chapter 2 resource masters -...
TRANSCRIPT
Chapter 2Resource 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 2 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: 0-07-828005-2 Algebra 2Chapter 2 Resource Masters
2 3 4 5 6 7 8 9 10 066 11 10 09 08 07 06 05 04 03
Glencoe/McGraw-Hill
© Glencoe/McGraw-Hill iii Glencoe Algebra 2
Contents
Vocabulary Builder . . . . . . . . . . . . . . . . vii
Lesson 2-1Study Guide and Intervention . . . . . . . . . 57–58Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . 59Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Reading to Learn Mathematics . . . . . . . . . . . 61Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Lesson 2-2Study Guide and Intervention . . . . . . . . . 63–64Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . 65Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Reading to Learn Mathematics . . . . . . . . . . . 67Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Lesson 2-3Study Guide and Intervention . . . . . . . . . 69–70Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . 71Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Reading to Learn Mathematics . . . . . . . . . . . 73Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Lesson 2-4Study Guide and Intervention . . . . . . . . . 75–76Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . 77Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Reading to Learn Mathematics . . . . . . . . . . . 79Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Lesson 2-5Study Guide and Intervention . . . . . . . . . 81–82Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . 83Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Reading to Learn Mathematics . . . . . . . . . . . 85Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Lesson 2-6Study Guide and Intervention . . . . . . . . . 87–88Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . 89Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Reading to Learn Mathematics . . . . . . . . . . . 91Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Lesson 2-7Study Guide and Intervention . . . . . . . . . 93–94Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . 95Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Reading to Learn Mathematics . . . . . . . . . . . 97Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Chapter 2 AssessmentChapter 2 Test, Form 1 . . . . . . . . . . . . . 99–100Chapter 2 Test, Form 2A . . . . . . . . . . . 101–102Chapter 2 Test, Form 2B . . . . . . . . . . . 103–104Chapter 2 Test, Form 2C . . . . . . . . . . . 105–106Chapter 2 Test, Form 2D . . . . . . . . . . . 107–108Chapter 2 Test, Form 3 . . . . . . . . . . . . 109–110Chapter 2 Open-Ended Assessment . . . . . . 111Chapter 2 Vocabulary Test/Review . . . . . . . 112Chapter 2 Quizzes 1 & 2 . . . . . . . . . . . . . . . 113Chapter 2 Quizzes 3 & 4 . . . . . . . . . . . . . . . 114Chapter 2 Mid-Chapter Test . . . . . . . . . . . . . 115Chapter 2 Cumulative Review . . . . . . . . . . . 116Chapter 2 Standardized Test Practice . . 117–118
Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A32
© Glencoe/McGraw-Hill iv Glencoe Algebra 2
Teacher’s Guide to Using theChapter 2 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 2 Resource Masters includes the core materials neededfor Chapter 2. 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 2-1.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 2Resource 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 106–107. 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 _____
22
© Glencoe/McGraw-Hill vii Glencoe Algebra 2
Voca
bula
ry B
uild
erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 2.As you study the chapter, complete each term’s definition or description.Remember to add the page number where you found the term. Add these pages toyour Algebra Study Notebook to review vocabulary at the end of the chapter.
Vocabulary Term Found on Page Definition/Description/Example
absolute value function
boundary
constant function
family of graphs
function
greatest integer function
identity function
linear equation
line of fit
one-to-one function
(continued on the next page)
© Glencoe/McGraw-Hill viii Glencoe Algebra 2
Vocabulary Term Found on Page Definition/Description/Example
parent graph
piecewise function
PEES·WYZ
point-slope form
prediction equation
pree·DIHK·shuhn
relation
scatter plot
slope
slope-intercept form
IHN·tuhr·SEHPT
standard form
step function
Reading to Learn MathematicsVocabulary Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
22
Study Guide and InterventionRelations and Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
2-12-1
© Glencoe/McGraw-Hill 57 Glencoe Algebra 2
Less
on
2-1
Graph Relations A relation can be represented as a set of ordered pairs or as anequation; the relation is then the set of all ordered pairs (x, y) that make the equation true.The domain of a relation is the set of all first coordinates of the ordered pairs, and therange is the set of all second coordinates.A function is a relation in which each element of the domain is paired with exactly oneelement of the range. You can tell if a relation is a function by graphing, then using thevertical line test. If a vertical line intersects the graph at more than one point, therelation is not a function.
Graph the equation y � 2x � 3 and find the domain and range. Doesthe equation represent a function?
Make a table of values to find ordered pairs that satisfy the equation. Then graph the ordered pairs.
The domain and range are both all real numbers. Thegraph passes the vertical line test, so it is function.
Graph each relation or equation and find the domain and range. Then determinewhether the relation or equation is a function.
1. {(1, 3), (�3, 5), 2. {(3, �4), (1, 0), 3. {(0, 4), (�3, �2),(�2, 5), (2, 3)} (2, �2), (3, 2)} (3, 2), (5, 1)}
D � {�3, �2, 1, 2}, D � {1, 2, 3}, D � {�3, 0, 3, 5},R � {3, 5}; yes R � {�4, �2, 0, 2}; no R � {�2, 1, 2, 4}; yes
4. y � x2 � 1 5. y � x � 4 6. y � 3x � 2
D � all reals, D � all reals, D � all reals,R � {yy � �1}; yes R � all reals; yes R � all reals; yes
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ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 58 Glencoe Algebra 2
Equations of Functions and Relations Equations that represent functions areoften written in functional notation. For example, y � 10 � 8x can be written as f(x) � 10 � 8x. This notation emphasizes the fact that the values of y, the dependentvariable, depend on the values of x, the independent variable.
To evaluate a function, or find a functional value, means to substitute a given value in thedomain into the equation to find the corresponding element in the range.
Given the function f(x) � x2 � 2x, find each value.
a. f(3)
f(x) � x2 � 2x Original function
f(3) � 32 � 2(3) Substitute.
� 15 Simplify.
b. f(5a)
f(x) � x2 � 2x Original function
f(5a) � (5a)2 � 2(5a) Substitute.
� 25a2 � 10a Simplify.
Find each value if f(x) � �2x � 4.
1. f(12) �20 2. f(6) �8 3. f(2b) �4b � 4
Find each value if g(x) � x3 � x.
4. g(5) 120 5. g(�2) �6 6. g(7c) 343c3 � 7c
Find each value if f(x) � 2x � and g(x) � 0.4x2 � 1.2.
7. f(0.5) 5 8. f(�8) �16 9. g(3) 2.4
10. g(�2.5) 1.3 11. f(4a) 8a � 12. g� � � 1.2
13. f � � 6 14. g(10) 38.8 15. f(200) 400.01
Let f(x) � 2x2 � 1.
16. Find the values of f(2) and f(5). f (2) � 7, f (5) � 49
17. Compare the values of f(2) � f(5) and f(2 � 5). f (2) � f (5) � 343, f (2 � 5) � 199
2�3
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Study Guide and Intervention (continued)
Relations and Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
2-12-1
ExampleExample
ExercisesExercises
Skills PracticeRelations and Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
2-12-1
© Glencoe/McGraw-Hill 59 Glencoe Algebra 2
Less
on
2-1
Determine whether each relation is a function. Write yes or no.
1. yes 2. no
3. yes 4. no
Graph each relation or equation and find the domain and range. Then determinewhether the relation or equation is a function.
5. {(2, �3), (2, 4), (2, �1)} 6. {(2, 6), (6, 2)}
D � {2}, R � {�3, �1, 4}; no D � {2, 6}, R � {2, 6}; yes
7. {(�3, 4), (�2, 4), (�1, �1), (3, �1)} 8. x � �2
D � {�3, �2, �1, 3}, D � {�2}, R � all reals; no R � {�1, 4}; yes
Find each value if f(x) � 2x � 1 and g(x) � 2 � x2.
9. f(0) �1 10. f(12) 23 11. g(4) �14
12. f(�2) �5 13. g(�1) 1 14. f(d) 2d � 1
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© Glencoe/McGraw-Hill 60 Glencoe Algebra 2
Determine whether each relation is a function. Write yes or no.
1. no 2. yes
3. yes 4. no
Graph each relation or equation and find the domain and range. Then determinewhether the relation or equation is a function.
5. {(�4, �1), (4, 0), (0, 3), (2, 0)} 6. y � 2x � 1
D � {�4, 0, 2, 4}, D � all reals, R � all reals; yesR � {�1, 0, 3}; yes
Find each value if f(x) � and g(x) � �2x � 3.
7. f(3) 1 8. f(�4) � 9. g� � 2
10. f(�2) undefined 11. g(�6) 15 12. f(m � 2)
13. MUSIC The ordered pairs (1, 16), (2, 16), (3, 32), (4, 32), and (5, 48) represent the cost ofbuying various numbers of CDs through a music club. Identify the domain and range ofthe relation. Is the relation a function? D � {1, 2, 3, 4, 5}, R � {16, 32, 48}; yes
14. COMPUTING If a computer can do one calculation in 0.0000000015 second, then thefunction T(n) � 0.0000000015n gives the time required for the computer to do ncalculations. How long would it take the computer to do 5 billion calculations? 7.5 s
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Practice (Average)
Relations and Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
2-12-1
Reading to Learn MathematicsRelations and Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
2-12-1
© Glencoe/McGraw-Hill 61 Glencoe Algebra 2
Less
on
2-1
Pre-Activity How do relations and functions apply to biology?
Read the introduction to Lesson 2-1 at the top of page 56 in your textbook.
• Refer to the table. What does the ordered pair (8, 20) tell you? For adeer, the average longevity is 8 years and the maximumlongevity is 20 years.
• Suppose that this table is extended to include more animals. Is it possibleto have an ordered pair for the data in which the first number is largerthan the second? Sample answer: No, the maximum longevitymust always be greater than the average longevity.
Reading the Lesson
1. a. Explain the difference between a relation and a function. Sample answer: Arelation is any set of ordered pairs. A function is a special kind ofrelation in which each element of the domain is paired with exactlyone element in the range.
b. Explain the difference between domain and range. Sample answer: The domainof a relation is the set of all first coordinates of the ordered pairs. Therange is the set of all second coordinates.
2. a. Write the domain and range of the relation shown in the graph.
D: {�3, �2, �1, 0, 3}; R: {�5, �4, 0, 1, 2, 4}
b. Is this relation a function? Explain. Sample answer: No, it is not a functionbecause one of the elements of the domain, 3, is paired with twoelements of the range.
Helping You Remember
3. Look up the words dependent and independent in a dictionary. How can the meaning ofthese words help you distinguish between independent and dependent variables in afunction? Sample answer: The variable whose values depend on, or aredetermined by, the values of the other variable is the dependent variable.
(0, 4)
(3, 1)
(3, –4)(–1, –5)
(–2, 0)
(–3, 2)
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© Glencoe/McGraw-Hill 62 Glencoe Algebra 2
MappingsThere are three special ways in which one set can be mapped to another. A setcan be mapped into another set, onto another set, or can have a one-to-onecorrespondence with another set.
State whether each set is mapped into the second set, onto the second set, or has a one-to-one correspondence with the second set.
1. 2. 3. 4.
into, onto into, onto into, onto, into, ontoone-to-one
5. 6. 7. 8.
into into, onto into, onto into, onto,one-to-one
9. Can a set be mapped onto a set with fewer elements than it has? yes
10. Can a set be mapped into a set that has more elements than it has? yes
11. If a mapping from set A into set B is a one-to-one correspondence, what can you conclude about the number of elements in A and B?The sets have the same number of elements.
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Into mappingA mapping from set A to set B where every element of A is mapped to one or more elements of set B, but never to an element not in B.
Onto mappingA mapping from set A to set B where each element of set B has at least one element of set A mapped to it.
One-to-one A mapping from set A onto set B where each element of set A is mapped to exactly one correspondence element of set B and different elements of A are never mapped to the same element of B.
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
2-12-1
Study Guide and InterventionLinear Equations
NAME ______________________________________________ DATE ____________ PERIOD _____
2-22-2
© Glencoe/McGraw-Hill 63 Glencoe Algebra 2
Less
on
2-2
Identify Linear Equations and Functions A linear equation has no operationsother than addition, subtraction, and multiplication of a variable by a constant. Thevariables may not be multiplied together or appear in a denominator. A linear equation doesnot contain variables with exponents other than 1. The graph of a linear equation is a line.
A linear function is a function whose ordered pairs satisfy a linear equation. Any linearfunction can be written in the form f(x) � mx � b, where m and b are real numbers.
If an equation is linear, you need only two points that satisfy the equation in order to graphthe equation. One way is to find the x-intercept and the y-intercept and connect these twopoints with a line.
Is f(x) � 0.2 � alinear function? Explain.
Yes; it is a linear function because it canbe written in the formf(x) � � x � 0.2.
Is 2x � xy � 3y � 0 alinear function? Explain.
No; it is not a linear function becausethe variables x and y are multipliedtogether in the middle term.
1�5
x�5
Find the x-intercept and they-intercept of the graph of 4x � 5y � 20.Then graph the equation.
The x-intercept is the value of x when y � 0.
4x � 5y � 20 Original equation
4x � 5(0) � 20 Substitute 0 for y.
x � 5 Simplify.
So the x-intercept is 5.Similarly, the y-intercept is �4. x
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Example 1Example 1 Example 3Example 3
Example 2Example 2
ExercisesExercises
State whether each equation or function is linear. Write yes or no. If no, explain.
1. 6y � x � 7 yes 2. 9x � No; the 3. f(x) � 2 � yes
variable y appears in the denominator.
Find the x-intercept and the y-intercept of the graph of each equation. Then graphthe equation.
4. 2x � 7y � 14 5. 5y � x � 10 6. 2.5x � 5y � 7.5 � 0
x-int: 7; y-int: 2 x-int: �10; y-int: 2 x-int: �3; y-int: 1.5
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18�y
© Glencoe/McGraw-Hill 64 Glencoe Algebra 2
Standard Form The standard form of a linear equation is Ax � By � C, where A, B, and C are integers whose greatest common factor is 1.
Write each equation in standard form. Identify A, B, and C.
Study Guide and Intervention (continued)
Linear Equations
NAME ______________________________________________ DATE ____________ PERIOD _____
2-22-2
ExampleExample
a. y � 8x � 5
y � 8x � 5 Original equation
�8x � y � �5 Subtract 8x from each side.
8x � y � 5 Multiply each side by �1.
So A � 8, B � �1, and C � 5.
b. 14x � �7y � 21
14x � �7y � 21 Original equation
14x � 7y � 21 Add 7y to each side.
2x � y � 3 Divide each side by 7.
So A � 2, B � 1, and C � 3.
ExercisesExercises
Write each equation in standard form. Identify A, B, and C.
1. 2x � 4y �1 2. 5y � 2x � 3 3. 3x � �5y � 22x � 4y � �1; A � 2, 2x � 5y � �3; A � 2, 3x � 5y � 2; A � 3,B � �4, C � �1 B � �5, C � �3 B � 5, C � 2
4. 18y � 24x � 9 5. y � x � 5 6. 6y � 8x � 10 � 0
8x � 6y � 3; A � 8, 8x � 9y � �60; A � 8, 4x � 3y � 5; A � 4,B � �6, C � 3 B � �9, C � �60 B � �3, C � 5
7. 0.4x � 3y � 10 8. x � 4y � 7 9. 2y � 3x � 62x � 15y � 50; A � 2, x � 4y � �7; A � 1, 3x � 2y � �6; A � 3,B � 15, C � 50 B � �4, C� �7 B � �2, C � �6
10. x � y �2 � 0 11. 4y � 4x � 12 � 0 12. 3x � �18
6x � 5y � 30; A � 6, x � y � �3; A � 1, x � �6; A � 1,B � 5, C � 30 B � 1, C � �3 B � 0, C � �6
13. x � � 7 14. 3y � 9x � 18 15. 2x � 20 � 8y
9x � y � 63; A � 9, 3x � y � 6; A � 3, x � 4y � 10; A � 1,B � �1, C � 63 B � �1, C � 6 B � 4, C � 10
16. � 3 � 2x 17. � � � y � 8 18. 0.25y � 2x � 0.75
8x � y � �12; A � 8, 10x � 3y � 32; A � 10, 8x � y � 3; A � 8,B � �1, C� �12 B � �3, C � 32 B � �1, C � 3
19. 2y� � 4 � 0 20. 1.6x � 2.4y � 4 21. 0.2x � 100 � 0.4y
x � 12y � �24; A � 1, 2x � 3y � 5; A � 2, x � 2y � 500; A � 1,B � �12, C � �24 B � �3, C � 5 B � 2, C � 500
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Skills PracticeLinear Equations
NAME ______________________________________________ DATE ____________ PERIOD _____
2-22-2
© Glencoe/McGraw-Hill 65 Glencoe Algebra 2
Less
on
2-2
State whether each equation or function is linear. Write yes or no. If no, explainyour reasoning.
1. y � 3x 2. y � �2 � 5x
yes yes
3. 2x � y � 10 4. f(x) � 4x2
yes No; the exponent of x is not 1.
5. � � y � 15 6. x � y � 8
No; x is in a denominator. yes
7. g(x) � 8 8. h(x) � �x� � 3
yes No; x is inside a square root.
Write each equation in standard form. Identify A, B, and C.
9. y � x x � y � 0; 1, �1, 0 10. y � 5x � 1 5x � y � �1; 5, �1, �1
11. 2x � 4 � 7y 2x � 7y � 4; 2, 7, 4 12. 3x � �2y � 2 3x � 2y � �2; 3, 2, �2
13. 5y � 9 � 0 5y � 9; 0, 5, 9 14. �6y � 14 � 8x 4x � 3y � 7; 4, 3, 7
Find the x-intercept and the y-intercept of the graph of each equation. Then graphthe equation.
15. y � 3x � 6 2, �6 16. y � �2x 0, 0
17. x � y � 5 5, 5 18. 2x � 5y � 10 5, 2
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© Glencoe/McGraw-Hill 66 Glencoe Algebra 2
State whether each equation or function is linear. Write yes or no. If no, explainyour reasoning.
1. h(x) � 23 yes 2. y � x yes
3. y � No; x is a denominator. 4. 9 � 5xy � 2 No; x and y are multiplied.
Write each equation in standard form. Identify A, B, and C.
5. y � 7x � 5 7x � y � 5; 7, �1, 5 6. y � x � 5 3x � 8y � �40; 3, �8, �40
7. 3y � 5 � 0 3y � 5; 0, 3, 5 8. x � � y � 28x � 8y � 21; 28, 8, 21
Find the x-intercept and the y-intercept of the graph of each equation. Then graphthe equation.
9. y � 2x � 4 �2, 4 10. 2x � 7y � 14 7, 2
11. y � �2x � 4 �2, �4 12. 6x � 2y � 6 1, 3
13. MEASURE The equation y � 2.54x gives the length in centimeters corresponding to alength x in inches. What is the length in centimeters of a 1-foot ruler? 30.48 cm
LONG DISTANCE For Exercises 14 and 15, use the following information.
For Meg’s long-distance calling plan, the monthly cost C in dollars is given by the linearfunction C(t) � 6 � 0.05t, where t is the number of minutes talked.
14. What is the total cost of talking 8 hours? of talking 20 hours? $30; $66
15. What is the effective cost per minute (the total cost divided by the number of minutestalked) of talking 8 hours? of talking 20 hours? $0.0625; $0.055
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Practice (Average)
Linear Equations
NAME ______________________________________________ DATE ____________ PERIOD _____
2-22-2
Reading to Learn MathematicsLinear Equations
NAME ______________________________________________ DATE ____________ PERIOD _____
2-22-2
© Glencoe/McGraw-Hill 67 Glencoe Algebra 2
Less
on
2-2
Pre-Activity How do linear equations relate to time spent studying?
Read the introduction to Lesson 2-2 at the top of page 63 in your textbook.
• If Lolita spends 2 hours studying math, how many hours will she have
to study chemistry? 1 hours• Suppose that Lolita decides to stay up one hour later so that she now has
5 hours to study and do homework. Write a linear equation that describesthis situation. x � y � 5
Reading the Lesson
1. Write yes or no to tell whether each linear equation is in standard form. If it is not,explain why it is not.
a. �x � 2y � 5 No; A is negative.
b. 9x � 12y � �5 yes
c. 5x � 7y � 3 yes
d. 2x � y � 1 No; B is not an integer.
e. 0x � 0y � 0 No; A and B are both 0.
f. 2x � 4y � 8 No; The greatest common factor of 2, 4, and 8 is 2, not 1.
2. How can you use the standard form of a linear equation to tell whether the graph is ahorizontal line or a vertical line? If A � 0, then the graph is a horizontal line. IfB � 0, then the graph is a vertical line.
Helping You Remember
3. One way to remember something is to explain it to another person. Suppose that you are studying this lesson with a friend who thinks that she should let x � 0 to find the x-intercept and let y � 0 to find the y-intercept. How would you explain to her how toremember the correct way to find intercepts of a line? Sample answer: The x-intercept is the x-coordinate of a point on the x-axis. Every point on the x-axis has y-coordinate 0, so let y � 0 to find an x-intercept. The y-intercept is the y-coordinate of a point on the y-axis. Every point on the y-axis has x-coordinate 0, so let x � 0 to find a y-intercept.
