Chapter 8 Resource Masters - No-IP

Chapter 8Resource Masters

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Copyright © by the McGraw-Hill Companies, Inc. All rights reserved. Permission is granted toreproduce the material contained herein on the condition that such material be reproduced only forclassroom use; be provided to students, teachers, and families without charge; and be used solelyin conjunction with Glencoe Algebra 2. Any other reproduction, for use or sale, is prohibited withoutprior written permission of the publisher.

Send all inquiries to:Glencoe/McGraw-Hill8787 Orion PlaceColumbus, OH 43240

ISBN13: 978-0-07-873978-1ISBN10: 0-07-873978-0 Algebra 2 CRM8

Printed in the United States of America

1 2 3 4 5 6 7 8 9 10 005 13 12 11 10 09 08 07 06

Consumable Workbooks Many of the worksheets contained in the Chapter Resource Masters areavailable as consumable workbooks in both English and Spanish.

ISBN10 ISBN13Study Guide and Intervention Workbook 0-07-877355-5 978-0-07-877355-6Skills Practice Workbook 0-07-877357-1 978-0-07-877357-0Practice Workbook 0-07-877358-X 978-0-07-877358-7Word Problem Practice Workbook 0-07-877360-1 978-0-07-877360-0

Spanish VersionsStudy Guide and Intervention Workbook 0-07-877356-3 978-0-07-877356-3Practice Workbook 0-07-877359-8 978-0-07-877359-4

Answers for Workbooks The answers for Chapter 8 of these workbooks can be found in the back ofthis Chapter Resource Masters booklet.

StudentWorks PlusTM This CD-ROM includes the entire Student Edition test along with the Englishworkbooks listed above.

TeacherWorks PlusTM All of the materials found in this booklet are included for viewing, printing, andediting in this CD-ROM.

Spanish Assessment Masters (ISBN10: 0-07-0-07-877361-X, ISBN13: 978-0-07-877361-7)These masters contain a Spanish version of Chapter 8 Test Form 2A and Form 2C.

Chapter 8 iii Glencoe Algebra 2

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Teacher’s Guide to Using the Chapter 8Resource Masters . . . . . . . . . . . . . . . . . . . . .iv

Chapter Resources Student-Built Glossary . . . . . . . . . . . . . . . . . . .1Anticipation Guide (English) . . . . . . . . . . . . . . .3Anticipation Guide (Spanish) . . . . . . . . . . . . . .4

Lesson 8-1Multiplying and Dividing Rational ExpressionsLesson Reading Guide . . . . . . . . . . . . . . . . . . .5Study Guide and Intervention . . . . . . . . . . . . . .6Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . .8Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9Word Problem Practice . . . . . . . . . . . . . . . . . .10Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . .11

Lesson 8-2Adding and Subtracting Rational ExpressionsLesson Reading Guide . . . . . . . . . . . . . . . . . .12Study Guide and Intervention . . . . . . . . . . . . .13Skills Practice . . . . . . . . . . . . . . . . . . . . . . . .15Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . .16Word Problem Practice . . . . . . . . . . . . . . . . . .17Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . .18

Lesson 8-3Graphing Rational FunctionsLesson Reading Guide . . . . . . . . . . . . . . . . . .19Study Guide and Intervention . . . . . . . . . . . . .20Skills Practice . . . . . . . . . . . . . . . . . . . . . . . .22Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . .23Word Problem Practice . . . . . . . . . . . . . . . . . .24Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . .25Graphing Calculator . . . . . . . . . . . . . . . . . . . .26

Lesson 8-4Direct, Joint, and Inverse VariationLesson Reading Guide . . . . . . . . . . . . . . . . . .27Study Guide and Intervention . . . . . . . . . . . . .28Skills Practice . . . . . . . . . . . . . . . . . . . . . . . .30Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . .31Word Problem Practice . . . . . . . . . . . . . . . . . .32Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . .33Spreadsheet . . . . . . . . . . . . . . . . . . . . . . . . .34

Lesson 8-5Classes of FunctionsLesson Reading Guide . . . . . . . . . . . . . . . . . .35Study Guide and Intervention . . . . . . . . . . . . .36Skills Practice . . . . . . . . . . . . . . . . . . . . . . . .38Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . .39Word Problem Practice . . . . . . . . . . . . . . . . . .40Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . .41

Lesson 8-6Classes of FunctionsLesson Reading Guide . . . . . . . . . . . . . . . . . .42Study Guide and Intervention . . . . . . . . . . . . .43Skills Practice . . . . . . . . . . . . . . . . . . . . . . . .45Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . .46Word Problem Practice . . . . . . . . . . . . . . . . . .47Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . .48

AssessmentStudent Recording Sheet . . . . . . . . . . . . . . . .49Rubric for Scoring Pre-AP . . . . . . . . . . . . . . .50Chapter 8 Quizzes 1 and 2 . . . . . . . . . . . . . . .51Chapter 8 Quizzes 3 and 4 . . . . . . . . . . . . . . .52Chapter 8 Mid-Chapter Test . . . . . . . . . . . . . .53Chapter 8 Vocabulary Test . . . . . . . . . . . . . . .54Chapter 8 Test, Form 1 . . . . . . . . . . . . . . . . . .55Chapter 8 Test, Form 2A . . . . . . . . . . . . . . . .57Chapter 8 Test, Form 2B . . . . . . . . . . . . . . . .59Chapter 8 Test, Form 2C . . . . . . . . . . . . . . . .61Chapter 8 Test, Form 2D . . . . . . . . . . . . . . . .63Chapter 8 Test, Form 3 . . . . . . . . . . . . . . . . . .65Chapter 8 Extended Response Test . . . . . . . .67Standardized Test Practice . . . . . . . . . . . . . . .68

Answers . . . . . . . . . . . . . . . . . . . . . .A1–A32

Chapter 8 iv Glencoe Algebra 2

Copyright ©

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panies, Inc.Teacher’s Guide to Using the Chapter 8 Resource Masters

The Chapter 8 Resource Masters includes the core materials needed for Chapter 8.These materials include worksheets, extensions, and assessment options. Theanswers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing onthe TeacherWorks PlusTM CD-ROM.

Chapter ResourcesStudent-Built Glossary (pages 1–2) These masters are a student study tool thatpresents up to twenty of the key vocabularyterms from the chapter. Students are torecord definitions and/or examples for eachterm. You may suggest that studentshighlight or star the terms with which theyare not familiar. Give this to studentsbefore beginning Lesson 8-1. Encouragethem to add these pages to theirmathematics study notebooks. Remindthem to complete the appropriate words asthey study each lesson.

Anticipation Guide (pages 3–4) Thismaster, presented in both English andSpanish, is a survey used before beginningthe chapter to pinpoint what students mayor may not know about the concepts in thechapter. Students will revisit this surveyafter they complete the chapter to see iftheir perceptions have changed.

Lesson ResourcesLesson Reading Guide Get Ready for theLesson extends the discussion from thebeginning of the Student Edition lesson.Read the Lesson asks students to interpretthe context of and relationships amongterms in the lesson. Finally, RememberWhat You Learned asks students tosummarize what they have learned usingvarious representation techniques. Use as astudy tool for note taking or as an informalreading assignment. It is also a helpful toolfor ELL (English Language Learners).

Study Guide and Intervention Thesemasters provide vocabulary, key concepts,additional worked-out examples and CheckYour Progress exercises to use as areteaching activity. It can also be used inconjunction with the Student Edition as aninstructional tool for students who havebeen absent.

Skills Practice This master focuses moreon the computational nature of the lesson.Use as an additional practice option or ashomework for second-day teaching of thelesson.

Practice This master closely follows thetypes of problems found in the Exercisessection of the Student Edition and includesword problems. Use as an additionalpractice option or as homework for second-day teaching of the lesson.

Word Problem Practice This masterincludes additional practice in solving wordproblems that apply the concepts of thelesson. Use as an additional practice or ashomework for second-day teaching of thelesson.

Enrichment These activities may extendthe concepts of the lesson, offer an historicalor multicultural look at the concepts, orwiden students’ perspectives on themathematics they are learning. They arewritten for use with all levels of students.

Chapter 8 v Glencoe Algebra 2

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Graphing Calculator, ScientificCalculator, or Spreadsheet ActivitiesThese activities present ways in whichtechnology can be used with the concepts insome lessons of this chapter. Use as analternative approach to some concepts or asan integral part of your lessonpresentation.

Assessment OptionsThe assessment masters in the Chapter 8Resource Masters offer a wide range ofassessment tools for formative (monitoring)assessment and summative (final)assessment.

Student Recording Sheet This mastercorresponds with the standardized testpractice at the end of the chapter.

Pre-AP Rubric This master providesinformation for teachers and students onhow to assess performance on open-endedquestions.

Quizzes Four free-response quizzes offerassessment at appropriate intervals in thechapter.

Mid-Chapter Test This 1-page testprovides an option to assess the first half ofthe chapter. It parallels the timing of theMid-Chapter Quiz in the Student Editionand includes both multiple-choice and free-response questions.

Vocabulary Test This test is suitable forall students. It includes a list of vocabularywords and 10 questions to assess students’knowledge of those words. This can also beused in conjunction with one of the leveledchapter tests.

Leveled Chapter Tests

• Form 1 contains multiple-choicequestions and is intended for use withbelow grade level students.

• Forms 2A and 2B contain multiple-choicequestions aimed at on grade levelstudents. These tests are similar informat to offer comparable testingsituations.

• Forms 2C and 2D contain free-responsequestions aimed at on grade levelstudents. These tests are similar informat to offer comparable testingsituations.

• Form 3 is a free-response test for usewith above grade level students.

All of the above mentioned tests include afree-response Bonus question.

Extended-Response Test Performanceassessment tasks are suitable for allstudents. Sample answers and a scoringrubric are included for evaluation.

Standardized Test Practice These threepages are cumulative in nature. It includesthree parts: multiple-choice questions withbubble-in answer format, griddablequestions with answer grids, and short-answer free-response questions.

Answers• The answers for the Anticipation Guide

and Lesson Resources are provided asreduced pages with answers appearing in red.

• Full-size answer keys are provided forthe assessment masters.

8 Student-Built Glossary

Chapter 8 1 Glencoe Algebra 2

This is an alphabetical list of the key vocabulary terms you will learn in Chapter 8.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to your AlgebraStudy Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

asymptoteA·suhm(p)·TOHT

complex fraction

constant of variation

continuityKAHN·tuhn·OO·uh·tee

direct variation

inverse (IHN·VUHRS)variation

joint variation

(continued on the next page)

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Vocabulary Term Found on Page Definition/Description/Example

point discontinuity

rational equation

rational expression

rational function

rational inequality

Student-Built Glossary

NAME ______________________________________________ DATE______________ PERIOD _____

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8 Anticipation GuideRational Expressions and Equations


Before you begin Chapter 8

• Read each statement.

• Decide whether you Agree (A) or Disagree (D) with the statement.

• Write A or D in the first column OR if you are not sure whether you agree or disagree,write NS (Not Sure).

After you complete Chapter 8

• Reread each statement and complete the last column by entering an A or a D.

• Did any of your opinions about the statements change from the first column?

• For those statements that you mark with a D, use a piece of paper to write an example of why you disagree.

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NAME ______________________________________________ DATE______________ PERIOD _____

Step 1

STEP 1 STEP 2A, D, or NS

StatementA or D

1. Since a denominator cannot equal 0, the expression is undefined for x � �5.

2. To divide two rational expressions, multiply by the reciprocal of the divisor.

3. The least common multiple of three monomials is found by multiplying the monomials together.

4. Before adding two rational expressions, a common denominator must be found.

5. The graph of a rational function containing an asymptote will be symmetric over the asymptote.

6. Since f(x ) � can be simplified to f(x ) � m � 2,

the graph of f(x ) will be the straight line defined by y � m � 2.

7. y � kxyz is an example of a joint variation if k, x, y, and z are all not equal to 0.

8. The shape of the graph of y � �3x2 � 2x � 4 can only be determined by graphing the function.

9. Because the graph of an absolute value function is in the shape of a V, the graph of y � � x � � 4 will also be in the shape of a V.

10. When solving rational equations, solutions that result in a zero in the denominator must be excluded.

(m � 4)(m � 2)��

m � 4

3x 2(x � 1)��

x � 5

Step 2

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NOMBRE ______________________________________ FECHA ____________ PERÍODO ___

Ejercicios preparatoriosExpresiones y ecuaciones racionales

Capítulo 8 4 Álgebra 2 de Glencoe

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8

PASO 1 Antes de comenzar el Capítulo 8

• Lee cada enunciado.

• Decide si estás de acuerdo (A) o en desacuerdo (D) con el enunciado.

• Escribe A o D en la primera columna O si no estás seguro(a) de la respuesta,escribe NS (No estoy seguro(a).

Después de completar el Capítulo 8

• Vuelve a leer cada enunciado y completa la última columna con una A o una D.

• ¿Cambió cualquiera de tus opiniones sobre los enunciados de la primera columna?

• En una hoja de papel aparte, escribe un ejemplo de por qué estás en desacuerdo con losenunciados que marcaste con una D.

PASO 2

PASO 1Enunciado

PASO 2A, D o NS A o D

1. Dado que un denominador no puede ser igual a 0, la expresión

es indefinida para x � �5.

2. Para dividir dos expresiones racionales, multiplica por el recíproco del divisor.

3. El mínimo común múltiplo de tres monomios se encuentra al multiplicar los monomios juntos.

4. Antes de sumar dos expresiones racionales, se debe hallar un denominador común.

5. La gráfica de una función racional que contiene una asíntota será simétrica sobre la asíntota.

6. Dado que f(x) � se puede reducir a f(x) � m � 2,

la gráfica de f(x) será la recta definida por y � m � 2.

7. y � kxyz es un ejemplo de una variación conjunta si k, x, y y zno son igual a 0.

8. La forma de la gráfica de y � �3x2 � 2x � 4 sólo puede determinarse al graficar la función.

9. Debido a que la gráfica de una función de valor absoluto tiene forma de V, la gráfica de y � � x � � 4 también tendrá forma de V.

10. Cuando se resuelven ecuaciones racionales, deben excluirse las soluciones que resulten en cero en el denominador.

(m � 4)(m � 2)��

m � 4

3x 2(x � 1)��

x � 5

8-1 Lesson Reading GuideMultiplying and Dividing Rational Expressions


Less

on

8-1

Get Ready for the LessonRead the introduction to Lesson 8-1 in your textbook.

• Suppose that the Goodie Shoppe also sells a candy mixture made with

4 pounds of chocolate mints and 3 pounds of caramels, then

of the mixture is mints and of the mixture is caramels.

• If the store manager adds another y pounds of mints to the mixture, what fraction of themixture will be mints?

Read the Lesson

1. a. In order to simplify a rational number or rational expression, the

numerator and and divide both of them by their

.

b. A rational expression is undefined when its is equal to .

To find the values that make the expression undefined, completely

the original and set each factor equal to .

2. a. To multiply two rational expressions, the andmultiply the denominators.

b. To divide two rational expressions, by the of

the .

3. a. Which of the following expressions are complex fractions?

i. ii. iii. iv. v.

b. Does a complex fraction express a multiplication or division problem?How is multiplication used in simplifying a complex fraction?

Remember What You Learned

4. One way to remember something new is to see how it is similar to something youalready know. How can your knowledge of division of fractions in arithmetic help you tounderstand how to divide rational expressions?

�r2 �

925

�

��r �

35

�

�z �

z1

�

�zr � 5�r � 5

�38

�

��156�

7�12

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NAME ______________________________________________ DATE______________ PERIOD _____

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8-1


Simplify Rational Expressions A ratio of two polynomial expressions is a rationalexpression. To simplify a rational expression, divide both the numerator and thedenominator by their greatest common factor (GCF).

Multiplying Rational Expressions For all rational expressions and , � � , if b � 0 and d � 0.

Dividing Rational Expressions For all rational expressions and , � � , if b � 0, c � 0, and d � 0.

Simplify each expression.

a.

� �

b. �

� � � �

c. �

� � �

� �


1. �(�220aabb

2

4)3

� 2. 3.

4. � 5. �

6. � 7. �

8. � 9. �4m2 � 1��4m � 8

2m � 1��m2 � 3m � 10

4p2 � 7p � 2��

7p516p2 � 8p � 1��

14p4

18xz2�5y

6xy4�25z3

m3 � 9m��

m2 � 9(m � 3)2

��m2 � 6m � 9

c2 � 4c � 5��c2 � 4c � 3

c2 � 3c�c2 � 25

4m5�m � 1

3m3 � 3m��

6m4

x2 � x � 6��x2 � 6x � 27

4x2 � 12x � 9��9 � 6x

x � 4�2(x � 2)

(x � 4)(x � 4)(x � 1)��2(x � 1)(x � 2)(x � 4)

x � 1��x2 � 2x � 8

x2 � 8x � 16��2x � 2

x2 � 2x � 8��x � 1

x2 � 8x � 16��2x � 2

x2 � 2x � 8��x � 1

x2 � 8x � 16��2x � 2

4s2�3rt2

2 � 2 � s � s��3 � r � t � t

3 � r � r � s � s � s � 2 � 2 � 5 � t � t��5 � t � t � t � t � 3 � 3 � r � r � r � s

20t2�9r3s

3r2s3�

5t4

20t2�9r3s

3r2s3�5t4

3a�2b2

2 � 2 � 2 � 3 � a � a � a � a � a � b � b��2 � 2 � 2 � 2 � a � a � a � a � b � b � b � b

24a5b2�(2ab)4

24a5b2�(2ab)4

ad�bc

c�d

a�b

c�d

a�b

ac�bd

c�d

a�b

c�d

a�b

NAME ______________________________________________ DATE______________ PERIOD _____

Study Guide and InterventionMultiplying and Dividing Rational Expressions

1 1 1 1

1 1

1 1

1 1 1

1 1 1 1 1 1 1

11 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1

Exercises

Example

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Study Guide and Intervention (continued)

Multiplying and Dividing Rational Expressions

NAME ______________________________________________ DATE______________ PERIOD _____


Less

on

8-1

Simplify Complex Fractions A complex fraction is a rational expression whosenumerator and/or denominator contains a rational expression. To simplify a complexfraction, first rewrite it as a division problem.

Simplify .

� � Express as a division problem.

� � Multiply by the reciprocal of the divisor.

� Factor.

� Simplify.

Simplify.

1. 2. 3.

4. 5.

6. 7.

8. 9.

x2 � x � 2��x3 � 6x2 � x � 30��x � 1

�x � 3

�b2 �

b �6b

2� 8

�

��b2

b2�

�b

1�6

2�

�2x2

x�

�9x

1� 9

�

��105xx

2

2��

179xx�

�26

�

�aa

2 ��

126

�

��aa

2

2��

3aa

��

24

�

�x2 �

x �6x

4� 9

�

��x2 �

3 �2x

x� 8

�

�b2 �

b2100�

��3b2 � 3

21bb � 10�

�3bb

2 ��

12

�

��3b2

b�

�b1� 2

�

�ax

2

2byc2

3�

��ca4xb

2

2

y�

�xa

3

2yb

2

2z

�

��a3

bx2

2y�

s3�s � 3

(3s � 1)s4��s(3s � 1)(s � 3)

s4��3s2 � 8s � 3

3s � 1�s

3s2 � 8s � 3��

s43s � 1�s

�3s

s� 1�

��3s2 �

s84s � 3�

�3s

s� 1�

��3s2 �

s84s � 3�

1

1 1

s3

8-1

Exercises

Example

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Skills PracticeMultiplying and Dividing Rational Expressions



1. 2.

3. 4.

5. 6.

7. 8. �

9. � 10. �

11. � 21g3 12. �

13. � 14. �

15. � 16. �

17. � (3x2 � 3x) 18. �

19. 20.�a2

4�a

b2�

��a

2�a

b�

�2cd

2

2�

�

��5cd

6�

4a � 5��a2 � 8a � 16

16a2 � 40a � 25��

3a2 � 10a � 8x2 � 5x � 4��2x � 8

2t � 2��t2 � 9t � 14

t2 � 19t � 84��4t � 4

w2 � 6w � 7��w � 3

w2 � 5w � 24��w � 1

q2 � 4�

3q2q2 � 2q�6q

3x�x2 � 4

3x2�x � 2

25y5�14z12v5

80y4�49z5v7

7g�y2

s � 2�10s5

5s2�s2 � 4

10(ef)3�

8e5f24e3�5f 2

n3�6

3m�2n

3a2 � 24a��3a2 � 12a

x2 � 4��(x � 2)(x � 1)

18�2x � 6

8y2(y6)3�

4y24(x6)3�(x3)4

5ab3�25a2b2

21x3y�14x2y2

NAME ______________________________________________ DATE______________ PERIOD _____

8-1

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PracticeMultiplying and Dividing Rational Expressions

NAME ______________________________________________ DATE______________ PERIOD _____

8-1


Less

on

8-1


1. 2. 3.

4. 5.

6. 7. �

8. � 9. �

10. � 11. �

12. � 13. �

14. � �3� 15. �

16. � 17. �

18. � 19.

20. 21.

22. GEOMETRY A right triangle with an area of x2 � 4 square units has a leg thatmeasures 2x � 4 units. Determine the length of the other leg of the triangle.

23. GEOMETRY A rectangular pyramid has a base area of square centimeters

and a height of centimeters. Write a rational expression to describe the

volume of the rectangular pyramid.

x2 � 3x��x2 � 5x � 6

x2 � 3x � 10��2x

�xx

2

3

��

22x

3�

��

�x2

(�x �

4x2

�)3

4�

�x2

4� 9�

��3 �

8x

�

�2x

x� 1�

��4 �

xx

�

2a � 6�5a � 10

9 � a2��a2 � 5a � 6

s2 � 10s � 25��s � 4

2s2 � 7s � 15��

(s � 4)26x2 � 12x��4x � 12

3x � 6�x2 � 9

x2 � y2�3

x � y�6

24x2�w5

2xy�w2

a3w2�w5y2

a5y3�wy7

25x2 � 1��x2 � 10x � 25

x � 5�10x � 2

5x2�8 � x

x2 � 5x � 24��6x � 2x2

w2 � n2�y � a

a � y�w � n

n2 � 6n�

n8n5

�n � 64

�y � aa � y�6

25x3�14u2y2

�2u3y�15xz5

x4 � x3 � 2x2��

x4 � x3

25 � v2��3v2 � 13v � 10

2k2 � k � 15��

k2 � 9

10y2 � 15y��35y2 � 5y

(2m3n2)3��18m5n4

9a2b3�27a4b4c

8-1


Word Problem Practice Multiplying and Dividing Rational Expressions

NAME ______________________________________________ DATE______________ PERIOD _____

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1. JELLY BEANS A large vat contains Ggreen jelly beans and R red jelly beans.A bag of 100 red and 100 green jellybeans is added to the vat. What is thenew ratio of red to green jelly beans inthe vat?

2. MILEAGE Beth’s car gets 15 miles pergallon in the city and 26 miles per gallonon the highway. Beth uses C gallons ofgas in the city and H gallons of gas onthe highway. Write an expression for the average number of miles per gallonthat Beth gets with her car in terms of C and H.

3. HEIGHT The front face of a Nordichouse is triangular. The surface area of the face is x2 � 3x � 10 where x is the base of the triangle.

What is the height of the triangle interms of x?

x

4. OIL SLICKS David was moving a drumof oil around his circular outdoor poolwhen the drum cracked, and oil spilledinto the pool. The oil spread itself evenlyover the surface of the pool. Let V denotethe volume of oil spilled and let r be theradius of the pool. Write an equation forthe thickness of the oil layer.

RUNNING For Exercises 5 and 6, usethe following information.

Harold runs to the local food mart to buy a gallon of soy milk. Because he is weigheddown on his return trip, he runs slower onthe way back. He travels S1 feet per secondon the way to the food mart and S2 feet per second on the way back. Let d be thedistance he has to run to get to the foodmart. Remember: distance � rate � time.

5. Write an equation that gives the totaltime Harold spent running for thiserrand.

6. What speed would Harold have to run if he wanted to maintain a constantspeed for the entire trip yet take thesame amount of time running?

8-28-1 EnrichmentDimensional Analysis


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NAME ______________________________________________ DATE______________ PERIOD _____

Scientists always express the units of measurement in their solution. It isinsufficient and ambiguous to state a solution regarding distance as 17;Seventeen what, feet, miles, meters? Often it is helpful to analyze the unitsof the quantities in a formula to determine the desired units of an output.For example, it is known that torque is the product of force and distance, butwhat are the units of force?

The units also depend on the measuring system. The two most commonlyused systems are the British system and the international system of units(SI). Some common units of the British system are inches, feet, miles,and pounds. Common SI units include meters, kilometers, Newtons, andgrams. Frequently conversion from one system to another is necessary and accomplished by multiplication by conversion factors.

Consider changing units from miles per hour to kilometers per hour. What is60 miles per hour in kilometers per hour? Use the conversion 1 ft � 30.5 cm.

