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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.
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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 ©
Glencoe/M
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he McG
raw-H
ill Com
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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8
Chapter 8 2 Glencoe Algebra 2
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
Chapter 8 3 Glencoe Algebra 2
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
Chapter 8 5 Glencoe Algebra 2
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
Chapter 8 6 Glencoe Algebra 2
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. �
� � �
� �
Simplify each expression.
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 _____
Chapter 8 7 Glencoe Algebra 2
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
Chapter 8 8 Glencoe Algebra 2
Simplify each expression.
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
Chapter 8 9 Glencoe Algebra 2
Less
on
8-1
Simplify each expression.
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
Chapter 8 10 Glencoe Algebra 2
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
Chapter 8 11 Glencoe Algebra 2
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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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Chapter 8 12 Glencoe Algebra 2
NAME ______________________________________________ DATE______________ PERIOD _____
Lesson Reading GuideAdding and Subtracting Rational Expressions
8-2
Get Ready for the LessonRead the introduction to Lesson 8-2 in your textbook.
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 .
Remember What You Learned
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
Chapter 8 13 Glencoe Algebra 2
Less
on
8-2
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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Study Guide and Intervention (continued)
Adding and Subtracting Rational Expressions
Chapter 8 14 Glencoe Algebra 2
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.
Simplify each expression.
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 _____
Chapter 8 15 Glencoe Algebra 2
Less
on
8-2
Find the LCM of each set of polynomials.
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
Simplify each expression.
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
Less
on
8-2
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PracticeAdding and Subtracting Rational Expressions
Chapter 8 16 Glencoe Algebra 2
Find the LCM of each set of polynomials.
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)
Simplify each expression.
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
Chapter 8 17 Glencoe Algebra 2
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8-2
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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Chapter 8 18 Glencoe Algebra 2
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
Chapter 8 19 Glencoe Algebra 2
Less
on
8-3
Get Ready for the LessonRead the introduction to Lesson 8-3 in your textbook.
• 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) �
Remember What You Learned
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
Chapter 8 20 Glencoe Algebra 2
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 _____
8-3
Chapter 8 21 Glencoe Algebra 2
Less
on
8-3
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
Study Guide and Intervention (continued)
Graphing Rational Functions
Exercises
Example
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he McG
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ill Com
panies, Inc.
Chapter 8 22 Glencoe Algebra 2
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) �
Graph each rational function.
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
Chapter 8 23 Glencoe Algebra 2
Less
on
8-3
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) �
Graph each rational function.
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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panies, Inc.
Chapter 8 24 Glencoe Algebra 2
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
Chapter 8 25 Glencoe Algebra 2
Less
on
8-3
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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-Hill, a division of T
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Chapter 8 26 Glencoe Algebra 2
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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NAME ______________________________________________ DATE______________ PERIOD _____
8-4
Chapter 8 27 Glencoe Algebra 2
Less
on
8-4
Get Ready for the LessonRead the introduction to Lesson 8-4 in your textbook.
• 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.
Remember What You Learned
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
on
8-4
Copyright ©
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-Hill, a division of T
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raw-H
ill Com
panies, Inc.
Chapter 8 28 Glencoe Algebra 2
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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NAME ______________________________________________ DATE______________ PERIOD _____
8-4
Chapter 8 29 Glencoe Algebra 2
Less
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8-4
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.
2. If y varies inversely as x and y � 100 when x � 38, find y when x � 76.
3. If y varies inversely as x and y � 32 when x � 42, find y when x � 24.
4. If y varies inversely as x and y � 36 when x � 10, find y when x � 30.
5. If y varies inversely as x and y � 18 when x � 124, find y when x � 93.
6. If y varies inversely as x and y � 90 when x � 35, find y when x � 50.
7. If y varies inversely as x and y � 42 when x � 48, find y when x � 36.
8. If y varies inversely as x and y � 44 when x � 20, find y when x � 55.
9. If y varies inversely as x and y � 80 when x � 14, find y when x � 35.
10. If y varies inversely as x and y � 3 when x � 8, find y when x � 40.
11. If y varies inversely as x and y � 16 when x � 42, find y when x � 14.
12. If y varies inversely as x and y � 23 when x � 12, find y when x � 15.
a2�12
8�4
a2�b1
a1�b2
k�x
Less
on
8-4
Study Guide and Intervention (continued)
Direct, Joint, and Inverse Variation
Exercises
Example
Copyright ©
Glencoe/M
cGraw
-Hill, a division of T
he McG
raw-H
ill Com
panies, Inc.
Chapter 8 30 Glencoe Algebra 2
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.
14. If y varies directly as x and y � 360 when x � 180, find y when x � 270.
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.
19. If y varies jointly as x and z and y � 120 when x � 4 and z � 6, find y when x � 3 and z � 2.
20. If y varies inversely as x and y � 2 when x � 2, find y when x � 1.
21. If y varies inversely as x and y � 6 when x � 5, find y when x � 10.
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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NAME ______________________________________________ DATE______________ PERIOD _____
8-4
Chapter 8 31 Glencoe Algebra 2
Less
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8-4
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.
9. If y varies directly as x and y � 8 when x � 2, find y when x � 6.
10. If y varies directly as x and y � �16 when x � 6, find x when y � �4.
11. If y varies directly as x and y � 132 when x � 11, find y when x � 33.
12. If y varies directly as x and y � 7 when x � 1.5, find y when x � 4.
13. If y varies jointly as x and z and y � 24 when x � 2 and z � 1, find y when x � 12 and z � 2.
14. If y varies jointly as x and z and y � 60 when x � 3 and z � 4, find y when x � 6 and z � 8.
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.
16. If y varies inversely as x and y � 16 when x � 4, find y when x � 3.
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
Less
on
8-4
PracticeDirect, Joint, and Inverse Variation
Copyright ©
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cGraw
-Hill, a division of T
he McG
raw-H
ill Com
panies, Inc.
Chapter 8 32 Glencoe Algebra 2
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 _____
Chapter 8 33 Glencoe Algebra 2
Less
on
8-4
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
Copyright ©
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-Hill, a division of T
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raw-H
ill Com
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Chapter 8 34 Glencoe Algebra 2
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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NAME ______________________________________________ DATE______________ PERIOD _____
8-5
Less
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8-5
Lesson Reading GuideClasses of Functions
Chapter 8 35 Glencoe Algebra 2
Less
on
8-5
Get Ready for the LessonRead the introduction to Lesson 8-5 in your textbook.
• 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.
Remember What You Learned
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
Copyright ©
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raw-H
ill Com
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Chapter 8 36 Glencoe Algebra 2
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
Chapter 8 37 Glencoe Algebra 2
Less
on
8-5
Less
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8-5
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)
Study Guide and Intervention (continued)
Classes of Functions
Copyright ©
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cGraw
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he McG
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ill Com
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Chapter 8 38 Glencoe Algebra 2
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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NAME ______________________________________________ DATE______________ PERIOD _____
8-5
Chapter 8 39 Glencoe Algebra 2
Less
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8-5
Identify the type of function represented by each graph.
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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Chapter 8 40 Glencoe Algebra 2
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
Chapter 8 41 Glencoe Algebra 2
Less
on
8-5
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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Chapter 8 42 Glencoe Algebra 2
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 _____
Chapter 8 43 Glencoe Algebra 2
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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Glencoe/M
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raw-H
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Chapter 8 44 Glencoe Algebra 2
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
Chapter 8 45 Glencoe Algebra 2
Less
on
8-6
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
Copyright ©
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-Hill, a division of T
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raw-H
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panies, Inc.
Chapter 8 46 Glencoe Algebra 2
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
Chapter 8 47 Glencoe Algebra 2
Less
on
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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Chapter 8 48 Glencoe Algebra 2
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.
Chapter 8 49 Glencoe Algebra 2
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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
Chapter 8 50 Glencoe Algebra 2
Rubric for Scoring Pre-AP(Use to score the Pre-AP question on page 495 of the Student Edition.)
NAME ______________________________________________ DATE______________ PERIOD _____
Copyright ©
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-Hill, a division of T
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ill Com
panies, Inc.
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)
Chapter 8 51 Glencoe Algebra 2
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ent
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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
Find the LCM of each set of polynomials.
6. 12a2, 15b3, 20ab2 7. 5x2 � 20, 3x � 6
8. 2t2 � 3t � 1, 2t2 � 7t � 4
Simplify each expression.
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 52 Glencoe Algebra 2
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 _____
Copyright ©
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-Hill, a division of T
he McG
raw-H
ill Com
panies, Inc.
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.
Identify the type of function represented by each graph.
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.
For Questions 2–5, simplify each expression.
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)
Chapter 8 53 Glencoe Algebra 2
Ass
essm
ent
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NAME ______________________________________________ DATE______________ PERIOD _____
xO
f (x)
Part II
8
Chapter 8 54 Glencoe Algebra 2
Chapter 8 Vocabulary Test SCORE
NAME ______________________________________________ DATE______________ PERIOD _____
Copyright ©
Glencoe/M
cGraw
-Hill, a division of T
he McG
raw-H
ill Com
panies, Inc.
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
Chapter 8 55 Glencoe Algebra 2
Ass
essm
ent
Cop
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/McG
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NAME ______________________________________________ DATE______________ PERIOD _____
Write the letter for the correct answer in the blank at the right of each question.
Simplify each expression.
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
Chapter 8 56 Glencoe Algebra 2
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 _____
Copyright ©
Glencoe/M
cGraw
-Hill, a division of T
he McG
raw-H
ill Com
panies, Inc.
8 Chapter 8 Test, Form 2A SCORE
Chapter 8 57 Glencoe Algebra 2
Ass
essm
ent
Write the letter for the correct answer in the blank at the right of each question.
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.
Simplify each expression.
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.
For Questions 7 and 8, find the LCM of each set of polynomials.
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
Chapter 8 58 Glencoe Algebra 2
11. Which rational function is graphed?
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.
14. If y varies inversely as x and y � 2 when x � 6, find y when x � 36.
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 _____
Copyright ©
Glencoe/M
cGraw
-Hill, a division of T
he McG
raw-H
ill Com
panies, Inc.
xO
f (x)
8 Chapter 8 Test, Form 2B SCORE
Chapter 8 59 Glencoe Algebra 2
Ass
essm
ent
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 �2xx2
2��
43xx��
42� undefined?
A. ��12�, 0, 2 B. ��
12�, 2 C. �2, �
12� D. ��
12� 1.
Simplify each expression.
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.
For Questions 7 and 8, find the LCM of each set of polynomials.
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
Chapter 8 60 Glencoe Algebra 2
11. Which rational function is graphed?
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.
14. If y varies inversely as x and y � 5 when x � 5, find y when x � 45.
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 _____
Copyright ©
Glencoe/M
cGraw
-Hill, a division of T
he McG
raw-H
ill Com
panies, Inc.
xO
f (x)
8 Chapter 8 Test, Form 2C SCORE
Chapter 8 61 Glencoe Algebra 2
Ass
essm
ent
1. For what value(s) of x is the expression �2x2x�
2 �3x
9� 9� 1.
undefined?
Simplify each expression.
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.
Find the LCM of each set of polynomials.
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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xO
f (x)
8
Chapter 8 62 Glencoe Algebra 2
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
Chapter 8 63 Glencoe Algebra 2
Ass
essm
ent
1. For what value(s) of x is the expression �2xx
2
2��
xx
��
610� 1.
undefined?