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© Glencoe/McGraw-Hill 68 Glencoe Algebra 2
Greatest Common FactorSuppose we are given a linear equation ax � by � c where a, b, and c are nonzerointegers, and we want to know if there exist integers x and y that satisfy theequation. We could try guessing a few times, but this process would be timeconsuming for an equation such as 588x � 432y � 72. By using the EuclideanAlgorithm, we can determine not only if such integers x and y exist, but also find them. The following example shows how this algorithm works.
Find integers x and y that satisfy 588x � 432y � 72.
Divide the greater of the two coefficients by the lesser to get a quotient andremainder. Then, repeat the process by dividing the divisor by the remainderuntil you get a remainder of 0. The process can be written as follows.
588 � 432(1) � 156 (1)432 � 156(2) � 120 (2)156 � 120(1) � 36 (3)120 � 36(3) � 12 (4)36 � 12(3)
The last nonzero remainder is the GCF of the two coefficients. If the constantterm 72 is divisible by the GCF, then integers x and y do exist that satisfy theequation. To find x and y, work backward in the following manner.
72 � 6 � 12� 6 � [120 � 36(3)] Substitute for 12 using (4)
� 6(120) � 18(36)� 6(120) � 18[156 � 120(1)] Substitute for 36 using (3)
� �18(156) � 24(120)� �18(156) � 24[432 � 156(2)] Substitute for 120 using (2)
� 24(432) � 66(156)� 24(432) � 66[588 � 432(1)] Substitute for 156 using (1)
� 588(�66) � 432(90)
Thus, x � �66 and y � 90.
Find integers x and y, if they exist, that satisfy each equation.
1. 27x � 65y � 3 2. 45x � 144y � 36
3. 90x � 117y � 10 4. 123x � 36y � 15
5. 1032x � 1001y � 1 6. 3125x � 3087y � 1
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
2-22-2
ExampleExample
Study Guide and InterventionSlope
NAME ______________________________________________ DATE ____________ PERIOD _____
2-32-3
© Glencoe/McGraw-Hill 69 Glencoe Algebra 2
Less
on
2-3
Slope
Slope m of a Line For points (x1, y1) and (x2, y2), where x1 � x2, m � �y2 � y1�x2 � x1
change in y��change in x
Determine the slope ofthe line that passes through (2, �1) and(�4, 5).
m � Slope formula
� (x1, y1) � (2, �1), (x2, y2) � (�4, 5)
� � �1 Simplify.
The slope of the line is �1.
6��6
5 � (�1)���4 � 2
y2 � y1�x2 � x1
Graph the line passingthrough (�1, �3) with a slope of .
Graph the ordered pair (�1, �3). Then,according to the slope, go up 4 unitsand right 5 units.Plot the new point(4,1). Connect thepoints and draw the line.
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Example 1Example 1 Example 2Example 2
ExercisesExercises
Find the slope of the line that passes through each pair of points.
1. (4, 7) and (6, 13) 3 2. (6, 4) and (3, 4) 0 3. (5, 1) and (7, �3) �2
4. (5, �3) and (�4, 3) � 5. (5, 10) and (�1,�2) 2 6. (�1, �4) and (�13, 2) �
7. (7, �2) and (3, 3) � 8. (�5, 9) and (5, 5) � 9. (4, �2) and (�4, �8)
Graph the line passing through the given point with the given slope.
10. slope � � 11. slope � 2 12. slope � 0
passes through (0, 2) passes through (1, 4) passes through (�2, �5)
13. slope � 1 14. slope � � 15. slope �
passes through (�4, 6) passes through (�3, 0) passes through (0, 0)
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© Glencoe/McGraw-Hill 70 Glencoe Algebra 2
Parallel and Perpendicular Lines
Study Guide and Intervention (continued)
Slope
NAME ______________________________________________ DATE______________ PERIOD _____
2-32-3
In a plane, nonvertical lines with thesame slope are parallel. All verticallines are parallel.
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In a plane, two oblique lines are perpendicular ifand only if the product of their slopes is �1. Anyvertical line is perpendicular to any horizontal line.
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ExampleExample Are the line passing through (2, 6) and (�2, 2) and the line passingthrough (3, 0) and (0, 4) parallel, perpendicular, or neither?
Find the slopes of the two lines.
The slope of the first line is � 1.
The slope of the second line is � � .
The slopes are not equal and the product of the slopes is not �1, so the lines are neitherparallel nor perpendicular.
Are the lines parallel, perpendicular, or neither?
1. the line passing through (4, 3) and (1, �3) and the line passing through (1, 2) and (�1, 3)perpendicular
2. the line passing through (2, 8) and (�2, 2) and the line passing through (0, 9) and (6, 0)neither
3. the line passing through (3, 9) and (�2, �1) and the graph of y � 2x parallel
4. the line with x-intercept �2 and y-intercept 5 and the line with x-intercept 2 and y-intercept �5 parallel
5. the line with x-intercept 1 and y-intercept 3 and the line with x-intercept 3 and y-intercept 1 neither
6. the line passing through (�2, �3) and (2, 5) and the graph of x � 2y � 10perpendicular
7. the line passing through (�4, �8) and (6, �4) and the graph of 2x � 5y � 5 parallel
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ExercisesExercises
Skills PracticeSlope
NAME ______________________________________________ DATE ____________ PERIOD _____
2-32-3
© Glencoe/McGraw-Hill 71 Glencoe Algebra 2
Less
on
2-3
Find the slope of the line that passes through each pair of points.
1. (1, 5), (�1, �3) 4 2. (0, 2), (3, 0) � 3. (1, 9), (0, 6) 3
4. (8, �5), (4, �2) � 5. (�3, 5), (�3, �1) undefined 6. (�2, �2), (10, �2) 0
7. (4, 5), (2, 7) �1 8. (�2, �4), (3, 2) 9. (5, 2), (�3, 2) 0
Graph the line passing through the given point with the given slope.
10. (0, 4), m � 1 11. (2, �4), m � �1
12. (�3, �5), m � 2 13. (�2, �1), m � �2
Graph the line that satisfies each set of conditions.
14. passes through (0, 1), perpendicular to 15. passes through (0, �5), parallel to the
a line whose slope is graph of y � 1
16. HIKING Naomi left from an elevation of 7400 feet at 7:00 A.M. and hiked to an elevationof 9800 feet by 11:00 A.M. What was her rate of change in altitude? 600 ft /h
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© Glencoe/McGraw-Hill 72 Glencoe Algebra 2
Find the slope of the line that passes through each pair of points.
1. (3, �8), (�5, 2) � 2. (�10, �3), (7, 2) 3. (�7, �6), (3, �6) 0
4. (8, 2), (8, �1) undefined 5. (4, 3), (7, �2) � 6. (�6, �3), (�8, 4) �
Graph the line passing through the given point with the given slope.
7. (0, �3), m � 3 8. (2, 1), m � �
9. (0, 2), m � 0 10. (2, �3), m �
Graph the line that satisfies each set of conditions.
11. passes through (3, 0), perpendicular 12. passes through (�3, �1), parallel to a line
to a line whose slope is whose slope is �1
DEPRECIATION For Exercises 13–15, use the following information.A machine that originally cost $15,600 has a value of $7500 at the end of 3 years. The samemachine has a value of $2800 at the end of 8 years.
13. Find the average rate of change in value (depreciation) of the machine between itspurchase and the end of 3 years. �$2700 per year
14. Find the average rate of change in value of the machine between the end of 3 years andthe end of 8 years. �$940 per year
15. Interpret the sign of your answers. It is negative because the value is decreasing.
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Practice (Average)
Slope
NAME ______________________________________________ DATE______________ PERIOD _____
2-32-3
Reading to Learn MathematicsSlope
NAME ______________________________________________ DATE ____________ PERIOD _____
2-32-3
© Glencoe/McGraw-Hill 73 Glencoe Algebra 2
Less
on
2-3
Pre-Activity How does slope apply to the steepness of roads?
Read the introduction to Lesson 2-3 at the top of page 68 in your textbook.
• What is the grade of a road that rises 40 feet over a horizontal distanceof 1000 feet? 4%
• What is the grade of a road that rises 525 meters over a horizontaldistance of 10 kilometers? (1 kilometer � 1000 meters) 5.25%
Reading the Lesson
1. Describe each type of slope and include a sketch.
Type of Slope Description of Graph Sketch
Positive The line rises to the right.
Zero The line is horizontal.
Negative The line falls to the right.
Undefined The line is vertical.
2. a. How are the slopes of two nonvertical parallel lines related? They are equal.
b. How are the slopes of two oblique perpendicular lines related? Their product is �1.
Helping You Remember
3. Look up the terms grade, pitch, slant, and slope. How can everyday meanings of thesewords help you remember the definition of slope? Sample answer: All these wordscan be used when you describe how much a thing slants upward ordownward. You can describe this numerically by comparing rise to run.
© Glencoe/McGraw-Hill 74 Glencoe Algebra 2
Aerial Surveyors and AreaMany land regions have irregular shapes. Aerial surveyors supply aerial mappers with lists of coordinates and elevations for the areas that need to be photographed from the air. These maps provide information about the horizontal and vertical features of the land.
Step 1 List the ordered pairs for the vertices in counterclockwise order, repeating the first ordered pair at the bottom of the list.
Step 2 Find D, the sum of the downward diagonal products (from left to right).D � (5 � 5) � (2 � 1) � (2 � 3) � (6 � 7)
� 25 � 2 � 6 � 42 or 75
Step 3 Find U, the sum of the upward diagonal products (from left to right).U � (2 � 7) � (2 � 5) � (6 � 1) � (5 � 3)
� 14 � 10 � 6 � 15 or 45
Step 4 Use the formula A � �12�(D � U) to find the area.
A � �12�(75 � 45)
� �12�(30) or 15
The area is 15 square units. Count the number of square units enclosed by the polygon. Does this result seem reasonable?
Use the coordinate method to find the area of each region in square units.
1. 2. 3.
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Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
2-32-3
Study Guide and InterventionWriting Linear Equations
NAME ______________________________________________ DATE ____________ PERIOD _____
2-42-4
© Glencoe/McGraw-Hill 75 Glencoe Algebra 2
Less
on
2-4
Forms of Equations
Slope-Intercept Form of a Linear Equation
y � mx � b, where m is the slope and b is the y-intercept
Point-Slope Form y � y1 � m(x � x1), where (x1, y1) are the coordinates of a point on the line and of a Linear Equation m is the slope of the line
Write an equation inslope-intercept form for the line thathas slope �2 and passes through thepoint (3, 7).
Substitute for m, x, and y in the slope-intercept form.
y � mx � b Slope-intercept form
7 � (�2)(3) � b (x, y ) � (3, 7), m � �2
7 � �6 � b Simplify.
13 � b Add 6 to both sides.
The y-intercept is 13. The equation in slope-intercept form is y � �2x � 13.
Write an equation inslope-intercept form for the line thathas slope and x-intercept 5.
y � mx � b Slope-intercept form
0 � � �(5) � b (x, y ) � (5, 0), m �
0 � � b Simplify.
� � b Subtract from both sides.
The y-intercept is � . The slope-intercept
form is y � x � .5�3
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Example 1Example 1 Example 2Example 2
ExercisesExercises
Write an equation in slope-intercept form for the line that satisfies each set ofconditions.
1. slope �2, passes through (�4, 6) 2. slope , y-intercept 4
y � �2x � 2 y � x � 4
3. slope 1, passes through (2, 5) 4. slope � , passes through (5, �7)
y � x � 3 y � � x � 6
Write an equation in slope-intercept form for each graph.
5. 6. 7.
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© Glencoe/McGraw-Hill 76 Glencoe Algebra 2
Parallel and Perpendicular Lines Use the slope-intercept or point-slope form to findequations of lines that are parallel or perpendicular to a given line. Remember that parallellines have equal slope. The slopes of two perpendicular lines are negative reciprocals, thatis, their product is �1.
Study Guide and Intervention (continued)
Writing Linear Equations
NAME ______________________________________________ DATE ____________ PERIOD _____
2-42-4
Write an equation of theline that passes through (8, 2) and isperpendicular to the line whose equation is y � � x � 3.
The slope of the given line is � . Since the
slopes of perpendicular lines are negativereciprocals, the slope of the perpendicularline is 2.Use the slope and the given point to writethe equation.y � y1 � m(x � x1) Point-slope form
y � 2 � 2(x � 8) (x1, y1) � (8, 2), m � 2
y � 2 � 2x � 16 Distributive Prop.
y � 2x � 14 Add 2 to each side.
An equation of the line is y � 2x � 14.
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Write an equation of theline that passes through (�1, 5) and isparallel to the graph of y � 3x � 1.
The slope of the given line is 3. Since theslopes of parallel lines are equal, the slopeof the parallel line is also 3.Use the slope and the given point to writethe equation.y �y1 � m(x � x1) Point-slope form
y � 5 � 3(x � (�1)) (x1, y1) � (�1, 5), m � 3
y � 5 � 3x � 3 Distributive Prop.
y � 3x � 8 Add 5 to each side.
An equation of the line is y � 3x � 8.
Example 1Example 1 Example 2Example 2
ExercisesExercises
Write an equation in slope-intercept form for the line that satisfies each set ofconditions.
1. passes through (�4, 2), parallel to the line whose equation is y � x � 5 y � x � 4
2. passes through (3, 1), perpendicular to the graph of y � �3x � 2 y � x
3. passes through (1, �1), parallel to the line that passes through (4, 1) and (2, �3)y � 2x � 3
4. passes through (4, 7), perpendicular to the line that passes through (3, 6) and (3, 15)y � 7
5. passes through (8, �6), perpendicular to the graph of 2x � y � 4 y � � x � 2
6. passes through (2, �2), perpendicular to the graph of x � 5y � 6 y � 5x � 12
7. passes through (6, 1), parallel to the line with x-intercept �3 and y-intercept 5
y � x � 9
8. passes through (�2, 1), perpendicular to the line y � 4x � 11 y � � x � 1�2
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Skills PracticeWriting Linear Equations
NAME ______________________________________________ DATE ____________ PERIOD _____
2-42-4
© Glencoe/McGraw-Hill 77 Glencoe Algebra 2
Less
on
2-4
State the slope and y-intercept of the graph of each equation.
1. y � 7x � 5 7, �5 2. y � � x � 3 � , 3
3. y � x , 0 4. 3x � 4y � 4 � , 1
5. 7y � 4x � 7 , �1 6. 3x � 2y � 6 � 0 , 3
7. 2x � y � 5 2, �5 8. 2y � 6 � 5x � , 3
Write an equation in slope-intercept form for each graph.
9. 10. 11.
y � 3x � 1 y � �1 y � �2x � 3
Write an equation in slope-intercept form for the line that satisfies each set ofconditions.
12. slope 3, passes through (1, �3) 13. slope �1, passes through (0, 0)
y � 3x � 6 y � �x
14. slope �2, passes through (0, �5) 15. slope 3, passes through (2, 0)
y � �2x � 5 y � 3x � 6
16. passes through (�1, �2) and (�3, 1) 17. passes through (�2, �4) and (1, 8)
y � � x � y � 4x � 4
18. x-intercept 2, y-intercept �6 19. x-intercept , y-intercept 5
y � 3x � 6 y � �2x � 5
20. passes through (3, �1), perpendicular to the graph of y � � x � 4. y � 3x � 101�3
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© Glencoe/McGraw-Hill 78 Glencoe Algebra 2
State the slope and y-intercept of the graph of each equation.
1. y � 8x � 12 8, 12 2. y � 0.25x � 1 0.25, �1 3. y � � x � , 0
4. 3y � 7 0, 5. 3x � �15 � 5y , 3 6. 2x � 3y � 10 , �
Write an equation in slope-intercept form for each graph.
7. 8. 9.
y � 2 y � x � 2 y � � x � 1
Write an equation in slope-intercept form for the line that satisfies each set ofconditions.
10. slope �5, passes through (�3, �8) 11. slope , passes through (10, �3)
y � �5x � 23 y � x � 11
12. slope 0, passes through (0, �10) 13. slope � , passes through (6, �8)
y � �10 y � � x � 4
14. passes through (3, 11) and (�6, 5) 15. passes through (7, �2) and (3, �1)
y � x � 9 y � � x �
16. x-intercept 3, y-intercept 2 17. x-intercept �5, y-intercept 7
y � � x � 2 y � x � 7
18. passes through (�8, �7), perpendicular to the graph of y � 4x � 3 y � � x � 9
19. RESERVOIRS The surface of Grand Lake is at an elevation of 648 feet. During thecurrent drought, the water level is dropping at a rate of 3 inches per day. If this trendcontinues, write an equation that gives the elevation in feet of the surface of Grand Lakeafter x days. y � �0.25x � 648
20. BUSINESS Tony Marconi’s company manufactures CD-ROM drives. The company willmake $150,000 profit if it manufactures 100,000 drives, and $1,750,000 profit if itmanufactures 500,000 drives. The relationship between the number of drivesmanufactured and the profit is linear. Write an equation that gives the profit P when n drives are manufactured. P � 4n � 250,000
1�4
7�5
2�3
1�4
1�4
2�3
2�3
2�3
4�5
4�5
2�3
3�2
x
y
O(3, –1)
(–3, 3)
x
y
O
(4, 4)
(0, –2)
x
y
O
(0, 2)
10�3
2�3
3�5
7�3
3�5
3�5
Practice (Average)
Writing Linear Equations
NAME ______________________________________________ DATE ____________ PERIOD _____
2-42-4
Reading to Learn MathematicsWriting Linear Equations
NAME ______________________________________________ DATE ____________ PERIOD _____
2-42-4
© Glencoe/McGraw-Hill 79 Glencoe Algebra 2
Less
on
2-4
Pre-Activity How do linear equations apply to business?
Read the introduction to Lesson 2-4 at the top of page 75 in your textbook.
• If the total cost of producing a product is given by the equation y � 5400 � 1.37x, what is the fixed cost? What is the variable cost (for each item produced)? $5400; $1.37
• Write a linear equation that describes the following situation:A company that manufactures computers has a fixed cost of $228,750 anda variable cost of $852 to produce each computer.y � 228,750 � 852x
Reading the Lesson
1. a. Write the slope-intercept form of the equation of a line. Then explain the meaning ofeach of the variables in the equation. y � mx � b; m is the slope and b is they-intercept. The variables x and y are the coordinates of any point onthe line.
b. Write the point-slope form of the equation of a line. Then explain the meaning of eachof the variables in the equation. y � y1 � m(x � x1); m is the slope. x and yare the coordinates of any point on the line. x1 and y1 are the coordinates of one specific point on the line.
2. Suppose that your algebra teacher asks you to write the point-slope form of the equationof the line through the points (�6, 7) and (�3, �2). You write y � 2 � �3(x � 3) andyour classmate writes y � 7 � �3(x � 6). Which of you is correct? Explain. You areboth correct. Either point may be used as (x1, y1) in the point-slope form.You used (�3, �2), and your classmate used (�6, 7).
3. You are asked to write an equation of two lines that pass through (3, �5), one of themparallel to and one of them perpendicular to the line whose equation is y � �3x � 4.The first step in finding these equations is to find their slopes. What is the slope of theparallel line? What is the slope of the perpendicular line? �3;
Helping You Remember
4. Many students have trouble remembering the point-slope form for a linear equation.How can you use the definition of slope to remember this form? Sample answer:
Write the definition of slope: m � . Multiply both sides of this
equation by x2 � x1. Drop the subscripts in y2 and x2. This gives thepoint-slope form of the equation of a line.
y2 � y1�x2 � x1
1�3
© Glencoe/McGraw-Hill 80 Glencoe Algebra 2
Two-Intercept Form of a Linear EquationYou are already familiar with the slope-intercept form of a linear equation,
y � mx � b. Linear equations can also be written in the form �ax
� � �by
� � 1 with x-intercept a and y-intercept b. This is called two-intercept form.
Draw the graph of ��
x3�
� �6y
� � 1.
The graph crosses the x-axis at �3 and the y-axis at 6. Graph (�3, 0) and (0, 6), then draw a straight line through them.
Write 3x � 4y � 12 in two-intercept form.
�132x� � �1
42y� � �
1122� Divide by 12 to obtain 1 on the right side.
�4x
� � �3y
� � 1 Simplify.
The x-intercept is 4; the y-intercept is 3.
Use the given intercepts a and b, to write an equation in two-intercept form. Then draw the graph. See students’ graphs.
1. a � �2, b � �4 2. a � 1, b � 8
3. a � 3, b � 5 4. a � 6, b � 9
Write each equation in two-intercept form. Then draw the graph.
5. 3x � 2y � �6 6. �12�x � �
14�y � 1 7. 5x � 2y � �10
x
y
Ox
y
Ox
y
O
x
y
O
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
2-42-4
Example 1Example 1
Example 2Example 2
Study Guide and InterventionModeling Real-World Data: Using Scatter Plots
NAME ______________________________________________ DATE ____________ PERIOD _____
2-52-5
© Glencoe/McGraw-Hill 81 Glencoe Algebra 2
Less
on
2-5
Scatter Plots When a set of data points is graphed as ordered pairs in a coordinateplane, the graph is called a scatter plot. A scatter plot can be used to determine if there isa relationship among the data.