60 � 60 � � � � � 96.62

1. The SI unit for force is a Newton (N) and the SI unit for distance ismeters or centimeters. The British unit for force is pounds and theBritish unit for distance is feet or inches. Using the formula for torque(Torque � Force times Distance), determine the SI unit and the Britishunit for torque.

2. The density of a fluid is given by the formula density � . Suppose

that a volume of a fluid in a cylindrical can is r2h, where r and hare measured in meters. Find an expression for the mass, given in

kilograms (kg), of gasoline, which has a known density of 680 .

3. Convert the following measurements.

a. 72 miles/hour to feet/second

b. 32 pounds/square inch to pounds per square foot

c. 100 kilometers/hour to miles per hour

kg�m3

mass�volume

km�h

1 km�1000 m

1 m�100 cm

30.5 cm�

1 ft5280 ft�

1 mimi�h

mi�h

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NAME ______________________________________________ DATE______________ PERIOD _____

Lesson Reading GuideAdding and Subtracting Rational Expressions

8-2


A person is standing 5 feet from a camera that has a lens with a focal length of 3 feet.Write an equation that you could solve to find how far the film should be from the lens to get a perfectly focused photograph.

Read the Lesson

1. a. In work with rational expressions, LCD stands for

and LCM stands for . The LCD is the ofthe denominators.

b. To find the LCM of two or more numbers or polynomials, each

number or . The LCM contains each the

number of times it appears as a .

2. To add and , you should first factor the of

each fraction. Then use the factorizations to find the of x2 � 5x � 6 and

x3 � 4x2 � 4x. This is the for the two fractions.

3. When you add or subtract fractions, you often need to rewrite the fractions as equivalentfractions. You do this so that the resulting equivalent fractions will each have a

denominator equal to the of the original fractions.

4. To add or subtract two fractions that have the same denominator, you add or subtract

their and keep the same .

5. The sum or difference of two rational expressions should be written as a polynomial or

as a fraction in .


6. Some students have trouble remembering whether a common denominator is needed toadd and subtract rational expressions or to multiply and divide them. How can yourknowledge of working with fractions in arithmetic help you remember this?

x � 4��x3 � 4x2 � 4x

x2 � 3��x2 � 5x � 6

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Example

Study Guide and InterventionAdding and Subtracting Rational Expressions

NAME ______________________________________________ DATE______________ PERIOD _____

8-2


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LCM of Polynomials To find the least common multiple of two or more polynomials,factor each expression. The LCM contains each factor the greatest number of times itappears as a factor.

Find the LCM of 16p2q3r,40pq4r2, and 15p3r4.

16p2q3r � 24 � p2 � q3 � r40pq4r2 � 23 � 5 � p � q4 � r2

15p3r4 � 3 � 5 � p3 � r4

LCM � 24 � 3 � 5 � p3 � q4 � r4

� 240p3q4r4

Find the LCM of 3m2 � 3m � 6 and 4m2 � 12m � 40.

3m2 � 3m � 6 � 3(m � 1)(m � 2)4m2 � 12m � 40 � 4(m � 2)(m � 5)LCM � 12(m � 1)(m � 2)(m � 5)

Find the LCM of each set of polynomials.

1. 14ab2, 42bc3, 18a2c 2. 8cdf3, 28c2f, 35d4f 2

3. 65x4y, 10x2y2, 26y4 4. 11mn5, 18m2n3, 20mn4

5. 15a4b, 50a2b2, 40b8 6. 24p7q, 30p2q2, 45pq3

7. 39b2c2, 52b4c, 12c3 8. 12xy4, 42x2y, 30x2y3

9. 56stv2, 24s2v2, 70t3v3 10. x2 � 3x, 10x2 � 25x � 15

11. 9x2 � 12x � 4, 3x2 � 10x � 8 12. 22x2 � 66x � 220, 4x2 � 16

13. 8x2 � 36x � 20, 2x2 � 2x � 60 14. 5x2 � 125, 5x2 � 24x � 5

15. 3x2 � 18x � 27, 2x3 � 4x2 � 6x 16. 45x2 � 6x � 3, 45x2 � 5

17. x3 � 4x2 � x � 4, x2 � 2x � 3 18. 54x3 � 24x, 12x2 � 26x � 12

Exercises

Example

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Adding and Subtracting Rational Expressions


Add and Subtract Rational Expressions To add or subtract rational expressions,follow these steps.

Step 1 If necessary, find equivalent fractions that have the same denominator.Step 2 Add or subtract the numerators.Step 3 Combine any like terms in the numerator.Step 4 Factor if possible.Step 5 Simplify if possible.

Simplify � .

�

� � Factor the denominators.

� � The LCD is 2(x � 3)(x � 2)(x � 2).

� Subtract the numerators.

� Distributive Property

� Combine like terms.

� Simplify.


1. � 2. �

3. � 4. �

5. � 6. �5x

��20x2 � 5

4��4x2 � 4x � 1

x � 1�x2 � 1

3x � 3��x2 � 2x � 1

4x � 5�3x � 6

3�x � 2

15b�5ac

4a�3bc

1�x � 1

2�x � 3

4y2�2y

�7xy�3x

x��(x � 3)(x � 2)(x � 2)

2x��2(x � 3)(x � 2)(x � 2)

6x � 12 � 4x � 12��2(x � 3)(x � 2)(x � 2)

6(x � 2) � 4(x � 3)��2(x � 3)(x � 2)(x � 2)

2 � 2(x � 3)��2(x � 3)(x � 2)(x � 2)

6(x � 2)��2(x � 3)(x � 2)(x � 2)

2��(x � 2)(x � 2)

6��2(x � 3)(x � 2)

2�x2 � 4

6��2x2 � 2x � 12

2�x2 � 4

6��2x2 � 2x � 12

NAME ______________________________________________ DATE______________ PERIOD _____

8-2

Exercises

Example

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Skills PracticeAdding and Subtracting Rational Expressions

NAME ______________________________________________ DATE______________ PERIOD _____


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1. 12c, 6c2d 2. 18a3bc2, 24b2c2

3. 2x � 6, x � 3 4. 5a, a � 1

5. t2 � 25, t � 5 6. x2 � 3x � 4, x � 1


7. � 8. �

9. � 4 10. �

11. � 12. �

13. � 14. �

15. � 16. �

17. � 18. �

19. � 20. �

21. � 22. �2

��y2 � 6y � 8

3��y2 � y � 12

2n � 2��n2 � 2n � 3

n�n � 3

4��x2 � 3x � 10

2x � 1�x � 5

x�x � 1

1��x2 � 2x � 1

z � 4�z � 1

4z�z � 4

m�n � m

m�m � n

5�x � 2

3t�2 � x

2�w2 � 9

3�w � 3

2�3bd

5�3b � d

3�2a

2�a � 2

3�4h2

7�4gh

2�5yz

12�5y2

5�n

2�m2n

2c � 7�3

5�4p2q

3�8p2q

5�y

3�x

8-2

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PracticeAdding and Subtracting Rational Expressions



1. x2y, xy3 2. a2b3c, abc4 3. x � 1, x � 3

4. g � 1, g2 � 3g � 4 5. 2r � 2, r2 � r, r � 1 6. 3, 4w � 2, 4w2 � 1

7. x2 � 2x � 8, x � 4 8. x2 � x � 6, x2 � 6x � 8 9. d2 � 6d � 9, 2(d2 � 9)


10. � 11. � 12. �

13. � 2 14. 2x � 5 � 15. �

16. � 17. � 18. �

19. � 20. � 21. � �

22. � � 23. 24.

25. GEOMETRY The expressions , , and represent the lengths of the sides of a

triangle. Write a simplified expression for the perimeter of the triangle.

26. KAYAKING Mai is kayaking on a river that has a current of 2 miles per hour. If rrepresents her rate in calm water, then r � 2 represents her rate with the current, and r � 2 represents her rate against the current. Mai kayaks 2 miles downstream and then

back to her starting point. Use the formula for time, t � , where d is the distance, to

write a simplified expression for the total time it takes Mai to complete the trip.

d�r

10�x � 4

20�x � 4

5x�2

�r �

r6

� � �r �

12

�

��r2

r�2 �

4r2�r

3�

�x �

2y

� � �x �

1y

�

��x �

1y

�

36�a2 � 9

2a�a � 3

2a�a � 3

7�10n

3�4

1�5n

5�p2 � 9

2p � 3��p2 � 5p � 6

20��x2 � 4x � 12

5�2x � 12

y��y2 � y � 2

y � 5��y2 � 3y � 10

4m � 5�9 � m

2 � 5m�m � 9

2�x � 4

16�x2 � 16

9�a � 5

4�a � 3

x � 8�x � 4

4m�3mn

3�4cd3

1�6c2d

1�5x2y3

5�12x4y

7�8a

5�6ab

NAME ______________________________________________ DATE______________ PERIOD _____

8-2

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Word Problem PracticeAdding and Subtracting Rational Expressions

NAME ______________________________________________ DATE______________ PERIOD _____

8-2


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1. SQUARES Susan’s favorite perfectsquare is s2 and Travis’ is t2, where sand t are whole numbers. What perfectsquare is guaranteed to be divisible byboth Susan’s and Travis’ favorite perfectsquares regardless of their specificvalue?

2. ELECTRIC POTENTIAL The electricalpotential function between two electronsis given by a formula that has the form

� . Simplify this expression.

3. TRAPEZOIDS The cross section of astand consists of two trapezoids stackedone on top of the other.

The total area of the cross section is x2

square units. Assuming the trapezoidshave the same height, write anexpression for the height of the stand interms of x. Put your answer in simplestform. (Recall that the area of a trapezoidwith height h and bases b1 and b2 is

given by h(b1 � b2).)1�2

x � 4

x � 2

x

1�1 � r

1�r

4. FRACTIONS In the seventeenthcentury, Lord Brouncker wrote down amost peculiar mathematical equation:

��4

� � 1 � 12

2 �32

2 �52

2 � �7∞

2�

This is an example of a continuedfraction. Simplify the continued fraction

n � .

RELAY RACE For Exercises 5-7, use thefollowing information.

Mark, Connell, Zack, and Moses run the 4by 400 meter relay together. Their averagespeeds were s, s � 0.5, s � 0.5, and s � 1meters per second, respectively.

5. What were their individual times fortheir own legs of the race?

6. Write an expression for their time as ateam. Write your answer as a ratio oftwo polynomials.

7. If s was 6 meters per second, what wasthe team’s time? Round your answer tothe nearest second.

1�n � �

n1

�

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Enrichment Zeno’s Paradox

NAME ______________________________________________ DATE______________ PERIOD _____

8-2

The Greek philosopher Zeno of Elea (born sometime between 495 and 480 B.C.) proposed four paradoxes to challenge the notions of space and time. Zeno’s first paradox works like this:

Suppose you are on your way to school. Assume you are able to cover half ofthe remaining distance each minute that you walk. You leave your house at7:45 A.M. After the first minute, you are half of the way to school. In the nextminute you cover half of the remaining distance to school, and at 7:47 A.M. youare three-quarters of the way to school. This pattern continues each minute.At what time will you arrive at school? Before 8:00 A.M.? Before lunch?

Since space is infinitely divisible, we can repeat this pattern forever. Thus,on the way to school you must reach an infinite number of ‘midpoints’ in afinite time. This is impossible, so you can never reach your goal. In general,according to Zeno anyone who wants to move from one point to another must meet these requirements, and motion is impossible. Therefore, what we perceive as motion is merely an illusion.

Addition of fractions can be defined by � � , similarly forsubtraction.

Assume your house is one mile from school. At 7:46 A.M., you have walked

half of a mile, so you have left 1 � , or a mile. At 7:47 A.M. you only have

� � of a mile to go.

To determine how far you have walked and how far away from the school you

are at 7:48 A.M., add the distances walked each minute, � � � of

a mile so far and you still have 1 � � of a mile to go.

1. Determine how far you have walked and how far away from the schoolyou are at 7:50 A.M.

2. Suppose instead of covering one-half the distance to school each minute,you cover three-quarters of the distance remaining to school each minute,now will you be able to make it to school on time? Determine how far youstill have left to go at 7:47 A.M.

3. Suppose that instead of covering one-half or three-quarters of the

distance to school each minute, you cover of the distance

remaining, where x is a whole number greater than 2. What is your distance from school at 7:46 A.M.?

1�x � 1

1�8

7�8

7�8

1�8

1�4

1�2

1�4

1�4

1�2

1�2

1�2

ad � bc�

bdc�d

a�b

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Lesson Reading GuideGraphing Rational Functions

NAME ______________________________________________ DATE______________ PERIOD _____

8-3


Less

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8-3


• If 15 students contribute to the gift, how much would each of them pay?

• If each student pays $5, how many students contributed?

Read the Lesson

1. Which of the following are rational functions?

A. f(x) � B. g(x) � �x� C. h(x) �

2. a. Graphs of rational functions may have breaks in . These may occur

as vertical or as point . The of a rational function is limited to values for which the function is defined.

b. The graphs of two rational functions are shown below.

I. II.

Graph I has a at x � .

Graph II has a at x � .

Match each function with its graph above.

f(x) � g(x) �


3. One way to remember something new is to see how it is related to something you alreadyknow. How can knowing that division by zero is undefined help you to remember how tofind the places where a rational function has a point discontinuity or an asymptote?

x2 � 4�x � 2

x�x � 2

x

y

Ox

y

O

x2 � 25��x2 � 6x � 9

1�x � 5

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Exercises

Study Guide and InterventionGraphing Rational Functions


Domain and Range

Rational Function an equation of the form f(x) � , where p(x) and q(x) are polynomial expressions and q(x) � 0

Domain The domain of a rational function is limited to values for which the function is defined.

Vertical Asymptote An asymptote is a line that the graph of a function approaches. If the simplified form of therelated rational expression is undefined for x � a, then x � a is a vertical asymptote.

Point Discontinuity Point discontinuity is like a hole in a graph. If the original related expression is undefined for x � a but the simplified expression is defined for x � a, then there is a hole in the graph at x � a.

Horizontal Often a horizontal asymptote occurs in the graph of a rational function where a value isAsymptote excluded from the range.

Determine the equations of any vertical asymptotes and the values

of x for any holes in the graph of f(x) � .

First factor the numerator and the denominator of the rational expression.

f(x) � �

The function is undefined for x � 1 and x � �1.

Since � , x � 1 is a vertical asymptote. The simplified expression is

defined for x � �1, so this value represents a hole in the graph.

Determine the equations of any vertical asymptotes and the values of x for anyholes in the graph of each rational function.

1. f(x) � 2. f(x) � 3. f(x) �

4. f(x) � 5. f(x) � 6. f(x) �

7. f(x) � 8. f(x) � 9. f(x) � x3 � 2x2 � 5x � 6��

x2 � 4x � 32x2 � x � 3��2x2 � 3x � 9

x � 1��x2 � 6x � 5

3x2 � 5x � 2��x � 3

x2 � 6x � 7��x2 � 6x � 7

3x � 1��3x2 � 5x � 2

x2 � x � 12��

x2 � 4x2x2 � x � 10��2x � 5

4��x2 � 3x � 10

4x � 3�x � 1

(4x � 3)(x � 1)��(x � 1)(x � 1)

(4x � 3)(x � 1)��(x � 1)(x � 1)

4x2 � x � 3��

x2 � 1

4x2 � x � 3��

x2 � 1

p(x)�q(x)

NAME ______________________________________________ DATE______________ PERIOD _____

8-3

Example

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NAME ______________________________________________ DATE______________ PERIOD _____

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Graph Rational Functions Use the following steps to graph a rational function.

Step 1 First see if the function has any vertical asymptotes or point discontinuities.Step 2 Draw any vertical asymptotes.Step 3 Make a table of values.Step 4 Plot the points and draw the graph.

Graph f(x) � .

� or

Therefore the graph of f(x) has an asymptote at x � �3 and a point discontinuity at x � 1.Make a table of values. Plot the points and draw the graph.

Graph each rational function.

1. f(x) � 2. f(x) � 3. f(x) �

4. f(x) � 5. f(x) � 6. f(x) �

xO

f (x)

xO

f (x)

xO

f (x)

x2 � 6x � 8��x2 � x � 2

x2 � x � 6��x � 3

2�(x � 3)2

xO

f (x)

4 8

8

4

–4

–8

–4–8xO

f (x)

xO

f (x)

2x � 1�x � 3

2�x

3�x � 1

x �2.5 �2 �1 �3.5 �4 �5

f(x) 2 1 0.5 �2 �1 �0.5

1�x � 3

x � 1��(x � 1)(x � 3)

x � 1��x2 � 2x � 3

x

f (x)

O

x � 1��x2 � 2x � 3


Graphing Rational Functions

Exercises

Example

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1. f(x) � 2. f(x) �

3. f(x) � 4. f(x) �

5. f(x) � 6. f(x) �


7. f(x) � 8. f(x) � 9. f(x) �

10. f(x) � 11. f(x) � 12. f(x) �

xO

f (x)

xO

f (x)

xO

f (x)

x2 � 4�x � 2

x�x � 2

2�x � 1

xO

f (x)

xO

f (x)

2

2

xO

f (x)

�4�x

10�x

�3�x

x2 � x � 12��x � 3

x2 � 8x � 12��x � 2

x � 1��x2 � 4x � 3

x � 12��x2 � 10x � 24

10��x2 � 13x � 36

3��x2 � 2x � 8

Skills PracticeGraphing Rational Functions

NAME ______________________________________________ DATE______________ PERIOD _____

8-3

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PracticeGraphing Rational Functions

NAME ______________________________________________ DATE______________ PERIOD _____

8-3


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1. f(x) � 2. f(x) � 3. f(x) �

4. f(x) � 5. f(x) � 6. f(x) �


7. f(x) � 8. f(x) � 9. f(x) �

10. PAINTING Working alone, Tawa can give the shed a coat of paint in 6 hours. It takes her father x hours working alone to give the

shed a coat of paint. The equation f (x) � describes the

portion of the job Tawa and her father working together can

complete in 1 hour. Graph f (x) � for x 0, y 0. If Tawa’s

father can complete the job in 4 hours alone, what portion of the job can they complete together in 1 hour? What domain and rangevalues are meaningful in the context of the problem?

11. LIGHT The relationship between the illumination an object receives from a light source of I foot-candles and the square of the distance d in feet of the object from the source can be

modeled by I(d) � . Graph the function I(d) � for

0 � I � 80 and 0 � d � 80. What is the illumination in foot-candles that the object receives at a distance of 20 feet from the light source? What domain and range values are meaningful in the context of the problem?

4500�

d24500�

d2

6 � x�6x

6 � x�6x

xO

f (x)

xO

f (x)

xO

f (x)

xO

f (x)

3x�(x � 3)2

x � 3�x � 2

�4�x � 2

x2 � 9x � 20��x � 5

x2 � 2x � 24��x � 6

x2 � 100��x � 10

x � 2��x2 � 4x � 4

x � 7��x2 � 10x � 21

6��x2 � 3x � 10

dO

III

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Word Problem PracticeGraphing Rational Expressions

NAME ______________________________________________ DATE______________ PERIOD _____

8-3

1. ROAD TRIP Robert and Sarah start off on a road trip from the same house.During the trip, Robert’s and Sarah’scars remain separated by a constantdistance. The graph shows the ratio ofthe distance Sarah has traveled to thedistance Robert has traveled. The dottedline shows how this graph would beextended to hypothetical negative valuesof x. What does the x-coordinate of thevertical asymptote represent?

2. GRAPHS Alma graphed the function

f(x) � below.

There is a problem with her graph.Explain how to correct it.

y

xO

x2 � 4x�x � 4

y

xO

3. FINANCE A quick way to get an idea of how many years before a savingsaccount will double at an interest rate of I percent compounded annually, is todivide I into 72. Sketch a graph of the

function f(I) � .

4. NEWTON Sir Isaac Newton studied the rational function

f(x) � .

Assuming that d � 0, where will therebe a vertical asymptote to the graph ofthis function?

BATTING AVERAGES For Exercises 5and 6, use the following information.

Josh has made 26 hits in 80 at bats for a batting average of .325. Josh goes on ahitting streak and makes x hits in the next2x at bats.

5. What function describes Josh’s battingaverage during this streak?

6. What is the equation of the horizontalasymptote to the graph of the functionyou wrote for Exercise 5? What is itsmeaning?

ax3 � bx2 � cx � d��

x

I

50

5O

f (I )

72�I

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EnrichmentCharacteristics of Rational Function Graphs

NAME ______________________________________________ DATE______________ PERIOD _____

8-3


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Use the information in the table to graph rational functions

A sign chart uses an x value from the left and right of each critical value to determine if the graph is positive or negative on that

interval. A sign chart for y � is shown below.

The graph of is shown

to the right.

Create a sign chart for y � . Use an x-value from the left and

right of each critical value to determine if the graph is positive ornegative on that interval. Then graph the function.

y

x�2 2

x � 1�x2 � 4

x � 1��x2 � x � 6

y

x�2 3

�3 �2 �1

� � � �

0 1 2 3 4

x � 1��x2 � x � 6

CHARACTERISTIC MEANING HOW TO FIND IT

Vertical asymptotes A vertical line at an x value where the Set the denominator equal to zero and rational function is undefined solve for x.

Horizontal asymptotes A horizontal line that the rational Study the end-behaviors.function

Right end-behavior How the graph behaves at large Evaluate the rational expression at positive values of x increasing positive values of x.

Left end-behavior How the graph behaves at large Evaluate the rational expression at negative values of x increasing negative values of x.

Roots, zeros, or x-intercepts Point(s) where the graph crosses the Set the numerator equal to zero and x-axis solve for x.

y-intercepts Point where the graph crosses the Set x = 0 to determine the y-intercept.y-axis

Example

Exercise

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NAME ______________________________________________ DATE______________ PERIOD _____

8-3

The line y � b is a horizontal asymptote for the rational function f(x) if f(x) → b as x → or as x → � . The horizontal asymptote can be found by using the TABLE feature of the graphing calculator.

Find the horizontal asymptote for each function.

a. f(x) � �x2 � 41x � 5�

Enter the function into Y1. Place [TblSet] in the Ask mode. Enter thenumbers 10,000, 100,000, 1,000,000, and 5,000,000 and their opposites inthe x-list.Keystrokes: 1 4 5 [TBLSET] [TABLE]. Then enter thevalues for x.

Notice that as x increases, y approaches 0. Thus, y � 0 is thehorizontal asymptote.

b. f(x) � �2x2 �3x

52

x � 6�

Enter the equation into Y1. Enter the numbers 10,000, 100,000,1,000,000, and 5,000,000 and their opposites in the x-list. Note the pattern. As x increases, y approaches 1.5. Thus, y � 1.5 is thehorizontal asymptote.

2ndENTER

2nd)—+x 2(�Y=

Find the horizontal asymptote for each function.

1. f(x) � �x2�x

1� 2. f(x) � �2x2x�

2

7�x

1� 12� 3. f(x) � �2x3 �

62xx3

2 � 2�

4. f(x) � �3x2 �25xx � 1� 5. f(x) � �

15x2 �x3

3x � 7� 6. f(x) �

7. f(x) � �5xx2

��

23

� 8. f(x) � �2x2 �6x

33

x � 6� 9. f(x) � �2x

2� 4�

x3 � 8x2 � 4x � 11��x4 � 3x3� 4x � 6

Graphing Calculator ActivityHorizontal Asymptotes and Tables

Exercises

Example

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• For each additional student who enrolls in a public college, the total

high-tech spending will (increase/decrease) by .

• For each decrease in enrollment of 100 students in a public college, the total high-tech spending will (increase/decrease) by .

Read the Lesson

1. Write an equation to represent each of the following variation statements. Use k as theconstant of variation.

a. m varies inversely as n.

b. s varies directly as r.

c. t varies jointly as p and q.

2. Which type of variation, direct or inverse, is represented by each graph?

a. b.


3. How can your knowledge of the equation of the slope-intercept form of the equation of aline help you remember the equation for direct variation?

x

y

Ox

y

O

Lesson Reading GuideDirect, Joint, and Inverse Variation

NAME ______________________________________________ DATE______________ PERIOD _____

Less

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8-4

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NAME ______________________________________________ DATE______________ PERIOD _____

8-4 Study Guide and InterventionDirect, Joint, and Inverse Variation

Direct Variation and Joint Variation

Direct Variationy varies directly as x if there is some nonzero constant k such that y � kx. k is called theconstant of variation.

Joint Variation y varies jointly as x and z if there is some number k such that y � kxz, where x � 0 and z � 0.

Find each value.

a. If y varies directly as x and y � 16when x � 4, find x when y � 20.

� Direct proportion

� y1 � 16, x1 � 4, and y2 � 20

16x2 � (20)(4) Cross multiply.

x2 � 5 Simplify.

The value of x is 5 when y is 20.

20�x2

16�4

y2�x2

y1�x1

b. If y varies jointly as x and z and y � 10when x � 2 and z � 4, find y when x � 4 and z � 3.

� Joint variation

� y1 � 10, x1 � 2, z1 � 4, x2 � 4, and z2 � 3

120 � 8y2 Simplify.

y2 � 15 Divide each side by 8.