Simplify each expression.
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.
Find the LCM of each set of polynomials.
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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8
Chapter 8 64 Glencoe Algebra 2
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.
16. Identify the type of function 16.represented by the graph.
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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ill Com
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8 Chapter 8 Test, Form 3 SCORE
Chapter 8 65 Glencoe Algebra 2
Ass
essm
ent
1. For what value(s) of x is the expression �6x23x�
2 �13
xx�2 �
105x� 1.
undefined?
For Questions 2–6, simplify each expression.
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��
����4
2�n
n2�
�34mm
��
43nn�
���34mm
��
43nn�
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xO
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8
Chapter 8 66 Glencoe Algebra 2
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.
14. Identify the type of function 14.represented by the graph.
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 ©
Glencoe/M
cGraw
-Hill, a division of T
he McG
raw-H
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
Chapter 8 67 Glencoe Algebra 2
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
Chapter 8 68 Glencoe Algebra 2
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 _____
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ill Com
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Z Y
W X
Part 1: Multiple Choice
Instructions: Fill in the appropriate circle for the best answer.
8 Standardized Test Practice (continued)
Chapter 8 69 Glencoe Algebra 2
Ass
essm
ent
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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
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
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
9
8
7
6
5
4
3
2
1
0
F G H J1�9
2�3
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
1�2
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
Chapter 8 70 Glencoe Algebra 2
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 _____
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-Hill, a division of T
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raw-H
ill Com
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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
eadi
ng G
uide
Mu
ltip
lyin
g a
nd
Div
idin
g R
atio
nal
Exp
ress
ion
s
Cha
pter
85
Gle
ncoe
Alg
ebra
2
Lesson 8-1
Get
Rea
dy
for
the
Less
on
Rea
d t
he
intr
odu
ctio
n t
o L
esso
n 8
-1 i
n y
our
text
boo
k.
•S
upp
ose
that
th
e G
oodi
e S
hop
pe a
lso
sell
s a
can
dy m
ixtu
re m
ade
wit
h
4 po
un
ds o
f ch
ocol
ate
min
ts a
nd
3 po
un
ds o
f ca
ram
els,
then
of t
he
mix
ture
is
min
ts a
nd
of t
he
mix
ture
is
cara
mel
s.
•If
th
e st
ore
man
ager
add
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er y
pou
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of
min
ts t
o th
e m
ixtu
re,w
hat
fra
ctio
n o
f th
em
ixtu
re w
ill
be m
ints
?
Rea
d t
he
Less
on
1.a.
In o
rder
to
sim
plif
y a
rati
onal
nu
mbe
r or
rat
ion
al e
xpre
ssio
n,
the
nu
mer
ator
an
d an
d di
vide
bot
h o
f th
em b
y th
eir
.
b.
A r
atio
nal e
xpre
ssio
n is
und
efin
ed w
hen
its
is e
qual
to
.
To
fin
d th
e va
lues
th
at m
ake
the
expr
essi
on u
nde
fin
ed,c
ompl
etel
y
the
orig
inal
an
d se
t ea
ch f
acto
r eq
ual
to
.
2.a.
To
mu
ltip
ly t
wo
rati
onal
exp
ress
ion
s,th
e an
dm
ult
iply
th
e de
nom
inat
ors.
b.
To
divi
de t
wo
rati
onal
exp
ress
ion
s,by
th
e of
the
.
3.a.
Wh
ich
of
the
foll
owin
g ex
pres
sion
s ar
e co
mpl
ex f
ract
ion
s?ii,
iv,v
i.ii
.ii
i.iv
.v.
b.
Doe
s a
com
plex
fra
ctio
n e
xpre
ss a
mu
ltip
lica
tion
or
divi
sion
pro
blem
?d
ivis
ion
How
is
mu
ltip
lica
tion
use
d in
sim
plif
yin
g a
com
plex
fra
ctio
n?
Sam
ple
an
swer
:To
div
ide
the
nu
mer
ato
r o
f th
e co
mp
lex
frac
tio
n b
y th
e d
eno
min
ato
r,m
ult
iply
th
e n
um
erat
or
by t
he
reci
pro
cal o
f th
e d
eno
min
ato
r.
Rem
emb
er W
hat
Yo
u L
earn
ed
4.O
ne
way
to
rem
embe
r so
met
hin
g n
ew i
s to
see
how
it
is s
imil
ar t
o so
met
hin
g yo
ual
read
y kn
ow.H
ow c
an y
our
know
ledg
e of
div
isio
n o
f fr
acti
ons
in a
rith
met
ic h
elp
you
to
un
ders
tan
d h
ow t
o di
vide
rat
ion
al e
xpre
ssio
ns?
Sam
ple
an
swer
:To
div
ide
rati
on
alex
pre
ssio
ns,
mu
ltip
ly t
he
firs
t ex
pre
ssio
n b
y th
e re
cip
roca
l of
the
seco
nd
.Th
is is
th
e sa
me
“inv
ert
and
mu
ltip
ly”
pro
cess
th
at is
use
d w
hen
div
idin
g a
rith
met
ic f
ract
ion
s.
�r2� 9
25�
� �r� 3
5�
�z� z
1�
�z
r�
5� r
�5
�3 8�
� � 15 6�
7 � 12
div
iso
rre
cip
roca
lm
ult
iply
nu
mer
ato
rsm
ult
iply
0d
eno
min
ato
rfa
cto
r0d
eno
min
ato
rg
reat
est
com
mo
n f
acto
rd
eno
min
ato
rfa
cto
r
4 �
y� 7
�y
�3 7�
�4 7�
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
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8A
ntic
ipat
ion
Gui
deR
atio
nal
Exp
ress
ion
s an
d E
qu
atio
ns
Cha
pter
83
Gle
ncoe
Alg
ebra
2
Bef
ore
you
beg
in C
ha
pte
r 8
•R
ead
each
sta
tem
ent.
•D
ecid
e w
het
her
you
Agr
ee (
A)
or D
isag
ree
(D)
wit
h t
he
stat
emen
t.
•W
rite
A o
r D
in
th
e fi
rst
colu
mn
OR
if
you
are
not
su
re w
het
her
you
agr
ee o
r di
sagr
ee,
wri
te N
S(N
ot S
ure
).
Aft
er y
ou c
omp
lete
Ch
ap
ter
8
•R
erea
d ea
ch s
tate
men
t an
d co
mpl
ete
the
last
col
um
n b
y en
teri
ng
an A
or
a D
.
•D
id a
ny
of y
our
opin
ion
s ab
out
the
stat
emen
ts c
han
ge f
rom
th
e fi
rst
colu
mn
?
•F
or t
hos
e st
atem
ents
th
at y
ou m
ark
wit
h a
D,u
se a
pie
ce o
f pa
per
to w
rite
an
exa
mpl
e of
wh
y yo
u d
isag
ree.
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
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____
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____
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____
__D
ATE
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__P
ER
IOD
____
_
Step
1
ST
EP
1S
TE
P 2
A,D
,or
NS
Sta
tem
ent
A o
r D
1.S
ince
a d
enom
inat
or c
ann
ot e
qual
0,t
he
expr
essi
on
Ais
un
defi
ned
for
x�
�5.
2.T
o di
vide
tw
o ra
tion
al e
xpre
ssio
ns,
mu
ltip
ly b
y th
e re
cipr
ocal
of
th
e di
viso
r.A
3.T
he
leas
t co
mm
on m
ult
iple
of
thre
e m
onom
ials
is
fou
nd
by
mu
ltip
lyin
g th
e m
onom
ials
tog
eth
er.
D
4.B
efor
e ad
din
g tw
o ra
tion
al e
xpre
ssio
ns,
a co
mm
on
den
omin
ator
mu
st b
e fo
un
d.A
5.T
he
grap
h o
f a
rati
onal
fu
nct
ion
con
tain
ing
an a
sym
ptot
e w
ill
be s
ymm
etri
c ov
er t
he
asym
ptot
e.D
6.S
ince
f(x
) �
can
be
sim
plifi
ed t
o f(
x) �
m�
2,
the
grap
hof
f(x)
wil
lbe
the
stra
igh
tli
ne
defi
ned
byy
�m
�2.
D
7.y
�kx
yzis
an
exa
mpl
e of
a jo
int
vari
atio
n i
f k,
x,y,
and
zar
e al
l n
ot e
qual
to
0.A
8.T
he
shap
e of
th
e gr
aph
of
y�
�3x
2�
2x�
4 ca
n o
nly
be
dete
rmin
ed b
y gr
aph
ing
the
fun
ctio
n.
D
9.B
ecau
se t
he
grap
h o
f an
abs
olu
te v
alu
e fu
nct
ion
is
in t
he
shap
e of
a V
,th
e gr
aph
of
y�
�x��
4 w
ill
also
be
in t
he
shap
e A
of a
V.
10.
Wh
en s
olvi
ng
rati
onal
equ
atio
ns,
solu
tion
s th
at r
esu
lt i
n a
ze
ro i
n t
he
den
omin
ator
mu
st b
e ex
clu
ded.
A
(m�
4)(m
�2)
��
m�
4
3x2(x
�1)
��
x�
5
Step
2
Chapter Resources
Chapter 8 A2 Glencoe Algebra 2
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Stud
y G
uide
and
Inte
rven
tion
(co
nti
nued
)
Mu
ltip
lyin
g a
nd
Div
idin
g R
atio
nal
Exp
ress
ion
s
NA
ME
____
____
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____
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__D
ATE
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__P
ER
IOD
____
_
Cha
pter
87
Gle
ncoe
Alg
ebra
2
Lesson 8-1
Sim
plif
y C
om
ple
x Fr
acti
on
sA
com
ple
x fr
acti
onis
a r
atio
nal
exp
ress
ion
wh
ose
nu
mer
ator
an
d/or
den
omin
ator
con
tain
s a
rati
onal
exp
ress
ion
.To
sim
plif
y a
com
plex
frac
tion
,fir
st r
ewri
te i
t as
a d
ivis
ion
pro
blem
.
Sim
pli
fy
.
��
Exp
ress
as
a di
visi
on p
robl
em.
��
Mul
tiply
by
the
reci
proc
al o
f th
e di
viso
r.
�Fa
ctor
.
�S
impl
ify.
Sim
pli
fy.
1.2.
3.(b
�1)
2
4.5.
6.a
�4
7.x
�3
8.9.
1� x
�5
x2�
x�
2�
��
x3�
6x2
�x
�30
��
�x
�1
� x�
3
b�
4�
�(b
�1)
(b�
2)
� b2�b
� 6b2 �
8�
��
�b2
b2��b
1� 62
�
�2x2
x��9x
1�9
�
��
�10 5x x2 2
� �1 79 xx �
�26
�
�a a2� �
1 26�
��
�a a2 2� �3 aa ��
24�
1�
�(x
�3)
(x�
2)
� x2�x
� 6x4 �
9�
��
�x2� 3
�2xx�
8�
2(b
�10
)�
�b
(3b
�1)
�b2� b210
0�
��
�3b2
�3 21 bb
�10
�
� 3b b2� �
1 2�
��
� 3b2b �
�b1 �
2�
ac7
� by
�a x2 2b yc 23�
� � ca 4 xb 22 y�
xyz
� a5
�x a3 2y b2 2z�
� �a3
bx 22 y �
s3� s
�3(3s
�1)
s4�
�s(
3s�
1)(s
�3)s4
��
3s2
�8s
�3
3s�
1�
s
3s2
�8s
�3
�� s4
3s�
1�
s
�3ss�
1�
��
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�s8 4
s�
3�
�3ss�
1�
��
�3s2
�
s8 4s
�3
�
1
11
s3
8-1
Exer
cise
s
Exam
ple
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
lyn
omia
l ex
pres
sion
s is
a r
atio
nal
exp
ress
ion
.To
sim
plif
y a
rati
onal
exp
ress
ion
,div
ide
both
th
e n
um
erat
or a
nd
the
den
omin
ator
by
thei
r gr
eate
st c
omm
on f
acto
r (G
CF
).
