (2) 2 19 a 84.5 lb b 27.6° 20 a 87.3° b 83.9 ft2 c 4 21 68° 22 3.9 mi 23 10.6° 24 a 658.9 ft b $607,279.90 25 a 5.5 兹2 in. b Two of the angles are 11° too large, two of the angles are 11° too small. c 7.0 in.
a 122.78° b 81.53°
Chapter Review
61
CHAPTER
Statistics
15.1 Gathering Data: Univariate Statistics (pages 488–489) 1 (4) shoppers at a mall 2 (1) people who received free samples of the candy 3 (2) the post office 4 (3) the first fifty people encountered on a city street 5 (3) calling the tenth person listed on each page of the city telephone directory
15 5 6 7 8
9 10
In 6–12, explanations will vary. 6 controlled experiment 7 observation 8 sample survey 9 population survey 10 controlled experiment 11 simulation 12 sample survey 13 sample survey 14 sample survey 15 population survey 16 sample survey 17 sample survey 18 Answers will vary.
11
15.2 Measures of Central Tendency
14
(pages 495–497) 1 (2) 74 2 (3) mean median 3 (4) 100 4 (2) 6
62
Chapter 15: Statistics
12
13
15
(4) mean mode (3) 3.5 (3) 34 (3) an equal number of players are taller than 6 feet 4 inches and shorter than 6 feet 4 inches mean $39,858,881.50; median $34,271,570.50; no mode a mean $188; median $200; mode $200 b Mode or mean; this value represents the highest-priced gown that will still be sellable to the majority of the girls. a mean 222.81; median 222.35; no mode b The mean and median are both representative since the distribution of values if fairly even on both sides of the mean. c 7.6 a mean $20.40; median $19; mode $17.50; Andrew should use the mean as the “average” price for yard work. b His dad could counter with the mode, saying “most” of the neighborhood pays this lesser amount. Answers will vary. Example: The median age will not be as skewed by outliers such as the very young and the very old. a mean $922.61 million; median $863.45 million b The median is a better average since the earnings of Gone With the Wind represent an outlier that skews the mean upward. Answers will vary.
16
17
Arrange the data in ascending order. The median is then either the middle number, if the number of data values is odd, or the average of the two middle numbers, if the number of data values is even. Repeat this procedure on the half of the data greater than the median to find the third quartile and again on the half of the data less than the median to find the first quartile. Changing any data value greater than the median such that it continues to be greater than the median will not affect the median. Likewise, changing any data value less than the median such that it continues to be less than the median will not affect the median. Any data value greater than the median may be changed to be less than the median without affecting the median itself so long as a second data value less than the median is changed to be greater than the median. The median is also unaffected if a pair of data points, one greater than the median and one less than the median, is changed to equal the median. Similarly, if a pair of data values shares the value of the median, one can be changed to be greater than the median and the other changed to be less than the median without affecting the median. It is also possible to change data values in a list without changing the mode or median.
15.3 Measures of Dispersion: Range, Interquartile Range, Variance, and Standard Deviation (pages 502–506) 1 (3) 10 2 (4) 4 3 (1) 256 4 (2) {14, 76, 56, 42, 86} 5 (3) 8 6 (2) 30 7 (3) there is no change 8 (2) 7.6 9 (3) 27.276 10 (1) be multiplied by 2
11
12 13 14 15 16
17
18 19
20 21 22 23 24 25
26 27
a 5 b 4 c 2 a 81.2 b 4.2 25 a 90 b 0 B; the data have smaller deviations from the mean. a 78.8 in. b 16 in. c median 80 in., Q1 75.5 in., Q3 82.5 in., IQR 7 in. a mean 16, median 17, Q1 8.5, Q3 21 b range 31, IQR 12.5 c Answers will vary. Either the mean, 68.6, or the median, 70, would be a good average. a 53.84 b 10.31 a 190 b Maurice appears to be a less consistent bowler than Monica. His interquartile range is 50, so his range will be greater than 50. a 2:33:09.4 b 19 minutes, 27 seconds 160°F 5,344 ft a 262.0 b 47.3 a 255.8 lb b 46.6 lb a 15.58 b 2.97 c Answers will vary. Examples: advertising agencies, concert promoters, vendors a $6.67 b $4.12 a 3.1 in. b 0.4 in.
15.4 Normal Distribution (pages 510–512) 1 (3) 80.5 2 (2) 2.3 3 (3) 163
15.4 Normal Distribution
63
4 5 6 7 8 9 10 11
(3) (4) (2) (1) (2) (4) (1) a b c d
12
13
14
a b c a b c
a b c d
12.6 90 29,120 50 300 40 5 15 days 2 days 50%: 15 days; 85%: between 17 and 18 days; 98%: between 19 and 20 days After 21 days, a person could be almost completely certain (99.9%) that he or she will not get the disease from the single exposure. 68.2 97.7 191 32.3° 5.2° (1) 32.3° (2) 21.9°–42.7° (3) close to 50% (4) 24.5° (5) 27.1° and 37.5° x 20.8, 0.24 21.28 20.56 and 21.04 443,300
15.5 Bivariate Statistics, Correlation Coefficients, and the Line of Best Fit (pages 517–518) 1 a 1 2 d 0.1 3 f 0.9 4 e 0.6 5 b 0.8 6 c 0.3 7 (4) 0.9 8 (4) If the correlation coefficient is negative, the line of best fit has a negative slope. 9 (1) 0.89 10 (3) 0.15
64
Chapter 15: Statistics
15.6 Linear Regression (pages 521–523) 1 (4) 0.9017 2 (3) y 1.65x 122.4 3 a y 806.4x 93892.2 b r 0.9965093864 c 99,537 d Answers will vary. Example: The model is not a close fit for years beyond 1994; the pattern of adoptions is changing. 4 a y 442.35x 743.27 b r 0.9597033424 c approximately 442 d 6,494 e Answers will vary. Example: Russian government restrictions on foreign adoptions slowed American adoptions beginning in 2005. f The model does not hold. 5 a
b y 217.14x 2,363.37 c r 0.991 d 1982: 9,312 stations; 2010: 15,392 stations 6 a
b y 0.526x 8.012 c r 0.9976339247; this means that there is a strong positive correlation between the year and the percentage of women over 25 in the population who have completed four or more years of college. d 24.9% e 2010: 29.1%; 2020: 34.3%; the greater the extrapolation, the less accurate the model. f 1977
7
a
12
a
13
b y 10.17515021(0.8453967676)x c 10.175 grams; the model does not exactly fit all data points. d 3.7145 grams e It will never disappear entirely, but will continue to get smaller and smaller. a, c
b y 3.6469x 27.4449 c 63.9% d 2009
15.7 Curve Fitting (pages 527–529) 1 (4) y 2(1.7)x 2 (1) y 5(0.4)x 3 (1) y 20log x 4 (3) logarithmic 5 (3) 23 6 (4) 3.5 7 (1) linear 8 (2) 0 9 a Approximately 4.68 hours per day b Approximately 5.77 hours per day c 24.48 weeks after school starts 10 f(x): vii; the graph appears linear with a positive slope and a positive y-intercept. g(x): i; the graph appears logarithmic without any reflections. h(x): iv; the graph appears exponential, and since the y-values decrease and the x-values increase, 0 x 1. 11 a
b c d e f
y 1,014.2434(1.0683) 6.83% $5,647.37 $4,440.73 Keep the money where it is. x
b logarithmic; the shape is comparable to a logarithmic function. c Approximately 26.8 weeks
15.8 More Curve Fitting (pages 535–537) 1 (2) y 0.73x 2.969 2 (4) cubic 3 y 3.93218146x 2.762206751 4 a
b y 1.184961951x 0.4183021777 c Approximately 2.32 seconds 2p d T 兹5 ⬇ 2.48 seconds, which is 兹32 only slightly longer than the time found in part c.
15.8 More Curve Fitting
65
5
a
9
b Exponential: y 99,348.09(1.03)x Power: y 99,764.55x 0.13 c Exponential: 2008; Power: 2024 6 a
a
b Exponential c y 19.50607505(1.026824147)x d y 0.0162946429x2 0.3163690476x 20.13333333 e Exponential: 50.59%; Quadratic: 52.64% FYI
7
b y 7.81952384x 0.6644168582 c 102,382 d 2004 a
8
b y 11.414 22.308ln x c y 13.489x 0.626 d The power regression more closely fits the data. e 85.3 a
b c d e f
66
y 9.136(1.072)x y 0.006x3 0.028x2 0.058x 10.258 $22.50 $28.26 Exponential: December 2008; Cubic: July 2008 Chapter 15: Statistics
(page 539) a y 22.22sin (0.53x 2.37) 55.42 b Midline 55.42. This represents the average temperature in Central Park. c Amplitude 22.22. This represents the maximum variation in temperature from the average. Chapter Review (pages 539–545) 1 (i) strong positive linear correlation 2 (iv) weak negative linear correlation 3 (ii) strong negative linear correlation 4 (v) linear correlation close to zero 5 (2) 30 6 (3) 83 7 (3) logarithmic 8 (4) 81.75–84.5 9 (1) 5 10 (3) 420 11 (3) 150 12 (1) 3.58 13 (3) y 3x 14 (4) 32.49 15 (2) ask drivers in the neighborhood 16 (2) median mean 17 (1) The median salary would not be affected. 18 (3) multiplied by 兹2 19 (2)
20 21
(3) divided by 4
22
23
a
24
b C(t) 199.001t 28.934 c C(0) 28.93, C(8) 1620.94, C(18) 3610.95; these would be the expected cost of tuition given the linear model. d Answers will vary. e C(t) 926.184(1.073)t f C(0) 926.18, C(8) 1628.77, C(18) 3298.47 g Answers will vary. h The exponential function more closely fits the data. i The linear function would mean a lower tuition than the exponential function. a
b
c y 2.398601399x 31.42715618 d Approximately 55.41 inches, close to the actual data e Approximately 65 inches f y 2.236013986x 32.68822844 g Girls; according to the equations, they grow approximately 2.399 inches per year, while boys grow approximately 2.236 inches per year. h Approximately 48.34 inches i Approximately 79.64 inches; no, this does not make sense because the average 21-year-old male is not 6 feet 8 inches tall. 25 a 7.9 b 1.1 c 1.1 d y
4.64 5.18 5.72 6.27 6.81 7.36 7.9 8.44 8.99 9.53 10.08 10.62 11.16
e 77 f 77% g No, the mean and the median are different from each other. However, the data are close to being normally distributed.