BASEBALL The table below shows the number of home runs andruns batted in for various baseball players who won the Most Valuable PlayerAward during the 1990s. Make a scatter plot of the data.
Source: New York Times Almanac
Make a scatter plot for the data in each table below.
1. FUEL EFFICIENCY The table below shows the average fuel efficiency in miles per gallon of new cars manufactured during the years listed.
Source: New York Times Almanac
2. CONGRESS The table below shows the number of women serving in the United States Congress during the years 1987�1999.
Source: Wall Street Journal Almanac
Congressional Session Number of Women
100 25
101 31
102 33
103 55
104 58
105 62
Session of Congress
Nu
mb
er o
f W
om
en
100 102 104
70
56
42
28
14
0
Women in Congress
Year Fuel Efficiency (mpg)
1960 15.5
1970 14.1
1980 22.6
1990 26.9 Year
Mile
s p
er G
allo
n
1960 1970 1980 1990
36
30
24
18
12
6
0
Average Fuel Efficiency
Home Runs
MVP HRs and RBIs
Ru
ns
Bat
ted
In
1260 24 3618 30 42 48
150
125
100
75
50
25
Home Runs Runs Batted In
33 114
39 116
40 130
28 61
41 128
47 144
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 82 Glencoe Algebra 2
Prediction Equations A line of fit is a line that closely approximates a set of datagraphed in a scatter plot. The equation of a line of fit is called a prediction equationbecause it can be used to predict values not given in the data set.
To find a prediction equation for a set of data, select two points that seem to represent thedata well. Then to write the prediction equation, use what you know about writing a linearequation when given two points on the line.
STORAGE COSTS According to a certain prediction equation, thecost of 200 square feet of storage space is $60. The cost of 325 square feet ofstorage space is $160.
a. Find the slope of the prediction equation. What does it represent?Since the cost depends upon the square footage, let x represent the amount of storagespace in square feet and y represent the cost in dollars. The slope can be found using the
formula m � . So, m � � � 0.8
The slope of the prediction equation is 0.8. This means that the price of storage increases80¢ for each one-square-foot increase in storage space.
b. Find a prediction equation.Using the slope and one of the points on the line, you can use the point-slope form to finda prediction equation.
y � y1 � m(x � x1) Point-slope form
y � 60 � 0.8(x � 200) (x1, y1) � (200, 60), m � 0.8
y � 60 � 0.8x � 160 Distributive Property
y � 0.8x � 100 Add 60 to both sides.
A prediction equation is y � 0.8x � 100.
SALARIES The table below shows the years of experience for eight technicians atLewis Techomatic and the hourly rate of pay each technician earns. Use the datafor Exercises 1 and 2.
Experience (years) 9 4 3 1 10 6 12 8
Hourly Rate of Pay $17 $10 $10 $7 $19 $12 $20 $15
1. Draw a scatter plot to show how years of experience are related to hourly rate of pay. Draw a line of fit. See graph.
2. Write a prediction equation to show how years of experience(x) are related to hourly rate of pay (y). Sample answerusing (1, 7) and (9, 17): y � 1.25x � 5.75
Experience (years)
Ho
url
y Pa
y ($
)
20 6 104 8 12 14
24
20
16
12
8
4
Technician Salaries
100�125
160 � 60��325 � 200
y2 � y1�x2 � x1
Study Guide and Intervention (continued)
Modeling Real-World Data: Using Scatter Plots
NAME ______________________________________________ DATE ____________ PERIOD _____
2-52-5
ExampleExample
ExercisesExercises
Skills PracticeModeling Real-World Data: Using Scatter Plots
NAME ______________________________________________ DATE ____________ PERIOD _____
2-52-5
© Glencoe/McGraw-Hill 83 Glencoe Algebra 2
Less
on
2-5
For Exercises 1–3, complete parts a–c for each set of data.
a. Draw a scatter plot.b. Use two ordered pairs to write a prediction equation.c. Use your prediction equation to predict the missing value.
1. 1a.
1b. Sample answer using (1, 1) and (8, 15): y � 2x � 11c. Sample answer: 19
2. 2a.
2b. Sample answer using (5, 9) and (40, 44): y � x � 42c. Sample answer: 54
3. 3a.
3b. Sample answer using (2, 16) and (7, 34): y � 3.6x � 8.83c. Sample answer: 19.6
1 3 5 72 4 6 8
36
30
24
18
12
6
0 x
yx y
1 16
2 16
3 ?
4 22
5 30
7 34
8 36
5 15 25 3510 20 30 40
40
32
24
16
8
0 x
yx y
5 9
10 17
20 22
25 30
35 38
40 44
50 ?
1 3 5 72 4 6 8
15
12
9
6
3
0 x
yx y
1 1
3 5
4 7
6 11
7 12
8 15
10 ?
© Glencoe/McGraw-Hill 84 Glencoe Algebra 2
For Exercises 1–3, complete parts a–c for each set of data.a. Draw a scatter plot.b. Use two ordered pairs to write a prediction equation.c. Use your prediction equation to predict the missing value.
1. FUEL ECONOMY The table gives the approximate weights in tons and estimates for overall fuel economy in miles per gallon for several cars.1b. Sample answer using (1.4, 24) and
(2.4, 15): y � �9x � 36.6
1c. Sample answer: 18.6 mi/gal
2. ALTITUDE In most cases, temperature decreases with increasing altitude. As Ancharadrives into the mountains, her car thermometer registers the temperatures (°F) shownin the table at the given altitudes (feet).
2b. Sample answer using (7500, 61) and (9700, 50): y � �0.005x � 98.5
2c. Sample answer: 38.5°F
3. HEALTH Alton has a treadmill that uses the time on the treadmill and the speed of walking or running to estimate the number of Calories he burns during a workout. Thetable gives workout times and Calories burned for several workouts.
3b. Sample answer using (24, 280) and(48, 440): y � 6.67x � 119.92
3c. Sample answer: about 520 calories
Time (min) 18 24 30 40 42 48 52 60
Calories Burned 260 280 320 380 400 440 475 ?
Altitude (ft)
Tem
per
atu
re (�
F)
0 7,000 8,000 9,000 10,000
65
60
55
50
45
TemperatureVersus Altitude
Altitude (ft) 7500 8200 8600 9200 9700 10,400 12,000
Temperature (�F) 61 58 56 53 50 46 ?
Weight (tons)
Fuel
Eco
no
my
(mi/
gal
)
0 0.5 1.0 1.5 2.0 2.5
30
25
20
15
10
5
Fuel Economy Versus Weight
Weight (tons) 1.3 1.4 1.5 1.8 2 2.1 2.4
Miles per Gallon 29 24 23 21 ? 17 15
Practice (Average)
Modeling Real-World Data: Using Scatter Plots
NAME ______________________________________________ DATE ____________ PERIOD _____
2-52-5
Reading to Learn MathematicsModeling Real-World Data: Using Scatter Plots
NAME ______________________________________________ DATE ____________ PERIOD _____
2-52-5
© Glencoe/McGraw-Hill 85 Glencoe Algebra 2
Less
on
2-5
Pre-Activity How can a linear equation model the number of Calories you burnexercising?
Read the introduction to Lesson 2-5 at the top of page 81 in your textbook.
• If a woman runs 5.5 miles per hour, about how many Calories will sheburn in an hour? Sample answer: 572 Calories
• If a man runs 7.5 miles per hour, about how many Calories will he burnin half an hour? Sample answer: 397 Calories
Reading the Lesson
1. Suppose that a set of data can be modeled by a linear equation. Explain the differencebetween a scatter plot of the data and a graph of the linear equation that models thatdata.Sample answer: The scatter plot is a discrete graph. It is made up just ofthe individual points that represent the data points. The linear equationhas a continuous graph that is the line that best fits the data points.
2. Suppose that tuition at a state college was $3500 per year in 1995 and has beenincreasing at a rate of $225 per year.
a. Write a prediction equation that expresses this information.y � 3500 � 225x
b. Explain the meaning of each variable in your prediction equation.x represents the number of year since 1995 and y represents thetuition in that year.
3. Use this model to predict the tuition at this college in 2007. $6200
Helping You Remember
4. Look up the word scatter in a dictionary. How can its definition help you to rememberthe meaning of the difference between a scatter plot and the graph of a linear equation?Sample answer: To scatter means to break up and go in many directions.The points on a scatter plot are broken up. In a scatter plot, the pointsare scattered or broken up. In the graph of a linear equation, the pointsare connected to form a continuous line.
© Glencoe/McGraw-Hill 86 Glencoe Algebra 2
Median-Fit Lines A median-fit line is a particular type of line of fit. Follow the steps below to find the equation of the median-fit line for the data.
Approximate Percentage of Violent Crimes Committed by Juveniles That Victims Reported to Law Enforcement
Year 1980 1982 1984 1986 1988 1990 1992 1994 1996
Offenders 36 35 33 32 31 30 29 29 30
Source: U.S. Bureau of Justice Statistics
1. Divide the data into three approximately equal groups. There should always be the same number of points in the first and third groups. In this case, there will be three data points in each group.
Group 1 Group 2 Group 3enders
2. Find x1, x2, and x3, the medians of the x values in groups 1, 2, and 3,respectively. Find y1, y2, and y3, the medians of the y values in groups 1, 2, and 3, respectively. 1982, 1988, 1994; 35, 31, 29
3. Find an equation of the line through (x1, y1) and (x3, y3). y � �0.5x � 1026
4. Find Y, the y-coordinate of the point on the line in Exercise 2 with an x-coordinate of x2. 32
5. The median-fit line is parallel to the line in Exercise 2, but is one-third
closer to (x2, y2). This means it passes through �x2, �23�Y � �
13�y2�. Find this
ordered pair. about (1988, 31.67)
6. Write an equation of the median-fit line. y � �0.5x � 1025.67
7. Use the median-fit line to predict the percentage of juvenile violent crime offenders in 2010 and 2020. 2010: about 21%; 2020: about16%
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
2-52-5
Study Guide and InterventionSpecial Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
2-62-6
© Glencoe/McGraw-Hill 87 Glencoe Algebra 2
Less
on
2-6
Step Functions, Constant Functions, and the Identity Function The chartbelow lists some special functions you should be familiar with.
Function Written as Graph
Constant f(x) � c horizontal line
Identity f(x) � x line through the origin with slope 1
Greatest Integer Function f(x) � �x�one-unit horizontal segments, with right endpoints missing, arranged like steps
The greatest integer function is an example of a step function, a function with a graph thatconsists of horizontal segments.
Identify each function as a constant function, the identity function,or a step function.
a. b.
a constant function a step function
Identify each function as a constant function, the identity function, a greatestinteger function, or a step function.
1. 2. 3.
a constant function a step function the identity function
x
f(x)
Ox
f(x)
Ox
f(x)
O
x
f(x)
Ox
f(x)
O
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 88 Glencoe Algebra 2
Absolute Value and Piecewise Functions Another special function is theabsolute value function, which is also called a piecewise function.
Absolute Value Function f(x ) � x two rays that are mirror images of each other and meet at a point, the vertex
To graph a special function, use its definition and your knowledge of the parent graph. Findseveral ordered pairs, if necessary.
Graph f(x) � 3x � 4.
Find several ordered pairs. Graph the points andconnect them. You would expect the graph to looksimilar to its parent function, f(x) � x .
Graph f(x) � �2x if x � 2x � 1 if x � 2.
First, graph the linear function f(x) � 2x for x � 2. Since 2 does notsatisfy this inequality, stop with a circle at (2, 4). Next, graph thelinear function f(x) � x � 1 for x � 2. Since 2 does satisfy thisinequality, begin with a dot at (2, 1).
Graph each function. Identify the domain and range.
1. g(x) � � � 2. h(x) � 2x � 1 3. h(x ) �
domain: all real domain: all real domain: all real numbers; range: numbers; range: numbers; range:all integers {yy � 0} {yy 1}
x
y
O
x
y
O
x
y
O
x�3
x
f(x)
O
x
f(x)
O
x 3x � 4
0 �4
1 �1
2 2
�1 �1
�2 2
Study Guide and Intervention (continued)
Special Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
2-62-6
ExercisesExercises
Example 1Example 1
Example 2Example 2
if x 0
2x � 6 if 0 � x � 21 if x � 2
x�3
Skills PracticeSpecial Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
2-62-6
© Glencoe/McGraw-Hill 89 Glencoe Algebra 2
Less
on
2-6
Identify each function as S for step, C for constant, A for absolute value, or P forpiecewise.
1. 2. 3.
S C A
Graph each function. Identify the domain and range.
4. f(x) � �x � 1� 5. f(x) � �x � 3�
D � all reals, R � all integers D � all reals, R � all integers
6. g(x) � 2 x 7. f(x) � x � 1
D � all reals, D � all reals, R � {yy � 1}R � nonnegative reals
8. f(x) � �x if x � 09. h(x) � �3 if x � �1
2 if x � 0 x � 1 if x > 1
D � all reals, D � {xx � �1 or x 1},R � {yy � 0 or y � 2} R � {yy 2}
x
h(x)
O
x
f(x)
O
x
f(x)
Ox
g(x)
O
x
f(x)
O
x
f(x)
O
x
y
O
x
y
Ox
y
O
© Glencoe/McGraw-Hill 90 Glencoe Algebra 2
Graph each function. Identify the domain and range.
1. f(x) � �0.5x� 2. f(x) � �x� � 2
D � all reals, R � all integers D � all reals, R � all integers3. g(x) � �2 x 4. f(x) � x � 1
D � all reals, D � all reals,R � nonpositive reals R � nonnegative reals
5. f(x) � �x � 2 if x � 26. h(x) � �4 � x if x 0
3x if x �2 �2x � 2 if x � 0
D � all reals, R � all reals D � all nonzero reals, R � all reals7. BUSINESS A Stitch in Time charges 8. BUSINESS A wholesaler charges a store $3.00
$40 per hour or any fraction thereof per pound for less than 20 pounds of candy andfor labor. Draw a graph of the step $2.50 per pound for 20 or more pounds. Draw afunction that represents this situation. graph of the function that represents this
situation.
Hours
Tota
l Co
st (
$)
10 3 52 4 6 7
280
240
200
160
120
80
40
Labor Costs
x
f(x)
O
x
g(x)
O
x
f(x)
Ox
f(x)
O
Practice (Average)
Special Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
2-62-6
Reading to Learn MathematicsSpecial Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
2-62-6
© Glencoe/McGraw-Hill 91 Glencoe Algebra 2
Less
on
2-6
Pre-Activity How do step functions apply to postage rates?
Read the introduction to Lesson 2-6 at the top of page 89 in your textbook.
• What is the cost of mailing a letter that weighs 0.5 ounce?$0.34 or 34 cents
• Give three different weights of letters that would each cost 55 cents tomail. Answers will vary. Sample answer: 1.1 ounces,1.9 ounces, 2.0 ounces
Reading the Lesson
1. Find the value of each expression.
a. �3 � ��3� �
b. 6.2 � �6.2� �
c. �4.01 � ��4.01� �
2. Tell how the name of each kind of function can help you remember what the graph looks like.
a. constant function Sample answer: Something is constant if it does notchange. The y-values of a constant function do not change, so thegraph is a horizontal line.
b. absolute value function Sample answer: The absolute value of a numbertells you how far it is from 0 on the number line. It makes no differencewhether you go to the left or right so long as you go the samedistance each time.
c. step function Sample answer: A step function’s graph looks like stepsthat go up or down.
d. identity function Sample answer: The x- and y-values are alwaysidentically the same for any point on the graph. So the graph is a linethrough the origin that has slope 1.
Helping You Remember
3. Many students find the greatest integer function confusing. Explain how you can use anumber line to find the value of this function for any real number. Answers will vary.Sample answer: Draw a number line that shows the integers. To find thevalue of the greatest integer function for any real number, place thatnumber on the number line. If it is an integer, the value of the function isthe number itself. If not, move to the integer directly to the left of thenumber you chose. This integer will give the value you need.
�54.01
66.2
�33
© Glencoe/McGraw-Hill 92 Glencoe Algebra 2
Greatest Integer FunctionsUse the greatest integer function � x� to explore some unusual graphs. It will be helpful to make a chart of values for each functions and to use a colored pen or pencil.
Graph each function.
1. y � 2x � � x� 2. y � ���xx
��
�
3. y � ���00..55xx
�
�
11
��
� 4. y � ��xx��
x
y
O 1–1–2–3–4 2 3 4
4
3
2
1
–1
–2
–3
–4
x
y
O 1–1–2–3–4 2 3 4
4
3
2
1
–1
–2
–3
–4
x
y
O 1–1–2–3–4 2 3 4
4
3
2
1
–1
–2
–3
–4
x
y
O 1–1–2–3–4 2 3 4
4
3
2
1
–1
–2
–3
–4
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
2-62-6
Study Guide and InterventionGraphing Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
2-72-7
© Glencoe/McGraw-Hill 93 Glencoe Algebra 2
Less
on
2-7
Graph Linear Inequalities. A linear inequality, like y � 2x � 1, resembles a linearequation, but with an inequality sign instead of an equals sign. The graph of the relatedlinear equation separates the coordinate plane into two half-planes. The line is theboundary of each half-plane.
To graph a linear inequality, follow these steps.
1. Graph the boundary, that is, the related linear equation. If the inequality symbol is or �, the boundary is solid. If the inequality symbol is � or , the boundary is dashed.
2. Choose a point not on the boundary and test it in the inequality. (0, 0) is a good point tochoose if the boundary does not pass through the origin.
3. If a true inequality results, shade the half-plane containing your test point. If a falseinequality results, shade the other half-plane.
Graph x � 2y � 4.
The boundary is the graph of x � 2y � 4.
Use the slope-intercept form, y � � x � 2, to graph the boundary line.
The boundary line should be solid.
Now test the point (0, 0).
0 � 2(0) �? 4 (x, y ) � (0, 0)
0 � 4 false
Shade the region that does not contain (0, 0).
Graph each inequality.
1. y � 3x � 1 2. y � x � 5 3. 4x � y �1
4. y � � 4 5. x � y 6 6. 0.5x � 0.25y � 1.5
x
y
O
x
y
O
x
y
O
x�2
x
y
O
x
y
O
x
y
O
1�2
x
y
O
ExercisesExercises
ExampleExample
© Glencoe/McGraw-Hill 94 Glencoe Algebra 2
Graph Absolute Value Inequalities Graphing absolute value inequalities is similarto graphing linear inequalities. The graph of the related absolute value equation is theboundary. This boundary is graphed as a solid line if the inequality is or �, and dashed ifthe inequality is � or . Choose a test point not on the boundary to determine which regionto shade.
Graph y 3x � 1.
First graph the equation y � 3 x � 1 .Since the inequality is , the graph of the boundary is solid.Test (0, 0).0 ? 3 0 � 1 (x, y) � (0, 0)
0 ? 3 �1 �1 � 1
0 3 true
Shade the region that contains (0, 0).
Graph each inequality.
1. y � x � 1 2. y 2x � 1 3. y � 2 x 3
4. y � � x � 3 5. x � y � 4 6. x � 1 � 2y � 0
7. 2 � x � y �1 8. y � 3 x � 3 9. y 1 � x � 4
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Study Guide and Intervention (continued)
Graphing Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
2-72-7
ExercisesExercises
ExampleExample
Skills PracticeGraphing Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
2-72-7
© Glencoe/McGraw-Hill 95 Glencoe Algebra 2
Less
on
2-7
Graph each inequality.
1. y 1 2. y x � 2 3. x � y 4
4. x � 3 � y 5. 2 � y � x 6. y � �x
7. x � y �2 8. 9x � 3y � 6 0 9. y � 1 � 2x
10. y � 7 �9 11. x �5 12. y x
x
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© Glencoe/McGraw-Hill 96 Glencoe Algebra 2
Graph each inequality.
1. y �3 2. x 2 3. x � y �4
4. y � �3x � 5 5. y � x � 3 6. y � 1 � �x
7. x � 3y 6 8. y x � 1 9. y �3 x � 1 � 2
COMPUTERS For Exercises 10–12, use the following information.
A school system is buying new computers. They will buy desktop computers costing $1000 per unit, andnotebook computers costing $1200 per unit. The total cost of the computers cannot exceed $80,000.
10. Write an inequality that describes this situation.1000d � 1200n 80,000
11. Graph the inequality.
12. If the school wants to buy 50 of the desktop computers and 25 of the notebook computers,will they have enough money? yes
Desktops
No
teb
oo
ks
100 30 5020 40 60 70 80 90 100
80
70
60
50
40
30
20
10
Computers Purchased
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Practice (Average)
Graphing Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
2-72-7
Reading to Learn MathematicsGraphing Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
2-72-7
© Glencoe/McGraw-Hill 97 Glencoe Algebra 2
Less
on
2-7
Pre-Activity How do inequalities apply to fantasy football?
Read the introduction to Lesson 2-7 at the top of page 96 in your textbook.
• Which of the combinations of yards and touchdowns listed would Danaconsider a good game? The first one: 168 yards and 3 touchdowns
• Suppose that in one of the games Dana plays, Moss gets 157 receivingyards. What is the smallest number of touchdowns he must get in orderfor Dana to consider this a good game? 3
Reading the Lesson
1. When graphing a linear inequality in two variables, how do you know whether to makethe boundary a solid line or a dashed line? If the symbol is � or , the line issolid. If the symbol is or �, the line is dashed.