Mu
ltip
lyin
g R
atio
nal
Exp
ress
ion
sFo
r al
l rat
iona
l exp
ress
ions
an
d ,
��
, if b
�0
and
d�
0.
Div
idin
g R
atio
nal
Exp
ress
ion
sFo
r al
l rat
iona
l exp
ress
ions
an
d ,
��
, if b
�0,
c�
0, a
nd d
�0.
Sim
pli
fy e
ach
exp
ress
ion
.
a.
��
b.
� ��
��
c.� �
��
�
�
Sim
pli
fy e
ach
exp
ress
ion
.
1.�(�
22 0a ab b2 4)3�
�2.
3.
4.�
2m2 (
m�
1)5.
�
6.�
m7.
�
8.�
9.�
4�
�(2
m�
1)(m
�5)
p(4
p�
1)�
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p�
2)
4m2
�1
��
4m�
82m
�1
��
m2
�3m
�10
4p2
�7p
�2
��
7p5
16p2
�8p
�1
��
14p4
y5
� 15z5
18xz
2�
5y6x
y4� 25
z3m
3�
9m�
�m
2�
9(m
�3)
2�
�m
2�
6m�
9
c� c
�5
c2�
4c�
5�
�c2
�4c
�3
c2�
3c� c2
�25
4m5
� m�
13m
3�
3m�
�6m
4
x�
2� x
�9
x2�
x�
6�
�x2
�6x
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3 �
2x�
34x
2�
12x
�9
��
9 �
6x2a
2 b2
�5
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�
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x2�
2x�
8�
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8x�
16�
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x2�
2x�
8�
�x
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x2�
8x�
16�
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4s2
� 3rt2
2 �
2 �
s�
s�
�3
�r
�t
�t
3 �
r�
r�
s�
s�
s�
2 �
2 �
5 �
t�
t�
��
�5
�t
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�t
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�r
�r
�r
�s
20t2
� 9r3 s
3r2 s
3�
5t4
20t2
� 9r3 s
3r2 s
3�
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�
b24
a5 b2
� (2ab
)4
24a
5 b2
� (2a
b)4
ad � bcc � d
a � bc � d
a � b
ac � bdc � d
a � bc � d
a � b
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
Stud
y G
uide
and
Inte
rven
tion
Mu
ltip
lyin
g a
nd
Div
idin
g R
atio
nal
Exp
ress
ion
s
11
11
11
11
11
1
11
11
11
1
11
11
11
11
1
11
11
11
11
1
Exer
cise
s
Exam
ple
Copyright ©
Glencoe/M
cGraw
-Hill, a division of T
he McG
raw-H
ill Com
panies, Inc.
Answers (Lesson 8-1)
Chapter 8 A3 Glencoe Algebra 2
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.
Answers (Lesson 8-1)
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Prac
tice
Mu
ltip
lyin
g a
nd
Div
idin
g R
atio
nal
Exp
ress
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
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__P
ER
IOD
____
_
8-1
Cha
pter
89
Gle
ncoe
Alg
ebra
2
Lesson 8-1
Sim
pli
fy e
ach
exp
ress
ion
.
1.2.
�3.
4.5.
�
6.7.
��
8.�
9.�
10.
�n
�w
11.
��
12.
�13
.�
14. �
�3�
15.
�
16.
�17
.�
18.
��
19.
20.
�2(
x�
3)21
.
22.G
EOM
ETRY
A r
igh
t tr
ian
gle
wit
h a
n a
rea
of x
2�
4 sq
uar
e u
nit
s h
as a
leg
th
atm
easu
res
2x�
4 u
nit
s.D
eter
min
e th
e le
ngt
h o
f th
e ot
her
leg
of
the
tria
ngl
e.x
�2
un
its
23.G
EOM
ETRY
A r
ecta
ngu
lar
pyra
mid
has
a b
ase
area
of
squ
are
cen
tim
eter
s
and
a h
eigh
t of
ce
nti
met
ers.
Wri
te a
rat
ion
al e
xpre
ssio
n t
o de
scri
be t
he
volu
me
of t
he
rect
angu
lar
pyra
mid
.cm
3x
� 5
�6
x2�
3x�
�x2
�5x
�6
x2�
3x�
10�
� 2x
x2
�2x
�4
��
x(x
�2)
� xx 23
��22 x3
�
��
� x2( �x
� 4x2 �)3
4�
�x2
4�9
�
� �3� 8
x�
2x�
1� 4
�x
�2xx�
1�
� �4� x
x�
5 � 22a
�6
� 5a�
109
�a2
��
a2�
5a�
6
2s�
3�
�(s
�4)
(s�
5)s2
�10
s�
25�
�s
�4
2s2
�7s
�15
��
(s�
4)2
2�
�x(
x�
3)6x
2�
12x
��
4x�
123x
�6
� x2�
9
1�
�2(
x�
y)x2
�y2
�3
x�
y�
6xy
3� 3w
24x2
�w
52x
y� w
2
a2 w
2�
y2
a3 w2
� w5 y
2a5 y
3� w
y75x
�1
��
2(x
�5)
25x2
�1
��
x2�
10x
�25
x�
5� 10
x�
2
5x � 25x
2� 8
�x
x2�
5x�
24�
�6x
�2x
2w
2�
n2
�y
�a
a�
y� w
�n
1 � n2
n2
�6n
�n
8n
5� n
�6
2 � 34
� y�
aa
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5ux
2� 21
yz5
25x3
� 14u
2 y2
�2u
3 y� 15
xz5
x�
2�
xx4
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2�
�x4
�x3
v�
5� 3v
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25 �
v2�
�3v
2�
13v
�10
2k�
5� k
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2k2
�k
�15
��
k2�
9
2y�
3� 7y
�1
10y2
�15
y�
�35
y2�
5y4m
4 n2
�9
(2m
3 n2 )
3�
��
18m
5 n4
1� 3a
2 bc
9a2 b
3� 27
a4 b4 c
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Skill
s Pr
acti
ceM
ult
iply
ing
an
d D
ivid
ing
Rat
ion
al E
xpre
ssio
ns
Cha
pter
88
Gle
ncoe
Alg
ebra
2
Sim
pli
fy e
ach
exp
ress
ion
.
1.2.
3.x
64.
5.6.
7.8.
�
9.�
6e10
.�
11.
�21
g312
.�
13.
�x
(x�
2)14
.�
15.
�16
.�
(w�
8)(w
�7)
17.
�(3
x2�
3x)
18.
�
19.
�20
.a
�b
�2
�a2
4� ab2
�
��a
2� ab
�
5� 2c
4 d
� 2c d2 2�
� �� 5c d6 �
(4a
�5)
(a�
4)�
�3a
�2
4a�
5�
�a2
�8a
�16
16a2
�40
a�
25�
��
3a2
�10
a�
81 � 6x
x2�
5x�
4�
�2x
�8
t�
12� 2(
t�
2)2t
�2
��
t2�
9t�
14t2
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t�
84�
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w2
�6w
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w�
3w
2�
5w�
24�
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q2
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�2)
q2�
4�
3q2
q2�
2q�
6q3x
� x2�
43x
2� x
�2
32z7
� 35v
2 y25
y5� 14
z12v5
80y4
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71
� 3g2 y
27g � y2
1�
�2s
3 (s
�2)
s�
2� 10
s55s
2� s2
�4
10(e
f)3
�8e
5 f24
e3� 5f
2
mn
2�
4n
3� 6
3m � 2na
�8
� a�
43a
2�
24a
��
3a2
�12
a
x�
2� x
�1
x2�
4�
�(x
�2)
(x�
1)9
� x�
318
� 2x�
6
2 � y4
8y2 (
y6 )3
�4y
24(x
6 )3
� (x3 )
4
b � 5a5a
b3� 25
a2 b2
3x � 2y21
x3 y� 14
x2 y2N
AM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
____
PE
RIO
D__
___
8-1
Chapter 8 A4 Glencoe Algebra 2
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
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
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
�1
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
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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
per
gal
lon
that
Bet
h g
ets
wit
h h
er c
ar i
n t
erm
s of
C
and
H.
3.H
EIG
HT
Th
e fr
ont
face
of
a N
ordi
ch
ouse
is
tria
ngu
lar.
Th
e su
rfac
e ar
ea
of t
he
face
is
x2�
3x�
10 w
her
e x
is
the
base
of
the
tria
ngl
e.
Wh
at i
s th
e h
eigh
t of
th
e tr
ian
gle
inte
rms
of x
?
h�
2x2
�6x
�20
�� x
x
15C
�26
H�
�C
�H
R�
100
� G�
100
4.O
IL S
LIC
KS
Dav
id w
as m
ovin
g a
dru
mof
oil
aro
un
d h
is c
ircu
lar
outd
oor
pool
wh
en t
he
dru
m c
rack
ed,a
nd
oil
spil
led
into
th
e po
ol.T
he
oil
spre
ad i
tsel
f ev
enly
over
th
e su
rfac
e of
th
e po
ol.L
et V
den
ote
the
volu
me
of o
il s
pill
ed a
nd
let
rbe
th
era
diu
s of
th
e po
ol.W
rite
an
equ
atio
n f
orth
e th
ickn
ess
of t
he
oil
laye
r.
h�
RU
NN
ING
For
Exe
rcis
es 5
an
d 6
,use
the
foll
owin
g in
form
atio
n.
Har
old
run
s to
th
e lo
cal
food
mar
t to
bu
y a
gall
on o
f so
y m
ilk.
Bec
ause
he
is w
eigh
eddo
wn
on
his
ret
urn
tri
p,h
e ru
ns
slow
er o
nth
e w
ay b
ack.
He
trav
els
S1
feet
per
sec
ond
on t
he
way
to
the
food
mar
t an
d S
2fe
et
per
seco
nd
on t
he
way
bac
k.L
et d
be t
he
dist
ance
he
has
to
run
to
get
to t
he
food
mar
t.R
emem
ber:
dist
ance
�ra
te �
tim
e.
5.W
rite
an
equ
atio
n t
hat
giv
es t
he
tota
lti
me
Har
old
spen
t ru
nn
ing
for
this
erra
nd.
t�
�
6.W
hat
spe
ed w
ould
Har
old
hav
e to
ru
n
if h
e w
ante
d to
mai
nta
in a
con
stan
tsp
eed
for
the
enti
re t
rip
yet
take
th
esa
me
amou
nt
of t
ime
run
nin
g?
2S1S
2� S
1�
S2
d � S2
d � S1V � �r2
Copyright ©
Glencoe/M
cGraw
-Hill, a division of T
he McG
raw-H
ill Com
panies, Inc.
Answers (Lesson 8-1)
Chapter 8 A5 Glencoe Algebra 2
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.