Chapter Review
67
x
26
a
b y 0.9995x0.6667 c 39.47 au d The regression equation is approximately the same as the statement of Kepler’s Third Law, i.e., x2 ky3 or 2 y cx 3. 27 a mean 81.79, median 85, mode 85 b No, the mean, median and mode are different from each other. c v 91.45; 9.56 d 20 e If the scores were normally distributed, 68.2% of them should lie within one standard deviation of the mean. As it stands, 71.4% of the scores lie within one standard deviation of the mean. 28 a y 0.2340282448x 0.4959650303 b Approximately 4.5 hours c Approximately 12.83 pounds d C(20) 5.1765299; it takes about 5 hours to cook a 20-pound turkey. e w 83.340517; a turkey weighing approximately 83.3 pounds needs to be cooked for 20 hours. f Part d makes sense because we have 20-pound turkeys. Part e does not make sense because we don’t have turkeys weighing 83 pounds. Additionally, we don’t have enough data to know if the trend holds for hypothetical turkeys of that size.
68
Chapter 15: Statistics
29
a
30
b c d e f a
Exponential y 102.3521(1.0379)x 6,095,719 Answers will vary. Answers will vary.
b Logarithmic or power c Since log 1 0, the 15.3902 represents the fuel efficiency when x 1, the year 1975. d 1993 e approximately 29.958 mpg f y 15.6226x0.1891 g 1993 h approximately 30.759 mpg
CHAPTER
Probability
16
16.1 Fundamental Counting Principle
4
(pages 547–549) 1 (4) 12 2 (3) 72 3 (2) 105 4 (3) 720 5 (4) 192 6 (1) 36 7 (4) 20 8 (4) 1,440 9 (2) 192 10 (4) 420 11 480 12 a 42 b 35 13 432 14 24 15 144 16 12 17 150 18 9
5 6
8 9 10 11 12 13 14 15 16 17 18
16.2 Permutations
16.3 Combinations
(pages 554–556) 1 a 24 b 6 c 18 d 18 e 4: 4,267; 4,627; 6,247; 6,427 2 a 362,880 b 15,120 c 1,680 d 210 3 720
(pages 559–561) 1 (3) 15P7 2 (2) 495 3 (3) 3,003 4 (2) 35 5 (1) 66 6 (2) 560 7 (4) 58,212 8 (4) 600 9 (3) 840 10 (4) 11,088
7
a 11! ⫽ 39,916,800 b 10! ⫽ 3,628,800 c 9! ⫽ 362,880 5,040 a 840 b 1,680 c 210 a 210 b 6,720 c 180 d 34,650 e 3,780 (3) 30 (2) 60,480 (1) 12 (1) 7P1 (2) 6 (3) 120 (3) 6,720 (1) 552 (4) 210 (2) 24 (4) 58,500
16.3 Combinations
69
11 12
13 14 15 16 17
18
19 20 21 22
1,400 a 7C3 • 11C3 b 7C2 • 11C4 c 11C6 d 1 • 17C5 a Combination; 330 b Permutation; 24 Permutation; 56 a Combination; 120 b Permutation; 5,040 Combination; 324 a Combination; 55 b Combination; 36 c Permutation; 9! ⫽ 362,880 a Permutation; 120 b P: 72 c P: 72 d P: 54 a Permutation; 200P3 b Permutation; 199P2 10! ⫽ 151,200 Permutation; 3!2!2! Combination; 6,000 a Combination; 5,400 b Combination; 1,080 c Permutation; 9! ⫽ 362,880
10 11 12
13
14 15
16
16.4 Probability (pages 567–569) 1 1 (1) 12 1 2 (4) 1,776 3 3 (4) 28 1 4 (2) 220 1 5 (1) 33 y ⫺ x2 6 (3) y 8C3 • 6C2 7 (1) 14C5 1 8 (4) 210 1 9 (1) 504
70
Chapter 16: Probability
17 18 19 20
1 4 11 ⫺ 2p (2) 11 5 a 8 3 b 4 1 a 28 1 b 28 3 c 7 1 5 2 a 15 2 b 15 28 c 45 7 d 15 9 16 29 8C6 ⫹ 6C6 ⫽ 134,596 24C6 C ⫹ C • C 13 3 3 3 2 12 1 ⫽ 455 15C3 1 • 15C4 5 ⫽ 16 16C5 1 1,320 (3)
16.5 The Binomial Theorem (page 574) 1 z3 ⫹ 15z2 ⫹ 75z ⫹ 125 2 a4 ⫺ 12a3b ⫹ 54a2b2 ⫺ 108ab3 ⫹ 81b4 3 1,024p5 ⫹ 2,560p4 ⫹ 2,560p3 ⫹ 1,280p2 ⫹ 320p ⫹ 32 4 41 ⫹ 38i 5 1,296x4 ⫺ 864x3 ⫹ 216x2 ⫺ 24x ⫹ 1 3 5 1 6 z ⫹ z ⫹ 15x 4 ⫹ 160z 3 ⫹ 960z 2 ⫹ 6 64 4 3,072z ⫹ 4,096 7 ⫺86,016
8 9 10 11 12 13 14 15 16 17 18 19 20
4,860x4y2 540 1,176c2d2 (1) 10p 2 (3) ⫺8sin3 x (2) ⫺720 (2) 256 (4) 216tan2 u (4) 35 (4) 8C5(4x)3(3)5 (1) 84 (3) the middle term (4) 160i
5
16.6 Binomial Probability (pages 577–580) 3 1 a 8 1 b 8 3 c 8 2 a .189 b .02835 c .036756909 d .2401 25 3 a 216 5 b 324 125 c 324 1 d 1,296 625 e 1,296 1 4 a 8 1 b 2 1 c 2 21 d 512 5 e 16 1 f 16
6
1 8 1 b 6 1 c 3 1 d 8 1 e 4 21 f 512 4 g 9 15 h 1,024 25 i 216 1 j 4,096 Answering 3 out of 5 on the test where each question has 5 choices has the greater probability, assuming you are guessing randomly. 1 4 3 1 P(4 of 5, 4 choices) ⫽ 5C4 4 4 15 ⫽ ⬇ .015 1,024 1 3 4 2 P(3 of 5, 5 choices) ⫽ 5C3 5 5 160 ⫽ ⬇ .051 3,125 125 (3) 216 4 (2) 9 3 (3) 8 (3) 10C1(.999)9(.001) 1 10 (3) 10 2 32 (2) 243 1 (1) 125 4 17 1 3 (4) 20C3 5 5 9 (4) 64 a
冢 冣冢 冣 冢 冣冢 冣
7 8 9 10 11 12 13 14 15
冢 冣
冢 冣冢 冣
16.6 Binomial Probability
71
16 17 18 19 20
216 625 108 (3) 343 (3)
16
冢 冣冢 冣
1 2 5 4 6 6 4 (1) 5C1(.99) (.01)1 16 (1) 625 (3) 6C2
17
16.7 At Least or at Most r Successes in n Trials (pages 582–584) 19 1 144 15 2 16 5 3 16 15 4 16 16 5 27 6 (2) 1 ⫺ 2x 3,125 7 (3) 8,192 14,375 8 (4) 16,807 215 9 (4) 216 5 10 (2) 16 11 (4) .973 12 (1) .94 13 (3) .457 189 14 (3) 256 15 The probability of randomly guessing the correct answer on a given question is .25 for the first-period class and .5 for the fifthperiod class. Therefore, students who randomly guess will tend to do better on the fifth-period quiz.
72
Chapter 16: Probability
18
19
20
This is a wise strategy. Without the marketing strategy, the resort would have 120 fullprice stays for February. The probability that there is snow on at least 2 days of a 20 3-day stay is . That means that the resort 27 can expect 115 full-price stays and 20 halfprice stays, which is the same as 125 fullprice stays. 2 a 27 3,773 b 4,096 13 c 16 20 a 27 19 b 27 5 a 16 1 b 2 a 56 b 120 c 24
16.8 Normal Approximation to the Binomial Distribution (page 586) 1 .052 2 .029 3 .922 4 .026 5 .804 6 a .215 b .207 c The normal approximation slightly underestimates the probability that the coin will land on heads. d .448 FYI
(page 587) a .3292 b .4219 c .2995 d .9997 e .0579
Chapter Review (pages 588–591) 1 (4) 720 2 (3) 960 3 (4) 24x2y2 1 4 (1) 2 67 5 (2) 256 6 (1) 28 2x 7 (2) 2x ⫹ 1 8 (4) ⫺20a3b3i 9 (4) 1,320 10 (4) .015625 11 (2) cos3 u 98 12 (1) 125 13 (2) 72 14 (1) ⫺40 297 15 (4) 625 16 (4) 54a2b2 27 17 (3) 64 2,125 18 (3) 4,096 19 (3) ⫺2,035 ⫹ 828i 347 20 (2) 2,048 6,250 21 (2) 16,807 32 22 (1) 243 23 (3) 336 24 a 19C5 b 8C2 • 11C3 C • C c 8 2 11 3 19C5 1 • 8C2 • 10C2 d 19C5 25 a3 ⫺ 6a2b ⫹ 12ab2 ⫺ 8b3 26 a 0.6561 b 0.0001 c 0.0037 d 0.9999
27
16
28
a b c d
29
a b c d
1 5 2 5 12 125 98 125 4 13 2 13
冢 冣冢 冣 11 2 C冢 冣冢 冣⫹ C冢 冣冢 冣 13 13 11 2 ⫹ C冢 冣冢 冣 13 13 12 1 12 1 C 冢 冣冢 冣 ⫹ C 冢 冣 冢 冣 13 13 13 13 23 26 11 13
3C2
3
1
2
2
3 26 2 13
3
0
3
3
32
33
0
3
3
3
2
3
⫺7 ⫹ 24i 99 31 a 6C1 100 99 b 10C0 100
30
2
2
2
e
1
3
冢 冣冢 冣 冢 冣冢 冣 5
1 1 100 10 1 100
0
c 1⫺
99 99 冣 冢 1001 冣 ⫹ C 冢 100 冣 冢 1001 冣 冥 冤 C 冢 100
d
10C1
冢 冣冢 冣
a b c d a b c
.552 .186 .138 .990 .878 .199 .344
7
8
1
1
99 100
8
8
9
1 100
1
0
0
冢 冣冢 冣
⫹ 10C0
99 100
10
Chapter Review
1 100
0
73
Cumulative Reviews Each review has a total of 48 points. Use the following chart, adapted from the Regents Examinations, to convert the student’s raw score to a scaled score. Raw Score 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32
Raw Score 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15
Scaled Score 100 99 97 96 94 92 91 89 87 86 84 82 80 79 77 76 74
Raw Score 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
Scaled Score 72 70 68 67 65 63 61 60 58 56 54 53 51 49 47 46 44
Scaled Score 42 41 39 37 35 34 32 30 28 26 21 16 10 5 0
Chapters 1–2 (pages 592–594) Part I Allow a total of 26 credits, 2 credits for each of the following. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 1 (2) {4, 6} 2 (1) x2 ⫹ 2x ⬍ 8 3 (3) {0, 4} 4 (2) ⫺1 or 1 5 (4) x3 ⫹ 3x2 ⫹ 3x ⫹ 1 2x ⫹ 1 6 (3) ⫺ 2⫹x
74
Cumulative Reviews
冦 冧
1 ,1 2 8 (4) 2a ⫹ 3b 9 (2) Its factors are (2x ⫹ 3)(x ⫹ 2)(x ⫺ 2). 4x ⫹ 3 10 (2) 2 x ⫺ 3x 7
(3)
11
(2) –1
12 13
0
(3) x2 ⫹ 15x ⫹ 17 x⫺1 (2) 2
3 2
Part II For each question, use the specific criteria to award a maximum of 2 credits. 14
15
16
17
Score
Explanation
2
For a fraction to be undefined, the denominator must equal zero. Since x2 ⱖ 0; the denominator x2 ⫹ 3 is always greater than 0.