2. How do you know which side of the boundary to shade? Sample answer: If the testpoint gives a true inequality, shade the region containing the test point. Ifthe test point gives a false inequality, shade the region not containingthe test point.
3. Match each inequality with its graph.
a. y 2x � 3 iii b. y � �2x � 3 iv c. y � 2x � 3 ii d. y � �2x � 3 i
i. ii. iii. iv.
Helping You Remember
4. Describe some ways in which graphing an inequality in one variable on a number line issimilar to graphing an inequality in two variables in a coordinate plane. How can whatyou know about graphing inequalities on a number line help you to graph inequalities ina coordinate plane? Sample answer: A boundary on a coordinate graph issimilar to an endpoint on a number line graph. A dashed line is similar toa circle on a number line: both are open and mean not included; theyrepresent the symbols and �. A solid line is similar to a dot on anumber line: both are closed and mean included; they represent thesymbols � and .
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© Glencoe/McGraw-Hill 98 Glencoe Algebra 2
Algebraic ProofThe following paragraph states a result you might be asked to prove in amathematics course. Parts of the paragraph are numbered.
01 Let n be a positive integer.
02 Also, let n1 � s(n1) be the sum of the squares of the digits in n.
03 Then n2 � s(n1) is the sum of the squares of the digits of n1, and n3 � s(n2)is the sum of the squares of the digits of n2.
04 In general, nk � s(nk � 1) is the sum of the squares of the digits of nk � 1.
05 Consider the sequence: n, n1, n2, n3, …, nk, ….
06 In this sequence either all the terms from some k on have the value 1,
07 or some term, say nj, has the value 4, so that the eight terms 4, 16, 37, 58, 89, 145, 42, and 20 keep repeating from that point on.
Use the paragraph to answer these questions.
1. Use the sentence in line 01. List the first five values of n.
2. Use 9246 for n and give an example to show the meaning of line 02.
3. In line 02, which symbol shows a function? Explain the function in a sentence.
4. For n � 9246, find n2 and n3 as described in sentence 03.
5. How do the first four sentences relate to sentence 05?
6. Use n � 31 and find the first four terms of the sequence.
7. Which sentence of the paragraph is illustrated by n � 31?
8. Use n � 61 and find the first ten terms.
9. Which sentence is illustrated by n � 61?
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
2-72-7
Write the letter for the correct answer in the blank at the right of each question. 1. Find the domain of the relation {(0, 0), (1, 1), (2, 0)}. Then determine
whether the relation is a function.A. {0, 1, 0}; function B. {0, 1, 0}; not a functionC. {0, 1, 2}; function D. {0, 1, 2}; not a function 1.
2. The table shows the annualized percent return of a mutual fund for several years. Find the range of the relation. Then determine whether the relation is a function.
A. {20.9, 22.8, 20.0, 20.5}; not a function B. {1, 3, 5, 10}; not a functionC. {20.9, 22.8, 20.0, 20.5}; function D. {1, 3, 5, 10}; function 2.
3. Find f(�1) if f(x) � �3x � 5.A. �9 B. �8 C. �2 D. 2 3.
4. Find f(0) if f(t) � t2 � 2t � 2.A. 2 B. �4 C. 0 D. �2 4.
5. Which equation is linear?A. xy � 60 B. 3x � 2y � 5C. y � x2 � 3x � 1 D. y2 � 1 � x 5.
6. Which function is a linear function?A. f(x) � x3 � x B. g(s) � 1 � 4s
C. h(t) � 2t � �1t�
D. f(r) � �r� 6.
7. Write y � 4x � 7 in standard form.A. 4x � y � �7 B. 4x � y � 7 C. y � 4x � 7 D. 4x � y � 7 7.
8. Find the x-intercept of the graph of �5x � 10y � 20.A. �2 B. 2 C. 4 D. �4 8.
9. Find the slope of the line that passes through (0, 2) and (8, 8).
A. 8 B. �43� C. �
34� D. �
54� 9.
10. If a line rises to the right, its slope is ___?____.A. zero B. positive C. negative D. undefined 10.
11. What is the slope of a line that is perpendicular to the graph of y � 2x � 5?
A. ��12� B. �
12� C. 2 D. �2 11.
12. Graph the line through (2, 3) that is parallel to the line with equation y � �1. Which point below also lies on that line?A. (2, 9) B. (�5, 3) C. (0, 1) D. (1, 4) 12.
Chapter 2 Test, Form 1
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 99 Glencoe Algebra 2
Ass
essm
ent
22
Year 1 3 5 10
Percent Return 20.9 22.8 20.0 20.5
© Glencoe/McGraw-Hill 100 Glencoe Algebra 2
Chapter 2 Test, Form 1 (continued)
13. Write an equation in slope-intercept form for the line that has a slope of
��45� and passes through (0, 7).
A. y � 7x B. y � 7x � �45� C. y � �
45�x � 7 D. y � ��
45�x � 7 13.
14. Write an equation for the line that passes through (0, 1) and is perpendicular to the line whose equation is y � 2x.
A. y � �2x � 1 B. y � 2x � 1 C. y � �12�x � 1 D. y � ��
12�x � 1 14.
15. Use a scatter plot to determine which data point is an outlier.A. (0, 2) B. (1, 3) C. (2, 10) D. (8, 18) 15.
16. The scatter plot shows the area of the floor and the price for certain tents.Which equation could be a prediction equation for this set of data?A. y � x � 50 B. y � 10x � 25C. y � 5x � 50 D. y � 5x � 22 16.
17. A banquet hall has tables that can seat 8 people. The number of tables needed depends on the number of guests.What type of special function models this situation?A. linear function B. step functionC. absolute value function D. constant function 17.
18. Identify the range of y � � x �.A. all real numbers B. {x � x � 0}C. {y � y � 0} D. {y � y � 0} 18.
19. The graph of the linear inequality y � 2x � 1 is the region __?___ the graph of the line y � 2x � 1.A. on or above B. on or below C. above D. below 19.
20. Which inequality is graphed at the right?A. y � � x � � 3 B. y � � x � � 3
C. y � � x � � 3 D. y � � x � � 3 20.
Bonus Find the value of k so that the slope of the line through (4, 2)
and (k, 3) is �16�. B:
NAME DATE PERIOD
22
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50100150200250300350400450500
12 18 24 30 36 42 48 54 60
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Floor Area (ft2)
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Chapter 2 Test, Form 2A
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 101 Glencoe Algebra 2
Ass
essm
ent
Write the letter for the correct answer in the blank at the right of each question.
1. Find the range of the relation {(�2, 3), (�1, 3), (�1, 5)}. Then determine whether the relation is a function.A. {�2, �1}; function B. {�2, �1}; not a functionC. {3, 5}; function D. {3, 5}; not a function 1.
2. Find f(�1) if f(x) � �xx2
��
24
�.
A. �5 B. �3 C. 1 D. 3 2.
3. Find f(a) if f(t) � t2 � 2t � 2.A. (t � a)2 � 2t � a � 2 B. (t � a)2 � 2(t � a) � 2C. a2 � 2t � 2 D. a2 � 2a � 2 3.
4. Which equation is linear?
A. y � x � 2 B. y � x2 C. y � 3 D. y2 � �12�x � 1 4.
5. Write 3y � �1 � 5x in standard form.A. 5x � 3y � �1 B. �5x � 3y � �1
C. y � ��53�x � 1 D. 3x � 5y � 1 � 0 5.
6. Find the x-intercept and the y-intercept of the graph of 3x � 2y � 12.A. (4, �6) B. 4; �6 C. (2, �3) D. �6; 4 6.
7. Find the slope of the line that passes through (2, 6) and (�7, 8).
A. ��52� B. ��
25� C. ��
29� D. ��
92� 7.
8. What is the slope of the line y � �2?
A. �2 B. 0 C. �12� D. undefined 8.
9. What is the slope of a line that is parallel to the graph of 2x � 3y � 5?
A. �32� B. ��
23� C. �
23� D. ��
32� 9.
10. The graph of the line through (2, 3) that is perpendicular to the line with equation y � �1 also goes through which point?A. (0, 1) B. (1, 4) C. (2, �4) D. (�2, 3) 10.
11. Write an equation in slope-intercept form for the line that has a slope of �4 and passes through (1, 2).A. y � �2x � 4 B. y � �4x � 6 C. y � �4x � 2 D. y � �4x � 9 11.
12. Write an equation in slope-intercept form for the line that passes through (1, �2) and (3, 7).
A. y � �92�x � �
123� B. y � �
92�x � �
527� C. y � �
29�x � �
193� D. y � �
29�x � �
139� 12.
22
© Glencoe/McGraw-Hill 102 Glencoe Algebra 2
Chapter 2 Test, Form 2A (continued)
13. Write an equation for the line that passes through (0, 5) and is parallel to the line whose equation is 4x � y � 3.
A. y � ��14�x � 5 B. y � 4x � 3 C. y � �
14�x � 5 D. y � 4x � 5 13.
14. The table shows the relationship between height and growing times for 8 plants of the same species. Use a scatter plot to determine which data point is an outlier.
A. (15, 6) B. (17, 14) C. (20, 18) D. (25, 24) 14.
15. Which equation could be a prediction equation for the data points shown in the scatter plot at the right?
A. y � �74�x � 400 B. y � �
151�x � 650
C. y � 5x � 600 D. y � �32�x � 800 15.
16. Evaluate f��34�� if f(x) � �1 � 2x�.
A. 0 B. �2 C. �1 D. 1 16.
17. Identify the range of y � � x � � 4.A. {x � x � 4} B. {y � y � �4}C. {y � y � 0} D. all real numbers 17.
18. Which is not part of the definition of the piecewise function shown?A. �3 if x � �2B. x � 2 if � 2 � x � 1C. x � 3 if x � �2D. �x � 1 if x � 1 18.
19. The graph of the linear inequality y � ��23�x � 2 is the region ___?___ the
graph of y � ��23�x � 2.
A. above B. below C. on or above D. on or below 19.
20. Which point satisfies the inequality y � � �x � 3 �?A. (3, 6) B. (�2, 4) C. (5, 7) D. (1, 4) 20.
Bonus Find the value of k so that the slope of the line through B:(2, �k) and (�1, 4) is �3.
NAME DATE PERIOD
22
y
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200400600800
100012001400160018002000
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)
Hard Drive Size (MB)
Height (inches) 15 17 18 19 20 22 23 25
Growing Time (weeks) 6 14 16 17 18 21 23 24
Chapter 2 Test, Form 2B
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 103 Glencoe Algebra 2
Ass
essm
ent
Write the letter for the correct answer in the blank at the right of each question.
1. Find the range of the relation {(�1, 4), (2, 5), (3, 5)}. Then determine whether the relation is a function.A. {�1, 2, 3}; function B. {�1, 2, 3}; not a functionC. {4, 5}; function D. {4, 5}; not a function 1.
2. Find f(�1) if f(x) � �xx2 �
�62x
�.
A. �5 B. ��53� C. �
73� D. 7 2.
3. Find f(a) if f(t) � 2t2 � t � 2.A. 2(t � a)2 � 2t � a � 2 B. 2(t � a)2 � 2(t � a) � 2C. 2a2 � a � 2 D. 4a2 � 2a � 2 3.
4. Which equation is linear?
A. x � �2 B. y � 3x2 � 1 C. y � 5x � 2 D. y2 � �12�x � 3 4.
5. Write �3y � �1 � 5x in standard form.
A. �5x � 3y � 1 B. 5x � 3y � 1 C. y � ��53�x � 1 D. 3x � 5y � 1 � 0 5.
6. Find the x-intercept and the y-intercept of the graph of 4x � 2y � 8.A. (2, �4) B. �4; 2 C. (4, �2) D. 2; �4 6.
7. Find the slope of a line that passes through (2, 4) and (�7, 8).
A. ��49� B. ��
45� C. �
54� D. ��
94� 7.
8. What is the slope of the line x � �2?
A. �2 B. 0 C. �12� D. undefined 8.
9. What is the slope of a line that is parallel to the graph of 2x � 3y � 6?
A. �32� B. ��
23� C. �
23� D. ��
32� 9.
10. The graph of the line through (2, 3) that is perpendicular to the line with equation x � �1 also goes through which point?A. (0, �1) B. (�2, 3) C. (2, �4) D. (1, 4) 10.
11. Write an equation in slope-intercept form for the line that has a slope of 3 and passes through (�1, 2).A. y � 3x � 1 B. y � 3x � 5 C. y � 5x � 3 D. y � 3x � 5 11.
12. Write an equation in slope-intercept form for the line that passes through (�1, �2) and (3, �7).
A. y � �54�x � �
34� B. y � ��
45�x � �
65� C. y � �
45�x � �
65� D. y � ��
54�x � �
143� 12.
22
© Glencoe/McGraw-Hill 104 Glencoe Algebra 2
Chapter 2 Test, Form 2B (continued)
13. Write an equation for the line that passes through (0, �2) and is parallel to the line whose equation is 3x � 5y � 3.
A. y � ��35�x � 2 B. y � 3x � 2 C. y � �
35�x � 2 D. y � �3x � 2 13.
14. The table shows the relationship between the number of hours practiced and the number of free throws made by 6 players. Use a scatter plot to determine which data point is an outlier.
A. (1, 0) B. (3, 4) C. (7, 16) D. (12, 18) 14.
15. Which equation could be a prediction equation for the data points shown in the scatter plot at the right?A. y � 10x � 6
B. y � ��110�
x � 6
C. y � x � 6
D. y � �110�
x � 6 15.
16. Evaluate f���34�� if f(x) � �2x � 1�.
A. 1 B. �3 C. �1 D. �2 16.
17. Identify the domain of y � 3� x � 2 �.A. all real numbers B. {x � x � 2}C. {y � y � 0} D. {y � y � 2} 17.
18. Which is not part of the definition of the piecewise function shown?A. 2 if x � �1B. x � 1 if �1 � x � 1C. �x � 1 if �1 � x � 1D. 2x if x � 1 18.
19. The graph of the linear inequality y � 3x � 1 is the region ___?___ the graph of y � 3x � 1.A. above B. below C. on or above D. on or below 19.
20. Which point satisfies the inequality y � �� x � 2 �?A. (�1, �1) B. (1, 0) C. (�4, �3) D. (3, 2) 20.
Bonus Find the value of k so that the slope of the line through B:(2, �k) and (�1, 4) is 4.
NAME DATE PERIOD
22
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1224364860728496
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Source: The World Almanac
400 600 800 1000
Men
's W
orl
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rd (
seco
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Distance Run (meters)
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Hours Practiced 1 3 4 6 7 12
Free Throws Made 0 4 6 9 16 18
Chapter 2 Test, Form 2C
© Glencoe/McGraw-Hill 105 Glencoe Algebra 2
1. Graph the relation {(�3, 3), (�3, 2), (�3, 1), (�3, 0)} and 1.find the domain and range. Then determine whether the relation is a function.
Determine whether each relation is a function.
2. 3.
Find each value if f(x) � 10x � 3x2 and g(x) � 5x2 � 8x.
4. f(�3) 5. g(a)
For Questions 6 and 7, state whether each equation or function is linear. If no, explain your reasoning.
6. f(x) � �x �1
3� 7. y � 3x � 10
8. Write the equation �52�x � 9 � 8y in standard form. Identify
A, B, and C.
9. Find the x-intercept and the y-intercept of the graph of 3y � 2x � 6.
For Questions 10–13, graph each equation or inequality.
10. y � 3x � 2
11. f(x) � �
12. x � 2 � �12�y
1 � x if x � 23 if x � 2
y
xO
y
xO
NAME DATE PERIOD
SCORE 22
Ass
essm
ent
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12. y
xO
xO
f (x )
y
xO
y
xO
© Glencoe/McGraw-Hill 106 Glencoe Algebra 2
Chapter 2 Test, Form 2C (continued)
13. y � � �2x � 2 �
14. Find the slope of the line that passes through (�7, 9) and (�6, �5).
15. Graph the line passing through (2, 4) that is perpendicular to the graph of y � �3.
16. Write an equation in slope-intercept form for the line that has a slope of 2 and passes through (1, �5).
17. Write an equation for the line that passes through (�2, 3) and is parallel to the line whose equation is 3x � 2y � 6.
For Questions 18 and 19, use the set of data in the table.
The table below shows the relationship between the number of field goals attempted and the number of points scored by one basketball player over a 6-game period.
18. Draw a scatter plot for the data.
19. Use two ordered pairs to write a prediction equation. Then use your prediction equation to predict the number of points scored when 20 field goals are attempted.
20. Determine whether the graph represents a step function,a constant function, the identity function, an absolute value function, or a piecewise function.Then identify the domain and range.
Bonus Find the value of k so that the slope of the line through B:(2, �k) and (�1, 4) is 1.
NAME DATE PERIOD
22
13.
14.
15.
16.
17.
18.
19.
20.
p
a80
91011121314151617
5 6 7 8 9 10 11 12 13 14
Poin
ts S
core
d
Field Goals Attempted
y
xO
y
xO
xO
y
Field Goals Attempted (a) 8 6 10 9 7 10
Points Scored (p) 12 9 14 14 11 15
Chapter 2 Test, Form 2D
© Glencoe/McGraw-Hill 107 Glencoe Algebra 2
1. Graph the relation {(0, 0), (2, 4), (�4, 0), (4, 0)} and find the domain and range. Then determine whether the relation is a function.
Determine whether each relation is a function.
2. 3.
Find each value if f(x) � �3x � 2x2 and g(x) � �4x2 � 2x � 3.
4. f(�2) 5. g(a)
For Questions 6 and 7, state whether each equation or function is linear. If no, explain your reasoning.
6. f(x) � 100x � 37 7. xy � 60 � 0
8. Write �2x7� 1� � 8y in standard form. Identify A, B, and C.
9. Find the x-intercept and the y-intercept of the graph of 4y � 12 � 3x.
For Questions 10–13, graph each equation or inequality.
10. 3y � 2x � 9
11. f(x) � �
12. x � 2y � 4
�2 if x � �2x � 3 if x � �2
y
xO
y
xO
NAME DATE PERIOD
SCORE 22
Ass
essm
ent
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12. y
xO
xO
f (x )
y
xO
y
xO
© Glencoe/McGraw-Hill 108 Glencoe Algebra 2
Chapter 2 Test, Form 2D (continued)
13. y � � x � 1 �
14. Find the slope of the line that passes through (2, 18) and (4, �2).
15. Graph the line passing through (3, �2) that is perpendicular to the graph of x � �3.
16. Write an equation in slope-intercept form for the line that has a slope of �1 that passes through (�4, 3).
17. Write an equation for the line that passes through (2, �5) and is parallel to the line whose equation is 5x � 2y � 6.
For Questions 18 and 19, use the set of data in the table.
The table below shows the relationship between the number of phone calls made and the number of tickets sold during a fundraising campaign by 6 callers.
18. Draw a scatter plot for the data.
19. Use two ordered pairs to write a prediction equation.Then use your prediction equation to predict the number of tickets sold when 16 calls are made.
20. Determine whether the graph represents a step function,a constant function, the identity function, an absolute value function, or a piecewise function.Then identify the domain and range.
Bonus Find the value of k so that the slope of the line through B:(2, �k) and (4, �1) is �2.
NAME DATE PERIOD
22
xO
f(x )
13.
14.
15.
16.
17.
18.
19.
20.
t
n0
12141618202224262830
6 7 8 9 10 11 12 13 14 15
Tick
ets
Sold
Calls Made
y
xO
y
xO
Calls Made (n) 8 9 7 8 6 12
Tickets Sold (t) 16 17 15 15 12 25
Chapter 2 Test, Form 3
© Glencoe/McGraw-Hill 109 Glencoe Algebra 2
1. Graph the relation x � 1 � y2 and find the domain and range. Then determine whether the relation is a function.
2. Determine whether the 2.relation shown at the right is a function.
3. Find f(�2) if f(x) � �2� 1x3
�
�
x42x�
�. 3.
4. If f(2b � 1) � 6b � 2, find f(x). 4.
5. State whether each equation or function is linear. 5.
A. f(x) � �3x �
514
� B. 3x � xy � y
6. Write �2.5x3� 0.3� � �
16� y in standard form. Identify A, B, and C. 6.
7. Find the x-intercept and the y-intercept of the graph of 7.2( y � 0.5) � 3.5x � 2y.
8. Determine whether the graph at the 8.right represents a step function, a constant function, an absolute value function, or a piecewise function.
For Questions 9 and 10, graph each equation.
9. 2y � 1 � 0.8x
10. y � ��12�x � 1�
11. Determine the value of t so that the line through (1.6, t)
and (2, 5) has slope ��32�.
12. The median weekly earnings for American workers in 1990 was $412 and in 1999 it was $549. Calculate the average rate of change between 1990 and 1999.
NAME DATE PERIOD
SCORE 22
Ass
essm
ent
y
xO
1.
9.
10.
11.
12.
y
xO
y
xO
y
xO
xO
f(x )
© Glencoe/McGraw-Hill 110 Glencoe Algebra 2
Chapter 2 Test, Form 3 (continued)
13. Write an equation for the line that passes through (5, �4) and is perpendicular to the graph of 5x � 2y � �6.
14. Write an equation in slope-intercept form of the line
through ���14�, �
53�� and ��
12�, ��
43��.