Answers (Lesson 8-2)
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exam
ple
Stud
y G
uide
and
Inte
rven
tion
Ad
din
g a
nd
Su
btr
acti
ng
Rat
ion
al E
xpre
ssio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
8-2
Cha
pter
813
Gle
ncoe
Alg
ebra
2
Lesson 8-2
LCM
of
Poly
no
mia
lsT
o fi
nd
the
leas
t co
mm
on m
ult
iple
of
two
or m
ore
poly
nom
ials
,fa
ctor
eac
h e
xpre
ssio
n.T
he
LC
M c
onta
ins
each
fac
tor
the
grea
test
nu
mbe
r of
tim
es i
tap
pear
s as
a f
acto
r.
Fin
d t
he
LC
M o
f 16
p2 q
3 r,
40p
q4 r
2 ,an
d 1
5p3 r
4 .
16p2
q3r
�24
�p2
�q3
�r
40pq
4 r2
�23
�5
�p
�q4
�r2
15p3
r4�
3 �
5 �
p3�
r4
LC
M �
24�
3 �
5 �
p3�
q4�
r4
�24
0p3 q
4 r4
Fin
d t
he
LC
M o
f 3m
2�
3m�
6 an
d 4
m2
�12
m�
40.
3m2
�3m
�6
�3(
m�
1)(m
�2)
4m2
�12
m�
40 �
4(m
�2)
(m�
5)L
CM
�12
(m�
1)(m
�2)
(m�
5)
Fin
d t
he
LC
M o
f ea
ch s
et o
f p
olyn
omia
ls.
1.14
ab2 ,
42bc
3 ,18
a2c
2.8c
df3
,28c
2 f,3
5d4 f
2
126a
2 b2 c
328
0c2 d
4 f3
3.65
x4y,
10x2
y2,2
6y4
4.11
mn
5 ,18
m2 n
3 ,20
mn
4
130x
4 y4
1980
m2 n
5
5.15
a4b,
50a2
b2,4
0b8
6.24
p7q,
30p2
q2,4
5pq3
600a
4 b8
360p
7 q3
7.39
b2c2
,52b
4 c,1
2c3
8.12
xy4 ,
42x2
y,30
x2y3
156b
4 c3
420x
2 y4
9.56
stv2
,24s
2 v2 ,
70t3
v310
.x2
�3x
,10x
2�
25x
�15
840s
2 t3 v
35x
(x�
3)(2
x�
1)
11.9
x2�
12x
�4,
3x2
�10
x�
812
.22x
2�
66x
�22
0,4x
2�
16(3
x�
2)2 (
x�
4)44
(x�
2)(x
�2)
(x�
5)
13.8
x2�
36x
�20
,2x2
�2x
�60
14.5
x2�
125,
5x2
�24
x�
54(
x�
5)(x
�6)
(2x
�1)
5(x
�5)
(x�
5)(5
x�
1)
15.3
x2�
18x
�27
,2x3
�4x
2�
6x16
.45x
2�
6x�
3,45
x2�
56x
(x�
3)2 (
x�
1)15
(5x
�1)
(3x
�1)
(3x
�1)
17.x
3�
4x2
�x
�4,
x2�
2x�
318
.54x
3�
24x,
12x2
�26
x�
12(x
�1)
(x�
1)(x
�3)
(x�
4)6x
(3x
�2)
(3x
�2)
(2x
�3)
Exer
cise
s
Exam
ple
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Cha
pter
812
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
Less
on R
eadi
ng G
uide
Ad
din
g a
nd
Su
btr
acti
ng
Rat
ion
al E
xpre
ssio
ns
8-2
Get
Rea
dy
for
the
Less
on
Rea
d t
he
intr
odu
ctio
n t
o L
esso
n 8
-2 i
n y
our
text
boo
k.
A p
erso
n i
s st
andi
ng
5 fe
et f
rom
a c
amer
a th
at h
as a
len
s w
ith
a f
ocal
len
gth
of
3 fe
et.
Wri
te a
n e
quat
ion
th
at y
ou c
ould
sol
ve t
o fi
nd
how
far
th
e fi
lm s
hou
ld b
e fr
om t
he
len
s to
get
a p
erfe
ctly
foc
use
d ph
otog
raph
.
��
Rea
d t
he
Less
on
1.a.
In w
ork
wit
h r
atio
nal
exp
ress
ion
s,L
CD
sta
nds
for
and
LC
M s
tan
ds f
or
.Th
e L
CD
is
the
ofth
e de
nom
inat
ors.
b.
To
fin
d th
e L
CM
of
two
or m
ore
nu
mbe
rs o
r po
lyn
omia
ls,
each
nu
mbe
r or
.T
he
LC
M c
onta
ins
each
th
e
nu
mbe
r of
tim
es i
t ap
pear
s as
a
.
2.T
o ad
d an
d ,y
ou s
hou
ld f
irst
fac
tor
the
of
each
fra
ctio
n.T
hen
use
th
e fa
ctor
izat
ion
s to
fin
d th
e of
x2
�5x
�6
and
x3�
4x2
�4x
.Th
is i
s th
e fo
r th
e tw
o fr
acti
ons.
3.W
hen
you
add
or
subt
ract
fra
ctio
ns,
you
oft
en n
eed
to r
ewri
te t
he
frac
tion
s as
equ
ival
ent
frac
tion
s.Yo
u d
o th
is s
o th
at t
he
resu
ltin
g eq
uiv
alen
t fr
acti
ons
wil
l ea
ch h
ave
a
den
omin
ator
equ
al t
o th
e of
th
e or
igin
al f
ract
ion
s.
4.T
o ad
d or
su
btra
ct t
wo
frac
tion
s th
at h
ave
the
sam
e de
nom
inat
or,y
ou a
dd o
r su
btra
ct
thei
r an
d ke
ep t
he
sam
e .
5.T
he
sum
or
diff
eren
ce o
f tw
o ra
tion
al e
xpre
ssio
ns
shou
ld b
e w
ritt
en a
s a
poly
nom
ial
or
as a
fra
ctio
n i
n
.
Rem
emb
er W
hat
Yo
u L
earn
ed
6.S
ome
stu
den
ts h
ave
trou
ble
rem
embe
rin
g w
het
her
a c
omm
on d
enom
inat
or i
s n
eede
d to
add
and
subt
ract
rat
ion
al e
xpre
ssio
ns
or t
o m
ult
iply
an
d di
vide
th
em.H
ow c
an y
our
know
ledg
e of
wor
kin
g w
ith
fra
ctio
ns
in a
rith
met
ic h
elp
you
rem
embe
r th
is?
Sam
ple
an
swer
:In
ari
thm
etic
,a c
om
mo
n d
eno
min
ato
r is
nee
ded
to
ad
dan
d s
ub
trac
t fr
acti
on
s,bu
t n
ot
to m
ult
iply
an
d d
ivid
e th
em.T
he
situ
atio
nis
th
e sa
me
for
rati
on
al e
xpre
ssio
ns.
sim
ple
st f
orm
den
om
inat
or
nu
mer
ato
rs
LC
D
LC
D
LC
M
den
om
inat
or
x�
4�
�x3
�4x
2�
4xx2
�3
��
x2�
5x�
6
fact
or
gre
ates
tfa
cto
rp
oly
no
mia
lfa
cto
r
LC
Mle
ast
com
mo
n m
ult
iple
leas
t co
mm
on
den
om
inat
or
1 � 51 � 3
1 � q
Chapter 8 A6 Glencoe Algebra 2
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Skill
s Pr
acti
ceA
dd
ing
an
d S
ub
trac
tin
g R
atio
nal
Exp
ress
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
Cha
pter
815
Gle
ncoe
Alg
ebra
2
Lesson 8-2
Fin
d t
he
LC
M o
f ea
ch s
et o
f p
olyn
omia
ls.
1.12
c,6c
2 d12
c2 d
2.18
a3bc
2 ,24
b2c2
72a
3 b2 c
2
3.2x
�6,
x�
32(
x�
3)4.
5a,a
�1
5a(a
�1)
5.t2
�25
,t�
5(t
�5)
(t�
5)6.
x2�
3x�
4,x
�1
(x�
4)(x
�1)
Sim
pli
fy e
ach
exp
ress
ion
.
7.�
8.�
9.�
410
.�
11.
�12
.�
13.
�14
.�
15.
�16
.�
17.
�18
.�
19.
�20
.�
21.
�22
.�
y�
12�
��
(y�
4)(y
�3)
(y�
2)n
�2
� n�
3
2�
�y2
�6y
�8
3�
�y2
�y
�12
2n�
2�
�n
2�
2n�
3n
� n�
3
2x2
�5x
�2
��
(x�
5)(x
�2)
4�
�x2
�3x
�10
2x�
1� x
�5
x2
�x
�1
��
(x�
1)2
x� x
�1
1�
�x2
�2x
�1
5z2
�4z
�16
��
(z�
4)(z
�1)
z�
4� z
�1
4z� z
�4
2m� m
�n
m� n
�m
m� m
�n
5 �
3t� x
�2
5� x
�2
3t� 2
�x
3w�
7�
�(w
�3)
(w�
3)2
� w2
�9
3� w
�3
15b
d�
6b�
2d�
�3b
d(3
b�
d)
2� 3b
d5
� 3b�
da
�6
��
2a(a
�2)
3 � 2a2
� a�
2
7h�
3g�
�4g
h2
3� 4h
27
� 4gh
12z
�2y
��
5y2 z
2� 5y
z12 � 5y
2
2 �
5m2
��
m2 n
5 � n2
� m2 n
2c�
5�
32c
�7
�3
13� 8p
2 q5
� 4p2 q
3� 8p
2 q5x
�3y
�xy
5 � y3 � x8-2
Lesson 8-2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Stud
y G
uide
and
Inte
rven
tion
(co
nti
nued
)
Ad
din
g a
nd
Su
btr
acti
ng
Rat
ion
al E
xpre
ssio
ns
Cha
pter
814
Gle
ncoe
Alg
ebra
2
Ad
d a
nd
Su
btr
act
Rat
ion
al E
xpre
ssio
ns
To
add
or s
ubt
ract
rat
ion
al e
xpre
ssio
ns,
foll
ow t
hes
e st
eps.
Ste
p 1
If ne
cess
ary,
fin
d eq
uiva
lent
fra
ctio
ns t
hat
have
the
sam
e de
nom
inat
or.
Ste
p 2
Add
or
subt
ract
the
num
erat
ors.
Ste
p 3
Com
bine
any
like
ter
ms
in t
he n
umer
ator
.S
tep
4Fa
ctor
if p
ossi
ble.
Ste
p 5
Sim
plify
if p
ossi
ble.
Sim
pli
fy
�.
�
��
Fact
or t
he d
enom
inat
ors.
��
The
LC
D is
2(x
�3)
(x�
2)(x
�2)
.
�S
ubtr
act
the
num
erat
ors.
�D
istr
ibut
ive
Pro
pert
y
�C
ombi
ne li
ke t
erm
s.
�S
impl
ify.
Sim
pli
fy e
ach
exp
ress
ion
.
1.�
�2.
