Chapter 9 Resource Masters - KTL MATH CLASSES · Chapter 9 Test, Form 3 . . . . . . . . . . ....
-
Upload
truongdiep -
Category
Documents
-
view
287 -
download
4
Transcript of Chapter 9 Resource Masters - KTL MATH CLASSES · Chapter 9 Test, Form 3 . . . . . . . . . . ....
Chapter 9Resource Masters
Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.
Study Guide and Intervention Workbook 0-07-828029-XSkills Practice Workbook 0-07-828023-0Practice Workbook 0-07-828024-9
ANSWERS FOR WORKBOOKS The answers for Chapter 9 of these workbookscan be found in the back of this Chapter Resource Masters booklet.
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teacher, and families without charge; and be used solely in conjunction with Glencoe’s Algebra 2. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.
Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027
ISBN: 0-07-828012-5 Algebra 2Chapter 9 Resource Masters
1 2 3 4 5 6 7 8 9 10 066 11 10 09 08 07 06 05 04 03 02
Glencoe/McGraw-Hill
© Glencoe/McGraw-Hill iii Glencoe Algebra 2
Contents
Vocabulary Builder . . . . . . . . . . . . . . . . vii
Lesson 9-1Study Guide and Intervention . . . . . . . . 517–518Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 519Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 520Reading to Learn Mathematics . . . . . . . . . . 521Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 522
Lesson 9-2Study Guide and Intervention . . . . . . . . 523–524Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 525Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 526Reading to Learn Mathematics . . . . . . . . . . 527Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 528
Lesson 9-3Study Guide and Intervention . . . . . . . . 529–530Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 531Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 532Reading to Learn Mathematics . . . . . . . . . . 533Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 534
Lesson 9-4Study Guide and Intervention . . . . . . . . 535–536Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 537Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 538Reading to Learn Mathematics . . . . . . . . . . 539Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 540
Lesson 9-5Study Guide and Intervention . . . . . . . . 541–542Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 543Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 544Reading to Learn Mathematics . . . . . . . . . . 545Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 546
Lesson 9-6Study Guide and Intervention . . . . . . . . 547–548Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 549Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 550Reading to Learn Mathematics . . . . . . . . . . 551Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 552
Chapter 9 AssessmentChapter 9 Test, Form 1 . . . . . . . . . . . . 553–554Chapter 9 Test, Form 2A . . . . . . . . . . . 555–556Chapter 9 Test, Form 2B . . . . . . . . . . . 557–558Chapter 9 Test, Form 2C . . . . . . . . . . . 559–560Chapter 9 Test, Form 2D . . . . . . . . . . . 561–562Chapter 9 Test, Form 3 . . . . . . . . . . . . 563–564Chapter 9 Open-Ended Assessment . . . . . . 565Chapter 9 Vocabulary Test/Review . . . . . . . 566Chapter 9 Quizzes 1 & 2 . . . . . . . . . . . . . . . 567Chapter 9 Quizzes 3 & 4 . . . . . . . . . . . . . . . 568Chapter 9 Mid-Chapter Test . . . . . . . . . . . . . 569Chapter 9 Cumulative Review . . . . . . . . . . . 570Chapter 9 Standardized Test Practice . . 571–572
Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A29
© Glencoe/McGraw-Hill iv Glencoe Algebra 2
Teacher’s Guide to Using theChapter 9 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 9 Resource Masters includes the core materials neededfor Chapter 9. These materials include worksheets, extensions, and assessment options.The answers for these pages appear at the back of this booklet.
All of the materials found in this booklet are included for viewing and printing in theAlgebra 2 TeacherWorks CD-ROM.
Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.
WHEN TO USE Give these pages tostudents before beginning Lesson 9-1.Encourage them to add these pages to theirAlgebra 2 Study Notebook. Remind them to add definitions and examples as theycomplete each lesson.
Study Guide and InterventionEach lesson in Algebra 2 addresses twoobjectives. There is one Study Guide andIntervention master for each objective.
WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.
Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.
WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.
Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.
WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.
Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.
WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.
Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.
WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.
© Glencoe/McGraw-Hill v Glencoe Algebra 2
Assessment OptionsThe assessment masters in the Chapter 9Resource Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.
Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions
and is intended for use with basic levelstudents.
• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.
• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.
• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.
All of the above tests include a free-response Bonus question.
• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.
• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.
Intermediate Assessment• Four free-response quizzes are included
to offer assessment at appropriateintervals in the chapter.
• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.
Continuing Assessment• The Cumulative Review provides
students an opportunity to reinforce andretain skills as they proceed throughtheir study of Algebra 2. It can also beused as a test. This master includes free-response questions.
• The Standardized Test Practice offerscontinuing review of algebra concepts invarious formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and quantitative-comparison questions. Bubble-in andgrid-in answer sections are provided onthe master.
Answers• Page A1 is an answer sheet for the
Standardized Test Practice questionsthat appear in the Student Edition onpages 518–519. This improves students’familiarity with the answer formats theymay encounter in test taking.
• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.
• Full-size answer keys are provided forthe assessment masters in this booklet.
Reading to Learn MathematicsVocabulary Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
99
Voca
bula
ry B
uild
erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 9.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
asymptote
A·suhm(p)·TOHT
complex fraction
constant of variation
continuity
KAHN·tuhn·OO·uh·tee
direct variation
inverse variation
IHN·VUHRS
joint variation
(continued on the next page)
© Glencoe/McGraw-Hill vii Glencoe Algebra 2
© Glencoe/McGraw-Hill viii Glencoe Algebra 2
Vocabulary Term Found on Page Definition/Description/Example
point discontinuity
rational equation
rational expression
rational function
rational inequality
Reading to Learn MathematicsVocabulary Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
99
Study Guide and InterventionMultiplying and Dividing Rational Expressions
NAME ______________________________________________ DATE ____________ PERIOD _____
9-19-1
© Glencoe/McGraw-Hill 517 Glencoe Algebra 2
Less
on
9-1
Simplify Rational Expressions A ratio of two polynomial expressions is a rationalexpression. To simplify a rational expression, divide both the numerator and thedenominator by their greatest common factor (GCF).
Multiplying Rational Expressions For all rational expressions and , � � , if b � 0 and d � 0.
Dividing Rational Expressions For all rational expressions and , � � , if b � 0, c � 0, and d � 0.
Simplify each expression.
a.
� �
b. �
� � � �
c. �
� � �
� �
Simplify each expression.
1. �(�220aabb
2
4)3
� � 2. 3.
4. � 2m2(m � 1) 5. �
6. � m 7. �
8. � 9. �
4��
p(4p � 1)��
4m2 � 1��4m � 8
2m � 1��m2 � 3m � 10
4p2 � 7p � 2��
7p516p2 � 8p � 1��
14p4
y5�18xz2
�5y6xy4�25z3
m3 � 9m��
m2 � 9(m � 3)2
��m2 � 6m � 9
c�c2 � 4c � 5
��c2 � 4c � 3
c2 � 3c�c2 � 25
4m5�m � 1
3m3 � 3m��
6m4
x � 2�x2 � x � 6
��x2 � 6x � 27
3 � 2x�4x2 � 12x � 9
��9 � 6x2a2b2�
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
ExampleExample
ExercisesExercises
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
© Glencoe/McGraw-Hill 518 Glencoe Algebra 2
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. (b � 1)2
4. 5.
6. a � 4 7. x � 3
8. 9.1
�
x2 � x � 2���x3 � 6x2 � x � 30���x � 1
�x � 3
b � 4��
�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
�
1��
�x2 �
x �6x
4� 9
�
���x2 �
3 �2x
x� 8
�
2(b � 10)��b(3b � 1)
�b2 �
b2100�
���3b2 � 3
21bb � 10�
�3bb
2 ��
12
�
���3b2
b�
�b1� 2
�
ac7�
�ax
2
2byc2
3�
��ca4xb
2
2
y�
xyz�
�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�
Study Guide and Intervention (continued)
Multiplying and Dividing Rational Expressions
NAME ______________________________________________ DATE ____________ PERIOD _____
9-19-1
ExampleExample
1
1 1
s3
ExercisesExercises
Skills PracticeMultiplying and Dividing Rational Expressions
NAME ______________________________________________ DATE ____________ PERIOD _____
9-19-1
© Glencoe/McGraw-Hill 519 Glencoe Algebra 2
Less
on
9-1
Simplify each expression.
1. 2.
3. x6 4.
5. 6.
7. 8. �
9. � 6e 10. �
11. � 21g3 12. �
13. � x (x � 2) 14. �
15. � 16. �
(w � 8)(w � 7)
17. � (3x2 � 3x) 18. �
19. � 20.a � b�
�a2
4�a
b2�
��a
2�a
b�
5�
�2cd
2
2�
�
��5cd
6�
(4a � 5)(a � 4)��4a � 5
��a2 � 8a � 16
16a2 � 40a � 25���
3a2 � 10a � 81
�x2 � 5x � 4��2x � 8
t � 12�2t � 2
��t2 � 9t � 14
t2 � 19t � 84��4t � 4
w2 � 6w � 7��w � 3
w2 � 5w � 24��w � 1
q2�
q2 � 4�
3q2q2 � 2q�6q
3x�x2 � 4
3x2�x � 2
32z7�
25y5�14z12v5
80y4�49z5v7
1�
7g�y2
1��s � 2
�10s5
5s2�s2 � 4
10(ef)3�
8e5f24e3�5f 2
mn2�n3
�63m�2n
a � 8�3a2 � 24a
��3a2 � 12a
x � 2�x2 � 4
��(x � 2)(x � 1)9
�18�2x � 6
2�
8y2(y6)3�
4y24(x6)3�(x3)4
b�5ab3
�25a2b2
3x�
21x3y�14x2y2
© Glencoe/McGraw-Hill 520 Glencoe Algebra 2
Simplify each expression.
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. 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.x � 2 units
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. cm3x � 5�
x2 � 3x��x2 � 5x � 6
x2 � 3x � 10��2x
x2 � 2x � 4��
�xx
2
3
��
22x
3�
��
�x2
(�x �
4x2
�)3
4�
�x2
4� 9�
��3 �
8x
�
2x � 1�
�2x
x� 1�
��4 �
xx
�
5�
2a � 6�5a � 10
9 � a2��a2 � 5a � 6
2s � 3��
s2 � 10s � 25��s � 4
2s2 � 7s � 15��
(s � 4)22
��6x2 � 12x��4x � 12
3x � 6�x2 � 9
1��
x2 � y2�3
x � y�6
xy3�
24x2�w5
2xy�w2
a2w2�
a3w2�w5y2
a5y3�wy7
5x � 1��
25x2 � 1��x2 � 10x � 25
x � 5�10x � 2
5x�
5x2�8 � x
x2 � 5x � 24��6x � 2x2
w2 � n2�y � a
a � y�w � n
1�
n2 � 6n�
n8n5
�n � 62�
4�y � a
a � y�6
5ux2�
25x3�14u2y2
�2u3y�15xz5
x � 2�
x4 � x3 � 2x2��
x4 � x3
v � 5�
25 � v2��3v2 � 13v � 10
2k � 5�
2k2 � k � 15��
k2 � 9
2y � 3�
10y2 � 15y��35y2 � 5y
4m4n2�
(2m3n2)3���18m5n4
1�
9a2b3�27a4b4c
Practice (Average)
Multiplying and Dividing Rational Expressions
NAME ______________________________________________ DATE ____________ PERIOD _____
9-19-1
Reading to Learn MathematicsMultiplying and Dividing Rational Expressions
NAME ______________________________________________ DATE ____________ PERIOD _____
9-19-1
© Glencoe/McGraw-Hill 521 Glencoe Algebra 2
Less
on
9-1
Pre-Activity How are rational expressions used in mixtures?
Read the introduction to Lesson 9-1 at the top of page 472 in your textbook.
• Suppose that the Goodie Shoppe also sells a candy mixture of chocolatemints and caramels. If this mixture is 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, whatfraction of the mixture will be mints?
Reading 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? ii, iv, v
i. ii. iii. iv. v.
b. Does a complex fraction express a multiplication or division problem? divisionHow is multiplication used in simplifying a complex fraction? Sample answer: To divide the numerator of the complex fraction by the denominator,multiply the numerator by the reciprocal of the denominator.
Helping You Remember
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? Sample answer: To dividerational expressions, multiply the first expression by the reciprocal ofthe second. This is the same “invert and multiply” process that is usedwhen dividing arithmetic fractions.
�r2 �
925
�
��r �
35
�
�z �
z1
�
�zr � 5�r � 5
�38
�
��156�
7�12
divisorreciprocalmultiply
numeratorsmultiply
0denominatorfactor
0denominatorgreatest common factor
denominatorfactor
4 � y�
�37
�
�47
�
© Glencoe/McGraw-Hill 522 Glencoe Algebra 2
Reading AlgebraIn mathematics, the term group has a special meaning. The followingnumbered sentences discuss the idea of group and one interesting example of a group.
01 To be a group, a set of elements and a binary operation must satisfy fourconditions: the set must be closed under the operation, the operationmust be associative, there must be an identity element, and everyelement must have an inverse.