1
Student attempts substitution of various values.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation x⫹2 x⫹2 miles, while Efrim traveled 3 4 miles. Dividing the same number by a smaller value produces a larger quotient.
2
Cody traveled farther. Cody traveled
1
Cody, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
x ⫽ 5 and appropriate work is shown.
1
Appropriate work is shown but one computational error is made or
1
x ⫽ 5 but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
x ⫽ ⫺1,
7 and appropriate work is shown. 3
1
x ⫽ ⫺1,
7 but no work is shown or 3
1
Only one solution is correct with appropriate work shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Chapters 1–2
75
Part III For each question, use the specific criteria to award a maximum of 4 credits. 18
19
Score
Explanation
4
x ⫽ 2, 6 and appropriate work is shown.
3
Appropriate work is shown with one computational error.
2
Appropriate work is shown with two computational errors or
2
Only one solution is correct with appropriate work shown.
1
x ⫽ 2, 6 but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
冢
冣
4
x⫹4 1 , x ⫽ ⫺2, ⫺ , 0, 7 and appropriate work is shown. x 2
3
An appropriate method is used but one factoring or computational error is made.
2
An appropriate method is used but two computational errors are made.
1
Multiple factoring or mathematical errors are made or
1
x⫹4 but no work is shown. x
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Part IV Use the specific criteria to award a maximum of 6 credits. 20
76
Score
Explanation
6
x ⫽ ⫺1, 4 and appropriate work is shown.
5
Appropriate work is shown, but one computational error is made.
4
Appropriate work is shown, but one conceptual error is made.
3
Only one solution is given with appropriate work shown.
2
Multiple errors are made in the algebraic solution.
1
x ⫽ ⫺1, 4 but no work is shown
1
Student shows minimal understanding/correct algebra.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Cumulative Reviews
or
Chapters 1–3 (pages 595–596) Part I Allow a total of 26 credits, 2 credits for each of the following. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 1 2 3 4 5
6 (1) 1 7 (4) 3a4b4 兹2b 8 (1) ⫺7 ⬍ x ⬍ 3 9 (3) 7兹2 10 (2) 7 only
(3) 3 ⫺ 兹5 1 (1) x⫹y (3) {⫺1, 4} (4) ⫺1 (4) 4
11 12 13
2兹5 ⫹ 5兹2 10 (1) 0 ⬍ x ⬍ 6 (3) 3 (4)
Part II For each question, use the specific criteria to award a maximum of 2 credits. 14
Score 2
5x and appropriate work is shown. x⫹1
1
Appropriate work is shown, but one computational error is made or
1
5x but no work is shown. x⫹1
0
15
16
Explanation
Score
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Explanation
2
y ⫽ ⫺1 and appropriate work is shown.
1
Appropriate work is shown, but one computational error is made or
1
Appropriate work is shown, but the solution y ⫽ ⫺
1
y ⫽ ⫺1 but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
5 is not rejected or 2
Explanation
2
x ⫽ 10 and appropriate work is shown.
1
Appropriate work is shown, but one computational error is made or
1
Appropriate work is shown, but the negative solution is not rejected or
1
x ⫽ 10 but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–3
77
17
Score
Explanation
2
冤⫺ 32 , 5冥 or ⫺ 32 ⱕ x ⱕ 5 and appropriate work is shown.
1
Appropriate work is shown, but one computational error is made or
1
Student correctly solves for the related equality
1
冤⫺ 32 , 5冥 or ⫺ 32 ⱕ x ⱕ 5 but no work is shown.
0
or
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Part III For each question, use the specific criteria to award a maximum of 4 credits. 18
78
Score
Explanation
4
x ⫽ 2 and appropriate work is shown, such as correctly solving the equation 兹x ⫹ 14 ⫽ x ⫹ 2 algebraically or graphically.
3
Appropriate work is shown, but one computational error is made.
2
Appropriate work is shown, but two or more computational errors are made or
2
Appropriate work is shown, but one conceptual error is made, such as incorrectly squaring both sides.
1
Appropriate work is shown, but one conceptual error and one computational error are made or
1
x ⫽ 2 but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Cumulative Reviews
19
Score
Explanation
4
a 兩 t ⫺ 110 兩 ⬍ 20 b 90 ⬍ t ⬍ 130 and appropriate work is shown, such as correctly solving the inequality 兩 t ⫺ 110 兩 ⬍ 20 algebraically or graphically.
3
Appropriate work is shown, but one computational error is made.
2
Appropriate work is shown, but two or more computational errors are made or
2
Appropriate work is shown, but one conceptual error is made when solving the inequality or
2
An incorrect, but similar, inequality is given and is solved correctly.
1
Appropriate work is shown, but one conceptual error and one computational error are made or
1
兩 t ⫺ 110 兩 ⬍ 20 and 90 ⬍ t ⬍ 130 but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Part IV Use the specific criteria to award a maximum of 6 credits. 20
Score
Explanation
6
x ⫽ 3 and appropriate work is shown.
5
Appropriate work is shown, but one computational error is made.
4
Appropriate work is shown, but student does not eliminate extraneous root.
3
A correct equation is solved but only one solution is given.
2
Multiple errors are made in the algebraic solution.
1
x ⫽ 3 but no work is shown or
1
Student shows minimal understanding of quadratics/correct algebra.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Chapters 1–3
79
Chapters 1–4 (pages 597–598) Part I Allow a total of 26 credits, 2 credits for each of the following. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 1 (2) 6兹5 2 (3) {0, 2} 3 (4) y2 ⫺ 3y ⫺ 2 ⫽ x 4 (2) 兩 140 ⫺ t 兩 ⬍ 5 5 (3) $210 6 (4) all real numbers except ⫾3 7 (1) 1
8
(3) f ⫺1(x) ⫽
冪
x⫹1 3
13
(4)
9 (4) 10 ⫺ 5兹3 10 (1) 120 ⫹ 31x兹3 ⫹ 6x 2 11 (2) (x ⫺ 1)(x ⫹ 1)(2x ⫹ 7) 12 (4) {0, 3}
Part II For each question, use the specific criteria to award a maximum of 2 credits. 14
15
16
80
Score
Explanation
2
7兹6 and appropriate simplification of radicals is shown.
1
One term is correctly simplified; others are not.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
Explanations will vary but should mention the use of the conjugate of the denominator. The result of the process should be ⫺10 ⫺ 5兹7.
1
A correct explanation or a correct solution is given but not both.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
(x ⫹ 3)2 ⫹ (y ⫺ 8)2 ⫽ 25
1
One term in the equation is incorrect.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Cumulative Reviews
17
Score
Explanation
2
x ⬍ ⫺2 or x ⬎ 2; these values cause the denominator of the fraction to be a real, nonzero number. Values between ⫺2 and 2 would produce a fraction that is undefined.
1
Correct domain but no explanation given or
1
Correct explanation but incorrect domain is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Part III For each question, use the specific criteria to award a maximum of 4 credits. 18
Score
冢
冣 and appropriate work is shown.
4
x⫺4 3 1 , x ⫽ ⫺4, ⫺ , 2 2 3
3
An appropriate method is used but one factoring error is made.
2
Appropriate work is shown with two factoring or algebraic errors or
2
Solution is not simplified.
1 0
19
Explanation
Score
x⫺4 but no work is shown. 2 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Explanation
4
9x2 ⫺ 18x ⫺ 3 and appropriate work is shown.
3
An appropriate method is used but one computational error is made.
2
An appropriate method is used but two computational errors are made or
2
An appropriate method is used but a conceptual error, such as the incorrect order of composition, is made.
1
Multiple factoring or mathematical errors are made or
1
9x2 ⫺ 18x ⫺ 3 but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Chapters 1–4
81
Part IV Use the specific criteria to award a maximum of 6 credits. 20
Score
Explanation
冢
冣
6
3 ⫺ 2x 5 3 , x ⫽ ⫺4, ⫺ , , 4 and appropriate work is shown. 2 3 2
5
Appropriate work is shown, but one factoring/algebraic error is made.
4
Appropriate work is shown, but two factoring/algebraic errors are made.
3
Some appropriate work is shown, but the factor of ⫺1 is not used.
2
Multiple errors are made in the algebraic solution.
1
Student shows minimal understanding/correct algebra.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Chapters 1–5 (pages 599–600) Part I Allow a total of 26 credits, 2 credits for each of the following. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 1 2 3 4 5 6
(3) (3) (4) (1) (1) (2)
4 ⫹ 兹15 ⫺5 ⫹ 12i 7 11 {x : ⫺1 ⬍ x ⬍ 5} ⫺1
a a⫹1 8 (4) ⫺3 ⬍ x ⬍ 3 9 (4) (2, 7) 2 1 x⫺ 10 (4) 3 3 7
(4)
11 12 13
(2) ⫺5 ⫺ 5i (3) real, irrational, and unequal (2) y ⫽ 3
Part II For each question, use the specific criteria to award a maximum of 2 credits. 14
82
Score
Explanation
2
34 ⫺ 42i and appropriate work is shown.
1
Appropriate work is shown, but one computational error is made or
1
Appropriate work is shown, but one simplification error is made.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Cumulative Reviews
15
16
17
Score
Explanation
2
x ⫽ 2 and appropriate work is shown.
1
Appropriate work is shown, but one computational error is made or
1
Appropriate work is shown, but one conceptual error is made, such as not rejecting x ⫽ ⫺1 or
1
x ⫽ 2, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
x ⫽ 8 and appropriate work is shown.
1
Appropriate work is shown, but one computational error is made.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
Mary Lu is 4 years old and Jane is 8 years old, and appropriate work is shown.