The value of y is 15 when x � 4 and z � 3.

y2�4 � 310

�2 � 4

y2�x2z2

y1�x1z1

Find each value.

1. If y varies directly as x and y � 9 when 2. If y varies directly as x and y � 16 when x � 6, find y when x � 8. x � 36, find y when x � 54.

3. If y varies directly as x and x � 15 4. If y varies directly as x and x � 33 when when y � 5, find x when y � 9. y � 22, find x when y � 32.

5. Suppose y varies jointly as x and z. 6. Suppose y varies jointly as x and z. Find yFind y when x � 5 and z � 3, if y � 18 when x � 6 and z � 8, if y � 6 when x � 4when x � 3 and z � 2. and z � 2.

7. Suppose y varies jointly as x and z. 8. Suppose y varies jointly as x and z. Find yFind y when x � 4 and z � 11, if y � 60 when x � 5 and z � 2, if y � 84 when when x � 3 and z � 5. x � 4 and z � 7.

9. If y varies directly as x and y � 39 10. If y varies directly as x and x � 60 whenwhen x � 52, find y when x � 22. y � 75, find x when y � 42.

11. Suppose y varies jointly as x and z. 12. Suppose y varies jointly as x and z. Find yFind y when x � 7 and z � 18, if when x � 5 and z � 27, if y � 480 when y � 351 when x � 6 and z � 13. x � 9 and z � 20.

Exercises

Example

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Inverse Variation

Inverse Variation y varies inversely as x if there is some nonzero constant k such that xy � k or y � .

If a varies inversely as b and a � 8 when b � 12, find a when b � 4.

� Inverse variation

� a1 � 8, b1 � 12, b2 � 4

8(12) � 4a2 Cross multiply.

96 � 4a2 Simplify.

24 � a2 Divide each side by 4.

When b � 4, the value of a is 24.

Find each value.

1. If y varies inversely as x and y � 12 when x � 10, find y when x � 15.












a2�12

8�4

a2�b1

a1�b2

k�x

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Direct, Joint, and Inverse Variation

Exercises

Example

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State whether each equation represents a direct, joint, or inverse variation. Thenname the constant of variation.

1. c � 12m 2. p � 3. A � bh

4. rw � 15 5. y � 2rst 6. f � 5280m

7. y � 0.2s 8. vz � �25 9. t � 16rh

10. R � 11. � 12. C � 2r

Find each value.

13. If y varies directly as x and y � 35 when x � 7, find y when x � 11.


15. If y varies directly as x and y � 540 when x � 10, find x when y � 1080.

16. If y varies directly as x and y � 12 when x � 72, find x when y � 9.

17. If y varies jointly as x and z and y � 18 when x � 2 and z � 3, find y when x � 5 and z � 6.

18. If y varies jointly as x and z and y � �16 when x � 4 and z � 2, find y when x � �1 and z � 7.




22. If y varies inversely as x and y � 3 when x � 14, find x when y � 6.

23. If y varies inversely as x and y � 27 when x � 2, find x when y � 9.

24. If y varies directly as x and y � �15 when x � 5, find x when y � �36.

1�3

a�b

8�w

1�2

4�q

NAME ______________________________________________ DATE______________ PERIOD _____

8-4NAME ______________________________________________ DATE______________ PERIOD _____

Skills PracticeDirect, Joint, and Inverse Variation

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State whether each equation represents a direct, joint, or inverse variation. Thenname the constant of variation.

1. u � 8wz 2. p � 4s 3. L � 4. xy � 4.5

5. � 6. 2d � mn 7. � h 8. y �

Find each value.


10. If y varies directly as x and y � �16 when x � 6, find x when y � �4.


12. If y varies directly as x and y � 7 when x � 1.5, find y when x � 4.



15. If y varies jointly as x and z and y � 12 when x � �2 and z � 3, find y when x � 4 and z � �1.


17. If y varies inversely as x and y � 3 when x � 5, find x when y � 2.5.

18. If y varies inversely as x and y � �18 when x � 6, find y when x � 5.

19. If y varies directly as x and y � 5 when x � 0.4, find x when y � 37.5.

20. GASES The volume V of a gas varies inversely as its pressure P. If V � 80 cubiccentimeters when P � 2000 millimeters of mercury, find V when P � 320 millimeters ofmercury.

21. SPRINGS The length S that a spring will stretch varies directly with the weight F thatis attached to the spring. If a spring stretches 20 inches with 25 pounds attached, howfar will it stretch with 15 pounds attached?

22. GEOMETRY The area A of a trapezoid varies jointly as its height and the sum of itsbases. If the area is 480 square meters when the height is 20 meters and the bases are28 meters and 20 meters, what is the area of a trapezoid when its height is 8 meters andits bases are 10 meters and 15 meters?

3�4x

1.25�g

C�d

5�k

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PracticeDirect, Joint, and Inverse Variation

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NAME ______________________________________________ DATE______________ PERIOD _____

8-4 Word Problem Practice Direct, Joint, and Inverse Variation

1. DIVING The height that a diver leapsabove a diving board varies directly withthe amount that the tip of the divingboard dips below its normal level. If adiver leaps 44 inches above the divingboard when the diving board tip dips 12inches, how high will the diver leapabove the diving board if the tip dips 18inches?

2. PARKING LOT DESIGN As a generalrule, the number of parking spaces in a parking lot for a movie theatercomplex varies directly with the numberof theaters in the complex. A typicaltheater has 30 parking spaces for eachtheater. A businessman wants to build a new cinema complex on a lot that has enough space for 210 parkingspaces. How many theaters should thebusinessman build in his complex?

3. RENT An apartment rents for m dollarsper month. If n students share the rentequally, how much would each studenthave to pay? How does the cost perstudent vary with the number ofstudents? If 2 students have to pay $700 each, how much money would each student have to pay if there were 5 students sharing the rent?

4. PAINTING The cost of painting a wallvaries directly with the area of the wall.Write a formula for the cost of paintinga rectangular wall with dimensions � byw. With respect to � and w, does the costvary directly, jointly, or inversely?

HYDROGEN For Exercises 5-7, use thefollowing information.

The cost of a hydrogen storage tank variesdirectly with the volume of the tank. Alaboratory wants to purchase a storage tankshaped like a block with dimensions L by Wby H.

5. Fill in the missing spaces in thefollowing table from a brochure ofvarious tank sizes.

6. The hydrogen tank must fit in a shelfthat has a fixed height and depth. Howdoes the cost of the hydrogen storagetank vary with the width of tank withfixed depth and height?

7. How much would a spherical tank ofradius 24 inches cost? (Recall that the

volume of a sphere is given by �r3,

where r is the radius.)

4�3

Hydrogen TankDimensions (inches) Cost

L W H

36 36 36

18 24 $150

24 24 72 $800

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8-4 EnrichmentGeosynchronous Satellites

NAME ______________________________________________ DATE______________ PERIOD _____


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Satellites circling the Earth are almost as common as the cell phones thatdepend on them. A geosynchronous satellite is one that maintains the sameposition above the Earth at all times. Geosynchronous satellites are used incell phone communications, transmitting signals from towers on Earth andto each other.

The speed at which they travel is very important. If the speed is too low,the satellite will be forced back down to Earth due to the Earth’s gravity.However, if it is too fast, it will overcome gravity’s force and escape intospace, never to return. Newton’s second law of motion says that force on anobject is equal to mass times acceleration or F � ma. It is also well knownthat the net gravitational force between two objects is inversely proportionalto the square of the distance between them. Therefore, there are twovariables on which the force depends: speed and height above the Earth.

In particular, Newton’s second law, F � ma, shows that force varies directlywith acceleration, where m is the constant taking the place of “k.”

1. Show that the net gravitational force providing a satellite with accelera-tion is inversely proportional to the square of the distance between themby expressing this variation as an equation.

2. Use your equation from Number 1 and equate it with Newton’s formulaabove to determine how the satellite’s acceleration varies with its heightabove the Earth.

3. Determine how the speed of a geosynchronous satellite varies with itsheight above the Earth by using the fact that speed is equal to distancedivided by time and the path of the satellite is circular.

Exercises

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NAME ______________________________________________ DATE______________ PERIOD _____

8-4

You have learned to solve problems involving direct, inverse, and joint variation.Many physical situations involve at least one of these types of variation. Forexample, according to Newton’s law of universal gravitation, the weight of amass near Earth depends on the distance between the mass and the center ofEarth. Study the spreadsheet below to determine the type of variation thatexists between the quantity of an astronaut’s weight and the distance of theastronaut from the center of Earth.

In the spreadsheet, the values for the astronaut’s weight in newtons are enteredin the cells in column A, and the values for the astronaut’s distance in metersfrom the center of Earth are entered in cells in column B. Column C contains theastronaut’s distance from Earth’s surface.

Spreadsheet ActivityVariation

1. Use the values in the spreadsheet to make a graph of the astronaut’s weight plotted against the astronaut’s distance from Earth’s center.

2. Based on your graph, is this an inverse or direct variation?

3. Write an equation that represents this situation. LetW represent the astronaut’s weight, k the constant ofvariation, and R the distance from Earth’s center.

4. Use the equation to find the weight of the astronaut at these distances from Earth’s surface. (Hint: Remember to add these values to the value in cell B2 to find the distance from Earth’s center.)a. 145,300,000 m b. 65 m c. 25,600 m

d. 300,800,700 m e. 6580 m f. 180,560 m

A1

32

4567

B C

Gravitation.xls

734.5843712.0675548.9825111.44062.642112

6,380,0006,480,0007,380,000

16,380,000106,380,000

0100

100010,000

100,000

Astronaut’s Weight (N) Distance from Earth’s Center (m) Distance from Earth’s Surface (km)

Sheet 1 Sheet 2 Sheet 3

Exercises

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8-5

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Lesson Reading GuideClasses of Functions


Less

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• Based on the graph, estimate the weight on Mars of a child who weighs 40 pounds on Earth.

• Although the graph does not extend far enough to the right to read it directly from the graph, use the weight you found above and your knowledge that this graph represents direct variation to estimate the weight on Mars of a woman who weighs 120 pounds on Earth.

Read the Lesson

1. Match each graph below with the type of function it represents. Some types may be usedmore than once and others not at all.I. square root II. quadratic III. absolute value IV. rationalV. greatest integer VI. constant VII. identity

a. b. c.

d. e. f.


2. How can the symbolic definition of absolute value that you learned in Lesson 1-4 helpyou to remember the graph of the function f(x) � |x |?

x

y

Ox

y

Ox

y

O

x

y

Ox

y

Ox

y

O

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Identify Graphs You should be familiar with the graphs of the following functions.

Function Description of Graph

Constant a horizontal line that crosses the y-axis at a

Direct Variation a line that passes through the origin and is neither horizontal nor vertical

Identity a line that passes through the point (a, a), where a is any real number

Greatest Integer a step function

Absolute Value V-shaped graph

Quadratic a parabola

Square Root a curve that starts at a point and curves in only one direction

Rational a graph with one or more asymptotes and/or holes

Inverse Variationa graph with 2 curved branches and 2 asymptotes, x � 0 and y � 0 (special case of rational function)

Identify the function represented by each graph.

1. 2. 3.

4. 5. 6.

7. 8. 9.

x

y

O

x

y

O

x

y

O

x

y

Ox

y

Ox

y

O

x

y

Ox

y

O

x

y

O

NAME ______________________________________________ DATE______________ PERIOD _____

8-5 Study Guide and InterventionClasses of Functions

Exercises

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Exercises

NAME ______________________________________________ DATE______________ PERIOD _____

8-5


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Less

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Identify Equations You should be able to graph the equations of the following functions.

Function General Equation

Constant y � a

Direct Variation y � ax

Greatest Integer equation includes a variable within the greatest integer symbol, � �

Absolute Value equation includes a variable within the absolute value symbol, | |

Quadratic y � ax2 � bx � c, where a � 0

Square Root equation includes a variable beneath the radical sign, ��

Rational y �

Inverse Variation y �

Identify the function represented by each equation. Then graph the equation.

1. y � 2. y � x 3. y � �

4. y � |3x| � 1 5. y � � 6. y �

7. y � �x � 2� 8. y � 3.2 9. y �

x

y

Ox

y

Ox

y

O

x2 � 5x � 6��x � 2

x

y

Ox

y

Ox

y

O

x�2

2�x

x

y

Ox

y

Ox

y

O

x2�2

4�3

6�x

a�x

p(x)�q(x)


Classes of Functions

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Identify the type of function represented by each graph.

1. 2. 3.

Match each graph with an equation below.

A. y � |x � 1| B. y � C. y � �1 � x� D. y � �x� � 1

4. 5. 6.

Identify the type of function represented by each equation. Then graph theequation.

7. y � 8. y � 2�x� 9. y � �3x

x

y

Ox

y

OxO

y

2�x

x

y

O

x

y

Ox

y

O

1�x � 1

x

y

O

x

y

Ox

y

O

8-5 Skills PracticeClasses of Functions

NAME ______________________________________________ DATE______________ PERIOD _____

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1. 2. 3.

Match each graph with an equation below.

A. y � |2x � 1 | B. y � �2x � 1� C. y � D. y � ��x�

4. 5. 6.

Identify the type of function represented by each equation. Then graph theequation.

7. y � �3 8. y � 2x2 � 1 9. y �

10. BUSINESS A startup company uses the function P � 1.3x2 � 3x � 7 to predict its profit orloss during its first 7 years of operation. Describe the shape of the graph of the function.

11. PARKING A parking lot charges $10 to park for the first day or part of a day. After that,it charges an additional $8 per day or part of a day. Describe the graph and find the cost

of parking for 6 days.1�2

x

y

Ox

y

O

x

y

O

x2 � 5x � 6��x � 2

x

y

O

x

y

Ox

y

O

x � 3�2

x

y

O

x

y

O

x

y

O

Less

on

8-5

PracticeClasses of Functions

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Word Problem Practice Classes of Functions

NAME ______________________________________________ DATE______________ PERIOD _____

8-5

1. STAIRS What type of a function has a graph that could be used to model a staircase?

2. GOLF BALLS The trajectory of a golfball hit by an astronaut on the moon is described by the function f(x) � �0.0125(x � 40)2 � 20.

Describe the shape of this trajectory.

3. RAVINE The graph shows the cross-section of a ravine.

What kind of function is represented bythe graph? Write the function.

4. LEAKY FAUCETS A leaky faucet leaks 1 milliliter of water every second.Write a function that gives the numberof milliliters leaked in t seconds as afunction of t. What type of function is it?

y

xO

y

x80

21

O

PUBLISHING For Exercises 5-8, use thefollowing information.

Kate has just finished writing a book thatexplains how to sew your own stuffedanimals. She hopes to make $1000 fromsales of the book because that is how muchit would cost her to go to the EuropeanSewing Convention. Each book costs $2 toprint and assemble. Let P be the sellingprice of the book. Let N be the number ofpeople who will buy the book.

5. Write an equation that relates P and Nif she earns exactly $1,000 from sales ofthe book.

6. Solve the equation you wrote forExercise 5 for P in terms of N.

7. What kind of function is P in terms ofN? Sketch a graph of P as a function ofN.

8. If Kate thinks that 125 people will buyher book, how much should she chargefor the book?

Sale

Pri

ce

0

50

100

Number of Buyers100

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EnrichmentPhysical Properties of Functions

NAME ______________________________________________ DATE______________ PERIOD _____

8-5


Less

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Mathematical functions are classified based on properties similar to how biologists classifyanimal species. Functions can be classified as continuous or non-continuous, increasing ordecreasing, polynomial or non-polynomial for example. The class of polynomials functionscan be further classified as linear, quadratic, cubic, etc., based on its degree.

Characteristics of functions include:• A function is bounded below if there exists a number that is less than any function

value.• A function is bounded above if a number exists that is greater than any function

value.• A function is symmetric (about a vertical axis) if it is a mirror image about that

vertical axis.• A function is continuous if it can be drawn without lifting your pencil.• A function is increasing if f (x) f (y) when x y. Continual growth from left to right.• A function is decreasing if f (x) � f (y) when x � y. Continual decay from left to right.

1. Sketch the graph of y � x2 � 5x � 6. List the characteristics of functions displayed by this graph.

2. What characteristics do absolute value functions and quadratic functions have in common? How do they differ?

3. Graph y � ⏐x � 3⏐. 4. Graph y � x2 � 8x � 7.

y

x

y

x

y

x

Exercises

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Lesson Reading Guide Solving Rational Equations and Inequalities

NAME ______________________________________________ DATE______________ PERIOD _____

8-6

Get Ready for the LessonRead the introduction to Lesson 8-6 in your textbook.• If you increase the number of songs that you download, will your total bill increase or

decrease?

• Will your actual cost per song increase or decrease?

Read the Lesson1. When solving a rational equation, any possible solution that results in

in the denominator must be excluded from the list of solutions.

2. Suppose that on a quiz you are asked to solve the rational inequality � 0.Complete the steps of the solution.

Step 1 The excluded values are and .

Step 2 The related equation is .

To solve this equation, multiply both sides by the LCD, which is .Solving this equation will show that the only solution is �4.

Step 3 Divide a number line into regions using the excluded values and thesolution of the related equation. Draw dashed vertical lines on the number linebelow to show these regions.

Consider the following values of � for various test values of z.

If z � �5, � � 0.2. If z � �3, � � �1.

If z � �1, � � 9. If z � 1, � � �5.

Using this information and your number line, write the solution of the inequality.

Remember What You Learned3. How are the processes of adding rational expressions with different denominators and of

solving rational expressions alike, and how are they different?

6�z

3�z � 2

6�z

3�z � 2

6�z

3�z � 2

6�z

3�z � 2

6�z

3�z � 2

�3�4�5�6 �2 �1 0 1 2 3 4 5 6

6�z

3�z � 2

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NAME ______________________________________________ DATE______________ PERIOD _____


Less

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8-6

Study Guide and InterventionSolving Rational Equations and Inequalities

8-6

Less

on

8-6

Solve Rational Equations A rational equation contains one or more rationalexpressions. To solve a rational equation, first multiply each side by the least commondenominator of all of the denominators. Be sure to exclude any solution that would producea denominator of zero.

Solve � � .

� � Original equation

10(x � 1)� � � � 10(x � 1)� � Multiply each side by 10(x � 1).

9(x � 1) � 2(10) � 4(x � 1) Multiply.

9x � 9 � 20 � 4x � 4 Distributive Property

5x � �25 Subtract 4x and 29 from each side.

x � �5 Divide each side by 5.

Check � � Original equation

� � x � �5

� � Simplify.

�

Solve each equation.

1. � � 2 2. � � 1 3. � �

4. � � 4 5. � 6. � � 10

7. NAVIGATION The current in a river is 6 miles per hour. In her motorboat Marissa cantravel 12 miles upstream or 16 miles downstream in the same amount of time. What isthe speed of her motorboat in still water? Is this a reasonable answer? Explain.

8. WORK Adam, Bethany, and Carlos own a painting company. To paint a particular house

alone, Adam estimates that it would take him 4 days, Bethany estimates 5 days, and

Carlos 6 days. If these estimates are accurate, how long should it take the three of them

to paint the house if they work together? Is this a reasonable answer?

1�2

4�x � 2

x�x � 2

x � 1�12

4�x � 1

2m � 1�2m

3m � 2�5m

1�2

x � 5�4

2x � 1�3

4 � 2t�3

4t � 3�5

y � 3�6

2y�3

2�5

2�5

2�5

10�20

18�20

2�5

2��5 � 1

9�10

2�5

2�x � 1

9�10

2�5

2�x � 1

9�10

2�5

2�x � 1

9�10

2�5

2�x � 1

9�10

Exercises

Example

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Solve Rational Inequalities To solve a rational inequality, complete the following steps.

Step 1 State the excluded values.Step 2 Solve the related equation.Step 3 Use the values from steps 1 and 2 to divide the number line into regions. Test a value in each region to

see which regions satisfy the original inequality.

Solve � .

Step 1 The value of 0 is excluded since this value would result in a denominator of 0.

Step 2 Solve the related equation.

� � Related equation

15n� � � � 15n� � Multiply each side by 15n.

10 � 12 � 10n Simplify.

22 � 10n Simplify.

2.2 � n Simplify.

Step 3 Draw a number with vertical lines at the excluded value and the solution to the equation.

Test n � �1. Test n � 1. Test n � 3.

� � �� is true. � � is not true. � � is true.

The solution is n � 0 or n � 2.2.

Solve each inequality.

1. � 3 2. � 4x 3. �

4. � 5. � � 2 6. � 1 2�x � 1

3�x2 � 1

5�x

4�x � 1

1�4

2�x

3�2x

2�3

4�5p

1�2p

1�x

3�a � 1

2�3

4�15

2�9

2�3

4�5

2�3

2�3

4�5

2�3

�3 �2 �1 0 1 22.2

3

2�3

4�5n

2�3n

2�3

4�5n

2�3n

2�3

4�5n

2�3n

NAME ______________________________________________ DATE______________ PERIOD _____

8-6 Study Guide and Intervention (continued)

Solving Rational Equations and Inequalities

Exercises

Example

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NAME ______________________________________________ DATE______________ PERIOD _____

8-6


Less

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Skills PracticeSolving Rational Equations and Inequalities

NAME ______________________________________________ DATE______________ PERIOD _____

Less

on

8-6

Solve each equation or inequality. Check your solutions.

1. � 2. 2 � �

3. � 4. 3 � z �

5. � 6. �

7. � 8. � � y � 7

9. � 10. � 0

11. 2 � � 12. n � �

13. � � � 14. � � 1

15. � � 9 16. � 4 �

17. 2 � � 18. 8 � �

19. � � 20. � �

21. � � 22. � �

23. � � 24. � �2

�t � 34

�t � 38

�t2 � 9

2�e � 2

1�e � 2

2e�e2 � 4

5�s � 4

3�s � 3

12s � 19��s2 � 7s � 12

2x � 3�x � 1

x�2x � 2

x � 8�2x � 2

4�w2 � 4

1�w � 2

1�w � 2

2�n � 3

5�n2 � 9

1�n � 3

8z � 8�z � 2

4�z

2q�q � 1

5�2q

b � 2�b � 1

3b � 2�b � 1

9x � 7�x � 2

15�x

2�x

1�2x

5�2

3�m

1�2m

12�n

3�n

5�v

3�v

4�3k

3�k

x � 1�x � 10

x � 2�x � 4

12�y

3�2

2x � 3�x � 1

8�s

s � 3�5

1�d � 2

2�d � 1

2�z

�6�2

9�3x

1�3

4�n

1�2

x�x � 1

Less

on

8-6

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NAME ______________________________________________ DATE______________ PERIOD _____

8-6

Solve each equation or inequality. Check your solutions.

1. � � 2. � 1 �

3. � 4. � s �

5. � � 1 6. � � 0

7. � 8. � �

9. � 10. 5 � �

11. � � 12. 8 �

13. � � 14. � �

15. g � � 16. b � � 1 �

17. � � 18. � 4 �

19. � � 20. � �

21. � � 22. � �

23. � � 24. 3 � �

27. BASKETBALL Kiana has made 9 of 19 free throws so far this season. Her goal is to make60% of her free throws. If Kiana makes her next x free throws in a row, the function

f(x) � represents Kiana’s new ratio of free throws made. How many successful free

throws in a row will raise Kiana’s percent made to 60%? Is this a reasonable answer?Explain.

28. OPTICS The lens equation � � relates the distance p of an object from a

lens, the distance q of the image of the object from the lens, and the focal length f of the lens. What is the distance of an object from a lens if the image of the object is 5 centimeters from the lens and the focal length of the lens is 4 centimeters? Is this a reasonable answer? Explain.

1�f

1�q

1�p

9 � x�19 � x

22�a � 5

6a � 1�2a � 7

r2 � 16�r2 � 16

4�r � 4

r�r � 4

2�x � 2

x�2 � x

x2 � 4�x2 � 4

14��y2 � 3y � 10

7�y � 5

y�y � 2

2��v2 � 3v � 2

5v�v � 2

4v�v � 1

25��k2 � 7k � 12

4�k � 4

3�k � 3

12��c2 � 2c � 3

c � 1�c � 3

3�n2 � 4

1�n � 2

1�n � 2

b � 3�b � 1

2b�b � 1

2�g � 2

g�g � 2

2�x � 1

4�x � 2

6�x � 1

1�5

1�3p

4�p

19�y

3�y

3�2x

1�10

4�5x

7�a

3�a

�1�w � 3

4�w � 2

3�h � 1

5�h

1�2h

9�2t � 1

5�t

5�x

1�3x � 2

y�y � 5

5�y � 5

5s � 8�s � 2

s�s � 2

4�p

p � 10�p2 � 2

x�2

x�x � 1

3�2

3�4

12�x

PracticeSolving Rational Equations and Inequalities

NAME ______________________________________________ DATE______________ PERIOD _____

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Word Problem PracticeSolving Rational Equations and Inequalities

NAME ______________________________________________ DATE______________ PERIOD _____

8-6


Less

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8-6

1. HEIGHT Serena can be described asbeing 8 inches shorter than her sisterMalia, or as being 12.5% shorter than

Malia. In other words, � ,

where H is Serena’s height in inches.How tall is Serena?