02 The following six functions form a group under the operation of
composition of functions: f1(x) � x, f2(x) � �1x�, f3(x) � 1 � x,
f4(x) � �(x �
x1)
�, f5(x) � �(x �x
1)�, and f6(x) � �(1 �1
x)�.
03 This group is an example of a noncommutative group. For example,f3 � f2 � f4, but f2 � f3 � f6.
04 Some experimentation with this group will show that the identityelement is f1.
05 Every element is its own inverse except for f4 and f6, each of which is theinverse of the other.
Use the paragraph to answer these questions.
1. Explain what it means to say that a set is closed under an operation. Is the set of positive integers closed under subtraction?
2. Subtraction is a noncommutative operation for the set of integers. Writean informal definition of noncommutative.
3. For the set of integers, what is the identity element for the operation ofmultiplication? Justify your answer.
4. Explain how the following statement relates to sentence 05:
(f6 � f4)(x) � f6[ f4(x)] � f6��(1 �1
x)�� � � x � f1(x).1���1 � (x
x� 1)�
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
9-19-1
Study Guide and InterventionAdding and Subtracting Rational Expressions
NAME ______________________________________________ DATE ____________ PERIOD _____
9-29-2
© Glencoe/McGraw-Hill 523 Glencoe Algebra 2
Less
on
9-2
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)
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find the LCM of each set of polynomials.
1. 14ab2, 42bc3, 18a2c 2. 8cdf3, 28c2f, 35d4f 2
126a2b2c3 280c2d4f 3
3. 65x4y, 10x2y2, 26y4 4. 11mn5, 18m2n3, 20mn4
130x4y4 1980m2n5
5. 15a4b, 50a2b2, 40b8 6. 24p7q, 30p2q2, 45pq3
600a4b8 360p7q3
7. 39b2c2, 52b4c, 12c3 8. 12xy4, 42x2y, 30x2y3
156b4c3 420x2y4
9. 56stv2, 24s2v2, 70t3v3 10. x2 � 3x, 10x2 � 25x � 15840s2t3v3 5x(x � 3)(2x � 1)
11. 9x2 � 12x � 4, 3x2 � 10x � 8 12. 22x2 � 66x � 220, 4x2 � 16(3x � 2)2(x � 4) 44(x � 2)(x � 2)(x � 5)
13. 8x2 � 36x � 20, 2x2 � 2x � 60 14. 5x2 � 125, 5x2 � 24x � 54(x � 5)(x � 6)(2x � 1) 5(x � 5)(x � 5)(5x � 1)
15. 3x2 � 18x � 27, 2x3 � 4x2 � 6x 16. 45x2 � 6x � 3, 45x2 � 56x(x � 3)2(x � 1) 15(5x � 1)(3x � 1)(3x � 1)
17. x3 � 4x2 � x � 4, x2 � 2x � 3 18. 54x3 � 24x, 12x2 � 26x � 12(x � 1)(x � 1)(x � 3)(x � 4) 6x(3x � 2)(3x � 2)(2x � 3)
© Glencoe/McGraw-Hill 524 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. ��2x2 � 9x � 4��(2x � 1)(2x � 1)2
5x��20x2 � 5
4��4x2 � 4x � 1
4�x � 1
�x2 � 1
3x � 3��x2 � 2x � 1
4x � 14�4x � 5
�3x � 63
�x � 24a2 � 9b2��15b
�5ac4a�3bc
x � 1��1
�x � 12
�x � 3y�
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
Study Guide and Intervention (continued)
Adding and Subtracting Rational Expressions
NAME ______________________________________________ DATE ____________ PERIOD _____
9-29-2
ExampleExample
ExercisesExercises
Skills PracticeAdding and Subtracting Rational Expressions
NAME ______________________________________________ DATE ____________ PERIOD _____
9-29-2
© Glencoe/McGraw-Hill 525 Glencoe Algebra 2
Less
on
9-2
Find the LCM of each set of polynomials.
1. 12c, 6c2d 12c2d 2. 18a3bc2, 24b2c2 72a3b2c2
3. 2x � 6, x � 3 2(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)
Simplify each expression.
7. � 8. �
9. � 4 10. �
11. � 12. �
13. � 14. �
15. � 16. �
17. � 18. �
19. � 20. �
21. � 22. �
y � 12���
n � 2�
2��y2 � 6y � 8
3��y2 � y � 12
2n � 2��n2 � 2n � 3
n�n � 3
2x2 � 5x � 2��
4��x2 � 3x � 10
2x � 1�x � 5
x2 � x � 1��
x�x � 1
1��x2 � 2x � 1
5z2 � 4z � 16��
z � 4�z � 1
4z�z � 4
2m�
m�n � m
m�m � n
5 � 3t�
5�x � 2
3t�2 � x
3w � 7��
2�w2 � 9
3�w � 3
15bd � 6b � 2d��
2�3bd
5�3b � d
a � 6��
3�2a
2�a � 2
7h � 3g��
3�4h2
7�4gh
12z � 2y��
2�5yz
12�5y2
2 � 5m2��
5�n
2�m2n
2c � 5�
2c � 7�3
13�
5�4p2q
3�8p2q
5x � 3y�
5�y
3�x
© Glencoe/McGraw-Hill 526 Glencoe Algebra 2
Find the LCM of each set of polynomials.
1. x2y, xy3 2. a2b3c, abc4 3. x � 1, x � 3x2y3 a2b3c4 (x � 1)(x � 3)
4. g � 1, g2 � 3g � 4 5. 2r � 2, r2 � r, r � 1 6. 3, 4w � 2, 4w2 � 1(g � 1)(g � 4) 2r(r � 1) 6(2w � 1)(2w � 1)
7. x2 � 2x � 8, x � 4 8. x2 � x � 6, x2 � 6x � 8 9. d2 � 6d � 9, 2(d2 � 9)(x � 4)(x � 2) (x � 2)(x � 4)(x � 3) 2(d � 3)(d � 3)2
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.
h4r��
d�r
5(x3 � 4x � 16)��
10�x � 4
20�x � 4
5x�2
r � 4�
3x � y�
12�
�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
3(6 � 5n)��
2p2 � 2p � 1���
5��
7�10n
3�4
1�5n
5�p2 � 9
2p � 3��p2 � 5p � 6
20��x2 � 4x � 12
5�2x � 12
2y � 1��
7 � 9m�
2�
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��
2(x � 3)(x � 2)��
2(2 � 3n)��
3n
9�a � 5
4�a � 3
x � 8�x � 4
4m�3mn
2d 2 � 9c��
25y2 � 12x2��
20 � 21b��
24ab
3�4cd3
1�6c2d
1�5x2y3
5�12x4y
7�8a
5�6ab
Practice (Average)
Adding and Subtracting Rational Expressions
NAME ______________________________________________ DATE ____________ PERIOD _____
9-29-2
Reading to Learn MathematicsAdding and Subtracting Rational Expressions
NAME ______________________________________________ DATE ____________ PERIOD _____
9-29-2
© Glencoe/McGraw-Hill 527 Glencoe Algebra 2
Less
on
9-2
Pre-Activity How is subtraction of rational expressions used in photography?
Read the introduction to Lesson 9-2 at the top of page 479 in your textbook.
A person is standing 5 feet from a camera that has a lens with a focallength of 3 feet. Write an equation that you could solve to find how far thefilm should be from the lens to get a perfectly focused photograph.
� �
Reading the Lesson
1. a. In work with rational expressions, LCD stands for
and LCM stands for . The LCD is the of the 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 .
Helping You Remember
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?
Sample answer: In arithmetic, a common denominator is needed to addand subtract fractions, but not to multiply and divide them. The situationis the same for rational expressions.
simplest form
denominatornumerators
LCD
LCD
LCM
denominatorx � 4��x3 � 4x2 � 4x
x2 � 3��x2 � 5x � 6
factorgreatestfactorpolynomial
factor
LCMleast common multipleleast common denominator
1�
1�
1�
© Glencoe/McGraw-Hill 528 Glencoe Algebra 2
SuperellipsesThe circle and the ellipse are members of an interesting family of curves that were first studied by the French physicist and mathematician Gabriel Lamé (1795–1870). The general equation for the family is
� �ax
� �n � � �by
� �n � 1, with a � 0, b � 0, and n � 0.
For even values of n greater than 2, the curves are called superellipses.
1. Consider two curves that are not superellipses.Graph each equation on the grid at the right.State the type of curve produced each time.
a. � �2x
� �2 � � �2y
� �2 � 1
b. � �3x
� �2 � � �2y
� �2 � 1
2. In each of the following cases you are given values of a, b, and n to use in the general equation. Write the resulting equation. Then graph. Sketch each graph on the grid at the right.
a. a � 2, b � 3, n � 4 b. a � 2, b � 3, n � 6 c. a � 2, b � 3, n � 8
3. What shape will the graph of � �2x
� �n � � �2y
� �napproximate for greater and greater even,whole-number values of n?
1–1–2–3 2 3
3
2
1
–1
–2
–3
1–1–2–3 2 3
3
2
1
–1
–2
–3
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
9-29-2
Study Guide and InterventionGraphing Rational Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
9-39-3
© Glencoe/McGraw-Hill 529 Glencoe Algebra 2
Less
on
9-3
Vertical Asymptotes and Point Discontinuity
Rational Function an equation of the form f(x) � , where p(x) and q(x) are polynomial expressions and q(x) � 0
Vertical Asymptote An asymptote is a line that the graph of a function approaches, but never crosses. of the Graph of a If the simplified form of the related rational expression is undefined for x � a, Rational Function 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 of the Graph of a for x � a but the simplified expression is defined for x � a, then there is a hole in the Rational Function graph at x � a.
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) �
asymptotes: x � 2, hole: x � asymptote: x � 0; x � �5 hole x � 4
4. f(x) � 5. f(x) � 6. f(x) �
asymptote: x � �2; asymptotes: x � 1, asymptote: x � �3
hole: x � x � �7
7. f(x) � 8. f(x) � 9. f(x) �
asymptotes: x � 1, asymptote: x � �3; holes: x � 1, x � 3 x � 5 hole: x �
3�
x3 � 2x2 � 5x � 6���
x2 � 4x � 32x2 � x � 3��2x2 � 3x � 9
x � 1��x2 � 6x � 5
1�
3x2 � 5x � 2��x � 3
x2 � 6x � 7��x2 � 6x � 7
3x � 1��3x2 � 5x � 2
5�
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)
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 530 Glencoe Algebra 2
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
NAME ______________________________________________ DATE ____________ PERIOD _____
9-39-3
ExampleExample
ExercisesExercises
Skills PracticeGraphing Rational Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
9-39-3
© Glencoe/McGraw-Hill 531 Glencoe Algebra 2
Less
on
9-3
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) �
asymptotes: x � 4, x � �2 asymptotes: x � 4, x � 9
3. f(x) � 4. f(x) �
asymptote: x � 2; hole: x � �12 asymptote: x � 3; hole: x � 1
5. f(x) � 6. f(x) �
hole: x � �2 hole: x � 3
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
© Glencoe/McGraw-Hill 532 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) �
asymptotes: x � 2, asymptote: x � 3; asymptote: x � �2x � �5 hole: x � 7
4. f(x) � 5. f(x) � 6. f(x) �
hole: x � �10 hole: x � 6 hole: x � �5
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?
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? 11.25 foot-candles
4500�
d24500�
d2
5�
6 � x�6x
6 � x�6x
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
Practice (Average)
Graphing Rational Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
9-39-3
Reading to Learn MathematicsGraphing Rational Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
9-39-3
© Glencoe/McGraw-Hill 533 Glencoe Algebra 2
Less
on
9-3
Pre-Activity How can rational functions be used when buying a group gift?
Read the introduction to Lesson 9-3 at the top of page 485 in your textbook.
• If 15 students contribute to the gift, how much would each of them pay?$10
• If each student pays $5, how many students contributed?30 students
Reading the Lesson
1. Which of the following are rational functions? A and C
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 .
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) � II g(x) � I
Helping You Remember
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?
Sample answer: A point discontinuity or vertical asymptote occurs wherethe function is undefined, that is, where the denominator of the relatedrational expression is equal to 0. Therefore, set the denominator equal tozero and solve for the variable.
x2 � 4�x � 2
x�x � 2
�2vertical asymptote
�2point discontinuity
x
y
Ox
y
O
discontinuitiesasymptotescontinuity
x2 � 25��x2 � 6x � 9
1�x � 5
© Glencoe/McGraw-Hill 534 Glencoe Algebra 2
Graphing with Addition of y-CoordinatesEquations of parabolas, ellipses, and hyperbolas that are “tipped” with respect to the x- and y-axes are more difficult to graph than the equations you have been studying.
Often, however, you can use the graphs of two simplerequations to graph a more complicated equation. For example, the graph of the ellipse in the diagram at the right is obtained by adding the y-coordinate of each point on the circle and the y-coordinate of the corresponding point of the line.
Graph each equation. State the type of curve for each graph.
1. y � 6 � x � �4 � x2� 2. y � x � �x�
Use a separate sheet of graph paper to graph these equations. State the type ofcurve for each graph.
