Holt Algebra 2

Holt Algebra 2

Know-It NotebookTM

Copyright © by Holt, Rinehart and Winston All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Requests for permission to make copies of any part of the work should be mailed to the following address: Permissions Department, Holt, Rinehart and Winston, 10801 N. MoPac Expressway, Building 3, Austin, Texas 78759. HOLT and the “Owl Design” are trademarks licensed to Holt, Rinehart and Winston, registered in the United States of America and/or other jurisdictions. Printed in the United States of America ISBN 0-03-078424-7 1 2 3 4 5 6 7 8 9

862

09 08 07 06

Copyright © by Holt, Rinehart and Winston. All rights reserved.

Contents Using the Know-It Notebook ........................................................................................................iv Note Taking Strategies ..................................................................................................................2 Chapter 1.......................................................................................................................................4 Chapter 2.....................................................................................................................................30 Chapter 3.....................................................................................................................................56 Chapter 4.....................................................................................................................................72 Chapter 5.....................................................................................................................................92 Chapter 6...................................................................................................................................120 Chapter 7...................................................................................................................................144 Chapter 8...................................................................................................................................166 Chapter 9...................................................................................................................................186 Chapter 10.................................................................................................................................202 Chapter 11.................................................................................................................................224 Chapter 12.................................................................................................................................244 Chapter 13.................................................................................................................................260 Chapter 14.................................................................................................................................280


iii

Algebra 2

USING THE KNOW-IT NOTEBOOK ™ This Know-It Notebook will help you take notes, organize your thinking, and study for quizzes and tests. There are Know-It Notes™ pages for every lesson in your textbook. These notes will help you identify important mathematical information that you will need later.

Know-It Notes Vocabulary One good note-taking practice is to keep a list of the new vocabulary. • Use the page references or the glossary in your textbook to find each definition and a clarifying example. • Write each definition and example on the lines provided. Lesson Objectives Another good note-taking practice is to know the objective the content covers. Key Concepts Key concepts from each lesson are included. These are indicated in your student book with the KIN logo. • Write each answer in the space provided. • Check your answers with your book. • Ask your teacher to help you with any concept that you don’t understand.

CHAPTER

Vocabulary

LESSON 1-1 CONTINUED

1 8. Interval notation (p. 7): ___________________________________________ a method of describing an interval by specifying all The table contains important vocabulary terms from Chapter 1. As you work through the chapter, fill in the page number, definition, and a clarifying example.

______________________________________________________________ numbers between two endpoints using the symbols [ and ] to include an ______________________________________________________________ endpoint and ( and ) to exclude an endpoint.

Term dependent variable

domain

function

Page 52

44

45

independent variable

52

parent function

67

principal root

radicand

21

21

Definition

Clarifying Example

9. Set-builder notation (p. 8): _________________________________________ a method of describing a set by using the

The output of a function; a variable whose value depends on the value of the input, or independent variable.

For y ⫽ 2x ⫹ 1, y is the dependent variable.

The set of all possible input values of a relation or function.

The domain of the function f(x) ⫽ 兹x 苶 is {x 冨 x ⱖ 0}.

______________________________________________________________ properties of the elements of the set.

Key Concepts 10. Real Numbers (p. 6): Real Numbers Rational Numbers (R)

0.5 –5

Function: {(0, 5), (1, 3), A relation in which (2, 1)} every input is paired with exactly one output. Not a function: {(0, 1), (0, 3), (2, 1)}

f (x) ⫽ x 2 is the parent function for g(x) ⫽ x 2 ⫹ 4 and h(x) ⫽ 5(x ⫹ 2)2 ⫺ 3

The positive root of a number, indicated by the radical sign.

36 has two square roots, 6 and ⫺6. The principal square root of 36 is 兹36 苶 ⫽ 6.


4

π

Whole Numbers (W) Natural Numbers (N)

兹2 苶

–2

兹7 苶 ᎏ 2

2

1

3

— 5.312

e

11. Methods of Representing Intervals (p. 8): WORDS Numbers less than 3 Numbers greater than or equal to ⫺2 Numbers between 2 and 4

The expression under a Expression: 兹x 苶 ⫹3 radical sign. Radicand: x ⫹ 3


ⴚ5兹3 苶

9

0

The input of a function; For y ⫽ 2x ⫹ 1, x is the a variable whose value independent variable. determines the value of the output, or dependent variable. The simplest function with the defining characteristics of the family. Functions in the same family are transformations of their parent function.

Irrational Numbers

7 2__

Integers (Z)

Numbers 1 through 3

Algebra 2


iv

NUMBER LINE

INEQUALITY

INTERVAL NOTATION

2

3

4

5

x⬍3

(⫺⬁, 3)

-4 -3 -2 -1

0

1

2

x ⱖ ⫺2

[⫺2, ⬁)

-1

-1

0

1

0

1

2

3

4

5

2⬍x⬍4

(2, 4)

-2 -1

0

1

2

3

4

1ⱕxⱕ3

[1, 3]

7

Algebra 2

Algebra 2

Chapter Review Complete Chapter Review problems that follow each lesson. This is a good review before you take the chapter test. • Write each answer in the space provided. • Check your answers with your teacher or another student. • Ask your teacher to help you understand any problem that you answered incorrectly. Big Ideas The Big Ideas have you summarize the important chapter concepts in your own words. Putting ideas in your words requires that you think about the ideas and understand them. This will also help you remember them. • Write each answer in the space provided. • Check your answers with your teacher or another student. • Ask your teacher to help you understand any question that you answered incorrectly.

CHAPTER

Chapter Review CHAPTER

1

Big Ideas

1 Answer these questions to summarize the important concepts from Chapter 1 in your own words.

1-1 Sets of Numbers Order the given numbers from least to greatest. Then classify each number by the subsets of the real numbers to which it belongs. 1

1. 7ᎏ4ᎏ, 兹21 苶, ⫺4.15, 3.6 苶6 苶 1

rational; real

2. 7ᎏ4ᎏ 4. 兹21 苶

irrational; real 1

6. ⫺兹10 苶, 7, ᎏ5ᎏ, ⫺3 7. ⫺兹10 苶

9. 7

1. Explain how the various sets of numbers are related.

៮, 兹21 ⫺4.15, 3.66 苶, 7ᎏ1 ᎏ 4 3. ⫺4.15 5. 3.6 苶6 苶

Real Numbers consist of Rational Numbers and Irrational Numbers. The Rational Numbers consist of Integers, Whole Numbers, and Natural Numbers

rational; real rational; real

2. Explain how the Additive Inverse Property differs from the Multiplicative Inverse Property.

⫺兹10 苶, ⫺3, ᎏ1 ᎏ, 7 5

irrational; real

whole; integer; rational; real

1

rational; real

8. ᎏ5ᎏ

The Additive Inverse Property states that the sum of a number and its opposite is 0. The Multiplicative Inverse Property states the product of a nonzero number and its reciprocal is 1.

integer; rational; real

10. ⫺3

3. Explain how to simplify an algebraic expression.

Rewrite each set in the indicated notation. 11. {x 冨 ⫺2 ⱕ x ⬍ 4};

12.

interval notation

–5

0

5

To simplify an algebraic expression, combine like terms by adding or subtracting the coefficients. Like terms have the same exponent raised to the same power.

;

set-builder notation {x 冨 ⫺3 ⬍ x ⱕ 2}

(⫺2, 4)

4. What makes a relation a function? Explain how the inputs and outputs of a function are related.

1-2 Properties of Real Numbers Identify the property demonstrated by each equation. 13. t ⫹ 4 ⫽ 4 ⫹ t Commutative Property of Addition 15. 2a ⫹ 2b ⫽ 2(a ⫹ b)

A relation in which the first coordinate is never repeated is called a function. A function has only one output for each input, so each element of the domain is mapped to exactly one element in the range. Even though a function cannot map a single input to more than one output, two or more different inputs can be mapped to the same output.

14. a ⫹ (6 ⫹ y) ⫽ (a ⫹ 6) ⫹ y Associative Property of Addition

For more review of Chapter 1:

16. 0 ⫹ 21 ⫽ 21

• Complete the Chapter 1 Study Guide and Review on pages 76–79 of your Distributive Property

Identity Property of Zero

textbook.

• Complete the Ready to Go On quizzes on pages 43 and 75 of your

17. Use mental math to find a 25% discount on an item that costs $160. Explain your steps. $40; Sample answer: 25% ⫽ 10% ⫹ 10% ⫹ 5%; 10% of $160 is $16 and 5% is $8. $16 ⫹ $16 ⫹ $8 ⫽ $40 Copyright © by Holt, Rinehart and Winston. 24 All rights reserved.

Copyright © by Holt, Rinehart and Winston.

textbook.

Algebra 2


1

29

Algebra 2

Algebra 2

Note Taking Strategies Taking good notes is very important in many of your classes and will be even more important when you take college classes. This Notebook was designed to help you get started. Here are some other steps that can help you take good notes. Getting Ready 1. Use a loose-leaf notebook. You can add pages to this as where and when you want to. It will help keep you organized. During the Lecture 2. If you are taking notes during a lecture, write the big ideas. Use abbreviations to save time. Do not worry about spelling or writing every word. Use headings to show changes in the topics discussed. Use numbering or bullets to organize supporting ideas under each topic heading. Leave space before each new heading so that you can fill in more information later. After the Lecture 3. As soon as possible after the lecture, read through your notes and add any information you can so that when you review your notes later, they make sense. You should also summarize the information into key words or key phrases. This will help your comprehension and will help you process the information. These key words and key phrases will be your memory cues when you are reviewing or taking a test. At this time you may also want to write questions to help clarify the meaning of the ideas and facts. 4. Read your notes out loud. As you do this, state the ideas in your own words and do as much as you can by memory. This will help you remember and will also help with your thinking process. It helps you think about and understand the information. 5. Reflect upon the information you have learned. Ask yourself how new information relates to information you already know. Ask how this relates to your personal experience. Ask how you can apply this information and why it is important. Before the Test 6. Review your notes. Don’t wait until the night before the test to do this review. Do frequent reviews. Don’t just read through your notes. Put the information in your notes into your own words. If you do this you will be able to connect the new material with material you already know. You will be better prepared for tests. You will have less test anxiety and will have better recall. 7. Summarize your notes. This should be in your own words and should only include the main points that you need to remember. This will help you internalize the information.


2

Algebra 2

CHAPTER

Vocabulary

1 The table contains important vocabulary terms from Chapter 1. As you work through the chapter, fill in the page number, definition, and a clarifying example. Term

Page

Definition

Clarifying Example

dependent variable

domain

function

independent variable

parent function

principal root

radicand


4

Algebra 2

CHAPTER 1 VOCABULARY CONTINUED

Term

Page

Definition

Clarifying Example

reflection

relation scientific notation

set set-builder notation

subset

transformation

translation


5

Algebra 2

LESSON

Sets of Numbers

1-1 Lesson Objectives (p. 6): ______________________________________________________________ ______________________________________________________________

Vocabulary 1. Set (p. 6): _____________________________________________________ ______________________________________________________________ 2. Element (p. 6): __________________________________________________ ______________________________________________________________ 3. Subset (p. 6): ___________________________________________________ ______________________________________________________________ 4. Empty set (p. 6): ________________________________________________ ______________________________________________________________ 5. Roster notation (p. 7): ____________________________________________ ______________________________________________________________ 6. Finite set (p. 7): _________________________________________________ ______________________________________________________________ 7. Infinite set (p. 7): ________________________________________________ ______________________________________________________________


6

Algebra 2


8. Interval notation (p. 7): ___________________________________________ ______________________________________________________________ ______________________________________________________________ 9. Set-builder notation (p. 8): _________________________________________ ______________________________________________________________

Key Concepts 10. Real Numbers (p. 6): Real Numbers Rational Numbers (R)

Irrational Numbers

Integers (Z) Whole Numbers (W) Natural Numbers (N)

11. Methods of Representing Intervals (p. 8): WORDS Numbers less than 3 Numbers greater than or equal to 2 Numbers between 2 and 4 Numbers 1 through 3


NUMBER LINE

-1

0

1

2

3

4

5

-4 -3 -2 -1

0

1

2

-1

0

1

2

3

4

5

-2 -1

0

1

2

3

4

INEQUALITY

7

INTERVAL NOTATION

Algebra 2


12. Methods of Set Notation (p. 9): WORDS

ROSTER NOTATION

INTERVAL NOTATION

SET-BUILDER NOTATION

All real numbers except 1

Positive odd numbers

Numbers within 3 units of 2

13. Get Organized Complete the table by showing the correct notation for each example. (p. 9). SET

ROSTER NOTATION

INTERVAL NOTATION

SET-BUILDER NOTATION

1, 2, 3, 4, and 5

2 n 2

Whole numbers less than 3


8

Algebra 2

LESSON

Properties of Real Numbers


Key Concepts 1. Properties of Real Numbers—Identities and Inverses (p. 14): WORDS

NUMBERS

ALGEBRA

Additive Identity Property

Multiplicative Identity Property

Additive Inverse Property:

Multiplicative Inverse Property


9

Algebra 2


2. Properties of Real Numbers—Addition and Multiplication (p. 15): WORDS

NUMBERS

ALGEBRA

Closure Property

Commutative Property

Associative Property

Distributive Property

3. Get Organized In each box, write an example of the property indicated. (p. 16). PROPERTY

ADDITION

MULTIPLICATION

Identity Inverse Associative Commutative Distributive


10

Algebra 2

LESSON

Square Roots


Vocabulary 1. Radical symbol (p. 21): ___________________________________________ ______________________________________________________________ 2. Radicand (p. 21): ________________________________________________ ______________________________________________________________ 3. Principal root (p. 21): _____________________________________________ ______________________________________________________________ 4. Rationalize the denominator (p. 22): _________________________________ ______________________________________________________________ ______________________________________________________________ 5. Like radical terms (p. 23): _________________________________________ ______________________________________________________________


11

Algebra 2


Key Concepts 6. Properties of Square Roots (p. 22): WORDS

NUMBERS

ALGEBRA

Product Property of Square Roots

Quotient Property of Square Roots

7. Get Organized Write examples of each operation with square roots. (p. 23).

Multiplying:

Adding:


Dividing:

Square Roots

Subtracting:

12

Algebra 2

LESSON

Simplifying Algebraic Expressions


Key Concepts 1. Get Organized In each box, write key words that may indicate each operation. (p. 29).

Addition:

Subtraction:

Key Words Division: Multiplication:


13

Algebra 2

LESSON

Properties of Exponents


Vocabulary 1. Scientific notation (p. 36): _________________________________________ ______________________________________________________________

Key Concepts 2. Zero and Negative Exponents (p. 35): WORDS

NUMBERS

ALGEBRA

Zero Exponent Property

Negative Exponent Property


14

Algebra 2


3. Properties of Exponents (p. 35): WORDS

NUMBERS

ALGEBRA

Product of Powers Property

Quotient of Powers Property

Power of a Powers Property

Power of a Product Property

Power of a Quotient Property

4. Get Organized Provide a numerical and algebraic example of each property. (p. 38). PROPERTY

NUMERICAL EXAMPLE ALGEBRAIC EXAMPLE

Product of Powers Quotient of Powers Power of Powers Power of a Product Power of a Quotient


15

Algebra 2

LESSON

Relations and Functions


Vocabulary 1. Relation (p. 44): _________________________________________________ ______________________________________________________________ 2. Domain (p. 44): _________________________________________________ ______________________________________________________________ 3. Range (p. 44): __________________________________________________ ______________________________________________________________ 4. Function (p. 45): ________________________________________________ ______________________________________________________________ ______________________________________________________________

Key Concepts 5. Vertical-Line Test (p. 46):


16

Algebra 2


6. Get Organized In each box, give an example of a table, a graph, and a set of ordered pairs. (p. 46).

Relation

Function


17

Algebra 2

LESSON

Function Notation


Vocabulary 1. Function notation (p. 51): _________________________________________ ______________________________________________________________ ______________________________________________________________ 2. Dependent variable (p. 52): ________________________________________ ______________________________________________________________ 3. Independent variable (p. 52): ______________________________________ ______________________________________________________________

Key Concepts 4. Get Organized In each blank, fill in the missing portion of the label. (p. 53).

put

((x, f (x))

variable


put

variable

18

Algebra 2

Exploring Transformations

LESSON


Vocabulary 1. Transformation (p. 59): ___________________________________________ ______________________________________________________________ 2. Translation (p. 59): ______________________________________________ ______________________________________________________________ 3. Reflection (p. 60): _______________________________________________ ______________________________________________________________ 4. Stretch (p. 61): _________________________________________________ ______________________________________________________________ 5. Compression (p. 61): _____________________________________________ ______________________________________________________________

Key Concepts 6. Translations (p. 59): HORIZONTAL TRANSLATION

VERTICAL TRANSLATION

y

y (1, 4)

4

4 (1, 2)

2 units

(4, 2)

2

2 3 units

0

2

(1, 2)

x 4


0

19

2

x 4

Algebra 2


7. Reflections (p. 60): REFLECTION ACROSS y-axis

REFLECTION ACROSS x-axis

y

y

(−1, 2)

(1, 2)

1 unit

1 unit x

−2

0

2

(1, 2) 2 units x

0 −2

2

2 units (1, −2)

8. Stretches and Compressions (p. 61): HORIZONTAL

VERTICAL

STRETCH

COMPRESSION


20

Algebra 2


9. Get Organized In each box, describe the transformations indicated by each rule. (p. 62).

(x, y )

(bx, y)

(x, y )

(x + h, y )


Transformations

21

(x, y)

(−x, y )

(x, y)

(x, ay )

Algebra 2

LESSON

Introduction to Parent Functions


Vocabulary 1. Parent function (p. 67): ___________________________________________ ______________________________________________________________

Key Concepts 2. Parent functions (p. 67): FAMILY

CONSTANT LINEAR QUADRATIC CUBIC SQUARE ROOT

RULE GRAPH

DOMAIN RANGE INTERSECTS y-axis


22

Algebra 2


3. Get Organized In each box, give the appropriate information for a translation of the parent function 3 units up. (p. 70). Transformed Parent Functions FAMILY

LINEAR

QUADRATIC

SQUARE ROOT

RULE GRAPH

DOMAIN RANGE INTERSECTS y-axis


23

Algebra 2

CHAPTER

Chapter Review

1 1-1 Sets of Numbers Order the given numbers from least to greatest. Then classify each number by the subsets of the real numbers to which it belongs. 1

1. 74, 21 , 4.15, 3.6 6 1

2. 74

3. 4.15

4. 21

5. 3.6 6 1

6. 10 , 7, 5, 3 7. 10

9. 7

1

8. 5

10. 3

Rewrite each set in the indicated notation. 11. {x 2 x 4};

12.

interval notation

–5

0

5

;

set-builder notation

1-2 Properties of Real Numbers Identify the property demonstrated by each equation. 13. t 4 4 t

14. a (6 y) (a 6) y

15. 2a 2b 2(a b)

16. 0 21 21

17. Use mental math to find a 25% discount on an item that costs $160. Explain your steps.


24

Algebra 2

CHAPTER 1 REVIEW CONTINUED

1-3 Square Roots 18. Margaret is putting baseboard around the bottom edge of a square-shaped room. The room is 196 ft 2. If the baseboard comes in lengths of 10 feet, how many pieces of baseboard should she buy to place baseboard around the entire room?