1
Appropriate work is shown, but one computational error is made or
1
Appropriate work is shown, but student answers Mary Lu is 8 years old and Jane is 4 years old or
1
Mary Lu is 4 years old and Jane is 8 years old, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Chapters 1–5
83
Part III For each question, use the specific criteria to award a maximum of 4 credits. 18
Score 4
Explanation a, b
yi 9
7 + 8i
8 7 6 5
2 + 5i
4 3
5 + 3i
2 1
–3 –2 –1 O 1 –1 –2 –3
c
19
84
2
3
4
5
6
7
8
9
x
7 ⫹ 8i, and appropriate work is shown.
3
Appropriate work is shown, but one computational or graphing error is made.
2
Appropriate work is shown, but two or more computational or graphing errors are made or
2
Appropriate work is shown, but one conceptual error is made.
1
Appropriate work is shown, but one conceptual error and one computational or graphing error are made or
1
7 ⫹ 8i but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
4
x ⫽ ⫺3, x ⫽ 6 and appropriate work is shown.
3
Appropriate work is shown, but one computational error is made or
3
Appropriate work is shown, but only one answer is given.
2
Appropriate work is shown, but two or more computational errors are made or
2
Appropriate work is shown, but one conceptual error is made, such as not finding an appropriate common denominator.
1
Appropriate work is shown, but one conceptual error and one computational error are made or
1
x ⫽ ⫺3, x ⫽ 6 but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Cumulative Reviews
Part IV Use the specific criteria to award a maximum of 6 credits. 20
Score
Explanation
6
{(⫺7, ⫺25), (5, 11)} and appropriate work is shown.
5
Appropriate work is shown, but one computational error is made.
4
Appropriate work is shown, but one conceptual error is made.
3
A correct equation is solved, but only one solution is given or
3
{(⫺7, ⫺25), (5, 11)}, but one solution is achieved without using algebra.
2
Multiple errors are made in the algebraic solution.
1
Student shows minimal understanding/correct algebra
1
{(⫺7, ⫺25), (5, 11)} but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
or
Chapters 1–6 (pages 601–603) Part I Allow a total of 26 credits, 2 credits for each of the following. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 1 2
3 4 5 6 7
(2) ⫺14 (1)
8 9
(3) (1) (4) (4) (1)
113.6 10 36 82 The product is a real, rational number. 1 (4)n (2) 2 (4) 4
10 11 12 13
3 4 3 ⫹ 24i (2) 13 (1) 1 ⫾ 2i (2) 2 (3) ⫾
Part II For each question, use the specific criteria to award a maximum of 2 credits. 14
Score
Explanation
2
second root: 3 ⫺ 2i; equation: x2 ⫺ 6x ⫹ 13 ⫽ 0 and appropriate work is shown.
1
The second root or the equation is correct and appropriate work is shown, but the other is not or
1
3 ⫺ 2i; x2 ⫺ 6x ⫹ 13 ⫽ 0 but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–6
85
15
16
17
Score
Explanation
2
Two real, unequal, irrational roots; explanations will vary but should include mention of the discriminant, 40.
1
Correct description of roots or appropriate explanation, but not both.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
x ⫽ 7 and appropriate work is shown.
1
One or more errors are made in the solution of the equation or
1
The equation is solved correctly, but the extraneous root is not discarded.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation 1 ⫾ 2i and appropriate work is shown. 2
2
x⫽
1
Equation is correctly solved but solution is not simplified
1
Computational errors are made in the solution of the quadratic equation or
1
x⫽
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
or
1 ⫾ 2i, but no work is shown. 2
Part III For each question, use the specific criteria to award a maximum of 4 credits. 18
86
Score
Explanation
4
sum ⫽ 399; 5th term: ⫺486; 6th term: 729, and appropriate work is shown.
3
Two of the three answers are correct and appropriate work is shown.
2
Appropriate work is shown with two formula or computational errors.
1
Only one of the three solutions is correct
1
sum ⫽ 399; 5th term: ⫺486; 6th term: 729 but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Cumulative Reviews
or
19
Score
Explanation
4
$2,800,000 in 2009, and appropriate work or graph is shown.
3
2009 or $2,800,000 is correct and appropriate work or graph is shown.
2
An appropriate method is used, but two computational errors are made.
1
Multiple errors are made
1
$2,800,000 in 2009, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
or
Part IV Use the specific criteria to award a maximum of 6 credits. 20
Score
Explanation
6
a an ⫽ 96(0.6)n⫺1 and appropriate work is shown. b 6 swings and appropriate work is shown.
5
The equation is correct and appropriate work is shown, but the number of swings is incorrect.
4
An appropriate formula is used, but one computational error is made.
3
Student uses arithmetic sequence formulas but does so incorrectly.
2
Multiple errors are made in the algebraic solution.
1
Student shows minimal understanding of geometric sequences/correct algebra
1
an ⫽ 96(0.6)n⫺1 and 6 swings but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
or
Chapters 1–7 (pages 604–606) Part I Allow a total of 26 credits, 2 credits for each of the following. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. (1) (2) (3) (4) (2) (4)
I 2 760 16 2a2b⫺2 (⫺4, ⫺1)
7
(3)
8
n
(3)
9 (4) 10 (1) 11 (2)
Game Owners
1 2 3 4 5 6
t
12
(4)
13
(4)
1 3 (x ⫺ 3)2 ⫹ (y ⫹ 6)2 ⫽ 25 I and II x 2 ⫹ 2x ⫺ 24 ⱕ 0 1 2 {8}
Time
Chapters 1–7
87
Part II For each question, use the specific criteria to award a maximum of 2 credits. 14
15
16
17
88
Score
Explanation
2
29 and appropriate work is shown.
1
Appropriate work is shown, but one computational error is made or
1
Student correctly multiplies by an incorrect conjugate with appropriate work shown or
1
29, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
domain: All real numbers or (⫺⬁, ⬁); range: {y : y ⱖ 3} or [3, ⬁), and appropriate work is shown.
1
Appropriate work is shown, but one computational error is made or
1
Appropriate work is shown, but only one correct answer is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
x ⫽ 3 and appropriate work is shown.
1
Appropriate work is shown, but one computational error is made or
1
Student sets the exponents 2x and x ⫺ 1 equal to each other and solves correctly with appropriate work shown or
1
x ⫽ 3, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
1 x ⫽ ⫺ , x ⫽ 3 and appropriate work is shown. 3
1
Appropriate work is shown, but one computational error is made or
1
1 x ⫽ ⫺ , x ⫽ 3 but no work is shown or 3
1
Appropriate work is shown, but only one correct answer is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Cumulative Reviews
Part III For each question, use the specific criteria to award a maximum of 4 credits. 18
19
Score
Explanation
4
x ⫽ 0, x ⫽ ⫺3 ⫹ 4i, x ⫽ ⫺3 ⫺ 4i, and appropriate work is shown, such as solving the cubic equation by factoring and using the quadratic formula or completing the square.
3
Appropriate work is shown, but one computational error is made.
2
Appropriate work is shown, but two or more computational errors are made or
2
Appropriate work is shown, but one conceptual error is made, such as not including the complex roots in the solution.
1
Appropriate work is shown, but one conceptual error and one computational error are made or
1
x ⫽ 0, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
4
a f(x) ⫽ 3x ⫹ 2 b f(x) ⫽ 3x⫺3
3
One of the equations f(x) ⫽ 3x ⫹ 2, f(x) ⫽ 3x⫺3 is correct, but the other one has an error, such as an error in the sign of the constant.
2
One of the equations f(x) ⫽ 3x ⫹ 2, f(x) ⫽ 3x⫺3 is correct, but the other equation is completely wrong or
2
Both of the equations f(x) ⫽ 3x ⫹ 2, f(x) ⫽ 3x⫺3 have errors in the signs of the constant.
1
One of the equations f(x) ⫽ 3x ⫹ 2, f(x) ⫽ 3x⫺3 is wrong, and the other one has an error, such as an error in the sign of the constant.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Chapters 1–7
89
Part IV Use the specific criteria to award a maximum of 6 credits. 20
Score
Explanation
6
a
5
Appropriate work is shown, but one computational or rounding error is made.
4
Appropriate work is shown, but only one of the explanations given for the meaning of N(4) or N(t) ⫽ 4 is correct or
4
Appropriate work is shown, but one conceptual error is made.
3
Only one of the evaluations of N(4) or N(t) ⫽ 4 is correct, along with its explanation.
2
Multiple errors are made in the algebraic solution.
1
Student shows minimal understanding of exponents/logarithms.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
N(4) ⫽ 5,247,000; four months after being introduced, the show has approximately 5,247,000 viewers. b t ⫽ 2.058; the show can be expected to have 4,000,000 viewers slightly more than 2 months after first airing.
Chapters 1–8 (pages 607–608) Part I Allow a total of 26 credits, 2 credits for each of the following. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 1 2 3 4
(4) imaginary 3 (1) 2 c⫺2 (3) c (3) are the same graph
5 6 7 8 9
(2) (4) (2) (3)
2 It is a complex number. {5} m ⫹ 2p 1 (1) 2x 2 ⫺ 5x ⫽ 2
10
(3) 3
11
(3) ⫺
12 13
(4) 4 (4) y ⫽ 1,000(1.04)18
64 81
Part II For each question, use the specific criteria to award a maximum of 2 credits. 14
90
Score
Explanation
2
4x2 ⫹ 4x ⫺ 3 and appropriate work is shown.
1
4x2 ⫹ 4x ⫺ 3, but no work is shown
1
4x2 ⫹ 8x ⫹ 9; student performed composition in wrong order or
1
One computational error is made in performing the composition.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Cumulative Reviews
or
15
16
17
Score
Explanation
2
The population is decreasing by (1 ⫺ 0.7882)% or 21.18% yearly. 17,432 is the initial population in 2009, and 0.7882 represents the decay in population.
1
One of the two explanations is incorrect.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
冢
冣
1 , 2 , and appropriate work is shown. 2
2
x, x ⫽ 0,
1
Appropriate work is shown but one factoring or computational error is made or
1
x, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
1 1 ⫾ i and appropriate work is shown. 3 3
1
1 1 ⫾ i, but no work is shown or 3 3
1
Equation is solved correctly but solution is not simplified.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Chapters 1–8
91
Part III For each question, use the specific criteria to award a maximum of 4 credits. 18
Score 4
Explanation a, b f –1(x) = 2x
f(x) = log2 x
19
92
3
Function and inverse are graphed over the wrong domain or
3
The function is graphed correctly but there is one error on the inverse.
2
Only the original function is graphed correctly or
2
The graph of the function is correct but the student does not graph the inverse or
2
The graph of the function is incorrect but the correct inverse of the student’s graph is shown.