15. Sweets Bakery charges $12 for each pie and $15 for each cake. Yesterday, the bakery took in no more than $360 for sales of pies and cakes. Write an inequality to represent the situation, where p is the number of pies sold and c is the number of cakes sold. Then graph the inequality.
16. Write an equation in standard form for the line that is
perpendicular to the graph of �15�x � �
25�y � 0.05 and has the
same y-intercept as the graph of �0.8x � 1.2y � �0.6.
For Questions 17 and 18, use the set of data in the table.The table below shows the relationship between distance traveled and elapsed time.
17. Draw a scatter plot for the data. Then identify any outliers.
18. Use two ordered pairs to write a prediction equation.Then use your prediction equation to predict the time for a distance of 160 kilometers. Compare your prediction to the one given in the table.
19. Write an absolute value inequality for the graph at the right.
20. Write the function shown in the graph at the right.
Bonus Find the value of k so that the graph of kx � 3y � 4 is parallel to the line through (2, �k) and (4, �1).
NAME DATE PERIOD
22
y
xO
xO
f (x )
13.
14.
15.
16.
17.
18.
19.
t
d0
20406080
100120140160
60 100 140 180 220Ti
me
(min
)Distance (km)
c
p
Distance d (km) 40 75 110 150 160 200
Time t (min) 30 60 80 110 150 150
20.
B:
Chapter 2 Open-Ended Assessment
© Glencoe/McGraw-Hill 111 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. Explain two ways to determine whether a relation is a function.Use specific examples. Then write a relation that is not afunction.
2. Give an example of a real-world situation for which there wouldbe a negative rate of change.
3. The point-slope form of the equation of a line is y � 2 � �12�(x � 6).
Write the equation in slope-intercept form, then write theequation in standard form. Which of the three forms of theequation is most useful? Explain your choice.
4. Suppose you are looking at a scatter plot and the graph of a lineof fit for the data points. The horizontal axis is labeled 1990,1991, …, 2000. The vertical axis is labeled 0, 10, …, 100. You usea prediction equation to predict values for the years 1994 and2005. Which prediction do you think would be more accurate?Why ?
5. Compare the graph of the parent function f(x) � � x � with thegraphs of the functions g(x) � � x � 2 � and h(x) � � x � 3 �. How arethe graphs similar? How are they different? How would thegraph of y � � x � 500 � compare with the graph of the parentfunction?
6. When graphing the linear inequality y � �2 � 5, Alessia firstgraphed the line y � �2x � 5. She then selected the test point(�1, 7) in order to complete her graph. Why did Alessia need atest point? What information did the point (�1, 7) give Alessiaabout her graph?
7. Is the graph of the relation y � � x � 3 � a function? Explain.
NAME DATE PERIOD
SCORE 22
Ass
essm
ent
© Glencoe/McGraw-Hill 112 Glencoe Algebra 2
Chapter 2 Vocabulary Test/Review
Write the letter of the term that best describes each example.
1. f(x) � 6
2. 3x � 5y � 2
3. f(x) � 4x � 3
4. y � �5x � 10
5. (�12, 8)
6. f(x) � �x� � 1
7. y � 5 � �2(x � 3)
8. f(x) � � x � 3 if x � 02 � x if x � 0
9. �38�
�(�
51)� � �
34�
10. {3, 4, 5} for the function {(0, 4), (2, 5), (3, 3)}
In your own words—Define each term.
11. vertical line test
12. linear function
absolute value functionboundaryCartesian coordinate planeconstant functiondependent variabledomainfamily of graphsfunctionfunctional notationgreatest integer functionidentity functionindependent variable
linear equationlinear functionline of fitmappingone-to-one functionordered pairparent graphpiecewise functionpoint-slope formprediction equationquadrantrange
rate of changerelationscatter plotslopeslope-intercept formstandard formstep functionvertical line testx-intercepty-intercept
NAME DATE PERIOD
SCORE 22
a. ordered pair
b. point-slope form
c. step function
d. range
e. constant function
f. piecewise function
g. slope-intercept form
h. absolute value function
i. standard form
j. slope
Chapter 2 Quiz (Lessons 2–1 and 2–2)
22
© Glencoe/McGraw-Hill 113 Glencoe Algebra 2
1. Find the domain and range of x � y � 1. Then determine 1.whether it is a function.
Determine whether the relation shown in the graph is a function.
2. 3. Find f(4) if f(x) � �xx2
��
19
�.
4. Write x � 2 � �15�y in standard form. Identify A, B, and C.
5. Find the x-intercept and the y-intercept of the graph of 3x � 4y � 12. Then graph the equation.
y
xO
NAME DATE PERIOD
SCORE
Chapter 2 Quiz (Lessons 2–3 and 2–4)
For Questions 1 and 2, find the slope of the line that passes through each pair of points. 1.
1. (2, 4), (4, 7) 2. ��12�, �4�, ��
12�, 5� 2.
3. Graph the line passing through (2, 4) that is parallel to the 3.graph of x � 3y � 6.
4. Standardized Test Practice Which is an equation of the
line that has a slope of ��23� and passes through (�1, 3)?
A. y � �32�x � �
73� B. y � ��
23�x � �
73�
C. y � ��23�x � 1 D. y � ��
23�x � �
131� 4.
5. Write an equation for the line that passes through (3, 5) 5.and is perpendicular to the line whose equation is
y � �12�(x � 2).
y
xO
NAME DATE PERIOD
SCORE 22
Ass
essm
ent
2.
3.
4.
5.
y
xO
© Glencoe/McGraw-Hill 114 Glencoe Algebra 2
For Questions 1 and 2, use the set of data in the table. 1.
1. Draw a scatter plot for the data. Then state which of the data points is an outlier.
2. Use two ordered pairs to write a prediction equation.Then use your prediction equation to predict the missing value.
3. f(x) � � x � 2 �. Identify the domain and range.
Chapter 2 Quiz (Lesson 2–7)
1. Write an inequality for the graph shown.
Graph each inequality.
2. y � �2
3. 6 � 2y � 3x
4. y � � � 2x �
y
xO
(0, 1)
(1, 4)
NAME DATE PERIOD
SCORE
Chapter 2 Quiz (Lessons 2–5 and 2–6)
22
NAME DATE PERIOD
SCORE
22
1.
2.
3. y
xO
y
x
20
1520253035404550
4 6 8 10 12 14 16 18 20
1.2.
3.
4. y
xO
y
xO
y
xO
x 2 5 10 15 20 30
y 1 25 21 32 41 ?
Chapter 2 Mid-Chapter Test (Lessons 2–1 through 2–4)
© Glencoe/McGraw-Hill 115 Glencoe Algebra 2
Write the letter for the correct answer in the blank at the right of each question.
1. Which of the following relations is not a function?A. B. C. D. 1.
2. Which equation or function is linear?
A. y � �x �3
1� B. f(x) � �23�(1 � x)2 C. 2y � �
2x4� 1� D. 3xy � 4 2.
3. Write an equation in standard form for the line that is parallel to the graph of �8x � 5 � 4y and has y-intercept �0.5.
A. x � 0.5y � 0.25 B. 10x � 5y � 2.5 C. 4x � 2y � 1 D. 2x � y � 1 3.
4. Find the slope of the line that passes through ��4.5, �72�� and (3, 3.5).
A. ��16� B. �6 C. undefined D. 0 4.
5. The graphs of which pair of lines are perpendicular?
A. 2x � 3y � 12, y � ��23�x � 5 B. 3x � 2y � 6, 2x � 3y � 7
C. y � 4x � 13, y � �14�x � 13 D. x � y � 1, 2y � �2x � 2 5.
Graph each relation and find the domain and range.Then determine whether the relation is a function.
6. {(2, 4), (4, �2), (1, 3), (0, 3)}
7. y � 2x � 3
For Questions 8 and 9, find each value if f(x) � �3x3 � 2x2.
8. f(�1) 9. f��12��
10. Write an equation in slope-intercept form for the line that
has a slope of ��13� and passes through (�6, 1).
Part II
y
xO
y
xO
y
xO
y
xO
Part I
NAME DATE PERIOD
SCORE 22
Ass
essm
ent
6.
7.
8.
9.
10.
y
xO
y
xO
© Glencoe/McGraw-Hill 116 Glencoe Algebra 2
Chapter 2 Cumulative Review (Chapters 1 and 2)
1. Evaluate �7aa2
��
2bc
� if a � 3, b � 2, and c � 5. (Lesson 1-1) 1.
2. Name the sets of numbers to which �42.1 belongs. (Lesson 1-2) 2.
3. Solve 3�7 � a� � 12. Check each solution. (Lesson 1-4) 3.
For Questions 4 and 5, solve each inequality. Graph the solution set.
4. 2(3x � 1) � 5x � 3 (Lesson 1-5) 4.
5. �6 � 2(y � 1) � 10 (Lesson 1-6) 5.
6. Find the domain and range of the relation. Then determine 6.whether the relation is a function.{(4, �7), (3, �7), (2, 0), (4, 0)} (Lesson 2-1)
7. Find f(�7) if f(x) � 2x2 � 3x. (Lesson 2-1) 7.
8. Find the x-intercept and the y-intercept of the graph of 8.3x � 4y � 8. (Lesson 2-2)
9. Find the slope of the line whose graph is perpendicular to 9.the graph of 2x � 5y � 7. (Lesson 2-3)
10. Write an equation in slope-intercept form for the line that 10.has a slope of �4 and passes through (3, �5). (Lesson 2-4)
11. The prediction equation y � 5.92x � 119.21 models the 11.median selling price, in thousands of dollars, of new homes in a certain area since 1995. Predict the median selling price in 2015. (Lesson 2-5)
12. Identify the domain and range of the piecewise function 12.
h(x) � � . (Lesson 2-6)
13. Graph y � ��45�x � 1. (Lesson 2-7) 13.
y
xO
x � 5 if x � �2�4x if x � �2
�1�2 0 1 2 4 5 63
�1�2�3�4 0 1 2 43
NAME DATE PERIOD
22
Standardized Test Practice (Chapters 1 and 2)
© Glencoe/McGraw-Hill 117 Glencoe Algebra 2
1. If the perimeter of a rectangle is 96 inches and the length is 4 inches longer than the width, what is the area?A. 22 in2 B. 26 in2 C. 230 in2 D. 572 in2 1.
2. For what values of a will 3a � 1 be equal to 3a � 10?E. all negative values F. 0G. all positive values H. no values 2.
3. In the figure at the right, if RSTV is a square with perimeter 24, what is the area of the circle with center R?A. 6 B. 36
C. 12 D. 144 3.
4. In a group of 20 students, 12 belonged to the band,7 belonged to the choir, and 5 belonged to both the band and the choir. How many students did not belong to either the band or the choir?E. 1 F. 2 G. 6 H. 14 4.
5. A point on the graph of 2x � 2y � 12 is __?___.A. (�3, �3) B. (�3, 3) C. (3, 3) D. (3, �3) 5.
6. If x � y � 6 and 3x � 10 � 2y, what is the value of y?E. �8 F. �4 G. 4 H. 8 6.
7. Which is equal to x3 � 8?A. (x � 2)(x2 � 4x � 4) B. (x � 2)(x2 � 2x � 4)C. (x � 2)(x2 � 4x � 4) D. (x � 2)(x2 � 2x � 4) 7.
8. In the sequence 1, 3, 12, 60, 360, ___, ___, ___, the eighth term is __?___.E. 2520 F. 2880 G. 20,160 H. 181,440 8.
9. If m�ABD � 65, m�EBC � 70, and m�ABC � 115, find m�EBD.A. 5° B. 20°C. 45° D. 50° 9.
10. 8 less than a is 6 more than c. Thus, cexpressed in terms of a is __?___.E. �
a �6
8� F. a � 2 G. a � 14 H. 2 � a 10. HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
Part 1: Multiple Choice
Instructions: Fill in the appropriate oval for the best answer.
NAME DATE PERIOD
22
Ass
essm
ent
V T
R S
B
C
AE D
© Glencoe/McGraw-Hill 118 Glencoe Algebra 2
Standardized Test Practice (continued)
11. A shoe salesperson sold 20 pairs of shoes for 11. 12.$640. A brown pair of shoes sells for $30 and a black pair for $35. How many brown pairs were sold?
12. How many integers between 299 and 501 are divisible by 2 or 5?
13. The histogram 13. 14.shows the distribution of mid-term exam scores for Ms. Hawkins’ three algebra classes.What percent of her students scored at least 70?
14. If 3x�2 � 81, what is the value of 22x�7?
Column A Column B
15. 15.
16. 8 � 7 � y
16.
17. 17.
18.
, a � b � c � 4a
18. DCBAbc
a˚ c˚b˚
DCBA(c)(c)(c)(c)�4(c � c)
DCBA4y � 102y � 2
DCBA8(4 � 3) � 63(6)(0) � 14
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
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.
99 9 987654321
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.
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NAME DATE PERIOD
22
NAME DATE PERIOD
10
15
5
0
25
20
Nu
mb
er o
f St
ud
ents
Exam Scores50 60 70 80 90 100
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
D
C
B
Standardized Test PracticeStudent Record Sheet (Use with pages 106–107 of the Student Edition.)
© Glencoe/McGraw-Hill A1 Glencoe Algebra 2
NAME DATE PERIOD
22
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 11–17, also enter your answer by writing each number or symbol ina box. Then fill in the corresponding oval for that number or symbol.
10 12 14 16
11 13 15 17
Select the best answer from the choices given and fill in the corresponding oval.
18 20 22
19 21 DCBADCBA
DCBADCBADCBA
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.
99 9 987654321
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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 2-1)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Rel
atio
ns
and
Fu
nct
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
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____
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____
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lenc
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Gle
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Alg
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2
Lesson 2-1
Gra
ph
Rel
atio
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A r
elat
ion
can
be
repr
esen
ted
as a
set
of
orde
red
pair
s or
as
aneq
uat
ion
;th
e re
lati
on i
s th
en t
he
set
of a
ll o
rder
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airs
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hat
mak
e th
e eq
uat
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e.T
he
dom
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of a
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n i
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fir
st c
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f th
e or
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and
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nct
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n i
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n i
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ent
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ou c
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Mak
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tabl
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nct
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Gle
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Alg
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2
Equ
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Fun
ctio
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and
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ctio
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his
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asiz
es t
he
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th
at t
he
valu
es o
f y,
the
dep
end
ent
vari
able
,dep
end
on t
he
valu
es o
f x,
the
ind
epen
den
t va
riab
le.
To
eval
uat
e a
fun
ctio
n,o
r fi
nd
a fu
nct
ion
al v
alu
e,m
ean
s to
su
bsti
tute
a g
iven
val
ue
in t
he
dom
ain
in
to t
he
equ
atio
n t
o fi
nd
the
corr
espo
ndi
ng
elem
ent
in t
he
ran
ge.
Giv
en t
he
fun
ctio
n f
(x)
�x2
�2x
,fin
d e
ach
val
ue.
a.f(
3)
f(x)
�x2
�2x
Orig
inal
fun
ctio
n
f(3)
�32
�2(
3)S
ubst
itute
.
�15
Sim
plify
.
b.
f(5a
)
f(x)
�x2
�2x
Orig
inal
fun
ctio
n
f(5a
) �
(5a)
2�
2(5a
)S
ubst
itute
.
�25
a2�
10a
Sim
plify
.
Fin
d e
ach
val
ue
if f
(x)
��
2x�
4.
1.f(
12)
�20
2.f(
6)�
83.
f(2b
)�
4b�
4
Fin
d e
ach
val
ue
if g
(x)
�x3
�x.
4.g(
5)12
05.
g(�
2)�
66.
g(7c
)34
3c3
�7c
Fin
d e
ach
val
ue
if f
(x)
�2x
�an
d g
(x)
�0.
4x2
�1.
2.
7.f(
0.5)
58.
f(�
8)�
169.
g(3)
2.4
10.g
(�2.
5)1.
311
.f(4
a)8a
�12
.g�
��
1.2
13.f
��6
14.g
(10)
38.8
15.f
(200
)40
0.01
Let
f(x
) �
2x2
�1.
16.F
ind
the
valu
es o
f f(
2) a
nd
f(5)
.f(
2) �
7,f(
5) �
49
17.C
ompa
re t
he
valu
es o
f f(
2) �
f(5)
an
d f(
2 �
5).
f(2)
�f(
5) �
343,
f(2
�5)
�19
9
2 � 31 � 3
b2
� 10b � 2
1 � 2a1 � 4
2 � x
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Rel
atio
ns
and
Fu
nct
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-1
2-1
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A3 Glencoe Algebra 2
An
swer
s
Answers (Lesson 2-1)
Skil
ls P
ract
ice
Rel
atio
ns
and
Fu
nct
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-1
2-1
©G
lenc
oe/M
cGra
w-H
ill59
Gle
ncoe
Alg
ebra
2
Lesson 2-1
Det
erm
ine
wh
eth
er e
ach
rel
atio
n i
s a
fun
ctio
n.W
rite
yes
or n
o.
1.ye
s2.
no
3.ye
s4.
no
Gra
ph
eac
h r
elat
ion
or
equ
atio
n a
nd
fin
d t
he
dom
ain
an
d r
ange
.Th
en d
eter
min
ew
het
her
th
e re
lati
on o
r eq
uat
ion
is
a fu
nct
ion
.
5.{(
2,�
3),(
2,4)
,(2,
�1)
}6.
{(2,
6),(
6,2)
}
D �
{2},
R �
{�3,
�1,
4};
no
D �
{2,6
},R
�{2
,6};
yes
7.{(
�3,
4),(
�2,
4),(
�1,
�1)
,(3,
�1)
}8.
x�
�2
D �
{�3,
�2,
�1,
3},
D �
{�2}
,R �
all r
eals
;n
o
R �
{�1,
4};
yes
Fin
d e
ach
val
ue
if f
(x)
�2x
�1
and
g(x
) �
2 �
x2.
9.f(
0)�
110
.f(1
2)23
11.g
(4)
�14
12.f
(�2)
�5
13.g
(�1)
114
.f(d
)2d
�1
x
y
O
( –2,
4)
( –3,
4)
( –1,
–1)
( 3, –
1)x
y
O
( 2, 6
) ( 6, 2
)
x
y
O
( 2, 4
)
( 2, –
1)
( 2, –
3)
x
y
O
x
y
O
xy
12
24
36
D 3
R 1 5
D 100
200
300
R 50
100
150
©G
lenc
oe/M
cGra
w-H
ill60
Gle
ncoe
Alg
ebra
2
Det
erm
ine
wh
eth
er e
ach
rel
atio
n i
s a
fun
ctio
n.W
rite
yes
or n
o.
1.n
o2.
yes
3.ye
s4.
no
Gra
ph
eac
h r
elat
ion
or
equ
atio
n a
nd
fin
d t
he
dom
ain
an
d r
ange
.Th
en d
eter
min
ew
het
her
th
e re
lati
on o
r eq
uat
ion
is
a fu
nct
ion
.
5.{(
�4,
�1)
,(4,
0),(
0,3)
,(2,
0)}
6.y
�2x
�1
D �
{�4,
0,2,
4},
D �
all r
eals
,R �
all r
eals
;ye
sR
�{�
1,0,
3};
yes
Fin
d e
ach
val
ue
if f
(x)
�an
d g
(x)
��
2x�
3.
7.f(
3)1
8.f(
�4)
�9.
g ��2
10.f
(�2)
un
def
ined
11.g
(�6)
1512
.f(m
�2)
13.M
USI
CT
he
orde
red
pair
s (1
,16)
,(2,
16),
(3,3
2),(
4,32
),an
d (5
,48)
rep
rese
nt
the
cost
of
buyi
ng
vari
ous
nu
mbe
rs o
f C
Ds
thro
ugh
a m
usi
c cl
ub.
Iden
tify
th
e do
mai
n a
nd
ran
ge o
fth
e re
lati
on.I
s th
e re
lati
on a
fu
nct
ion
?D
�{1
,2,3
,4,5
},R
�{1
6,32
,48}
;ye
s
14.C
OM
PUTI
NG
If a
com
pute
r ca
n d
o on
e ca
lcu
lati
on i
n 0
.000
0000
015
seco
nd,
then
th
efu
nct
ion
T(n
) �
0.00
0000
0015
ngi
ves
the
tim
e re
quir
ed f
or t
he
com
pute
r to
do
nca
lcu
lati
ons.
How
lon
g w
ould
it
take
th
e co
mpu
ter
to d
o 5
bill
ion
cal
cula
tion
s?7.
5 s
5 � m
1 � 25 � 2
5� x
�2
x
y
O
( –4,
–1)
( 2, 0
)
( 0, 3
)
( 4, 0
) x
y
O
x
y
O
xy
�3
0
�1
�1
00
2�
2
34
D 5 10 15
R 105
110
D 2 8
R 21 25 30
Pra
ctic
e (
Ave
rag
e)
Rel
atio
ns
and
Fu
nct
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
2-1
2-1
© Glencoe/McGraw-Hill A4 Glencoe Algebra 2
Answers (Lesson 2-1)
Readin
g t
o L
earn
Math
em
ati
csR
elat
ion
s an
d F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-1
2-1
©G
lenc
oe/M
cGra
w-H
ill61
Gle
ncoe
Alg
ebra
2
Lesson 2-1
Pre-
Act
ivit
yH
ow d
o re
lati
ons
and
fu
nct
ion
s ap
ply
to
bio
logy
?