Simplify each expression. 80 19. 5

20. 72

21. 18 28

22. 950 42

1-4 Simplifying Algebraic Expressions Evaluate each expression for the given values of the variables. 23. 12ab ab 2 for a 3 and b 4

2

2ab 24. for a 2 and b 3 2 5a b

Simplify each expression. 25. 7x 2 5y 9x 2 y

26. 3(2x y) 5x 6y

1-5 Properties of Exponents Simplify each expression. Assume all variables are nonzero.

2

5 3

27. x 8y8

6a b 28. 2

29. 6(m 2n3)3

x 2y4 30. 10

3ab

3

y

31. One parasec is about 3.26 light-years and 1 light-year is about 5.88 1012 miles. Find the number of miles in one parasec. Copyright © by Holt, Rinehart and Winston. All rights reserved.

25

Algebra 2


1-6 Relations and Functions Give the domain and range for each relation. Then tell whether the relation is a function. 32.

33. 2

1

4 2

6 8

34.

Perimeter of Square 4 8 12 16

Area of Square 1 4 9 16

y 4 2 –4 –2 –2

x 2

4

–4

1-7 Function Notation For each function, determine f(–1), f(0), and f (2). 35. f(x) x 2 4

36. f(x) 8 x 3

37. f(x) 3x 4

38. A wood planer costs $1.50 to turn on and $0.75 per minute of use. a. Write a function to represent the cost of the wood planer per number of minutes used.


26

Algebra 2


b. Graph the function. y 4 2

x

–4 –2 –2

2

4

–4

c. Give the value of the function for an input of 12 and explain its real-world meaning.

1-8 Exploring Transformations 39. Use a table to perform the transformation of y f(x). Graph the transformed function on the same coordinate plane as the original function.

y 6 4 2

translation up 3 units –2

40. The graph shows the gross pay that you would make working a particular number of hours per week. Sketch a graph to represent an hourly rate increase of $1 per hour and identify the transformation of the original graph that it represents.

400

2

4

6

y

320 240 160 80 0


x

27

x 8 16 24 32 40

Algebra 2


1-9 Introduction to Parent Functions Identify the parent function for g from its equation. Then graph g on your calculator and describe what transformation of the parent function it represents. 41. g(x) 3x 2

42. g(x) x 3 1

1 43. g(x) 2x 2 3

44. Graph the relationship between the number of minutes spent studying and the score on the math quiz. Identify which parent function best describes the data. Then use the graph to estimate the score on a quiz when 40 minutes are spent studying. Minutes Studying

10 60 50 0

100

y

80 60 40 20

35 25

0

x 20

40

60

Score of 65 95 85 50 80 70 Math Quiz


28

Algebra 2

CHAPTER

Big Ideas

1 Answer these questions to summarize the important concepts from Chapter 1 in your own words. 1. Explain how the various sets of numbers are related.

2. Explain how the Additive Inverse Property differs from the Multiplicative Inverse Property.

3. Explain how to simplify an algebraic expression.

4. What makes a relation a function? Explain how the inputs and outputs of a function are related.


• Complete the Chapter 1 Study Guide and Review on pages 76–79 of your textbook.

• Complete the Ready to Go On quizzes on pages 43 and 75 of your textbook.


29

Algebra 2

CHAPTER

Vocabulary


Page

Definition

Clarifying Example

absolute value function

correlation

identity

indirect measurement

line of best fit


30

Algebra 2


Term

Page

Definition

Clarifying Example

linear function

proportion rate

scale factor

slope

y-intercept


31

Algebra 2

Solving Linear Equations and 2-1 Inequalities

LESSON

Lesson Objectives (p. 90): ______________________________________________________________ ______________________________________________________________

Vocabulary 1. Equation (p. 90): ________________________________________________ ______________________________________________________________ 2. Solution set of an equation (p. 90): __________________________________ ______________________________________________________________ 3. Linear equation in one variable (p. 90): ______________________________ ______________________________________________________________ 4. Identity (p. 92): _________________________________________________ ______________________________________________________________ 5. Contradiction (p. 92): _____________________________________________ ______________________________________________________________ 6. Inequality (p. 90): _______________________________________________ ______________________________________________________________


32

Algebra 2


Key Concepts 7. Properties of Equality (p. 90): WORDS

NUMBERS

ALGEBRA

Addition

Subtraction

Multiplication

Division

8. Inequalities—Multiplying or Dividing by a Negative Number (p. 93): WORDS


NUMBERS

33

ALGEBRA

Algebra 2


9. Get Organized Note the similarities and differences in the properties and methods you use. (p. 93).

Similarities:


Solving Equations and Inequalities

34

Differences:

Algebra 2

LESSON

Proportional Reasoning


Vocabulary 1. Ratio (p. 97): ___________________________________________________ ______________________________________________________________ 2. Proportion (p. 97): _______________________________________________ ______________________________________________________________ 3. Rate (p. 98): ___________________________________________________ ______________________________________________________________ 4. Similar (p. 99): __________________________________________________ ______________________________________________________________ 5. Indirect measurement (p. 99): ______________________________________ ______________________________________________________________

Key Concepts 6. Cross Product Property (p. 97): WORDS


NUMBERS

35

ALGEBRA

Algebra 2


5. Get Organized In each box, write examples of each item that relate to the concept of proportions. (p. 100).

Nonproportions:

Proportions:

Ratios and Proportions Similar figures:


Indirect measurement:

36

Algebra 2

LESSON

Graphing Linear Functions


Vocabulary 1. Linear function (p. 105): __________________________________________ ______________________________________________________________ 2. Slope (p. 106): __________________________________________________ ______________________________________________________________ 3. y-intercept (p. 106): ______________________________________________ ______________________________________________________________ 4. x-intercept (p. 106): ______________________________________________ ______________________________________________________________ 5. slope-intercept form (p. 107): ______________________________________ ______________________________________________________________


37

Algebra 2


Key Concepts 6. Vertical and Horizontal Lines (p.108): VERTICAL LINES

HORIZONTAL LINES

y

y

x

x

6. Get Organized Complete the graphic organizer for linear functions. (p. 109).

Characteristics:

Definition:

Linear Function Nonexamples:

Examples:


38

Algebra 2

LESSON

Writing Linear Functions


Vocabulary 1. Point-slope form (p. 116): _________________________________________ ______________________________________________________________

Key Concepts 2. Slope Formula (p. 116): WORDS

ALGEBRA

GRAPH

3. Point-Slope Form (p. 117):


39

Algebra 2


4. Parallel and Perpendicular Lines (p. 119): WORDS

GRAPH

Parallel Lines

ALGEBRA 8

y

x –8

8

Perpendicular Lines 4

y

x –4

4

–4

5. Get Organized In each box, write any appropriate formulas and examples of equations. (p. 120).

Slope-intercept form:

Point-slope form:

Lines

Perpendicular:

Parallel:


40

Algebra 2

LESSON

Linear Inequalities In Two Variables


Vocabulary 1. Linear inequality (p. 124): _________________________________________ ______________________________________________________________ 2. Boundary line (p. 124): ___________________________________________ ______________________________________________________________

Key Concepts 3. Get Organized Give examples of inequalities that are solved for y and those in other forms. (p. 127). DASHED LINE SHADE ABOVE


DASHED LINE SHADE BELOW

SOLID LINE SHADE ABOVE

41

SOLID LINE SHADE BELOW

Algebra 2

LESSON

Transforming Linear Functions


Key Concepts 1. Translations and Reflections (p. 134): TRANSLATIONS AND REFLECTIONS Translations

Reflections


42

Algebra 2


2. Stretches and Compressions (p. 135): STRETCHES AND COMPRESSIONS Horizontal

Vertical

3. Get Organized In each box, give an example of the indicated transformation of the parent function f(x) x. Include an equation and a graph. (p. 137)

Reflection: across the x-axis g(x) = –x

Translation: g(x) = x + 3

f (x ) = x Compression: vertical g(x) = 0.5x

Stretch: vertical g(x) = 2x


43

Algebra 2

LESSON

Curve Fitting Using Linear Models


Vocabulary 1. Regression (p. 142): _____________________________________________ ______________________________________________________________ 2. Correlation (p. 142): _____________________________________________ ______________________________________________________________ 3. Line of best fit (p. 142): ___________________________________________ ______________________________________________________________ 4. Correlation coefficient (p. 143): _____________________________________ ______________________________________________________________

Key Concepts 5. Properties of the Correlation Coefficient r (p. 143):


44

Algebra 2


5. Get Organized Make a scatter plot for each type of correlation and estimate the r-value. (p. 145). CORRELATION

SCATTER PLOT

ESTIMATED r-VALUE

Strong positive

Weak positive

No correlation

Weak negative

Strong negative


45

Algebra 2

Solving Absolute-Value Equations and 2-8 Inequalities

LESSON


Vocabulary 1. Disjunction (p. 150): _____________________________________________ ______________________________________________________________ 2. Conjunction (p. 150): _____________________________________________ ______________________________________________________________ 3. Absolute value (p. 151): __________________________________________ ______________________________________________________________

Key Concepts 4. Absolute Value (p. 151): WORDS

NUMBERS

ALGEBRA

5. Absolute-Value Equations and Inequalities (p. 151): For all real numbers x and all positive real numbers a:


46

Algebra 2


6. Solving an Absolute Value Inequality (p. 152): TO SOLVE AN ABSOLUTE-VALUE INEQUALITY 1. 2. 3. 7. Get Organized Use the flowchart to explain the decisions and steps needed to solve an absolute-value equation or inequality. (p. 153).

Equation or inequality? Equation Disjunction or conjunction? Disjunction


Conjunction

47

Algebra 2

LESSON

Absolute Value Functions


Vocabulary 1. Absolute-value function (p. 158): ____________________________________ ______________________________________________________________

Key Concepts 2. Absolute-Value Parent Function (p. 158): THE ABSOLUTE-VALUE PARENT FUNCTION f(x) x Domain:

y x

x

4

y

Range: x –4

4

Vertex: –4

3.

Vertex of an Absolute-Value Function (p. 159):


48

Algebra 2


4. Get Organized Fill in the table with examples of absolute-value transformations. (p. 160). TRANSFOR- ABSOLUTEMATION VALUE FUNCTION FUNCTION

TRANSFORMED

GRAPH

Vertical translation

Horizontal translation

(h, k) translation

Stretch

Compression


49

Algebra 2

CHAPTER

Chapter Review

2 2-1 Solving Linear Equations and Inequalities Solve. 4

1. 18 6x 4x

2. 3(5x 1) 8

3. 14 17x 27 7x

4. 3(x 2) 5(x 5) 3

Solve and graph. 5. 32 8 6y

6. 3 9x 39

7. 4x 3(5x 12) 8

8. 8(2t 1) 4(7t 7)

Write an equation or inequality, and solve. 9. A barbeque catering company charges a $45 set up fee plus $18 per person. The cost of the annual picnic cannot exceed $720. How many people can attend the barbecue?

2-2 Proportional Reasoning Solve each proportion. x

4x 8 11. 9 11

9

10. 8 3 5.2

3.2

12. x 6

4 5 13. 5 3x 2

14. If a 4-feet tall child standing on the beach casts a 6 foot shadow, how long a shadow would a 10-feet high lifeguard station cast at the same time of day?


50

Algebra 2

LESSON CONTINUED CHAPTER2-1 2 REVIEW CONTINUED

2-3 Graphing Linear Functions Find the intercepts. Then graph. 15. 3x 2y 6

16. 4x 2y 20 y

y

4

8

2

4

x

–4 –2 –2

2

4

x

–8 –4 –4

–4

4

8

–8

2 17. 3x 2y 6

5

18. x y 2 y

y

8

4

4

2

x

–8 –4 –4

4

8

x

–4 –2 –2

–8

2

4

–4

Write each function in slope-intercept form. Then graph. 19. y 4x 12

20. 3x 4y 12 y

y

12

4

8

2

4 –8 –4 –4


x 4

–4 –2 –2

8

x 2

4

–4

51

Algebra 2


2-4 Writing Linear Functions Write an equation in slope-intercept form for each line. 21. through (4, 19) and (6, 31) 2

22. passing through (3, 2) and slope 3 1

23. through (4, 7) and parallel to y 2x 3 24. through ( 7, 1) and perpendicular to 7x 2y 14

2-5 Linear Inequalities in Two Variables Solve for y. Graph the solution. 25. y 3 6

26. 6x 2y 12 y

y

8

8

4

4

x

–8 –4 –4

4

8

–8 –4 –4

–8

4

8

–8

27. 6x 4y 3x 8

28. 2(2x 3) y 4x 8 y

y

8

8

4 –8 –4 –4

4

x 4

8

–8 –4 –4

–8


x

x 4

8

–8

52

Algebra 2


2-6 Transforming Linear Functions Let g(x) be the indicated transformation of f (x). Write the rule for g(x). 29. f(x) 2x vertical translation 3 units up 30. f(x) 4x vertical stretch by a factor of 6 1 31. f(x) x 4 vertical compression by a factor of 4 followed by a horizontal translation right 3 units

32. f(x) 2x 4 vertical translation 6 units up followed 2 by a horizontal stretch with a factor of 3

2-7 Curve-Fitting with Linear Models 33. Find the following for the given table of data. Source: http://www.tinet.ita.doc.gov/view/f-2000-04-001/index.html

a. Make a scatter plot of the data using years as the independent variable.

Year x

1994 1995 1996 1997 1998 1999 2000

Foreign Travelers to U.S. (in millions) 4.48 4.33 4.65 4.78 4.64 4.85 5.09

b. Find the correlation coefficient r and the line of best fit for the data.


53

Algebra 2


2-8 Solving Absolute-Value Equations and Inequalities Solve. 34. 18 6x 30

35. 5x 3 37

Solve each inequality. Then graph the solution. 36. 3x 9 15

x 2 4 37.

38. 34x 6 2 4

39. 9x 2 3 17

2-9 Absolute Value Functions Translate f (x) x so that the vertex is at the given point. 40. ( 1, 4)

42. (3, 4)

41. (2, 0)

Perform each transformation. Then graph. 43. f(x) x 4 reflected across the y-axis

44. f(x) 3x 1 compressed vertically 1 by 3

y

y

8

4

4 –8 –4 –4

2

x 4

8

–4 –2 –2

–8


x 2

4

–4

54

Algebra 2

CHAPTER

Big Ideas

2 Answer these questions to summarize the important concepts from Chapter 2 in your own words. 1. Explain how inverse operations are used to solve equations.

2. Why do absolute-value equations sometimes have no solution or two solutions?

3. Compare ratios, rates, and proportions.

1 4. Explain why 0.25, one quarter, 25%, and all represent the same value. 4





55

Algebra 2

CHAPTER

Vocabulary


Page

Definition

Clarifying Example

consistent system constraint

dependent system elimination

feasible region

inconsistent system independent system


56

Algebra 2


Term

Page

Definition

Clarifying Example

linear programming

ordered triple

substitution

system of equations system of linear inequalities


57

Algebra 2

LESSON

Lesson Title


Vocabulary 1. Systems of equations (p. 182): _____________________________________ ______________________________________________________________ 2. Linear system (p. 182): ___________________________________________ ______________________________________________________________ 3. Consistent system (p. 183): _______________________________________ ______________________________________________________________ 4. Inconsistent system (p. 183): ______________________________________ ______________________________________________________________ 5. Independent system (p. 184): ______________________________________ ______________________________________________________________ 6. Dependent system (p. 184): _______________________________________ ______________________________________________________________


58

Algebra 2


Key Concepts 7. Classifying Linear Systems (p. 184): EXACTLY ONE SOLUTION

INFINITELY MANY SOLUTIONS

NO SOLUTION

8. Get Organized In each box, give information about or examples of each solution type. (p. 185). EXACTLY ONE SOLUTION

INFINITELY MANY SOLUTIONS

NO SOLUTION

Example

Graph Art: A207TEC03L01.A01

Slopes

y-intercept


59

Algebra 2

LESSON

Lesson Title


Vocabulary 1. Substitution (p. 190): _____________________________________________ ______________________________________________________________ 2. Elimination (p. 191): _____________________________________________ ______________________________________________________________

Key Concepts 3. Get Organized In each box, show an example of the given method of solving a linear system. (p. 194).

Graphing:

Solving Linear Systems

Substitution:


Elimination:

60

Algebra 2

LESSON

Lesson Title


Vocabulary 1. System of linear inequalities (p. 199): ________________________________ ______________________________________________________________

Key Concepts 2. Get Organized For the region, write the system of inequalities whose solution it represents. (p. 201).

Region 4

Region 1

4

y

x –4

0

–4

2

Region 3

Region 2


61

Algebra 2

LESSON

Lesson Title


Vocabulary 1. Linear programming (p. 205): ______________________________________ ______________________________________________________________ 2. Constraint (p. 205): ______________________________________________ ______________________________________________________________ 3. Feasible region (p. 205): __________________________________________ ______________________________________________________________ 4. Objective function (p. 206): ________________________________________ ______________________________________________________________

Key Concepts 5. Vertex Principle of Linear Programming (p. 206):


62

Algebra 2


6. Get Organized In each box, write an example of the given characteristic, using data from Examples 1 and 2. (p. 208).

Feasible region:

Constraints:

Vertices:


Linear Programming

63

Objective function:

Algebra 2

LESSON

Lesson Title


Vocabulary 1. Three-dimensional coordinate system (p. 214):_________________________ ______________________________________________________________ 2. Ordered triple (p. 214): ___________________________________________ ______________________________________________________________ 3. z-axis (p. 214): __________________________________________________ ______________________________________________________________

Key Concepts 4. Get Organized Label each axis, plane and point shown. (p. 216).