2. CRANES For a wedding, Paula wants tofold 1000 origami cranes.

She does not want to make anyone foldmore than 15 cranes. In other words, ifN is the number of people enlisted to

fold cranes, Paula wants � 15.

What is the minimum number of peoplethat will satisfy this inequality?

3. RENTAL Carlos and his friends rent acar. They split the $200 rental fee evenly.Carlos, together with just two of hisfriends, decide to rent a portable DVDplayer as well, and split the $30 rentalfee for the DVD player evenly amongthemselves. Carlos ends up spending$50 for these rentals. Write an equationinvolving N, the number of friendsCarlos has, using this information. Solvethe equation for N.

1000�

N

1�8

8�H � 8

4. PROJECTILES A projectile target is launched into the air. A rocketinterceptor is fired at the target. Theratio of the altitude of the rocket to thealtitude of the projectile t seconds afterthe launch of the rocket is given by the

formula . At what time

are the target and interceptor at thesame altitude?

FLIGHT TIME For Exercises 5 and 6, usethe following information.

The distance between New York City andLos Angeles is about 2500 miles. Let S bethe airspeed of a jet. The wind speed is 100miles per hour. Because of the wind, it takeslonger to fly one way than the other.

5. Write an equation for S if it takes 2 hours and 5 minutes longer to flybetween New York and Los Angelesagainst the wind versus flying with the wind.

6. Solve the equation you wrote in Exercise 5 for S.

7. Write an equation and find how muchlonger to fly between New York and LosAngeles if the wind speed increases to150 miles per hour and the airspeed ofthe jet is 525 miles per hour.

333t��32t2 � 420t � 27

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Oblique AsymptotesThe graph of y � ax � b, where a � 0, is called an oblique asymptote of y � f(x) if the graph of f comes closer and closer to the line as x → ∞ or x → �∞. ∞ is themathematical symbol for infinity, which means endless.

For f(x) � 3x � 4 � �2x�, y � 3x � 4 is an oblique asymptote because

f(x) � 3x � 4 � �2x�, and �

2x� → 0 as x → ∞ or �∞. In other words, as | x |

increases, the value of �2x� gets smaller and smaller approaching 0.

Find the oblique asymptote for f(x) � �x2 �

x8�

x2� 15

�.

�2 1 8 15 Use synthetic division.

�2 �121 6 3

y � �x2 �

x8�x

2� 15� � x � 6 � �x �

32�

As | x | increases, the value of �x �3

2� gets smaller. In other words, since

�x �3

2� → 0 as x → ∞ or x → �∞, y � x � 6 is an oblique asymptote.

Use synthetic division to find the oblique asymptote for each function.

1. y � �8x2 �

x �4x

5� 11

�

2. y � �x2 �

x3�x

2� 15�

3. y � �x2 �

x2�x

3� 18�

4. y � �ax2

x�

�bx

d� c

�

5. y � �ax2

x�

�bx

d� c

�

NAME ______________________________________________ DATE______________ PERIOD _____

Enrichment

NAME ______________________________________________ DATE______________ PERIOD _____

8-6

Example

Read each question. Then fill in the correct answer.

8 Student Recording SheetUse this recording sheet with pages 494–495 of the Student Edition.


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NAME ______________________________________________ DATE______________ PERIOD _____

1.

2.

3. Record your answer and fill in thebubbles in the grid below. Be sure to use the correct place value.

4.

5.

6. A B C D

F G H J

A B C D

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

.

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

F G H J

A B C D 7.

8.

9.

10.

11. Record your answer and fill in thebubbles in the grid below. Be sure to use the correct place value.

Record your answers for Question 12on the back of this paper.

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

.

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

A B C D

F G H J

A B C D

F G H J

Ass

essm

ent

Pre-AP

8


Rubric for Scoring Pre-AP(Use to score the Pre-AP question on page 495 of the Student Edition.)

NAME ______________________________________________ DATE______________ PERIOD _____

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General Scoring Guidelines

• If a student gives only a correct numerical answer to a problem but does not show howhe or she arrived at the answer, the student will be awarded only 1 credit. All extended-response questions require the student to show work.

• A fully correct answer for a multiple-part question requires correct responses for allparts of the question. For example, if a question has three parts, the correct response to one or two parts of the question that required work to be shown is not considered a fully correct response.

• Students who use trial and error to solve a problem must show their method. Merelyshowing that the answer checks or is correct is not considered a complete response forfull credit.

Exercise 12 Rubric

Score Specific Criteria

4 Part a shows an understanding that the situation is an inverse situationbecause the smaller gear has fewer teeth and makes more revolutions. Thestudent shows that if the larger gear makes 36 revolutions the smaller gearwill make 26 revolutions.

3 A generally correct solution, but may contain minor flaws in reasoning orcomputation.

2 A partially correct interpretation and/or solution to the problem.

1 A correct solution with no supporting evidence or explanation.

0 An incorrect solution indicating no mathematical understanding of theconcept or task, or no solution is given.

For Questions 1–4, determine the equations of any vertical asymptotes and the values of x for any holes in the graph of each rational function.

1. f(x) � �x2 �3x � 2� 2. f(x) � �x2 �

x �2x

3� 3�

3. f(x) � �x2 �

x �2x

4� 8

� 4. f(x) �

5. Graph f(x) � �x �4

3�.

x2 � 3x�x � 3

1.

2.

3.

4.

5.

8 Chapter 8 Quiz 1 SCORE

(Lessons 8–1 and 8–2)


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NAME ______________________________________________ DATE______________ PERIOD _____

8 Chapter 8 Quiz 2(Lesson 8–3)

NAME ______________________________________________ DATE______________ PERIOD _____

For Questions 1–4, simplify each expression.

1. �1x22an3

4n

� � �96ax7

5nn

5

2� 2. �x2

3�x �

6x1�2

8� � �x2 �

x2

5�x

4� 6�

3. �2x2

x��

x4� 3

� � �x2 �

x2�x

1� 24� 4.

5. MULTIPLE CHOICE For what value(s) of x is the

expression �xx2

2��

57xx

��

1140� undefined?

A. �5, 2 B. 0, 2, 5 C. �2 D. 0, 2 E. �5, �2


6. 12a2, 15b3, 20ab2 7. 5x2 � 20, 3x � 6

8. 2t2 � 3t � 1, 2t2 � 7t � 4


9. �m72n�

� �5m2

n� 10. �y25�y

3y� � �3 �7

y�

�p2p�

2 �6p

3�p

9��

�4p2�0

12�

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

xO

f (x)

8


Chapter 8 Quiz 3 SCORE

(Lessons 8–4 and 8–5)

NAME ______________________________________________ DATE______________ PERIOD _____

8 Chapter 8 Quiz 4(Lesson 8–6)

NAME ______________________________________________ DATE______________ PERIOD _____

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1. State whether rt � 30 represents a direct, joint, or inverse 1.variation. Then name the constant of variation.

2. Suppose y varies jointly as x and z. Find y when x � 1 and 2.z � 4, if y � 96 when x � 4 and z � 8.


3. 4. 3.

5. Identify the type of function represented by y � 3� x � � 2.Then graph the equation.

xO

yy

xO

12

For Questions 1–4, solve each equation or inequality.

1. �x �6

2� � �xx

��

72� � �

14� 1.

2. �tt

��

53� � �

tt

��

33� � �t �

13� 2.

3. 3 � �2t�

�8t�

3.

4. �m6� 5� 2 4.

5. NUMBER THEORY The ratio of two less than a number to six more than that number is 2 to 3. Find the number. 5.

4.

5.y

xO

Write the letter for the correct answer in the blank at the right of each question.

1. For what value(s) of x is the expression �(x �2x

4(x)(

�x2

3�)

9)� undefined?

A. �4, 9 B. �4, �3, 0, 3 C. �4, 0, 3, 9 D. �4, �3, 3 1.


2. �92yy2

��

11

� � �13y

��

21y

�

F. �3y � 1 G. 3y � 1 H. �3y � 1 J. 3y � 1 2.

3. �cc2

2��

c6c

��205� � �

c32

c��

136

�

A. �c �3

4� B. �c �3

4� C. �c �

34

� D. �c �

34

� 3.

4.

F. �169m(m

2(m�

�2)

2)� G. �m(mm2

��2

4)� H. m � 2 J. �

4(m3� 2)� 4.

5. �15� � �4

3w�

� �103w�

A. �4w

20�w

21� B. �

4w20

�w

9� C. �20

1w�

D. ��41w�

5.

6. Simplify �x2 �xx � 6� � �x2 � 6

1x � 8�. 6.

For Questions 7 and 8, find the LCM for each set 7.of polynomials.

7. 12s3, 18s2t, 24t4 8. 9c � 15, 21c � 358.

9. Determine the equations of any vertical asymptotes and the 9.

values of x for any holes in the graph of f(x) � �x2 �x �

x �3

12�.10.

10. Graph f(x) � �(x �4

2)2�.

11. If y varies inversely as x and x � 16 when y � 4,find x when y � 8.

12. If y varies directly as x and y � 1 when x � 3,find y when x � 21. 11.

12.

�43mm

2

2

��

81m2

��8m

6m2 �

�1162m�

Part I

8 Chapter 8 Mid-Chapter Test SCORE

(Lessons 8–1 through 8–4)


Ass

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ent

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NAME ______________________________________________ DATE______________ PERIOD _____

xO

f (x)

Part II

8


Chapter 8 Vocabulary Test SCORE

NAME ______________________________________________ DATE______________ PERIOD _____

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Underline or circle the correct word or phrase to complete each sentence.

1. The equation y � �3x� is an example of (direct variation, inverse variation).

2. r(x) � �xx

2

2��

65xx

��

96� is an example of a (rational function, rational expression).

3. The graph of y � �x �3

5� has a(n) (asymptote, point discontinuity).

4. Adding or subtracting rational expressions requires you to find a(n) (least common denominator, asymptote).

5. The formula for simple interest, I � Prt, is an example of (direct variation, joint variation).

6. The graph of y � �xx

��

53� has a break in (discontinuity, continuity)

at x � 3.

7. �2t�

� �t32� � 1 is an example of a (rational inequality, rational equation).

8. If you walk at a steady speed, your speed and the time it takes towalk 1 mile are (asymptotes, inversely proportional) to each other.

9. The equation C � d gives the circumference of a circle in terms of itsdiameter. Here, is called the (constant of variation, point discontinuity).

10. If the rational expression in a rational function is not written in lowest terms, the graph of the function may have a (constant of variation,point discontinuity).

Define each term in your own words.

11. rational expression

12. complex fraction

asymptotecomplex fractionconstant of variation

continuitydirect variationinverse variation

joint variationpoint discontinuityrational equation

rational expressionrational functionrational inequality

8 Chapter 8 Test, Form 1 SCORE


Ass

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ent

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NAME ______________________________________________ DATE______________ PERIOD _____



1. �2148mm

n2�

A. �34mn�

B. �4m

3n

� C. �34mn� D. �

43� 1.

2. �6a �

512

� � �a1�0

2�

F. 12 G. 24 H. 12a � 12 J. 24a 2.

3. �x2 �y

y2� � �xy�

2

y�

A. �y(x1� y)� B. C. �

x �y

y� D. �y(x

1� y)� 3.

4.

F. 5mn G. �5mn�

H. �15�mn J. �

mn

2� 4.

5. �p10

q��

4q�

A. �10

p�q2

4p� B. �q(p

1�4

1)� C. �10p

pq� 4� D. �

10p�q

4p� 5.

6. �k �4

1� � �2(k9� 1)�

F. �2(k1�3

1)� G. �2(k1�7

1)� H. �k1�1

1� J. �89� 6.

For Questions 7 and 8, find the LCM of each set of polynomials.

7. 10x2, 30xy2

A. 30x2y2 B. 300x3y2 C. 10x D. 40x2y2 7.

8. 3z � 12, 6z � 24F. 18(z � 4) G. 3(z � 4) H. 6(z � 4) J. z � 4 8.

9. Which is an equation of the vertical asymptote of the graph of f(x) � �xx

��

12�?

A. y � 1 B. y � 2 C. x � 2 D. x � 1 9.

10. Which rational function is graphed?

F. f(x) � �x �2

1� H. f(x) � �x �2

1�

G. f(x) � �x �x

1� J. f(x) � �x �x

1� 10.

�5mn

2

3��

�nm

2�

y3��x3 � x2y � xy2 � y3

xO

f (x)

8


11. The equation z � 30x represents a(n) __?___ variation.A. direct B. joint C. inverse D. combined 11.

12. Suppose y varies jointly as x and z. If y � 24 when x � 2 and z � 3, find ywhen x � 1 and z � 5.F. 5 G. 20 H. 10 J. 4 12.

13. The equation m � �n4

� represents a(n) __?___ variation.

A. direct B. joint C. inverse D. reverse 13.

14. If y varies inversely as x and y � 2 when x � 10, find y when x � 5.F. 1 G. 4 H. 25 J. 100 14.

For Questions 15 and 16, identify the function represented by each graph.

15. A. absolute valueB. greatest integerC. direct variationD. quadratic 15.

16. F. identityG. constantH. inverse variationJ. rational 16.

17. Identify the type of function represented by y � �16x�.A. direction variation C. inverse variationB. quadratic D. square root 17.

18. Solve �x �x

2� � �75�.

F. �7 G. 5 H. 7 J. ��57� 18.

19. Solve y � 4 � �5y�.

A. �5, 1 B. �1, 5 C. �1 D. � 19.

20. Solve �m9� 5� � 3.

F. m � 5 or m 8 H. m � �2 or m 5G. �2 � m � 5 J. 5 � m � 8 20.

Bonus Determine the equations of any vertical asymptotes and B:

the values of x for any holes in the graph of f(x) � �xx2

2

��

39x�.

y

xO

y

xO

Chapter 8 Test, Form 1 (continued)

NAME ______________________________________________ DATE______________ PERIOD _____

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8 Chapter 8 Test, Form 2A SCORE


Ass

essm

ent


1. For what value(s) of m is the expression �m2m2

2�

�2mm

��

13� undefined?

A. ��32�, 0, 1 B. �1, �

32� C. � �

32�, 1 D. �

32� 1.


2. �xx2

2��

52xx

��

41� � �

2xx��

42

�

F. �12� G. 2 H. �2

((xx��

41))2

2� J. �2(xx��

41)� 2.

3. �a �

3b

� � �a2

1�2

b2�

A. �4(aa

2��

bb2)� B. �a �

4b� C. �a �

4b� D. �

4a(2a

��

bb2)

� 3.

4.

F. �s

1�2

3� G. 12s � 36 H. �

ss

��

33� J. 3 4.

5. �n26�n

9� � �n �3

3�

A. �n �3

3� B. �n �3

3� C. �n26�n

n�

�3

12� D. �6nn2 �

�93

� 5.

6. �mm� 5� � �5 �

2m�

F. �m2�m

5� G. �mm

��

25� H. �

mm

��

25� J. �(m

2�m

5)2� 6.


7. 5p � 20, 15p � 60A. 75(p � 4) B. 15(p � 4) C. p � 4 D. 5(p � 4) 7.

8. t2 � 8t � 15, t2 � t � 20F. (t � 3)(t � 5)(t � 4) H. (t � 3)(t � 5)(t � 4)G. (t � 3)(t � 5)(t � 4) J. (t � 3)(t � 5)(t � 4) 8.

9. Determine the equations of any vertical asymptotes of the graph of

f(x) � �x2 �

x �5x

1� 6

�.

A. x � 1 C. x � �2, x � �3B. x � �2 D. y � 1 9.

10. Determine the values of x for any holes in the graph of f(x) � �x2 �x �

6x5� 5�.

F. x � 5 H. x � �5G. x � 1 J. x � �1, x � �5 10.

�84ss2

2

��

2346s�

��122ss2 �

�63s6

�

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NAME ______________________________________________ DATE______________ PERIOD _____

8



A. f(x) � �x �3

2� C. f(x) � �x �x

2�

B. f(x) � �x �3

2� D. f(x) � �x �x

2� 11.

12. If y varies directly as x and y � 4 when x � �2, find y when x � 30.

F. ��145�

G. 60 H. �60 J. �145�

12.

13. The area A of a triangle varies jointly as the lengths of its base b and height h. If A � 75 when b � 15 and h � 10, find A when b � 8 and h � 6.A. 12 B. 48 C. 24 D. 96 13.


F. �16� G. 6 H. 3 J. �

13� 14.

15. The distance a car can travel on a certain amount of fuel varies inversely with its speed. If a car traveling 50 miles per hour can travel 300 miles on 10 gallons of fuel, how far could the car travel on 10 gallons of fuel at 60 miles per hour?A. 250 mi B. 360 mi C. 275 mi D. 300 mi 15.

16. Identify the type of function represented by y � (x � 1)2 � 4.F. square root H. rationalG. inverse variation J. quadratic 16.

17. Identify the type of function represented by y � �xx2

��

39

�.

A. quadratic C. inverse variationB. rational D. direct variation 17.

18. Solve �n �n

4� � n � �12n

��

44n

�.

F. �4, 3 G. �3, 4 H. �4 J. 3 18.

19. Solve 4 � �1b� � �

3b�.

A. b 0 B. b � 0 or b 1 C. 0 � b � 1 D. b � 1 19.

20. Tomas can do a job in 4 hours. Julia can do the same job in 6 hours. How many hours will it take the two of them to do the job if they work together?F. 3.5 G. 2.4 H. 5 J. 2 20.

Bonus Simplify . B:1 � �

3x�

��1 � �

4x� � �x

32�

Chapter 8 Test, Form 2A (continued)

NAME ______________________________________________ DATE______________ PERIOD _____

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xO

f (x)

8 Chapter 8 Test, Form 2B SCORE


Ass

essm

ent


1. For what value(s) of x is the expression �2xx2

2��

43xx��

42� undefined?

A. ��12�, 0, 2 B. ��

12�, 2 C. �2, �

12� D. ��

12� 1.


2. �t2 �

t22�t

1� 3

� � �t2 �3t

4�t �

33�

F. �t2 �

3t6�t �

39

� G. �3t(2t

��

13)

� H. 3 J. �t �3

1� 2.

3. �m �

62n

� � �m2

1�04n2�

A. �3(m5� 2n)� C. �m �

42n�

B. �3(m5� 2n)� D. 3.

4.

F. �bb

��

22� G. b � 2 H. 2b � 4 J. b � 2 4.

5. �m23�0

25� � �m3� 5�

A. �3mm2 �

�2255

� B. �m23�3

25� C. �m3� 5� D. �(m

3�(m

5)�(m

15�)

5)� 5.

6. �m7� 6� � �6 �

mm�

F. �7m

��

m6� G. �

mm

��

76� H. �

mm

��

76� J. �6 �

7m�

6.


7. 7m � 21, 14m � 42A. m � 3 B. 98(m � 3) C. 7(m � 3) D. 14(m � 3) 7.

8. t2 � t � 12, t2 � 2t � 24F. (t � 3)(t � 4)(t � 6) H. (t � 3)(t � 4)(t � 6)G. (t � 3)(t � 4)(t � 6) J. (t � 3)(t � 4)(t � 6) 8.

9. Determine the equations of any vertical asymptotes of the graph of

f(x) � �x22�x

2�x

3� 3�.

A. x � �1 B. x � 3 C. x � �3, x � 1 D. y � 2 9.

10. Determine the values of x for any holes in the graph of f(x) � �x2 �x �

5x3� 6�.

F. x � �3 G. x � 3 H. x � �2, x � �3 J. x � �2 10.

�63bb2

2

��

1122b�

��10

5bb2��

1200b�

m3 � 4mn2 � 2m2n � 8n3��60

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NAME ______________________________________________ DATE______________ PERIOD _____

8



A. f(x) � �xx

��

31� C. f(x) � �

xx

��

31�

B. f(x) � �(x � 3)3(x � 1)� D. f(x) � �(x � 3)

3(x � 1)� 11.

12. If y varies jointly as x and z and y � 60 when x � 10 and z � �3, find y when x � 8 and z � 15.F. �240 G. 15 H. 240 J. �15 12.

13. SALES An appliance store manager noted that weekly sales varied directly with the amount of money spent on advertising. If last week’s sales were $10,000 and $2000 was spent on advertising, what should sales be during a week that $1200 was spent on advertising?A. $4800 B. $6000 C. $16,667 D. $50,000 13.


F. �32� G. �

23� H. �

59� J. �

95� 14.

15. The distance a car can travel on a certain amount of fuel varies inversely with its speed. If a car traveling 50 miles per hour can travel 336 miles on 10 gallons of fuel, how far could the car travel on 10 gallons of fuel at 60 miles per hour?A. 315 mi B. 320 mi C. 403.2 mi D. 280 mi 15.

16. Identify the type of function represented by y � � x � 5 �.F. direct variation H. absolute valueG. inverse variation J. constant 16.

17. Identify the type of function represented by y � 4.A. greatest integer C. constantB. direct variation D. identity 17.

18. Solve �n �n

3� � n � �7nn

��

138

�.

F. 3 G. 6 H. 3, 6 J. �3, 6 18.

19. Solve 7 � �m3� �

1m8�.

A. m � 0 or m 3 C. m 3B. 0 � m � 3 D. m � 0 19.

20. The sum of a number and 16 times its reciprocal is 10. Find the number(s).F. �8 or �2 G. 2 or 8 H. 4 J. �4 20.

Bonus Simplify . B:1 � �

2x�

��1 � �

1x� � �x

22�

Chapter 8 Test, Form 2B (continued)

NAME ______________________________________________ DATE______________ PERIOD _____

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panies, Inc.

xO

f (x)

8 Chapter 8 Test, Form 2C SCORE


Ass

essm

ent

1. For what value(s) of x is the expression �2x2x�

2 �3x

9� 9� 1.

undefined?


2. �x2 �x3

64� � �xx�

2

8� 2.

3. �3bb2

2

��

63bb

��

56

� � �6bb2 �

�2152� 3.

4. 4.

5. �x �2

2� � �x28� 4� 5.

6. �3m5� 1� � �1 �

23m� 6.


7. 4m3n, 9mn4, 18m4n2 7.

8. n2 � 2n � 8, n2 � 2n � 24 8.

For Questions 9 and 10, determine the equations of any vertical asymptotes and the values of x for any holes in the graph of each rational function.

9. f(x) � �xx

��

13� 9.

10. f(x) � �x2 �

x �2x

2� 8

� 10.

11. Graph the rational function f(x) � �xx

��

32�. 11.

12. If y varies jointly as x and z and y � 6 when x � 4 and z � 12, find y when x � 24 and z � 5. 12.

�63mm

2

2

��

3705m�

��9m

4m2 �

�4250m�

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NAME ______________________________________________ DATE______________ PERIOD _____

xO

f (x)

8


13. PHOTOGRAPHS A film-developing company noted that, in a particular town, the number of customers requesting online delivery of their vacation pictures varied directly with the number of households having high-speed Internet access. Currently, 5000 households in the town have high-speed Internet access and 80 customers request online delivery of their photographs. If this trend continues, how many customers should the film-developing company expect to request online delivery when 12,000 households have high-speed Internet access? 13.

14. If y varies inversely as x and y � 25 when x � 6, find ywhen x � 150. 14.

15. WILDFIRES Firefighters battling wildfires in western states noted that the percentage P of the fire remaining uncontained varied inversely with the amount of precipitation A that fell the previous day. If k is the constant of variation, write an equation that expresses P as a function of A. 15.

16. Identify the type of function 16.represented by the graph.

17. Identify the type of function represented by y � ��23�x. 17.

For Questions 18 and 19, solve each equation or inequality.

18. x � �x2�x

2� � �3xx��

22

� 18.

19. 9 � �m2� �

4m7� 19.

20. PAINTING Alice can paint a room in 8 hours. Her assistant can paint the same room in 12 hours. How long will it take if the two of them work together? 20.

Bonus Solve � 1. B:�x �

12� � �x �

13�

��x �

12� � �x �

13�

y

xO

Chapter 8 Test, Form 2C (continued)

NAME ______________________________________________ DATE______________ PERIOD _____

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8 Chapter 8 Test, Form 2D SCORE


Ass

essm

ent

1. For what value(s) of x is the expression �2xx

2

2��

xx

��

610� 1.

undefined?


2. �x2 �x4

25� � �xx�

2

5� 2.

3. �3mm

2

2

��

125mm

��

812�� 8m

4m2 �

2 �16

4m� 3.

4. 4.

5. �x �3

3� � �x21�8

9� 5.

6. �2n3� 1� � �1 �

22n� 6.


7. 7s2t, 6st4, 14s3t2 7.

8. n2 � 6n � 5, n2 � 3n � 10 8.

For Questions 9 and 10, determine the equations of any vertical asymptotes and the values of x for any holes in the graph of each rational function.

9. f(x) � �x2 �x

2�x

6� 24� 9.

10. f(x) � �x2 � 73x � 10� 10.

11. Graph the rational function f(x) � �x �x

2�. 11.