�
3.�
4.�
5.�
6.�
�2x
2�
9x�
4�
�(2
x�
1)(2
x�
1)2
5x�
�20
x2�
54
��
4x2
�4x
�1
4� x
�1
x�
1� x2
�1
3x�
3�
�x2
�2x
�1
4x�
14� 3x
�6
4x�
5� 3x
�6
3� x
�2
4a2
�9b
2�
�3a
bc
15b
� 5ac
4a � 3bc
x�
1�
�(x
�1)
(x�
3)1
� x�
12
� x�
3y � 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�
�2x
2�
2x�
12
2� x2
�4
6�
�2x
2�
2x�
12
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
8-2
Exer
cise
s
Exam
ple
Copyright ©
Glencoe/M
cGraw
-Hill, a division of T
he McG
raw-H
ill Com
panies, Inc.
Answers (Lesson 8-2)
Chapter 8 A7 Glencoe Algebra 2
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.
Answers (Lesson 8-2)
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Wor
d Pr
oble
m P
ract
ice
Ad
din
g a
nd
Su
btr
acti
ng
Rat
ion
al E
xpre
ssio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
8-2
Cha
pter
817
Gle
ncoe
Alg
ebra
2
Lesson 8-2
1.SQ
UA
RES
Su
san
’s f
avor
ite
perf
ect
squ
are
is s
2an
d T
ravi
s’ i
s t2
,wh
ere
san
d t
are
wh
ole
nu
mbe
rs.W
hat
per
fect
squ
are
is g
uar
ante
ed t
o be
div
isib
le b
ybo
th S
usa
n’s
an
d T
ravi
s’ f
avor
ite
perf
ect
squ
ares
reg
ardl
ess
of t
hei
r sp
ecifi
cva
lue?
s2 t
2
2.EL
ECTR
IC P
OTE
NTI
AL
Th
e el
ectr
ical
pote
nti
al f
un
ctio
n b
etw
een
tw
o el
ectr
ons
is g
iven
by
a fo
rmu
la t
hat
has
th
e fo
rm
�.S
impl
ify
this
exp
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
�b 2
).)
x2
� 2x�
3
1 � 2
x �
4
x �
2
x
1� r(
1 �
r)1� 1
�r
1 � r
4.FR
AC
TIO
NS
In t
he
seve
nte
enth
cen
tury
,Lor
d B
rou
nck
er w
rote
dow
n a
mos
t pe
culi
ar m
ath
emat
ical
equ
atio
n:
� �4 ��
1 �
12
2 �
32
2 �
52
2 �
�7 ∞2 �
Th
is i
s an
exa
mpl
e of
a c
onti
nu
edfr
acti
on.S
impl
ify
the
con
tin
ued
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
4by
400
met
er r
elay
tog
eth
er.T
hei
r av
erag
esp
eeds
wer
e s,
s�
0.5,
s�
0.5,
and
s�
1m
eter
s pe
r se
con
d,re
spec
tive
ly.
5.W
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
7.If
sw
as 6
met
ers
per
seco
nd,
wh
at w
asth
e te
am’s
tim
e? R
oun
d yo
ur
answ
er t
oth
e n
eare
st s
econ
d.
281
seco
nd
s
16s
3�
12s
2�
2s�
1�
��
4s4
�4s
3�
s2
�s
400
� s�
140
0� s
��1 2�
400
� s�
�1 2�
400
�sn3
�2n
� n2
�11
� n�
� n1 �
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Prac
tice
Ad
din
g a
nd
Su
btr
acti
ng
Rat
ion
al E
xpre
ssio
ns
Cha
pter
816
Gle
ncoe
Alg
ebra
2
Fin
d t
he
LC
M o
f ea
ch s
et o
f p
olyn
omia
ls.
1.x2
y,xy
32.
a2b3
c,ab
c43.
x�
1,x
�3
x2 y
3a
2 b3 c
4(x
�1)
(x�
3)
4.g
�1,
g2�
3g�
45.
2r�
2,r2
�r,
r�
16.
3,4w
�2,
4w2
�1
(g�
1)(g
�4)
2r(r
�1)
6(2w
�1)
(2w
�1)
7.x2
�2x
�8,
x�
48.
x2�
x�
6,x2
�6x
�8
9.d
2�
6d�
9,2(
d2
�9)
(x�
4)(x
�2)
(x�
2)(x
�4)
(x�
3)2(
d�
3)(d
�3)
2
Sim
pli
fy e
ach
exp
ress
ion
.
10.
�11
.�
12.
�
13.
�2
14.2
x�
5 �
15.
�
16.
�17
.�
18.
�
19.
�20
.�
21.
��
22.
��
23.
24.
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
.
26.K
AYA
KIN
GM
ai i
s ka
yaki
ng
on a
riv
er t
hat
has
a c
urr
ent
of 2
mil
es p
er h
our.
If r
repr
esen
ts h
er r
ate
in c
alm
wat
er,t
hen
r�
2 re
pres
ents
her
rat
e w
ith
th
e cu
rren
t,an
d r
�2
repr
esen
ts h
er r
ate
agai
nst
th
e cu
rren
t.M
ai k
ayak
s 2
mil
es d
own
stre
am a
nd
then
back
to
her
sta
rtin
g po
int.
Use
th
e fo
rmu
la f
or t
ime,
t�
,wh
ere
dis
th
e di
stan
ce,t
o
wri
te a
sim
plif
ied
expr
essi
on f
or t
he
tota
l ti
me
it t
akes
Mai
to
com
plet
e th
e tr
ip.
h4r
��
(r�
2)(r
�2)
d � r
5(x3
�4x
�16
)�
�2(
x�
4)(x
�4)
10� x
�4
20� x
�4
5x � 2
r�
4� r
�1
3x�
y� x
�y
12� a
�3
�r� r
6�
�� r
�12
�
��
�r2
r� 2�4r
2� r3
�
� x�2
y�
�� x
�1y
�
��
� x�1
y�
36� a2
�9
2a� a
�3
2a� a
�3
3(6
�5n
)�
�20
n2p
2�
2p�
1�
��
(p�
2)(p
�3)
(p�
3)5
��
2(x
�2)
7� 10
n3 � 4
1 � 5n5
� p2�
92p
�3
��
p2�
5p�
620
��
x2�
4x�
125
� 2x�
12
2y�
1�
�(y
�2)
(y�
1)7
�9m
� m�
92
� x�
4
y�
�y2
�y
�2
y�
5�
�y2
�3y
�10
4m�
5� 9
�m
2 �
5m� m
�9
2� x
�4
16� x2
�16
13a
�47
��
(a�
3)(a
�5)
2(x
�3)
(x�
2)�
�x
�4
2(2
�3n
)�
�3n
9� a
�5
4� a
�3
x�
8� x
�4
4m � 3mn
2d2
�9c
��
12c
2 d3
25y
2�
12x
2�
�60
x4 y
320
�21
b�
�24
ab
3� 4c
d3
1� 6c
2 d1
� 5x2 y
35
� 12x4 y
7 � 8a5
� 6ab
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
8-2
Chapter 8 A8 Glencoe Algebra 2
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Less
on R
eadi
ng G
uide
Gra
ph
ing
Rat
ion
al F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
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
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Cha
pter
818
Gle
ncoe
Alg
ebra
2
Enri
chm
ent
Z
eno
’s P
arad
ox
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
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
-Hill, a division of T
he McG
raw-H
ill Com
panies, Inc.
Answers (Lessons 8-2 and 8-3)
Chapter 8 A9 Glencoe Algebra 2
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.
Answers (Lesson 8-3)
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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
xO
f(x)
xO
f(x)
2x�
1� x
�3
2 � x3
� x�
1
x�
2.5
�2
�1
�3.
5�
4�
5
f(x
)2
10.
5�
2�
1�
0.51� x
�3
x�
1�
�(x
�1)
(x�
3)x
�1
��
x2�
2x�
3
x
f(x)
O
x�
1�
�x2
�2x
�3
Stud
y G
uide
and
Inte
rven
tion
(co
nti
nued
)
Gra
ph
ing
Rat
ion
al F
un
ctio
ns
Exer
cise
s
Exam
ple
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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.
Ho
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.
Sin
ce
�,x
�1
is a
ver
tica
l as
ympt
ote.
Th
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
.
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,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
�6
��
�x2
�4x
�3
2x2
�x
�3
��
2x2
�3x
�9
x�
1�
�x2
�6x
�5
1 � 3
3x2
�5x
�2
��
x�
3x2
�6x
�7
��
x2�
6x�
73x
�1
��
3x2
�5x
�2
5 � 2
x2�
x�
12�
�x2
�4x
2x2
�x
�10
��
2x�
54
��
x2�
3x�
10
4x�
3� x
�1
(4x
�3)
(x�
1)�
�(x
�1)
(x�
1)
(4x
�3)
(x�
1)�
�(x
�1)
(x�
1)4x
2�
x�
3�
�x2
�1
4x2
�x
�3
��
x2�
1
p(x
)� q
(x)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
8-3
Exam
ple
Chapter 8 A10 Glencoe Algebra 2
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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
�3)
2x
�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
�10
x�
216
��
x2�
3x�
10
2040
Dist
ance
(ft)
Illu
min
atio
n
Illumination (foot-candles)
60
60 40 20
dOIII
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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)
�4
�x
10 � x�
3�
x
x2�
x�
12�
�x
�3
x2�
8x�
12�
�x
�2
x�
1�
�x2
�4x
�3
x�
12�
�x2
�10
x�
24
10�
�x2
�13
x�
363
��
x2�
2x�
8
Skill
s Pr
acti
ceG
rap
hin
g R
atio
nal
Fu
nct
ion
s
NA
ME
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__D
ATE
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__P
ER
IOD
____
_
8-3
Copyright ©
Glencoe/M
cGraw
-Hill, a division of T
he McG
raw-H
ill Com
panies, Inc.
Answers (Lesson 8-3)
Chapter 8 A11 Glencoe Algebra 2
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.
Answers (Lesson 8-3)
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Enri
chm
ent
Ch
arac
teri
stic
s o
f R
atio
nal
Fu
nct
ion
Gra
ph
s
NA
ME
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__D
ATE
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__P
ER
IOD
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_
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
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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
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__D
ATE
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__P
ER
IOD
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_
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
ion
?
x�
0
BA
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
oes
no
t ch
ang
e
26 �
x� 80
�2x
ax3
�bx
2�
cx�
d�
��
x
I
50
5O
f(I)
72 � I
Chapter 8 A12 Glencoe Algebra 2
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
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__D
ATE
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__P
ER
IOD
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_
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
e of
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
slo
pe-
inte
rcep
t fo
rm o
f th
e eq
uat
ion
of
a lin
e is
y�
mx
�b
.In
dir
ect
vari
atio
n,i
f o
ne
of
the
qu
anti
ties
is 0
,th
e o
ther
qu
anti
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
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____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
Lesson 8-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Cha
pter
826
Gle
ncoe
Alg
ebra
2
NA
ME
____
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____
____
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____
____
__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�
y�
3
4.f(
x)�� 3x
2�
2 5x x�
1�
y�
05.
f(x)
��15
x2� x33x
�7
�y
�0
6.f(
x)�
y�
0
7.f(
x)�
�5 xx2 ��23
�n
on
e8.
f(x)
�� 2x
2�6x
33 x�
6�
no
ne
9.f(
x)�
�2x2�
4�
no
ne
x3�
8x2
�4x
�11
��
�x4
�3x
3 �4x
�6
Gra
phin
g Ca
lcul
ator
Act
ivit
yH
ori
zon
tal A
sym
pto
tes
and
Tab
les
Exer
cise
s
Exam
ple
Copyright ©
Glencoe/M
cGraw
-Hill, a division of T
he McG
raw-H
ill Com
panies, Inc.