1
Student demonstrates minimal understanding of functions and their graphs.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
4
a 2.1425 and appropriate work is shown. b 0.643 and appropriate work is shown.
3
One solution is correct but one computational error is made in the other.
2
Appropriate use of logarithms is shown, but two computational errors are made.
1
Multiple computational errors are made
1
Minimal understanding of logarithms is displayed
1
2.1425 and 0.643, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Cumulative Reviews
or or
Part IV Use the specific criteria to award a maximum of 6 credits. 20
Score
Explanation
6
6.3 hours and appropriate work is shown.
5
Appropriate work is shown, but a rounding or computational error is made.
4
Appropriate work is shown, but one conceptual error is made.
3
Student expresses 6.3 without a unit of measure or with an incorrect unit of measure, such as milligrams.
2
Multiple errors are made in the algebraic solution.
1
Student shows minimal understanding of appropriate methodology or
1
6.3 hours, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Chapters 1–9 (pages 609–610) Part I Allow a total of 26 credits, 2 credits for each of the following. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 1 (1) 0° 2 (3) 73 3 (3) 28i 4 (4) imaginary 5 (4) 4,882,812
6 7 8 9 10
(4) (2) (3) (1) (3)
15 330° 8 ⫺6 210°
11 (1) {1} 12 (4) y ⫽ log2 x 13 (3) 8
Part II For each question, use the specific criteria to award a maximum of 2 credits. 14
Score
Explanation
2
10x ⫹ 26 and appropriate work is shown. x2 ⫺ 9
1
Appropriate work is shown, but one computational error is made or
1
Student adds length and width correctly, and appropriate work is shown or
1
10x ⫹ 26 but no work is shown. x2 ⫺ 9
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Chapters 1–9
93
15
16
17
94
Score
Explanation
2
k ⫽ 3 and appropriate work is shown.
1
Appropriate work is shown, but one computational error is made or
1
k ⫽ 3 but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation xy 2 and appropriate work is shown. z
2
log
1
Appropriate work is shown, but one computational error is made or
1
Appropriate work is shown, but one incorrect logarithmic rule is used or
1
log
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
xy 2 but no work is shown. z
Explanation
2
x ⫽ 0 and appropriate work is shown.
1
Appropriate work is shown, but one computational error is made or
1
x ⫽ 0 but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Cumulative Reviews
Part III For each question, use the specific criteria to award a maximum of 4 credits. 18
19
Score
Explanation
4
15 hours and appropriate work is shown.
3
Appropriate work is shown, but one computational error is made.
2
Appropriate work is shown, but two or more computational errors are made or
2
Student uses an incorrect equation using the given information, but solves it correctly.
1
Appropriate work is shown, but one conceptual error and one computational error are made or
1
15 hours, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
4
⫺1 and appropriate work is shown.
3
Appropriate work is shown, but one computational or factoring error is made.
2
Appropriate work is shown, but two or more computational or factoring errors are made or
2
Appropriate work is shown, but one conceptual error is made.
1
Appropriate work is shown, but one conceptual error and one computational error are made or
1
⫺1 but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Chapters 1–9
95
Part IV Use the specific criteria to award a maximum of 6 credits. 20
Score
Explanation
6
a A(t) ⫽ 500(0.98)t b A(50) ⫽ 182.08 grams and appropriate work is shown. c 34.31 years and appropriate work is shown.
5
Appropriate work is shown, but one or more rounding error is made or
5
The correct equation is solved incorrectly, and that answer is used correctly for the other parts of the problem.
4
An incorrect equation is solved correctly, and that answer is used correctly for the other parts of the problem or
4
Appropriate work is shown, but one conceptual error is made.
3
The initial equation is written correctly, but no other work is correct.
2
Only one of the three required answers is correct, and no work is shown for the other two parts.
1
A(t) ⫽ 500(0.98)t; A(50) ⫽ 182.08; approximately 34.31 years, and no work is shown or
1
Student shows minimal understanding of exponential functions.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Chapters 1–10 (pages 611–612) Part I Allow a total of 26 credits, 2 credits for each of the following. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 1 2 3 4 5
96
(3) y ⫽ 兹1 ⫺ x 2 125 (2) 16 (1) 1 (4) $120.00 (4) 315°
Cumulative Reviews
(2) 兩 58 ⫺ h 兩 ⬍ 4 11p 7 (3) 6 8 (4) 12 9 (2) a ⫹ 3b 10 (1) 4x2 ⫺ 4x ⫹ 17 ⫽ 0 6
11 12 13
(1) 42°, 48° (2) 3.04 (2) 2
Part II For each question, use the specific criteria to award a maximum of 2 credits. 14
15
16
17
Score
Explanation
2
An appropriate explanation is given, such as the sine function is represented by the y-coordinate on the unit circle and in the y-coordinate is positive in Quadrants I and II.
1
An incomplete explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
16.1 years, and appropriate work is shown.
1
16.1 years and no work is shown
1
Appropriate work is shown, but one computational error is made.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
or
Explanation
2
S(30) ⫽ 47,531.44, which will be Kelly Ann’s salary after having worked at the job for three years.
1
Appropriate work is shown, but one computational error is made or
1
S(30) ⫽ 47,531.44, but no work is shown or no explanation is provided.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
1
Graph of the parabola is correct but no shading or incorrect shading is drawn or
1
Shading is appropriate, but there is an error on the parabola.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Chapters 1–10
97
Part III For each question, use the specific criteria to award a maximum of 4 credits. 18
19
Score
Explanation
4
a1 ⫽ 20; d ⫽ 7, and appropriate work is shown.
3
Appropriate work is shown, but one computational error is made.
2
Only the common difference or the first term is correct.
1
a1 ⫽ 20; d ⫽ 7, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation 1 and appropriate work is shown. 2
4
x⫽⫺
3
An appropriate method is used, but one computational error is made.
2
An appropriate method is used, but two computational errors are made.
1
Only minimal understanding of logarithmic equations is displayed
1
x⫽⫺
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
or
1 but no work is shown. 2
Part IV Use the specific criteria to award a maximum of 6 credits. 20
98
Score
Explanation
6
a $12,845.16 and appropriate work is shown. b $20,911.84 and appropriate work is shown.
5
A correct exponential equation is shown, but one computational or rounding error is made.
4
Appropriate work is shown, but one conceptual error is made or
4
Appropriate work is shown, but two computational or rounding errors are made.
3
An incorrect equation is solved correctly.
2
Multiple errors are made in the algebraic solution.
1
Student shows minimal understanding of exponential functions.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Cumulative Reviews
Chapters 1–11 (pages 613–615) Part I Allow a total of 26 credits, 2 credits for each of the following. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4.
2 3 4
(3) 5 ⫹ 6i 兹3 , 1 (3) 2 2 (4) 兩 32 ⫺ t 兩 ⬍ 2 (2) ⫺1
冢
冣
(1)
5
6
d
Distance
1
t
Time
(2)
兹2
2 7 (4) ⫺2 ⬍ y ⬍ 4 8 (1) 20.5 3 9 (1) ⫺ 2 10 (1) a 11 (1) {⫺5, 5} 5 12 (2) ⫺ 4 13 (2) y ⫽ sec x
Part II For each question, use the specific criteria to award a maximum of 2 credits. 14
Score 2
x 2 ⫺ 3x ⫹ 2 and appropriate work is shown. x2 ⫺ x ⫺ 6
1
Appropriate work is shown, but one computational error is made or
1
x 2 ⫺ 3x ⫹ 2 but no work is shown. x2 ⫺ x ⫺ 6
0
15
Explanation
Score
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Explanation
2
150 people, and appropriate work is shown.
1
Appropriate work is shown, but one computational error is made or
1
150 people but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Chapters 1–11
99
16
17
Score
Explanation
2
x ⫽ ⫺3, x ⫽ 6, and appropriate work is shown.
1
Appropriate work is shown, but one computational error is made or
1
x ⫽ ⫺3, x ⫽ 6, but no work is shown
1
Appropriate work is shown, but only one correct answer is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
or
Explanation
2
50° and appropriate work is shown.
1
Appropriate work is shown, but one computational error is made or
1
Appropriate work is shown, but the coterminal angle given is not the smallest positive acute angle or
1
50°, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Part III For each question, use the specific criteria to award a maximum of 4 credits. 18
100
Score
Explanation
4
5.7 feet and appropriate work is shown, either algebraic or graphical.
3
Appropriate work is shown, but one computational or rounding error is made.
2
Appropriate work is shown, but two or more computational or rounding errors are made or
2
Appropriate work is shown, but one conceptual error is made.
1
Appropriate work is shown, but one conceptual error and one computational or rounding error are made or
1
5.7 feet, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Cumulative Reviews
19
Score
Explanation
4
25 indicates that Mathland had a population of 25,000 when it was founded; 1.03 indicates that the population is growing at a rate of 3% per year; 46.9 years, and appropriate work is shown.
3
Appropriate work is shown, but one computational or rounding error is made or
3
Appropriate work is shown, but one of the interpretations of 25 and 1.03 is incorrect.
2
Appropriate work is shown, but two or more computational or rounding errors are made or
2
Appropriate work is shown, but both of the interpretations of the 25 and 1.03 are incorrect.
1
Appropriate work is shown, but one conceptual and one computational or rounding error are made or
1
46.9 years, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Part IV Use the specific criteria to award a maximum of 6 credits. 20
Score
Explanation
6
h(t) ⫽ 1.5cos
冢 p6 t冣 ⫹ 2.1 or h(t) ⫽ ⫺1.5sin 冢 p6 (x ⫺ 3)冣 ⫹ 2.1; 4.6 hours and
appropriate work/graph is shown. 5
The amplitude, frequency, or midline is incorrect, but the resulting hour is correct for the student’s equation or
5
The equation is incorrectly rounded, but the resulting hour is correct for the incorrect equation.
4
The equation is correct, but no other work is done
4
Two of the values for amplitude, frequency, or midline are incorrect, and the resulting hour is correct for the student’s equation.
3
Two of the values of amplitude, frequency, or midline are incorrect, and the resulting hour is also incorrect for the student’s equation.
2
None of the values for amplitude, frequency, or midline are correct; however, the resulting hour is correct for the student’s equation.