64

Algebra 2

LESSON

Lesson Title


Key Concepts 1. Get Organized In each box, describe the similarities and differences between 2-by-2 and 3-by-3 systems. (p. 224).

Systems of Equations

2-by-2:


3-by-3:

65

Algebra 2

CHAPTER

Chapter Review

3 3-1 Solving Linear Systems by Using Graphs and Tables Solve each system by using a graph and a table. Check your answer. 1.

x2xyy54

2.

x2xy 3y 1 0

y

3.

3xx 2yy 45

y

y

4

4

4

2

2

2

–2

2

4

6

x

–4

–2

2

4

x –4

–2

2

–2

–2

–2

–4

–4

–4

4

x

Classify each system, and determine the number of solutions. 4.

2y 2 6x 9x 3y 1

5.

y6 2x 3x 4y 4

6.

2x 3y 6

3-2 Solving Linear Systems by Using Algebraic Methods Use substitution to solve each system of equations. 7.

y2x53y 4x13


8.

x3x7 2y 11

66

9.

x4xy 3y 7 26

Algebra 2


Use elimination to solve each system of equations. 10.

3y 3 9x 2x 3y 8

11.

7y 16 5x 2x 8y 26

12.

3y 24 5x 3x 5y 28

3-3 Solving Systems of Linear Inequalities Graph each system of inequalities.

y 2x 3 15. y 2x 1 y5

xy4 13. xy4

yx 14. y 2x 4 y

y

4

4

2

2

y

4

–4

–2

2

4

2

x –4

–2

2

–2

–2

–4

–4

4

x –4

–2

2

4

x

–2

16. Pamela is selling necklaces and bracelets at a craft show. She only has enough beads to make a total of 40 items. Therefore, she can sell no more than a total of 40 necklaces and bracelets. Each necklace sells for $5.00 and each bracelet sells for $3.50. Pamela needs at least $150 in sales to meet her goal. Write and graph a system of inequalities that models this situation.


67

y

40 30 20 10

x 10

20

30

40

Algebra 2


3-4 Linear Programming Graph each feasible region, and maximize or minimize the objective function shown for each exercise. 17. P(x) 7x 8y

18. P(x) 2y 3x

3x 2y 12 2y x 4 maximize; x0 y0

y 2x 1 y 2x 3 minimize; x3

y

y 4

6 2 4 –4

2

–2

2

4

x

–2 2

4

6

x –4

19. A landscaper is working on a design for a customer. The customer has a budget of $700. The landscaper makes a $10 profit on each shrub he sells and $18 profit on each tree. The customer has room to a maximum of 10 plants. The landscaper must plant at least 3 shrubs to help with drainage. The shrubs cost $40 each and the trees cost $100 each. Find the number of shrubs and trees that produces the maximum profit.

3-5 Linear Equations in Three Dimensions Graph each point in three-dimensional space. 21. (3, 4, 3)

20. (4, 3, 2) z

22. (4, 2, 4) z

z

x


y

y

y

x

x

68

Algebra 2


Graph each linear equation in three-dimensional space. 23. x y z 1

24. x 2y z 1

25. 2x y 2z 1

z

z

z

y

y

y

x

x

x

Use the following information and the table for Exercises 26 and 27. A hobby store sells train engines and cars. The stores charges $125 for each engine, $100 for each boxcar and $75 for each gondola car. The store’s total income from the sale of the engines and train cars was exactly $3500 for each of the days shown in the table.

Day Thursday Friday Saturday Sunday

Boxcars 15 9 11

Engines Gondolas 10 23 8 16 17

26. Write a linear equation in three variables to represent this situation. 27. Complete the table for the possible number of item sales each day.

3-6 Solving Linear Systems in Three Variables Use elimination to solve each system of equations.

28.

2x y z 10 xyz6 4x 2y 3z 10 29. 2x 3y 2z 2 x 3y 2z 8 3x 5y 4z 4


69

2x y 3z 1 2x y z 9 30. x 2y 4z 17

Algebra 2


Use the following information and the table for Exercises 31 and 32. Sam’s Tea House has three different sizes of tea, small, medium, and large. The table shows the total revenues for three hours on a particular afternoon. Time 3:00 P.M. - 4:00 P.M. 4:00 P.M. - 5:00 P.M. 5:00 P.M. - 6:00 P.M.

Small 4 9 12

Medium 3 8 2

Large 2 7 8

Revenue $26 $71 $64

31. Write a system in three variables to represent the data in the table. 32. How much does each size tea cost?

Classify each system as consistent or inconsistent, and determine the number of solutions.

y 3z 4 33. x y 2z 0 x 2y z 1


xyz4 4x y z 17 34. 5x 2y 3z 2 35. x 3y 2z 8 4x 3y 4z 2 5x 2y 3z 5

70

Algebra 2

Big Ideas

CHAPTER

3 Answer these questions to summarize the important concepts from Chapter 3 in your own words. 1. How can you check your solution to a system of linear equations in either two or three variables?

2. You would like to minimize the amount of work required to solve a system of equations. Tell whether you would solve each system using substitution or elimination and why. 4x y 6 3x y 7 3x 2y 0 A. B. C. y 2x x y 5 9x 8y 7

3. Explain how to determine which region to shade to indicate the solution set of a system of linear inequalities.

4. Why is it necessary to eliminate the same variable from two equations when solving a system of three equations and three variables?



• Complete the Ready to Go On quizzes on pages 213 and 229 of your textbook. Copyright © by Holt, Rinehart and Winston. All rights reserved.

71

Algebra 2

CHAPTER

Vocabulary


Page

Definition

Clarifying Example

augmented matrix

coefficient matrix

constant matrix

determinant

dimension of a matrix

matrix equation


72

Algebra 2


Term

Page

Definition

Clarifying Example

matrix product

multiplicative identity matrix

reflection matrix

rotation matrix

square matrix


73

Algebra 2

LESSON

Matrices and Data


Vocabulary 1. Matrix (p. 246): _________________________________________________ ______________________________________________________________ 2. Dimensions (p. 246): _____________________________________________ ______________________________________________________________ 3. Entry (p. 246): __________________________________________________ ______________________________________________________________ 4. Address (p. 246): ________________________________________________ ______________________________________________________________ 5. Scalar (p. 248): _________________________________________________ ______________________________________________________________

Key Concepts 6. Adding and Subtracting Matrices (p. 247): WORDS


NUMBERS

74

ALGEBRA

Algebra 2


7. Properties of Equality for Matrices (p. 249): WORDS

NUMBERS

ALGEBRA


Associative Property

Additive Property

Additive Inverse

8. Get Organized Give examples for matrices and real numbers. (p. 249). PROPERTY OR OPERATION

REAL NUMBERS

MATRICES

Addition Subtraction Multiplication by a number


75

Algebra 2

LESSON

Multiplying Matrices


Vocabulary 1. Matrix product (p. 253): ___________________________________________ ______________________________________________________________ 2. Square matrix (p. 255): ___________________________________________ ______________________________________________________________ 3. Main diagonal (p. 255): ___________________________________________ ______________________________________________________________ 4. Multiplicative identity matrix (p. 255): ________________________________ ______________________________________________________________ ______________________________________________________________

Key Concepts 5. Multiplying Matrices—Rules (p. 253):

• Matrices A and B can be multiplied only .

• The product of an m n and an n p matrix is

.

6. Multiplying Matrices (p. 254): WORDS


NUMBERS

76

ALGEBRA

Algebra 2


7. Multiplicative Identity Matrix (p. 255): The multiplicative identity matrix is any square matrix, named with the letter I, that has ____________________________________________________________ ____________________________________________________________ 8. Get Organized In the decision diamond, enter a question to determine whether AB is defined. Then give the general procedure for finding AB, if it is defined. (p. 256).

For A = [m × n], B = [p × x]. . .

No

Yes Dimensions of AB:

AB . . .

To find AB . . .


77

Algebra 2

Using Matrices to Transform Geometric 4-3 Figures

LESSON


Vocabulary 1. Translation matrix (p. 262): ________________________________________ ______________________________________________________________ 2. Reflection matrix (p. 263): _________________________________________ ______________________________________________________________ 3. Rotation matrix (p. 264): __________________________________________ ______________________________________________________________

Key Concepts 4. Get Organized Complete the summary by filling in a matrix expression. Q is a triangle represented by its 2 3 coordinate matrix. (p. 264). TRANSFORMATION

MATRIX OPERATION

Translate Q vertically

Translate Q horizontally

Enlarge or reduce Q

Reflect Q across the x-axis or y-axis Rotate Q 90° clockwise or counterclockwise


78

Algebra 2

LESSON

Determinants and Cramer’s Rule


Vocabulary 1. Determinant (p. 270): ____________________________________________ ______________________________________________________________ 2. Coefficient matrix (p. 271): ________________________________________ ______________________________________________________________ 3. Cramer’s rule (p. 271): ___________________________________________ ______________________________________________________________

Key Concepts 4. Determinant of a 2 2 Matrix (p. 270): WORDS

NUMBERS

ALGEBRA

5. Cramer’s Rule for Two Equations (p. 271):


79

Algebra 2


6. Solutions of Systems (p. 271) Solutions of Systems

7. Cramer’s Rule for Three Equations (p. 273):


80

Algebra 2


8. Get Organized In each box, write the appropriate formula. (p. 274). 2 2 MATRIX

3 3 MATRIX

Determinant

Cramer’s Rule


81

Algebra 2

LESSON

Matrix Inverses and Solving Systems


Vocabulary 1. Multiplicative inverse matrix (p. 278): ________________________________ ______________________________________________________________ ______________________________________________________________ 2. Matrix equation (p. 279): __________________________________________ ______________________________________________________________ 3. Variable matrix (p. 279): __________________________________________ ______________________________________________________________ 4. Constant matrix (p. 279): _________________________________________ ______________________________________________________________

Key Concepts 5. Inverse of a 2 2 Matrix (p. 278):


82

Algebra 2


6. Get Organized Compare multiplicative inverses of real numbers and matrices. (p. 281). MULTIPLICATIVE INVERSE REAL NUMBERS

MATRICES

Notation and Example

How to Show That It is the Multiplicative Inverse



83

Algebra 2

Row Operations and Augmented 4-6 Matrices

LESSON


Vocabulary 1. Augmented matrix (p. 287): ________________________________________ ______________________________________________________________ 2. Row operation (p. 288): ___________________________________________ ______________________________________________________________ 3. Row reduction (p. 288): ___________________________________________ ______________________________________________________________ 4. Reduced row-echelon form (p. 288): _________________________________ ______________________________________________________________

Key Concepts 5. Elementary Row Operations (p. 288): ELEMENTARY ROW OPERATIONS


84

Algebra 2


6. Get Organized Fill in the augmented matrix for a three-equation system. Then write an example of the given operation in each box. Tell whether the operation produces an equivalent system. (p. 290). SYSTEM OF EQUATIONS

AUGMENTED MATRIX

Interchange rows or equations

Replace a row or equation with a multiple.

Replace a row or equation with a sum or difference.

Combine the above.


85

Algebra 2

CHAPTER

Chapter Review

4 4-1 Matrices and Data Use the table for Exercises 1–4. 1. Display the data in the form of a matrix M.

Coffee Shop Muffin Orders (dozens) Barb’s Dugan’s Gonzalez Banana nut 8 15 9 Blueberry 6 10 7 Cinnamon 4 3 5

2. What are the dimensions of M?

3. What is the value of the matrix entry with the address M23? What does it represent?

4. What is the address of the entry that has the value 6? Use the matrices below for Exercises 5–8. Evaluate, if possible. 1 6 2 1 4 6 3 1 1 3 5 A B C 0 D 1 1 2 2 0 3 2 4 5 2

5. A C

6. 2B

7. 3A C

8. 2C D


1

86

Algebra 2


4-2 Multiplying Matrices Use the matrices named below for Exercises 9–12. Tell whether each product is defined. If so, give its dimensions. P43, Q33, R34, and S32 9. PQ

10. QR

11. RS

12. PS

Use the matrices below for Exercises 13–16. Evaluate, if possible. 1 1 1 2 3 1 3 M [0.5 1 0.25] N 3 1 P L 4 3 0 3 0 1 1 2 0 1

13. LM

14. MP

15. PN

16. N 2

4-3 Using Matrices to Transform Geometric Figures For Exercises 17–21, use polygon ABCD with coordinates A(0, 1), B(–1, 5), C(–3, 6), D(–4, 3). Give the coordinates of the image and graph. 17. Translate polygon ABCD 1 unit right and 2 units up.

y

C B

18. Enlarge polygon ABCD by a factor of 2.

D –4

6

12

4

8

2

2

A –2

x 2

4

–2

4

x

x –2

4

y

y 8

–4

6

2

4


–8

87

–4

4

8

Algebra 2


19. Use

10 01 to rotate polygon

20. Use

01

1 to rotate polygon ABCD. 0

ABCD. Describe the image.

Describe the image. y

y 8 6 4

x

4 –8

2

–4

x –4

–2

4

2

4

8

–4 –8

21. How does multiplying by y

03 30 transform polygon ABCD?

16 8

x –16

–8

8

16

–8 –16

4-4 Determinants and Cramer’s Rule Find the determinant of each matrix. 22.

14 14

23.

1 3

0

2 4 3

0.2 1.5 24. 0.4 4.0

1 2 4 25. 3 2 3 2 1 5

Use Cramer’s rule to solve. 26.

xy yx 46 0

27.

5y 14 2x y7x

29.

5 yx 3x 3y 1

3x y z 7 28. 4x 2y 3z 2 zx2


88

Algebra 2


4-5 Matrix Inverses and Solving Systems Find the inverse matrix of each matrix, if it is defined. 30.

32.

35 47

1 2

1

3 12

31.

1 3 1 3

1 1

0 2 0 33. 3 3 2 2 5 1

Write the matrix equation for the system, and solve, if possible. 34.

4y 13 3x 2x 3y 14

6x 7y 16 36. 12x 3y 12

35.

3y) 9 4(x x 3y 9

37.

(x y) 3z 1 2x y 9 z 3x y 8 4z

38. You are writing three proposals for office furniture and copiers as a system of equations. Use x as the price for a file cabinet, y as the price for a desk, and z as the price for a copier. What is the price for each type of office furniture or a copier?

4x 8y 2z 2920 2x 3y z 1270 5x 9y 2x 3285

4-6 Row Operations and Augmented Matrices Write the augmented matrix, and use row reduction to solve, if possible. 39.

x6y3y2x83

40.

6y 4 2x 3x 9y 6

41.

x2x4y5y95

42.

4y 7 0 3x 2y 5x 10


89

Algebra 2


43. The system of equations represents the costs of three different types of bread at a bakery. Use a to represent the cost of a loaf of honey wheat bread, b the cost of a loaf of pumpernickel, and c the cost of a loaf of raisin bread. Find the cost of each type of bread.

3a 3b c 1.10 20.60 4a 3b 2c 1.10 25.25 5a 4b 3c 1.10 33.25


90

Algebra 2

CHAPTER

Big Ideas

4 Answer these questions to summarize the important concepts from Chapter 4 in your own words. 1. Explain how you can add or subtract two matrices.

2. Explain how you can tell if two matrices can be multiplied.

3. Explain how determinants and Cramer’s rule are used.

4. Explain how to solve a system of equations using the inverse of a matrix.


• Complete the Chapter 4 Study Guide and Review on pages xx–xx of your textbook.

• Complete the Ready to Go On quizzes on pages xx and xx of your textbook.


91

Algebra 2

CHAPTER

Vocabulary


Page

Definition

Clarifying Example

absolute value of a complex number

axis of symmetry

binomial

complex conjugate

complex number

imaginary number


92

Algebra 2


Term

Page

Definition

Clarifying Example

maximum value (of a function)

minimum value (of a function)

parabola

quadratic function

root of an equation standard form (of a quadratic equation) trinomial zero of a function


93

Algebra 2

Using Transformations to Graph 5-1 Quadratic Functions

LESSON


Vocabulary 1. Quadratic function (p. 315): _______________________________________ ______________________________________________________________ 2. Parabola (p. 315): _______________________________________________ ______________________________________________________________ 3. Vertex of a parabola (p. 318): ______________________________________ ______________________________________________________________ 4. Vertex form (p. 318): _____________________________________________ ______________________________________________________________

Key Concepts 5. Linear and Quadratic Parent Functions (p. 315): ALGEBRA

NUMBERS

GRAPH

Linear Parent Function f(x) x Quadratic Parent Function f(x) x 2


94

Algebra 2


6. Translations of Quadratic Functions (p. 316): HORIZONTAL TRANSLATIONS

VERTICAL TRANSLATIONS

Horizontal Shift of h Units

Vertical Shift of h Units

7. Reflections, Stretches, and Compressions of Quadratic Functions (p. 317): REFLECTIONS Reflection Across y-axis

Reflection Across x-axis

STRETCHES AND COMPRESSIONS Horizontal Stretch/Compression by a Factor of b


Vertical Stretch/Compression by a Factor of a

95

Algebra 2


8. Vertex Form of Quadratic Functions (p. 318):

f (x) a(x h)2 k

9. Get Organized In each row, write an equation that represents the indicated transformation of the quadratic parent function, and show its graph. (p. 319). TRANSFORMATION

EQUATION

GRAPH



Reflection

Vertical stretch

Vertical compression


96

Algebra 2

Properties of Quadratic Functions in 5-2 Standard Form

LESSON


Vocabulary 1. Axis of symmetry (p. 323): ________________________________________ ______________________________________________________________ 2. Standard form (p. 324): ___________________________________________ ______________________________________________________________ 3. Minimum value (p. 326): __________________________________________ ______________________________________________________________ 4. Maximum value (p. 326): __________________________________________ ______________________________________________________________

Key Concepts 5. Axis of Symmetry—Quadratic Functions (p. 323): WORDS


ALGEBRA

GRAPH

97

Algebra 2


6. Properties of a Parabola (p. 324):

7. Minimum and Maximum Values (p. 326):


98

Algebra 2


8. Get Organized In each box, write the criteria or equation to find each property of the parabola for f(x) ax 2 bx c. (p. 327).