3. y � 2x � �7 � 6�x � x2� 4. y � �2x � ��2x�
1 4 5 6 72 3
8
7
6
5
4
3
2
1
–1
–2
y � x
y � ���x
x
y
O
1 4 5–1–2 2 3
9
8
7
6
5
4
3
2
1
–1
–2y � �����4x � x2
y � 6 � x
x
y
O
x
y
O
y � �����4x � x2
y � x � 6 � ����4x � x2
y � x � 6
A
B�
A�
B
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
9-39-3
Study Guide and InterventionDirect, Joint, and Inverse Variation
NAME ______________________________________________ DATE ____________ PERIOD _____
9-49-4
© Glencoe/McGraw-Hill 535 Glencoe Algebra 2
Less
on
9-4
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.ExampleExample
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
ExercisesExercises
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. 12 x � 36, find y when x � 54. 24
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. 27 y � 22, find x when y � 32. 48
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. 45 and z � 2. 36
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. 176 x � 4 and z � 7. 30
9. If y varies directly as x and y � 14 10. If y varies directly as x and x � 200 whenwhen x � 35, find y when x � 12. 4.8 y � 50, find x when y � 1000. 4000
11. If y varies directly as x and y � 39 12. If y varies directly as x and x � 60 whenwhen x � 52, find y when x � 22. 16.5 y � 75, find x when y � 42. 33.6
13. Suppose y varies jointly as x and z. 14. Suppose y varies jointly as x and z. Find yFind y when x � 6 and z � 11, if when x � 5 and z � 10, if y � 12 when y � 120 when x � 5 and z � 12. 132 x � 8 and z � 6. 12.5
15. Suppose y varies jointly as x and z. 16. 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. 567 x � 9 and z � 20. 360
© Glencoe/McGraw-Hill 536 Glencoe Algebra 2
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. 8
2. If y varies inversely as x and y � 9 when x � 45, find y when x � 27. 15
3. If y varies inversely as x and y � 100 when x � 38, find y when x � 76. 50
4. If y varies inversely as x and y � 32 when x � 42, find y when x � 24. 56
5. If y varies inversely as x and y � 36 when x � 10, find y when x � 30. 12
6. If y varies inversely as x and y � 75 when x � 12, find y when x � 10. 90
7. If y varies inversely as x and y � 18 when x � 124, find y when x � 93. 24
8. If y varies inversely as x and y � 90 when x � 35, find y when x � 50. 63
9. If y varies inversely as x and y � 42 when x � 48, find y when x � 36. 56
10. If y varies inversely as x and y � 44 when x � 20, find y when x � 55. 16
11. If y varies inversely as x and y � 80 when x � 14, find y when x � 35. 32
12. If y varies inversely as x and y � 3 when x � 8, find y when x � 40. 0.6
13. If y varies inversely as x and y � 16 when x � 42, find y when x � 14. 48
14. If y varies inversely as x and y � 9 when x � 2, find y when x � 5. 3.6
15. If y varies inversely as x and y � 23 when x � 12, find y when x � 15. 18.4
a2�12
8�4
a2�b1
a1�b2
k�x
Study Guide and Intervention (continued)
Direct, Joint, and Inverse Variation
NAME ______________________________________________ DATE ____________ PERIOD _____
9-49-4
ExampleExample
ExercisesExercises
Skills PracticeDirect, Joint, and Inverse Variation
NAME ______________________________________________ DATE ____________ PERIOD _____
9-49-4
© Glencoe/McGraw-Hill 537 Glencoe Algebra 2
Less
on
9-4
State whether each equation represents a direct, joint, or inverse variation. Thenname the constant of variation.
1. c � 12m direct; 12 2. p � inverse; 4 3. A � bh joint;
4. rw � 15 inverse; 15 5. y � 2rst joint; 2 6. f � 5280m direct; 5280
7. y � 0.2s direct; 0.2 8. vz � �25 inverse; �25 9. t � 16rh joint; 16
10. R � inverse; 8 11. � direct; 12. C � 2�r direct; 2�
Find each value.
13. If y varies directly as x and y � 35 when x � 7, find y when x � 11. 55
14. If y varies directly as x and y � 360 when x � 180, find y when x � 270. 540
15. If y varies directly as x and y � 540 when x � 10, find x when y � 1080. 20
16. If y varies directly as x and y � 12 when x � 72, find x when y � 9. 54
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. 90
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. 14
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. 30
20. If y varies inversely as x and y � 2 when x � 2, find y when x � 1. 4
21. If y varies inversely as x and y � 6 when x � 5, find y when x � 10. 3
22. If y varies inversely as x and y � 3 when x � 14, find x when y � 6. 7
23. If y varies inversely as x and y � 27 when x � 2, find x when y � 9. 6
24. If y varies directly as x and y � �15 when x � 5, find x when y � �36. 12
1�
1�3
a�b
8�w
1�
1�2
4�q
© Glencoe/McGraw-Hill 538 Glencoe Algebra 2
State whether each equation represents a direct, joint, or inverse variation. Thenname the constant of variation.
1. u � 8wz joint; 8 2. p � 4s direct; 4 3. L � inverse; 5 4. xy � 4.5 inverse; 4.5
5. � � 6. 2d � mn 7. � h 8. y �
direct; � joint; inverse; 1.25 inverse;
Find each value.
9. If y varies directly as x and y � 8 when x � 2, find y when x � 6. 24
10. If y varies directly as x and y � �16 when x � 6, find x when y � �4. 1.5
11. If y varies directly as x and y � 132 when x � 11, find y when x � 33. 396
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. 288
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. 240
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. 8
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. 6
18. If y varies inversely as x and y � �18 when x � 6, find y when x � 5. �21.6
19. If y varies directly as x and y � 5 when x � 0.4, find x when y � 37.5. 3
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. 500 cm3
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? 12 in.
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? 100 m2
64�
56�
3�
1�
3�4x
1.25�g
C�d
5�k
Practice (Average)
Direct, Joint, and Inverse Variation
NAME ______________________________________________ DATE ____________ PERIOD _____
9-49-4
Reading to Learn MathematicsDirect, Joint, and Inverse Variation
NAME ______________________________________________ DATE ____________ PERIOD _____
9-49-4
© Glencoe/McGraw-Hill 539 Glencoe Algebra 2
Less
on
9-4
Pre-Activity How is variation used to find the total cost given the unit cost?
Read the introduction to Lesson 9-4 at the top of page 492 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
.
Reading 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. m �
b. s varies directly as r. s � kr
c. t varies jointly as p and q. t � kpq
2. Which type of variation, direct or inverse, is represented by each graph?
a. inverse b. direct
Helping You Remember
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?
Sample answer: The graph of an equation expressing direct variation isa line. The slope-intercept form of the equation of a line is y � mx � b. Indirect variation, if one of the quantities is 0, the other quantity is also 0,so b � 0 and the line goes through the origin. The equation of a linethrough the origin is y � mx, where m is the slope. This is the same asthe equation for direct variation with k � m.
x
y
Ox
y
O
k�
$14,900decrease
$149increase
© Glencoe/McGraw-Hill 540 Glencoe Algebra 2
Expansions of Rational ExpressionsMany rational expressions can be transformed into power series. A powerseries is an infinite series of the form A � Bx � Cx2 � Dx3 � …. Therational expression and the power series normally can be said to have thesame values only for certain values of x. For example, the following equationholds only for values of x such that �1 x 1.
�1 �1
x� � 1 � x � x2 � x3 � … for �1 x 1
Expand in ascending powers of x.
Assume that the expression equals a series of the form A � Bx � Cx2 � Dx3 � ….Then multiply both sides of the equation by the denominator 1 � x � x2.
�1 �
2 �
x �
3xx2� � A � Bx � Cx2 � Dx3 � …
2 � 3x � (1 � x � x2)(A � Bx � Cx2 � Dx3 � …)2 � 3x � A � Bx � Cx2 � Dx3 � …
� Ax � Bx2 � Cx3 � …� Ax2 � Bx3 � …
2 � 3x � A � (B � A)x � (C � B � A)x2 � (D � C � B)x3 � …
Now, match the coefficients of the polynomials.2 � A3 � B � A0 � C � B � A0 � D � C � B � A
Finally, solve for A, B, C, and D and write the expansion.A � 2, B � 1, C � �3, and D � 0
Therefore, �1 �
2 �
x �
3xx2� � 2 � x � 3x2 � …
Expand each rational expression to four terms.
1. �1 �
1x�
�
xx2�
2. �1 �2
x�
3. �1 �1
x�
2 � 3x��1 � x � x2
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
9-49-4
ExampleExample
Study Guide and InterventionClasses of Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
9-59-5
© Glencoe/McGraw-Hill 541 Glencoe Algebra 2
Less
on
9-5
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.
quadratic rational direct variation4. 5. 6.
constant absolute value greatest integer7. 8. 9.
identity square root inverse variation
x
y
O
x
y
O
x
y
O
x
y
Ox
y
Ox
y
O
x
y
Ox
y
O
x
y
O
ExercisesExercises
© Glencoe/McGraw-Hill 542 Glencoe Algebra 2
Identify Equations You should be able to graph the equations of the following functions.
Function General Equation
Constant y � a
Direct Variation y � ax
Identity y � x
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 � inverse variation 2. y � x direct variation 3. y � � quadratic
4. y � |3x| � 1 absolutevalue 5. y � � inverse variation6. y � greatest
integer
7. y � �x � 2� square root 8. y � 3.2 constant 9. y � rational
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
NAME ______________________________________________ DATE ____________ PERIOD _____
9-59-5
ExercisesExercises
Skills PracticeClasses of Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
9-59-5
© Glencoe/McGraw-Hill 543 Glencoe Algebra 2
Less
on
9-5
Identify the type of function represented by each graph.
1. 2. 3.
constant direct variation quadratic
Match each graph with an equation below.
A. y � |x � 1| B. y � C. y � �1 � x� D. y � �x� � 1
4. B 5. C 6. A
Identify the type of function represented by each equation. Then graph theequation.
7. y � 8. y � 2�x� 9. y � �3x
inverse variation greatest integer direct variationor rational
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
© Glencoe/McGraw-Hill 544 Glencoe Algebra 2
Identify the type of function represented by each graph.
1. 2. 3.
rational square root absolute value
Match each graph with an equation below.
A. y � |2x � 1 | B. y � �2x � 1� C. y � D. y � ��x�
4. D 5. C 6. A
Identify the type of function represented by each equation. Then graph theequation.
7. y � �3 8. y � 2x2 � 1 9. y �
constant quadratic rational
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.The graph is U-shaped; it is a parabola.
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. The graph looks like a series of steps, similar to a greatest integer function, but with open circles on the left and closedcircles on the right; $58.
1�2
x
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
Practice (Average)
Classes of Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
9-59-5
Reading to Learn MathematicsClasses of Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
9-59-5
© Glencoe/McGraw-Hill 545 Glencoe Algebra 2
Less
on
9-5
Pre-Activity How can graphs of functions be used to determine a person’sweight on a different planet?
Read the introduction to Lesson 9-5 at the top of page 499 in your textbook.
• Based on the graph, estimate the weight on Mars of a child who weighs40 pounds on Earth.about 15 pounds
• Although the graph does not extend far enough to the right to read itdirectly from the graph, use the weight you found above and yourknowledge that this graph represents direct variation to estimate theweight on Mars of a woman who weighs 120 pounds on Earth.about 45 pounds
Reading 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. III b. I c. VI
d. II e. IV f. V
Helping You Remember
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 |? Sample answer: Using thedefinition of absolute value, f(x) � x if x 0 and f(x) � �x if x 0.Therefore, the graph is made up of pieces of two lines, one with slope 1and one with slope �1, meeting at the origin. This forms a V-shapedgraph with “vertex” at the origin.
x
y
Ox
y
Ox
y
O
x
y
Ox
y
Ox
y
O
© Glencoe/McGraw-Hill 546 Glencoe Algebra 2
Partial FractionsIt is sometimes an advantage to rewrite a rational expression as the sum oftwo or more fractions. For example, you might do this in a calculus coursewhile carrying out a procedure called integration.
You can resolve a rational expression into partial fractions if two conditionsare met:(1) The degree of the numerator must be less than the degree of the
denominator; and(2) The factors of the denominator must be known.
Resolve �x3
3� 1� into partial fractions.
The denominator has two factors, a linear factor, x � 1, and a quadraticfactor, x2 � x � 1. Start by writing the following equation. Notice that thedegree of the numerators of each partial fraction is less than itsdenominator.
�x3
3� 1� � �x �
A1� � �
x2B�
x �
x �
C1
�
Now, multiply both sides of the equation by x3 � 1 to clear the fractions andfinish the problem by solving for the coefficients A, B, and C.
�x3
3� 1� � �x �
A1� � �
x2B�
x �
x �
C1
�
3 � A(x2 � x � 1) � (x � 1)(Bx � C)3 � Ax2 � Ax � A � Bx2 � Cx � Bx � C3 � (A � B)x2 � (B � C � A)x � (A � C)
Equating each term, 0x2 � (A � B)x2
0x � (B � C � A)x3 � (A � C)
Therefore, A � 1, B � �1, C � 2, and �x3
3� 1� � �x �
11� � �
x2�
�
xx�
�
21
�.