Answers (Lessons 8-3 and 8-4)
Chapter 8 A13 Glencoe Algebra 2
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.
Answers (Lesson 8-4)
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Cha
pter
828
Gle
ncoe
Alg
ebra
2
NA
ME
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__D
ATE
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__P
ER
IOD
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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
Chapter 8 A14 Glencoe Algebra 2
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
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__D
ATE
____
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__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
wh
en x
�5,
fin
d x
wh
en y
�2.
5.6
18.I
f y
vari
es i
nve
rsel
y as
xan
d y
��
18 w
hen
x�
6,fi
nd
yw
hen
x�
5.�
21.6
19.I
f y
vari
es d
irec
tly
as x
and
y�
5 w
hen
x�
0.4,
fin
d x
wh
en y
�37
.5.
3
20.G
ASE
ST
he
volu
me
Vof
a g
as v
arie
s in
vers
ely
as i
ts p
ress
ure
P.I
f V
�80
cu
bic
cen
tim
eter
s w
hen
P�
2000
mil
lim
eter
s of
mer
cury
,fin
d V
wh
en P
�32
0 m
illi
met
ers
ofm
ercu
ry.
500
cm3
21.S
PRIN
GS
Th
e le
ngt
h S
that
a s
prin
g w
ill
stre
tch
var
ies
dire
ctly
wit
h t
he
wei
ght
Fth
atis
att
ach
ed t
o th
e sp
rin
g.If
a s
prin
g st
retc
hes
20
inch
es w
ith
25
pou
nds
att
ach
ed,h
owfa
r w
ill
it s
tret
ch w
ith
15
pou
nds
att
ach
ed?
12 in
.
22.G
EOM
ETRY
Th
e ar
ea A
of a
tra
pezo
id v
arie
s jo
intl
y as
its
hei
ght
and
the
sum
of
its
base
s.If
th
e ar
ea i
s 48
0 sq
uar
e m
eter
s w
hen
th
e h
eigh
t is
20
met
ers
and
the
base
s ar
e28
met
ers
and
20 m
eter
s,w
hat
is
the
area
of
a tr
apez
oid
wh
en i
ts h
eigh
t is
8 m
eter
s an
dit
s ba
ses
are
10 m
eter
s an
d 15
met
ers?
100
m2
64 � 3
56 � 3
3 � 41 � 2
3 � 4x1.
25�
gC � d
5 � k
Lesson 8-4
Prac
tice
Dir
ect,
Join
t,an
d In
vers
e V
aria
tio
n
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Cha
pter
830
Gle
ncoe
Alg
ebra
2
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.c
�12
md
irec
t;12
2.p
�in
vers
e;4
3.A
�bh
join
t;
4.rw
�15
inve
rse;
155.
y�
2rst
join
t;2
6.f
�52
80m
dir
ect;
5280
7.y
�0.
2sd
irec
t;0.
28.
vz�
�25
inve
rse;
�25
9.t
�16
rhjo
int;
16
10.R
�in
vers
e;8
11.
�d
irec
t;12
.C�
2r
dir
ect;
2�
Fin
d e
ach
val
ue.
13.I
f y
vari
es d
irec
tly
as x
and
y�
35 w
hen
x�
7,fi
nd
yw
hen
x�
11.
55
14.I
f y
vari
es d
irec
tly
as x
and
y�
360
wh
en x
�18
0,fi
nd
yw
hen
x�
270.
540
15.I
f y
vari
es d
irec
tly
as x
and
y�
540
wh
en x
�10
,fin
d x
wh
en y
�10
80.
20
16.I
f y
vari
es d
irec
tly
as x
and
y�
12 w
hen
x�
72,f
ind
xw
hen
y�
9.54
17.I
f y
vari
es jo
intl
y as
xan
d z
and
y�
18 w
hen
x�
2 an
d z
�3,
fin
d y
wh
en x
�5
and
z�
6.90
18.I
f y
vari
es jo
intl
y as
xan
d z
and
y�
�16
wh
en x
�4
and
z�
2,fi
nd
yw
hen
x�
�1
and
z�
7.14
19.I
f y
vari
es jo
intl
y as
xan
d z
and
y�
120
wh
en x
�4
and
z�
6,fi
nd
yw
hen
x�
3 an
d z
�2.
30
20.I
f y
vari
es i
nve
rsel
y as
xan
d y
�2
wh
en x
�2,
fin
d y
wh
en x
�1.
4
21.I
f y
vari
es i
nve
rsel
y as
xan
d y
�6
wh
en x
�5,
fin
d y
wh
en x
�10
.3
22.I
f y
vari
es i
nve
rsel
y as
xan
d y
�3
wh
en x
�14
,fin
d x
wh
en y
�6.
7
23.I
f y
vari
es i
nve
rsel
y as
xan
d y
�27
wh
en x
�2,
fin
d x
wh
en y
�9.
6
24.I
f y
vari
es d
irec
tly
as x
and
y�
�15
wh
en x
�5,
fin
d x
wh
en y
��
36.
12
1 � 31 � 3
a � b8 � w
1 � 21 � 2
4 � q
NA
ME
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____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
8-4
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
Skill
s Pr
acti
ceD
irec
t,Jo
int,
and
Inve
rse
Var
iati
on
Copyright ©
Glencoe/M
cGraw
-Hill, a division of T
he McG
raw-H
ill Com
panies, Inc.
Answers (Lesson 8-4)
Chapter 8 A15 Glencoe Algebra 2
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.
Answers (Lesson 8-4)
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
8-4
Enri
chm
ent
Geo
syn
chro
no
us
Sat
ellit
es
NA
ME
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____
____
____
____
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____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
Cha
pter
833
Gle
ncoe
Alg
ebra
2
Lesson 8-4
Sat
elli
tes
circ
lin
g th
e E
arth
are
alm
ost
as c
omm
on a
s th
e ce
ll p
hon
es t
hat
depe
nd
on t
hem
.A g
eosy
nch
ron
ous
sate
llit
e is
on
e th
at m
ain
tain
s th
e sa
me
posi
tion
abo
ve t
he
Ear
th a
t al
l ti
mes
.Geo
syn
chro
nou
s sa
tell
ites
are
use
d in
cell
ph
one
com
mu
nic
atio
ns,
tran
smit
tin
g si
gnal
s fr
om t
ower
s on
Ear
th a
nd
to e
ach
oth
er.
Th
e sp
eed
at w
hic
h t
hey
tra
vel
is v
ery
impo
rtan
t.If
th
e sp
eed
is t
oo l
ow,
the
sate
llit
e w
ill
be f
orce
d ba
ck d
own
to
Ear
th d
ue
to t
he
Ear
th’s
gra
vity
.H
owev
er,i
f it
is
too
fast
,it
wil
l ov
erco
me
grav
ity’
s fo
rce
and
esca
pe i
nto
spac
e,n
ever
to
retu
rn.N
ewto
n’s
sec
ond
law
of
mot
ion
say
s th
at f
orce
on
an
obje
ct i
s eq
ual
to
mas
s ti
mes
acc
eler
atio
n o
r F
�m
a.It
is
also
wel
l kn
own
that
th
e n
et g
ravi
tati
onal
for
ce b
etw
een
tw
o ob
ject
s is
in
vers
ely
prop
orti
onal
to t
he
squ
are
of t
he
dist
ance
bet
wee
n t
hem
.Th
eref
ore,
ther
e ar
e tw
ova
riab
les
on w
hic
h t
he
forc
e de
pen
ds:s
peed
an
d h
eigh
t ab
ove
the
Ear
th.
In p
arti
cula
r,N
ewto
n’s
sec
ond
law
,F�
ma,
show
s th
at f
orce
var
ies
dire
ctly
wit
h a
ccel
erat
ion
,wh
ere
mis
th
e co
nst
ant
taki
ng
the
plac
e of
“k.
”
1.S
how
th
at t
he
net
gra
vita
tion
al f
orce
pro
vidi
ng
a sa
tell
ite
wit
h a
ccel
era-
tion
is
inve
rsel
y pr
opor
tion
al t
o th
e sq
uar
e of
th
e di
stan
ce b
etw
een
th
emby
exp
ress
ing
this
var
iati
on a
s an
equ
atio
n.
F�
,wh
ere
his
th
e h
eig
ht
of
the
sate
llite
ab
ove
the
surf
ace
of
the
Ear
th.
2.U
se y
our
equ
atio
n f
rom
Nu
mbe
r 1
and
equ
ate
it w
ith
New
ton
’s f
orm
ula
abov
e to
det
erm
ine
how
th
e sa
tell
ite’
s ac
cele
rati
on v
arie
s w
ith
its
hei
ght
abov
e th
e E
arth
.
ma
�⇒
a�
��
,th
eref
ore
it v
arie
s in
vers
ely
wit
h
the
squ
are
of
the
hei
gh
t.
3.D
eter
min
e h
ow t
he
spee
d of
a g
eosy
nch
ron
ous
sate
llit
e va
ries
wit
h i
tsh
eigh
t ab
ove
the
Ear
th b
y u
sin
g th
e fa
ct t
hat
spe
ed i
s eq
ual
to
dist
ance
divi
ded
by t
ime
and
the
path
of
the
sate
llit
e is
cir
cula
r.
Dir
ect
vari
atio
n.s
pee
d�
⇒sp
eed
�,w
her
e
r�
h�
Rad
ius
of
the
Ear
th.
2�r
� day
dis
tan
ce�
�ti
me
K � h2
1 � h2
k � mk � h
2
k � h2
Exer
cise
s
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Cha
pter
832
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
8-4
Wor
d Pr
oble
m P
ract
ice
D
irec
t,Jo
int,
and
Inve
rse
Var
iati
on
1.D
IVIN
GT
he
hei
ght
that
a d
iver
lea
psab
ove
a di
vin
g bo
ard
vari
es d
irec
tly
wit
hth
e am
oun
t th
at t
he
tip
of t
he
divi
ng
boar
d di
ps b
elow
its
nor
mal
lev
el.I
f a
dive
r le
aps
44 i
nch
es a
bove
th
e di
vin
gbo
ard
wh
en t
he
divi
ng
boar
d ti
p di
ps 1
2in
ches
,how
hig
h w
ill
the
dive
r le
apab
ove
the
divi
ng
boar
d if
th
e ti
p di
ps 1
8in
ches
?
66 in
ches
2.PA
RK
ING
LO
T D
ESIG
NA
s a
gen
eral
rule
,th
e n
um
ber
of p
arki
ng
spac
es i
n
a pa
rkin
g lo
t fo
r a
mov
ie t
hea
ter
com
plex
var
ies
dire
ctly
wit
h t
he
nu
mbe
rof
th
eate
rs i
n t
he
com
plex
.A t
ypic
alth
eate
r h
as 3
0 pa
rkin
g sp
aces
for
eac
hth
eate
r.A
bu
sin
essm
an w
ants
to
buil
d a
new
cin
ema
com
plex
on
a l
ot t
hat
h
as e
nou
gh s
pace
for
210
par
kin
gsp
aces
.How
man
y th
eate
rs s
hou
ld t
he
busi
nes
sman
bu
ild
in h
is c
ompl
ex?