12. If y varies jointly as x and z and y � 12 when x � 18 and 12.z � 6, find y when x � 81 and z � 7.

xO

f (x)

�182y2y2

��1468y�

��49yy2��

188y�

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NAME ______________________________________________ DATE______________ PERIOD _____

8


13. RESTAURANTS In a certain county, the planning commission noted that the number of restaurant permits renewed each year varied directly with the number of tourists visiting the county during the previous year. Last year, 400,000 tourists visited the county and 1200 restaurants renewed their permits. This year, 350,000 tourists are projected to visit the county. How many restaurant permits will be renewed if the trend continues? 13.

14. If y varies inversely as x and y � 12 when x � 6, find ywhen x � 8. 14.

15. GOVERNMENT Part of a model used by a state government indicates that revenue R varies inversely with the percentage of eligible workers who are unemployed U. If the constant of variation is k, write an equation that expresses R as a function of U. 15.


17. Identify the type of function represented by 17.

y � �1x1�.

For Questions 18 and 19, solve each equation or inequality.

18. �x2�x

3� � �12� � �2x

2� 6� 18.

19. �8r

r� 3� � �

4r5� 19.

20. GARDENING Joyce can plant a garden in 120 minutes,and Jim can do the same job in 80 minutes. How long will it take to plant the garden if both of them work together? 20.

Bonus Solve � 1. B:�x �

15� � �x �

11�

��

�x �1

5� � �x �1

1�

y

xO

Chapter 8 Test, Form 2D (continued)

NAME ______________________________________________ DATE______________ PERIOD _____

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8 Chapter 8 Test, Form 3 SCORE


Ass

essm

ent

1. For what value(s) of x is the expression �6x23x�

2 �13

xx�2 �

105x� 1.

undefined?


2. �3x22

x�2 �

12xx��

612

��3x34�x2

x2�

�9

10x� 2.

3. �g2

5�g

5�g

5� 4

� � �gg2

2��

8gg

��

1126

� 3.

4. 4.

5. �99aa

2

2��

44bb

2

2� � �2b3�a

3a� � �3a2�b

2b� 5.

6. 6.

7. Find the LCM of c2 � 2cd � d2, c2 � d2, and c � d. 7.

For Questions 8 and 9, determine the equations of any vertical asymptotes and the values of x for any holes in the graph of each rational function. Then graph each function.

8. f(x) � �(x��

23)2�

8.

9. f(x) � �2xx2 �

�44� 9.

10. If y varies jointly as x and z and y � �15� when x � �

13� and 10.

z � 15, find y when x � 10 and z � �14�.

(2 � n)��12� � �n

1��

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2�n

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NAME ______________________________________________ DATE______________ PERIOD _____

xO

f (x)

xO

f (x)

8


TELECOMMUNICATIONS For Questions 11 and 12,use the information below and in the table.

The average number of daily phone calls C between two cities is directly proportional to the product of the populations P1 and P2 of the cities and inversely proportional to the square of the distance d

between the cities. That is, C � �kP

d12

P2�.

11. Atlanta and Charleston are located approximately 324 miles apart and the average number of daily phone calls between the cities is 7700. Find the constant of variation k to the nearest hundredth. 11.

12. About 17,100 calls are made each day between Atlanta and Tallahassee. Find the distance between the cities to the nearest mile. 12.

13. The current I in an electrical circuit varies inversely with the resistance R in the circuit. If the current is 1.2 when the resistance is 6, write an equation relating the current and the resistance. Then find the current when the resistance is 0.18. 13.


15. Identify the type of function 15.represented by xy � 0.3.

For Questions 16–19, solve each equation or inequality.

16. �y �5

3� � �y2 �1y0

� 6� � �y �y

2� 16.

17. �n �2

5� � �n2 �3n

3�n �

110� � �n �

12� 17.

18. �61x�

� �32x�

� �59� 19. �1 �

4z� z � 3 18.

20. NUMBER THEORY A fraction has a value of �35�. If the

numerator is decreased by 8 and the denominator is

increased by 3, its value is �14�. Find the original fraction.

Bonus Simplify � and state any value(s) of x

B:for which the expression is undefined.

�x32� � �

2x�

��x32� � �x

23�

2 � �3x�

��2x� � �x

32�

Chapter 8 Test, Form 3 (continued)

NAME ______________________________________________ DATE______________ PERIOD _____

Copyright ©

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ill Com

panies, Inc.

CityPopulation

in 2000

Atlanta 416,000

Charleston 97,000

Raleigh 276,000

Tallahassee 151,000

y

xO

19.

20.

8 Chapter 8 Extended-Response Test SCORE


Ass

essm

ent

Demonstrate your knowledge by giving a clear, concise solutionto each problem. Be sure to include all relevant drawings andjustify your answers. You may show your solutions in more thanone way or investigate beyond the requirements of the problem.

1. Write three different rational expressions that are equivalent to theexpression �a �

a5�.

2. The volume of the rectangular box shown is given by V � (2x3 � 26x2 � 60x) cubic inches.a. Explain how to find an expression in terms

of x for the height h of the box.b. In terms of x, h � _______?________ in simplest form.c. Explain how you could check the expression you found

in part b. Then check your expression.

3. Write two polynomials for which the LCM is 3y2 � 12.

4. Compare and contrast the graphs of the rational functions

f(x) � �(x �

x2�)(x

2� 3)

� and g(x) � �(x �

x(x2)

�(x

2�)

3)�.

5. You decide to invest 10% of your before-tax income in a retirement fund,so you have your employer deduct this money from your weekly paycheck.a. Write an equation to represent the amount deducted from your paycheck

d for investment in your retirement fund for a week during which youworked h hours at r dollars per hour.

b. Is your equation a direct, joint, or inverse variation? Explain your choice.c. If you earn $9.50 per hour and worked 36 hours last week, explain how to

determine the amount deducted last week for your retirement fund.

6. The Franklin Electronics Company has determined that, after its first 50 CDplayers are produced, the average cost of producing one CD player can be

approximated by the function C(x) � �60x

x��

1570,000

�, where x represents the

number of CD players produced. Consumer research has indicated that thecompany should charge the consumer $80 per CD player in order to maximizeits profit. Thus, the revenue from the sale of each CD player can be representedby the function R(x) � 80.a. Identify the function represented by C(x). Explain your choice.b. Identify the function represented by R(x). Explain your choice.c. The company wants to determine how many CD players must be produced

and sold in order to ensure that the revenue from each one is greater thanthe average cost of producing each one. Write an inequality whose solutionrepresents the information for which the company is looking.

d. Solve your inequality and interpret your solution in the context of the problem.

2x in.h

(x � 10) in.

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8


1. If 6 more than the product of a number and �2 is greater than 10, which of the following could be that number?A �3 B �2 C 0 D 3 1.

2. If the diameter of a circle is doubled, then the area is multiplied by _______.F 2 G 4 H 8 J 16 2.

3. Which represents an irrational number?

A ��13� B 1 C �2� D �9� 3.

4. If a � 0, which of the following must be true?F a � 2 � 2 � a H �2a � a2

G a � 2 � 2a J a2 a � 2 4.

5. A cube is equal in volume to a rectangular solid with edges that measure 4, 6, and 9. What is the measure of an edge of the cube?A 216 B 36 C 108 D 6 5.

6. If abc � 30 and b � c, then a equals which of the following?

F �3c02� G �

1c5� H 30c2 J 15c 6.

7. What is the value of (a � b)3 if b � a � 2?A �8 B �6 C 6 D 8 7.

8. In the figure, WXZ and XYZ are isosceles right triangles. If XY � 8, find the perimeter of quadrilateral WXYZ.F 16 � 16�2� H 24 � 8�2� G 32 � 8�2� J 32 � 16�2� 8.

9. In a 30-day month, how many weekend days fall on dates that are prime numbers if the first day of the month is Thursday?A 2 B 3 C 4 D 5 9.

10. Sonia purchased 5 pencils and 2 pens for $5.10. Wai purchased 8 of the same type of pencil and 6 of the same type of pen, and spent $13.20. What is the cost of 2 pencils and one pen?F $2.10 G $3.90 H $1.80 J $2.40 10. F G H J

A B C D

F G H J

A B C D

F G H J

A B C D

F G H J

A B C D

F G H J

A B C D

Standardized Test Practice(Chapters 1–8)

NAME ______________________________________________ DATE______________ PERIOD _____

Copyright ©

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panies, Inc.

Z Y

W X

Part 1: Multiple Choice

Instructions: Fill in the appropriate circle for the best answer.

8 Standardized Test Practice (continued)


Ass

essm

ent

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NAME ______________________________________________ DATE______________ PERIOD _____

11. 3 is 12% of what number?A 36 B 0.36 C 15 D 25 11.

12. If w � 4x, y � 10z, x � 3, and z � , what is the value of � ?

F G � H J � 12.

13. Evaluate . Express the result in scientific notation.

A 0.3 � 10 B 3 � 102 C 0.3 � 10�3 D 3 � 10�4 13.

14. Simplify (5 � 2�3�)(2 � 4�3�).

F 10 � 8�3� G �62 � 16�3� H �14 J �14 � 16�3� 14.

15. Solve �3 y � 3� � 6 � 4.A 1003 B 103 C �5 D 11 15.

16. The quadratic equation 9x2 � 6x � 1 � 9 is to be solved by completing the square. Which equation would not be a step in that solution?

F �x � �13

��2

� 1 H x � ��13

� � 1

G 9x2 � 6x � 8 � 0 J x2 � x � � 1 16.

17. How many rectangles can be 17.found in the figure shown?

18. What is the value of a in 18.the figure shown?

9

8

7

6

5

4

3

2

1

0

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8

7

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F G H J1�9

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A B C D

F G H J

A B C D

6 � 10�2��20 � 10�5

F G H J1�7

3�20

3�20

13�30

3�w

2�y

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A B C D

Part 2: Griddable

Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate circle that corresponds to that entry.

a�

61�

25�18�

8


19. Determine whether C � � and D � � are 19.

inverses.

20. Simplify the expression �w�13��

�25�

. 20.

21. Solve x2 � 2x � 2 � 0 by completing the square. 21.

22. Graph y � x2 � 4x. 22.

23. Use synthetic substitution to find f(3) for 23.f(x) � 3x3 � 7x2 � 5x � 10.

24. List all of the possible rational zeros of 2x4 � 5x3 � 3x2 � 12x � 6. 24.

25. Simplify . 25.

26. Suppose y varies jointly as x and z. Find y when x � 16 and 26.z � 5, if y � 9 when x � 3 and z � 12.

27. Evita adds a 75% acid solution to 8 milliliters of solution 27.that is 15% acid. The function that represents the percent

of acid in the resulting solution is f(x) ��8(0.15

8) �

�xx(0.75)

�,

where x is the amount of 75% acid solution added. How much 75% acid solution shouldbe added to create a solution that is 50% acid?

28. In order to remain hydrated, a 150-pound human requires 80 ounces of water each day.

a. Write an equation to represent the amount of water needed to hydrate x 150-pound humans for d days. 28a.

b. Is your equation a direct, joint, or inverse variation? 28b.

c. How much water is needed for four 150-pound humans during the month of May? 28c.

�59yy2

2

��

1306y�

��10

6yy2��

1220y�

y

xO

5� ��116�

�1 ��156�

1 5�3 1

Standardized Test Practice (continued)

NAME ______________________________________________ DATE______________ PERIOD _____

Copyright ©

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raw-H

ill Com

panies, Inc.

Part 3: Short Answer

Instructions: Write your answers in the space provided.

Chapter 8 A1 Glencoe Algebra 2

An

swer

s

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Answers (Anticipation Guide and Lesson 8-1)

8-1

Less

on R

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atio

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s

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Lesson 8-1

Get

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-1 i

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then

of t

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to

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fin

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e va

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.

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To

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ly t

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To

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frac

tio

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it

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an y

our

know

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f fr

acti

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NA

ME

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83

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2

Bef

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f(x

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Step

2

Chapter Resources



Stud

y G

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and

Inte

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(co

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)

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NA

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____

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IOD

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_

Cha

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87

Gle

ncoe

Alg

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2

Lesson 8-1

Sim

plif

y C

om

ple

x Fr

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Exam

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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

8-1

Cha

pter

86

Gle

ncoe

Alg

ebra

2

Sim

plif

y R

atio

nal

Exp

ress

ion

sA

rat

io o

f tw

o po

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nal

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sim

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rati

onal

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ide

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e n

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CF

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d ,

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NA

ME

____

____

____

____

____

____

____

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____

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Stud

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s

11

11

11

11

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1

11

11

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1

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11

11

11

1

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11

11

1

Exer

cise

s

Exam

ple

Copyright ©

Glencoe/M

cGraw


he McG

raw-H

ill Com

panies, Inc.

Answers (Lesson 8-1)


An

swer

s

Cop

yrig

ht ©

Gle

ncoe

/McG

raw

-Hill

, a

divi

sion

of T

he M

cGra

w-H

ill C

ompa

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, In

c.



Prac

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IOD

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8-1

Cha

pter

89

Gle

ncoe

Alg

ebra

2

Lesson 8-1

Sim

pli

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ach

exp

ress

ion

.

1.2.

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4.5.

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6.7.

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EOM

ETRY

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rea

of x

2�

4 sq

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nit

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as a

leg

th

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4 u

nit

s.D

eter

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e th

e le

ngt

h o

f th

e ot

her

leg

of

the

tria

ngl

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�2

un

its

23.G

EOM

ETRY

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ecta

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lar

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mid

has

a b

ase

area

of

squ

are

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eter

s

and

a h

eigh

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ers.

Wri

te a

rat

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xpre

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o de

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be t

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volu

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he

rect

angu

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88

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ncoe

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2

Sim

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AM

E__

____

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____

____

____

____

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____

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PE

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___

8-1


8-2

8-1

Enri

chm

ent

Dim

ensi

on

al A

nal

ysis

Cha

pter

811

Gle

ncoe

Alg

ebra

2

Lesson 8-1


NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

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ER

IOD

____

_

Sci

enti

sts

alw

ays

expr

ess

the

un

its

of m

easu

rem

ent

in t

hei

r so

luti

on.I

t is

insu

ffici

ent

and

ambi

guou

s to

sta

te a

sol

uti

on r

egar

din

g di

stan

ce a

s 17

;S

even

teen

wh

at,f

eet,

mil

es,m

eter

s? O

ften

it

is h

elpf

ul

to a

nal

yze

the

un

its

of t

he

quan

titi

es i

n a

for

mu

la t

o de

term

ine

the

desi

red

un

its

of a

n o

utp

ut.

For

exa

mpl

e,it

is

know

n t

hat

tor

que

is t

he

prod

uct

of

forc

e an

d di

stan

ce,b

ut

wh

at a

re t

he

un

its

of f

orce

?

Th

e u

nit

s al

so d

epen

d on

th

e m

easu

rin

g sy

stem

.Th

e tw

o m

ost

com

mon

lyu

sed

syst

ems

are

the

Bri

tish

sys

tem

an

d th

e in

tern

atio

nal

sys

tem

of

un

its

(SI)

.Som

e co

mm

on u

nit

s of

th

e B

riti

sh s

yste

m a

re i

nch

es,f

eet,

mil

es,

and

pou

nds

.Com

mon

SI

un

its

incl

ude

met

ers,

kilo

met

ers,

New

ton

s,an

dgr

ams.

Fre

quen

tly

con

vers

ion

fro

m o

ne

syst

em t

o an

oth

er i

s n

eces

sary

an

d ac

com

plis

hed

by

mu

ltip

lica

tion

by

con

vers

ion

fac

tors

.

Con

side

r ch

angi

ng

un

its

from

mil

es p

er h

our

to k

ilom

eter

s pe

r h

our.

Wh

at i

s60

mil

es p

er h

our

in k

ilom

eter

s pe

r h

our?

Use

th

e co

nve

rsio

n 1

ft

�30

.5 c

m.

60�

60�

��

��

96.6

2

1.T

he

SI

un

it f

or f

orce

is

a N

ewto

n (

N)

and

the

SI

un

it f

or d

ista

nce

is

met

ers

or c

enti

met

ers.

Th

e B

riti

sh u

nit

for

for

ce i

s po

un

ds a

nd

the

Bri

tish

un

it f

or d

ista

nce

is

feet

or

inch

es.U

sin

g th

e fo

rmu

la f

or t

orqu

e(T

orqu

e �

For

ce t

imes

Dis

tan

ce),

dete

rmin

e th

e S

I u

nit

an

d th

e B

riti

shu

nit

for

tor

que.

Po

ssib

le a

nsw

ers

are

N�

man

d f

t�

lbo

r N

�cm

and

inch

�lb

2.T

he

den

sity

of

a fl

uid

is

give

n b

y th

e fo

rmu

la d

ensi

ty�

.Su

ppos

e

that

a v

olu

me

of a

flu

id i

n a

cyl

indr

ical

can

is

r2

h,w

her

e r

and

har

e m

easu

red

in m

eter

s.F

ind

an e

xpre

ssio

n f

or t

he

mas

s,gi

ven

in

kilo

gram

s (k

g),o

f ga

soli

ne,

wh

ich

has

a k

now

n d

ensi

ty o

f 68

0.

680�

r2h

kg

3.C

onve

rt t

he

foll

owin

g m

easu

rem

ents

.

a.72

mil

es/h

our

to f

eet/

seco

nd

105.

6 fe

et/s

eco

nd

b.

32 p

oun

ds/s

quar

e in

ch t

o po

un

ds p

er s

quar

e fo

ot

4608

po

un

ds

per

sq

uar

e fo

ot

c.10

0 ki

lom

eter

s/h

our

to m

iles

per

hou

r

62.1

mile

s p

er h

ou

r

kg � m3

mas

s� vo

lum

ekm � h1

km� 10

00 m

1 m

� 100

cm30

.5 c

m�

1 ft

5280

ft

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mi

mi

� hm

i� h

8-1

Cha

pter

810

Gle

ncoe

Alg

ebra

2

Wor

d Pr

oble

m P

ract

ice

M

ult

iply

ing

an

d D

ivid

ing

Rat

ion

al E

xpre

ssio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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ATE

____

____

____

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ER

IOD

____

_


1.JE

LLY

BEA

NS

A l

arge

vat

con

tain

s G

gree

n je

lly

bean

s an

d R

red

jell

y be

ans.

A b

ag o

f 10

0 re

d an

d 10

0 gr

een

jell

ybe

ans

is a

dded

to

the

vat.

Wh

at i

s th

en

ew r

atio

of

red

to g

reen

jell

y be

ans

inth

e va

t?

2.M

ILEA

GE

Bet

h’s

car

get

s 15

mil

es p

erga

llon

in

th

e ci

ty a

nd

26 m

iles

per

gal

lon

on t

he

hig

hw

ay.B

eth

use

s C

gall

ons

ofga

s in

th

e ci

ty a

nd

Hga

llon

s of

gas

on

the

hig

hw

ay.W

rite

an

exp

ress

ion

for

th

e av

erag

e n

um

ber

of m

iles

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Glencoe/M

cGraw


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raw-H

ill Com

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yrig

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Gle

ncoe

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raw

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Copyright ©

Glencoe/M

cGraw


he McG

raw-H

ill Com

panies, Inc.



An

swer

s

Cop

yrig

ht ©

Gle

ncoe

/McG

raw

-Hill

, a

divi

sion

of T

he M

cGra

w-H

ill C

ompa

nies

, In

c.



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d Pr

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hat

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th

e fo

rm

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impl

ify

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ress

ion

.

3.TR

APE

ZOID

ST

he

cros

s se

ctio

n o

f a

stan

d co

nsi

sts

of t

wo

trap

ezoi

ds s

tack

edon

e on

top

of

the

oth

er.

Th

e to

tal

area

of

the

cros

s se

ctio

n i

s x2

squ

are

un

its.

Ass

um

ing

the

trap

ezoi

dsh

ave

the

sam

e h

eigh

t,w

rite

an

expr

essi

on f

or t

he

hei

ght

of t

he

stan

d in

term

s of

x.P

ut

you

r an

swer

in

sim

ples

tfo

rm.(

Rec

all

that

th

e ar

ea o

f a

trap

ezoi

dw

ith

hei

ght

han

d ba

ses

b 1an

d b 2

is

give

n b

y h

(b1

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AC

TIO

NS

In t

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seve

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rote

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n a

mos

t pe

culi

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ath

emat

ical

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atio

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2 �

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s an

exa

mpl

e of

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onti

nu

edfr

acti

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impl

ify

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con

tin

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fra

ctio

n

n�

.

REL

AY

RA

CE

For

Exe

rcis

es 5

-7,u

se t

he

foll

owin

g in

form

atio

n.

Mar

k,C

onn

ell,

Zac

k,an

d M

oses

ru

n t

he

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er r

elay

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eth

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hei

r av

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esp

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s�

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s�

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r se

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spec

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hat

wer

e th

eir

indi

vidu

al t

imes

for

thei

r ow

n l

egs

of t

he

race

?

,,

,

6.W

rite

an

exp

ress

ion

for

th

eir

tim

e as

ate

am.W

rite

you

r an

swer

as

a ra

tio

oftw

o po

lyn

omia

ls.

400

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sw

as 6

met

ers

per

seco

nd,

wh

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asth

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am’s

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e? R

oun

d yo

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er t

oth

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eare

st s

econ

d.

281

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nd

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btr

acti

ng

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xpre

ssio

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pter

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ncoe

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ebra

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d t

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ch s

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25. G

EOM

ETRY

The

exp

ress

ions

,

,and

re

pres

ent

the

leng

ths

of t

he s

ides

of

a

tria

ngle

.Wri

te a

sim

plif

ied

expr

essi

on f

or t

he p

erim

eter

of

the

tria

ngle

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26.K

AYA

KIN

GM

ai i

s ka

yaki

ng

on a

riv

er t

hat

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a c

urr

ent

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mil

es p

er h

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esen

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er r

ate

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ith

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er r

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th

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ai k

ayak

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own

stre

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nd

then

back

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her

sta

rtin

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int.

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th

e fo

rmu

la f

or t

ime,

t�

,wh

ere

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th

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stan

ce,t

o

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te a

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plif

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me

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akes

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ME

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____

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____

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____

____

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ATE

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ER

IOD

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_

8-2



Less

on R

eadi

ng G

uide

Gra

ph

ing

Rat

ion

al F

un

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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ATE

____

____

____

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ER

IOD

____

_

8-3

Cha

pter

819

Gle

ncoe

Alg

ebra

2

Lesson 8-3

Get

Rea

dy

for

the

Less

on

Rea

d t

he

intr

odu

ctio

n t

o L

esso

n 8

-3 i

n y

our

text

boo

k.

•If

15

stu

den

ts c

ontr

ibu

te t

o th

e gi

ft,h

ow m

uch

wou

ld e

ach

of

them

pay

?$1

0•

If e

ach

stu

den

t pa

ys $

5,h

ow m

any

stu

den

ts c

ontr

ibu

ted?

30 s

tud

ents

Rea

d t

he

Less

on

1.W

hic

h o

f th

e fo

llow

ing

are

rati

onal

fu

nct

ion

s?A

an

d C

A.

f(x)

�B

.g(x

) �

�x�

C.h

(x)

�

2.a.

Gra

phs

of r

atio

nal

fu

nct

ion

s m

ay h

ave

brea

ks i

n

.Th

ese

may

occ

ur

as v

erti

cal

or a

s po

int

.Th

e of

a r

atio

nal

fu

nct

ion

is

lim

ited

to

valu

es f

or w

hic

h t

he

fun

ctio

n i

s de

fin

ed.

b.

Th

e gr

aph

s of

tw

o ra

tion

al f

un

ctio

ns

are

show

n b

elow

.

I.II

.

Gra

ph I

has

a

at x

�.

Gra

ph I

I h

as a

at

x�

.

Mat

ch e

ach

fu

nct

ion

wit

h i

ts g

raph

abo

ve.

f(x)

�II

g(x)

�I

Rem

emb

er W

hat

Yo

u L

earn

ed

3.O

ne w

ay t

o re

mem

ber

som

ethi

ng n

ew i

s to

see

how

it

is r

elat

ed t

o so

met

hing

you

alr

eady

know

.How

can

kn

owin

g th

at d

ivis

ion

by

zero

is

un

defi

ned

hel

p yo

u t

o re

mem

ber

how

to

fin

d th

e pl

aces

wh

ere

a ra

tion

al f

un

ctio

n h

as a

poi

nt

disc

onti

nu

ity

or a

n a

sym

ptot

e?