Resolve each rational expression into partial fractions.
1. �x2
5�
x2�
x3� 3
� � �x �A
1� � �x �B
3�
2. �(6xx�
�
27)2� � �x �
A2� � �
(x �
B2)2�
3. � �Ax� � �
xB2� � �x �
C1� � �
(x �
D1)2�
4x3 � x2 � 3x � 2���
x2(x � 1)2
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
9-59-5
ExampleExample
Study Guide and InterventionSolving Rational Equations and Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
9-69-6
© Glencoe/McGraw-Hill 547 Glencoe Algebra 2
Less
on
9-6
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.
� � Simplify.
� Simplify.
�
Solve each equation.
1. � � 2 5 2. � � 1 2 3. � � �
4. � � 4 � 5. � �7 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? 42 mph
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? about 1 days2�
1�2
8�4
�x � 2x
�x � 2x � 1�12
4�x � 1
1�2m � 1
�2m3m � 2�5m
13�1
�2x � 5�4
2x � 1�3
4 � 2t�3
4t � 3�5
y � 3�6
2y�3
2�5
2�5
2�5
8�20
2�5
10�20
18�20
2�5
2��4
9�10
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
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 548 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. � �
�1 a � 0 x � � or 0 x � 0 p
4. � � 5. � 2 6. � 1 �
�2 x 0 x 0 or x 1 x �1 or 0 x 1
or x � 5 or x � 2
1�
2�x � 1
3�x2 � 1
5�x
4�x � 1
1�4
2�x
3�2x
39�
1�
1�
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
Study Guide and Intervention (continued)
Solving Rational Equations and Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
9-69-6
ExampleExample
ExercisesExercises
Skills PracticeSolving Rational Equations and Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
9-69-6
© Glencoe/McGraw-Hill 549 Glencoe Algebra 2
Less
on
9-6
Solve each equation or inequality. Check your solutions.
1. � �1 2. 2 � �
3. � �1 4. 3 � z � 1, 2
5. � 5 6. � �5, 8
7. � �3 8. � � y � 7 3, 4
9. � 8 10. � � 0 k � 0
11. 2 � 0 v 4 12. n � n �3 or 0 n 3
13. � � 0 m 1 14. � 1 0 x
15. � � 9 3 16. � 4 � 4
17. 2 � � �5 18. 8 � �
19. � � �4 20. � �
21. � � 22. � � 2
23. � � �6 24. � � 52�t � 3
4�t � 3
8�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
2�8z � 8
�z � 24�z
2q�q � 1
5�2q
b � 2�b � 1
3b � 2�b � 1
9x � 7�x � 2
15�x
3�2
�x1
�2x5�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
12�1
�34�n
1�2
x�x � 1
© Glencoe/McGraw-Hill 550 Glencoe Algebra 2
Solve each equation or inequality. Check your solutions.
1. � � 16 2. � 1 � �1, 2
3. � � , 4 4. � s � 4
5. � � 1 all reals except 5 6. � � 0
7. t �5 or � t 0 8. � �
9. � �2 10. 5 � 0 a 2
11. � 0 x 7 12. 8 � � y 0 or y � 2
13. � p 0 or p � 14. � �
15. g � � �1 16. b � � 1 � �2
17. 2 � � 18. 5 � � 6
19. � � 20. � 4 � ��53
�, 5
21. � � 7 22. � � �1, �2
23. � � 0 24. � �
25. � � 26. 3 � � �2
all reals except �4 and 4
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%? 6
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 centimetersfrom the lens and the focal length of the lens is 4 centimeters? 20 cm
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�3
�n2 � 41
�n � 21
�n � 2
2d � 4�d � 2
3d � 2�d � 1
14�x � 2
�x � 6x � 2�x � 3
b � 3�b � 1
2b�b � 1
2�g � 2
g�g � 2
2�x � 1
4�x � 2
6�x � 1
65�1
�51
�3p4�p
19�y
3�y
3�2x
1�10
4�5x
7�a
3�a
�1�w � 3
4�w � 2
11�3
�h � 15�h
1�2h
1�9
�2t � 15�t
5�5
�x1
�3x � 2y
�y � 55
�y � 5
5s � 8�s � 2
s�s � 2
2�4
�pp � 10�p2 � 2
x�2
x�x � 1
3�2
3�4
12�x
Practice (Average)
Solving Rational Equations and Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
9-69-6
Reading to Learn MathematicsSolving Rational Equations and Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
9-69-6
© Glencoe/McGraw-Hill 551 Glencoe Algebra 2
Less
on
9-6
Pre-Activity How are rational equations used to solve problems involving unitprice?Read the introduction to Lesson 9-6 at the top of page 505 in your textbook.
• If you increase total number of minutes of long-distance calls from Marchto April, will your long-distance phone bill increase or decrease? increase
• Will your actual cost per minute increase or decrease? decrease
Reading the Lesson1. When solving a rational equation, any possible solution that results in 0 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 � � 0
.
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.
z �4 or �2 z 0
Helping You Remember3. How are the processes of adding rational expressions with different denominators and of
solving rational expressions alike, and how are they different? Sample answer:They are alike because both use the LCD of all the rational expressionsin the problem. They are different because in an addition problem, theLCD remains after the fractions are added, while in solving a rationalequation, the LCD is eliminated.
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
4
z (z � 2)
6�
3�
0�2
6�z
3�z � 2
© Glencoe/McGraw-Hill 552 Glencoe Algebra 2
LimitsSequences of numbers with a rational expression for the general term oftenapproach some number as a finite limit. For example, the reciprocals of thepositive integers approach 0 as n gets larger and larger. This is written usingthe notation shown below. The symbol ∞ stands for infinity and n → ∞ meansthat n is getting larger and larger, or “n goes to infinity.”
1, �12�, �
13�, �
14�, …, �n
1�, … lim
n→∞�n1
� � 0
Find limn→∞
�(n �
n2
1)2�
It is not immediately apparent whether the sequence approaches a limit ornot. But notice what happens if we divide the numerator and denominator ofthe general term by n2.
�(n �
n2
1)2� � �n2 �
n2
2
n � 1�
�
�
The two fractions in the denominator will approach a limit of 0 as n getsvery large, so the entire expression approaches a limit of 1.
Find the following limits.
1. limn→∞
�nn
3
4�
�56n
� 2. limn→∞
�1
n�
2n
�
3. limn→∞
�2(n
2�
n �1)
1� 1
� 4. limn→∞
�21n�
�
3n1
�
1��1 � �n
2� � �
n12�
�nn
2
2�
���nn
2
2� � �2
nn2� � �
n12�
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
9-69-6
ExampleExample
Chapter 9 Test, Form 1
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 553 Glencoe Algebra 2
Ass
essm
ent
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�
A. 12 B. 24 C. 12a � 12 D. 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.
A. 5mn B. �5mn�
C. �15�mn D. �
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)�
A. �2(k1�3
1)� B. �2(k1�7
1)� C. �k1�1
1� D. �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 � 24A. 18(z � 4) B. 3(z � 4) C. 6(z � 4) D. 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?
A. f(x) � �x �2
1� B. f(x) � �x �2
1�
C. f(x) � �x �x
1� D. f(x) � �x �x
1� 10.
�5mn
2
3��
�nm
2�
y3���x3 � x2y � xy2 � y3
99
xO
f (x)
© Glencoe/McGraw-Hill 554 Glencoe Algebra 2
Chapter 9 Test, Form 1 (continued)
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.A. 5 B. 20 C. 10 D. 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.A. 1 B. 4 C. 25 D. 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. A. identityB. constantC. inverse variationD. rational 16.
17. Identify the type of function represented by y � �16x�.A. direction variation B. quadraticC. inverse variation D. square root 17.
18. Solve �x �x
2� � �75�.
A. �7 B. 5 C. 7 D. ��57� 18.
19. Solve y � 4 � �5y�.
A. �5, 1 B. �1, 5 C. �1 D. � 19.
20. Solve �m9� 5� 3.
A. m 5 or m 8 B. m �2 or m 5C. �2 m 5 D. 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
NAME DATE PERIOD
99
Chapter 9 Test, Form 2A
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 555 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
�
A. �12� B. 2 C. �2
((xx��
41))2
2� D. �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.
A. �s
1�2
3� B. 12s � 36 C. �
ss
��
33� D. 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�
A. �m2�m
5� B. �mm
��
25� C. �
mm
��
25� D. �(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 � 20A. (t � 3)(t � 5)(t � 4) B. (t � 3)(t � 5)(t � 4)C. (t � 3)(t � 5)(t � 4) D. (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 B. x � �2C. x � �2, x � �3 D. y � 1 9.
10. Determine the values of x for any holes in the graph of f(x) � �x2 �x �
6x5� 5�.
A. x � 5 B. x � �5C. x � 1 D. x � �1, x � �5 10.
�84ss2
2
��
2346s�
���122ss2 �
�63s6
�
99
© Glencoe/McGraw-Hill 556 Glencoe Algebra 2
Chapter 9 Test, Form 2A (continued)
11. Which rational function is graphed?
A. f(x) � �x �3
2� B. f(x) � �x �3
2�
C. f(x) � �x �x
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.
A. ��145�
B. 60 C. �60 D. �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.
A. �16� B. 6 C. 3 D. �
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.A. square root B. rationalC. inverse variation D. quadratic 16.
17. Identify the type of function represented by y � �xx2
��
39
�.
A. quadratic B. rationalC. inverse variation D. direct variation 17.
18. Solve �n �n
4� � n � �12n
��
44n
�.
A. �4, 3 B. �3, 4 C. �4 D. 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?A. 3.5 B. 2.4 C. 5 D. 2 20.
Bonus Simplify . B:1 � �
3x�
��1 � �
4x� � �x
32�
NAME DATE PERIOD
99
xO
f (x)
Chapter 9 Test, Form 2B
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 557 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�
A. �t2 �
3t6�t �
39
� B. �3t(2t
��
13)
� C. 3 D. �t �3
1� 2.
3. �m �
62n
� � �m2
1�04n2�
A. �3(m5� 2n)� B. �3(m
5� 2n)�
C. �m �4
2n� D. 3.
4.
A. �bb
��
22� B. b � 2 C. 2b � 4 D. 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�
A. �7m
��
m6� B. �
mm
��
76� C. �
mm
��
76� D. �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 � 24A. (t � 3)(t � 4)(t � 6) B. (t � 3)(t � 4)(t � 6)C. (t � 3)(t � 4)(t � 6) D. (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�.
A. x � �3 B. x � 3 C. x � �2, x � �3 D. x � �2 10.
�63bb2
2
��
1122b�
���10
5bb2��
1200b�
m3 � 4mn2 � 2m2n � 8n3����60
99
© Glencoe/McGraw-Hill 558 Glencoe Algebra 2
Chapter 9 Test, Form 2B (continued)
11. Which rational function is graphed?
A. f(x) � �xx
��
31� B. f(x) � �(x � 3)
3(x � 1)�
C. f(x) � �xx
��
31� 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.A. �240 B. 15 C. 240 D. �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.
A. �32� B. �
23� C. �
59� D. �
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 �.A. direct variation B. absolute valueC. inverse variation D. constant 16.
17. Identify the type of function represented by y � 4.A. greatest integer B. direct variationC. constant D. identity 17.
18. Solve �n �n
3� � n � �7nn
��
138
�.
A. 3 B. 6 C. 3, 6 D. �3, 6 18.
19. Solve 7 � �m3� �
1m8�.
A. m 0 or m 3 B. 0 m 3C. m 3 D. m 0 19.
20. The sum of a number and 16 times its reciprocal is 10. Find the number(s).A. �8 or �2 B. 2 or 8 C. 4 D. �4 20.
Bonus Simplify . B:1 � �
2x�
��1 � �
1x� � �x
22�
NAME DATE PERIOD
99
xO
f (x)
Chapter 9 Test, Form 2C
© Glencoe/McGraw-Hill 559 Glencoe Algebra 2
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 12.z � 12, find y when x � 24 and z � 5.
�63mm
2
2
��
3705m�
���9m
4m2 �
�4250m�
NAME DATE PERIOD
SCORE 99
Ass
essm
ent
xO
f (x)
© Glencoe/McGraw-Hill 560 Glencoe Algebra 2
Chapter 9 Test, Form 2C (continued)
13. PHOTOGRAPHS A film-developing company noted that, in 13.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?
14. If y varies inversely as x and y � 25 when x � 6, find y 14.when x � 150.
15. WILDFIRES Firefighters battling wildfires in western states 15.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.
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 20.can paint the same room in 12 hours. How long will it take if the two of them work together?
Bonus Solve � 1. B:�x �
12� � �x �
13�
���x �
12� � �x �
13�
y
xO
NAME DATE PERIOD
99
Chapter 9 Test, Form 2D
© Glencoe/McGraw-Hill 561 Glencoe Algebra 2
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�
NAME DATE PERIOD
SCORE 99
Ass
essm
ent
© Glencoe/McGraw-Hill 562 Glencoe Algebra 2
Chapter 9 Test, Form 2D (continued)
13. RESTAURANTS In a certain county, the planning 13.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?
14. If y varies inversely as x and y � 12 when x � 6, find y 14.when x � 8.
15. GOVERNMENT Part of a model used by a state government 15.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.
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, 20.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?