1
h(t) ⫽ 1.5cos
or
冢 p6 t冣 ⫹ 2.1 or h(t) ⫽ ⫺1.5sin 冢 p6 (x ⫺ 3)冣 ⫹ 2.1; 4.6 hours, but no
work/graph is shown or 1
Student shows minimal understanding of trigonometric graphs.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Chapters 1–11
101
Chapters 1–12 (pages 616–618) Part I Allow a total of 26 credits, 2 credits for each of the following. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 1
(2) y ⫽ 3cos
2
(3)
3 4
1 x⫺1 2
7p 6 (2) x ⫽ 3, y ⫽ 6 (3) 14
5 6 7 8 9
(3) 14.2 hundred thousand dollars (2) 2 (4) 6.8% (1) 1 (2) {6}
10 11 12 13
(1) (3) (3) (1)
1 500 52 ⫺30°
Part II For each question, use the specific criteria to award a maximum of 2 credits. 14
15
16
102
Score
Explanation
2
82 and 100 and appropriate work is shown.
1
82 and 100, but no work is shown
1
Only one of the two arithmetic means is correct.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
or
Explanation
2
y ⫽ 2(x ⫺ 2)2 ⫹ 5 and appropriate work is shown.
1
y ⫽ 2(x ⫺ 2)2 ⫹ 5, but no work is shown
1
Appropriate work is shown, but errors are made in completing the square.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
or
Explanation
2
July 1; 28 blankets sold, and appropriate work is shown.
1
Only one of the two required answers is correct
1
Appropriate work is shown, but computational errors are made or
1
July 1; 28 blankets sold, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Cumulative Reviews
or
17
Score
Explanation
2
2 and appropriate work is shown. 3
1
Appropriate work is shown, but computational errors are made or
1
2 but no work is shown. 3
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Part III For each question, use the specific criteria to award a maximum of 4 credits. 18
19
Score
Explanation
4
a 3.25 seconds and appropriate work is shown. b 181 feet and appropriate work is shown.
3
Appropriate work is shown, but one rounding or computational error is made.
2
Appropriate work is shown, but two computational or rounding errors are made or
2
Only time is correct or distance is correct but not both.
1
3.25 seconds and 181 feet, but no work is shown
1
Minimal understanding of quadratic functions is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
or
Explanation
4
x ⫽ 16.034 and appropriate work is shown.
3
Logarithmic equation is correctly displayed, but one computational error is made.
2
Logarithms are used, but two computational or rounding errors are made or
2
x ⫽ 16.034, but the problem is not solved using logarithms.
1
Multiple computational errors are made
1
x ⫽ 16.034, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
or
Chapters 1–12
103
Part IV Use the specific criteria to award a maximum of 6 credits. Score
20
6
Explanation a
The sequence is geometric because it has a constant ratio, r ⫽
b a n ⫽ 324 c
冢 冣 2 3
2 . 3
n⫺1
and appropriate work is shown.
2,660 2 or 866 and appropriate work is shown. 3 3
5
Appropriate work is shown, but one computational error is made or
5
Appropriate work is shown, but no explanation is provided for the type of sequence.
4
Appropriate work is shown, but only two of the solutions are correct.
3
Student solves for an arithmetic sequences but does so incorrectly or
3
Student identifies the sequence as geometric, but no explanation is provided, and solves only one of the other parts of the question.
2
a n ⫽ 324
2
Multiple errors are made in the algebraic solution.
1
Student shows minimal understanding of sequences.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
冢 冣 2 3
n⫺1
and
2 2,660 or 866 , but no work is shown or 3 3
Chapters 1–13 (pages 619–620) Part I Allow a total of 26 credits, 2 credits for each of the following. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 1 2 3 4
(2) 2 (2) ⫺7 ⫹ 24i p (2) 2 a2 ⫹ 2a ⫹ 2 (2) a⫹2
104
Cumulative Reviews
(4) 2cos u (1) 32 3p 7 (3) 4 8 (3) N(t) ⫽ 5(1.08)t 9 (2) 44.444 5 6
(1) 512 3p 11 (4) 2 12 (2) P(t) ⫽ 250e1.3t 13 (3) 3 10
Part II For each question, use the specific criteria to award a maximum of 2 credits. 14
15
16
17
Score
Explanation
2
1 ⫾ 3i and appropriate work is shown.
1
Appropriate work is shown, but one computational error is made or
1
1 ⫾ 3i, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
⫺2 ⬍ n ⬍ 6 and appropriate work is shown.
1
Appropriate work is shown, but one computational error is made or
1
Student solves related equality correctly and appropriate work is shown or
1
Appropriate work is shown, but only one portion of the correct inequality is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
Moisha is correct. The discriminant, 25, is a perfect square, which indicates that the roots are real, rational, and unequal.
1
Appropriate work is shown, but one computational error is made or
1
Appropriate work is shown, but one conceptual error is made or
1
Stating that the discriminant is 25 or that the roots are real, rational, and unequal, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
No, and an appropriate explanation is given, such as the graph fails the vertical line test.
1
An answer of “no” without an explanation or
1
An answer of “no” with an incorrect explanation.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Chapters 1–13
105
Part III For each question, use the specific criteria to award a maximum of 4 credits. 18
19
Score
Explanation
4
a 2 seconds and appropriate work is shown. b 1.9 seconds and appropriate work is shown.
3
Appropriate work is shown, but one computational or rounding error is made.
2
Appropriate work is shown, but two or more computational or rounding errors are made or
2
Appropriate work is shown, but one conceptual error is made or
2
2 seconds and 1.9 seconds, but no work is shown.
1
Appropriate work is shown, but one conceptual error and one computational error or rounding error are made or
1
2 seconds or 1.9 seconds, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score 4
Explanation a
y 3 2
y = 2sin 3x
1 p 3
–1
2p 3
x
–2 –3
y = –2sin 3x
b y ⫽ ⫺2sin 3x c Reflecting in the y-axis would produce the curve y ⫽ 2sin (⫺3x) and, since sin (⫺x) ⫽ ⫺sin x, we see that y ⫽ 2sin (⫺3x) ⫽ ⫺2sin 3x.
106
3
Appropriate work is shown, but either the graph or its equation or the answer or explanation about the reflection in the y-axis is incorrect.
2
Appropriate work is shown, but either the graph and its equation or the answer and explanation about the reflection in the y-axis are incorrect.
1
Appropriate work is shown, but only one of the following is correct: the graph or its equation or the answer or explanation about the reflection in the y-axis.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Cumulative Reviews
Part IV Use the specific criteria to award a maximum of 6 credits. 20
Score
Explanation
6
u ⫽ 90°, 210°, 330° and appropriate work is shown.
5
Appropriate work is shown, but one computational error is made or
5
Appropriate work is shown, but either 210° or 330° is not given.
4
Appropriate work is shown, but two computational errors are made or
4
Appropriate work is shown, but either 90° or both 210° and 330° are not given or
4
Appropriate work is shown, and all three answers are correct but are given in radian measure.
3
Appropriate work is shown, but one conceptual error is made or
3
Appropriate work is shown, but three computational errors are made.
2
Appropriate work is shown, but one conceptual and one computational error are made or
2
90° or 210° or 330° and appropriate work is shown.
1
90°, 210°, 330°, but no work is shown
1
Student shows minimal understanding of trigonometric equations.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
or
Chapters 1–14 (pages 621–623) Part I Allow a total of 26 credits, 2 credits for each of the following. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 1 2 3 4
5 2 (1) ⫺45° 3 (1) ⫺ 4 (2) Its axis of symmetry is x ⫽ ⫺2. (3) ⫺
5 6 7 8 9 10
(4) 5.4 (3) ⫺cos 32° x⫺1 (2) 2 (1) 25 (3) 1,158 (4) 36
(4) 2.963 (4) The solution is a portion of the number line between 0 and 2 but excluding 0 and 2. 13 (4) {270°}
11 12
Chapters 1–14
107
Part II For each question, use the specific criteria to award a maximum of 2 credits. 14
15
16
17
108
Score
Explanation
2
⫺6,465 and appropriate work is shown.
1
Appropriate work is shown, but one computational error is made or
1
⫺6,465, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
4 ⫾ 2i and appropriate work is shown.
1
Appropriate work is shown, but solution is not simplified or
1
Quadratic is set up correctly, but algebraic errors are made or
1
4 ⫾ 2i, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
Decreasing exponential function, and appropriate explanation is given.
1
Decreasing exponential function, and no appropriate explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation 24 and appropriate work is shown. 25
2
⫺
1
Appropriate work is shown, but student solves for b only, not sin 2b or
1
⫺
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
24 , but no work is shown. 25
Cumulative Reviews
Part III For each question, use the specific criteria to award a maximum of 4 credits. 18
Score 4
冦(⫺2, 0), 冢 72 , ⫺ 774 冣冧 and appropriate work is shown.
3
Appropriate work is shown, but student finds only the x-values.
2
Appropriate work is shown, but student finds only one solution.
1
Multiple factoring or mathematical errors are made or
1
冦(⫺2, 0), 冢 72 , ⫺ 774 冣冧, but student uses a graphic solution
1
冦(⫺2, 0), 冢 72 , ⫺ 774 冣冧, but no work is shown.
0
19
Explanation
Score
or
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Explanation
4
Brandon is correct; the measurements are not accurate. Appropriate justification should include the Law of Cosines.
3
An appropriate method is used to justify that Brandon is correct, but one computational error is made.
2
Student correctly solves a problem using Law of Sines.
1
Student demonstrates minimal understanding of trigonometry
1
Brandon is correct, but no appropriate justification is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
or
Chapters 1–14
109
Part IV Use the specific criteria to award a maximum of 6 credits. 20
Score
Explanation
6
a 0.924 ⫽ 0.999879t b Approximately 653 years old, and appropriate work is shown. c Picasso did not paint the picture because he was not alive 653 years ago.
5
A correct equation is solved with appropriate work shown, but one computational error is made or
5
0.924 ⫽ 0.999879t and approximately 653 years old, but student omits discussion of Picasso’s artistry.
4
A correct exponential equation is solved correctly but using a method other than logarithms.
3
An incorrect exponential equation is solved correctly using logarithms.
2
0.924 ⫽ 0.999879t, but no solution or discussion of Picasso’s artistry is given or
2
Multiple errors are made in the solution of the question.
1
Student shows minimal understanding of exponential functions and logarithms.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Chapters 1–15 (pages 624–626) Part I Allow a total of 26 credits, 2 credits for each of the following. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 1 (1) 15 2 (2) a ⫽ 1, b ⫽ 7 3 (2) linear 4 (1) 1 5 (3) 27
110
Cumulative Reviews
6 7 8 9
(1) 10 (3) 3 (4) ⫺0.9 63 (2) 65
10 11 12 13
(2) (2) (3) (2)
8 II 75–95 2
Part II For each question, use the specific criteria to award a maximum of 2 credits. 14
Score
16
冢
冣 and appropriate work is shown.