Sam

ple

an

swer

:A

po

int

dis

con

tin

uit

y o

r ve

rtic

al a

sym

pto

te o

ccu

rsw

her

e th

e fu

nct

ion

is u

nd

efin

ed,t

hat

is,w

her

e th

e d

eno

min

ato

r o

f th

ere

late

d r

atio

nal

exp

ress

ion

is e

qu

al t

o 0

.Th

eref

ore

,set

th

e d

eno

min

ato

req

ual

to

zer

o a

nd

so

lve

for

the

vari

able

.

x2�

4� x

�2

x� x

�2

�2

vert

ical

asy

mp

tote

�2

po

int

dis

con

tin

uit

y

x

y Ox

y

O

do

mai

nd

isco

nti

nu

itie

sas

ymp

tote

sco

nti

nu

ity

x2�

25�

�x2

�6x

�9

1� x

�5


Cha

pter

818

Gle

ncoe

Alg

ebra

2

Enri

chm

ent

Z

eno

’s P

arad

ox

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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ATE

____

____

____

__P

ER

IOD

____

_

8-2

Th

e G

reek

ph

ilos

oph

er Z

eno

of E

lea

(bor

n s

omet

ime

betw

een

495

an

d 48

0 B

.C.)

pro

pose

d fo

ur

para

doxe

s to

ch

alle

nge

th

e n

otio

ns

of s

pace

an

d ti

me.

Zen

o’s

firs

t pa

rado

x w

orks

lik

e th

is:

Su

ppos

e yo

u a

re o

n y

our

way

to

sch

ool.

Ass

um

e yo

u a

re a

ble

to c

over

hal

f of

the

rem

ain

ing

dist

ance

eac

h m

inu

te t

hat

you

wal

k.Yo

u l

eave

you

r h

ouse

at

7:45

A.M

.Aft

er t

he

firs

t m

inu

te,y

ou a

re h

alf

of t

he

way

to

sch

ool.

In t

he

nex

tm

inu

teyo

uco

ver

hal

fof

the

rem

ain

ing

dist

ance

tosc

hoo

l,an

dat

7:47

A.M

.you

are

thre

e-qu

arte

rs o

f th

e w

ay t

o sc

hoo

l.T

his

pat

tern

con

tin

ues

eac

h m

inu

te.

At

wh

at t

ime

wil

l yo

u a

rriv

e at

sch

ool?

Bef

ore

8:00

A.M

.? B

efor

e lu

nch

?

Sin

ce s

pace

is

infi

nit

ely

divi

sibl

e,w

e ca

n r

epea

t th

is p

atte

rn f

orev

er.T

hu

s,on

th

e w

ay t

o sc

hoo

l yo

u m

ust

rea

ch a

n i

nfi

nit

e n

um

ber

of ‘m

idpo

ints

’ in

afi

nit

e ti

me.

Th

is i

s im

poss

ible

,so

you

can

nev

er r

each

you

r go

al.I

n g

ener

al,

acco

rdin

g to

Zen

o an

yon

e w

ho

wan

ts t

o m

ove

from

on

e po

int

to a

not

her

m

ust

mee

t th

ese

requ

irem

ents

,an

d m

otio

n i

s im

poss

ible

.Th

eref

ore,

wh

at

we

perc

eive

as

mot

ion

is

mer

ely

an i

llu

sion

.

Add

itio

n o

f fr

acti

ons

can

be

defi

ned

by

��

,sim

ilar

ly f

orsu

btra

ctio

n.

Ass

um

e yo

ur

hou

se i

s on

e m

ile

from

sch

ool.

At

7:46

A.M

.,yo

u h

ave

wal

ked

hal

f of

a m

ile,

so y

ou h

ave

left

1 �

,or

a m

ile.

At

7:47

A.M

.you

on

ly h

ave

��

of a

mil

e to

go.

To

dete

rmin

e h

ow f

ar y

ou h

ave

wal

ked

and

how

far

aw

ay f

rom

th

e sc

hoo

l yo

u

are

at 7

:48

A.M

.,ad

d th

e di

stan

ces

wal

ked

each

min

ute

,�

��

of

a m

ile

so f

ar a

nd

you

sti

ll h

ave

1 �

�of

a m

ile

to g

o.

1.D

eter

min

e h

ow f

ar y

ou h

ave

wal

ked

and

how

far

aw

ay f

rom

th

e sc

hoo

lyo

u a

re a

t 7:

50 A

.M.

You

hav

e w

alke

d

of

a m

ile,a

nd

will

be

of

a m

ile a

way

fro

m s

cho

ol.

2.S

upp

ose

inst

ead

of c

over

ing

one-

hal

f th

e di

stan

ce t

o sc

hoo

l ea

ch m

inu

te,

you

cov

er t

hre

e-qu

arte

rs o

f th

e di

stan

ce r

emai

nin

g to

sch

ool

each

min

ute

,n

ow w

ill

you

be

able

to

mak

e it

to

sch

ool

on t

ime?

Det

erm

ine

how

far

you

stil

l h

ave

left

to

go a

t 7:

47 A

.M.

No

.Yo

u w

ill h

ave

of

a m

ile r

emai

nin

g a

t 7:

47 A

.M.

3.S

upp

ose

that

in

stea

d of

cov

erin

g on

e-h

alf

or t

hre

e-qu

arte

rs o

f th

e

dist

ance

to

sch

ool

each

min

ute

,you

cov

er

of t

he

dist

ance

rem

ain

ing,

wh

ere

xis

a w

hol

e n

um

ber

grea

ter

than

2.W

hat

is

you

r di

stan

ce f

rom

sch

ool

at 7

:46

A.M

.?

You

are

o

f a

mile

fro

m s

cho

ol a

t 7:

46 A

.M.

x2

� (x �

1)2

1� x

�1

1 � 16

1 � 3231 � 32

1 � 87 � 8

7 � 81 � 8

1 � 41 � 2

1 � 41 � 4

1 � 2

1 � 21 � 2

ad�

bc�

bdc � d

a � b

Copyright ©

Glencoe/M

cGraw


he McG

raw-H

ill Com

panies, Inc.

Answers (Lessons 8-2 and 8-3)


An

swer

s

Cop

yrig

ht ©

Gle

ncoe

/McG

raw

-Hill

, a

divi

sion

of T

he M

cGra

w-H

ill C

ompa

nies

, In

c.



NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-3

Cha

pter

821

Gle

ncoe

Alg

ebra

2

Lesson 8-3

Gra

ph

Rat

ion

al F

un

ctio

ns

Use

th

e fo

llow

ing

step

s to

gra

ph a

rat

ion

al f

un

ctio

n.

Ste

p 1

Firs

t se

e if

the

func

tion

has

any

vert

ical

asy

mpt

otes

or

poin

t di

scon

tinui

ties.

Ste

p 2

Dra

w a

ny v

ertic

al a

sym

ptot

es.

Ste

p 3

Mak

e a

tabl

e of

val

ues.

Ste

p 4

Plo

t th

e po

ints

and

dra

w t

he g

raph

.

Gra

ph

f(x

) �

.

�or

Th

eref

ore

the

grap

h o

f f(

x) h

as a

n a

sym

ptot

e at

x�

�3

and

a po

int

disc

onti

nu

ity

at x

�1.

Mak

e a

tabl

e of

val

ues

.Plo

t th

e po

ints

an

d dr

aw t

he

grap

h.

Gra

ph

eac

h r

atio

nal

fu

nct

ion

.

1.f(

x) �

2.f(

x) �

3.f(

x) �

4.f(

x) �

5.f(

x) �

6.f(

x) �

xO

f(x)

xO

f(x)

xO

f(x)

x2�

6x�

8�

�x2

�x

�2

x2�

x�

6�

�x

�3

2� (x

�3)

2

xO

f(x)

48

8 4 –4 –8

–4–8

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��

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x

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Stud

y G

uide

and

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rven

tion

(co

nti

nued

)

Gra

ph

ing

Rat

ion

al F

un

ctio

ns

Exer

cise

s

Exam

ple


Exer

cise

s

Stud

y G

uide

and

Inte

rven

tion

Gra

ph

ing

Rat

ion

al F

un

ctio

ns

Cha

pter

820

Gle

ncoe

Alg

ebra

2

Do

mai

n a

nd

Ran

ge

Rat

ion

al F

un

ctio

nan

equ

atio

n of

the

form

f(x

) �

, w

here

p(x

) an

d q

(x)

are

poly

nom

ial e

xpre

ssio

ns a

nd

q(x

) �

0

Do

mai

nT

he d

omai

n of

a r

atio

nal f

unct

ion

is li

mite

d to

val

ues

for

whi

ch t

he f

unct

ion

is d

efin

ed.

Ver

tica

l Asy

mp

tote

A

n as

ympt

ote

is a

line

tha

t th

e gr

aph

of a

fun

ctio

n ap

proa

ches

.If

the

sim

plifi

ed fo

rm o

f th

ere

late

d ra

tiona

l exp

ress

ion

is u

ndef

ined

for

x�

a, t

hen

x�

ais

a v

ertic

al a

sym

ptot

e.

Po

int

Dis

con

tin

uit

y P

oint

dis

cont

inui

ty is

like

a h

ole

in a

gra

ph.I

f th

e or

igin

al r

elat

ed e

xpre

ssio

n is

und

efin

ed

for

x�

abu

t th

e si

mpl

ified

exp

ress

ion

is d

efin

ed fo

r x

�a,

the

n th

ere

is a

hol

e in

the

gr

aph

at x

�a.

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rizo

nta

l O

ften

a ho

rizon

tal a

sym

ptot

e oc

curs

in t

he g

raph

of

a ra

tiona

l fun

ctio

n w

here

a v

alue

isA

sym

pto

teex

clud

ed f

rom

the

ran

ge.

Det

erm

ine

the

equ

atio

ns

of a

ny

vert

ical

asy

mp

tote

s an

d t

he

valu

es

of x

for

any

hol

es i

n t

he

grap

h o

f f(

x) �

.

Fir

st f

acto

r th

e n

um

erat

or a

nd

the

den

omin

ator

of

the

rati

onal

exp

ress

ion

.

f(x)

��

Th

e fu

nct

ion

is

un

defi

ned

for

x�

1 an

d x

��

1.

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ce

�,x

�1

is a

ver

tica

l as

ympt

ote.

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e si

mpl

ifie

d ex

pres

sion

is

defi

ned

for

x�

�1,

so t

his

val

ue

repr

esen

ts a

hol

e in

th

e gr

aph

.

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erm

ine

the

equ

atio

ns

of a

ny

vert

ical

asy

mp

tote

s an

d t

he

valu

es o

f x

for

any

hol

es i

n t

he

grap

h o

f ea

ch r

atio

nal

fu

nct

ion

.

1.f(

x) �

2.f(

x) �

3.f(

x) �

asym

pto

tes:

x�

2,h

ole

:x

�as

ymp

tote

:x

�0;

x�

�5

ho

le x

�4

4.f(

x) �

5.f(

x) �

6.f(

x) �

asym

pto

te:

x�

�2;

asym

pto

tes:

x�

1,as

ymp

tote

:x

��

3

ho

le:

x�

x�

�7

7.f(

x) �

8.f(

x) �

9.f(

x) �

asym

pto

tes:

x�

1,as

ymp

tote

:x

��

3;h

ole

s:x

�1,

x�

3 x

�5

ho

le:

x�

3 � 2

x3�

2x2

�5x

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��

�x2

�4x

�3

2x2

�x

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��

2x2

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�9

x�

1�

�x2

�6x

�5

1 � 3

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��

x�

3x2

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�7

��

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6x�

73x

�1

��

3x2

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5 � 2

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x�

12�

�x2

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2x2

�x

�10

��

2x�

54

��

x2�

3x�

10

4x�

3� x

�1

(4x

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(x�

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�(x

�1)

(x�

1)

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(x�

1)�

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�1)

(x�

1)4x

2�

x�

3�

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4x2

�x

�3

��

x2�

1

p(x

)� q

(x)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-3

Exam

ple



Prac

tice

Gra

ph

ing

Rat

ion

al F

un

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-3

Cha

pter

823

Gle

ncoe

Alg

ebra

2

Lesson 8-3

Det

erm

ine

the

equ

atio

ns

of a

ny

vert

ical

asy

mp

tote

s an

d t

he

valu

es o

f x

for

any

hol

es i

n t

he

grap

h o

f ea

ch r

atio

nal

fu

nct

ion

.

1.f(

x) �

2.f(

x) �

3.f(

x) �

asym

pto

tes:

x�

2,as

ymp

tote

:x

�3;

asym

pto

te:

x�

�2

x�

�5

ho

le:

x�

74.

f(x)

�5.

f(x)

�6.

f(x)

�

ho

le:

x�

�10

ho

le:

x�

6h

ole

:x

��

5

Gra

ph

eac

h r

atio

nal

fu

nct

ion

.

7.f(

x) �

8.f(

x) �

9.f(

x) �

10.P

AIN

TIN

GW

orki

ng

alon

e,T

awa

can

giv

e th

e sh

ed a

coa

t of

pai

nt

in 6

hou

rs.I

t ta

kes

her

fat

her

xh

ours

wor

kin

g al

one

to g

ive

the

shed

a c

oat

of p

ain

t.T

he

equ

atio

n f

(x)

�de

scri

bes

the

port

ion

of

the

job

Taw

a an

d h

er f

ath

er w

orki

ng

toge

ther

can

com

plet

e in

1 h

our.

Gra

ph f

(x)

�fo

r x

0,

y

0.If

Taw

a’s

fath

er c

an c

ompl

ete

the

job

in 4

hou

rs a

lon

e,w

hat

por

tion

of

the

job

can

th

ey c

ompl

ete

toge

ther

in

1 h

our?

Wh

at d

omai

n a

nd

ran

geva

lues

are

mea

nin

gfu

l in

th

e co

nte

xt o

f th

e pr

oble

m?

;S

amp

le a

nsw

er:T

he

nu

mb

er o

f h

ou

rs it

tak

es h

er f

ath

er t

o g

ive

the

shed

a c

oat

of

pai

nt

sho

uld

be

po

siti

ve.T

her

efo

re,o

nly

val

ues

of

x

gre

ater

th

an 0

an

d v

alu

es o

f f(

x) g

reat

er t

han

ar

e m

ean

ing

ful.

11.L

IGH

TT

he

rela

tion

ship

bet

wee

n t

he

illu

min

atio

n a

n o

bjec

t re

ceiv

es f

rom

a l

igh

t so

urc

e of

Ifo

ot-c

andl

es a

nd

the

squ

are

of

the

dist

ance

din

fee

t of

th

e ob

ject

fro

m t

he

sou

rce

can

be

mod

eled

by

I(d

) �

.Gra

ph t

he

fun

ctio

n I

(d)

�fo

r

0 �

I�

80 a

nd

0 �

d�

80.W

hat

is

the

illu

min

atio

n i

n

foot

-can

dles

th

at t

he

obje

ct r

ecei

ves

at a

dis

tan

ce o

f 20

fee

t fr

om t

he

ligh

t so

urc

e?W

hat

dom

ain

an

d ra

nge

val

ues

are

m

ean

ingf

ul

in t

he

con

text

of

the

prob

lem

? 11

.25

foo

t-ca

nd

les;

Sam

ple

an

swer

:T

he

dis

tan

ce o

f th

e o

bje

ct f

rom

th

e so

urc

e sh

ou

ld b

e p

osi

tive

.Th

eref

ore

,o

nly

val

ues

of

dg

reat

er t

han

0 a

nd

val

ues

of

I(d

) g

reat

er t

han

0 a

rem

ean

ing

ful.

4500

�d

245

00�

d2

1 � 16

5 � 12

6 �

x�

6x

6 �

x�

6x

xO

f(x)

xOf(

x)

xO

f(x)

xO

f(x)

3x� (x

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�3

� x�

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20�

�x

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�x

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3x�

10

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ance

(ft)

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min

atio

n

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60

60 40 20

dOIII


Cha

pter

822

Gle

ncoe

Alg

ebra

2

Det

erm

ine

the

equ

atio

ns

of a

ny

vert

ical

asy

mp

tote

s an

d t

he

valu

es o

f x

for

any

hol

es i

n t

he

grap

h o

f ea

ch r

atio

nal

fu

nct

ion

.

1.f(

x) �

2.f(

x) �

asym

pto

tes:

x�

4,x

��

2as

ymp

tote

s:x

�4,

x�

9

3.f(

x) �

4.f(

x) �

asym

pto

te:

x�

2;h

ole

:x

��

12as

ymp

tote

:x

�3;

ho

le:

x�

1

5.f(

x) �

6.f(

x) �

ho

le:

x�

�2

ho

le:

x�

3

Gra

ph

eac

h r

atio

nal

fu

nct

ion

.

7.f(

x) �

8.f(

x) �

9.f(

x) �

10.f

(x)

�11

.f(x

) �

12.f

(x)

�

xO

f(x)

xO

f(x)

xO

f(x)

x2�

4� x

�2

x� x

�2

2� x

�1

xO

f(x)

xO

f(x) 2

2

xO

f(x)

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10 � x�

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x

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x�

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8x�

12�

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x�

12�

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�10

x�

24

10�

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x�

363

��

x2�

2x�

8

Skill

s Pr

acti

ceG

rap

hin

g R

atio

nal

Fu

nct

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-3

Copyright ©

Glencoe/M

cGraw


he McG

raw-H

ill Com

panies, Inc.



An

swer

s

Cop

yrig

ht ©

Gle

ncoe

/McG

raw

-Hill

, a

divi

sion

of T

he M

cGra

w-H

ill C

ompa

nies

, In

c.



Enri

chm

ent

Ch

arac

teri

stic

s o

f R

atio

nal

Fu

nct

ion

Gra

ph

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-3

Cha

pter

825

Gle

ncoe

Alg

ebra

2

Lesson 8-3

Use

th

e in

form

atio

n i

n t

he

tabl

e to

gra

ph r

atio

nal

fu

nct

ion

s

A s

ign

ch

art

use

s an

xva

lue

from

th

e le

ft a

nd

righ

t of

ea

ch c

riti

cal

valu

e to

det

erm

ine

if t

he

grap

h i

s po

siti

ve o

r n

egat

ive

on t

hat

inte

rval

.A s

ign

ch

art

for

y�

is s

how

n b

elow

.

Th

e gr

aph

of

is s

how

n

to t

he

righ

t.

Cre

ate

a si

gn c

har

t fo

r y

�.U

se a

n x

-val

ue

from

th

e le

ft a

nd

righ

t of

eac

h c

riti

cal

valu

e to

det

erm

ine

if t

he

grap

h i

s p

osit

ive

orn

egat

ive

on t

hat

in

terv

al.T

hen

gra

ph

th

e fu

nct

ion

.

y

x�

22

�3

�2

�1

��

��

01

23

x�

1� x

2�

4

x�

1�

�x2

�x

�6

y

x�

23

�3

�2

�1

��

��

01

23

4

x�

1�

�x2

�x

�6

CH

AR

AC

TE

RIS

TIC

ME

AN

ING

HO

W T

O F

IND

IT

Ver

tica

l asy

mp

tote

sA

ver

tical

line

at

an x

valu

e w

here

the

S

et t

he d

enom

inat

or e

qual

to

zero

and

ra

tiona

l fun

ctio

n is

und

efin

edso

lve

for

x.

Ho

rizo

nta

l asy

mp

tote

sA

hor

izon

tal l

ine

that

the

rat

iona

l S

tudy

the

end

-beh

avio

rs.

func

tion

Rig

ht

end

-beh

avio

rH

ow t

he g

raph

beh

aves

at

larg

e E

valu

ate

the

ratio

nal e

xpre

ssio

n at

po

sitiv

e va

lues

of

xin

crea

sing

pos

itive

val

ues

of x

.

Lef

t en

d-b

ehav

ior

How

the

gra

ph b

ehav

es a

t la

rge

Eva

luat

e th

e ra

tiona

l exp

ress

ion

at

nega

tive

valu

es o

f x

incr

easi

ng n

egat

ive

valu

es o

f x.

Ro

ots

,zer

os,

or

x-i

nte

rcep

tsP

oint

(s)

whe

re t

he g

raph

cro

sses

the

S

et t

he n

umer

ator

equ

al t

o ze

ro a

nd

x-ax

isso

lve

for

x.

y-i

nte

rcep

tsP

oint

whe

re t

he g

raph

cro

sses

the

S

et x

= 0

to

dete

rmin

e th

e y-

inte

rcep

t.y-

axis

Exam

ple

Exer

cise


Cha

pter

824

Gle

ncoe

Alg

ebra

2

Wor

d Pr

oble

m P

ract

ice

Gra

ph

ing

Rat

ion

al E

xpre

ssio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-3

1.R

OA

D T

RIP

Rob

ert

and

Sar

ah s

tart

of

f on

a r

oad

trip

fro

m t

he

sam

e h

ouse

.D

uri

ng

the

trip

,Rob

ert’s

an

d S

arah

’sca

rs r

emai

n s

epar

ated

by

a co

nst

ant

dist

ance

.Th

e gr

aph

sh

ows

the

rati

o of

the

dist

ance

Sar

ah h

as t

rave

led

to t

he

dist

ance

Rob

ert

has

tra

vele

d.T

he

dott

edli

ne

show

s h

ow t

his

gra

ph w

ould

be

exte

nde

d to

hyp

oth

etic

al n

egat

ive

valu

esof

x.W

hat

doe

s th

e x-

coor

din

ate

of t

he

vert

ical

asy

mpt

ote

repr

esen

t?

the

dis

tan

ce b

y w

hic

h S

arah

trai

ls R

ob

ert

2.G

RA

PHS

Alm

a gr

aph

ed t

he

fun

ctio

n

f(x)

�be

low

.

Th

ere

is a

pro

blem

wit

h h

er g

raph

.E

xpla

in h

ow t

o co

rrec

t it

.

Th

e p

oin

t (4

,4)

nee

ds

to b

eer

ased

an

d a

sm

all c

ircl

e p

ut

aro

un

d it

.

y

xO

x2�

4x�x

�4

y

xO

3.FI

NA

NC

EA

qu

ick

way

to

get

an i

dea

of h

ow m

any

year

s be

fore

a s

avin

gsac

cou

nt

wil

l do

ubl

e at

an

in

tere

st r

ate

of I

perc

ent

com

pou

nde

d an

nu

ally

,is

todi

vide

Iin

to 7

2.S

ketc

h a

gra

ph o

f th

e

fun

ctio

n f

(I)

�.

4.N

EWTO

NS

ir I

saac

New

ton

stu

died

th

e ra

tion

al f

un

ctio

n

f(x)

�.

Ass

um

ing

that

d�

0,w

her

e w

ill

ther

ebe

a v

erti

cal

asym

ptot

e to

th

e gr

aph

of

this

fu

nct

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?

x�

0

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TTIN

G A

VER

AG

ESF

or E

xerc

ises

5an

d 6

,use

th

e fo

llow

ing

info

rmat

ion

.

Josh

has

mad

e 26

hit

s in

80

at b

ats

for

a ba

ttin

g av

erag

e of

.325

.Jos

h g

oes

on a

hit

tin

g st

reak

an

d m

akes

xh

its

in t

he

nex

t2x

at b

ats.

5.W

hat

fu

nct

ion

des

crib

es J

osh

’s b

atti

ng

aver

age

duri

ng

this

str

eak?

f(x)

�

6.W

hat

is

the

equ

atio

n o

f th

e h

oriz

onta

las

ympt

ote

to t

he

grap

h o

f th

e fu

nct

ion

you

wro

te f

or E

xerc

ise

5? W

hat

is

its

mea

nin

g?

y�

0.5;

0.5

rep

rese

nts

an

up

per

bo

un

d o

n J

osh

’s b

atti

ng

ave

rag

eif

his

hit

rat

e d

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no

t ch

ang

e

26 �

x� 80

�2x

ax3

�bx

2�

cx�

d�

��

x

I

50

5O

f(I)

72 � I



NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-4

Cha

pter

827

Gle

ncoe

Alg

ebra

2

Lesson 8-4

Get

Rea

dy

for

the

Less

on

Rea

d t

he

intr

odu

ctio

n t

o L

esso

n 8

-4 i

n y

our

text

boo

k.

•F

or e

ach

add

itio

nal

stu

den

t w

ho

enro

lls

in a

pu

blic

col

lege

,th

e to

tal

hig

h-t

ech

spe

ndi

ng

wil

l (i

ncr

ease

/dec

reas

e) b

y .

•F

or e

ach

dec

reas

e in

en

roll

men

t of

100

stu

den

ts i

n a

pu

blic

col

lege

,th

e to

tal

hig

h-t

ech

spe

ndi

ng

wil

l (i

ncr

ease

/dec

reas

e) b

y .

Rea

d t

he

Less

on

1.W

rite

an

equ

atio

n t

o re

pres

ent

each

of

the

foll

owin

g va

riat

ion

sta

tem

ents

.Use

kas

th

eco

nst

ant

of v

aria

tion

.

a.m

vari

es i

nve

rsel

y as

n.

m�

b.

sva

ries

dir

ectl

y as

r.

s�

kr

c.t

vari

es jo

intl

y as

pan

d q.

t�

kpq

2.W

hic

h t

ype

of v

aria

tion

,dir

ect

or i

nve

rse,

is r

epre

sen

ted

by e

ach

gra

ph?

a.in

vers

eb

.d

irec

t

Rem

emb

er W

hat

Yo

u L

earn

ed

3.H

ow c

an y

our

know

ledg

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th

e eq

uat

ion

of

the

slop

e-in

terc

ept

form

of

the

equ

atio

n o

f a

lin

e h

elp

you

rem

embe

r th

e eq

uat

ion

for

dir

ect

vari

atio

n?