Opens upward or downward

Axis of symmetry

Properties of Parabolas

y-intercept


Vertex

99

Algebra 2

Solving Quadratic Equations by 5-3 Graphing and Factoring

LESSON


Vocabulary 1. Zero of a function (p. 333): ________________________________________ ______________________________________________________________ 2. Root of an equation (p. 334): ______________________________________ ______________________________________________________________ 3. Binominal (p. 336): ______________________________________________ ______________________________________________________________ 4. Trinomial (p. 336): _______________________________________________ ______________________________________________________________

Key Concepts 5. Zero Product Property (p. 334): For all real numbers a and b, WORDS


NUMBERS

100

ALGEBRA

Algebra 2


6. Special Products and Factors (p. 336): DIFFERENCE OF TWO SQUARES

PERFECT-SQUARE TRINOMIAL

7. Get Organized In each box, give information about special products and factors. (p. 337). NAME

RULE

EXAMPLE

GRAPH

Difference of Two Squares

PerfectSquare Trinomial


101

Algebra 2

LESSON

Completing the Square


Vocabulary 1. Completing the square (p. 342): ____________________________________ ______________________________________________________________

Key Concepts 2. Square-Root Property (p. 341): WORDS

NUMBERS

ALGEBRA

3. Completing the Square (p.342): WORDS


NUMBERS

102

ALGEBRA

Algebra 2


4. Solving Quadratic Equations by Completing the Square (p. 343): 1. 2. 3. 4. 5. 5. Get Organized Compare and contrast two methods of solving quadratic equations. (p. 344).

Using Square-Root Property vs. Completing the Square

Similarities:


Differences:

103

Algebra 2

LESSON

Complex Numbers and Roots


Vocabulary 1. Imaginary unit (p. 350): ___________________________________________ ______________________________________________________________ 2. Imaginary number (p. 350): ________________________________________ ______________________________________________________________ 3. Complex number (p. 351): _________________________________________ ______________________________________________________________ 4. Real part (p. 351): ______________________________________________ ______________________________________________________________ 5. Imaginary part (p. 351): __________________________________________ ______________________________________________________________ 6. Complex conjugate (p. 352): _______________________________________ ______________________________________________________________


104

Algebra 2


Key Concepts 7. Imaginary Numbers (p. 350): WORDS

NUMBERS

ALGEBRA

8. Get Organized In each box or oval, give a definition and examples of each type of number. (p. 352).

Complex Numbers

Real Numbers


Imaginary Numbers

105

Algebra 2

LESSON

The Quadratic Formula


Vocabulary 1. Discriminant (p. 357): ____________________________________________ ______________________________________________________________

Key Concepts 2. The Quadratic Formula (p. 356):

3. Discriminant (p. 358):


106

Algebra 2


4. Summary of Solving Quadratic Equations (p. 360): METHOD

WHEN TO USE. . .

EXAMPLES

Graphing

Factoring

Square roots

Completing the square

Quadratic formula


107

Algebra 2


5. Get Organized Describe the possible solution methods for each value of the discriminant. (p. 360). VALUE OF DISCRIMINANT

TYPE OF SOLUTIONS

POSSIBLE SOLUTION METHODS

Negative

Zero

Positive


108

Algebra 2

LESSON

Solving Quadratic Inequalities


Vocabulary 1. Quadratic inequalities in two variables (p. 366): ________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Key Concepts 2. Graphing Quadratic Inequalities (p. 366): TO GRAPH A QUADRATIC INEQUALITY 1. 2. 3.

3. Get Organized Compare the solutions of quadratic equations and inequalities. (p. 370). EQUATION ()

“LESS THAN” INEQUALITY ( or )

“GREATER THAN” INEQUALITY ( or )

Example Graph Solution Set


109

Algebra 2

LESSON

Curve Fitting with Quadratic Models


Vocabulary 1. Quadratic model (p. 376): _________________________________________ ______________________________________________________________ 2. Quadratic regression (p. 376): _____________________________________ ______________________________________________________________

Key Concepts 3. Get Organized Compare the different quadratic models presented in the lesson. (p. 377). QUADRATIC MODEL

WHEN APPROPRIATE

PROCEDURE

Exact model

Approximate model


110

Algebra 2

LESSON

Operations with Complex Numbers


Vocabulary 1. Complex plane (p. 382): __________________________________________ ______________________________________________________________ 2. Absolute value of a complex number (p. 382): _________________________ ______________________________________________________________

Key Concepts 3. Absolute Value of a Complex Number (p. 382): WORDS


ALGEBRA

NUMBERS

111

GRAPH

Algebra 2


4. Get Organized In each box, give an example. (p. 385).

Absolute value

Adding

Complex Numbers

Multiplying


Conjugates

112

Algebra 2

CHAPTER

Chapter Review

5 5-1 Using Transformations to Graph Quadratic Functions Using the graph of f(x) x 2 as a guide, describe the transformations, and then graph each function. 1. g(x) (x 3)2 2

1

2. g(x) 3(x 5)2

3. g(x) 2x 2 4

y

y

y

4 6

6

4

4

2

2

2

x 2

4

6

x x

–6 –4 –2

–4 –2 –2

2

4

–4

Use the description to write each quadratic function in vertex form. 4. f(x)2 is vertically stretched by a factor of 5 and translated 4 units right to create g(x).


5. f(x)2 is reflected across the x-axis, shifted 3 units right, and translated 2 units down to create g(x).

113

Algebra 2


5-2 Properties of Quadratic Functions in Standard Form For each function, (a) determine whether the graph opens upward or downward, (b) find the axis of symmetry, (c) find the vertex, (d) find the y-intercept, and (e) graph the function. 6. f(x) x 2 8x 12

7. g(x) x 2 2x 8

8. h(x) x 2 3x

a)

a)

a)

b)

b)

b)

c)

c)

c)

d)

d)

d)

e)

e)

y

e)

y

y

4

8

4

2

6

2

x

–4

x

4

–6 –4 –2 –2

–4 –2 –2

2 –4 –2

2

4

x

2

4

–4

9. A baseball player hits a baseball whose height is modeled by the function h(x) 0.03x 2 2.4x 2 where x is the horizontal distance in feet that the ball travels. Find the maximum height of the ball to the nearest foot.

5-3 Solving Quadratic Equations by Graphing and Factoring Find the roots of each equation by factoring. 10. x 2 x 20


11. x 2 36 0

114

12. 7x 2 49x 0

Algebra 2


5-4 Completing the Square Solve each equation by completing the square. 13. x 2 2x 63

14. x 2 10x 14

15. x 2 12x 9

Write each function in vertex form, and identify its vertex. 16. f(x) x 2 8x 14

17. g(x) x 2 12x 10818. h(x) 4x 2 24x 39

5-5 Complex Numbers and Roots Solve each equation. 19. 6x 2 150 0

20. x 2 8x 18

21. x 2 x 19

5-6 The Quadratic Formula Find the zeros of each function by using the Quadratic Formula. 22. f(x) 2x 2 8x 24 23. g(x) 3x 2 6x 8

24. h(x) x 2 4x 77

Find the type and number of solutions for each equation. 25. x 2 81 18x


26. x 2 9x 36

115

27. x 2 100 0

Algebra 2


5-7 Solving Quadratic Inequalities Graph each inequality. 28. y x 2 2x 8

29. y x 2 3x y

y x –4 –2 –2

2

4

4

2 x

–4 –6

–4 –2 –2

–8

–4

2

4

Solve each inequality by using tables or graphs. 30. x 2 10x 20 4

31. 2x 2 4x 27 3

Solve each inequality by algebra. 32. x 2 5x 0

33. x 2 4x 27 6

34. The height of an object thrown upwards off of a cliff is modeled by the function h(x) 16t 2 25t 40, where t is the time. For what range of time will the object have a height of at least 40 feet?


116

Algebra 2


5-8 Curve Fitting with Quadratic Models Determine whether each data set could represent a quadratic function. Explain. 35.

36. x

0

1

2

3

4

x

1

3

5

7

9

y

4

5

4

1

4

y

3

1

5

9

13

Write a quadratic function that fits each set of points. 37. (0, 6), (1, 0), and (2, 8)

38. (0, 3), (2, –5), and (4, –21)

For Exercises 43–45, use the table of the number of normal temperature highs for Anchorage, Alaska. DAY # DATE TEMPERATURE 39. Use the data to find a quadratic 31 Jan 31 22 regression equation to model the normal temperature high 120 April 30 47 given the day. 212 July 31 64 304

October 31

37

40. Use your model to predict the normal temperature high on June 1 (day 181).

41. Use your model to predict the normal temperature high on November 24 (day 328).


117

Algebra 2


5-9 Operations with Complex Numbers Find each absolute value. 42. 8i

43. 2 5i

44. 4 i

Perform each indicated operation, and write the result in the form a bi. 45. (4 2i ) (5 3i )

46. (7 2i ) (6 5i )

47. 2i (8 2i )

48. (7 3i )(4 8i )

49. (3 9i )(3 9i )

50. 6i 18

9 5i

51. i


2i 52. 7 4i

118

Algebra 2

CHAPTER

Big Ideas

5 Answer these questions to summarize the important concepts from Chapter 5 in your own words. 1. Explain how to convert a quadratic equation in vertex form to standard form.

2. Explain how to convert a quadratic equation in standard form to vertex form.

3. Explain how the quadratic formula relates to the process of completing the square.

4. What is the relationship between the roots of a quadratic equation and the graph of the quadratic equation?



• Complete the Ready to Go On quizzes on pages 329 and 365 of your textbook. Copyright © by Holt, Rinehart and Winston. All rights reserved.

119

Algebra 2

CHAPTER

Vocabulary


Page

Definition

Clarifying Example

degree of a monomial

degree of a polynomial

end behavior

leading coefficient

local maximum

local minimum


120

Algebra 2


Term

Page

Definition

Clarifying Example

monomial

multiplicity

polynomial

synthetic division

turning point


121

Algebra 2

LESSON

Polynomials


Vocabulary 1. Monomial (p. 406): ______________________________________________ ______________________________________________________________ 2. Polynomial (p. 406): _____________________________________________ ______________________________________________________________ 3. Degree of a monomial (p. 406): ____________________________________ ______________________________________________________________ 4. Degree of a polynomial (p. 406): ____________________________________ ______________________________________________________________ 5. Leading coefficient (p. 406): _______________________________________ ______________________________________________________________ 6. Binomial (p. 407): _______________________________________________ ______________________________________________________________ 7. Trinomial (p. 407): _______________________________________________ ______________________________________________________________


122

Algebra 2


8. Polynomial function (p. 408): ______________________________________________________________ ______________________________________________________________

Key Concepts 9. Classifying Polynomials by Degree (p. 407): NAME

DEGREE

EXAMPLE

Constant Linear Quadratic Cubic Quartic Quintic 10. Get Organized Complete the graphic organizer. (p. 409).

Definition

Characteristics

Polynomials

Examples


Nonexamples

123

Algebra 2

LESSON

Multiplying Polynomials


Key Concepts 1. Binomial Expansion (p. 416):

2. Get Organized In each box, write an example and find the product. (p. 417).

Binomial × trinomial (horizontal method)

Monomial × trinomial

Binomial × trinomial (vertical method)

Multiplying Polynomials

Trinomial × trinomial

Expand a binomial


124

Algebra 2

LESSON

Dividing Polynomials


Vocabulary 1. Synthetic division (p. 423): ________________________________________ ______________________________________________________________

Key Concepts 2. Synthetic Division Method. (p. 423) Divide (2x 2 7x 9) by (x 2) by using synthetic division. WORDS

NUMBERS

Step 1

Step 2

Step 3


125

Algebra 2


3. Remainder Theorem (p. 424): THEOREM

EXAMPLE

4. Get Organized Complete the graphic organizer. (p. 425).

Long Division and Synthetic Division

Similarities


Differences

126

Algebra 2

LESSON

Factoring Polynomials


Key Concepts 1. Factor Theorem (p. 430): THEOREM

EXAMPLE

2. Factoring the Sum and Difference of Two Cubes (p. 431): METHOD

ALGEBRA

3. Get Organized For each method, give an example of a polynomial and its factored form. (p. 432). METHOD

POLYNOMIAL

FACTORED FORM

Difference of Two Squares Difference of Two Cubes Sum of Two Cubes


127

Algebra 2

Finding Real Roots of Polynomial 6-5 Equations

LESSON


Vocabulary 1. Multiplicity (p. 439): ______________________________________________ ______________________________________________________________

Key Concepts 2. Rational Root Theorem (p. 439):

3. Irrational Root Theorem (p. 441):


128

Algebra 2


4. Get Organized Give roots that satisfy each theorem and write a polynomial equation that has those roots. (p. 442). THEOREM

ROOTS

POLYNOMIAL

Rational Root Theorem

Irrational Root Theorem


129

Algebra 2

LESSON

Fundamental Theorem of Algebra

6-6 Lesson Objectives (p. 455): ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Key Concepts 1. Properties of Polynomials (p. 445):

2. The Fundamental Theorem of Algebra (p. 446):

3. Complex Conjugate Root Theorem (p. 447):


130

Algebra 2


4. Get Organized Give an example of a polynomial with each type of root. (p. 448). Rational

Irrational

Polynomial Roots

Complex

Real


131

Algebra 2

Investigating Graphs of Polynomial 6-7 Equations

LESSON

Lesson Objectives (p. 453): ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Vocabulary 1. End behavior (p. 453): ____________________________________________ ______________________________________________________________ 2. Turning point (p. 455): ____________________________________________ ______________________________________________________________ 3. Local maximum (p. 455): __________________________________________ ______________________________________________________________ ______________________________________________________________ 4. Local minimum (p. 455): __________________________________________ ______________________________________________________________ ______________________________________________________________


132

Algebra 2


Key Concepts 5. Polynomial End Behavior (p. 453): P(x) has …

ODD DEGREE

EVEN DEGREE

Leading coefficient a0

Leading coefficient a0

6. Local Maxima and Minima (p. 455):


133

Algebra 2


7. Get Organized In each box, sketch a graph that fits the description. (p. 456). ODD DEGREE

EVEN DEGREE

Positive Leading Coefficient

Negative Leading Coefficient


134

Algebra 2

LESSON

Transforming Polynomial Functions


Key Concepts 1. Transformations of f(x) (p. 460): TRANSFORMATION

f (x) NOTATION

EXAMPLES

Vertical Transformation Horizontal Transformation Vertical Stretch/Compression Horizontal Stretch/Compression

Reflection

2. Get Organized Complete the graphic organizer. (p. 463). TRANSFORMATION VERTICAL HORIZONTAL VERTICAL VERTICAL SHIFT SHIFT STRETCH COMPRESSION Example


135

Algebra 2

LESSON

Curve Fitting with Polynomial Models


Key Concepts 1. Finite Differences of Polynomials (p. 466): FUNCTION TYPE

DEGREE

CONSTANT FINITE DIFFERENCES

Linear Quadratic Cubic Quartic Quintic

2. Get Organized For each type of function, indicate the degree and the constant differences and give an example of a data set. (p. 468). Linear

Quadratic

Polynomial Models

Cubic Copyright © by Holt, Rinehart and Winston. All rights reserved.

Quartic 136

Algebra 2

CHAPTER

Chapter Review

6 6-1 Polynomials Rewrite each polynomial in standard form. Then identify the leading coefficient, degree, and number of terms. Name the polynomial. 1. 3x 2 3x 4 x 3 2 Leading coefficient: Degree: Number of Terms: Name:

3. 2x 3 x 3 3x 5 Leading coefficient: Degree: Number of Terms: Name:

2. 14 3x 2 x

Leading coefficient: Degree: Number of Terms: Name: 4. 22 6x

Leading coefficient: Degree: Number of Terms: Name:

Add or subtract. Write your answer in standard form. 5. (2x 2 3x 6) (5x 2 4x 6) 6. (3x 3 8x 2) (6x 3 3x 2 7x)

7. (14 2x x 2) (7 5x 9x 2) 8. The cost on x-units of a product can be modeled by C(x) x 3 18x 12. Evaluate C(x) for x 50, and describe what the value represents. Graph each polynomial function on a calculator. Describe the graph, and identify the number of real zeros. 9. g(x) x 3 7x 6


137

Algebra 2


Graph each polynomial function on a calculator. Describe the graph, and identify the number of real zeros. 10. h(x) x 5 x

11. f(x) x 4 x 3 2x 4

6-2 Multiplying Polynomials Find each product. 12. 7x(4x 8x 3)

13. (x y)(x 2 y 2)

14. 2x 5

15. (3x 2y)(5x 2 x 6)

1

2

Expand each expression. 16. (x 5)3

17. (x 2y)4

18. (2x 1)4 19. Find the polynomial expression in terms of x for the volume of the rectangular prism shown. 3x – 1 4x + 2 2x


138

Algebra 2


6-3 Dividing Polynomials Divide. 20. (18x 2 3x 10) (3x 2)

21. (2x 3 18x 2 33x 35) (x 7)

Use synthetic substitution to evaluate the polynomial for the given value. 22. P(x) x 3 9x 2 3x 7 for x 2

23. P(x) x 4 x 3 10x 2 10x 5 for x 1

6-4 Factoring Polynomials Factor each expression. 24. 9x 2 25

25. 2x 3 8x 2 24x

26. a3 6a 2 3a 18

27. t 9 64

y

28. The volume of a box is modeled by the function V(x) x 3 4x 2 7x 10. Identify the values of x for which the volume is 0 and use the graph to factor V(x).

12 8 4 –4

–2

–4

2

4

x

–8 –12 –16 –20

6-5 Finding Real Roots of Polynomial Equations 29. The yearly profit of a company in thousands of dollars can be modeled by P(t) x 4 34x 2 225, where t is the number of years since 1999. Factor to find the years in which the profit was 0.


139

Algebra 2


Identify the roots of each equation. State the multiplicity of each root. 30. x 3 3x 2 72x 324 0

31. 2x 3 2x 2 28x 48 0

32. x 4 2x 3 7x 2 4x 0

6-6 Fundamental Theorem of Algebra Write the simplest polynomial function with the given roots. 33. 1, 2, 3

34. i, i, 0

6-7 Investigating Graphs of Polynomials Functions 35. Solve x 4 5x 3 8x 2 20x 16 0 by finding all roots.