Bonus Solve � 1. B:�x �
15� � �x �
11�
��
�x �1
5� � �x �1
1�
y
xO
NAME DATE PERIOD
99
Chapter 9 Test, Form 3
© Glencoe/McGraw-Hill 563 Glencoe Algebra 2
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��
��
�34mm
��
43nn�
���34mm
��
43nn�
NAME DATE PERIOD
SCORE 99
Ass
essm
ent
xO
f (x)
xO
f (x)
© Glencoe/McGraw-Hill 564 Glencoe Algebra 2
Chapter 9 Test, Form 3 (continued)
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 11.324 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.
12. About 17,100 calls are made each day between Atlanta and 12.Tallahassee. Find the distance between the cities to the nearest mile.
13. The current I in an electrical circuit varies inversely with 13.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.
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�
NAME DATE PERIOD
99
CityPopulation
in 2000
Atlanta 416,000
Charleston 97,000
Raleigh 276,000
Tallahassee 151,000
19.
20.
y
xO
Chapter 9 Open-Ended Assessment
© Glencoe/McGraw-Hill 565 Glencoe Algebra 2
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.
NAME DATE PERIOD
SCORE 99
Ass
essm
ent
© Glencoe/McGraw-Hill 566 Glencoe Algebra 2
Chapter 9 Vocabulary Test/Review
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,
joint variation).
2. r(x) � �xx
2
2��
65xx
��
96� is an example of a (complex fraction, rational function,
rational expression).
3. The graph of y � �x �3
5� has a(n) (asymptote, point discontinuity,
constant of variation).
4. Adding or subtracting rational expressions requires you to find a(n) (least common denominator, asymptote, complex fraction).
5. The formula for simple interest, I � Prt, is an example of (direct variation, inverse variation, joint variation).
6. The graph of y � �xx
��
53� has a break in (asymptote, discontinuity, continuity)
at x � 3.
7. �2t�
� �t32� 1 is an example of a (rational inequality, rational equation,
rational function).
8. If you walk at a steady speed, your speed and the time it takes towalk 1 mile are (asymptotes, inversely proportional, direct variations)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,asymptote).
10. If the rational expression in a rational function is not written in lowest terms, the graph of the function may have a (continuity,constant of variation, point discontinuity).
In your own words—Define each term.
11. rational expression
12. complex fraction
asymptotecomplex fractionconstant of variation
continuitydirect variationinverse variation
joint variationpoint discontinuityrational equation
rational expressionrational functionrational inequality
NAME DATE PERIOD
SCORE 99
Chapter 9 Quiz (Lessons 9–1 and 9–2)
99
© Glencoe/McGraw-Hill 567 Glencoe Algebra 2
NAME DATE PERIOD
SCORE
Chapter 9 Quiz (Lesson 9–3)
For Questions 1–3, 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 �
2x3� 3�
3. f(x) � �x2 �
x �2x
4� 8
�
4. Graph f(x) � �x �4
3�.
NAME DATE PERIOD
SCORE 99
Ass
essm
ent
9
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. Standardized Test Practice 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.
1.
2.
3.4.
xO
f (x)
© Glencoe/McGraw-Hill 568 Glencoe Algebra 2
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. 5.Then graph the equation.
xO
yy
xO
Chapter 9 Quiz (Lesson 9–6)
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 5.to six more than that number is 2 to 3. Find the number.
NAME DATE PERIOD
SCORE
Chapter 9 Quiz (Lessons 9–4 and 9–5)
99
NAME DATE PERIOD
SCORE
99
4.
y
xO
Chapter 9 Mid-Chapter Test (Lessons 9–1 through 9–3)
© Glencoe/McGraw-Hill 569 Glencoe Algebra 2
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
�
A. �3y � 1 B. 3y � 1 C. �3y � 1 D. 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.
A. �169m(m
2(m�
�2)
2)� B. �m(mm2
��2
4)� C. m � 2 D. �
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 of polynomials.
7. 12s3, 18s2t, 24t4 7.
8. 9c � 15, 21c � 35 8.
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. Graph f(x) � �(x �4
2)2�. 10.
�43mm
2
2
��
81m2
����8m
6m2 �
�1162m�
Part I
NAME DATE PERIOD
SCORE 99
Ass
essm
ent
xO
f (x)
Part II
© Glencoe/McGraw-Hill 570 Glencoe Algebra 2
Chapter 9 Cumulative Review (Chapters 1–9)
1. Determine whether C � � � and D � � � are 1.
inverses. (Lesson 4-7)
2. Simplify the expression �w�13��
�25�
. (Lesson 5-7) 2.
3. Solve x2 � 2x � 2 � 0 by completing the square. (Lesson 6-4) 3.
4. Graph y � x2 � 4x. (Lesson 6-7) 4.
5. Use synthetic substitution to find f(3) for 5.f(x) � 3x3 � 7x2 � 5x � 10. (Lesson 7-4)
6. List all of the possible rational zeros of 6.2x4 � 5x3 � 3x2 � 12x � 6. (Lesson 7-6)
7. Write an equation for a circle with center at (0, �3) that 7.passes through (5, 7). (Lessons 8-1 and 8-3)
8. Write an equation for the ellipse whose major axis is 8.10 units long and parallel to the x-axis, whose minor axis is 6 units long, and whose center is at (1, �2). (Lesson 8-4)
9. State whether the graph of 5x2 � 5y2 � 10x � 15y � 10 is an 9.ellipse, circle, parabola, or hyperbola. (Lesson 8-6)
10. Simplify . (Lesson 9-1) 10.
11. Suppose y varies jointly as x and z. Find y when x � 16 and 11.z � 5, if y � 9 when x � 3 and z � 12. (Lesson 9-4)
12. Evita adds a 75% acid solution to 8 milliliters of solution 12.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 should be added to create a solution that is 50% acid? (Lesson 9-6)
�59yy2
2
��
1306y�
���10
6yy2��
1220y�
y
xO
�116�
��156�
1 5�3 1
NAME DATE PERIOD
99
Standardized Test Practice (Chapters 1–9)
© Glencoe/McGraw-Hill 571 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 _______.E. 2 F. 4 G. 8 H. 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?E. a � 2 2 � a F. �2a a2
G. a � 2 2a H. 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?
E. �3c02� F. �
1c5� G. 30c2 H. 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.E. 16 � 16�2� F. 24 � 8�2� G. 32 � 8�2� H. 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?E. $2.10 F. $3.90 G. $1.80 H. $2.40 10. HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
NAME DATE PERIOD
99
Ass
essm
ent
Part 1: Multiple Choice
Instructions: Fill in the appropriate oval for the best answer.
Z Y
W X
© Glencoe/McGraw-Hill 572 Glencoe Algebra 2
Standardized Test Practice (continued)
11. 3 is 12% of what number? 11. 12.
12. If w � 4x, y � 10z, x � 3, and z � �12�, what
is the value of �2y� � �w
3�?
13. How many rectangles can be found in the figure shown?
13. 14.
14. What is the value of a in the figure shown?
Column A Column B
15. R h � 1 15.
16.m � n � p � 3m
16.
17. �3t�
� 2; �2s� � 3 17.
18. y 0 18.
�3y3
y�2
y2�3y � 1
DCBA
st
DCBA
pn
DCBAm �
p �n �
hR
DCBA
Part 3: Quantitative Comparison
Instructions: Compare the quantities in columns A and B. Shade in if the quantity in column A is greater; if the quantity in column B is greater; if the quantities are equal; or if the relationship cannot be determined from the information given.
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
NAME DATE PERIOD
99
NAME DATE PERIOD
Part 2: Grid In
Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.
a�
61�
25�18�
A
D
C
B
Standardized Test PracticeStudent Record Sheet (Use with pages 518–519 of the Student Edition.)
© Glencoe/McGraw-Hill A1 Glencoe Algebra 2
NAME DATE PERIOD
99
An
swer
s
Select the best answer from the choices given and fill in the corresponding oval.
1 4 7
2 5 8
3 6 9
Solve the problem and write your answer in the blank.
For Questions 14–20, also enter your answer by writing each number or symbol ina box. Then fill in the corresponding oval for that number or symbol.
10 15 17 19
11
12
13
14 16 18 20
Select the best answer from the choices given and fill in the corresponding oval.
21 23 25
22 24 DCBADCBA
DCBADCBADCBA
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
DCBADCBADCBA
DCBADCBADCBA
DCBADCBADCBA
Part 2 Short Response/Grid InPart 2 Short Response/Grid In
Part 1 Multiple ChoicePart 1 Multiple Choice
Part 3 Quantitative ComparisonPart 3 Quantitative Comparison
© Glencoe/McGraw-Hill A2 Glencoe Algebra 2
Answers (Lesson 9-1)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Mu
ltip
lyin
g a
nd
Div
idin
g R
atio
nal
Exp
ress
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-1
9-1
©G
lenc
oe/M
cGra
w-H
ill51
7G
lenc
oe A
lgeb
ra 2
Lesson 9-1
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
sF
or a
ll ra
tiona
l exp
ress
ions
an
d ,
��
, if b
�0
and
d�
0.
Div
idin
g R
atio
nal
Exp
ress
ion
sF
or a
ll ra
tiona
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)�
�2(
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
�27
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�
�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
� 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
Exam
ple
Exam
ple
Exer
cises
Exer
cises
11
11
11
11
11
1
11
11
11
1
11
11
11
11
1
11
11
11
11
1
©G
lenc
oe/M
cGra
w-H
ill51
8G
lenc
oe A
lgeb
ra 2
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.
�F
acto
r.
�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�
��
�3s2
�s8 4
s�
3�
�3ss�
1�
��
�3s2
�
s8 4s
�3
�
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Mu
ltip
lyin
g a
nd
Div
idin
g R
atio
nal
Exp
ress
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-1
9-1
Exam
ple
Exam
ple
1
11
s3
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A3 Glencoe Algebra 2
An
swer
s
Answers (Lesson 9-1)
Skil
ls P
ract
ice
Mu
ltip
lyin
g a
nd
Div
idin
g R
atio
nal
Exp
ress
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-1
9-1
©G
lenc
oe/M
cGra
w-H
ill51
9G
lenc
oe A
lgeb
ra 2
Lesson 9-1
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
�19
t�
84�
�4t
�4
w2
�6w
�7
��
w�
3w
2�
5w�
24�
�w
�1
q2
� 2(q
�2)
q2�
4�
3q2
q2�
2q�
6q3x
� x2�
43x
2� x
�2
32z7
� 35v
2 y25
y5� 14
z12v5
80y4
� 49z5 v
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 y2
©G
lenc
oe/M
cGra
w-H
ill52
0G
lenc
oe A
lgeb
ra 2
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
�y
�6
5ux
2� 21
yz5
25x3
� 14u
2 y2
�2u
3 y� 15
xz5
x�
2�
xx4
�x3
�2x
2�
�x4
�x3
v�
5� 3v
�2
25 �
v2�
�3v
2�
13v
�10
2k�
5� k
�3
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
Pra
ctic
e (
Ave
rag
e)
Mu
ltip
lyin
g a
nd
Div
idin
g R
atio
nal
Exp
ress
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-1
9-1
© Glencoe/McGraw-Hill A4 Glencoe Algebra 2
Answers (Lesson 9-1)
Readin
g t
o L
earn
Math
em
ati
csM
ult
iply
ing
an
d D
ivid
ing
Rat
ion
al E
xpre
ssio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-1
9-1
©G
lenc
oe/M
cGra
w-H
ill52
1G
lenc
oe A
lgeb
ra 2
Lesson 9-1
Pre-
Act
ivit
yH
ow a
re r
atio
nal
exp
ress
ion
s u
sed
in
mix
ture
s?
Rea
d th
e in
trod
uct
ion
to
Les
son
9-1
at
the
top
of p
age
472
in y
our
text
book
.
•S
upp
ose
that
th
e G
oodi
e S
hop
pe a
lso
sell
s a
can
dy m
ixtu
re o
f ch
ocol
ate
min
ts a
nd
cara
mel
s.If
th
is m
ixtu
re i
s m
ade
wit
h 4
pou
nds
of
choc
olat
e
min
ts a
nd
3 po
un
ds o
f ca
ram
els,
then
of
th
e m
ixtu
re i
s
min
ts a
nd
of t
he
mix
ture
is
cara
mel
s.
•If
th
e st
ore
man
ager
add
s an
oth
er y
pou
nds
of
min
ts t
o th
e m
ixtu
re,w
hat
frac
tion
of
the
mix
ture
wil
l be
min
ts?
Rea
din
g 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.