7
3.R
ENT
An
apa
rtm
ent
ren
ts f
or m
doll
ars
per
mon
th.I
f n
stu
den
ts s
har
e th
e re
nt
equ
ally
,how
mu
ch w
ould
eac
h s
tude
nt
hav
e to
pay
? H
ow d
oes
the
cost
per
stu
den
t va
ry w
ith
th
e n
um
ber
ofst
ude
nts
? If
2 s
tude
nts
hav
e to
pay
$7
00 e
ach
,how
mu
ch m
oney
wou
ld
each
stu
den
t h
ave
to p
ay i
f th
ere
wer
e 5
stu
den
ts s
har
ing
the
ren
t?
Eac
h s
tud
ent
pay
s d
olla
rs.
Th
e co
st p
er s
tud
ent
vari
esin
vers
ely
wit
h t
he
nu
mb
er o
fst
ud
ents
,so
eac
h s
tud
ent
wo
uld
pay
$28
0.
m � n
4.PA
INTI
NG
Th
e co
st o
f pa
inti
ng
a w
all
vari
es d
irec
tly
wit
h t
he
area
of
the
wal
l.W
rite
a f
orm
ula
for
th
e co
st o
f pa
inti
ng
a re
ctan
gula
r w
all
wit
h d
imen
sion
s �
byw
.Wit
h r
espe
ct t
o �
and
w,d
oes
the
cost
vary
dir
ectl
y,jo
intl
y,or
in
vers
ely?
C�
k�w
,wh
ere
Cis
th
e co
st a
nd
kis
a c
on
stan
t.C
vari
es jo
intl
yw
ith
�an
d w
.
HY
DR
OG
ENF
or E
xerc
ises
5-7
,use
th
efo
llow
ing
info
rmat
ion
.
Th
e co
st o
f a
hyd
roge
n s
tora
ge t
ank
vari
esdi
rect
ly w
ith
th
e vo
lum
e of
th
e ta
nk.
Ala
bora
tory
wan
ts t
o pu
rch
ase
a st
orag
e ta
nk
shap
ed l
ike
a bl
ock
wit
h d
imen
sion
s L
by W
by H
.
5.F
ill
in t
he
mis
sin
g sp
aces
in
th
efo
llow
ing
tabl
e fr
om a
bro
chu
re o
fva
riou
s ta
nk
size
s.
6.T
he
hyd
roge
n t
ank
mu
st fi
t in
a s
hel
fth
at h
as a
fixe
d h
eigh
t an
d de
pth
.How
does
th
e co
st o
f th
e h
ydro
gen
sto
rage
tan
k va
ry w
ith
th
e w
idth
of
tan
k w
ith
fixe
d de
pth
an
d h
eigh
t?
Th
e co
st v
arie
s d
irec
tly
wit
h t
he
wid
th.
7.H
ow m
uch
wou
ld a
sph
eric
al t
ank
ofra
diu
s 24
in
ches
cos
t? (
Rec
all
that
th
e
volu
me
of a
sph
ere
is g
iven
by
�r3
,
wh
ere
ris
th
e ra
diu
s.)
$1,1
17.0
1
4 � 3
Hyd
rog
en T
ank
Dim
ensi
on
s (i
nch
es)
Co
st
LW
H
3636
36$9
0018
1824
$150
2424
72$8
00
Chapter 8 A16 Glencoe Algebra 2
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
8-5
Lesson 8-5
Less
on R
eadi
ng G
uide
Cla
sses
of
Fu
nct
ion
s
Cha
pter
835
Gle
ncoe
Alg
ebra
2
Lesson 8-5
Get
Rea
dy
for
the
Less
on
Rea
d t
he
intr
odu
ctio
n t
o L
esso
n 8
-5 i
n y
our
text
boo
k.
•B
ased
on
th
e gr
aph
,est
imat
e th
e w
eigh
t on
Mar
s of
a c
hil
d w
ho
wei
ghs
40 p
oun
ds o
n E
arth
.ab
ou
t 15
po
un
ds
•A
lth
ough
th
e gr
aph
doe
s n
ot e
xten
d fa
r en
ough
to
the
righ
t to
rea
d it
dir
ectl
y fr
om t
he
grap
h,u
se t
he
wei
ght
you
fou
nd
abov
e an
d yo
ur
know
ledg
e th
at t
his
gr
aph
rep
rese
nts
dir
ect
vari
atio
n t
o es
tim
ate
the
wei
ght
on M
ars
of a
wom
an
wh
o w
eigh
s 12
0 po
un
ds o
n E
arth
.ab
ou
t 45
po
un
ds
Rea
d t
he
Less
on
1.M
atch
eac
h g
raph
bel
ow w
ith
th
e ty
pe o
f fu
nct
ion
it
repr
esen
ts.S
ome
type
s m
ay b
e u
sed
mor
e th
an o
nce
an
d ot
her
s n
ot a
t al
l.I.
squ
are
root
II.
quad
rati
cII
I.ab
solu
te v
alu
eIV
.ra
tion
alV.
grea
test
in
tege
rV
I.co
nst
ant
VII
.ide
nti
ty
a.III
b.
Ic.
VI
d.
IIe.
IVf.
V
Rem
emb
er W
hat
Yo
u L
earn
ed
2.H
ow c
an t
he
sym
boli
c de
fin
itio
n o
f ab
solu
te v
alu
e th
at y
ou l
earn
ed i
n L
esso
n 1
-4 h
elp
you
to
rem
embe
r th
e gr
aph
of
the
fun
ctio
n f
(x)
�|x
|?S
amp
le a
nsw
er:
Usi
ng
th
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
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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
-Hill, a division of T
he McG
raw-H
ill Com
panies, Inc.
Answers (Lessons 8-4 and 8-5)
Chapter 8 A17 Glencoe Algebra 2
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.
Answers (Lesson 8-5)
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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
Chapter 8 A18 Glencoe Algebra 2
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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
-Hill, a division of T
he McG
raw-H
ill Com
panies, Inc.
Answers (Lesson 8-5)
Chapter 8 A19 Glencoe Algebra 2
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.
Answers (Lesson 8-5)
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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
____
____
____
__P
ER
IOD
____
_
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
.
1000
�N
(P �
2)
6.S
olve
th
e eq
uat
ion
you
wro
te f
orE
xerc
ise
5 fo
r P
in t
erm
s of
N.
P�
7.W
hat
kin
d of
fu
nct
ion
is
Pin
ter
ms
ofN
? S
ketc
h a
gra
ph o
f P
as a
fu
nct
ion
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
Chapter 8 A20 Glencoe Algebra 2
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
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)
��
��10
(x�
1)�
�Mul
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�
4D
istr
ibut
ive
Pro
pert
y
5x�
�25
Sub
trac
t 4x
and
29
from
eac
h si
de.
x�
�5
Div
ide
each
sid
e by
5.
Ch
eck
��
Orig
inal
equ
atio
n
��
x�
�5
��
Sim
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
� x�
2x
�1
�12
4� x
�1
1 � 242m
�1
�2m
3m�
2�
5m
13 � 51 � 2
x�
5�
42x
�1
�3
4 �
2t�
34t
�3
�5
y�
3�
62y � 3
2 � 52 � 5
2 � 510 � 20
18 � 20
2 � 52
� �5
�1
9 � 10
2 � 52
� x�
19 � 10
2 � 52
� x�
19 � 10
2 � 52
� x�
19 � 10
2 � 52
� x�
19 � 10
Exer
cise
s
Exam
ple
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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,�
�9.
If z
�1,
��
�5.
Usi
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�
�4
or
�2
�z
�0
Rem
emb
er W
hat
Yo
u L
earn
ed3.
How
are
th
e pr
oces
ses
of a
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
ion
s in
th
ep
rob
lem
.Th
ey a
re d
iffe
ren
t b
ecau
se in
an
ad
dit
ion
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
aleq
uat
ion
,th
e L
CD
is e
limin
ated
.
6 � z3
� z�
26 � z
3� z
�2
6 � z3
� z�
26 � z
3� z
�2
6 � z3
� z�
2
�3
�4
�5
�6
�2
�1
01
23
45
64
z(z
�2)
6 � z3
� z�
2
0�
2
6 � z3
� z�
2
0
Copyright ©
Glencoe/M
cGraw
-Hill, a division of T
he McG
raw-H
ill Com
panies, Inc.
Answers (Lesson 8-6)
Chapter 8 A21 Glencoe Algebra 2
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.
Answers (Lesson 8-6)
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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
2.2
��
3.�
�1
4.3
�z
�1,
2
5.�
56.
��
5,8
7.�
�3
8.�
�y
�7
3,4
9.�
810
.�
0
k�
0
11.2
��
0 �
v�
412
.n�
�n
��
3 o
r 0
�n
�3
13.
��
�0
�m
�1
14.
��
10
�x
�
15.
��
93
16.
�4
�4
17.2
��
�5
18.8
��
19.
��
�4
20.
��
�
21.
��
�22
.�
�2
23.
��
�6
24.
��
52
� t�
34
� t�
38
� t2�
92
� e�
21
� e�
22e
� e2�
4
5� s
�4
3� s
�3
12s
�19
��
s2�
7s�
122x
�3
� x�
1x
� 2x�
2x
�8
� 2x�
2
4� w
2�
41
� w�
21
� w�
22
� n�
35
� n2
�9
1� n
�3
2 � 58z
�8
� z�
24 � z
2q� q
�1
5 � 2q
b�
2� b
�1
3b�
2� b
�1
9x�
7� x
�2
15 � x
3 � 22 � x
1 � 2x5 � 2
3 � m1
� 2m
12 � n3 � n
5 � v3 � v
4 � 3k3 � k
x�
1� x
�10
x�
2� x
�4
12 � y3 � 2
2x�
3� x
�1
8 � ss
�3
�5
1� d
�2
2� d
� 1
2 � z�
6�2
9 � 3x
12 � 51 � 3
4 � n1 � 2
x� x
�1
Lesson 8-6
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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�
�M
ultip
ly e
ach
side
by
15n.
10 �
12 �
10n
Sim
plify
.
22 �
10n
Sim
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
.
Tes
t n
��
1.T
est
n�
1.T
est
n�
3.
��
����
is t
rue.
��
is n
ottr
ue.
��
is t
rue.
Th
e so
luti
on i
s n
�0
or n
�2.
2.
Sol
ve e
ach
in
equ
alit
y.
1.�
32.
�4x
3.�
�1
�a
0
x
�o
r 0
�x
0
�p
�
4.�
5.
��
26.
�1
�2
�x
�0
x�
0 o
r �
x�
1x
��
1 o
r 0
�x
�1
or
x�
5o
r x
�2
1 � 2
2� x
�1
3� x2
�1
5 � x4
� x�
11 � 4
2 � x3 � 2x
39 � 201 � 2
1 � 2
2 � 34 � 5p
1 � 2p1 � x
3� a
�1
2 � 34 � 15
2 � 92 � 3
4 � 52 � 3
2 � 34 � 5
2 � 3
�3
�2
�1
01
22.2 3
2 � 34 � 5n
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
Chapter 8 A22 Glencoe Algebra 2
Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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
Cha
pter
847
Gle
ncoe
Alg
ebra
2
Lesson 8-6
1.H
EIG
HT
Ser
ena
can
be
desc
ribe
d as
bein
g 8
inch
es s
hor
ter
than
her
sis
ter
Mal
ia,o
r as
bei
ng
12.5
% s
hor
ter
than
Mal
ia.I
n o
ther
wor
ds,
�,
wh
ere
His
Ser
ena’
s h
eigh
t in
in
ches
.H
ow t
all
is S
eren
a?