Graph each function. 36. f (x) x 3 2x 2 3x 4

37. f(x) x 4 3x 3 20

y

y

10

15

8

10

6

5

4 –4

2 –4

–2

–2

2

4

x

–2

–5

4

x

–10 –15

–4

–20

–6

–25


2

140

Algebra 2


Identify whether the function graphed has an odd or even degree and a positive or negative leading coefficient. y

38.

y

39.

y

40.

x x

x

6-8 Transforming Polynomial Functions Let f(x) x 3 2x 2 – 3x – 1. Write a function g(x) that performs each transformation. 41. Reflect f(x) across x-axis.

42. Reflect f(x) across the y-axis.

Let f(x) 2x 4 4x 2 2x 3. Graph f(x) and g(x) on the same coordinate plane. Describe g(x) as a transformation of f(x). 47. g(x ) 2f(x)

48. g(x) f(x 2)

y

49. g(x) f(2x )

y

y

8 12

12

8

8

4

4

4

–4

–2

2

4

x

–4 –8


–4

–2

–2

2

141

4

x

–4

–2

–2

2

4

x

Algebra 2


6-9 Curve Fitting with Polynomial Models 46. The table shows the population of butterflies in a butterfly house. Write a polynomial function for the data.


142

Time (h)

1

2

3

4

5

Number of butterflies

26

72

180

370

665

Algebra 2

CHAPTER

Big Ideas

6 Answer these questions to summarize the important concepts from Chapter 6 in your own words. 1. Explain the importance of factoring polynomials.

2. Explain how to tell if a function is increasing or decreasing.

3. Explain the difference between the graphs of f(x) and f(x a).

4. Explain what is meant by expanding a polynomial (x a)b.





143

Algebra 2

CHAPTER

Vocabulary


Page

Definition

Clarifying Example

base of an exponential function

common logarithm exponential decay

exponential equation

exponential function

exponential growth


144

Algebra 2


Term

Page

Definition

Clarifying Example

inverse function

inverse relation

logarithm

logarithmic equation logarithmic function

natural logarithm


145

Algebra 2

Exponential Functions, Growth, and 7-1 Decay

LESSON


Vocabulary 1. Exponential function (p. 490): ______________________________________ ______________________________________________________________ 2. Base (p. 490): __________________________________________________ ______________________________________________________________ 3. Asymptote (p. 490): ______________________________________________ ______________________________________________________________ 4. Exponential growth (p. 490): _______________________________________ ______________________________________________________________ 5. Exponential decay (p. 490): _______________________________________ ______________________________________________________________

Key Concepts 6. Get Organized Compare exponential growth and decay. (p. 493). f(x) abx, where a 0

GROWTH

DECAY

Value of b General shape of graph

What happens to f(x) as x increases? What happens to f(x) as x decreases?


146

Algebra 2

LESSON

Inverses of Relations and Functions


Vocabulary 1. Inverse relation (p. 498): __________________________________________ ______________________________________________________________ 2. Inverse function (p. 499): __________________________________________ ______________________________________________________________

Key Concepts 3. Get Organized Show a possible input value, inverse function, and output value for a function f(x). (p. 501).

Input

→

→

f(x)

↑ Output


output

↓ ←

←

f 1(x)

147

Input

Algebra 2

LESSON

Logarithmic Functions


Vocabulary 1. Logarithm (p. 505): ______________________________________________ ______________________________________________________________ 2. Common logarithm (p. 506): _______________________________________ ______________________________________________________________ 3. Logarithmic function (p. 507): ______________________________________ ______________________________________________________________

Key Concepts 4. Special Properties of Logarithms. (p. 506) For any base b such that b 0 and b 1, LOGARITHMIC FORM

EXPONENTIAL FORM

EXAMPLE

Logarithm of Base b

Logarithm of 1


148

Algebra 2


5. Get Organized Use your own words to explain a logarithmic function. (p. 508). Definition:

Characteristics:

Logarithmic Function Examples:


Nonexamples:

149

Algebra 2

LESSON

Properties of Logarithms


Key Concepts 1. Product Property of Logarithms (p. 512): For any positive numbers m, n, and b (b 1), WORDS

NUMBERS

ALGEBRA

2. Quotient Property of Logarithms (p. 513): For any positive numbers m, n, and b (b 1), WORDS

NUMBERS

ALGEBRA

3. Power Property of Logarithms (p. 513): For any real number p and positive numbers a and b (b 1), WORDS


NUMBERS

ALGEBRA

150

Algebra 2


4. Inverse Properties of Logarithms and Exponents (p. 514): For any base b such that b 0 and b 1, ALGEBRA

EXAMPLE

5. Change of Base Formula (p. 514): For a 0 and a 1 and any base b such that b 0 and b 1, ALGEBRA

EXAMPLE

6. Get Organized Use your own words to show related properties of exponents and logarithms. (p. 515). PROPERTY OF EXPONENTS


PROPERTY OF LOGARITHMS

151

Algebra 2

Exponential and Logarithmic Equations 7-5 and Inequalities

LESSON


Vocabulary 1. Exponential equation (p. 522): _____________________________________ ______________________________________________________________ 2. Logarithmic equation (p. 523): ______________________________________ ______________________________________________________________

Key Concepts 3. Get Organized Write the strategies and points to remember in your own words for both exponential and logarithmic equations. (p. 525).

Equation

Exponential

Strategies to solve:


Logarithmic

Points to remember:

Strategies to solve:

152

Points to remember:

Algebra 2

LESSON

The Natural Base


Vocabulary 1. Natural logarithm (p. 531): ________________________________________ ______________________________________________________________ 2. Natural logarithmic function (p. 532): _________________________________ ______________________________________________________________

Key Concepts 3. Natural Logarithmic Function (p. 532):

4. Get Organized Fill in each box to compare and contrast the two kinds of logarithms. Give general forms and examples. Simplify, if appropriate. (p. 533). COMMON LOGARITHMS NATURAL LOGARITHMS BASE LOGARITHMIC FORM EXPONENTIAL FORM logb1 logbb logbbx blogbx Copyright © by Holt, Rinehart and Winston. All rights reserved.

153

Algebra 2

Transforming Exponential and 7-7 Logarithmic Functions

LESSON


Key Concepts 1. Transformations of Exponential Functions (p. 537): TRANSFORMATION

f(x) NOTATION EXAMPLES



Vertical stretch or compression

Horizontal stretch or compression

Reflection


154

Algebra 2


2. Transformations of Logarithmic Functions (p. 538): TRANSFORMATION





Horizontal stretch or compression

Reflection


155

Algebra 2


3. Get Organized Give an example of an indicated transformation for both types of exponential and logarithmic functions. Remember, e is a constant. (p. 541).

Transformation

f(x) 5x f(x) e x

f(x) logbx f(x) ln x

Vertical translation Horizontal translation Reflection Vertical stretch Vertical compression


156

Algebra 2

Curve Fitting with Exponential and 7-8 Logarithmic Models

LESSON


Vocabulary 1. Exponential regression (p. 546): ____________________________________ ______________________________________________________________ 2. Logarithmic regression (p. 547): ____________________________________ ______________________________________________________________

Key Concepts 3. Get Organized Fill in each line to show the steps for finding an exponential or logarithmic model. (p. 547).

Regression

Exponential:


Logarithmic:

157

Algebra 2

CHAPTER

Chapter Review

7 7-1 Exponential Functions, Growth, and Decay Tell whether the function shows growth or decay. Then graph. 1. f(x) 2x 1

1

2. f(x) 4(0.25)x y

–4

y

6

6

4

4

2

2

–2

2

4

x

x –4

–2

2

4. f(x) 3.214x 1

3. f(x) 12(1.5)x y

–6

–4

4

y

6

6

4

4

2

2

–2

x –8

–4

5. Suppose that the number of bacteria in a culture was 1200 on Sunday and the number has been increasing at a rate of 75% per day since then. a. Write a function representing the growth of the culture per day.

4

x

8

y

60,000 45,000 30,000

b. Graph the function, and use the graph to predict the number of bacteria in the culture the following Sunday.


158

15,000

x 2

4

6

8

Algebra 2


7-2 Inverses of Relations and Functions Graph each relation. Then graph its inverse. 6.

x

1 0

1

2

3

y

5 2

1

4

7

7.

x

2 1

0

1

2

y

8 1

0

1

8

y

y

8

8

4

4

x –4

–2

2

–8

4

–4

4

x

8

–4 –8

–8

Graph each function. Then write and graph the inverse. y

8. f(x) x 3.4 4

4

2

2

x –4

–2

2

–4

4

–2

–2

–2

–4

–4

10. f(x) 3x 2

1

11. f(x) 4(x 3)

y

–4

y

1

9. f(x) 2 x

4

2

2

2

4

4

2

4

x

y

4

–2

2

x

x –4

–2

–2

–2

–4

–4

12. Junie’s washing machine repair bill includes $150 for parts and $45 per hour for labor. Her bill can be expressed as a function of hour x by f(x) 150 45x. Find the inverse function. Use it to find the number of hours of labor she was charged if her bill was $262.50.


159

Algebra 2


7-3 Logarithmic Functions Write the exponential function in logarithmic form. 13. 42 16

15. 23 0.125

14. 14.30 1

16. 0.5x 8

Write the logarithmic function in exponential form. 17. log2512 9

18. log 36 2 1 6

19. log0.450 1

20. loge x 7

y

21. Use the given x-values to graph f(x) = 2x; x 2, 1, 0, 1, 2. Then graph the inverse function. 1

4 2

x –4

–2

2

4

–2 –4

7-4 Properties of Logarithms Express as a single logarithm. Simplify, if possible. 22. log216 log24

1

1

log3 23. log3 27 81

24. log 64 + log 16 1

1 4

1 4

Simplify each expression. 25. log32432

1 25

26. log 216

27. 5log

29. log

30. log12816

5

1 6

Evaluate. 28. log1632


1 1000

100

160

Algebra 2


7-5 Exponential and Logarithmic Equations and Inequalities Solve. 1

31. 5x 625

32. 36x2 63x

33. 102x1 72

34. 32x2 30

35. log3(x 2) 4

36. log4x 3

37. log25 logx 3

38. 2log3x log3 (x 2) 2

2 3

39. Suppose you deposit $1000 into an account that pays 1.8% compounded quarterly. The equation A P(1 r)n gives the amount A in the account after n quarters for an initial investment of P that earns interest at a rate of r. Use logarithms to solve for n to find how long it will take for the account to contain at least $1200.

7-6 The Natural Base, e Graph. 40. f(x) e x 4

41. f(x) 4 e x

y

y 4

6 2 4

x

2

–4

–2 –2

–4

2

4


–2

2

4

–2

x

–4

161

Algebra 2


ex

42. f(x) 4

43. f(x) 4(e x 1) y

y 4

6 2 4 –4

2

–2

–2

2

4

x

–2

x –4

2

4

–4

Simplify. 45. ln ex

44. ln e3

46. e ln(2x5)

47. ln e x3

48. An accident at a nuclear power plant released 12 grams of radioactive plutonium-239 into the atmosphere. The half-life of plutonium-239 is 24,360 years. 1

a. Use the formula 2 ekt to find the value of the decay constant for plutonium-239. b. Use the decay function N t N0ekt to determine how much of the 12 grams of plutonium-239 will remain after 500 years.

7-7 Transforming Exponential and Logarithmic Functions Graph the function. Find the y-intercept and asymptote. Describe how the graph is transformed from the graph of the parent function. 49. f(x) 0.2(2x )

50. g(x) e(3x)

y

y 4

6 2 4

x

2

–4

–2

2

4


2

4

–2

x –4

–2

–4

162

Algebra 2


51. h(x) 2.1log(x 2)

52. p(x) ln(x 1)

y

–4

y

4

4

2

2

–2

2

x

4

–4

–2

2

–2

–2

–4

–4

x

4

Write the transformed function. 53. f(x) 2x is reflected across the y-axis and translated 2 units to the right.

7-8 Curve Fitting: Exponential and Logarithmic Models Determine whether y is an exponential function of x. If so, find the constant ratio. Then use exponential regression to find a function that models the data. 54.

x

0

1

2

3

4

5

y

1 1

5

11

19

29


55. x y

163

0

1

2

3

4

5

0.5

1.5

4.5

13.5

40.5

121.5

Algebra 2

CHAPTER

Big Ideas

7 Answer these questions to summarize the important concepts from Chapter 7 in your own words. 1. Explain how to determine if an exponential function is a growth equation or a decay equation.

2. Explain how to find the inverse of a function, if it exists.

3. Explain why the ln e 1 by converting to logarithmic form.

4. Explain how to solve an exponential equation.





164

Algebra 2

CHAPTER

Vocabulary


Page

Definition

Clarifying Example

combined variation constant of variation

continuous function

direct variation

discontinuous function

extraneous solution

inverse variation


166

Algebra 2


Term

Page

Definition

Clarifying Example

joint variation

radical function

radical inequality rational equation rational exponent

rational expression

rational function rational inequality square-root function


167

Algebra 2

LESSON

Variation Functions

8-1 Lesson Objectives (p. 569): ______________________________________________________________

Vocabulary 1. Direct variation (p. 569): __________________________________________ ______________________________________________________________ 2. Constant of variation (p. 569): ______________________________________ ______________________________________________________________ 3. Joint variation (p. 570): ___________________________________________ ______________________________________________________________ 4. Inverse variation (p. 570): _________________________________________ ______________________________________________________________ 5. Combined variation (p. 572): _______________________________________ ______________________________________________________________

Key Concepts 6. Get Organized In each box, write the general variation equation, draw a graph, or give an example. (p. 573). TYPE OF VARIATION

EQUATION

GRAPH

EXAMPLE

Direct

Joint

Inverse


168

Algebra 2

Multiplying and Dividing Rational 8-2 Expressions

LESSON


Vocabulary 1. Rational expression (p. 577): ______________________________________ ______________________________________________________________

Key Concepts 2. Multiplying Rational Expressions. (p. 578) MULTIPLYING RATIONAL EXPRESSIONS 1. 2. 3. 4.

3. Get Organized In each box, write a worked out example. (p. 580).

[A207TE-C08-L02-A01


169

Algebra 2

Adding and Subtracting Rational 8-3 Expressions

LESSON


Vocabulary 1. Complex fraction (p. 586): _________________________________________ ______________________________________________________________

Key Concepts 2. Least Common Multiple (LCM) of Polynomials. (p. 584) LEAST COMMON MULTIPLE OF POLYNOMIALS to find the LCM of polynomials: 1. 2.

3. Get Organized In each box, write an example and show how to simplify it.

[ART: A207TE-C08-L03-A01]


170

Algebra 2

LESSON

Rational Functions


Vocabulary 1. Rational function (p. 592): _________________________________________ ______________________________________________________________ 2. Discontinuous function (p. 593): ____________________________________ ______________________________________________________________ 3. Continuous function (p. 593): ______________________________________ ______________________________________________________________ 4. Hole (in a graph) (p. 596): _________________________________________ ______________________________________________________________

Key Concepts 5. Rational Function (p. 592):

f(x)


a k xh

171

Algebra 2


6. Rational Functions (p. 593):

7. Zeros and Vertical Asymptotes (p. 594):

8. Horizontal Asymptotes (p. 594):

9. Holes in Graphs (p. 596):


172

Algebra 2


10. Get Organized In each box, write the formula or method for identifying the characteristics of graphs of rational functions. (p. 596). Zeros

Vertical asymptotes

p(x)

f(x) q(x)

Horizontal asymptotes


Holes

173

Algebra 2

Solving Rational Equations and 8-5 Inequalities

LESSON


Vocabulary 1. Rational equation (p. 600): ________________________________________ ______________________________________________________________ 2. Extraneous solution (p. 600): ______________________________________ ______________________________________________________________ 3. Rational inequality (p. 603): _______________________________________ ______________________________________________________________

Key Concepts 4. Get Organized In each box, write the appropriate information related to rational equations. (p. 604). Definition

Characteristics

Rational Equations

Examples


Nonexamples

174

Algebra 2

Rational Expressions and Rational 8-6 Exponents

LESSON


Vocabulary 1. Index (p. 610): __________________________________________________ ______________________________________________________________ 2. Rational exponent (p. 611): ________________________________________ ______________________________________________________________

Key Concepts 3. Properties of nth Roots (p. 611): For a 0 and b 0, WORDS

NUMBERS

ALGEBRA

Product Property of Roots

Quotient Property of Roots

4. Rational Exponents (p. 611): For any natural number n and integer m, WORDS


NUMBERS

175

ALGEBRA

Algebra 2


5. Properties of Rational Exponents (p. 612): For all nonzero real numbers a and b and integers m and n, WORDS

NUMBERS

ALGEBRA

Product of Powers Property

Quotient of Powers Property

Power of a Power Property

Power of a Product Property

Power of a Quotient Property

6. Get Organized In each box, give a numeric and algebraic example of the given property of rational exponents. (p. 614).

[ART: A207TE-C08-L06-A01]


176

Algebra 2

LESSON

Radical Functions


Vocabulary 1. Radical function (p. 619): _________________________________________ ______________________________________________________________ 2. Square-root function (p. 619): ______________________________________ ______________________________________________________________

Key Concepts 3. Transformations of Square Root Function f(x ) x (p. 620): TRANSFORMATION





Horizontal stretch/ compression

Reflection


177

Algebra 2


4. Get Organized In each box, give an example of the transformation of the square-root function f(x) x . (p. 623). TRANSFORMATION

EQUATION

GRAPH [ART: A207TE-C08-L07-A10]


Horizontal translation [ART: A207TE-C08-L07-A11]

Reflection

[ART: A207TE-C08-L07-A12]

Vertical stretch


[ART: A207TE-C08-L07-A13]

178

Algebra 2

Solving Radical Equations and 8-8 Inequalities

LESSON


Vocabulary 1. Radical equation (p. 628): _________________________________________ ______________________________________________________________ 2. Radical Inequality (p. 630): ________________________________________ ______________________________________________________________

Key Concepts 3. Solving Radical Equations (p. 611): SOLVING RADICAL EQUATIONS 1.

2.

3.

4. Get Organized In each box, write a step needed to solve a radical equation with extraneous solutions. (p. 632).

[ART: A207TE-C08-L08A01]


179

Algebra 2

CHAPTER

Chapter Review

8 8-1 Variation Functions 1. The cost c to fill a sandbox varies directly as the depth of the sand s. If a sandbox filled with 6 inches of sand cost $80 to fill, what is the cost to fill a sandbox to a depth of 9 inches? 2. The time t in hours needed to paint a house varies inversely with the number of painter’s p. If 4 painters can paint a 3000 square foot house in 48 hours, how many hours will it take 12 painters to paint the house?