Hel
pin
g Y
ou
Rem
emb
er
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�
©G
lenc
oe/M
cGra
w-H
ill52
2G
lenc
oe A
lgeb
ra 2
Rea
din
g A
lgeb
raIn
mat
hem
atic
s,th
e te
rm g
rou
ph
as a
spe
cial
mea
nin
g.T
he
foll
owin
gn
um
bere
d se
nte
nce
s di
scu
ss t
he
idea
of
grou
p an
d on
e in
tere
stin
g ex
ampl
e of
a g
rou
p.
01T
o be
a g
rou
p,a
set
of e
lem
ents
an
d a
bin
ary
oper
atio
n m
ust
sat
isfy
fou
rco
ndi
tion
s:th
e se
t m
ust
be
clos
ed u
nde
r th
e op
erat
ion
,th
e op
erat
ion
mu
st b
e as
soci
ativ
e,th
ere
mu
st b
e an
ide
nti
ty e
lem
ent,
and
ever
yel
emen
t m
ust
hav
e an
in
vers
e.
02T
he
foll
owin
g si
x fu
nct
ion
s fo
rm a
gro
up
un
der
the
oper
atio
n o
f
com
posi
tion
of
fun
ctio
ns:
f 1(x
) �
x,f 2
(x)
��1 x� ,
f 3(x
) �
1 �
x,
f 4(x
) �
�(x� x
1)�
,f5(
x) �
� (x�x
1)�
,an
d f 6
(x)
�� (1
�1x)
�.
03T
his
gro
up
is a
n e
xam
ple
of a
non
com
mu
tati
ve g
rou
p.F
or e
xam
ple,
f 3�
f 2�
f 4,b
ut
f 2�
f 3�
f 6.
04S
ome
expe
rim
enta
tion
wit
h t
his
gro
up
wil
l sh
ow t
hat
th
e id
enti
tyel
emen
t is
f1.
05E
very
ele
men
t is
its
ow
n i
nve
rse
exce
pt f
or f
4an
d f 6
,eac
h o
f w
hic
h i
s th
ein
vers
e of
th
e ot
her
.
Use
th
e p
arag
rap
h t
o an
swer
th
ese
qu
esti
ons.
1.E
xpla
in w
hat
it
mea
ns
to s
ay t
hat
a s
et i
s cl
osed
un
der
an o
pera
tion
.Is
the
set
of p
osit
ive
inte
gers
clo
sed
un
der
subt
ract
ion
?P
erfo
rmin
g t
he
op
er-
atio
n o
n a
ny t
wo
ele
men
ts o
f th
e se
t re
sult
s in
an
ele
men
t o
f th
esa
me
set.
No
,3 a
nd
4 a
re p
osi
tive
inte
ger
s bu
t 3
� 4
is n
ot.
2.S
ubt
ract
ion
is
a n
onco
mm
uta
tive
ope
rati
on f
or t
he
set
of i
nte
gers
.Wri
tean
in
form
al d
efin
itio
n o
f n
onco
mm
uta
tive
.T
he
ord
er in
wh
ich
th
eel
emen
ts a
re u
sed
wit
h t
he
op
erat
ion
can
aff
ect
the
resu
lt.
3.F
or t
he
set
of i
nte
gers
,wh
at i
s th
e id
enti
ty e
lem
ent
for
the
oper
atio
n o
fm
ult
ipli
cati
on?
Just
ify
you
r an
swer
.1,
bec
ause
,fo
r ev
ery
inte
ger
a,a
�1
�a
and
1 �
a�
a.
4.E
xpla
in h
ow t
he
foll
owin
g st
atem
ent
rela
tes
to s
ente
nce
05:
(f6
�f 4
)(x)
�f 6
[f4(
x)]
�f 6�� (1
�1x)
���
�x
�f 1
(x).
It s
ho
ws
that
f4
is t
he
inve
rse
of
f 6.1
��
�1�
(x x�
1)�
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-1
9-1
© Glencoe/McGraw-Hill A5 Glencoe Algebra 2
An
swer
s
Answers (Lesson 9-2)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Ad
din
g a
nd
Su
btr
acti
ng
Rat
ion
al E
xpre
ssio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-2
9-2
©G
lenc
oe/M
cGra
w-H
ill52
3G
lenc
oe A
lgeb
ra 2
Lesson 9-2
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 .16
p2q3
r�
24�
p2�
q3�
r40
pq4 r
2�
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)
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
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)
©G
lenc
oe/M
cGra
w-H
ill52
4G
lenc
oe A
lgeb
ra 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
4F
acto
r if
poss
ible
.S
tep
5S
impl
ify if
pos
sibl
e.
Sim
pli
fy
�.
�
��
Fac
tor
the
deno
min
ator
s.
��
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
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Ad
din
g a
nd
Su
btr
acti
ng
Rat
ion
al E
xpre
ssio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-2
9-2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A6 Glencoe Algebra 2
Answers (Lesson 9-2)
Skil
ls P
ract
ice
Ad
din
g a
nd
Su
btr
acti
ng
Rat
ion
al E
xpre
ssio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-2
9-2
©G
lenc
oe/M
cGra
w-H
ill52
5G
lenc
oe A
lgeb
ra 2
Lesson 9-2
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 � x
©G
lenc
oe/M
cGra
w-H
ill52
6G
lenc
oe A
lgeb
ra 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
Pra
ctic
e (
Ave
rag
e)
Ad
din
g a
nd
Su
btr
acti
ng
Rat
ion
al E
xpre
ssio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-2
9-2
© Glencoe/McGraw-Hill A7 Glencoe Algebra 2
An
swer
s
Answers (Lesson 9-2)
Readin
g t
o L
earn
Math
em
ati
csA
dd
ing
an
d S
ub
trac
tin
g R
atio
nal
Exp
ress
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-2
9-2
©G
lenc
oe/M
cGra
w-H
ill52
7G
lenc
oe A
lgeb
ra 2
Lesson 9-2
Pre-
Act
ivit
yH
ow i
s su
btr
acti
on o
f ra
tion
al e
xpre
ssio
ns
use
d i
n p
hot
ogra
ph
y?
Rea
d th
e in
trod
uct
ion
to
Les
son
9-2
at
the
top
of p
age
479
in y
our
text
book
.
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.W
rite
an
equ
atio
n t
hat
you
cou
ld s
olve
to
fin
d h
ow f
ar t
he
film
sh
ould
be
from
th
e le
ns
to g
et a
per
fect
ly f
ocu
sed
phot
ogra
ph.
��
Rea
din
g 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
of t
he
den
omin
ator
s.
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
.
Hel
pin
g Y
ou
Rem
emb
er
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
©G
lenc
oe/M
cGra
w-H
ill52
8G
lenc
oe A
lgeb
ra 2
Su
per
ellip
ses
Th
e ci
rcle
an
d th
e el
lips
e ar
e m
embe
rs o
f an
in
tere
stin
g fa
mil
y of
cu
rves
th
at w
ere
firs
t st
udi
ed b
y th
e F
ren
ch p
hys
icis
t an
d m
ath
emat
icia
n G
abri
el
Lam
é (1
795–
1870
).T
he
gen
eral
equ
atio
n f
or t
he
fam
ily
is
�� ax � �n�
�� by � �n�
1,w
ith
a�
0,b
�0,
and
n�
0.
For
eve
n v
alu
es o
f n
grea
ter
than
2,t
he
curv
es a
re c
alle
d su
per
elli
pse
s.
1.C
onsi
der
two
curv
es t
hat
are
not
supe
rell
ipse
s.G
raph
eac
h e
quat
ion
on
th
e gr
id a
t th
e ri
ght.
Sta
te t
he
type
of
curv
e pr
odu
ced
each
tim
e.
a.�� 2x � �2
��� 2y � �2
�1
circ
le
b.
�� 3x � �2�
�� 2y � �2�
1el
lipse
2.In
eac
h o
f th
e fo
llow
ing
case
s yo
u a
re
give
n v
alu
es o
f a,
b,an
d n
to u
se i
n t
he
gen
eral
equ
atio
n.W
rite
th
e re
sult
ing
equ
atio
n.T
hen
gra
ph.S
ketc
h e
ach
gra
ph
on t
he
grid
at
the
righ
t.
a.a
�2,
b�
3,n
�4
b.
a�
2,b
�3,
n�
6 c.
a�
2,b
�3,
n�
8
See
stu
den
ts’g
rap
hs.
3.W
hat
sh
ape
wil
l th
e gr
aph
of
�� 2x � �n�
�� 2y � �n
appr
oxim
ate
for
grea
ter
and
grea
ter
even
,w
hol
e-n
um
ber
valu
es o
f n
?
a re
ctan
gle
th
at is
6 u
nit
s lo
ng
an
d4
un
its
wid
e,ce
nte
red
at
the
ori
gin
1–1
–2–3
23
3 2 1 –1 –2 –3
1–1
–2–3
23
3 2 1 –1 –2 –3
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-2
9-2
© Glencoe/McGraw-Hill A8 Glencoe Algebra 2
Answers (Lesson 9-3)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Gra
ph
ing
Rat
ion
al F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-3
9-3
©G
lenc
oe/M
cGra
w-H
ill52
9G
lenc
oe A
lgeb
ra 2
Lesson 9-3
Ver
tica
l Asy
mp
tote
s an
d P
oin
t D
isco
nti
nu
ity
Rat
ion
al F
un
ctio
nan
equ
atio
n of
the
for
m f
(x)
�,
whe
re p
(x)
and
q(x
) ar
e po
lyno
mia
l exp
ress
ions
and
q
(x)
�0
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
, bu
t ne
ver
cros
ses.
o
f th
e G
rap
h o
f a
If th
e si
mpl
ified
for
m o
f th
e re
late
d ra
tiona
l exp
ress
ion
is u
ndef
ined
for
x�
a,
Rat
ion
al F
un
ctio
nth
en x
�a
is a
ver
tical
asy
mpt
ote.
Po
int
Dis
con
tin
uit
y P
oint
dis
cont
inui
ty is
like
a h
ole
in a
gra
ph.
If th
e or
igin
al r
elat
ed e
xpre
ssio
n is
und
efin
ed
of
the
Gra
ph
of
a fo
r x
�a
but
the
sim
plifi
ed e
xpre
ssio
n is
def
ined
for
x�
a, t
hen
ther
e is
a h
ole
in t
he
Rat
ion
al F
un
ctio
ngr
aph
at x
�a.
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)
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill53
0G
lenc
oe A
lgeb
ra 2
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
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Gra
ph
ing
Rat
ion
al F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-3
9-3
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A9 Glencoe Algebra 2
An
swer
s
Answers (Lesson 9-3)
Skil
ls P
ract
ice
Gra
ph
ing
Rat
ion
al F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-3
9-3
©G
lenc
oe/M
cGra
w-H
ill53
1G
lenc
oe A
lgeb
ra 2
Lesson 9-3
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) �
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
©G
lenc
oe/M
cGra
w-H
ill53
2G
lenc
oe A
lgeb
ra 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) �
3.f(
x) �
asym
pto
tes:
x�
2,as
ymp
tote
:x
�3;
asym
pto
te:
x�
�2
x�
�5
ho
le:
x�
7
4.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 a
lone
,Taw
a ca
n gi
ve t
he s
hed
a co
at o
f pa
int
in 6
hou
rs.I
t ta
kes
her
fath
er x
hou
rs w
orki
ng
alon
e to
giv
e th
e
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
rsal
one,
wh
at p
orti
on o
f th
e jo
b ca
n t
hey
com
plet
e to
geth
er i
n 1
hou
r?
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?11
.25
foo
t-ca
nd
les
4500
�d
245
00�
d2
2040
Dist
ance
(ft)
Illu
min
atio
n
Illumination (foot-candles)
60
60 40 20
dOI
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
Pra
ctic
e (
Ave
rag
e)
Gra
ph
ing
Rat
ion
al F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-3
9-3
© Glencoe/McGraw-Hill A10 Glencoe Algebra 2
Answers (Lesson 9-3)
Readin
g t
o L
earn
Math
em
ati
csG
rap
hin
g R
atio
nal
Fu
nct
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-3
9-3
©G
lenc
oe/M
cGra
w-H
ill53
3G
lenc
oe A
lgeb
ra 2
Lesson 9-3
Pre-
Act
ivit
yH
ow c
an r
atio
nal
fu
nct
ion
s b
e u
sed
wh
en b
uyi
ng
a gr
oup
gif
t?
Rea
d th
e in
trod
uct
ion
to
Les
son
9-3
at
the
top
of p
age
485
in y
our
text
book
.
•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
din
g 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
.
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
Hel
pin
g Y
ou
Rem
emb
er
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
dis
con
tin
uit
ies
asym
pto
tes
con
tin
uit
y
x2�
25�
�x2
�6x
�9
1� x
�5
©G
lenc
oe/M
cGra
w-H
ill53
4G
lenc
oe A
lgeb
ra 2
Gra
ph
ing
wit
h A
dd
itio
n o
f y-
Co
ord
inat
esE
quat
ion
s of
par
abol
as,e
llip
ses,
and
hyp
erbo
las
that
are
“t
ippe
d”w
ith
res
pect
to
the
x- a
nd
y-ax
es a
re m
ore
diff
icu
lt
to g
raph
th
an t
he
equ
atio
ns
you
hav
e be
en s
tudy
ing.