2
3x ⫹ 1 1 , x ⫽ 0, 3x 3
1
Appropriate work is shown, but one computational error is made or
1
Appropriate work is shown, but one simplification error is made or
1
3x ⫹ 1 1 , but no work is shown. , x ⫽ 0, 3x 3
0
15
Explanation
Score
冢
冣
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Explanation
2
length: 10.5 feet; width: 8.5 feet, and appropriate work is shown.
1
Appropriate work is shown, but one computational error is made or
1
Appropriate work is shown, but only one dimension is given or
1
length: 10.5 feet; width: 8.5 feet, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
No, and an appropriate explanation is given, such as two people could have the same height but different weights.
1
An answer of “no” without an explanation or
1
An answer of “no” with an incorrect explanation.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Chapters 1–15
111
17
Score
Explanation
2
u ⫽ 60°, 90° and appropriate work is shown.
1
Appropriate work is shown, but one computational error is made or
1
Appropriate work is shown, but only one angle is given or
1
u ⫽ 60°, 90°, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Part III For each question, use the specific criteria to award a maximum of 4 credits. 18
19
Score 4
69.51° and appropriate work is shown.
3
Appropriate work is shown, but one computational or rounding error is made.
2
Appropriate work is shown, but two or more computational or rounding errors are made or
2
Appropriate work is shown, but one conceptual error is made, such as solving for the wrong angle.
1
Appropriate work is shown, but one conceptual error and one computational or rounding error are made or
1
69.51°, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation p , D ⫽ 300, and appropriate work is shown. 6
4
A ⫽ ⫺200, B ⫽
3
Appropriate work is shown, but one computational or rounding error is made.
2
Appropriate work is shown, but two or more computational or rounding errors are made or
2
Appropriate work is shown, but one conceptual error is made.
1
Appropriate work is shown, but one conceptual error and one computational or rounding error are made or p A ⫽ ⫺200, B ⫽ , D ⫽ 300, but no work is shown. 6
1 0
112
Explanation
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Cumulative Reviews
Part IV Use the specific criteria to award a maximum of 6 credits. 20
Score 6
Explanation a
y ⫽ 20.125x ⫹ 199.536
b Approximately 501 students, and appropriate work is shown. c 2028; students’ explanations will vary but should mention the probable inaccuracy due to extrapolation; appropriate work is shown. 5
Appropriate work is shown, but a rounding error is made or
5
Appropriate work is shown, but no discussion of the accuracy of 2028 is given.
4
Appropriate work is shown, but no scatter plot is drawn or
4
Only two of the three parts of the problem are completed or
4
Appropriate work is shown for all parts, but one conceptual error is made.
3
The scatter plot and regression equation are correct, but no other work is done.
2
Only one of the three parts of the problem is completed with work shown.
1
Student shows minimal understanding of regression equations
1
y ⫽ 20.125x ⫹ 199.536; approximately 501 students; 2028, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
or
Chapters 1–16 (pages 627–629) Part I Allow a total of 26 credits, 2 credits for each of the following. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 1
(2)
冦⫺ 12 , 3冧
6 7
2 3 4 5
(2) (2) (4) (1)
11.4 30 195° ⫺26 ⫹ 18i
8 9
(1) 0 ⬍ r ⱕ 1 (3) The rate of change is 83 percent. (3) 12 (4) every fifth person who enters the local mall
10 11 12 13
(2) 3.72 (1) 0 (3) 60 4 p (2) 3
Chapters 1–16
113
Part II For each question, use the specific criteria to award a maximum of 2 credits. 14
Score
Explanation
2
yi 5 4
1 + 3i
3 2 1 O –5 –4 –3 –2 –1 –1
–2 –3 –4 –5
15
16
114
1
2
3
4
x
5
5 – 2i
4 – 5i
1
Graph is correctly labeled, but sum is incorrect.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
x ⫽ 3.6; s ⫽ 4.695, and appropriate work is shown.
1
Appropriate work is shown, but rounding errors are made or
1
Appropriate work is shown, but only one of two values is correct or
1
x ⫽ 3.6; s ⫽ 4.695, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
y ⫽ 4.2sin 2x ⫺ 3, and appropriate work is shown.
1
An error is made in the amplitude, period, or midline.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Cumulative Reviews
17
Score
Explanation
2
720a3b2 and appropriate work is shown.
1
Binomial expansion is attempted, but an error is made or
1
720a3b2, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Part III For each question, use the specific criteria to award a maximum of 4 credits. 18
19
Score
Explanation
4
u ⫽ {63.4°, 126.9°, 243.4°, 306.9°} and appropriate work is shown.
3
Equation is solved for u, but additional angle measures in interval are not found or
3
Appropriate work is shown, but rounding errors are made.
2
Appropriate work is shown, but student finds only two solutions.
1
u ⫽ {63.4°, 126.9°, 243.4°, 306.9°}, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
4
a 58° and appropriate work is shown. b 58.5 inches and appropriate work is shown.
3
One or both solutions are incorrectly rounded or
3
An appropriate method is used, but one computational error is made.
2
Student uses an incorrect formula, but solves it correctly.
1
Minimal understanding of trigonometry is demonstrated
1
58° and 58.5 inches, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
or
Chapters 1–16
115
Part IV Use the specific criteria to award a maximum of 6 credits. 20
Score
Explanation
6
a y ⫽ 0.474t ⫹ 11.391 and appropriate work is shown. b 30.35% and appropriate work is shown. c approximately 81.5 years; explanation is given, such as it is unreasonable because it is unlikely that the model would be valid for that period of time.
5
Appropriate work is shown, but there is a rounding error or
5
Appropriate work is shown, but there is no discussion of the reliability of the extrapolated value.
4
Only two of the three parts of the question are answered.
3
Student finds a nonlinear regression equation and bases answers on that.
2
y ⫽ 0.474t ⫹ 11.391, but no other solutions or discussions are given or
2
Only one of the three parts of the question is answered.
1
Student shows minimal understanding of linear regression.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Practice Algebra 2 and Trigonometry Regents Examinations Practice Regents Examination One (pages 633–638) Part I Allow a total of 54 credits, 2 credits for each of the following. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 1 (4) 4 2 (4) x ⫽ 0 3 (3) 3 inches
116
4 5
(1) 77 1 2 (3) 4x 3y 3
Practice Algebra 2 and Trigonometry Regents Examination
6 7
5p 6 (3) ⫺1 ⫹ 2i (2)
8 9 10 11 12 13 14 15 16
(2) 64 (3) sec x y⫺x (3) y (4) trigonometric (4) 16 (2) ⫺432a3b (2) 5,700 3 (3) z⫺3 (1) (⫺1, 8)
(2) {(1, 2), (1, 3), (5, 4), (7, 5), (9, 6)} 18 (1) {⫺2, 5} 19 (4) 3兹5 ⫺ 6 20 (2) (y ⫺ 3)2 ⫽ 19 21 (1) ⫺1.25 22 (2) 2 p 3p , 23 (3) 4 4 119 24 (2) ⫺ 169 17
25 26 27
(2) (9, 10) (4) y ⫽ 3sin px y (3)
x
Part II For each question, use the specific criteria to award a maximum of 2 credits. 28
29
Score
Explanation
2
x ⫺ 3, (x ⫽ 0, ⫺3) and appropriate work is shown.
1
Appropriate work is shown, but one computational error is made or
1
x ⫺ 3 but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation 1
2
b 2兹a a 2b 2 log or log and appropriate work is shown. c c
1
b 2兹a a 2b 2 or but no work is shown or c c
1
b 2兹a a 2b 2 log or log but no work is shown. c c
1
1
0
30
Score
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Explanation
2
f ⫺1(x) ⫽ log2 x and appropriate work is shown.
1
f (x) ⫽ log2 x and appropriate work is shown
1
f ⫺1(x) ⫽ log2 x but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
or
Practice Regents Examination One
117
31
32
Score
Explanation
2
x ⫽ 9 and appropriate work is shown.
1
冦⫺ 32 , 9冧 and appropriate work is shown
1
x ⫽ 9 but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation d
Distance
2
or
t
Time
33
1
Time and distance are transposed or
1
One error on graph but some work is correct.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score 2
3 兹15 i and appropriate work is shown. ⫾ 4 4
1
The quadratic formula is used correctly, but one computational error is made or
1
15 3 ⫾ 兹 i but no work is shown. 4 4
0
118
Explanation
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Practice Algebra 2 and Trigonometry Regents Examinations
34
35
Score
Explanation
2
55.8% and appropriate work is shown.
1
28.3%, the probability of exactly two, is found and appropriate work is shown or
1
Appropriate work is shown, but one computational error is made or
1
55.8% but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score 2
Explanation a
yi 8 7
Z1
6 5 4
Z2
3 2
Z1 – Z2
1
–3 –2 –1 O 1 –1 –2
2
3
4
5
6
7
x
b 4 ⫹ 3i 1
4 ⫹ 3i without a graph or
1
Correct graph but no 4 ⫹ 3i or
1
8 ⫹ 13i and a graph depicting addition of vectors.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Practice Regents Examination One
119
Part III For each question, use the specific criteria to award a maximum of 4 credits. 36
37
120
Score
Explanation
4
No, it would take approximately 14.2 years. Appropriate work is shown. Explanations will vary, but should make use of the equation 2 ⫽ (1.05)t.
3
Appropriate work and explanation are shown, but one computational or rounding error is made or
3
14.2 years with appropriate work but no explanation.
2
Appropriate work and explanation are shown, but two computational or rounding errors are made.
1
14.2 years but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
4
240.6 pounds and appropriate work is shown.
3
An appropriate method is used, such as the Law of Cosines, but one computational or rounding error is made.
2
An appropriate method is used, but two computational errors are made.
1
Multiple mathematical errors are made
1
240.6 pounds, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
or
Practice Algebra 2 and Trigonometry Regents Examinations
38
Score 4
Explanation a
y ⫽ ⫺2cos
冢 2p5 x冣 ⫹ 2 and appropriate work is shown.
b 1.7 seconds and appropriate work is shown. 3
One of the values of a, b, or c in the equation y ⫽ acos bx ⫹ c is incorrect
3
The equation is correct, but the time is incorrect.
2
Two of the values of a, b, or c in the equation y ⫽ acos bx ⫹ c are incorrect or
2
Time is incorrect, and one of the values of a, b, or c in the equation y ⫽ acos bx ⫹ c is incorrect.
1
Only one value of a, b, or c in the equation y ⫽ acos bx ⫹ c is correct or
1
Only 1.7 seconds is correct.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
or
Part IV Use the specific criteria to award a maximum of 6 credits. 39
Score
Explanation
6
a y ⫽ 29.08x ⫹ 492.62 and appropriate work is shown. b 842 and appropriate work is shown. c 1994 and appropriate work is shown.
5
Appropriate work is shown, but one computational or rounding error is made.