8-2 Multiplying and Dividing Rational Expressions Simplify. Identify any x-values for which the expression is undefined. x x 12 4. 2

3

x 1 5. 2

2

6x 3. 2 12x 6x

x 9x 20

x 4x 5

Multiply or divide. Assume that all expressions are defined. x 2 25

2x 6

6. x3 x5

16x 8y 2

x

30x y

6x

4x 3 8x 2 x 4x 12

7. 3 3 6

x 2 5x 14 x 36

8. 2 2

8-3 Adding and Subtracting Rational Expressions Add or subtract. Identify any x-values for which the expression is undefined. 4x 7

x2

9. x3 x3


x 2 7x

4

10. x6 x 2 36

180

x

1

11. x4 x4

Algebra 2


12. A hot air balloon traveled from Austin, TX to a private island. The balloon averaged 10 mi/h. On the return trip the balloon averaged 12 mi/h. To the nearest mile per hour, what is the balloons average speed for the entire trip?

8-4 Rational Functions Identify the zeros and asymptotes of each function. Then graph. x2 9

3 x 14. f(x) 2

13. f(x) x4

x 9

y

–8

y

8

8

4

4

–4

4

8

x –8

–4

4

–4

–4

–8

–8

8

x

8-5 Solving Rational Equations and Inequalities Solve each equation. 24

15. x x 2

3x 1

6x 5

16. x4 2x 7

3

14

21 17. 2 x2 x2 x 4

18. Marty and Carla Johnson work on refinishing tables. Working alone Carla can complete a table in 7 hours. If the two work together, the job takes 5 hours. How long will it take Mary to refinish the table working alone?


181

Algebra 2


8-6 Radical Expressions and Rational Exponents Simplify each expression. Assume that all variables are positive. 19. 75x 3

8

a 21. 8

20. 27y15z 9 3

4

Write each expression in radical form, and simplify. 3

2

22. 36 2

2

23. 27 3

24. (125) 3

Write each expression by using rational exponents. 26. 164

74 25. 3

5

27. 100

2

3

5

28. In an experiment involving bacteria growth, the initial population is 250. The growth of the population can be modeled by the t function n(t ) 250 2 50 , where n is the number of bacteria and t is the time in hours. Based on this model, what is the population of bacteria after 2 weeks?

8-7 Radical Functions Graph each function, and identify its domain and range. 3

29. f(x) x 2

30. f(x) x2

y

y 4

2

–1

2

4

6

x

2

–2

–4

–2

2

–4

–2

–6

–4


182

4

x

Algebra 2


31. Oil is draining from a tank connected to two pipes. The speed f in feet per second at which oil drains through the first pipe can be modeled by f(x) 36(x , 3) where x is the depth of the oil in the tank in feet. The graph of the corresponding function for the second pipe is a translation of f 5 units right. Write a corresponding function g, and use it to estimate the speed at which oil drains through the second pipe when the depth of the water is 12 ft. 32. Use the description to write the square-root function g. The parent function f(x) x is vertically stretched by a factor of 2 and then translated 3 units left and 2 units up. Graph each function, and identify its domain and range. 33. y x 3

34. y x 4

y

y

6 6 4 4 2 2 –4

–2

2

4

x –1

–2

2

4

6

x

8-8 Solving Radical Equations and Inequalities Solve each equation. 3

35. 4 x 1 4


36. 9 xx3

183

4

4

37. a 8 2a

Algebra 2


38. The formula d

4w relates the 0.028 47 3

average diameter d of a cultured pearl in millimeters to its weight w in carats. To the nearest tenth of a carat, what is the weight of a cultured pearl with an average diameter of 9 mm? Solve each inequality. 39. x 75


3

40. 3x 6

41. x 5 12 5

184

Algebra 2

CHAPTER

Big Ideas

8 Answer these questions to summarize the important concepts from Chapter 8 in your own words. 1. Explain how to multiply and divide rational expressions.

2. Explain how to add and subtract rational expressions.

3. Explain how rational exponents and radicals are related.

4. Explain how to solve a radical equation.





185

Algebra 2

CHAPTER

Vocabulary


Page

Definition

Clarifying Example

composition of functions

one-to-one function


186

Algebra 2


Term

Page

Definition

Clarifying Example

piecewise function

step function


187

Algebra 2

LESSON

Multiple Representations of Functions


Key Concepts 1. Translating Between Multiple Representations. (p. 656) TRANSLATING BETWEEN MULTIPLE REPRESENTATIONS When given a(n). . .

Try to. . .

Table

Graph

Equation

Verbal Description


188

Algebra 2


2. Get Organized In each box, give an example. (p. 658).

Words

Graph

Multiple Representations

Function notation

Table


189

Algebra 2

LESSON

Piecewise Functions


Vocabulary 1. Piecewise function (p. 662): _______________________________________ ______________________________________________________________ 2. Step function (p. 663): ____________________________________________ ______________________________________________________________

Key Concepts 3. Get Organized Describe the domain and range for each function. Then include an example. (p. 665). FUNCTION DOMAIN

RANGE

EXAMPLE

Piecewise

Step


190

Algebra 2

LESSON

Transforming Functions


Key Concepts 1. Transformations of f(x). (p. 672) TRANSFORMATIONS OF f(x) Horizontal Translation

Vertical Translation

Reflection Across y-axis


Horizontal Stretch/Compression

Vertical Stretch/Compression


191

Algebra 2


2. Effects of Transformations on Intercepts of f(x). (p. 673) TRANSFORMATIONS OF f (x) Horizontal Stretch or Compression 1 by a Factor of b T

Vertical Stretch or Compression by a Factor of a

Reflection Across y-axis


3. Get Organized In each box, write an example and show how to simplify it. (p. 676). TRANSFORMATION

x-intercepts

y-intercepts

Horizontal stretch or compression by a factor of b Vertical stretch or compression by a factor of a Reflection across y-axis Reflection across x-axis


192

Algebra 2

LESSON

Operations with Functions


Vocabulary 1. Composition of functions (p. 683): __________________________________ ______________________________________________________________

Key Concepts 2. Notation of Function Operations (p. 682): OPERATION

NOTATION

Addition Subtraction Multiplication Division

3. Composition of Functions (p. 683):

4. Get Organized Write the correct notation for each function operation. (p. 685). OPERATION

NOTATION

Addition Subtraction Multiplication Division Composition


193

Algebra 2

LESSON

Functions and Their Inverses


Vocabulary 1. One-to-one function (p. 691): ______________________________________ ______________________________________________________________

Key Concepts 2. Horizontal-line Tests (p. 690): WORDS

EXAMPLES

3. Identifying Inverse Functions (p. 692): WORDS


ALGEBRA

194

EXAMPLE

Algebra 2


4. Get Organized Describe how each method or characteristic is used to find or verify inverses. (p. 693). Composition

Vertical/Horizontal Line Test

Inverses of Functions

Symmetry about y x


Switching x and y

195

Algebra 2

LESSON

Modeling Real-World Data


Key Concepts 1. Families of Functions (p. 698): FAMILY

LINEAR

QUADRATIC

EXPONENTIAL

SQUARE ROOT

Rule Graph

Constant Differences or Ratios

2. Get Organized Explain how each method can help you determine which model best fits a data set. (p. 701).

Identifying Models

Common differences or ratios: Constant first differences: linear; constant second differences: quadratic; constant third differences: cubic; constant ratios: exponential

Scatter plots: Compare the shape of the data points to the graphs of the parent functions.


Coefficient of determination: The closer the value of r 2 is to 1, the better the fit is.

196

Algebra 2

CHAPTER

Chapter Review

9 1. Jose is climbing down a 1200-foot cliff at a rate of 15 feet per second. Create a table, a graph and an equation to represent the number of feet Jose has left to climb down the cliff with relation to time.

2. The height of a rocket at different times after it was fired is shown in the table.

Time(s) Height (ft)

0 1 120 168

2 184

3 4 5 168 120 40

a. Find an appropriate model for the height of the rocket. b. Find the maximum height of the rocket. c. How long will the rocket stay in the air?

9-2 Piecewise Functions Graph each function. 3. f(x )

42x 1

if x 0 if x 0

4. f(x) 4 x 1 2x

y

if x 2 if x 2

y

6

6

4

4

2

2 x

–4

–2

0

2

x –4

4


197

–2

0

2

4

Algebra 2


5. The cost of renting cross-country skis is $45 for the first 4 hours and $5 for each each additional hour. Sketch a graph of the cost of renting cross-country skis for 0 to 8 hours. Then write the piecewise function for the graph.

y 100 90 80 70 60 50 40 30 20 10 0

x 1 2 3 4 5 6 7 8 9 10

Write a piecewise function for each graph. 6.

4

7.

y

2

8.

y

4 2

–4

–2

2

4

y

4

x 0

8

x –4

–2

–2

0

2

–2

4

x –8

–4

0

4

8

–4 –8

9-3 Transforming Functions Identify the x- and y-intercepts of f(x). Without graphing g(x), identify its x- and y-intercepts. 9. f(x ) 3x 6 and g(x ) 2f(x )


10. f(x) x 2 16 and g(x) f(x )

198

Algebra 2


Given f(x), graph g(x). 11. f(x ) x 2 and g(x ) 2f(x) 1 12. f(x) x 2 3 and g(x) 2f(x ) 6

y

8

4

4 x

2

–8

x –4

–2

0

y

2

–4

0

4

8

–4

4

–2

–8

9-4 Operations with Functions 4

, g(x) x 7, and h(x) x 2 4x 21, find each Given f(x) x1 function or value.

13. (f g)(5)

14. (g h)(x)

g 15. (5) h

h 16. g (x)

17. (gh)(3)

18. (gf )(x)

19. g(f (3))

20. h(g(x))

21. Find (g f ). State the domain of the composite function. 22. The local clothing store is having a 30% off sale. Preferred customers receive a coupon worth an additional 10% off. Write a composite function for the price a preferred customer pays for an item with an original price of p dollars.


199

Algebra 2


9-5 Functions and Their Inverses State whether the inverse of each relation is a function. 23.

4

24.

y

4

2

y

2 x

–4

0

–2

2

x –4

4

–2

0

–2

–2

–4

–4

2

4

Write the rule for the inverse of each function. Then state the domain and range of the inverse. 7

1

25. f(x ) 2x 6

26. h(x) x3

27. h(x ) x 2 25

28. h(x) x 3 4

9-6 Modeling Real-World Data 29. Use finite differences or ratios to determine which parent function would best model this set of data. x y

0 1

1 0

2 3 4 1 4 9

30. The table shows the mass g in grams of a radioactive substance remaining in a container t days after the beginning of the experiment. Find a model for the amount of radioactive substance remaining. Time (days) Mass (g)

0

1

2000 1834.08


2

3

1683.24

1544.2

200

4

5

1416.67 1299.67

6 1192.28

Algebra 2

CHAPTER

Big Ideas

9 Answer these questions to summarize the important concepts from Chapter 9 in your own words. 1. Explain a piecewise function.

2. Explain how to transform a piecewise function.

3. Explain how to perform operations on functions.

4. Explain how determine if the inverse of a function is a relation and how to write the rule for an inverse function.





201

Algebra 2

CHAPTER

Vocabulary


Page

Definition

Clarifying Example

circle

conic section

directrix

ellipse


202

Algebra 2


Term

Page

Definition

Clarifying Example

hyperbola

major axis

minor axis

nonlinear system of equations tangent


203

Algebra 2

LESSON

Introduction to Conic Sections


Vocabulary 1. Conic section (p. 722): ___________________________________________ ______________________________________________________________ ______________________________________________________________

Key Concepts 2. Midpoint and Distance Formulas. (p. 724) FORMULA


EXAMPLE

204

GRAPH

Algebra 2


3. Get Organized List the types of conic sections, and sketch an example of each. (p. 725).

Conic Sections


205

Algebra 2

LESSON

Circles


Vocabulary 1. Circle (p. 729): __________________________________________________ ______________________________________________________________ 2. Tangent (p. 731): ________________________________________________ ______________________________________________________________

Key Concepts 3. Equation of a Circle (p. 729): EQUATION


EXAMPLE

206

GRAPH

Algebra 2


4. Get Organized Sketch each circle, and give its equation. (p. 731). r1

r3

Center (0, 0)

Center (1, 2)


207

Algebra 2

LESSON

Ellipses


Vocabulary 1. Ellipse (p. 736): _________________________________________________ ______________________________________________________________ ______________________________________________________________ 2. Foci of an ellipse (p. 736): _________________________________________ ______________________________________________________________ 3. Major axis (p. 736): ______________________________________________ ______________________________________________________________ 4. Vertices of an ellipse (p. 736): _____________________________________ ______________________________________________________________ 5. Minor axis (p. 736): ______________________________________________ ______________________________________________________________ 6. Co-vertices of an ellipse (p. 736): ___________________________________ ______________________________________________________________

Key Concepts 7. Standard Form for the Equation of an Ellipse (Center at (0, 0)). (p. 737) MAJOR AXIS

HORIZONTAL

VERTICAL

Equation Vertices Foci Co-vertices


208

Algebra 2


8. Standard Form for the Equation of an Ellipse (Center at (h, k)). (p. 738) MAJOR AXIS

HORIZONTAL

VERTICAL

Equation Vertices Foci Co-vertices

9. Get Organized Give an equation for each type of ellipse. (p. 739).

Horizontal major axis

Vertical major axis

Ellipses

Center (h, k)

Center (0, 0)


209

Algebra 2

LESSON

Hyperbolas


Vocabulary 1. Hyperbola (p. 744): ______________________________________________ ______________________________________________________________ 2. Foci of a hyperbola (p. 744): _______________________________________ ______________________________________________________________ 3. Branch of a hyperbola (p. 744): ____________________________________ ______________________________________________________________ 4. Traverse axis (p. 744): ___________________________________________ ______________________________________________________________ 5. Vertices of a hyperbola (p. 744): ___________________________________ ______________________________________________________________ 6. Conjugate axis (p. 744): __________________________________________ ______________________________________________________________ 7. Co-vertices of an ellipse (p. 744): __________________________________ ______________________________________________________________


210

Algebra 2


Key Concepts 8. Standard Form for the Equation of a Hyperbola (Center at (0, 0)). (p. 745) TRANSVERSE AXIS

HORIZONTAL

VERTICAL

Equation Vertices Foci Co-vertices Asymptotes

9. Standard Form for the Equation of a Hyperbola (Center at (h, k)). (p. 746) TRANSVERSE AXIS

HORIZONTAL

VERTICAL

Equation Vertices Foci Co-vertices Asymptotes


211

Algebra 2


10. Get Organized Label all of the parts of the hyperbola. (p. 747).

y

x


212

Algebra 2

LESSON

Parabolas


Vocabulary 1. Focus of a parabola (p. 751): ______________________________________ ______________________________________________________________ ______________________________________________________________ 2. Directrix (p. 751): ________________________________________________ ______________________________________________________________ ______________________________________________________________

Key Concepts 3. Standard Form for the Equation of a Parabola (Vertex at (0, 0)). (p. 752) AXIS OF SYMMETRY

HORIZONTAL y 0 VERTICAL x 0

Equation Direction Focus Directrix Graph


213

Algebra 2


4. Standard Form for the Equation of a Parabola (Vertex at (h, k)). (p. 752) AXIS OF SYMMETRY

HORIZONTAL y k

VERTICAL x h

Equation Direction Focus Directrix Graph

5. Get Organized Sketch an example and give an equation for each type of parabola. (p. 754). Opens upward

Opens right

Parabola

Opens downward


Opens left

214

Algebra 2

LESSON

Identifying Conic Sections


Key Concepts 1. Standard Forms of the Conic Sections with Center (h, k) (p. 760): Circle HORIZONTAL AXIS

VERTICAL AXIS

Ellipse Hyperbola Parabola

2. Classifying Conic Sections (p. 761): For an equation of the form Ax 2 Bxy Cy 2 Dx Ey F 0 (A, B, and C do not all equal 0.) CONIC SECTION

COEFFICIENTS

Circle Ellipse Hyperbola Parabola


215

Algebra 2


3. Get Organized Give an example of coefficients for each conic section in the general form. (p. 763).

Ellipse

Circle

Coefficients of Ax + Bxy + Cy 2 + Dx + Ey + F = 0 2

Hyperbola


Parabola

216

Algebra 2

LESSON

Solving Nonlinear Systems


Vocabulary 1. Nonlinear system of equations (p. 768): ______________________________ ______________________________________________________________

Key Concepts 2. Get Organized Use the table to record information on the intersection of a hyperbola and a circle. (p. 771). GRAPH

EXAMPLE

No solution

One solution

Two solutions

Three solutions

Four solutions


217

Algebra 2

CHAPTER

Chapter Review

10 10-1 Introduction to Conic Sections 1. A delivery area of a pizza parlor extends to the locations (6, 2) and (6, 8). Write an equation for the delivery area of the pizza parlor if a line between the locations represents a diameter of the delivery area. Identify and describe each conic section. (x 1)2

(y 3)2

2. 36 1 36

3. 4x 2 9y 2 36

4. x y 4

x2 y2 5. 1 25 36

2

10-2 Circles Write the equation of each circle. 6. center (2, 5) and radius r 4

7. center (3, 2) and containing the point (11, 2)

8. Write the equation of the line that is tangent to x 2 y 2 25 at (3, 4).


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10-3 Ellipses Find the center, vertices, co-vertices, and foci of each ellipse. Then graph. x2

y2

9. 1 25 9

10. 25(x 1)2 9(y 1)2 225

y

y 8

4

4

2

x

x

–4 –2 –2

2

–8 –4 –4

4

4

8

–8

–4

11. A child models a semi-elliptical bridge for a science fair project. The bridge is 12 inches wide and 4 inches high at its highest point. Write an equation for a cross section of the bridge.

10-4 Hyperbolas Find the center, vertices, co-vertices, foci, and asymptotes for each hyperbola. Then graph. x2

(y 2)2

y2

12. 4 9 1

(x 2)2

13. 9 16 1

y

y 8

4 2 –4 –2 –2

4

x 2

–8 –4 –4

4

4

8

–8

–4


x

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14. Write the equation of a hyperbola with vertices (–2, 0) and (–2, 6) and co-vertices (3, 3) and (1, 3).