Sam

ple

an

swer

:Th

e g

rap

h o

f an

eq

uat

ion

exp

ress

ing

dir

ect

vari

atio

n is

alin

e.T

he

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pe-

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rm o

f th

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y�

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.In

dir

ect

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atio

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ne

of

the

qu

anti

ties

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e o

ther

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ty is

als

o 0

,so

b�

0 an

d t

he

line

go

es t

hro

ug

h t

he

ori

gin

.Th

e eq

uat

ion

of

a lin

eth

rou

gh

th

e o

rig

in is

y�

mx,

wh

ere

mis

th

e sl

op

e.T

his

is t

he

sam

e as

the

equ

atio

n f

or

dir

ect

vari

atio

n w

ith

k�

m.

x

y Ox

y

O

k � n

$20,

300

dec

reas

e

$203

incr

ease

Less

on R

eadi

ng G

uide

Dir

ect,

Join

t,an

d In

vers

e V

aria

tio

n

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

Lesson 8-4


Cha

pter

826

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-3

Th

e li

ne

y�

bis

a h

oriz

onta

l as

ympt

ote

for

the

rati

onal

fu

nct

ion

f(x

) if

f(

x)→

bas

x→

or

as

x→

�

.Th

e h

oriz

onta

l as

ympt

ote

can

be

fou

nd

by

usi

ng

the

TA

BL

Efe

atu

re o

f th

e gr

aph

ing

calc

ula

tor.

Fin

d t

he

hor

izon

tal

asym

pto

te f

or e

ach

fu

nct

ion

.

a.f(

x)�� x2

�41 x

�5

�

En

ter

the

fun

ctio

n in

to Y

1.P

lace

[T

blS

et]

in t

he

Ask

mod

e.E

nte

r th

en

um

bers

10,

000,

100,

000,

1,00

0,00

0,an

d 5,

000,

000

and

thei

r op

posi

tes

inth

e x-

list

.K

eyst

roke

s:1

4 5

[TB

LS

ET

] [T

AB

LE

].T

hen

en

ter

the

valu

es f

or x

.

Not

ice

that

as

xin

crea

ses,

yap

proa

ches

0.T

hu

s,y

�0

is t

he

hor

izon

tal a

sym

ptot

e.

b.f

(x)

�� 2x

2�3x

52 x�

6�

En

ter

the

equ

atio

n in

to Y

1.E

nte

r th

e n

um

bers

10,

000,

100,

000,

1,00

0,00

0,an

d 5,

000,

000

and

thei

r op

posi

tes

in t

he

x-li

st.N

ote

the

patt

ern

.As

xin

crea

ses,

yap

proa

ches

1.5

.Th

us,

y�

1.5

is t

he

hor

izon

tal a

sym

ptot

e.

2nd

EN

TER

2nd

)—

+x

2(

�Y

=

Fin

d t

he

hor

izon

tal

asym

pto

te f

or e

ach

fu

nct

ion

.

1.f(

x)�

� x2 �x

1�

y�

22.

f(x)

�� 2x

2x �2

7� x1 �

12�

y�

�1 2�3.

f(x)

�� 2x

3�

6 2x x3

2�

2�

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3

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x)�� 3x

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2 5x x�

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x)�

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0

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x)�

�5 xx2 ��23

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on

e8.

f(x)

�� 2x

2�6x

33 x�

6�

no

ne

9.f(

x)�

�2x2�

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no

ne

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3 �4x

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Gra

phin

g Ca

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ator

Act

ivit

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ori

zon

tal A

sym

pto

tes

and

Tab

les

Exer

cise

s

Exam

ple

Copyright ©

Glencoe/M

cGraw


he McG

raw-H

ill Com

panies, Inc.



An

swer

s

Cop

yrig

ht ©

Gle

ncoe

/McG

raw

-Hill

, a

divi

sion

of T

he M

cGra

w-H

ill C

ompa

nies

, In

c.



NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-4

Cha

pter

829

Gle

ncoe

Alg

ebra

2

Lesson 8-4

Inve

rse

Var

iati

on

Inve

rse

Var

iati

on

yva

ries

inve

rsel

y as

xif

ther

e is

som

e no

nzer

o co

nsta

nt k

such

tha

t xy

�k

or y

�.

If a

vari

es i

nve

rsel

y as

ban

d a

�8

wh

en b

�12

,fin

d a

wh

en b

�4.

�In

vers

e va

riatio

n

�a 1

�8,

b1

�12

, b 2

�4

8(12

) �

4a2

Cro

ss m

ultip

ly.

96 �

4a2

Sim

plify

.

24 �

a 2D

ivid

e ea

ch s

ide

by 4

.

Wh

en b

�4,

the

valu

e of

ais

24.

Fin

d e

ach

val

ue.

1.If

yva

ries

in

vers

ely

as x

and

y�

12 w

hen

x�

10,f

ind

yw

hen

x�

15.

8

2.If

yva

ries

in

vers

ely

as x

and

y�

100

wh

en x

�38

,fin

d y

wh

en x

�76

.50

3.If

yva

ries

in

vers

ely

as x

and

y�

32 w

hen

x�

42,f

ind

yw

hen

x�

24.

56

4.If

yva

ries

in

vers

ely

as x

and

y�

36 w

hen

x�

10,f

ind

yw

hen

x�

30.

12

5.If

yva

ries

in

vers

ely

as x

and

y�

18 w

hen

x�

124,

fin

d y

wh

en x

�93

.24

6.If

yva

ries

in

vers

ely

as x

and

y�

90 w

hen

x�

35,f

ind

yw

hen

x�

50.

63

7.If

yva

ries

in

vers

ely

as x

and

y�

42 w

hen

x�

48,f

ind

yw

hen

x�

36.

56

8.If

yva

ries

in

vers

ely

as x

and

y�

44 w

hen

x�

20,f

ind

yw

hen

x�

55.

16

9.If

yva

ries

in

vers

ely

as x

and

y�

80 w

hen

x�

14,f

ind

yw

hen

x�

35.

32

10.I

f y

vari

es i

nve

rsel

y as

xan

d y

�3

wh

en x

�8,

fin

d y

wh

en x

�40

.0.

6

11.I

f y

vari

es i

nve

rsel

y as

xan

d y

�16

wh

en x

�42

,fin

d y

wh

en x

�14

.48

12.I

f y

vari

es i

nve

rsel

y as

xan

d y

�23

wh

en x

�12

,fin

d y

wh

en x

�15

.18

.4

a 2� 12

8 � 4

a 2� b 1

a 1� b 2

k � x

Lesson 8-4

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nued

)

Dir

ect,

Join

t,an

d In

vers

e V

aria

tio

n

Exer

cise

s

Exam

ple


Cha

pter

828

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-4

Stud

y G

uide

and

Inte

rven

tion

Dir

ect,

Join

t,an

d In

vers

e V

aria

tio

nD

irec

t V

aria

tio

n a

nd

Jo

int

Var

iati

on

Dir

ect V

aria

tio

ny

varie

s di

rect

ly a

s x

if th

ere

is s

ome

nonz

ero

cons

tant

ksu

ch t

hat

y�

kx.k

is c

alle

d th

eco

nsta

nt o

f va

riatio

n.

Join

t Var

iati

on

yva

ries

join

tly a

s x

and

zif

ther

e is

som

e nu

mbe

r k

such

tha

t y

�kx

z, w

here

x�

0 an

d z

�0.

Fin

d e

ach

val

ue.

a.If

yva

ries

dir

ectl

y as

xan

d y

�16

wh

en x

�4,

fin

d x

wh

en y

�20

.

�D

irect

pro

port

ion

�y 1

�16

, x 1

�4,

and

y2

�20

16x 2

�(2

0)(4

)C

ross

mul

tiply

.

x 2�

5S

impl

ify.

Th

e va

lue

of x

is 5

wh

en y

is 2

0.

20 � x 2

16 � 4

y 2� x 2

y 1� x 1

b.

If y

vari

es j

oin

tly

as x

and

zan

d y

�10

wh

en x

�2

and

z �

4,fi

nd

yw

hen

x

�4

and

z�

3.

�Jo

int

varia

tion

�y 1

�10

, x 1

�2,

z 1

�4,

x2

�4,

an

d z 2

�3

120

�8y

2S

impl

ify.

y 2�

15D

ivid

e ea

ch s

ide

by 8

.

Th

e va

lue

of y

is 1

5 w

hen

x�

4 an

d z

�3.

y 2� 4

�3

10� 2

�4

y 2� x 2

z 2

y 1� x 1z

1

Fin

d e

ach

val

ue.

1.If

yva

ries

dir

ectl

y as

xan

d y

�9

wh

en

2.If

yva

ries

dir

ectl

y as

xan

d y

�16

wh

en

x�

6,fi

nd

yw

hen

x�

8.12

x�

36,f

ind

yw

hen

x�

54.

24

3.If

yva

ries

dir

ectl

y as

xan

d x

�15

4.

If y

vari

es d

irec

tly

as x

and

x�

33 w

hen

w

hen

y�

5,fi

nd

xw

hen

y�

9.27

y�

22,f

ind

xw

hen

y�

32.

48

5.S

upp

ose

yva

ries

join

tly

as x

and

z.6.

Su

ppos

e y

vari

es jo

intl

y as

xan

d z.

Fin

d y

Fin

d y

wh

en x

�5

and

z�

3,if

y�

18

wh

en x

�6

and

z�

8,if

y�

6 w

hen

x�

4w

hen

x�

3 an

d z

�2.

45an

d z

�2.

36

7.S

upp

ose

yva

ries

join

tly

as x

and

z.8.

Su

ppos

e y

vari

es jo

intl

y as

xan

d z.

Fin

d y

Fin

d y

wh

en x

�4

and

z�

11,i

f y

�60

w

hen

x�

5 an

d z

�2,

if y

�84

wh

en

wh

en x

�3

and

z�

5.17

6x

�4

and

z�

7.30

9.If

yva

ries

dir

ectl

y as

xan

d y

�39

10

.If

yva

ries

dir

ectl

y as

xan

d x

�60

wh

enw

hen

x�

52,f

ind

yw

hen

x�

22.

16.5

y�

75,f

ind

xw

hen

y�

42.

33.6

11.S

upp

ose

yva

ries

join

tly

as x

and

z.12

.Su

ppos

e y

vari

es jo

intl

y as

xan

d z.

Fin

d y

Fin

d y

wh

en x

�7

and

z�

18,i

f w

hen

x�

5 an

d z

�27

,if

y�

480

wh

en

y�

351

wh

en x

�6

and

z�

13.

567

x�

9 an

d z

�20

.36

0

Exer

cise

s

Exam

ple



NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-4

Cha

pter

831

Gle

ncoe

Alg

ebra

2

Lesson 8-4

Sta

te w

het

her

eac

h e

qu

atio

n r

epre

sen

ts a

dir

ect,

join

t,or

in

vers

eva

riat

ion

.Th

enn

ame

the

con

stan

t of

var

iati

on.

1.u

�8w

zjo

int;

82.

p�

4sd

irec

t;4

3.L

�

inve

rse;

54.

xy�

4.5

inve

rse;

4.5

5.�

6.

2d�

mn

7.�

h8.

y�

dir

ect;

�jo

int;

inve

rse;

1.25

inve

rse;

Fin

d e

ach

val

ue.

9.If

yva

ries

dir

ectl

y as

xan

d y

�8

wh

en x

�2,

fin

d y

wh

en x

�6.

24

10.I

f y

vari

es d

irec

tly

as x

and

y�

�16

wh

en x

�6,

fin

d x

wh

en y

��

4.1.

5

11.I

f y

vari

es d

irec

tly

as x

and

y�

132

wh

en x

�11

,fin

d y

wh

en x

�33

.39

6

12.I

f y

vari

es d

irec

tly

as x

and

y�

7 w

hen

x�

1.5,

fin

d y

wh

en x

�4.

13.I

f y

vari

es jo

intl

y as

xan

d z

and

y�

24 w

hen

x�

2 an

d z

�1,

fin

d y

wh

en x

�12

an

d z

�2.

288

14.I

f y

vari

es jo

intl

y as

xan

d z

and

y�

60 w

hen

x�

3 an

d z

�4,

fin

d y

wh

en x

�6

and

z�

8.24

0

15.I

f y

vari

es jo

intl

y as

xan

d z

and

y�

12 w

hen

x�

�2

and

z�

3,fi

nd

yw

hen

x�

4 an

d z

��

1.8

16.I

f y

vari

es i

nve

rsel

y as

xan

d y

�16

wh

en x

�4,

fin

d y

wh

en x

�3.

17.I

f y

vari

es i

nve

rsel

y as

xan

d y

�3

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Skill

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Copyright ©

Glencoe/M

cGraw


he McG

raw-H

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panies, Inc.



An

swer

s

Cop

yrig

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Gle

ncoe

/McG

raw

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, a

divi

sion

of T

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cGra

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, In

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8-4

Enri

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forc

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plac

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how

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how

th

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ed

efin

itio

n o

f ab

solu

te v

alu

e,f(

x)

�x

if x

�0

and

f(x

) �

�x

if x

�0.

Th

eref

ore

,th

e g

rap

h is

mad

e u

p o

f p

iece

s o

f tw

o li

nes

,on

e w

ith

slo

pe

1an

d o

ne

wit

h s

lop

e �

1,m

eeti

ng

at

the

ori

gin

.Th

is f

orm

s a

V-sh

aped

gra

ph

wit

h “

vert

ex”

at t

he

ori

gin

.

x

y

Ox

y Ox

y O

x

y

Ox

y

Ox

y

O


Cha

pter

834

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-4

You

have

lear

ned

to s

olve

pro

blem

s in

volv

ing

dire

ct,i

nver

se,a

nd jo

int

vari

atio

n.M

any

phys

ical

sit

uat

ion

s in

volv

e at

lea

st o

ne

of t

hes

e ty

pes

of v

aria

tion

.For

exam

ple,

acco

rdin

g to

New

ton

’s l

aw o

f u

niv

ersa

l gr

avit

atio

n,t

he

wei

ght

of a

mas

s n

ear

Ear

th d

epen

ds o

n t

he

dist

ance

bet

wee

n t

he

mas

s an

d th

e ce

nte

r of

Ear

th.S

tudy

th

e sp

read

shee

t be

low

to

dete

rmin

e th

e ty

pe o

f va

riat

ion

th

atex

ists

bet

wee

n t

he

quan

tity

of

an a

stro

nau

t’s w

eigh

t an

d th

e di

stan

ce o

f th

eas

tron

aut

from

th

e ce

nte

r of

Ear

th.

In t

he

spre

adsh

eet,

the

valu

es f

or t

he

astr

onau

t’s w

eigh

t in

new

ton

s ar

e en

tere

din

th

e ce

lls

in c

olu

mn

A,a

nd

the

valu

es f

or t

he

astr

onau

t’s d

ista

nce

in

met

ers

from

the

cen

ter

of E

arth

are

ent

ered

in c

ells

in c

olum

n B

.Col

umn

C c

onta

ins

the

astr

onau

t’s d

ista

nce

fro

m E

arth

’s s

urf

ace.

Spre

adsh

eet

Act

ivit

yV

aria

tio

n

1.U

se t

he

valu

es i

n t

he

spre

adsh

eet

to m

ake

a gr

aph

of

the

astr

onau

t’s w

eigh

t pl

otte

d ag

ain

st t

he

astr

onau

t’s

dist

ance

fro

m E

arth

’s c

ente

r.

2.B

ased

on

you

r gr

aph

,is

this

an

in

vers

e or

dir

ect

vari

atio

n?

inve

rse

3.W

rite

an

equ

atio

n t

hat

rep

rese

nts

th

is s

itua

tion

.Let

Wre

pres

ent

the

astr

onau

t’sw

eigh

t,k

the

con

stan

t of

vari

atio

n,a

nd

R t

he

dist

ance

fro

m E

arth

’s c

ente

r.

W�

� RK2�

4.U

se t

he

equ

atio

n t

o fi

nd

the

wei

ght

of t

he

astr

onau

t at

th

ese

dist

ance

s fr

om E

arth

’s s

urf

ace.

(Hin

t:R

emem

ber

to a

dd t

hes

e va

lues

to

the

valu

e in

cel

l B

2 to

fin

d th

e di

stan

ce f

rom

Ear

th’s

cen

ter.

)a.

145,

300,

000

mb

.65

mc.

25,6

00 m

1.29

9615

N73

4.54

94 N

728.

7047

N

d.

300,

800,

700

me.

6580

mf.

180,

560

m0.

3168

72 N

733.

0515

N69

4.68

73 N

A1 32 4 5 6 7

BC

Gra

vita

tio

n.x

ls 734.

5843

712.

0675

548.

9825

111.

4406

2.64

2112

6,38

0,00

06,

480,

000

7,38

0,00

016

,380

,000

106,

380,

000

010

010

0010

,000

100,

000

Ast

rona

ut’s

Wei

ght (

N)

Dis

tanc

e fr

om E

arth

’s C

ente

r (m

)D

ista

nce

from

Ear

th’s

Sur

face

(km

)

Sh

eet

1S

hee

t 2

Sh

eet

3

Exer

cise

s

Weight (N)

200

300

100 0

400

500

600

700

800

Dis

tan

ce (

mill

ion

s o

f m

eter

s)20

4060

100

Copyright ©

Glencoe/M

cGraw


he McG

raw-H

ill Com

panies, Inc.



An

swer

s

Cop

yrig

ht ©

Gle

ncoe

/McG

raw

-Hill

, a

divi

sion

of T

he M

cGra

w-H

ill C

ompa

nies

, In

c.



Exer

cise

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-5

Cha

pter

837

Gle

ncoe

Alg

ebra

2

Lesson 8-5 Lesson 8-5

Iden

tify

Eq

uat

ion

sYo

u sh

ould

be

able

to

grap

h th

e eq

uati

ons

of t

he f

ollo

win

g fu

ncti

ons.

Fu

nct

ion

Gen

eral

Eq

uat

ion

Co

nst

ant

y�

a

Dir

ect V

aria

tio

ny

�ax

Gre

ates

t In

teg

ereq

uatio

n in

clud

es a

var

iabl

e w

ithin

the

gre

ates

t in

tege

r sy

mbo

l, ��

Ab

solu

te V

alu

eeq

uatio

n in

clud

es a

var

iabl

e w

ithin

the

abs

olut

e va

lue

sym

bol,

||

Qu

adra

tic

y�

ax2

�bx

�c,

whe

re a

�0

Sq

uar

e R

oo

teq

uatio

n in

clud

es a

var

iabl

e be

neat

h th

e ra

dica

l sig

n, �

�

Rat

ion

aly

�

Inve

rse

Var

iati

on

y�

Iden

tify

th

e fu

nct

ion

rep

rese

nte

d b

y ea

ch e

qu

atio

n.T

hen

gra

ph

th

e eq

uat

ion

.

1.y

�in

vers

e va

riat

ion

2.y

�x

dir

ect

vari

atio

n3.

y�

�q

uad

rati

c

4.y

�| 3

x|�

1ab

solu

teva

lue

5.y

��

inve

rse

vari

atio

n6.

y�

gre

ates

tin

teg

er

7.y

��

x�

2�

squ

are

roo

t8.

y�

3.2

con

stan

t9.

y�

rati

on

al

x

y

Ox

y

Ox

y O

x2�

5x�

6�

�x

�2

x

y

Ox

y

Ox

y

O

x � 22 � x

x

y Ox

y

Ox

y

O

x2� 2

4 � 36 � x

a � xp(x

)� q

(x)

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nued

)

Cla

sses

of

Fu

nct

ion

s


Cha

pter

836

Gle

ncoe

Alg

ebra

2

Iden

tify

Gra

ph

sYo

u s

hou

ld b

e fa

mil

iar

wit

h t

he

grap

hs

of t

he

foll

owin

g fu

nct

ion

s.

Fu

nct

ion

Des

crip

tio

n o

f G

rap

h

Co

nst

ant

a ho

rizon

tal l

ine

that

cro

sses

the

y-a

xis

at a

Dir

ect V

aria

tio

na

line

that

pas

ses

thro

ugh

the

orig

in a

nd is

nei

ther

hor

izon

tal n

or v

ertic

al

Iden

tity

a lin

e th

at p

asse

s th

roug

h th

e po

int

(a,

a),

whe

re a

is a

ny r

eal n

umbe

r

Gre

ates

t In

teg

era

step

fun

ctio

n

Ab

solu

te V

alu

eV

-sha

ped

grap

h

Qu

adra

tic

a pa

rabo

la

Sq

uar

e R

oo

ta

curv

e th

at s

tart

s at

a p

oint

and

cur

ves

in o

nly

one

dire

ctio

n

Rat

ion

ala

grap

h w

ith o

ne o

r m

ore

asym

ptot

es a

nd/o

r ho

les

Inve

rse

Var

iati

on

a gr

aph

with

2 c

urve

d br

anch

es a

nd 2

asy

mpt

otes

, x

�0

and

y�

0 (s

peci

al c

ase

of r

atio

nal f

unct

ion)

Iden

tify

th

e fu

nct

ion

rep

rese

nte

d b

y ea

ch g

rap

h.

1.2.

3.

qu

adra

tic

rati

on

ald

irec

t va

riat

ion

4.5.

6.

con

stan

tab

solu

te v

alu

eg

reat

est

inte

ger

7.8.

9.

iden

tity

squ

are

roo

tin

vers

e va

riat

ionx

y

O

x

y O

x

y O

x

y

Ox

y

Ox

y

O

x

y

Ox

y O

x

y ONA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-5

Stud

y G

uide

and

Inte

rven

tion

Cla

sses

of

Fu

nct

ion

s

Exer

cise

s



NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-5

Cha

pter

839

Gle

ncoe

Alg

ebra

2

Lesson 8-5

Iden

tify

th

e ty

pe

of f

un

ctio

n r

epre

sen

ted

by

each

gra

ph

.

1.2.

3.

rati

on

alsq

uar

e ro

ot

abso

lute

val

ue

Mat

ch e

ach

gra

ph

wit

h a

n e

qu

atio

n b

elow

.

A.y

�| 2

x�

1|

B.y

��2

x�

1�C

.y�

D.y

��

�x

�

4.D

5.C

6.A

Iden

tify

th

e ty

pe

of f

un

ctio

n r

epre

sen

ted

by

each

eq

uat

ion

.Th

en g

rap

h t

he

equ

atio

n.

7.y

��

38.

y�

2x2

�1

9.y

�

con

stan

tq

uad

rati

cra

tio

nal

10.B

USI

NES

SA

sta

rtup

com

pany

use

s th

e fu

ncti

on P

�1.

3x2

�3x

�7

to p

redi

ct it

s pr

ofit

or

loss

du

rin

g it

s fi

rst

7 ye

ars

of o

pera

tion

.Des

crib

e th

e sh

ape

of t

he

grap

h o

f th

e fu

nct

ion

.T

he

gra

ph

is U

-sh

aped

;it

is a

par

abo

la.

11.P

AR

KIN

GA

par

kin

g lo

t ch

arge

s $1

0 to

par

k fo

r th

e fi

rst

day

or p

art

of a

day

.Aft

er t

hat

,it

ch

arge

s an

add

itio

nal

$8

per

day

or p

art

of a

day

.Des

crib

e th

e gr

aph

an

d fi

nd

the

cost

of p

arki

ng

for

6da

ys.

Th

e g

rap

h lo

oks

like

a s

erie

s o

f st

eps,

sim

ilar

to a

g

reat

est

inte

ger

fu

nct

ion

,bu

t w

ith

op

en c

ircl

es o

n t

he

left

an

d c

lose

dci

rcle

s o

n t

he

rig

ht;

$58.

1 � 2

x

y

Ox

y O

x

y

O

x2�

5x�

6�

�x

�2

x

y

O

x

y

Ox

y O

x�

3�

2

x

y

O

x

y O

x

y

O

Lesson 8-5

Prac

tice

Cla

sses

of

Fu

nct

ion

s


Cha

pter

838

Gle

ncoe

Alg

ebra

2

Iden

tify

th

e ty

pe

of f

un

ctio

n r

epre

sen

ted

by

each

gra

ph

.

1.2.

3.

con

stan

td

irec

t va

riat

ion

qu

adra

tic

Mat

ch e

ach

gra

ph

wit

h a

n e

qu

atio

n b

elow

.

A.

y�

|x�

1|B

.y�

C.y

��

1 �

x�

D.y

��x

��

1

4.B

5.C

6.A

Iden

tify

th

e ty

pe

of f

un

ctio

n r

epre

sen

ted

by

each

eq

uat

ion

.Th

en g

rap

h t

he

equ

atio

n.

7.y

�8.

y�

2�x�

9.y

��

3x

inve

rse

vari

atio

n

gre

ates

t in

teg

erd

irec

t va

riat

ion

or

rati

on

al

x

y

Ox

y

Ox

O

y

2 � x

x

y

O

x

y Ox

y

O

1� x

�1

x

y O

x

y Ox

y O

8-5

Skill

s Pr

acti

ceC

lass

es o

f F

un

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_ Copyright ©

Glencoe/M

cGraw


he McG

raw-H

ill Com

panies, Inc.