Oft
en,h
owev
er,y
ou c
an u
se t
he
grap
hs
of t
wo
sim
pler
equ
atio
ns
to g
raph
a m
ore
com
plic
ated
equ
atio
n.F
or
exam
ple,
the
grap
h o
f th
e el
lips
e in
th
e di
agra
m a
t th
e ri
ght
is o
btai
ned
by
addi
ng
the
y-co
ordi
nat
e of
eac
h p
oin
t on
th
e ci
rcle
an
d th
e y-
coor
din
ate
of t
he
corr
espo
ndi
ng
poin
t of
th
e li
ne.
Gra
ph
eac
h e
qu
atio
n.S
tate
th
e ty
pe
of c
urv
e fo
r ea
ch g
rap
h.
1.y
�6
�x
��
4 �
x2�
ellip
se2.
y�
x�
�x�
par
abo
la
Use
a s
epar
ate
shee
t of
gra
ph
pap
er t
o gr
aph
th
ese
equ
atio
ns.
Sta
te t
he
typ
e of
curv
e fo
r ea
ch g
rap
h.
3.y
�2x
��
7 �
6�
x�
x2�
ellip
se4.
y�
�2x
��
�2x
�p
arab
ola
See
stu
den
ts’g
rap
hs.
See
stu
den
ts’g
rap
hs.
y �
x �
��x
14
56
72
3
8 7 6 5 4 3 2 1 –1 –2
y �
x
y �
��
�x
x
y
O
y �
6 �
x �
��
��
4x �
x2
14
5–1
–22
3
9 8 7 6 5 4 3 2 1 –1 –2y
� �
��
��
4x �
x2
y �
6 �
x
x
y
O
x
y
O
y �
��
��
�4x
� x
2
y �
x �
6 �
��
��
4x �
x2
y �
x �
6
A
B� A
�
B
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-3
9-3
© Glencoe/McGraw-Hill A11 Glencoe Algebra 2
An
swer
s
Answers (Lesson 9-4)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Dir
ect,
Join
t,an
d In
vers
e V
aria
tio
n
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-4
9-4
©G
lenc
oe/M
cGra
w-H
ill53
5G
lenc
oe A
lgeb
ra 2
Lesson 9-4
Dir
ect
Var
iati
on
an
d J
oin
t V
aria
tio
n
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.
kis
cal
led
the
cons
tant
of
varia
tion.
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.
Exam
ple
Exam
ple
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
Exer
cises
Exer
cises
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
�14
10
.If
yva
ries
dir
ectl
y as
xan
d x
�20
0 w
hen
wh
en x
�35
,fin
d y
wh
en x
�12
.4.
8y
�50
,fin
d x
wh
en y
�10
00.
4000
11.I
f y
vari
es d
irec
tly
as x
and
y�
39
12.I
f y
vari
es d
irec
tly
as x
and
x�
60 w
hen
wh
en x
�52
,fin
d y
wh
en x
�22
.16
.5y
�75
,fin
d x
wh
en y
�42
.33
.6
13.S
upp
ose
yva
ries
join
tly
as x
and
z.14
.Su
ppos
e y
vari
es jo
intl
y as
xan
d z.
Fin
d y
Fin
d y
wh
en x
�6
and
z�
11,i
f
wh
en x
�5
and
z�
10,i
f y
�12
wh
en
y�
120
wh
en x
�5
and
z�
12.
132
x�
8 an
d z
�6.
12.5
15.S
upp
ose
yva
ries
join
tly
as x
and
z.16
.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
©G
lenc
oe/M
cGra
w-H
ill53
6G
lenc
oe A
lgeb
ra 2
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�
9 w
hen
x�
45,f
ind
yw
hen
x�
27.
15
3.If
yva
ries
in
vers
ely
as x
and
y�
100
wh
en x
�38
,fin
d y
wh
en x
�76
.50
4.If
yva
ries
in
vers
ely
as x
and
y�
32 w
hen
x�
42,f
ind
yw
hen
x�
24.
56
5.If
yva
ries
in
vers
ely
as x
and
y�
36 w
hen
x�
10,f
ind
yw
hen
x�
30.
12
6.If
yva
ries
in
vers
ely
as x
and
y�
75 w
hen
x�
12,f
ind
yw
hen
x�
10.
90
7.If
yva
ries
in
vers
ely
as x
and
y�
18 w
hen
x�
124,
fin
d y
wh
en x
�93
.24
8.If
yva
ries
in
vers
ely
as x
and
y�
90 w
hen
x�
35,f
ind
yw
hen
x�
50.
63
9.If
yva
ries
in
vers
ely
as x
and
y�
42 w
hen
x�
48,f
ind
yw
hen
x�
36.
56
10.I
f y
vari
es i
nve
rsel
y as
xan
d y
�44
wh
en x
�20
,fin
d y
wh
en x
�55
.16
11.I
f y
vari
es i
nve
rsel
y as
xan
d y
�80
wh
en x
�14
,fin
d y
wh
en x
�35
.32
12.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
13.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
14.I
f y
vari
es i
nve
rsel
y as
xan
d y
�9
wh
en x
�2,
fin
d y
wh
en x
�5.
3.6
15.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
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Dir
ect,
Join
t,an
d In
vers
e V
aria
tio
n
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-4
9-4
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A12 Glencoe Algebra 2
Answers (Lesson 9-4)
Skil
ls P
ract
ice
Dir
ect,
Join
t,an
d In
vers
e V
aria
tio
n
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-4
9-4
©G
lenc
oe/M
cGra
w-H
ill53
7G
lenc
oe A
lgeb
ra 2
Lesson 9-4
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.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�
2�r
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
©G
lenc
oe/M
cGra
w-H
ill53
8G
lenc
oe A
lgeb
ra 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.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
Pra
ctic
e (
Ave
rag
e)
Dir
ect,
Join
t,an
d In
vers
e V
aria
tio
n
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-4
9-4
© Glencoe/McGraw-Hill A13 Glencoe Algebra 2
An
swer
s
Answers (Lesson 9-4)
Readin
g t
o L
earn
Math
em
ati
csD
irec
t,Jo
int,
and
Inve
rse
Var
iati
on
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-4
9-4
©G
lenc
oe/M
cGra
w-H
ill53
9G
lenc
oe A
lgeb
ra 2
Lesson 9-4
Pre-
Act
ivit
yH
ow i
s va
riat
ion
use
d t
o fi
nd
th
e to
tal
cost
giv
en t
he
un
it c
ost?
Rea
d th
e in
trod
uct
ion
to
Les
son
9-4
at
the
top
of p
age
492
in y
our
text
book
.
•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
tota
l h
igh
-tec
h s
pen
din
g w
ill
(in
crea
se/d
ecre
ase)
by
.
Rea
din
g 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
Hel
pin
g Y
ou
Rem
emb
er
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
$14,
900
dec
reas
e
$149
incr
ease
©G
lenc
oe/M
cGra
w-H
ill54
0G
lenc
oe A
lgeb
ra 2
Exp
ansi
on
s o
f R
atio
nal
Exp
ress
ion
sM
any
rati
onal
exp
ress
ion
s ca
n b
e tr
ansf
orm
ed i
nto
pow
er s
erie
s.A
pow
erse
ries
is
an i
nfi
nit
e se
ries
of
the
form
A�
Bx
�C
x2�
Dx3
�…
.Th
era
tion
al e
xpre
ssio
n a
nd
the
pow
er s
erie
s n
orm
ally
can
be
said
to
hav
e th
esa
me
valu
es o
nly
for
cer
tain
val
ues
of
x.F
or e
xam
ple,
the
foll
owin
g eq
uat
ion
hol
ds o
nly
for
val
ues
of
xsu
ch t
hat
�1
x
1.
� 1�1
x�
�1
�x
�x2
�x3
�…
for
�1
x
1
Exp
and
in
asc
end
ing
pow
ers
of x
.
Ass
um
e th
at t
he
expr
essi
on e
qual
s a
seri
es o
f th
e fo
rm A
�B
x�
Cx2
�D
x3�
….
Th
en m
ult
iply
bot
h s
ides
of
the
equ
atio
n b
y th
e de
nom
inat
or 1
�x
�x2
.
� 1�2
� x�3x
x2�
�A
�B
x�
Cx2
�D
x3�
…
2 �
3x�
(1 �
x�
x2)(
A�
Bx
�C
x2�
Dx3
�…
)2
�3x
�A
�B
x�
Cx2
�D
x3�
…�
Ax
�B
x2�
Cx3
�…
�A
x2�
Bx3
�…
2 �
3x�
A�
(B�
A)x
�(C
�B
�A
)x2
�(D
�C
�B
)x3
�…
Now
,mat
ch t
he
coef
fici
ents
of
the
poly
nom
ials
.2
�A
3 �
B�
A0
�C
�B
�A
0 �
D�
C�
B�
A
Fin
ally
,sol
ve f
or A
,B,C
,an
d D
and
wri
te t
he
expa
nsi
on.
A�
2,B
�1,
C�
�3,
and
D�
0
Th
eref
ore,� 1
�2� x
�3xx2
��
2 �
x�
3x2
�…
Exp
and
eac
h r
atio
nal
exp
ress
ion
to
fou
r te
rms.
1.� 1
�1x�
�xx2
�1
�2x
�x2
�x3
�…
2.� 1
�2x
�2
�2x
�2x
2�
2x3
� …
3.� 1
�1x
�1
�x
�x
2�
x3
�…
2 �
3x�
�1
�x
�x
2
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-4
9-4
Exam
ple
Exam
ple
© Glencoe/McGraw-Hill A14 Glencoe Algebra 2
Answers (Lesson 9-5)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Cla
sses
of
Fu
nct
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-5
9-5
©G
lenc
oe/M
cGra
w-H
ill54
1G
lenc
oe A
lgeb
ra 2
Lesson 9-5
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 O
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill54
2G
lenc
oe A
lgeb
ra 2
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
Iden
tity
y�
x
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)
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Cla
sses
of
Fu
nct
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-5
9-5
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A15 Glencoe Algebra 2
An
swer
s
Answers (Lesson 9-5)
Skil
ls P
ract
ice
Cla
sses
of
Fu
nct
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-5
9-5
©G
lenc
oe/M
cGra
w-H
ill54
3G
lenc
oe A
lgeb
ra 2
Lesson 9-5
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
©G
lenc
oe/M
cGra
w-H
ill54
4G
lenc
oe A
lgeb
ra 2
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
OPra
ctic
e (
Ave
rag
e)
Cla
sses
of
Fu
nct
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-5
9-5
© Glencoe/McGraw-Hill A16 Glencoe Algebra 2
Answers (Lesson 9-5)
Readin
g t
o L
earn
Math
em
ati
csC
lass
es o
f F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-5
9-5
©G
lenc
oe/M
cGra
w-H
ill54
5G
lenc
oe A
lgeb
ra 2
Lesson 9-5
Pre-
Act
ivit
yH
ow c
an g
rap
hs
of f
un
ctio
ns
be
use
d t
o d
eter
min
e a
per
son
’sw
eigh
t on
a d
iffe
ren
t p
lan
et?
Rea
d th
e in
trod
uct
ion
to
Les
son
9-5
at
the
top
of p
age
499
in y
our
text
book
.
•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
dire
ctly
fro
m 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
gra
ph r
epre
sen
ts d
irec
t va
riat
ion
to
esti
mat
e th
ew
eigh
t on
Mar
s of
a w
oman
wh
o w
eigh
s 12
0 po
un
ds o
n E
arth
.ab
ou
t 45
po
un
ds
Rea
din
g 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
Hel
pin
g Y
ou
Rem
emb
er
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
©G
lenc
oe/M
cGra
w-H
ill54
6G
lenc
oe A
lgeb
ra 2
Par
tial
Fra
ctio
ns
It i
s so
met
imes
an
adv
anta
ge t
o re
wri
te a
rat
ion
al e
xpre
ssio
n a
s th
e su
m o
ftw
o or
mor
e fr
acti
ons.
For
exa
mpl
e,yo
u m
igh
t do
th
is i
n a
cal
culu
s co
urs
ew
hil
e ca
rryi
ng
out
a pr
oced
ure
cal
led
inte
grat
ion
.
You
can
res
olve
a r
atio
nal
exp
ress
ion
in
to p
arti
al f
ract
ion
s if
tw
o co
ndi
tion
sar
e m
et:
(1)
Th
e de
gree
of
the
nu
mer
ator
mu
st b
e le
ss t
han
th
e de
gree
of
the
den
omin
ator
;an
d(2
)T
he
fact
ors
of t
he
den
omin
ator
mu
st b
e kn
own
.
Res
olve
� x3
3 �1
�in
to p
arti
al f
ract
ion
s.
Th
e de
nom
inat
or h
as t
wo
fact
ors,
a li
nea
r fa
ctor
,x�
1,an
d a
quad
rati
cfa
ctor
,x2
�x
�1.
Sta
rt b
y w
riti
ng
the
foll
owin
g eq
uat
ion
.Not
ice
that
th
ede
gree
of
the
nu
mer
ator
s of
eac
h p
arti
al f
ract
ion
is
less
th
an i
tsde
nom
inat
or.