10-5 Parabolas Find the vertex, value of p, axis of symmetry, focus, and directrix for each parabola. Then graph. 1

1

15. x 4y 2

16. y 4(x 2)2

y

y

4

8

2 –4 –2 –2

4

x 2

4

–8 –4 –4

–4

x 4

8

–8

17. Write an equation of the parabola with focus (0, 3) and directrix x 4. 18. A fabricator fabricates a taillight for an automobile. If the depth of the parabolic taillight is 4 inches and 5 inches in diameter, what is the distance d the bulb should be from the vertex in order for the beam of light to shine straight ahead?

10-6 Identifying Conic Sections Identify the conic section that each equation represents. 19. (x 5)2 (y 3)2 16


x2

y2

20. 1 16 4

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Algebra 2


21. 9x 2 36x 16y 2 64y 44 0 22. 2x y 2 4y 12

23. (x 1)2 (y 2)

(x 1)2

24. 9x 2 18x 4y 2 16y 43 0

(y 1)2

25. 16 1 16

26. 4x 2 9y 2 36

Write each equation in the form Ax 2 Bxy Cy 2 Dx Ey F 0. 27. (x 4) (y 1) 2 2

2

(x 2)2 (y 3)2 28. 9 4 1

Find the standard form of each equation by completing the square. Then identify the conic. 29. 3y 2 24y 2x 2 12x 24 0

30. 9x 2 18x 4y 2 8y 23 0

31. 2x y 2 4y 12

32. x 2 6x y 2 4y 3 0


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Algebra 2


10-7 Solving Nonlinear Systems Solve each system of equations by graphing. 33.

x2 y2 9 2x 2 3y 2 18

34.

y2 x 3 2y x 4

y

x 2 2y 2 16 4x 2 y 2 4

y

4

y 4

4

2 –4 –2 –2

35.

2

x 2

4

–4 –2 –2

–4

2

x 2

4

x

–4 –2 –2

2

4

–4

–4

Solve each system of equations by using the substitution or elimination method. 36.

x2 y2 4 2x 2 y 2 8

37.

2x 2 2y 2 8 4x 2 9y 2 36

38.

x2 y2 9 9x 2 y 2 9

39. Two ice-skaters are giving a performance. The paths of the skaters are shown in the graph. During the performance, the lead skater moves in a path that 1 can be modeled by the equation y 4(x 2)2 1. The otherskater glides in formation along the equation x y 2 y 6. At what point(s) are the skaters in danger of colliding?

40. Find n so that the system


y 8 4 –8 –4 –4

x 4

8

–8

x 2 y 2 16 has exactly 3 solutions. x y2 n

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Algebra 2

CHAPTER

Big Ideas

10 Answer these questions to summarize the important concepts from Chapter 10 in your own words. 1. Name the four types of conic sections discussed in this chapter.

2. Describe the axes of an ellipsis in terms of the vertices, co-vertices, and foci.

3. Describe how to classify an equation in standard form, just by looking at it, in regards to squared terms.

4. Describe how the graph of a parabola is translated.





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Algebra 2

CHAPTER

Vocabulary


Page

Definition

Clarifying Example

Binomial Theorem

compound event

dependent events

equally likely outcomes

Fundamental Counting Principle


224

Algebra 2


Term

Page

Definition

Clarifying Example

outcome

permutation

probability

standard deviation

theoretical probability


225

Algebra 2

LESSON

Permutations and Combinations


Vocabulary 1. Fundamental Counting Principle (p. 794): ____________________________ ______________________________________________________________ ______________________________________________________________ 2. Permutation (p. 795): ____________________________________________ ______________________________________________________________ 3. Factorial (p. 795): _______________________________________________ ______________________________________________________________ 4. Combination (p. 796): ____________________________________________ ______________________________________________________________

Key Concepts 5. Fundamental Counting Principle. (p. 794)

6. n Factorial (p. 795): WORDS


NUMBERS

ALGEBRA

226

Algebra 2


7. Permutations (p. 795): NUMBERS

ALGEBRA

8. Combinations (p. 797): NUMBERS

ALGEBRA

9. Get Organized Complete the graphic organizer. (p. 797). FUNDAMENTAL COUNTING PRINCIPLE

PERMUTATIONS

COMBINATIONS

Formulas

Examples


227

Algebra 2

LESSON

Theoretical and Experimental Probability


Vocabulary 1. Probability (p. 802): ______________________________________________ ______________________________________________________________ 2. Outcome (p. 802): _______________________________________________ ______________________________________________________________ 3. Sample space (p. 802): ___________________________________________ ______________________________________________________________ 4. Event (p. 802): __________________________________________________ ______________________________________________________________ 5. Equally likely outcomes (p. 802): ___________________________________ ______________________________________________________________ 6. Favorable outcomes (p. 802): ______________________________________ ______________________________________________________________ 7. Theoretical probability (p. 802): ____________________________________ ______________________________________________________________ 8. Complement (p. 803): ____________________________________________ ______________________________________________________________ 9. Geometric probability (p. 804): _____________________________________ ______________________________________________________________ 10. Experiment (p. 805): _____________________________________________ ______________________________________________________________ 11. Trial (p. 805): a repetition of an experiment. ___________________________ ______________________________________________________________


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Algebra 2


12. Experimental probability (p. 805): ___________________________________ ______________________________________________________________

Key Concepts 13. Theoretical Probability. (p. 802)

14. Complement (p. 803):

15. Experimental Probability (p. 805):

16. Get Organized Give an example of each type of item. (p. 806). Experimental

Theoretical

Probability

Geometric


229

Algebra 2

LESSON

Independent and Dependent Events


Vocabulary 1. Independent events (p. 811): _______________________________________ ______________________________________________________________ 2. Dependent events (p. 812): ________________________________________ ______________________________________________________________ 3. Conditional probability (p. 812): _____________________________________ ______________________________________________________________

Key Concepts 4. Probability of Independent Events (p. 811):

5. Probability of Dependent Events (p. 812):


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Algebra 2


6. Get Organized In each box, compare independent and dependent events and their related probabilities. (p. 814).

Probability of Independent Events vs. Probability of Dependent Events

Similarities


Differences

231

Algebra 2

LESSON

Compound Events


Vocabulary 1. Simple Event (p. 819): ____________________________________________ ______________________________________________________________ 2. Compound event (p. 819): _________________________________________ ______________________________________________________________ 3. Mutually exclusive events (p. 819): __________________________________ ______________________________________________________________ 4. Inclusive events (p. 820): __________________________________________ ______________________________________________________________

Key Concepts 5. Mutually Exclusive Events (p. 819): WORDS


ALGEBRA

EXAMPLE

232

Algebra 2


6. Inclusive Events. (p. 820) WORDS

ALGEBRA EXAMPLE

7. Get Organized Give at least one example for each. (p. 822).

Adding Probabilities

Mutually Exclusive Events

Multiplying Probabilities

Inclusive Events Probabilities

Compound Events


233

Algebra 2

Measures of Central Tendency and 11-5 Variation LESSON


Vocabulary 1. Expected value (p. 828): __________________________________________ ______________________________________________________________ 2. Probability distribution (p. 828): ____________________________________ ______________________________________________________________ 3. Variance (p. 830): _______________________________________________ ______________________________________________________________ 4. Standard deviation (p. 830): _______________________________________ ______________________________________________________________ 5. Outlier (p. 831): _________________________________________________ ______________________________________________________________

Key Concepts 6. Finding Variance and Standard Deviation (p. 830): Finding Variance and Standard Deviation Step 1.

Step 2.

Step 3.

Step 4.


234

Algebra 2


7. Get Organized In each box, define and give an example of each measure. (p. 832).

Range

Variance

Measures of Variability

Interquartile Range


Standard Deviation

235

Algebra 2

LESSON

Binomial Distributions


Vocabulary 1. Binomial Theorem (p. 828): ________________________________________ ______________________________________________________________ 2. Binomial experiment (p. 838): ______________________________________ ______________________________________________________________ 3. Binomial probability (p. 838): ______________________________________ ______________________________________________________________

Key Concepts 4. Binomial Theorem (p. 838):

5. Binomial Probability (p. 838):


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Algebra 2


6. Get Organized Solve each problem that you include. (p. 840). BINOMIAL EXPERIMENTS PROBABILITY

EXAMPLE

Probability of r successes in n trials Probability of at least r successes Probability of at most r successes Probability using a complement


237

Algebra 2

CHAPTER

Chapter Review

11 11-1 Permutations and Combinations 1. Frank’s access code for his garage door consists of 4 digits from 0 through 9. How many possible access codes are there if no digit can be repeated? 2. Kara picks a dozen flowers from her back yard. How many ways can she choose 3 flowers from the bouquet? 3. Find the number of ways to arrange 3 compact discs from a selection of 7 compact discs in a CD player.

11-2 Theoretical and Experimental Probability 4. A candy jar contains 13 peppermint candies, 9 strawberry candies, 4 lemon candies, and 6 root beer flavored candies. If a child selects a candy from the jar without looking, what is the probability that the child will select a lemon candy? 5. Weston has 7 strands of holiday lights in a box. Two of the strands do not work. If he selects 2 strands from the box, what is the probability that both strands do not work? 6. Sue tosses a beanbag onto a rectangular rug. If the beanbag does not touch a line, what is the probability that the beanbag landed in a shaded area? 7. A number cube is rolled 48 times, and a 6 is rolled 14 times. Find the experimental probability of not rolling a 6.

11-3 Independent and Dependent Events 8. Explain why the event of rolling one die and getting a 6 on two turns in a row while playing a board game are independent. Then find the probability.


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Algebra 2


9. A golf bag contains 4 while golf balls and 2 yellow golf balls. Explain why the events of “choosing a white golf ball then choosing a yellow golf ball” without replacing the white golf ball are dependent, and find the probability.

10. The table shows the breakdown by age of preschoolers that were screened by a pediatric optometrist that wear glasses. Find the probability that a 3-year-old preschooler wears glasses.

Preschoolers by Age Wears Glasses

Does Not Wear Glasses

3 yr

2

43

4 yr

3

36

5 yr

5

47

11. A bag contains 15 marbles; 10 red and 5 blue. Determine whether the events “a red marble is selected, not replaced, then a blue marble is selected” is independent or dependent, and find the probability.

11-4 Compound Events A standard deck of cards is in a pile face down. One card is drawn. Find each probability. 12. drawing a 9 or a queen

13. drawing a red card or an ace

14. A dog pound currently has 35 dogs; 20 are puppies and 15 have collars. Half of the puppies have collars. What is the probability that an adult dog does not have a collar?


239

Algebra 2


11-5 Measures of Central Tendency and Dispersion 15. In each of the last five days, Jerome has driven 14, 17, 26, 17, and 16 miles. Find the mean, median, and mode of the data set. 16. At a hardware store there are 10 key chains on a rack. On the back of each key chain is a dollar amount that can be saved from your purchase. Use the data provided to Saving Amounts, n $1 find, find the expected savings for a Probability of n Savings 0.7 purchase.

$2

$5

0.2

0.1

17. Make a box-and-whisker plot of the data. Find the interquartile range. Hours worked each week at a summer job: 29, 32, 40, 31, 33, 39, 27, and 42.

18. The test scores for a science test, in percents, are given below. Find the percents within 1 standard deviation of the mean. Test scores: 82, 84, 87, 82, 85, 97, 68, 96, 99, and 60.


240

Algebra 2


The data set shows the average miles per gallon, rounded to the nearest mile, for six consecutive automobiles filling at a fueling station. 26, 30, 16, 28, 29, 27 19. Find the mean and standard deviation of the data.

20. Identify the outlier, and describe how it affects the mean and standard deviation.

11-6 Binomial Distributions 21. Use the binomial theorem to expand (3x y)3.

At a toy store, 1 out of every 5 action figures contains a free DVD. 22. What is the probability that Jack will get at least 2 DVD’s, if he purchases 4 action figures? 23. What is the probability that Jack will get at least 3 DVD’s, if he purchases 4 action figures?


241

Algebra 2


A festival game contains a spinning wheel in which is divided into 4 equal sections. Only one section is labeled winner. A person plays the game 10 times. Find the probability of each. 24. The player will win 5 times. 25. The player will win at least 1 time. 26. The player will win at most 7 times. 27. The player wins at most 1 time.


242

Algebra 2

CHAPTER

Big Ideas

11 Answer these questions to summarize the important concepts from Chapter 11 in your own words. 1. Tell how you decide when to use a permutation or a combination.

2. Describe the difference between independent events and dependent events.

3. Describe the difference between mutually exclusive events and inclusive events.

4. Tell what five key points a box-and-whisker plot displays.





243

Algebra 2

CHAPTER

Vocabulary


Page

Definition

Clarifying Example

arithmetic sequence

explicit formula

finite sequence

geometric mean

geometric sequence

infinite geometric series


244

Algebra 2


Term

Page

Definition

Clarifying Example

infinite sequence limit

partial sum

recursive formula

sequence series

summation notation

term of a sequence


245

Algebra 2

LESSON

Introduction to Sequences


Vocabulary 1. Sequence (p. 862): ______________________________________________ ______________________________________________________________ 2. Term of a sequence (p. 862): ______________________________________ ______________________________________________________________ 3. Infinite sequence (p. 862): ________________________________________ ______________________________________________________________ 4. Finite sequence (p. 862): _________________________________________ ______________________________________________________________ 5. Recursive formula (p. 862): ________________________________________ ______________________________________________________________ 6. Explicit formula (p. 863): __________________________________________ ______________________________________________________________ 7. Iteration (p. 864): ________________________________________________ ______________________________________________________________


246

Algebra 2


Key Concepts 8. Get Organized Summarize what you have learned about sequences. (p. 865).

Definition

Two types of sequences

Probability

Two possible formulas

Examples


247

Algebra 2

LESSON

Series and Summation Notation


Vocabulary 1. Series (p. 870): _________________________________________________ ______________________________________________________________ 2. Partial sum (p. 870): _____________________________________________ ______________________________________________________________ 3. Summation notation (p. 870): ______________________________________ ______________________________________________________________

Key Concepts 4. Summation Formulas. (p. 871) CONSTANT SERIES

LINEAR SERIES

QUADRATIC SERIES

5. Get Organized Write the general notation and an example for each term. (p. 873). SEQUENCE

SERIES

Notation

Example


248

Algebra 2

LESSON

Arithmetic Sequences and Series


Vocabulary 1. Arithmetic sequence (p. 879): ______________________________________ ______________________________________________________________ 2. Arithmetic series (p. 882): _________________________________________ ______________________________________________________________

Key Concepts 3. General Rule for Arithmetic Sequences (p. 880):

4. Sum of the First n Terms of an Arithmetic Series (p. 882): WORDS


NUMBERS

249

ALGEBRA

Algebra 2


5. Get Organized Write in each rectangle to summarize your understanding of arithmetic sequences. (p. 883). Definition

Characteristics

Probability

Examples


Formulas

250

Algebra 2

LESSON

Geometric Sequences and Series


Vocabulary 1. Geometric sequence (p. 890): _____________________________________ ______________________________________________________________ 2. Geometric mean (p. 892): _________________________________________ ______________________________________________________________ 3. Geometric series (p. 893): ________________________________________ ______________________________________________________________

Key Concepts 4. General Rule for Geometric Sequences (p. 891):

5. Geometric Mean (p. 892):

6. Sum of the First n Terms of a Geometric Series (p. 893):


251

Algebra 2


7. Get Organized In each box, summarize your understanding of geometric sequences. (p. 894).

Definition

Characteristics

Probability

Examples


Formulas

252

Algebra 2

Mathematical Induction and Infinite 12-5 Geometric Series LESSON


Vocabulary 1. Infinite geometric series (p. 900): ___________________________________ ______________________________________________________________ 2. Converge (p. 900): ______________________________________________ ______________________________________________________________ 3. Limit (p. 900): __________________________________________________ ______________________________________________________________ 4. Diverge (p. 900): ________________________________________________ ______________________________________________________________ 5. Mathematical induction (p. 831): ____________________________________ ______________________________________________________________

Key Concepts 6. Sum of an Infinite Geometric Series (p. 901)

7. Proof by Mathematical Induction (p. 902)


253

Algebra 2


8. Get Organized Summarize the different infinite geometric series. (p. 903). EXAMPLE

COMMON RATIO

SUM

Convergent Series

Divergent Series


254

Algebra 2

CHAPTER

Chapter Review

12 12-1 Introduction to Sequences Find the first 5 terms of each sequence. 4

1. an 5n

2. an 3n1

n

3. an n1

4. an n 2 3n

Write a possible explicit rule for the nth term of each sequence. 5. 1, 3, 5, 7, ...

6. 2, 4, 6, 8, ...

7. 400, 200, 100, 50, 25, ...

8. 366, 344, 322, 300, 278, ...

9. A car traveling at 65 mi/h passes a mile marker that reads mile 23. If the car maintains this speed for 5 hours, what mile marker should the car pass? Graph the sequence for n hours, and describe its pattern. y 350 315 280 245 210 175 140 105 70 35

x 1 2 3 4 5 6 7 8 9 10

12-2 Series and Summation Notation Expand each series and evaluate. 7

10. (2k 1) k1


4

5

12. (k 2 1)

k 11. k 1 k=1

k 1

255

Algebra 2


Evaluate each series. 6

13. 3 1

k1

10

20

14. k 3

15. k

k1

k1

16. The first row of an auditorium has 28 seats, and each of the following rows has 4 more seats than the preceding row. How many seats are in the first 12 rows?

12-3 Arithmetic Sequences and Series Find the 8th term of each arithmetic sequence. 17. 8.01, 8.02, 8.03, 8.04, …

18. 2, 8, 14, 20, …

19. a3 14 and a6 29

20. a3 8 and a7 32

Find the missing terms in each arithmetic sequence. 21. 46, ___, ___, –178

22. 62, ___, ___, ___, 158

Find the indicated sum for each arithmetic series. 5

23. S10 for 80 60 40 20 …

24. k 3 k1

8

25. (0.75k 1.25) k1


26. S9 for 8, 3, 2, 7, …

256

Algebra 2


27. Suppose that you make a bank deposit for $2.50 the first week in January, $3.00 the second week, $3.50 the third week, and so on. How much will you contribute to the account on the last week of the year (52nd week)? What is the total amount that you have deposited in the bank after one year?