4
y ⫽ 29.08x ⫹ 492.62 and 842 and appropriate work is shown
4
y ⫽ 29.08x ⫹ 492.62 and 1994 and appropriate work is shown.
3
A correct equation is shown, but parts b and c are missing or incorrect
3
The equation is incorrect, but all further work is appropriate.
2
The equation is incorrect, but some further work is appropriate.
1
An incorrect equation is shown, but part b or c is correct.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
or
or
Practice Regents Examination One
121
Practice Regents Examination Two (pages 639–645) Part I Allow a total of 54 credits, 2 credits for each of the following. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 1 (4) {4} 2 (4) x2 ⫹ 4x ⫺ 12 ⱖ 0 3 (1) ⫺7 2 4 (1) 2⫺a 5 (2) 2 6 (2)
y = 2x2 – 5
7 8 9 10
(2) (3) (3) (3)
11
(4)
12
(4)
13
(2)
14
(3)
15
(1)
16
(2)
$1,333.94 8,943.9 III y ⫽ log3 x 3 2 interquartile range p ,1 2 3 3 t⫽⫺ 2 increases, then decreases
冢
冣
17
(1)
18
(2)
19 20 21
(3) (1) (4)
22
(2)
23 24 25 26 27
(1) (2) (2) (4) (3)
p 4 1 16 19 I and II 330° 3 ⫺ 4 35.66° 48.65 1.29 120 54c2h2
Part II For each question, use the specific criteria to award a maximum of 2 credits. 28
29
Score 2
4兹x 3y 5 and appropriate work is shown.
1
Appropriate work is shown, but one computational error is made or
1
4兹x 3y 5 but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
冢
冣
2
1 2 2 , x ⫽ ⫺3, ⫺ , , 3 and appropriate work is shown. 3x ⫺ 2 3 3
1
Student multiplied instead of divided but factored correctly or
1
Factoring errors are present, which produce an incorrect answer
1
1 but no work is shown. 3x ⫺ 2
0
122
Explanation
or
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Practice Algebra 2 and Trigonometry Regents Examinations
30
31
32
33
Score
Explanation 2 i and appropriate work is shown. 3
2
3⫾
1
Fractions are not reduced or in a ⫹ bi form
1
The quadratic formula is used correctly, but one computational error is made.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
or
Explanation
2
5, ⫺1 and appropriate work is shown.
1
Appropriate work is shown with one computational error or
1
5, ⫺1 but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
a 7.9 and appropriate work is shown. b 2 and appropriate work is shown.
1
Student uses an appropriate method, but makes one computational or rounding error or
1
Student uses population standard deviation correctly or
1
7.9 and 2 but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation 77 and appropriate work is shown. 85
2
⫺
1
Identity is used correctly but one computational error is made or
1
Incorrect identity is used with correct trigonometric values or
1
⫺
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
77 but no work is shown. 85
Practice Regents Examination Two
123
34
35
Score
Explanation
2
x ⫽ 25 and appropriate work is shown.
1
Appropriate work is shown, but one computational error is made or
1
x ⫽ 25 but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
{210°, 330°} and appropriate work is shown.
1
Equation is solved with one computational error or
1
Equation is solved correctly, but only one angle is given or
1
{210°, 330°} but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Part III For each question, use the specific criteria to award a maximum of 4 credits. 36
124
Score
Explanation
4
a Price of $8.99: $14,240; price of $10.50: $16,065, and appropriate work is shown. b $12.90 and appropriate work or graph is shown. c $16,641, and appropriate work or graph is shown.
3
Appropriate work or graph is shown with one incorrect solution or
3
$14,240, $16,065, $12.90, and $16,641 with a rough sketch but no labels or explanation.
2
Appropriate work or graph is shown, but two computational or rounding errors are made.
1
$14,240, $16,065, $12.90, and $16,641, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Practice Algebra 2 and Trigonometry Regents Examinations
37
Score 4
25 and appropriate work is shown. 729
3
25 and appropriate method is indicated, but insufficient work is shown or 729
3
Appropriate work is shown, but one computational error is made.
2
24 , student finds exactly two rather than at least two. 729
1
An appropriate method is indicated, but multiple mathematical errors are made or
1
25 but no work is shown. 729
0
38
Explanation
Score
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Explanation
4
66.4 feet and appropriate work is shown.
3
An appropriate method is used, such as the Law of Sines, but one computational or rounding error is made.
2
The length of the hypotenuse is found, but no other correct work is shown.
1
The initial setup for the problem is correct, but multiple errors are made
1
Student uses the Pythagorean theorem to find the height of the flagpole or
1
66.4 feet but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
or
Practice Regents Examination Two
125
Part IV Use the specific criteria to award a maximum of 6 credits. Score
39
6
Explanation a, c g(x) = 2cos x – p
(
2
)
f (x) = 2cos x
冢
b
g(x) ⫽ 2cos x ⫺
d
g(x) ⫽ 2sin x x h(x) ⫽ 4cos 3
e
p 2
冣
5
Appropriate graph is shown, but one equation is incorrect.
4
Appropriate graph is shown, but two equations are incorrect.
3
Graphs of f(x) and g(x) are correct, but no other work is correct.
2
The graph of f(x) or g(x) is correct, but all further work is incorrect or
2
One equation is incorrect, and graphs are incorrect or not shown.
1
An incorrect graph is correctly shifted, but no other work is correct or
1
Two equations are incorrect, and graphs are incorrect or not shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Practice Regents Examination Three (pages 646–652) Part I Allow a total of 54 credits, 2 credits for each of the following. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. 1 2 3 4 5 6 7 8
(4) 164 (3) real, irrational, and unequal (3) 95 (2) 2 p (2) 2 (2) 29 (3) 0.6% (1) 2sin2 u ⫹ sin u ⫺ 1 ⫽ 0
126
9 10
(2) {12} (4)
11
(4)
12
(1)
13
(3)
14
(4)
兹2 ⫺ 2 exponential 4 ⫺ 5 1⬍y⬍6 21 ⫹ 8兹5 11
Practice Algebra 2 and Trigonometry Regents Examinations
15 (3) ⫺3c ⫺ 2 16 (1) f ⫺1(x) ⫽ log3 x 3
17
(2)
兺 (3t 2 ⫺ 4t ⫹ 12)
t⫽1
18 19 20 21
(1) 780 1 a (2) 2 (3) III (4) acute or obtuse
22 23
10 p 3 (2) ⫺527 ⫹ 336i (1)
24 25
(4) ⫺
26 27
兹3
2 (3) {⫺2, ⫺1, 1, 2}
(2) 0 and 4 (4) $36
Part II For each question, use the specific criteria to award a maximum of 2 credits. 28
29
30
31
Score
Explanation
2
x2 ⫺ 6x ⫹ 10 ⫽ 0 and appropriate work is shown.
1
Appropriate work is shown, but one computational error is made or
1
x2 ⫺ 6x ⫹ 10 and appropriate work is shown or
1
x2 ⫺ 6x ⫹ 10 but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
$1,074.50 and appropriate work is shown.
1
Appropriate work is shown, but one computational or rounding error is made or
1
$1,074 but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
g(x) ⫽ 兩 x ⫹ 3 兩 ⫺ 4 and appropriate work is shown.
1
Student has either the vertical shift or horizontal shift correct but not both or
1
g(x) ⫽ 兩 x ⫹ 3 兩 ⫺ 4 but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
{0, 5} and appropriate work is shown.
1
x ⫽ 0 or x ⫽ 5 but not both
1
{0, 5} but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
or
Practice Regents Examination Three
127
32
33
34
Score
Explanation 2 . 3 b arithmetic; there is a common difference of ⫺1.25.
2
a
1
Student correctly identifies type of sequence but does not give explanations or
1
One sequence is correctly identified and explained, but the other is not.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
Katie is correct. The cosine function is symmetric about the y-axis.
1
Katie is correct, but not explanation is provided.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score 2
Explanation a b
128
28 and appropriate work is shown. 55 26 and appropriate work is shown. 33 28 26 or and appropriate work is shown or 55 33
1
Either
1
28 26 and but no work is shown. 55 33
0
35
geometric; there is a common ratio of
Score
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure. Explanation
2
1.7 seconds and appropriate work or graph is shown.
1
Student sets up a correct quadratic equation, but one computational error is made or
1
1.7 seconds but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Practice Algebra 2 and Trigonometry Regents Examinations
Part III For each question, use the specific criteria to award a maximum of 4 credits. 36
37
Score
Explanation
4
a 15,145 people and appropriate work is shown. b 36.68 years and appropriate work is shown.
3
Appropriate work is shown, but one computational or rounding error is made or
3
15,145 people with appropriate work or diagram, and 36.68 years without an algebraic solution.
2
15,145 people or 36.68 years, but not both, and appropriate work is shown
2
Appropriate work is shown, but two computational or rounding errors are made.
1
15,145 people and 36.68 years, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Score
or
Explanation
4
u ⫽ {0°, 45°, 153.4°, 180°, 225°, 333.4°} and appropriate work is shown.
3
An appropriate method is used, but one computational error is made or
3
An appropriate method is used, but one or two angles are missing.
2
An appropriate method is used, but two computational errors are made or
2
An appropriate method is used, but three or four angles are missing.
1
Multiple computational/formula errors are made
1
u ⫽ {0°, 45°, 153.4°, 180°, 225°, 333.4°} but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
or
Practice Regents Examination Three
129
38
Score
Explanation
4
a 6.3 million and appropriate work is shown. b 0.7 million and appropriate work is shown. c Range ⫽ 1.9 million, Interquartile range ⫽ 1.4 million and appropriate work is shown. d Answers will vary, but student should indicate some knowledge of how statistics, although correct, can lead people to come to false conclusion. Alternatively, student may provide a regression analysis.
3
One of the parts a, b, c, or d is incorrect.
2
Two of the parts a, b, c, or d are incorrect.
1
Only one of the parts a, b, c, or d is correct.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Part IV Use the specific criteria to award a maximum of 6 credits. 39
130
Score
Explanation
6
CT ⫽ 2.3 miles, TH ⫽ 2.6 miles and appropriate work is shown. Police cars from point C will arrive first.
5
Appropriate work is shown, but one computational or rounding error is made.
4
CT ⫽ 2.3 miles and TH ⫽ 2.6 miles and appropriate work is shown, but student omits point C.
3
Appropriate work is shown, but two computational or rounding errors are made.
2
Appropriate work is shown, but only point C is correct.
1
Student shows minimal understanding of trigonometric relationships
1
CT ⫽ 2.3, TH ⫽ 2.6 miles, point C, but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response that was obtained by an obviously incorrect procedure.
Practice Algebra 2 and Trigonometry Regents Examinations
or