56 in
ches
2.C
RA
NES
For
a w
eddi
ng,
Pau
la w
ants
to
fold
100
0 or
igam
i cr
anes
.
Sh
e do
es n
ot w
ant
to m
ake
anyo
ne
fold
mor
e th
an 1
5 cr
anes
.In
oth
er w
ords
,if
Nis
th
e n
um
ber
of p
eopl
e en
list
ed t
o
fold
cra
nes
,Pau
la w
ants
�
15.
Wh
at i
s th
e m
inim
um
nu
mbe
r of
peo
ple
that
wil
l sa
tisf
y th
is i
neq
ual
ity?
67
3.R
ENTA
LC
arlo
s an
d h
is f
rien
ds r
ent
aca
r.T
hey
spl
it t
he
$200
ren
tal
fee
even
ly.
Car
los,
toge
ther
wit
h ju
st t
wo
of h
isfr
ien
ds,d
ecid
e to
ren
t a
port
able
DV
Dpl
ayer
as
wel
l,an
d sp
lit
the
$30
ren
tal
fee
for
the
DV
D p
laye
r ev
enly
am
ong
them
selv
es.C
arlo
s en
ds u
p sp
endi
ng
$50
for
thes
e re
nta
ls.W
rite
an
equ
atio
nin
volv
ing
N,t
he
nu
mbe
r of
fri
ends
Car
los
has
,usi
ng
this
in
form
atio
n.S
olve
the
equ
atio
n f
or N
.
��
50;
N�
430 � 3
200
� N�
1
1000
�N
1 � 88
� H�
8
4.PR
OJE
CTI
LES
A p
roje
ctil
e ta
rget
is
lau
nch
ed i
nto
th
e ai
r.A
roc
ket
inte
rcep
tor
is fi
red
at t
he
targ
et.T
he
rati
o of
th
e al
titu
de o
f th
e ro
cket
to
the
alti
tude
of
the
proj
ecti
le t
seco
nds
aft
erth
e la
un
ch o
f th
e ro
cket
is
give
n b
y th
e
form
ula
.A
t w
hat
tim
e
are
the
targ
et a
nd
inte
rcep
tor
at t
he
sam
e al
titu
de?
at t
�3
seco
nd
s
FLIG
HT
TIM
EF
or E
xerc
ises
5 a
nd
6,u
seth
e fo
llow
ing
info
rmat
ion
.
Th
e di
stan
ce b
etw
een
New
Yor
k C
ity
and
Los
An
gele
s is
abo
ut
2500
mil
es.L
et S
beth
e ai
rspe
ed o
f a
jet.
Th
e w
ind
spee
d is
100
mil
es p
er h
our.
Bec
ause
of
the
win
d,it
tak
eslo
nge
r to
fly
one
way
th
an t
he
oth
er.
5.W
rite
an
equ
atio
n f
or S
if i
t ta
kes
2 h
ours
an
d 5
min
ute
s lo
nge
r to
fly
betw
een
New
Yor
k an
d L
os A
nge
les
agai
nst
th
e w
ind
vers
us
flyi
ng
wit
h
the
win
d.
��
2
6.S
olve
th
e eq
uat
ion
you
wro
te i
n
Exe
rcis
e 5
for
S.
500
mp
h
7.W
rite
an
equ
atio
n a
nd
fin
d h
ow m
uch
lon
ger
to fl
y be
twee
n N
ew Y
ork
and
Los
An
gele
s if
th
e w
ind
spee
d in
crea
ses
to15
0 m
iles
per
hou
r an
d th
e ai
rspe
ed o
fth
e je
t is
525
mil
es p
er h
our.
��
x;
x�
�2.
96 h
400
� 135
2500
��
525
�15
025
00�
�52
5�
150
1 � 1225
00� S
�10
025
00� S
�10
0
333t
��
��
32t2
�42
0t�
27
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Cha
pter
846
Gle
ncoe
Alg
ebra
2
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
8-6
Sol
ve e
ach
eq
uat
ion
or
ineq
ual
ity.
Ch
eck
you
r so
luti
ons.
1.�
�16
2.�
1 �
�1,
2
3.�
�,4
4.�
s�
4
5.�
�1
all r
eals
exc
ept
56.
��
0
7.�
t�
�5
or
��
t�
08.
��
9.�
�2
10.5
��
0 �
a�
2
11.
��
0 �
x�
712
.8 �
y
�0
or
y�
2
13.
��
p�
0 o
r p
�14
.�
��
15.g
��
�1
16.b
��
1 �
�2
17.
��
18.
�4
��
�5 3� ,5
19.
��
720
.�
��
1,�
2
21.
��
022
.�
��
23.
��
24.3
��
�2
all r
eals
exc
ept
�4
and
4
27.B
ASK
ETB
ALL
Kia
na h
as m
ade
9 of
19
free
thr
ows
so f
ar t
his
seas
on.H
er g
oal
is t
o m
ake
60%
of
her
fre
e th
row
s.If
Kia
na
mak
es h
er n
ext
xfr
ee t
hro
ws
in a
row
,th
e fu
nct
ion
f(x)
�re
pres
ents
Kia
na’s
new
rat
io o
f fr
ee t
hrow
s m
ade.
How
man
y su
cces
sful
fre
e
thro
ws
in a
row
wil
l ra
ise
Kia
na’
s pe
rcen
t m
ade
to 6
0%?
Is t
his
a r
easo
nab
le a
nsw
er?
Exp
lain
.6;
Sam
ple
an
swer
:It
is a
rea
son
able
an
swer
.Sh
e w
ill h
ave
mad
e15
ou
t o
f 25
fre
e th
row
s,w
hic
h is
eq
uiv
alen
t to
60%
.
28.O
PTIC
ST
he l
ens
equa
tion
�
�re
late
s th
e di
stan
ce p
of a
n ob
ject
fro
m a
lens
,th
e di
stan
ce q
of t
he
imag
e of
th
e ob
ject
fro
m t
he
len
s,an
d th
e fo
cal
len
gth
fof
th
e le
ns.
Wh
at i
s th
e di
stan
ce o
f an
obj
ect
from
a l
ens
if t
he
imag
e of
th
e ob
ject
is
5 ce
nti
met
ers
from
th
e le
ns
and
the
foca
l le
ngt
h o
f th
e le
ns
is 4
cen
tim
eter
s? I
s th
is
a re
ason
able
an
swer
? E
xpla
in.
20 c
m;
Sam
ple
an
swer
:It
is a
rea
son
able
an
swer
,sin
ce
��
.1 � 4
1 � 51 � 20
1 � f1 � 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 � 23
� 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
65 � 31 � 5
1 � 3p4 � p
19 � y3 � y
3 � 2x1 � 10
4 � 5x
7 � a3 � a
�1
� w�
34
� w�
2
11 � 53
� h�
15 � h
1 � 2h1 � 2
9� 2t
�1
5 � t
5 � 85 � x
1� 3x
�2
y� y
�5
5� y
�5
5s�
8� s
�2
s� s
�2
2 � 34 � p
p�
10� p2
�2
x � 2x
� x�
13 � 2
3 � 412 � x
Prac
tice
So
lvin
g R
atio
nal
Eq
uat
ion
s an
d In
equ
alit
ies
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_ Copyright ©
Glencoe/M
cGraw
-Hill, a division of T
he McG
raw-H
ill Com
panies, Inc.
Answers (Lesson 8-6)
Chapter 8 A23 Glencoe Algebra 2
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.
Answers (Lesson 8-6)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Cha
pter
848
Gle
ncoe
Alg
ebra
2
Ob
liqu
e A
sym
pto
tes
Th
e gr
aph
of
y�
ax�
b,w
her
e a
� 0
,is
call
ed a
n o
bliq
ue
asym
ptot
e of
y�
f(x)
if
th
e gr
aph
of
fco
mes
clo
ser
and
clos
er t
o th
e li
ne
as x
→ ∞
or x
→ �
∞.∞
is t
he
mat
hem
atic
al s
ymbo
l fo
r in
fin
ity,
wh
ich
mea
ns
end
less
.
For
f(x
) �
3x�
4 �
�2 x� ,y
�3x
�4
is a
n o
bliq
ue
asym
ptot
e be
cau
se
f(x)
�3x
�4
��2 x� ,
and
�2 x�→
0 a
s x
→ ∞
or �
∞.I
n o
ther
wor
ds,a
s |x
|
incr
ease
s,th
e va
lue
of �2 x�
gets
sm
alle
r an
d sm
alle
r ap
proa
chin
g 0.
Fin
d t
he
obli
qu
e as
ymp
tote
for
f(x
) ��x2
� x8 �x
2�15
�.
�2
18
15U
se s
ynth
etic
div
isio
n.
�2
�12
16
3
y��x2
�x
8 �x2�
15�
�x
�6
�� x
�32
�
As
|x|i
ncr
ease
s,th
e va
lue
of � x
�32
�ge
ts s
mal
ler.
In o
ther
wor
ds,s
ince
� x�3
2�
→ 0
as
x →
∞or
x →
�∞
,y�
x�
6 is
an
obl
iqu
e as
ympt
ote.
Use
syn
thet
ic d
ivis
ion
to
fin
d t
he
obli
qu
e as
ymp
tote
for
eac
h f
un
ctio
n.
1.y
��8x
2� x
�4x5�
11�
y�
8x�
44
2.y
��x2
�x
3 �x2�
15�
y�
x�
5
3.y
��x2
�x
2 �x3�
18�
y�
x�
1
4.y
��ax
2
x��bx
d�
c�
y�
ax�
b�
ad
5.y
��ax
2
x��bx
d�
c�
y�
ax�
b�
ad
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
Enri
chm
ent
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
8-6
Exam
ple
Chapter 8 A24 Glencoe Algebra 2
Copyright ©
Glencoe/M
cGraw
-Hill, a division of T
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)
Chapter 8 A25 Glencoe Algebra 2
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. 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
Chapter 8 A26 Glencoe Algebra 2
Copyright ©
Glencoe/M
cGraw
-Hill, a division of T
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
Chapter 8 A27 Glencoe Algebra 2
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
Chapter 8 A28 Glencoe Algebra 2
Copyright ©
Glencoe/M
cGraw
-Hill, a division of T
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: �
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
Chapter 8 A29 Glencoe Algebra 2
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)
Chapter 8 A30 Glencoe Algebra 2
Copyright ©
Glencoe/M
cGraw
-Hill, a division of T
he McG
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.
Chapter 8 A31 Glencoe Algebra 2
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. 1 0 4
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
9
8
7
6
5
4
3
2
1
0
1 3
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
9
8
7
6
5
4
3
2
1
0
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
F G H J
A B C D
F G H J
A B C D
F G H J
A B C D
Chapter 8 Assessment Answer KeyStandardized Test PracticePage 68 Page 69
Chapter 8 A32 Glencoe Algebra 2
Copyright ©
Glencoe/M
cGraw
-Hill, a division of T
he McG
raw-H
ill Com
panies, Inc.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28a.
28b.
28c.9920 ounces or
77.5 gallons
joint variation
h � 80xd
11.2 mL
20
3(y � 2)
1, 2, 3, 6, �12
�, �32
�
23
y
xO
�1 �i
w�125�
no
Chapter 8 Assessment Answer KeyStandardized Test PracticePage 70
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