Rea
d th
e in
trod
uct
ion
to
Les
son
2-1
at
the
top
of p
age
56 i
n y
our
text
book
.
•R
efer
to
the
tabl
e.W
hat
doe
s th
e or
dere
d pa
ir (
8,20
) te
ll y
ou?
Fo
r a
dee
r,th
e av
erag
e lo
ng
evit
y is
8 y
ears
an
d t
he
max
imu
mlo
ng
evit
y is
20
year
s.•
Su
ppos
e th
at t
his
tab
le i
s ex
ten
ded
to i
ncl
ude
mor
e an
imal
s.Is
it
poss
ible
to h
ave
an o
rder
ed p
air
for
the
data
in
wh
ich
th
e fi
rst
nu
mbe
r is
lar
ger
than
th
e se
con
d?S
amp
le a
nsw
er:
No
,th
e m
axim
um
lon
gev
ity
mu
st a
lway
s b
e g
reat
er t
han
th
e av
erag
e lo
ng
evit
y.
Rea
din
g t
he
Less
on
1.a.
Exp
lain
th
e di
ffer
ence
bet
wee
n a
rel
atio
n a
nd
a fu
nct
ion
.S
amp
le a
nsw
er:
Are
lati
on
is a
ny s
et o
f o
rder
ed p
airs
.A f
un
ctio
n is
a s
pec
ial k
ind
of
rela
tio
n in
wh
ich
eac
h e
lem
ent
of
the
do
mai
n is
pai
red
wit
h e
xact
lyo
ne
elem
ent
in t
he
ran
ge.
b.
Exp
lain
th
e di
ffer
ence
bet
wee
n d
omai
n a
nd
ran
ge.
Sam
ple
an
swer
:Th
e d
om
ain
of
a re
lati
on
is t
he
set
of
all f
irst
co
ord
inat
es o
f th
e o
rder
ed p
airs
.Th
era
ng
e is
th
e se
t o
f al
l sec
on
d c
oo
rdin
ates
.
2.a.
Wri
te t
he
dom
ain
an
d ra
nge
of
the
rela
tion
sh
own
in
th
e gr
aph
.
D:
{�3,
�2,
�1,
0,3}
;R
:{�
5,�
4,0,
1,2,
4}
b.
Is t
his
rel
atio
n a
fu
nct
ion
? E
xpla
in.
Sam
ple
an
swer
:N
o,i
t is
no
t a
fun
ctio
nb
ecau
se o
ne
of
the
elem
ents
of
the
do
mai
n,3
,is
pai
red
wit
h t
wo
elem
ents
of
the
ran
ge.
Hel
pin
g Y
ou
Rem
emb
er
3.L
ook
up
the
wor
ds d
epen
den
tan
d in
dep
end
ent
in a
dic
tion
ary.
How
can
th
e m
ean
ing
ofth
ese
wor
ds h
elp
you
dis
tin
guis
h b
etw
een
in
depe
nde
nt
and
depe
nde
nt
vari
able
s in
afu
nct
ion
?S
amp
le a
nsw
er:T
he
vari
able
wh
ose
val
ues
dep
end
on
,or
are
det
erm
ined
by,
the
valu
es o
f th
e o
ther
var
iab
le is
th
e d
epen
den
t va
riab
le.
( 0, 4
)
( 3, 1
)
( 3, –
4)( –
1, –
5)
( –2,
0)
( –3,
2)
x
y
O
©G
lenc
oe/M
cGra
w-H
ill62
Gle
ncoe
Alg
ebra
2
Map
pin
gs
Th
ere
are
thre
e sp
ecia
l w
ays
in w
hic
h o
ne
set
can
be
map
ped
to a
not
her
.A s
etca
n b
e m
appe
d in
toan
oth
er s
et,o
nto
anot
her
set
,or
can
hav
e a
one-
to-o
ne
corr
espo
nd
ence
wit
h a
not
her
set
.
Sta
te w
het
her
eac
h s
et i
s m
app
ed i
nto
th
e se
con
d s
et,o
nto
th
e se
con
d
set,
or h
as a
on
e-to
-on
e co
rres
pon
den
ce w
ith
th
e se
con
d s
et.
1.2.
3.4.
into
,on
toin
to,o
nto
into
,on
to,
into
,on
too
ne-
to-o
ne
5.6.
7.8.
into
into
,on
toin
to,o
nto
into
,on
to,
on
e-to
-on
e
9.C
an a
set
be
map
ped
onto
a se
t w
ith
few
er e
lem
ents
th
an i
t h
as?
yes
10.C
an a
set
be
map
ped
into
a se
t th
at h
as m
ore
elem
ents
th
an i
t h
as?
yes
11.I
f a
map
pin
g fr
om s
et A
into
set
Bis
a o
ne-
to-o
ne
corr
espo
nde
nce
,wh
at
can
you
con
clu
de a
bou
t th
e n
um
ber
of e
lem
ents
in
Aan
d B
?T
he
sets
hav
e th
e sa
me
nu
mb
er o
f el
emen
ts.
–2 9 12 5
1 4 –7 0
–2 9 12 5
1 4 –7 0
–3
15 10 2
–2 9 12 5
1 4 –7 0
10 –6 24 2
3
1 3 7 9–
5
a g k l q
0–
3 9 7
4 12 6
2 4 –1 –4
7 0 2
Into
map
pin
gA
map
ping
fro
m s
et A
to s
et B
whe
re e
very
ele
men
t of
Ais
map
ped
to o
ne o
r m
ore
elem
ents
of
set
B,
but
neve
r to
an
elem
ent
not
in B
.
On
to m
app
ing
Am
appi
ng f
rom
set
Ato
set
Bw
here
eac
h el
emen
t of
set
Bha
s at
leas
t on
e el
emen
t of
se
t A
map
ped
to it
.
On
e-to
-on
e A
map
ping
fro
m s
et A
onto
set
Bw
here
eac
h el
emen
t of
set
Ais
map
ped
to e
xact
ly o
ne
corr
esp
on
den
ceel
emen
t of
set
Ban
d di
ffere
nt e
lem
ents
of
Aar
e ne
ver
map
ped
to t
he s
ame
elem
ent
of B
.
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-1
2-1
© Glencoe/McGraw-Hill A5 Glencoe Algebra 2
An
swer
s
Answers (Lesson 2-2)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Lin
ear
Eq
uat
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-2
2-2
©G
lenc
oe/M
cGra
w-H
ill63
Gle
ncoe
Alg
ebra
2
Lesson 2-2
Iden
tify
Lin
ear
Equ
atio
ns
and
Fu
nct
ion
sA
lin
ear
equ
atio
nh
as n
o op
erat
ion
sot
her
th
an a
ddit
ion
,su
btra
ctio
n,a
nd
mu
ltip
lica
tion
of
a va
riab
le b
y a
con
stan
t.T
he
vari
able
s m
ay n
ot b
e m
ult
ipli
ed t
oget
her
or
appe
ar i
n a
den
omin
ator
.A l
inea
r eq
uat
ion
doe
sn
ot c
onta
in v
aria
bles
wit
h e
xpon
ents
oth
er t
han
1.T
he
grap
h o
f a
lin
ear
equ
atio
n i
s a
lin
e.
A l
inea
r fu
nct
ion
is a
fu
nct
ion
wh
ose
orde
red
pair
s sa
tisf
y a
lin
ear
equ
atio
n.A
ny
lin
ear
fun
ctio
n c
an b
e w
ritt
en i
n t
he
form
f(x
) �
mx
�b,
wh
ere
man
d b
are
real
nu
mbe
rs.
If a
n e
quat
ion
is
lin
ear,
you
nee
d on
ly t
wo
poin
ts t
hat
sat
isfy
th
e eq
uat
ion
in
ord
er t
o gr
aph
the
equ
atio
n.O
ne
way
is
to f
ind
the
x-in
terc
ept
and
the
y-in
terc
ept
and
con
nec
t th
ese
two
poin
ts w
ith
a l
ine. Is
f(x
) �
0.2
�a
lin
ear
fun
ctio
n?
Exp
lain
.
Yes;
it i
s a
lin
ear
fun
ctio
n b
ecau
se i
t ca
nbe
wri
tten
in
th
e fo
rmf(
x) �
�x
�0.
2. Is 2
x�
xy�
3y�
0 a
lin
ear
fun
ctio
n?
Exp
lain
.
No;
it i
s n
ot a
lin
ear
fun
ctio
n b
ecau
seth
e va
riab
les
xan
d y
are
mu
ltip
lied
toge
ther
in
th
e m
iddl
e te
rm.
1 � 5
x � 5F
ind
th
e x-
inte
rcep
t an
d t
he
y-in
terc
ept
of t
he
grap
h o
f 4x
�5y
�20
.T
hen
gra
ph
th
e eq
uat
ion
.
Th
e x-
inte
rcep
t is
th
e va
lue
of x
wh
en y
�0.
4x�
5y�
20O
rigin
al e
quat
ion
4x�
5(0)
�20
Sub
stitu
te 0
for
y.
x�
5S
impl
ify.
So
the
x-in
terc
ept
is 5
.S
imil
arly
,th
e y-
inte
rcep
t is
�4.
x
y
O
Exam
ple1
Exam
ple1
Exam
ple3
Exam
ple3
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Sta
te w
het
her
eac
h e
qu
atio
n o
r fu
nct
ion
is
lin
ear.
Wri
te y
esor
no.
If n
o,ex
pla
in.
1.6y
�x
�7
yes
2.9x
�N
o;
the
3.f(
x) �
2 �
yes
vari
able
yap
pea
rs
in t
he
den
om
inat
or.
Fin
d t
he
x-in
terc
ept
and
th
e y-
inte
rcep
t of
th
e gr
aph
of
each
eq
uat
ion
.Th
en g
rap
hth
e eq
uat
ion
.
4.2x
�7y
�14
5.5y
�x
�10
6.2.
5x�
5y�
7.5
�0
x-in
t:7;
y-in
t:2
x-in
t:�
10;
y-in
t:2
x-in
t:�
3;y-
int:
1.5
x
y
Ox
y
Ox
y
O
x � 1118 � y
©G
lenc
oe/M
cGra
w-H
ill64
Gle
ncoe
Alg
ebra
2
Stan
dar
d F
orm
Th
e st
and
ard
for
mof
a l
inea
r eq
uat
ion
is
Ax
�B
y�
C,w
her
e A
,B,a
nd
Car
e in
tege
rs w
hos
e gr
eate
st c
omm
on f
acto
r is
1.
Wri
te e
ach
eq
uat
ion
in
sta
nd
ard
for
m.I
den
tify
A,B
,an
d C
.
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Lin
ear
Eq
uat
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-2
2-2
Exam
ple
Exam
ple
a.y
�8x
�5
y�
8x�
5O
rigin
al e
quat
ion
�8x
�y
��
5S
ubtr
act
8xfr
om e
ach
side
.
8x�
y�
5M
ultip
ly e
ach
side
by
�1.
So
A�
8,B
��
1,an
d C
�5.
b.
14x
��
7y�
21
14x
��
7y�
21O
rigin
al e
quat
ion
14x
�7y
�21
Add
7y
to e
ach
side
.
2x�
y�
3D
ivid
e ea
ch s
ide
by 7
.
So
A�
2,B
�1,
and
C�
3.
Exer
cises
Exer
cises
Wri
te e
ach
eq
uat
ion
in
sta
nd
ard
for
m.I
den
tify
A,B
,an
d C
.
1.2x
�4y
�1
2.5y
�2x
�3
3.3x
��
5y�
22x
�4y
��
1;A
�2,
2x�
5y�
�3;
A�
2,3x
�5y
�2;
A�
3,B
��
4,C
��
1B
��
5,C
��
3B
�5,
C�
2
4.18
y�
24x
�9
5.y
�x
�5
6.6y
�8x
�10
�0
8x�
6y�
3;A
�8,
8x�
9y�
�60
;A
�8,
4x�
3y�
5;A
�4,
B�
�6,
C�
3B
��
9,C
��
60
B�
�3,
C�
5
7.0.
4x�
3y�
108.
x�
4y�
79.
2y�
3x�
62x
�15
y�
50;
A�
2,x
�4y
��
7;A
�1,
3x�
2y�
�6;
A�
3,B
�15
,C�
50B
��
4,C
��
7 B
��
2,C
��
6
10.
x�
y�
2 �
011
.4y
�4x
�12
�0
12.3
x�
�18
6x�
5y�
30;
A�
6,x
�y
��
3;A
�1,
x�
�6;
A�
1,B
�5,
C�
30B
�1,
C�
�3
B�
0,C
��
6
13.x
��
714
.3y
�9x
�18
15.2
x�
20 �
8y
9x�
y�
63;
A�
9,3x
�y
�6;
A�
3,x
�4y
�10
;A
�1,
B �
�1,
C�
63B
��
1,C
�6
B�
4,C
�10
16.
�3
�2x
17. �
��y
�8
18.0
.25y
�2x
�0.
75
8x�
y�
�12
;A
�8,
10x
�3y
�32
;A
�10
,8x
�y
�3;
A�
8,B
��
1,C
��
12B
��
3,C
�32
B�
�1,
C�
3
19.2
y��
4 �
020
.1.6
x�
2.4y
�4
21.0
.2x
�10
0 �
0.4y
x�
12y
��
24;
A�
1,2x
�3y
�5;
A�
2,x
�2y
�50
0;A
�1,
B�
�12
,C �
�24
B�
�3,
C�
5 B
�2,
C �
500
x � 6
3 � 45x � 2
y � 4
y � 9
1 � 32 � 5
2 � 33 � 4
© Glencoe/McGraw-Hill A6 Glencoe Algebra 2
Answers (Lesson 2-2)
Skil
ls P
ract
ice
Lin
ear
Eq
uat
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-2
2-2
©G
lenc
oe/M
cGra
w-H
ill65
Gle
ncoe
Alg
ebra
2
Lesson 2-2
Sta
te w
het
her
eac
h e
qu
atio
n o
r fu
nct
ion
is
lin
ear.
Wri
te y
esor
no.
If n
o,ex
pla
inyo
ur
reas
onin
g.
1.y
�3x
2.y
��
2 �
5x
yes
yes
3.2x
�y
�10
4.f(
x) �
4x2
yes
No
;th
e ex
po
nen
t o
f x
is n
ot
1.
5.�
�y
�15
6.x
�y
�8
No
;x
is in
a d
eno
min
ato
r.ye
s
7.g(
x) �
88.
h(x
) �
�x��
3
yes
No
;x
is in
sid
e a
squ
are
roo
t.
Wri
te e
ach
eq
uat
ion
in
sta
nd
ard
for
m.I
den
tify
A,B
,an
d C
.
9.y
�x
x�
y�
0;1,
�1,
010
.y�
5x�
15x
�y
��
1;5,
�1,
�1
11.2
x�
4 �
7y2x
�7y
�4;
2,7,
412
.3x
��
2y�
23x
�2y
��
2;3,
2,�
2
13.5
y�
9 �
05y
�9;
0,5,
914
.�6y
�14
�8x
4x�
3y�
7;4,
3,7
Fin
d t
he
x-in
terc
ept
and
th
e y-
inte
rcep
t of
th
e gr
aph
of
each
eq
uat
ion
.Th
en g
rap
hth
e eq
uat
ion
.
15.y
�3x
�6
2,�
616
.y�
�2x
0,0
17.x
�y
�5
5,5
18.2
x�
5y�
105,
2
( 5, 0
)
( 0, 2
)
x
y
O( 5
, 0)
( 0, 5
)
x
y
O
( 0, 0
)x
y
O
( 2, 0
)
( 0, –
6)
x
y
O
1 � 33 � x
©G
lenc
oe/M
cGra
w-H
ill66
Gle
ncoe
Alg
ebra
2
Sta
te w
het
her
eac
h e
qu
atio
n o
r fu
nct
ion
is
lin
ear.
Wri
te y
esor
no.
If n
o,ex
pla
inyo
ur
reas
onin
g.
1.h
(x)
�23
yes
2.y
�x
yes
3.y
�N
o;
xis
a d
eno
min
ato
r.4.
9 �
5xy
�2
No
;x
and
yar
e m
ult
iplie
d.
Wri
te e
ach
eq
uat
ion
in
sta
nd
ard
for
m.I
den
tify
A,B
,an
d C
.
5.y
�7x
�5
7x�
y�
5;7,
�1,
56.
y�
x�
5 3x
�8y
��
40;3
,�8,
�40
7.3y
�5
�0
3y�
5;0,
3,5
8.x
��
y�
28x
�8y
�21
;28
,8,2
1
Fin
d t
he
x-in
terc
ept
and
th
e y-
inte
rcep
t of
th
e gr
aph
of
each
eq
uat
ion
.Th
en g
rap
hth
e eq
uat
ion
.
9.y
�2x
�4
�2,
410
.2x
�7y
�14
7,2
11.y
��
2x�
4�
2,�
412
.6x
�2y
�6
1,3
13.M
EASU
RE
Th
e eq
uat
ion
y�
2.54
xgi
ves
the
len
gth
in
cen
tim
eter
s co
rres
pon
din
g to
ale
ngt
h x
in i
nch
es.W
hat
is
the
len
gth
in
cen
tim
eter
s of
a 1
-foo
t ru
ler?
30.4
8 cm
LON
G D
ISTA
NC
EF
or E
xerc
ises
14
and
15,
use
th
e fo
llow
ing
info
rmat
ion
.
For
Meg
’s l
ong-
dist
ance
cal
lin
g pl
an,t
he
mon
thly
cos
t C
in d
olla
rs i
s gi
ven
by
the
lin
ear
fun
ctio
n C
(t)
�6
�0.
05t,
wh
ere
tis
th
e n
um
ber
of m
inu
tes
talk
ed.
14.W
hat
is
the
tota
l co
st o
f ta
lkin
g 8
hou
rs?
of t
alki
ng
20 h
ours
?$3
0;$6
6
15.W
hat
is
the
effe
ctiv
e co
st p
er m
inu
te (
the
tota
l co
st d
ivid
ed b
y th
e n
um
ber
of m
inu
tes
talk
ed)
of t
alki
ng
8 h
ours
? of
tal
kin
g 20
hou
rs?
$0.0
625;
$0.0
55
( 1, 0
)
( 0, 3
)
x
y
O
x
y
(–2,
0)
( 0, –
4)
O
( 7, 0
)
( 0, 2
)
x
y
O( –
2, 0
)
( 0, 4
)
x
y
O
3 � 42 � 7
3 � 8
5 � x
2 � 3
Pra
ctic
e (
Ave
rag
e)
Lin
ear
Eq
uat
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-2
2-2
© Glencoe/McGraw-Hill A7 Glencoe Algebra 2
An
swer
s
Answers (Lesson 2-2)
Readin
g t
o L
earn
Math
em
ati
csL
inea
r E
qu
atio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-2
2-2
©G
lenc
oe/M
cGra
w-H
ill67
Gle
ncoe
Alg
ebra
2
Lesson 2-2
Pre-
Act
ivit
yH
ow d
o li
nea
r eq
uat
ion
s re
late
to
tim
e sp
ent
stu
dyi
ng?
Rea
d th
e in
trod
uct
ion
to
Les
son
2-2
at
the
top
of p
age
63 i
n y
our
text
book
.
•If
Lol
ita
spen
ds 2
hou
rs s
tudy
ing
mat
h,h
ow m
any
hou
rs w
ill
she
hav
e
to s
tudy
ch
emis
try?
1h
ou
rs•
Su
ppos
e th
at L
olit
a de
cide
s to
sta
y u
p on
e h
our
late
r so
th
at s
he
now
has
5 h
ours
to
stu
dy a
nd
do h
omew
ork.
Wri
te a
lin
ear
equ
atio
n t
hat
des
crib
esth
is s
itu
atio
n.
x�
y�
5
Rea
din
g t
he
Less
on
1.W
rite
yes
or n
oto
tel
l w
het
her
eac
h l
inea
r eq
uat
ion
is
in s
tan
dard
for
m.I
f it
is
not
,ex
plai
n w
hy
it i
s n
ot.
a.�
x�
2y�
5N
o;
Ais
neg
ativ
e.
b.
9x�
12y
��
5ye
s
c.5x
�7y
�3
yes
d.
2x�
y�
1N
o;
Bis
no
t an
inte
ger
.
e.0x
�0y
�0
No
;A
and
Bar
e b
oth
0.
f.2x
�4y
�8
No
;Th
e g
reat
est
com
mo
n f
acto
r o
f 2,
4,an
d 8
is 2
,no
t 1.
2.H
ow c
an y
ou u
se t
he
stan
dard
for
m o
f a
lin
ear
equ
atio
n t
o te
ll w
het
her
th
e gr
aph
is
ah
oriz
onta
l li
ne
or a
ver
tica
l li
ne?
If A
�0,
then
th
e g
rap
h is
a h
ori
zon
tal l
ine.
IfB
�0,
then
th
e g
rap
h is
a v
erti
cal l
ine.