� x33 �
1�
�� x
�A1
��� x2B �x
� x�C
1�
Now
,mu
ltip
ly b
oth
sid
es o
f th
e eq
uat
ion
by
x3�
1 to
cle
ar t
he
frac
tion
s an
dfi
nis
h t
he
prob
lem
by
solv
ing
for
the
coef
fici
ents
A,B
,an
d C
.
� x33 �
1�
�� x
�A1
��� x2B �x
� x�C
1�
3 �
A(x
2�
x�
1) �
(x�
1)(B
x�
C)
3 �
Ax2
�A
x�
A�
Bx2
�C
x�
Bx
�C
3 �
(A�
B)x
2�
(B�
C�
A)x
�(A
�C
)
Equ
atin
g ea
ch t
erm
,0x2
�(A
�B
)x2
0x�
(B�
C�
A)x
3 �
(A�
C)
Th
eref
ore,
A�
1,B
��
1,C
�2,
and
� x33 �
1�
�� x
�11
��� x2� �x
x�
�21
�.
Res
olve
eac
h r
atio
nal
exp
ress
ion
in
to p
arti
al f
ract
ion
s.
1.� x2
5 �
x2� x
3 �3
��
� x�A
1�
�� x
�B3
�A
�2,
B�
3
2.� (6 xx ��
27 )2�
�� x
�A2
��
� (x�B
2)2
�A
�6,
B�
�5
3.�
�A x��
� xB 2��
� x�C
1�
�� (x
�D1)
2�
A�
1,B
��
2,C
�3,
D�
�4
4x3
�x2
�3x
�2
��
�x2
(x�
1)2
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-5
9-5
Exam
ple
Exam
ple
© Glencoe/McGraw-Hill A17 Glencoe Algebra 2
An
swer
s
Answers (Lesson 9-6)
Stu
dy G
uid
e a
nd I
nte
rven
tion
So
lvin
g R
atio
nal
Eq
uat
ion
s an
d In
equ
alit
ies
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-6
9-6
©G
lenc
oe/M
cGra
w-H
ill54
7G
lenc
oe A
lgeb
ra 2
Lesson 9-6
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)�
�M
ultip
ly e
ach
side
by
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 �
�O
rigin
al e
quat
ion
��
x�
�5
��
Sim
plify
.
��
Sim
plify
.
�S
impl
ify.
�
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
d of
her
mot
orbo
at i
n s
till
wat
er?
42 m
ph
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?
abo
ut
1d
ays
2 � 3
1 � 2
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 � 58 � 20
2 � 510 � 20
18 � 20
2 � 52
� �4
9 � 10
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
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill54
8G
lenc
oe A
lgeb
ra 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
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
So
lvin
g R
atio
nal
Eq
uat
ion
s an
d In
equ
alit
ies
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-6
9-6
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A18 Glencoe Algebra 2
Answers (Lesson 9-6)
Skil
ls P
ract
ice
So
lvin
g R
atio
nal
Eq
uat
ion
s an
d In
equ
alit
ies
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-6
9-6
©G
lenc
oe/M
cGra
w-H
ill54
9G
lenc
oe A
lgeb
ra 2
Lesson 9-6
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
©G
lenc
oe/M
cGra
w-H
ill55
0G
lenc
oe A
lgeb
ra 2
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
17.2
��
18.5
��
6
19.
��
20.
�4
��
�5 3� ,5
21.
��
722
.�
��
1,�
2
23.
��
024
.�
�
25.
��
26.3
��
�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%?
6
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
len
s,th
e
dist
ance
qof
th
e im
age
of t
he
obje
ct f
rom
th
e le
ns,
and
the
foca
l le
ngt
h f
of t
he
len
s.W
hat
is
the
dist
ance
of
an o
bjec
t fr
om a
len
s if
th
e im
age
of t
he
obje
ct i
s 5
cen
tim
eter
sfr
om t
he
len
s an
d th
e fo
cal
len
gth
of
the
len
s is
4 c
enti
met
ers?
20 c
m
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
2d�
4� d
�2
3d�
2� d
�1
14 � 3x
�2
� x�
6x
�2
� x�
3
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
Pra
ctic
e (
Ave
rag
e)
So
lvin
g R
atio
nal
Eq
uat
ion
s an
d In
equ
alit
ies
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-6
9-6
© Glencoe/McGraw-Hill A19 Glencoe Algebra 2
An
swer
s
Answers (Lesson 9-6)
Readin
g t
o L
earn
Math
em
ati
csS
olv
ing
Rat
ion
al E
qu
atio
ns
and
Ineq
ual
itie
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-6
9-6
©G
lenc
oe/M
cGra
w-H
ill55
1G
lenc
oe A
lgeb
ra 2
Lesson 9-6
Pre-
Act
ivit
yH
ow a
re r
atio
nal
eq
uat
ion
s u
sed
to
solv
e p
rob
lem
s in
volv
ing
un
itp
rice
?R
ead
the
intr
oduc
tion
to
Les
son
9-6
at t
he t
op o
f pa
ge 5
05 i
n yo
ur t
extb
ook.
•If
you
in
crea
se t
otal
nu
mbe
r of
min
ute
s of
lon
g-di
stan
ce c
alls
fro
m M
arch
to A
pril
,wil
l you
r lo
ng-d
ista
nce
phon
e bi
ll in
crea
se o
r de
crea
se?
incr
ease
•W
ill
you
r ac
tual
cos
t pe
r m
inu
te i
ncr
ease
or
decr
ease
?d
ecre
ase
Rea
din
g 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
0 i
n t
he
den
omin
ator
mu
st b
e ex
clu
ded
from
th
e li
st o
f so
luti
ons.
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
Hel
pin
g Y
ou
Rem
emb
er3.
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
©G
lenc
oe/M
cGra
w-H
ill55
2G
lenc
oe A
lgeb
ra 2
Lim
its
Seq
uen
ces
of n
um
bers
wit
h a
rat
ion
al e
xpre
ssio
n f
or t
he
gen
eral
ter
m o
ften
appr
oach
som
e n
um
ber
as a
fin
ite
lim
it.F
or e
xam
ple,
the
reci
proc
als
of t
he
posi
tive
in
tege
rs a
ppro
ach
0 a
s n
gets
lar
ger
and
larg
er.T
his
is
wri
tten
usi
ng
the
not
atio
n s
how
n b
elow
.Th
e sy
mbo
l ∞
stan
ds f
or i
nfi
nit
y an
d n
→∞
mea
ns
that
nis
get
tin
g la
rger
an
d la
rger
,or
“ngo
es t
o in
fin
ity.
”
1,�1 2� ,
�1 3� ,�1 4� ,
…,�
n1 � ,…
lim
n→
∞� n1 �
�0
Fin
d l
imn
→∞
� (n�n
2 1)2
�
It i
s n
ot i
mm
edia
tely
app
aren
t w
het
her
th
e se
quen
ce a
ppro
ach
es a
lim
it o
rn
ot.B
ut
not
ice
wh
at h
appe
ns
if w
e di
vide
th
e n
um
erat
or a
nd
den
omin
ator
of
the
gen
eral
ter
m b
y n
2 .
� (n�n
2 1)2
��
� n2
�n 22 n
�1
�
� �
Th
e tw
o fr
acti
ons
in t
he
den
omin
ator
wil
l ap
proa
ch a
lim
it o
f 0
as n
gets
very
lar
ge,s
o th
e en
tire
exp
ress
ion
app
roac
hes
a l
imit
of
1.
Fin
d t
he
foll
owin
g li
mit
s.
1.li
mn
→∞�n n3 4� �
5 6n�
02.
lim
n→
∞�1
n�2
n�
0
3.li
mn
→∞�2
(n2� n
�1)1�
1�
14.
lim
n→
∞�2 1n �
�
3n1
��
�2 3�
1�
�1
�� n2 �
�� n1 2�
�n n
2 2�
��
�n n
2 2��
�2 nn 2��
� n1 2�
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-6
9-6
Exam
ple
Exam
ple
© Glencoe/McGraw-Hill A20 Glencoe Algebra 2
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. B
A
D
B
C
B
A
D
B
C
asymptote: x � 0;hole: x � 3
A
A
C
D
B
A
B
C
B
A
D
C
C
A
B
D
B
A
A
C
Chapter 9 Assessment Answer Key Form 1 Form 2APage 553 Page 554 Page 555
(continued on the next page)
© Glencoe/McGraw-Hill A21 Glencoe Algebra 2
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
�
B
A
B
C
B
D
C
B
A
B
A
C
A
D
B
C
D
A
D
B
�x �
x1
�
B
C
A
B
D
A
D
C
C
A
Chapter 9 Assessment Answer Key Form 2A (continued) Form 2BPage 556 Page 557 Page 558
An
swer
s
© Glencoe/McGraw-Hill A22 Glencoe Algebra 2
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 9 Assessment Answer Key Form 2CPage 559 Page 560
xO
f (x) f (x) � x � 3x � 2
© Glencoe/McGraw-Hill A23 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B: �
48 min
��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 9 Assessment Answer Key Form 2DPage 561 Page 562
An
swer
s
© Glencoe/McGraw-Hill A24 Glencoe Algebra 2
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 9 Assessment Answer Key Form 3Page 563 Page 564
xO
f (x) �
f (x)
�2 (x � 3)2
xO
f (x) � x2�42x � 4
f (x)
Chapter 9 Assessment Answer KeyPage 565, Open-Ended Assessment
Scoring Rubric
© Glencoe/McGraw-Hill A25 Glencoe Algebra 2
Score General Description Specific Criteria
• Shows thorough understanding of the concepts ofsimplifying rational expressions, determining verticalasymptotes and point discontinuity of rational functions,solving joint variation problems, identifying equations asdifferent types of functions, and solving rational equationsand inequalities.
• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Goes beyond requirements of some or all problems.
• Shows an understanding of the concepts of simplifyingrational expressions, determining vertical asymptotes andpoint discontinuity of rational functions, solving jointvariation problems, identifying equations as different typesof functions, and solving rational equations andinequalities.
• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Satisfies all requirements of problems.
• Shows an understanding of most of the concepts ofsimplifying rational expressions, determining verticalasymptotes and point discontinuity of rational functions,solving joint variation problems, identifying equations asdifferent types of functions, and solving rational equationsand inequalities.
• May not use appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are satisfactory.• Satisfies the requirements of most of the problems.
• Final computation is correct.• No written explanations or work is shown to substantiate
the final computation.• Satisfies minimal requirements of some of the problems.
• Shows little or no understanding of most of the concepts ofsimplifying rational expressions, determining verticalasymptotes and point discontinuity of rational functions,solving joint variation problems, identifying equations asdifferent types of functions, and solving rational equationsand inequalities.
• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Does not satisfy requirements of problems.• No answer may be given.
0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given
1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation
2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem
3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation
4 SuperiorA correct solution that is supported by well-developed, accurateexplanations
An
swer
s
© Glencoe/McGraw-Hill A26 Glencoe Algebra 2
Chapter 9 Assessment Answer Key Page 565, Open-Ended Assessment
Sample Answers
1. Each student response must includethree expressions which, when
simplified, “reduce 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 A25, the following sample answers may be used as guidance in evaluating open-ended assessment items.
© Glencoe/McGraw-Hill A27 Glencoe Algebra 2
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 0.
12. Sample answer: Acomplex fraction isa fraction in whichthe numerator,denominator, orboth, containfractions.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Quiz (Lesson 9–3)
Page 567
1.
2.
3.4.
1.
2.
3.
4.
5.
Quiz (Lesson 9–6)
Page 568
1.
2.
3.
4.
5. 18
�5 � m � �2
t � 0 or t � 2
9
10
absolute value
greatest integer
quadratic
12
inverse; 30
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 9 Assessment Answer Key Vocabulary Test/Review Quiz (Lessons 9–1 and 9–2) Quiz (Lessons 9–4 and 9–5)
Page 566 Page 567 Page 568
An
swer
s
xO
f (x)f (x) � 4
x � 3
y
xO
y � 3�x � � 2
© Glencoe/McGraw-Hill A28 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12. 11.2 mL
20
3(y � 2)
circle
�(x �
251)2� � �
(y �9
2)2� � 1
x2 � (y � 3)2 � 125
1, 2, 3, 6, �12
�, �32
�
23
y
xO
�1 �i
w�125�
yes
asymptote: x � 4;hole: x � �3
21(3c � 5)
72s3t4
B
C
A
A
D
Chapter 9 Assessment Answer Key Mid-Chapter Test Cumulative ReviewPage 569 Page 570
xO
f (x) � 4(x � 2)2
f (x)
x2 � 5x � 3���(x � 3)(x � 2)(x � 4)
© Glencoe/McGraw-Hill A29 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11. 12.
13. 14.
15.
16.
17.
18. DCBA
DCBA
DCBA
DCBA
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
1 0 4
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
1 3
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
3 0/ 2
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
2 5
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
An
swer
s
Chapter 9 Assessment Answer KeyStandardized Test Practice
Page 571 Page 572