12-4 Geometric Sequences and Series Find the 8th term of each geometric sequence. 1 1 1 , , ... 28. 1, 4, 16 64

29. 2, 6, 18, 54, ...

30. 16, 8, 4, 2, ...

31. 6, 36, 216, 1296, ...

Find the 10th term of each geometric sequence with the given terms. 32. a2 20 and a4 5

1 33. a2 5 and a4 5

34. a2 10 and a9 781,250

35. a3 48 and a6 486

Find the geometric mean of each pair of numbers. 1 1 36. 6 and 24

1

37. 25 and 121

38. 108 and 3

Find the indicated sum for each geometric series. 39. S6 for 1 4 16 64 ...

10

41. 3k k0


1 1 1 ... 40. S9 for 18 6 2

9

42. 122k 1

k0

257

Algebra 2


43. A small business has spent $100,000 for gas. The cost of the gas is expected to increase at an annual rate of 3%. a. What are the gas costs in year 15? b. How much in total will be paid for gas over the first 15 years?

12-5 Mathematical Induction and Infinite Geometric Series Find the sum of each infinite series, if it exists. 44. 4 2 1 …

∞

45. 8 40 200 …

∞

46. 2k1

47. 500(1.11)k

1

k1

k0

Use mathematical induction to prove 4 8 12 … 4n 2n(n 1). 48. Step 1

49. Step 2

50. Step 3

51. A ping pong ball is dropped from a height of 16 feet. The ball rebounds to 25% of its previous height after each bounce. a. Write an infinite geometric series to represent the distance that the ball travels after it initially hits the ground. (Hint: The ball travels up and down on each bounce.) b. What is the total distance that the ball travels after it initially hits the ground?


258

Algebra 2

CHAPTER

Big Ideas

12 Answer these questions to summarize the important concepts from Chapter 12 in your own words. 1. Explain the difference between an arithmetic and geometric sequence.

2. How is sigma notation used?

3. What is the difference between a series and a sequence?

4. What is an infinite geometric series?





259

Algebra 2

Vocabulary

CHAPTER


Page

Definition

Clarifying Example

angle of rotation

cosecant

cosine

cotangent

coterminal angle

initial side


260

Algebra 2


Term

Page

Definition

Clarifying Example

radian

reference angle

secant

sine

standard position

tangent

terminal side


261

Algebra 2

LESSON

Right-Angle Trigonometry


Vocabulary 1. Trigonometric function (p. 929): ____________________________________ ______________________________________________________________ 2. Sine (p. 929): ___________________________________________________ ______________________________________________________________ 3. Cosine (p. 929): _________________________________________________ ______________________________________________________________ 4. Tangent (p. 929): ________________________________________________ ______________________________________________________________ 5. Cosecant (p. 932): _______________________________________________ ______________________________________________________________ 6. Secant (p. 932): _________________________________________________ ______________________________________________________________ 7. Cotangent (p. 932): ______________________________________________ ______________________________________________________________


262

Algebra 2


Key Concepts 8. Trigonometric Functions. (p. 929) WORDS

NUMBERS

SYMBOLS

9. Trigonometric Ratios of Special Right Triangles. (p. 930) DIAGRAM

SINE

COSINE

TANGENT

ART: a207se_c13 101_t06

ART: a207se_c13 101_t07


263

Algebra 2


10. Reciprocal Trigonometric Functions. (p. 932) WORDS

NUMBERS

SYMBOLS

11. Get Organized For each trigonometric function, give the name and the reciprocal function. (p. 932). SIN

COS

TAN

Function Name

Side Length Ratio Reciprocal Function


264

Algebra 2

LESSON

Angles of Rotation


Vocabulary 1. Standard position (p. 936): ________________________________________ ______________________________________________________________ 2. Initial side (p. 936): ______________________________________________ ______________________________________________________________ 3. Terminal side (p. 936): ___________________________________________ ______________________________________________________________ 4. Angle of rotation (p. 936): _________________________________________ ______________________________________________________________ 5. Coterminal angles (p. 937): ________________________________________ ______________________________________________________________ 6. Reference angle (p. 937): _________________________________________ ______________________________________________________________

Key Concepts 7. Standard Position and Rotations. (p. 932) Positive Rotation


Negative Rotation

265

Algebra 2


8. Trigonometric Functions. (p. 938) For a point P(x, y) on the terminal side of θ in standard position and r x 2 y 2, SINE

COSINE

TANGENT

9. Get Organized In each box, describe how to determine the given angle or position for an angle θ. (p. 938). Standard position

Reference angle

Angle θ

Positive coterminal angle


Negative coterminal angle

266

Algebra 2

LESSON

The Unit Circle


Vocabulary 1. Radian (p. 943): ________________________________________________ ______________________________________________________________ 2. Unit circle (p. 944): ______________________________________________ ______________________________________________________________

Key Concepts 3. Converting Angle Measures (p. 943): DEGREES TO RADIANS

RADIANS TO DEGREES

4. Trigonometric Functions and Reference Angles (p. 944): To find the sine, cosine, or tangent of θ: Step 1 Step 2 Step 3


267

Algebra 2


5. Arc Length Formula. (p. 945)

6. Get Organized In each box, give an expression that can be used to determine the value of the trigonometric function. (p. 946). ACUTE ANGLE OF ANGLE OF RIGHT TRIANGLE ROTATION WITH P(x, y)

ANGLE WITH P(x, y) ON UNIT CIRCLE

sin

cos

tan


268

Algebra 2

LESSON

Inverses of Trigonometric Functions


Vocabulary 1. Inverse sine function (p. 951): ______________________________________ ______________________________________________________________ 2. Inverse cosine function (p. 951): ____________________________________ ______________________________________________________________ 3. Inverse tangent function (p. 951): ___________________________________ ______________________________________________________________

Key Concepts 4. Inverse Trigonometric Functions (p. 891): WORDS

SYMBOL

DOMAIN

RANGE

The inverse sine function:

The inverse cosine function:

The inverse tangent function:


269

Algebra 2


5. Get Organized In each box, give the indicated property of the inverse trigonometric functions. (p. 953). Symbols

Domains

Inverse Trigonometric Functions

Associated Quadrants


Ranges

270

Algebra 2

LESSON

The Law of Sines


Key Concepts 1. Area of a Triangle (p. 958)

2. Law of Sines (p. 959)

3. Solving a Triangle (p. 960) Solving a Triangle Given a, b, and mA 1. 2. 3.


271

Algebra 2


4. Get Organized In each box, give the conditions for which the ambiguous case results in zero, one, or two triangles. (p. 962). SSA: Given a, b, and A ANGLE A

0 TRIANGLE

1 TRIANGLE

2 TRIANGLES

Obtuse

Acute


272

Algebra 2

LESSON

The Law of Cosines


Key Concepts 1. Sum of an Infinite Geometric Series (p. 966)

2. Heron’s Formula (p. 969)

3. Get Organized List the types of triangles that can be solved by using each law. Consider the following types of triangles: ASA, AAS, SAS, SSA, and SSS. (p. 970).

Law of Sines


Law of Cosines

273

Algebra 2

CHAPTER

Chapter Review

13 13-1 Right-Angle Trigonometry Find the values of the six trigonometric functions for . 1.

2.

8

15

3

17

1

Use a trigonometric function to find the value of x. 3.

4.

50 60°

35

x 45°

x


274

Algebra 2


5. A conservationist whose eye level is 5.5 feet above the ground measures the angle of elevation to the top of the tree to be 15°. If the conservationist is standing 100 feet away from the tree base, what is the height of the tree to the nearest foot?

13-2 Angles of Rotation Draw an angle with the given measure in standard position. 6. 105°

7. 510°

Point P is a point on the terminal side of in standard position. Find the exact value of the six trigonometric functions for . 8. P(2, 5)

9. P(3, 2)

13-3 The Unit Circle Convert each measure from degrees to radians or from radians to degrees. 10. 210°

11. 12°


3 12. 4

275

5 13. 6

Algebra 2


Use the unit circle to find the exact value of each trigonometric function. 14. cos 315°

2

5 17. tan 3

16. cos 3

15. tan 180°

18. A CD rotates through an angle of 20 radians in 1 second. At this speed, how many revolutions does the CD make in 1 hour?

13-4 Inverses of Trigonometric Functions Evaluate each inverse trigonometric function. Give your answer in both radians and degrees. 2 19. Cos1 2

20. Tan1 3

21. A hospital wants to make its picnic area wheel chair accessible. A 12 ft ramp is installed to reach a deck that is 3 ft off the ground. To the nearest degree, what angle does the ramp make with the ground?

13-5 The Law of Sines Find the area of each triangle. Round to the nearest tenth. 22.

23. 51 yd 120° 84 yd

17

80°


276

65°

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Solve each triangle. Round to the nearest tenth. 24.

25.

C

C

29° 43 in.

B

62°

46°

A B

52°

A

Catherine is designing wooden triangular sections for a cutting board. Determine the number of different triangles that she can form using the given measurements. Then solve the triangles. Round to the nearest tenth. 26. a 27, b 32, mA 104°

27. a 15.6, b 8.14, mA 43°

28. A hot air balloon is observed from two points A and B 800 feet apart. The angle of elevation is 63° from point A and 39° from point B. What is the distance from each point to the balloon?

A


277

63°

39° 800 ft

B

Algebra 2


13-6 The Law of Cosines Use the given measurements to solve each triangle. Round to the nearest tenth. 29.

30.

B

B 6

3 18

A

C 54°

C

5

20

A

31. An architect is designing a subdivision. There are three roads near completion as shown. To the nearest degree, what is the measure of the angle that Pacific road will make with State street?

?

State

16 Pacific

120° 28 Union

32. A horticulture class is designing a triangular flower garden for a customer. Its sides measure 29 feet, 42.5 feet, and 38 feet. What is the area of the flower garden to the nearest square foot?


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LESSON

Big Ideas

13 Answer these questions to summarize the important concepts from Chapter 13 in your own words. 1. Describe the six trigonometric functions in terms of a right triangle and the relationship between inverses.

2. Describe the angle of rotation in terms of how the terminal side is rotated.

3. Describe the relationship between degrees and radians.

4. Explain why and how the Law of Sines and Law of Cosines are used.





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CHAPTER

Vocabulary


Page

Definition

Clarifying Example

amplitude

cycle

frequency

period


280

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Term

Page

Definition

Clarifying Example

periodic function

phase shift

rotation matrix


281

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LESSON

Graphs of Sine and Cosines


Vocabulary 1. Periodic function (p. 990): _________________________________________ ______________________________________________________________ 2. Cycle (p. 990): __________________________________________________ ______________________________________________________________ 3. Period (p. 990): _________________________________________________ ______________________________________________________________ 4. Amplitude (p. 991): ______________________________________________ ______________________________________________________________ 5. Frequency (p. 992): ______________________________________________ ______________________________________________________________ 6. Phase shift (p. 993): _____________________________________________ ______________________________________________________________

Key Concepts 7. Characteristics of the Graphs of Sine and Cosine. (p. 991) FUNCTION

y sin x

y cos x

Graph

Domain Range Period Amplitude Copyright © by Holt, Rinehart and Winston. All rights reserved.

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8. Transformations of Sine and Cosine Graphs. (p. 991)

9. Get Organized For each type of transformation, give an example and state the period. (p. 994). Vertical Compression

Horizontal Stretch

Cosine Graphs

Reflection


Phase Shift

283

Algebra 2

LESSON

Graphs of Other Trigonometric Functions


Key Concepts 1. Characteristics of the Graphs of Tangent and Cotangent (p. 998): FUNCTION

y tan x

y cot x

Graph

Domain Range Period Amplitude 2. Transformations of Tangent Graphs (p. 998):


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3. Transformations of Cotangent Graphs (p. 999):

4. Characteristics of the Graphs of Secant and Cosecant (p. 1000): FUNCTION

y sec x

y csc x

Graph

Domain Range Period Amplitude


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5. Get Organized Complete the graphic organizer. (p. 1001). FUNCTION

ZEROS

ASYMPTOTES

PERIOD

y sec x y csc x y cot x y tan x


286

Algebra 2

LESSON

Fundamental Trigonometric Identities


Key Concepts 1. Fundamental Trigonometric Identities (p. 1008): RECIPROCAL IDENTITIES

TANGENT AND COTANGENT RATIO IDENTITIES

PYTHAGOREAN IDENTITIES

NEGATIVEANGLE IDENTITIES

2. Get Organized Write the three Pythagorean identities in the boxes. (p. 1010).

Pythagorean Identities


287

Algebra 2

LESSON

Sums and Difference Identities


Vocabulary 1. Rotation matrix (p. 1016): _________________________________________ ______________________________________________________________

Key Concepts 2. Sum and Difference Identities (p. 1014): SUM IDENTITIES

DIFFERENCE IDENTITIES

3. Using a Rotation Matrix (p. 1016):


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Algebra 2


4. Get Organized For each type of function, give the sum and difference identity and an example. (p. 1017).

Tangent

Sum and Difference Identities

Sine


Cosine

289

Algebra 2

LESSON

Double-Angle and Half-Angle Identities


Key Concepts 1. Double-Angle Identities (p. 1020): DOUBLE-ANGLE IDENTITIES

2. Half-Angle Identity (p. 1022) HALF-ANGLE IDENTITIES

3. Get Organized In each box, write one of the identities. (p. 1023).

Double-Angle Identity for Cosine


290

Algebra 2

LESSON

Solving Trigonometric Equations


Key Concepts 3. Get Organized Write when each method is most useful, and give an example. (p. 1030). FUNCTION

MOST USEFUL WHEN . . .

EXAMPLE

Graphing

Solving linear equations

Factoring

Quadratic Formula

Identity substitution


291

Algebra 2

CHAPTER

Chapter Review

14 14-1 Graphs of Sine and Cosine Identify whether each function is periodic. If the function is periodic, give the period. 1.

2.

3.

4.


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Algebra 2


Using f(x) sin x or f (x) cos x as a guide, graph each function. Identify the amplitude and period. 5. f(x) 2 cos x

7. h(x) 2sin2πx 1

6. g(x) sin 3x

y

y

4

y

4

2

x

–6.28 –3.14 –2

3.14 6.28

–4

4

2

x

–6.28 –3.14 –2

3.14 6.28

–4

2

x

–6.28 –3.14 –2

3.14 6.28

–4

Using f(x) sin x or f(x) cos x as a guide, graph each function. Identify the x-intercepts and phase shift.

8. f (x) sin x 2

y

x 3.14 6.28

–4


4

2 –6.28 –3.14 –2

y

4

2

3

10. h(x) sin x 4

y

4

–6.28 –3.14 –2

9. g(x) cos(x )

x 3.14 6.28

–4

2 –6.28 –3.14 –2

x 3.14 6.28

–4

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Algebra 2


11. The torque applied to a bolt is given by (x) Fr cos x, where r is the length of the wrench in meters, F is the applied force in newtons, and x is the angle between F and r in radians. Graph the torque for a 0.3 meter wrench and a force of 250 Newtons for 0 x 2. What is the torque for an angle of 4?

14-2 Graphs of Other Trigonometric Functions Using f(x) tan x as a guide, graph each function. Identify the period, x-intercepts, and asymptotes. 1

12. f(x) 3 tan 2x

14. h(x) tan 4x

y

y 4

y 4

4

2 –6.28 –3.14 –2

1

13. g(x) 4 tan 4x

x 3.14 6.28

–4


2 –6.28 –3.14 –2

x 3.14 6.28

–4

2 –6.28 –3.14 –2

x 3.14 6.28

–4

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Algebra 2


Using f(x) cot x as a guide, graph each function. Identify the period, x-intercepts and asymptotes. 1

15. f(x) cot 2x y

17. h(x) 2 cot 4x

y

4

y

4

2 –6.28 –3.14 –2

1

16. g(x) cot 2x

2

x 3.14 6.28

4

–6.28 –3.14 –2

–4

2

x 3.14 6.28

–6.28 –3.14 –2

–4

x 3.14 6.28

–4

14-3 Fundamental Trigonometric Identities Prove each trigonometric identity. 18. cot 2 x(sec2 x) 1


2

tan x 19. sin2 x 2 1 tan x

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Algebra 2


1

sec x tan x 20. sec x tan x

Rewrite each expression in terms of a single trigonometric function. cot x

21. csc x

22. (sin x cos x)2 (sin x cos x)2

cos2 x

23. sin x cos x

14-4 Sum and Difference Identities Find the exact value of each expression. 24. sin15°


7

25. cos 12

26. tan105°

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Algebra 2


15

1

with 90° A 180° and if cosB with Find each value if sinA 17 5 270° B 360°.

27. sin(A B)

28. sin(A B)

29. cos(A B)

30. Find the coordinates to the nearest hundredth, of the vertices of figure ABCD with A(1, 2), B(0, 0), C(1, 2), and D(4, 0) after a 60° rotation about the origin.

y 4 2 –4

–2

x 2

4

–2 –4

14-5 Double-Angle and Half-Angle Identities 3

Find each expression if cos 5 and 270° 360°. 31. sin2

34. sin2

32. cos2

33. tan2

35. cos2

36. tan2

37. Use the half-angle identities to find the exact value of sin 15°.


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Algebra 2


14-6 Solving Trigonometric Equations 38. Find all solutions of 1 2cos 0 where is in radians. Solve each equation for 0° 360°. 39. tan2x 3tanx 2

40. cos2x 5sin2x 5sinx 3

Use trigonometric identities to solve each equation for 0 2. 1

41. sin2x 2cos2x

42. sinx cosx 2

43. The average daily minimum temperature for Houston, Texas, can be modeled by

T(x) 15.85cos6(x 1) 76.85, where T is the temperature in degrees Fahrenheit, x is the time in months, and x 0 is January 1. On what date is the temperature 70° F? 90° F?


298

Algebra 2

CHAPTER

Big Ideas

14 Answer these questions to summarize the important concepts from Chapter 14 in your own words. 1. What does it mean for a function to be periodic?

2. Show how the sum identity cos(A B) cosAcosB sinAsinB can be used to verify the double angle identity cos2A cos2A sin2A.

3. How are the amplitude, period and phase shift determined from the function f (x) a sinb(x c)?

4. Explain why the graph of f(x) tanx has an asymptote at 2.





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