Hel
pin
g Y
ou
Rem
emb
er
3.O
ne
way
to
rem
embe
r so
met
hin
g is
to
expl
ain
it
to a
not
her
per
son
.Su
ppos
e th
at y
ou
are
stu
dyin
g th
is l
esso
n w
ith
a f
rien
d w
ho
thin
ks t
hat
sh
e sh
ould
let
x�
0 to
fin
d th
e x-
inte
rcep
t an
d le
t y
�0
to f
ind
the
y-in
terc
ept.
How
wou
ld y
ou e
xpla
in t
o h
er h
ow t
ore
mem
ber
the
corr
ect
way
to
fin
d in
terc
epts
of
a li
ne?
Sam
ple
an
swer
:Th
e x-
inte
rcep
t is
th
e x-
coo
rdin
ate
of
a p
oin
t o
n t
he
x-ax
is.E
very
po
int
on
th
e x-
axis
has
y-c
oo
rdin
ate
0,so
let
y�
0 to
fin
d a
n x
-in
terc
ept.
Th
e y-
inte
rcep
t is
th
e y-
coo
rdin
ate
of
a p
oin
t o
n t
he
y-ax
is.E
very
po
int
on
th
e y-
axis
has
x-c
oo
rdin
ate
0,so
let
x�
0 to
fin
d a
y-i
nte
rcep
t.
4 � 7
1 � 2
1 � 2
©G
lenc
oe/M
cGra
w-H
ill68
Gle
ncoe
Alg
ebra
2
Gre
ates
t C
om
mo
n F
acto
rS
uppo
se w
e ar
e gi
ven
a li
near
equ
atio
n ax
�by
�c
whe
re a
,b,a
nd c
are
nonz
ero
inte
gers
,and
we
wan
t to
kno
w if
the
re e
xist
int
eger
s x
and
yth
at s
atis
fy t
heeq
uati
on.W
e co
uld
try
gues
sing
a f
ew t
imes
,but
thi
s pr
oces
s w
ould
be
tim
eco
nsum
ing
for
an e
quat
ion
such
as
588x
�43
2y�
72.B
y us
ing
the
Euc
lide
anA
lgor
ithm
,we
can
dete
rmin
e no
t on
ly if
suc
h in
tege
rs x
and
yex
ist,
but
also
fin
d th
em.T
he f
ollo
win
g ex
ampl
e sh
ows
how
thi
s al
gori
thm
wor
ks.
Fin
d i
nte
gers
xan
d y
that
sat
isfy
588
x�
432y
�72
.
Div
ide
the
grea
ter
of t
he
two
coef
fici
ents
by
the
less
er t
o ge
t a
quot
ien
t an
dre
mai
nde
r.T
hen
,rep
eat
the
proc
ess
by d
ivid
ing
the
divi
sor
by t
he
rem
ain
der
un
til
you
get
a r
emai
nde
r of
0.T
he
proc
ess
can
be
wri
tten
as
foll
ows.
588
�43
2(1)
�15
6(1
)43
2 �
156(
2) �
120
(2)
156
�12
0(1)
�36
(3)
120
�36
(3)
�12
(4)
36 �
12(3
)
Th
e la
st n
onze
ro r
emai
nde
r is
th
e G
CF
of
the
two
coef
fici
ents
.If
the
con
stan
tte
rm 7
2 is
div
isib
le b
y th
e G
CF,
then
in
tege
rs x
and
ydo
exi
st t
hat
sat
isfy
th
eeq
uat
ion
.To
fin
d x
and
y,w
ork
back
war
d in
th
e fo
llow
ing
man
ner
.
72�
6 �
12�
6 �
[120
�36
(3)]
Sub
stitu
te f
or 1
2 us
ing
(4)
�6(
120)
�18
(36)
�6(
120)
�18
[156
�12
0(1)
]S
ubst
itute
for
36
usin
g (3
)
��
18(1
56)
�24
(120
)�
�18
(156
) �
24[4
32 �
156(
2)]
Sub
stitu
te f
or 1
20 u
sing
(2)
�24
(432
) �
66(1
56)
�24
(432
) �
66[5
88 �
432(
1)]
Sub
stitu
te f
or 1
56 u
sing
(1)
�58
8(�
66)
�43
2(90
)
Th
us,
x�
�66
an
d y
�90
.
Fin
d i
nte
gers
xan
d y
,if
they
exi
st,t
hat
sat
isfy
eac
h e
qu
atio
n.
1.27
x�
65y
�3
2.45
x�
144y
�36
x�
�36
an
d y
�15
x�
�12
an
d y
�4
3.90
x�
117y
�10
4.12
3x�
36y
�15
no
inte
gra
l so
luti
on
s ex
ist
x�
25 a
nd
y�
�85
5.10
32x
�10
01y
�1
6.31
25x
�30
87y
�1
x�
�22
6 an
d y
�23
3x
��
1381
an
d y
�13
98
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-2
2-2
Exam
ple
Exam
ple
© Glencoe/McGraw-Hill A8 Glencoe Algebra 2
Answers (Lesson 2-3)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Slo
pe
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-3
2-3
©G
lenc
oe/M
cGra
w-H
ill69
Gle
ncoe
Alg
ebra
2
Lesson 2-3
Slo
pe
Slo
pe
mo
f a
Lin
eF
or p
oint
s (x
1, y
1) a
nd (
x 2,
y 2),
whe
re x
1�
x 2,
m�
�y 2
�y 1
� x 2�
x 1
chan
ge in
y�
�ch
ange
in x
Det
erm
ine
the
slop
e of
the
lin
e th
at p
asse
s th
rou
gh (
2,�
1) a
nd
(�4,
5).
m�
Slo
pe f
orm
ula
�(x
1, y
1) �
(2,
�1)
, (x
2, y
2) �
(�4,
5)
��
�1
Sim
plify
.
Th
e sl
ope
of t
he
lin
e is
�1.
6� �
6
5 �
(�1)
��
�4
�2
y 2�
y 1� x 2
�x 1
Gra
ph
th
e li
ne
pas
sin
gth
rou
gh (
�1,
�3)
wit
h a
slo
pe
of
.
Gra
ph t
he
orde
red
pair
(�
1,�
3).T
hen
,ac
cord
ing
to t
he
slop
e,go
up
4 u
nit
san
d ri
ght
5 u
nit
s.P
lot
the
new
poi
nt
(4,1
).C
onn
ect
the
poin
ts a
nd
draw
th
e li
ne.
x
y
O
4 � 5
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Fin
d t
he
slop
e of
th
e li
ne
that
pas
ses
thro
ugh
eac
h p
air
of p
oin
ts.
1.(4
,7)
and
(6,1
3)3
2.(6
,4)
and
(3,4
)0
3.(5
,1)
and
(7,�
3)�
2
4.(5
,�3)
an
d (�
4,3)
�5.
(5,1
0) a
nd
(�1,
�2)
26.
(�1,
�4)
and
(�
13,2
)�
7.(7
,�2)
an
d (3
,3)
�8.
(�5,
9) a
nd
(5,5
)�
9.(4
,�2)
an
d (�
4,�
8)
Gra
ph
th
e li
ne
pas
sin
g th
rou
gh t
he
give
n p
oin
t w
ith
th
e gi
ven
slo
pe.
10.s
lope
��
11.s
lope
�2
12.s
lope
�0
pass
es t
hro
ugh
(0,
2)pa
sses
th
rou
gh (
1,4)
pass
es t
hro
ugh
(�
2,�
5)
13.s
lope
�1
14.s
lope
��
15.s
lope
�
pass
es t
hro
ugh
(�
4,6)
pass
es t
hro
ugh
(�
3,0)
pass
es t
hro
ugh
(0,
0) x
y
O
x
y
O
x
y
O
1 � 53 � 4
x
y
O
x
y
Ox
y
O
1 � 3
3 � 42 � 5
5 � 4
1 � 22 � 3
©G
lenc
oe/M
cGra
w-H
ill70
Gle
ncoe
Alg
ebra
2
Para
llel a
nd
Per
pen
dic
ula
r Li
nes
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Slo
pe
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-3
2-3
In a
pla
ne,
non
vert
ical
lin
es w
ith
th
esa
me
slop
e ar
e p
aral
lel.
All
ver
tica
lli
nes
are
par
alle
l.
x
y
O
slop
e �
m
slop
e �
m
In a
pla
ne,
two
obli
que
lin
es a
re p
erp
end
icu
lar
ifan
d on
ly i
f th
e pr
odu
ct o
f th
eir
slop
es i
s �
1.A
ny
vert
ical
lin
e is
per
pen
dicu
lar
to a
ny
hor
izon
tal
lin
e.
x
y
O
slop
e �
m
slop
e �
�1 m
Exam
ple
Exam
ple
Are
th
e li
ne
pas
sin
g th
rou
gh (
2,6)
an
d (
�2,
2) a
nd
th
e li
ne
pas
sin
gth
rou
gh (
3,0)
an
d (
0,4)
par
alle
l,p
erp
end
icu
lar,
or n
eith
er?
Fin
d th
e sl
opes
of
the
two
lin
es.
Th
e sl
ope
of t
he
firs
t li
ne
is
�1.
Th
e sl
ope
of t
he
seco
nd
lin
e is
�
�.
Th
e sl
opes
are
not
equ
al a
nd
the
prod
uct
of
the
slop
es i
s n
ot �
1,so
th
e li
nes
are
nei
ther
para
llel
nor
per
pen
dicu
lar.
Are
th
e li
nes
par
alle
l,p
erp
end
icu
lar,
or n
eith
er?
1.th
e li
ne
pass
ing
thro
ugh
(4,
3) a
nd
(1,�
3) a
nd
the
lin
e pa
ssin
g th
rou
gh (
1,2)
an
d (�
1,3)
per
pen
dic
ula
r
2.th
e li
ne
pass
ing
thro
ugh
(2,
8) a
nd
(�2,
2) a
nd
the
lin
e pa
ssin
g th
rou
gh (
0,9)
an
d (6
,0)
nei
ther
3.th
e li
ne
pass
ing
thro
ugh
(3,
9) a
nd
(�2,
�1)
an
d th
e gr
aph
of
y�
2xp
aral
lel
4.th
e li
ne
wit
h x
-in
terc
ept
�2
and
y-in
terc
ept
5 an
d th
e li
ne
wit
h x
-in
terc
ept
2 an
d y-
inte
rcep
t �
5p
aral
lel
5.th
e li
ne
wit
h x
-in
terc
ept
1 an
d y-
inte
rcep
t 3
and
the
lin
e w
ith
x-i
nte
rcep
t 3
and
y-in
terc
ept
1n
eith
er
6.th
e li
ne
pass
ing
thro
ugh
(�
2,�
3) a
nd
(2,5
) an
d th
e gr
aph
of
x�
2y�
10p
erp
end
icu
lar
7.th
e li
ne
pass
ing
thro
ugh
(�
4,�
8) a
nd
(6,�
4) a
nd
the
grap
h o
f 2x
�5y
�5
par
alle
l
4 � 34
�0
� 0 �
3
6 �
2�
�2
�(�
2)
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A9 Glencoe Algebra 2
An
swer
s
Answers (Lesson 2-3)
Skil
ls P
ract
ice
Slo
pe
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-3
2-3
©G
lenc
oe/M
cGra
w-H
ill71
Gle
ncoe
Alg
ebra
2
Lesson 2-3
Fin
d t
he
slop
e of
th
e li
ne
that
pas
ses
thro
ugh
eac
h p
air
of p
oin
ts.
1.(1
,5),
(�1,
�3)
42.
(0,2
),(3
,0)
�3.
(1,9
),(0
,6)
3
4.(8
,�5)
,(4,
�2)
�5.
(�3,
5),(
�3,
�1)
un
def
ined
6.(�
2,�
2),(
10,�
2)0
7.(4
,5),
(2,7
)�
18.
(�2,
�4)
,(3,
2)9.
(5,2
),(�
3,2)
0
Gra
ph
th
e li
ne
pas
sin
g th
rou
gh t
he
give
n p
oin
t w
ith
th
e gi
ven
slo
pe.
10.(
0,4)
,m�
111
.(2,
�4)
,m�
�1
12.(
�3,
�5)
,m�
213
.(�
2,�
1),m
��
2
Gra
ph
th
e li
ne
that
sat
isfi
es e
ach
set
of
con
dit
ion
s.
14.p
asse
s th
rou
gh (
0,1)
,per
pen
dicu
lar
to15
.pas
ses
thro
ugh
(0,
�5)
,par
alle
l to
th
e
a li
ne
wh
ose
slop
e is
gr
aph
of
y�
1
16.H
IKIN
GN
aom
i le
ft f
rom
an
ele
vati
on o
f 74
00 f
eet
at 7
:00
A.M
.an
d h
iked
to
an e
leva
tion
of 9
800
feet
by
11:0
0 A.M
.Wh
at w
as h
er r
ate
of c
han
ge i
n a
ltit
ude
?60
0 ft
/h
(0 ,–
5)
x
y
O(0
,1)
x
y
O
1 � 3
(–2,
–1)
x
y
O
(–3,
–5)
x
y
O
( 2, –
4)x
y
O( 0
, 4)
x
y
O
6 � 5
3 � 4
2 � 3
©G
lenc
oe/M
cGra
w-H
ill72
Gle
ncoe
Alg
ebra
2
Fin
d t
he
slop
e of
th
e li
ne
that
pas
ses
thro
ugh
eac
h p
air
of p
oin
ts.
1.(3
,�8)
,(�
5,2)
�2.
(�10
,�3)
,(7,
2)3.
(�7,
�6)
,(3,
�6)
0
4.(8
,2),
(8,�
1)u
nd
efin
ed5.
(4,3
),(7
,�2)
�6.
(�6,
�3)
,(�
8,4)
�
Gra
ph
th
e li
ne
pas
sin
g th
rou
gh t
he
give
n p
oin
t w
ith
th
e gi
ven
slo
pe.
7.(0
,�3)
,m�
38.
(2,1
),m
��
9.(0
,2),
m�
010
.(2,
�3)
,m�
Gra
ph
th
e li
ne
that
sat
isfi
es e
ach
set
of
con
dit
ion
s.
11.p
asse
s th
rou
gh (
3,0)
,per
pen
dicu
lar
12.p
asse
s th
rou
gh (
�3,
�1)
,par
alle
l to
a l
ine
to a
lin
e w
hos
e sl
ope
is
wh
ose
slop
e is
�1
DEP
REC
IATI
ON
For
Exe
rcis
es 1
3–15
,use
th
e fo
llow
ing
info
rmat
ion
.A
mac
hin
e th
at o
rigi
nal
ly c
ost
$15,
600
has
a v
alu
e of
$75
00 a
t th
e en
d of
3 y
ears
.Th
e sa
me
mac
hin
e h
as a
val
ue
of $
2800
at
the
end
of 8
yea
rs.
13.F
ind
the
aver
age
rate
of
chan
ge i
n v
alu
e (d
epre
ciat
ion
) of
th
e m
ach
ine
betw
een
its
purc
has
e an
d th
e en
d of
3 y
ears
.�
$270
0 p
er y
ear
14.F
ind
the
aver
age
rate
of
chan
ge i
n v
alu
e of
th
e m
ach
ine
betw
een
th
e en
d of
3 y
ears
an
dth
e en
d of
8 y
ears
.�
$940
per
yea
r
15.I
nter
pret
the
sig
n of
you
r an
swer
s.It
is n
egat
ive
bec
ause
th
e va
lue
is d
ecre
asin
g.
( –3,
–1)
x
y
O
(3, 0
)x
y
O
3 � 2
( 2, –
3)
xO
y
( 0, 2
)
x
y
O
4 � 5
(2, 1
)
x
y
O
(0, –
3)
x
y
O
3 � 4
7 � 25 � 3
5 � 175 � 4
Pra
ctic
e (
Ave
rag
e)
Slo
pe
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-3
2-3
© Glencoe/McGraw-Hill A10 Glencoe Algebra 2
Answers (Lesson 2-3)
Readin
g t
o L
earn
Math
em
ati
csS
lop
e
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-3
2-3
©G
lenc
oe/M
cGra
w-H
ill73
Gle
ncoe
Alg
ebra
2
Lesson 2-3
Pre-
Act
ivit
yH
ow d
oes
slop
e ap
ply
to
the
stee
pn
ess
of r
oad
s?
Rea
d th
e in
trod
uct
ion
to
Les
son
2-3
at
the
top
of p
age
68 i
n y
our
text
book
.
•W
hat
is
the
grad
e of
a r
oad
that
ris
es 4
0 fe
et o
ver
a h
oriz
onta
l di
stan
ceof
100
0 fe
et?
4%•
Wh
at i
s th
e gr
ade
of a
roa
d th
at r
ises
525
met
ers
over
a h
oriz
onta
ldi
stan
ce o
f 10
kil
omet
ers?
(1
kilo
met
er �
1000
met
ers)
5.25
%
Rea
din
g t
he
Less
on
1.D
escr
ibe
each
typ
e of
slo
pe a
nd
incl
ude
a s
ketc
h.
Typ
e o
f S
lop
eD
escr
ipti
on
of
Gra
ph
Ske
tch
Pos
itive
Th
e lin
e ri
ses
to t
he
rig
ht.
Zer
oT
he
line
is h
ori
zon
tal.
Neg
ativ
eT
he
line
falls
to
th
e ri
gh
t.
Und
efin
edT
he
line
is v
erti
cal.
2.a.
How
are
th
e sl
opes
of
two
non
vert
ical
par
alle
l li
nes
rel
ated
?T
hey
are
eq
ual
.
b.
How
are
the
slo
pes
of t
wo
obli
que
perp
endi
cula
r li
nes
rela
ted?
Th
eir
pro
du
ct is
�1.
Hel
pin
g Y
ou
Rem
emb
er
3.L
ook
up
the
term
s gr
ade,
pitc
h,s
lan
t,an
d sl
ope.
How
can
eve
ryda
y m
ean
ings
of
thes
ew
ords
hel
p yo
u r
emem
ber
the
defi
nit
ion
of
slop
e?S
amp
le a
nsw
er:
All
thes
e w
ord
sca
n b
e u
sed
wh
en y
ou
des
crib
e h
ow
mu
ch a
th
ing
sla
nts
up
war
d o
rd
ow
nw
ard
.Yo
u c
an d
escr
ibe
this
nu
mer
ical
ly b
y co
mp
arin
g r
ise
to r
un
.
x
y
O
x
y
O
x
y
O
x
y
O
©G
lenc
oe/M
cGra
w-H
ill74
Gle
ncoe
Alg
ebra
2
Aer
ial S
urv
eyo
rs a
nd
Are
aM
any
lan
d re
gion
s h
ave
irre
gula
r sh
apes
.Aer
ial
surv
eyor
s
supp
ly a
eria
l m
appe
rs w
ith
lis
ts o
f co
ordi
nat
es a
nd
elev
atio
ns
for
the
area
s th
at n
eed
to b
e ph
otog
raph
ed f
rom
th
e ai
r.T
hes
e m
aps
prov
ide
info
rmat
ion
abo
ut
the
hor
izon
tal
and
vert
ical
fe
atu
res
of t
he
lan
d.
Ste
p 1
Lis
t th
e or
dere
d pa
irs
for
the
vert
ices
in
co
un
terc
lock
wis
e or
der,
repe
atin
g th
e fi
rst
orde
red
pair
at
the
bott
om o
f th
e li
st.
Ste
p 2
Fin
d D
,th
e su
m o
f th
e do
wn
war
d di
agon
al p
rodu
cts
(fro
m l
eft
to r
igh
t).
D�
(5 �
5) �
(2 �
1) �
(2 �
3) �
(6 �
7)�
25 �
2 �
6 �
42 o
r 75
Ste
p 3
Fin
d U
,th
e su
m o
f th
e u
pwar
d di
agon
al p
rodu
cts
(fro
m l
eft
to r
igh
t).
U�
(2 �
7) �
(2 �
5) �
(6 �
1) �
(5 �
3)�
14 �
10 �
6 �
15 o
r 45
Ste
p 4
Use
th
e fo
rmu
la A
��1 2� (
D�
U)
to f
ind
the
area
.
A�
�1 2� (75
�45
)
��1 2� (
30)
or 1
5
Th
e ar
ea i
s 15
squ
are
un
its.
Cou
nt
the
nu
mbe
r of
squ
are
un
its
encl
osed
by
the
poly
gon
.Doe
s th
is r
esu
lt s
eem
rea
son
able
?
Use
th
e co
ord
inat
e m
eth
od t
o fi
nd
th
e ar
ea o
f ea
ch r
egio
n i
n s
qu
are
un
its.
1.2.
3.