An

swer

s

Cop

yrig

ht ©

Gle

ncoe

/McG

raw

-Hill

, a

divi

sion

of T

he M

cGra

w-H

ill C

ompa

nies

, In

c.



Enri

chm

ent

Phy

sica

l Pro

per

ties

of

Fu

nct

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-5

Cha

pter

841

Gle

ncoe

Alg

ebra

2

Lesson 8-5

Mat

hem

atic

al f

un

ctio

ns

are

clas

sifi

ed b

ased

on

pro

pert

ies

sim

ilar

to

how

bio

logi

sts

clas

sify

anim

al s

peci

es.F

un

ctio

ns

can

be

clas

sifi

ed a

s co

nti

nu

ous

or n

on-c

onti

nu

ous,

incr

easi

ng

orde

crea

sin

g,po

lyn

omia

l or

non

-pol

ynom

ial

for

exam

ple.

Th

e cl

ass

of p

olyn

omia

ls f

un

ctio

ns

can

be

furt

her

cla

ssifi

ed a

s li

nea

r,qu

adra

tic,

cubi

c,et

c.,b

ased

on

its

deg

ree.

Ch

arac

teri

stic

s of

fu

nct

ion

s in

clu

de:

•A

fu

nct

ion

is

bou

nd

ed b

elow

if t

her

e ex

ists

a n

um

ber

that

is

less

th

an a

ny

fun

ctio

nva

lue.

•A

fu

nct

ion

is

bou

nd

ed a

bov

eif

a n

um

ber

exis

ts t

hat

is

grea

ter

than

an

y fu

nct

ion

valu

e.•

A f

un

ctio

n i

s sy

mm

etri

c(a

bou

t a

vert

ical

axi

s) i

f it

is

a m

irro

r im

age

abou

t th

atve

rtic

al a

xis.

•A

fu

nct

ion

is

con

tin

uou

sif

it

can

be

draw

n w

ith

out

lift

ing

you

r pe

nci

l.•

A f

un

ctio

n i

s in

crea

sin

gif

f(x

)

f(y)

wh

en x

y.

Con

tin

ual

gro

wth

fro

m l

eft

to r

igh

t.•

A f

un

ctio

n i

s d

ecre

asin

gif

f(x

) �

f(y)

wh

en x

�y.

Con

tin

ual

dec

ay f

rom

lef

t to

rig

ht.

1.S

ketc

h t

he

grap

h o

f y

�x2

�5x

�6.

Lis

t th

e ch

arac

teri

stic

s of

fu

nct

ion

s di

spla

yed

by t

his

gra

ph.

So

me

pro

per

ties

incl

ud

e:sy

mm

etri

c,co

nti

nu

ou

s,b

ou

nd

ed b

elo

w.

2.W

hat

ch

arac

teri

stic

s do

abs

olu

te v

alu

e fu

nct

ion

s an

d qu

adra

tic

fun

ctio

ns

hav

e in

com

mon

? H

ow d

o th

ey d

iffe

r?

Co

mm

on

:S

ymm

etri

c,co

nti

nu

ou

s,b

ou

nd

ed

bel

ow

(o

r ab

ove)

.Dif

fer:

On

e is

U s

hap

ed

and

th

e o

ther

V s

hap

ed,o

ne

is s

mo

oth

an

d o

ne

has

a r

igid

co

rner

,an

d o

ne

incr

ease

s m

ore

rap

idly

th

an t

he

oth

er.

3.G

raph

y�

⏐x�

3 ⏐.

4.G

raph

y�

x2�

8x�

7.

y

x

y

x

y

x

Exer

cise

s


Cha

pter

840

Gle

ncoe

Alg

ebra

2

Wor

d Pr

oble

m P

ract

ice

C

lass

es o

f F

un

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

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ER

IOD

____

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8-5

1.ST

AIR

SW

hat

typ

e of

a f

un

ctio

n h

as

a gr

aph

th

at c

ould

be

use

d to

mod

el

a st

airc

ase?

the

gre

ates

t in

teg

er f

un

ctio

n

2.G

OLF

BA

LLS

Th

e tr

ajec

tory

of

a go

lfba

ll h

it b

y an

ast

ron

aut

on t

he

moo

n

is d

escr

ibed

by

the

fun

ctio

n

f(x)

��

0.01

25(x

�40

)2�

20.

Des

crib

e th

e sh

ape

of t

his

tra

ject

ory.

a p

arab

ola

3.R

AV

INE

Th

e gr

aph

sh

ows

the

cros

s-se

ctio

n o

f a

ravi

ne.

Wh

at k

ind

of f

un

ctio

n i

s re

pres

ente

d by

the

grap

h?

Wri

te t

he

fun

ctio

n.

an a

bso

lute

val

ue

fun

ctio

n;

f(x)

�⏐x

�2 ⏐

�2

4.LE

AK

Y F

AU

CET

SA

lea

ky f

auce

t le

aks

1 m

illi

lite

r of

wat

er e

very

sec

ond.

Wri

te a

fu

nct

ion

th

at g

ives

th

e n

um

ber

of m

illi

lite

rs l

eake

d in

tse

con

ds a

s a

fun

ctio

n o

f t.

Wh

at t

ype

of f

un

ctio

n i

s it

?

f(t)

�t;

an id

enti

ty f

un

ctio

n

y

xO

y

x80

21 O

PUB

LISH

ING

For

Exe

rcis

es 5

-8,u

se t

he

foll

owin

g in

form

atio

n.

Kat

e h

as ju

st fi

nis

hed

wri

tin

g a

book

th

atex

plai

ns

how

to

sew

you

r ow

n s

tuff

edan

imal

s.S

he

hop

es t

o m

ake

$100

0 fr

omsa

les

of t

he

book

bec

ause

th

at i

s h

ow m

uch

it w

ould

cos

t h

er t

o go

to

the

Eu

rope

anS

ewin

g C

onve

nti

on.E

ach

boo

k co

sts

$2 t

opr

int

and

asse

mbl

e.L

et P

be t

he

sell

ing

pric

e of

th

e bo

ok.L

et N

be t

he

nu

mbe

r of

peop

le w

ho

wil

l bu

y th

e bo

ok.

5.W

rite

an

equ

atio

n t

hat

rel

ates

Pan

d N

if s

he

earn

s ex

actl

y $1

,000

fro

m s

ales

of

the

book

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1000

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6.S

olve

th

e eq

uat

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you

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te f

orE

xerc

ise

5 fo

r P

in t

erm

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hat

kin

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fu

nct

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ter

ms

ofN

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ketc

h a

gra

ph o

f P

as a

fu

nct

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of

N.

rati

on

al;

8.If

Kat

e th

inks

th

at 1

25 p

eopl

e w

ill

buy

her

boo

k,h

ow m

uch

sh

ould

sh

e ch

arge

for

the

book

?

$10Sale Price

050100

Nu

mb

er o

f B

uye

rs10

0

2N�

1000

�� N



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ME

____

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____

____

____

____

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ATE

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Cha

pter

843

Gle

ncoe

Alg

ebra

2

Lesson 8-6

Stud

y G

uide

and

Inte

rven

tion

So

lvin

g R

atio

nal

Eq

uat

ion

s an

d In

equ

alit

ies

8-6

Lesson 8-6

Solv

e R

atio

nal

Eq

uat

ion

sA

rat

ion

al e

qu

atio

nco

nta

ins

one

or m

ore

rati

onal

expr

essi

ons.

To

solv

e a

rati

onal

equ

atio

n,f

irst

mu

ltip

ly e

ach

sid

e by

th

e le

ast

com

mon

den

omin

ator

of

all

of t

he

den

omin

ator

s.B

e su

re t

o ex

clu

de a

ny

solu

tion

th

at w

ould

pro

duce

a de

nom

inat

or o

f ze

ro.

Sol

ve

��

.

��

Orig

inal

equ

atio

n

10(x

�1)

��

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(x�

1)�

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tiply

eac

h si

de b

y 10

(x�

1).

9(x

�1)

�2(

10)

�4(

x�

1)M

ultip

ly.

9x�

9 �

20 �

4x�

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istr

ibut

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pert

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5x�

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Sub

trac

t 4x

and

29

from

eac

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de.

x�

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ide

each

sid

e by

5.

Ch

eck

��

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inal

equ

atio

n

��

x�

�5

��

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plify

.

�

Sol

ve e

ach

eq

uat

ion

.

1.�

�2

52.

��

12

3.�

��

4.�

�4

�5.

�

76.

��

10

7.N

AV

IGA

TIO

NT

he

curr

ent

in a

riv

er i

s 6

mil

es p

er h

our.

In h

er m

otor

boat

Mar

issa

can

trav

el 1

2 m

iles

ups

trea

m o

r 16

mil

es d

own

stre

am i

n t

he

sam

e am

oun

t of

tim

e.W

hat

is

the

spee

dof

her

mot

orbo

atin

stil

lwat

er?

Isth

isa

reas

onab

lean

swer

?E

xpla

in.

42m

ph

;S

amp

le a

nsw

er:T

he

answ

er is

rea

son

able

.Th

e b

oat

will

tra

vel

48 m

ph

on

e w

ay a

nd

36

mp

h t

he

oth

er w

ay.T

her

efo

re it

will

tak

e o

f an

ho

ur

to t

rave

l 16

mile

s an

d 1

2 m

iles,

resp

ecti

vely

.

8.W

OR

KA

dam

,Bet

han

y,an

d C

arlo

s ow

n a

pai

nti

ng

com

pan

y.T

o pa

int

a pa

rtic

ula

r h

ouse

alon

e,A

dam

est

imat

es t

hat

it

wou

ld t

ake

him

4 d

ays,

Bet

han

y es

tim

ates

5da

ys,a

nd

Car

los

6 da

ys.I

f th

ese

esti

mat

es a

re a

ccu

rate

,how

lon

g sh

ould

it

take

th

e th

ree

of t

hem

to p

ain

t th

e h

ouse

if

they

wor

k to

geth

er?

Isth

isa

reas

onab

lean

swer

?ab

ou

t 1

day

s;S

amp

le a

nsw

er:

It is

a r

easo

nab

le a

nsw

er.I

t w

ill t

ake

each

per

son

abo

ut

5 d

ays

to p

ain

t th

e h

ou

se a

lon

e,so

it s

ho

uld

tak

e ab

ou

t o

f th

eti

me

to p

ain

t th

e h

ou

se t

og

eth

er.

1 � 3

2 � 3

1 � 2

1 � 3

8 � 34

� x�

2x

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2x

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x�

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42x

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19 � 10

2 � 52

� x�

19 � 10

2 � 52

� x�

19 � 10

Exer

cise

s

Exam

ple


Cha

pter

842

Gle

ncoe

Alg

ebra

2

Less

on R

eadi

ng G

uide

S

olv

ing

Rat

ion

al E

qu

atio

ns

and

Ineq

ual

itie

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-6

Get

Rea

dy

for

the

Less

on

Rea

d t

he

intr

odu

ctio

n t

o L

esso

n 8

-6 i

n y

our

text

boo

k.

•If

you

in

crea

se t

he

nu

mbe

r of

son

gs t

hat

you

dow

nlo

ad,w

ill

you

r to

tal

bill

in

crea

se o

rde

crea

se?

incr

ease

•W

ill

you

r ac

tual

cos

t pe

r so

ng

incr

ease

or

decr

ease

?d

ecre

ase

Rea

d t

he

Less

on

1.W

hen

sol

vin

g a

rati

onal

equ

atio

n,a

ny

poss

ible

sol

uti

on t

hat

res

ult

s in

in

th

e de

nom

inat

or m

ust

be

excl

ude

d fr

om t

he

list

of

solu

tion

s.

2.S

upp

ose

that

on

a q

uiz

you

are

ask

ed t

o so

lve

the

rati

onal

in

equ

alit

y �

0.

Com

plet

e th

e st

eps

of t

he

solu

tion

.

Ste

p 1

Th

e ex

clu

ded

valu

es a

re

and

.

Ste

p 2

Th

e re

late

d eq

uat

ion

is

��

0 .

To s

olve

thi

s eq

uati

on,m

ulti

ply

both

sid

es b

y th

e L

CD

,whi

ch i

s .

Sol

vin

g th

is e

quat

ion

wil

l sh

ow t

hat

th

e on

ly s

olu

tion

is

�4.

Ste

p 3

Div

ide

a n

um

ber

lin

e in

to

regi

ons

usi

ng

the

excl

ude

d va

lues

an

d th

eso

luti

on o

f th

e re

late

d eq

uat

ion

.Dra

w d

ash

ed v

erti

cal

lin

es o

n t

he

nu

mbe

r li

ne

belo

w t

o sh

ow t

hes

e re

gion

s.

Con

side

r th

e fo

llow

ing

valu

es o

f �

for

vari

ous

test

val

ues

of

z.

If z

��

5,�

�0.

2.If

z�

�3,

��

�1.

If z

��

1,�

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If z

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��

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ng

this

in

form

atio

n a

nd

you

r n

um

ber

lin

e,w

rite

th

e so

luti

on o

f th

e in

equ

alit

y.

z�

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or

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emb

er W

hat

Yo

u L

earn

ed3.

How

are

th

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oces

ses

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ddin

g ra

tion

al e

xpre

ssio

ns

wit

h d

iffe

ren

t de

nom

inat

ors

and

ofso

lvin

g ra

tion

al e

xpre

ssio

ns

alik

e,an

d h

ow a

re t

hey

dif

fere

nt?

Sam

ple

an

swer

:Th

eyar

e al

ike

bec

ause

bo

th u

se t

he

LC

D o

f al

l th

e ra

tio

nal

exp

ress

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s in

th

ep

rob

lem

.Th

ey a

re d

iffe

ren

t b

ecau

se in

an

ad

dit

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pro

ble

m,t

he

LC

Dre

mai

ns

afte

r th

e fr

acti

on

s ar

e ad

ded

,wh

ile in

so

lvin

g a

rat

ion

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uat

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CD

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limin

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23

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6 � z3

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2

6 � z3

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0

Copyright ©

Glencoe/M

cGraw


he McG

raw-H

ill Com

panies, Inc.



An

swer

s

Cop

yrig

ht ©

Gle

ncoe

/McG

raw

-Hill

, a

divi

sion

of T

he M

cGra

w-H

ill C

ompa

nies

, In

c.



NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-6

Cha

pter

845

Gle

ncoe

Alg

ebra

2

Lesson 8-6

Skill

s Pr

acti

ceS

olv

ing

Rat

ion

al E

qu

atio

ns

and

Ineq

ual

itie

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

Lesson 8-6

Sol

ve e

ach

eq

uat

ion

or

ineq

ual

ity.

Ch

eck

you

r so

luti

ons.

1.�

�1

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3.�

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4.3

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2

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56.

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3 o

r 0

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5 � v3 � v

4 � 3k3 � k

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4 � n1 � 2

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Lesson 8-6


Cha

pter

844

Gle

ncoe

Alg

ebra

2

Solv

e R

atio

nal

Ineq

ual

itie

sTo

sol

ve a

rat

iona

l ine

qual

ity,

com

plet

e th

e fo

llow

ing

step

s.

Ste

p 1

Sta

te t

he e

xclu

ded

valu

es.

Ste

p 2

Sol

ve t

he r

elat

ed e

quat

ion.

Ste

p 3

Use

the

val

ues

from

ste

ps 1

and

2 t

o di

vide

the

num

ber

line

into

reg

ions

.Tes

t a

valu

e in

eac

h re

gion

to

see

whi

ch r

egio

ns s

atis

fy t

he o

rigin

al in

equa

lity.

Sol

ve

�

.

Ste

p 1

Th

e va

lue

of 0

is

excl

ude

d si

nce

th

is v

alu

e w

ould

res

ult

in

a d

enom

inat

or o

f 0.

Ste

p 2

Sol

ve t

he

rela

ted

equ

atio

n.

��

Rel

ated

equ

atio

n

15n�

��

15n�

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ultip

ly e

ach

side

by

15n.

10 �

12 �

10n

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plify

.

22 �

10n

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plify

.

2.2

�n

Sim

plify

.

Ste

p 3

Dra

w a

nu

mbe

r w

ith

ver

tica

l li

nes

at

the

ex

clu

ded

valu

e an

d th

e so

luti

on t

o th

e eq

uat

ion

.

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t n

��

1.T

est

n�

1.T

est

n�

3.

��

��

is t

rue.

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is n

ottr

ue.

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rue.

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e so

luti

on i

s n

�0

or n

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2.

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ve e

ach

in

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alit

y.

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r 0

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r �

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r 0

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r x

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1 � 2

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� x�

11 � 4

2 � x3 � 2x

39 � 201 � 2

1 � 2

2 � 34 � 5p

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2 � 92 � 3

4 � 52 � 3

2 � 34 � 5

2 � 3

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2 � 3n

2 � 34 � 5n

2 � 3n

2 � 34 � 5n

2 � 3n

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-6

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nued

)

So

lvin

g R

atio

nal

Eq

uat

ion

s an

d In

equ

alit

ies

Exer

cise

s

Exam

ple



Wor

d Pr

oble

m P

ract

ice

So

lvin

g R

atio

nal

Eq

uat

ion

s an

d In

equ

alit

ies

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-6

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Glencoe/M

cGraw


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raw-H

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ncoe

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raw

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, a

divi

sion

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cGra

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Cha

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2

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sym

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e gr

aph

of

y�

ax�

b,w

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call

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y�

f(x)

if

th

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aph

of

fco

mes

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as x

→ ∞

or x

→ �

∞.∞

is t

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mat

hem

atic

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ymbo

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r in

fin

ity,

wh

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mea

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less

.

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f(x

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3x�

4 �

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�3x

�4

is a

n o

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asym

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e be

cau

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f(x)

�3x

�4

��2 x� ,

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0 a

s x

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ympt

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____

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IOD

____

_

8-6

Exam

ple


Copyright ©

Glencoe/M

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he McG

raw-H

ill Com

panies, Inc.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Quiz 2 (Lesson 8–3)

Page 51

1.

2.

3.

4.5.

1.

2.

3.

4.

5.

Quiz 4 (Lesson 8–6)

Page 52

1.

2.

3.

4.

5.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12. 7

8

asymptote: x � 4;hole: x � �3

21(3c � 5)

72s3t4

B

H

A

F

D

18

�5 � m � �2

t � 0 or t � 2

9

10

absolute value

greatest integer

quadratic

12

inverse; 30

hole: x � 3

hole: x � �4

asymptote: x � 1;hole: x � �3

asymptotes:x � �2, x � 1

�y

1�2

3�

�35

5m�

22nm

�

(t � 1)(2t � 1)(t � 4)

15(x � 2)(x � 2)60a2b3

E

�p5

�

(2x � 3)(x � 6)

�x �

33

�

�8ax2

5�

Chapter 8 Assessment Answer Key Quiz 1 (Lessons 8–1 and 8–2) Quiz 3 (Lessons 8–4 and 8–5) Mid-Chapter TestPage 51 Page 52 Page 53

xO

f (x) � 4(x � 2)2

f (x)

xO

f (x)f (x) � 4

x � 3

y

xO

y � 3 x � 2 x2 � 5x � 3��(x � 3)(x � 2)(x � 4)


An

swer

s

Cop

yrig

ht ©

Gle

ncoe

/McG

raw

-Hill

, a

divi

sion

of T

he M

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w-H

ill C

ompa

nies

, In

c.

1. inverse variation

2. rational function

3. asymptote

4. least commondenominator

5. joint variation

6. continuity

7. rational inequality

8. inverselyproportional

9. constant ofvariation

10. point discontinuity

11. Sample answer: Arational expressionis the ratio of twopolynomials. Thedenominator cannotbe equal to 0.

12. Sample answer: Acomplex fraction isa fraction in whichthe numerator,denominator, orboth, containfractions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:asymptote: x � 0;

hole: x � 3

F

A

H

D

G

A

G

C

G

A

J

C

H

A

G

D

G

A

F

C

Chapter 8 Assessment Answer Key Vocabulary Test Form 1Page 54 Page 55 Page 56


Copyright ©

Glencoe/M

cGraw


he McG

raw-H

ill Com

panies, Inc.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:�x �

x1

�

G

A

G

C

H

D

H

B

F

B

F

C

F

D

G

C

J

A

J

B

�x �

x1

�

G

C

F

B

J

A

J

C

H

A

H

A

J

B

H

B

F

D

G

C

Chapter 8 Assessment Answer Key Form 2A Form 2BPage 57 Page 58 Page 59 Page 60


An

swer

s

Cop

yrig

ht ©

Gle

ncoe

/McG

raw

-Hill

, a

divi

sion

of T

he M

cGra

w-H

ill C

ompa

nies

, In

c.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B: �

4.8 h

m � 0 or m � 5

1

direct variation

square root

P � �Ak

�

1

192 customers

15

hole: x � �2

asymptote: x � 3

(n � 2)(n � 4)(n � 6)

36m4n4

�3m

7� 1�

�x �

22

�

�9(m

8� 5)�

�b �

25

�

�x �

x8

�

��32

�, 3

Chapter 8 Assessment Answer Key Form 2CPage 61 Page 62

xO

f (x) f (x) � x � 3x � 2


Copyright ©

Glencoe/M

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raw-H

ill Com

panies, Inc.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B: �

48 min

0 � r � 6

��13

�

inverse variation

constant

R � �Uk

�

9

1050 permits

63

xO

f (x)f (x) � x

x � 2

asymptotes:x � �5, x � �2

asymptote: x � �4;hole: x � 6

(n � 1)(n � 5)(n � 2)

42s3t4

�2n

5� 1�

�x �

33

�

�2(y

3� 2)�

�m

6�m

1�

�x

x�

2

5�

�2, �52

�

Chapter 8 Assessment Answer Key Form 2DPage 63 Page 64


An

swer

s

Cop

yrig

ht ©

Gle

ncoe

/McG

raw

-Hill

, a

divi

sion

of T

he M

cGra

w-H

ill C

ompa

nies

, In

c.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

��x2

3(2xx��

23)

�;

x � ��23

�, 0, �32

�

�1255�

z � �1 or �1� z � 1

x � 0 or x � �32

�

�

10

inverse variation

rational

I � �7R.2�; 40

271 mi

0.02

�110�

hole: x � �2

asymptote: x � 3

(c � d)(c � d)2

�1

0

�44mm

��

33nn

�

�g �

53

�

�3x((23xx

��

35))

�

��13

�, 0, �52

�

Chapter 8 Assessment Answer Key Form 3Page 65 Page 66

xO

f (x) �

f (x)

�2 (x � 3)2

xO

f (x) � x2�42x � 4

f (x)


Copyright ©

Glencoe/M

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raw-H

ill Com

panies, Inc.Chapter 8 Assessment Answer Key

Page 67, Extended-Response TestSample Answers

1. Each student response must includethree expressions which, when

simplified, are equivalent to �a �a

5�.

Sample answer: �3a3�a

15�, �a2a�

2

5a�,

�(a �a(a

5)�(a

1�)

1)�.

2a. Students should explain that the heightcan be found by dividing the volume bythe product of the length and width ofthe box.

2b. (x � 3) in.2c. Sample answer: Substitute a value for x

in each of the given expressions for thelength, width, and volume, and the samevalue for x in the expression found for h,and then check that V � �wh.CHECK: For x � 5,length � (5) � 10 � 15 in.width � 2(5) � 10 in.volume � 2(5)3 � 26(5)2 � 60(5)

� 1200 in3

height � (5) � 3 � 8 in.Verify V � �wh: 1200 � (15)(10)(8) ✓

3. Each student response must include twopolynomials in which 3, y � 2, and y � 2each appears as a factor of at least oneof those polynomials, but which have noother factor. Sample answer:y2 � 4, 3(y � 2).

4. Student responses should indicate thatthe graph of f(x) has a hole at x � �2,but no vertical asymptote. Its graph is astraight line with a hole in it at(�2, �5). The graph of g(x) also has ahole at x � �2, but has a verticalasymptote at x � 0. Its graph is not astraight line, but two curves having a

hole in the graph at ��2, �52��.

5a. d � 0.10hr5b. joint variation; the amount deducted

varies directly as the product of two quantities, the hourly wage and thenumber of hours worked.

5c. Students should indicate that theyshould substitute r � 9.50 and h � 36 in the formula they wrote inpart a.The amount deducted was $34.20.

6a. Students should conclude that C(x)is a rational function since it is of

the form y � �pq(

(xx))

�, where

p(x) � 60x � 17,000 and q(x) � x � 50are polynomial functions.

6b. Students should indicate that R(x) is aconstant function since it is of the formy � a, where a is any number.

6c. 80 �60x

x��

1570,000

�

6d. x 1050; The company must produceand sell at least 1050 CD players inorder to ensure that the revenue fromeach one is greater than the averagecost of producing each one.

In addition to the scoring rubric found on page 50, the following sample answers may be used as guidance in evaluating open-ended assessment items.


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Chapter 8 Assessment Answer KeyStandardized Test PracticePage 68 Page 69


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Chapter 8 Assessment Answer KeyStandardized Test PracticePage 70

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