20 u
nit
s214
un
its2
34 u
nit
s2
x
y
O
x
y
Ox
y
O
(5, 7
)
(2, 5
)
(2, 1
)
(6, 3
)
(5, 7
)
x
y
O
(2, 1
)
(2, 5
)
(5, 7
)
(6, 3
)
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-3
2-3
© Glencoe/McGraw-Hill A11 Glencoe Algebra 2
An
swer
s
Answers (Lesson 2-4)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Wri
tin
g L
inea
r E
qu
atio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-4
2-4
©G
lenc
oe/M
cGra
w-H
ill75
Gle
ncoe
Alg
ebra
2
Lesson 2-4
Form
s o
f Eq
uat
ion
s
Slo
pe-
Inte
rcep
t F
orm
o
f a
Lin
ear
Eq
uat
ion
y�
mx
�b,
whe
re m
is t
he s
lope
and
bis
the
y-in
terc
ept
Po
int-
Slo
pe
Fo
rm
y�
y 1�
m(x
�x 1
), w
here
(x 1
, y 1
) ar
e th
e co
ordi
nate
s of
a p
oint
on
the
line
and
of
a L
inea
r E
qu
atio
nm
is t
he s
lope
of
the
line
Wri
te a
n e
qu
atio
n i
nsl
ope-
inte
rcep
t fo
rm f
or t
he
lin
e th
ath
as s
lop
e �
2 an
d p
asse
s th
rou
gh t
he
poi
nt
(3,7
).
Su
bsti
tute
for
m,x
,an
d y
in t
he
slop
e-in
terc
ept
form
.y
�m
x�
bS
lope
-inte
rcep
t fo
rm
7 �
(�2)
(3)
�b
(x,
y)
�(3
, 7)
, m
��
2
7 �
�6
�b
Sim
plify
.
13 �
bA
dd 6
to
both
sid
es.
Th
e y-
inte
rcep
t is
13.
Th
e eq
uat
ion
in
sl
ope-
inte
rcep
t fo
rm i
s y
��
2x�
13.
Wri
te a
n e
qu
atio
n i
nsl
ope-
inte
rcep
t fo
rm f
or t
he
lin
e th
ath
as s
lop
e an
d x
-in
terc
ept
5.
y�
mx
�b
Slo
pe-in
terc
ept
form
0 �
��(5
) �
b(x
, y
) �
(5,
0),
m�
0 �
�b
Sim
plify
.
��
bS
ubtr
act
from
bot
h si
des.
Th
e y-
inte
rcep
t is
�.T
he
slop
e-in
terc
ept
form
is
y�
x�
.5 � 3
1 � 3
5 � 3
5 � 35 � 3
5 � 3
1 � 31 � 3
1 � 3
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Wri
te a
n e
qu
atio
n i
n s
lop
e-in
terc
ept
form
for
th
e li
ne
that
sat
isfi
es e
ach
set
of
con
dit
ion
s.
1.sl
ope
�2,
pass
es t
hro
ugh
(�
4,6)
2.sl
ope
,y-i
nte
rcep
t 4
y�
�2x
�2
y�
x�
4
3.sl
ope
1,pa
sses
th
rou
gh (
2,5)
4.sl
ope
�,p
asse
s th
rou
gh (
5,�
7)
y�
x�
3y
��
x�
6
Wri
te a
n e
qu
atio
n i
n s
lop
e-in
terc
ept
form
for
eac
h g
rap
h.
5.6.
7.
y�
�3x
�9
y�
xy
�x
�1
4 � 91 � 9
5 � 4
x
y
O
( –4,
1)
( 5, 2
)
x
y O
( 4, 5
)
( 0, 0
)
x
y
O
( 1, 6
)
( 3, 0
)
13 � 513 � 5
3 � 23 � 2
©G
lenc
oe/M
cGra
w-H
ill76
Gle
ncoe
Alg
ebra
2
Para
llel a
nd
Per
pen
dic
ula
r Li
nes
Use
th
e sl
ope-
inte
rcep
t or
poi
nt-
slop
e fo
rm t
o fi
nd
equ
atio
ns
of l
ines
th
at a
re p
aral
lel
or p
erpe
ndi
cula
r to
a g
iven
lin
e.R
emem
ber
that
par
alle
lli
nes
hav
e eq
ual
slo
pe.T
he
slop
es o
f tw
o pe
rpen
dicu
lar
lin
es a
re n
egat
ive
reci
proc
als,
that
is,t
hei
r pr
odu
ct i
s �
1.
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Wri
tin
g L
inea
r E
qu
atio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-4
2-4
Wri
te a
n e
qu
atio
n o
f th
eli
ne
that
pas
ses
thro
ugh
(8,
2) a
nd
is
per
pen
dic
ula
r to
th
e li
ne
wh
ose
equ
atio
n i
s y
��
x�
3.
Th
e sl
ope
of t
he
give
n l
ine
is �
.Sin
ce t
he
slop
es o
f pe
rpen
dicu
lar
lin
es a
re n
egat
ive
reci
proc
als,
the
slop
e of
th
e pe
rpen
dicu
lar
lin
e is
2.
Use
th
e sl
ope
and
the
give
n p
oin
t to
wri
teth
e eq
uat
ion
.y
� y
1�
m(x
�x 1
)P
oint
-slo
pe f
orm
y�
2 �
2(x
�8)
(x1,
y1)
�(8
, 2)
, m
�2
y�
2 �
2x�
16D
istr
ibut
ive
Pro
p.
y�
2x�
14A
dd 2
to
each
sid
e.
An
equ
atio
n o
f th
e li
ne
is y
�2x
�14
.
1 � 2
1 � 2
Wri
te a
n e
qu
atio
n o
f th
eli
ne
that
pas
ses
thro
ugh
(�
1,5)
an
d i
sp
aral
lel
to t
he
grap
h o
f y
�3x
�1.
Th
e sl
ope
of t
he
give
n l
ine
is 3
.Sin
ce t
he
slop
es o
f pa
rall
el l
ines
are
equ
al,t
he
slop
eof
th
e pa
rall
el l
ine
is a
lso
3.U
se t
he
slop
e an
d th
e gi
ven
poi
nt
to w
rite
the
equ
atio
n.
y�
y 1�
m(x
�x 1
)P
oint
-slo
pe f
orm
y�
5 �
3(x
�(�
1))
(x1,
y1)
�(�
1, 5
), m
�3
y�
5 �
3x�
3D
istr
ibut
ive
Pro
p.
y�
3x�
8A
dd 5
to
each
sid
e.
An
equ
atio
n o
f th
e li
ne
is y
�3x
�8.
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Wri
te a
n e
qu
atio
n i
n s
lop
e-in
terc
ept
form
for
th
e li
ne
that
sat
isfi
es e
ach
set
of
con
dit
ion
s.
1.pa
sses
th
rou
gh (
�4,
2),p
aral
lel
to t
he
lin
e w
hos
e eq
uat
ion
is
y�
x�
5y
�x
�4
2.pa
sses
th
rou
gh (
3,1)
,per
pen
dicu
lar
to t
he
grap
h o
f y
��
3x�
2y
�x
3.pa
sses
th
rou
gh (
1,�
1),p
aral
lel
to t
he
lin
e th
at p
asse
s th
rou
gh (
4,1)
an
d (2
,�3)
y�
2x�
3
4.pa
sses
th
rou
gh (
4,7)
,per
pen
dicu
lar
to t
he
lin
e th
at p
asse
s th
rou
gh (
3,6)
an
d (3
,15)
y�
7
5.pa
sses
th
rou
gh (
8,�
6),p
erpe
ndi
cula
r to
th
e gr
aph
of
2x�
y�
4y
��
x�
2
6.pa
sses
th
rou
gh (
2,�
2),p
erpe
ndi
cula
r to
th
e gr
aph
of
x�
5y�
6y
�5x
�12
7.pa
sses
th
rou
gh (
6,1)
,par
alle
l to
th
e li
ne
wit
h x
-in
terc
ept
�3
and
y-in
terc
ept
5
y�
x�
9
8.pa
sses
th
rou
gh (
�2,
1),p
erpe
ndi
cula
r to
th
e li
ne
y�
4x�
11y
��
x�
1 � 21 � 4
5 � 3
1 � 2
1 � 3
1 � 21 � 2
© Glencoe/McGraw-Hill A12 Glencoe Algebra 2
Answers (Lesson 2-4)
Skil
ls P
ract
ice
Wri
tin
g L
inea
r E
qu
atio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-4
2-4
©G
lenc
oe/M
cGra
w-H
ill77
Gle
ncoe
Alg
ebra
2
Lesson 2-4
Sta
te t
he
slop
e an
d y
-in
terc
ept
of t
he
grap
h o
f ea
ch e
qu
atio
n.
1.y
�7x
�5
7,�
52.
y�
�x
�3
�,3
3.y
�x
,04.
3x�
4y�
4�
,1
5.7y
�4x
�7
,�1
6.3x
�2y
�6
�0
,3
7.2x
�y
�5
2,�
58.
2y�
6 �
5x�
,3
Wri
te a
n e
qu
atio
n i
n s
lop
e-in
terc
ept
form
for
eac
h g
rap
h.
9.10
.11
.
y�
3x�
1y
��
1y
��
2x�
3
Wri
te a
n e
qu
atio
n i
n s
lop
e-in
terc
ept
form
for
th
e li
ne
that
sat
isfi
es e
ach
set
of
con
dit
ion
s.
12.s
lope
3,p
asse
s th
rou
gh (
1,�
3)13
.slo
pe �
1,pa
sses
th
rou
gh (
0,0)
y�
3x�
6y
��
x
14.s
lope
�2,
pass
es t
hro
ugh
(0,
�5)
15.s
lope
3,p
asse
s th
rou
gh (
2,0)
y�
�2x
�5
y�
3x�
6
16.p
asse
s th
rou
gh (
�1,
�2)
an
d (�
3,1)
17.p
asse
s th
rou
gh (
�2,
�4)
an
d (1
,8)
y�
�x
�y
�4x
�4
18.x
-in
terc
ept
2,y-
inte
rcep
t �
619
.x-i
nte
rcep
t ,y
-in
terc
ept
5
y�
3x�
6y
��
2x�
5
20.p
asse
s th
rou
gh (
3,�
1),p
erpe
ndi
cula
r to
th
e gr
aph
of
y�
�x
�4.
y�
3x�
101 � 3
5 � 2
7 � 23 � 2
x
y
O
( 0, 3
)
( 3, –
3)
x
y
O( –
3, –
1)( 4
, –1)
x
y
O
( –1,
–4)
( 1, 2
)
5 � 2
3 � 24 � 7
3 � 42 � 3
2 � 3
3 � 53 � 5
©G
lenc
oe/M
cGra
w-H
ill78
Gle
ncoe
Alg
ebra
2
Sta
te t
he
slop
e an
d y
-in
terc
ept
of t
he
grap
h o
f ea
ch e
qu
atio
n.
1.y
�8x
�12
8,12
2.y
�0.
25x
�1
0.25
,�1
3.y
��
x�
,0
4.3y
�7
0,5.
3x�
�15
�5y
,36.
2x�
3y�
10,�
Wri
te a
n e
qu
atio
n i
n s
lop
e-in
terc
ept
form
for
eac
h g
rap
h.
7.8.
9.
y�
2y
�x
�2
y�
�x
�1
Wri
te a
n e
qu
atio
n i
n s
lop
e-in
terc
ept
form
for
th
e li
ne
that
sat
isfi
es e
ach
set
of
con
dit
ion
s.
10.s
lope
�5,
pass
es t
hro
ugh
(�
3,�
8)11
.slo
pe
,pas
ses
thro
ugh
(10
,�3)
y�
�5x
�23
y�
x�
11
12.s
lope
0,p
asse
s th
rou
gh (
0,�
10)
13.s
lope
�,p
asse
s th
rou
gh (
6,�
8)
y�
�10
y�
�x
�4
14.p
asse
s th
rou
gh (
3,11
) an
d (�
6,5)
15.p
asse
s th
rou
gh (
7,�
2) a
nd
(3,�
1)
y�
x�
9y
��
x�
16.x
-in
terc
ept
3,y-
inte
rcep
t 2
17.x
-in
terc
ept
�5,
y-in
terc
ept
7
y�
�x
�2
y�
x�
7
18.p
asse
s th
rou
gh (
�8,
�7)
,per
pen
dicu
lar
to t
he
grap
h o
f y
�4x
�3
y�
�x
�9
19.R
ESER
VO
IRS
Th
e su
rfac
e of
Gra
nd
Lak
e is
at
an e
leva
tion
of
648
feet
.Du
rin
g th
ecu
rren
t dr
ough
t,th
e w
ater
lev
el i
s dr
oppi
ng
at a
rat
e of
3 i
nch
es p
er d
ay.I
f th
is t
ren
dco
nti
nu
es,w
rite
an
equ
atio
n t
hat
giv
es t
he
elev
atio
n i
n f
eet
of t
he
surf
ace
of G
ran
d L
ake
afte
r x
days
.y
��
0.25
x�
648
20.B
USI
NES
ST
ony
Mar
con
i’s c
ompa
ny
man
ufa
ctu
res
CD
-RO
M d
rive
s.T
he
com
pan
y w
ill
mak
e $1
50,0
00 p
rofi
t if
it
man
ufa
ctu
res
100,
000
driv
es,a
nd
$1,7
50,0
00 p
rofi
t if
it
man
ufa
ctu
res
500,
000
driv
es.T
he
rela
tion
ship
bet
wee
n t
he
nu
mbe
r of
dri
ves
man
ufa
ctu
red
and
the
prof
it i
s li
nea
r.W
rite
an
equ
atio
n t
hat
giv
es t
he
prof
it P
wh
en
ndr
ives
are
man
ufa
ctu
red.
P�
4n�
250,
000
1 � 4
7 � 52 � 3
1 � 41 � 4
2 � 3
2 � 32 � 3
4 � 54 � 5
2 � 33 � 2
x
y
O( 3
, –1)
( –3,
3)
x
y
O
( 4, 4
)
( 0, –
2)
x
y
O
( 0, 2
)
10 � 32 � 3
3 � 57 � 3
3 � 53 � 5
Pra
ctic
e (
Ave
rag
e)
Wri
tin
g L
inea
r E
qu
atio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-4
2-4
© Glencoe/McGraw-Hill A13 Glencoe Algebra 2
An
swer
s
Answers (Lesson 2-4)
Readin
g t
o L
earn
Math
em
ati
csW
riti
ng
Lin
ear
Eq
uat
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-4
2-4
©G
lenc
oe/M
cGra
w-H
ill79
Gle
ncoe
Alg
ebra
2
Lesson 2-4
Pre-
Act
ivit
yH
ow d
o li
nea
r eq
uat
ion
s ap
ply
to
bu
sin
ess?
Rea
d th
e in
trod
uct
ion
to
Les
son
2-4
at
the
top
of p
age
75 i
n y
our
text
book
.
•If
th
e to
tal
cost
of
prod
uci
ng
a pr
odu
ct i
s gi
ven
by
the
equ
atio
n
y�
5400
�1.
37x,
wh
at i
s th
e fi
xed
cost
? W
hat
is
the
vari
able
cos
t (f
or e
ach
ite
m p
rodu
ced)
?$5
400;
$1.3
7•
Wri
te a
lin
ear
equ
atio
n t
hat
des
crib
es t
he
foll
owin
g si
tuat
ion
:A
com
pan
y th
at m
anu
fact
ure
s co
mpu
ters
has
a f
ixed
cos
t of
$22
8,75
0 an
da
vari
able
cos
t of
$85
2 to
pro
duce
eac
h c
ompu
ter.
y�
228,
750
�85
2x
Rea
din
g t
he
Less
on
1.a.
Wri
te t
he
slop
e-in
terc
ept
form
of
the
equ
atio
n o
f a
lin
e.T
hen
exp
lain
th
e m
ean
ing
ofea
ch o
f th
e va
riab
les
in t
he
equ
atio
n.
y�
mx
�b
;m
is t
he
slo
pe
and
bis
th
ey-
inte
rcep
t.T
he
vari
able
s x
and
yar
e th
e co
ord
inat
es o
f an
y p
oin
t o
nth
e lin
e.
b.
Wri
te t
he
poin
t-sl
ope
form
of
the
equ
atio
n o
f a
lin
e.T
hen
exp
lain
th
e m
ean
ing
of e
ach
of t
he
vari
able
s in
th
e eq
uat
ion
.y
�y 1
�m
(x�
x 1);
mis
th
e sl
op
e.x
and
yar
e th
e co
ord
inat
es o
f an
y p
oin
t o
n t
he
line.
x 1an
d y
1ar
e th
e co
ord
inat
es o
f o
ne
spec
ific
po
int
on
th
e lin
e.
2.S
upp
ose
that
you
r al
gebr
a te
ach
er a
sks
you
to
wri
te t
he
poin
t-sl
ope
form
of
the
equ
atio
nof
th
e li
ne
thro
ugh
th
e po
ints
(�
6,7)
an
d (�
3,�
2).Y
ou w
rite
y�
2 �
�3(
x�
3) a
nd
you
r cl
assm
ate
wri
tes
y�
7 �
�3(
x�
6).W
hic
h o
f yo
u i
s co
rrec
t?E
xpla
in. Y
ou
are
bo
th c
orr
ect.
Eit
her
po
int
may
be
use
d a
s (x
1,y 1
) in
th
e p
oin
t-sl
op
e fo
rm.
You
use
d (
�3,
�2)
,an
d y
ou
r cl
assm
ate
use
d (
�6,
7).
3.Yo
u a
re a
sked
to
wri
te a
n e
quat
ion
of
two
lin
es t
hat
pas
s th
rou
gh (
3,�
5),o
ne
of t
hem
para
llel
to
and
one
of t
hem
per
pen
dicu
lar
to t
he
lin
e w
hos
e eq
uat
ion
is
y�
�3x
�4.
Th
e fi
rst
step
in
fin
din
g th
ese
equ
atio
ns
is t
o fi
nd
thei
r sl
opes
.Wh
at i
s th
e sl
ope
of t
he
para
llel
lin
e? W
hat
is
the
slop
e of
th
e pe
rpen
dicu
lar
lin
e?�
3;
Hel
pin
g Y
ou
Rem
emb
er
4.M
any
stu
den
ts h
ave
trou
ble
rem
embe
rin
g th
e po
int-
slop
e fo
rm f
or a
lin
ear
equ
atio
n.
How
can
you
use
th
e de
fin
itio
n o
f sl
ope
to r
emem
ber
this
for
m?
Sam
ple
an
swer
:
Wri
te t
he
def
init
ion
of
slo
pe:
m�
.Mu
ltip
ly b
oth
sid
es o
f th
is
equ
atio
n b
y x 2
�x 1
.Dro
p t
he
sub
scri
pts
in y
2an
d x
2.T
his
giv
es t
he
po
int-
slo
pe
form
of
the
equ
atio
n o
f a
line.
y 2�
y 1� x 2
�x 1
1 � 3
©G
lenc
oe/M
cGra
w-H
ill80
Gle
ncoe
Alg
ebra
2
Two
-In
terc
ept
Fo
rm o
f a
Lin
ear
Eq
uat
ion
You
are
alre
ady
fam
ilia
r w
ith
the
slop
e-in
terc
ept
form
of
a li
near
equ
atio
n,
y�
mx
�b.
Lin
ear
equa
tion
s ca
n al
so b
e w
ritt
en i
n th
e fo
rm � ax �
�� by �
�1
wit
h
x-in
terc
ept
aan
d y-
inte
rcep
t b.
Thi
s is
cal
led
two-
inte
rcep
t fo
rm.
Dra
w t
he
grap
h o
f � �x 3�
�� 6y �
�1.
The
gra
ph c
ross
es t
he x
-axi
s at
�3
and
the
y-ax
is a
t 6.
Gra
ph
(�3,
0) a
nd (
0,6)
,the
n dr
aw a
str
aigh
t li
ne t
hrou
gh t
hem
.
Wri
te 3
x�
4y�
12 i
n t
wo-
inte
rcep
t fo
rm.
� 13 2x ��
� 14 2y ��
�1 12 2�D
ivid
e by
12
to o
btai
n 1
on t
he r
ight
sid
e.
� 4x ��
� 3y ��
1S
impl
ify.
Th
e x-
inte
rcep
t is
4;t
he
y-in
terc
ept
is 3
.
Use
th
e gi
ven
in
terc
epts
a a
nd
b,t
o w
rite
an
eq
uat
ion
in
tw
o-in
terc
ept
form
.Th
en d
raw
th
e gr
aph
.S
ee s
tud
ents
’gra
ph
s.
1.a
��
2,b
��
4� �x 2�
�� �y 4�
�1
2.a
�1,
b�
8�x 1�
��y 8�
�1
3.a
�3,
b�
5�x 3�
��y 5�
�1
4.a
�6,
b�
9�x 6�
��y 9�
�1
Wri
te e
ach
eq
uat
ion
in
tw
o-in
terc
ept
form
.Th
en d
raw
th
e gr
aph
.
5.3x
�2y
��
66.
�1 2� x�
�1 4� y�
17.
5x�
2y�
�10
� �x 2��
�y 3��
1�x 2�
��y 4�
�1
� �x 2��
� �y 5��
1
x
y
Ox
y
Ox
y
O
x
y O
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-4
2-4
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
© Glencoe/McGraw-Hill A14 Glencoe Algebra 2
Answers (Lesson 2-5)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Mo
del
ing
Rea
l-W
orl
d D
ata:
Usi
ng
Sca
tter
Plo
ts
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-5
2-5
©G
lenc
oe/M
cGra
w-H
ill81
Gle
ncoe
Alg
ebra
2
Lesson 2-5
Scat
ter
Plo
tsW
hen
a s
et o
f da
ta p
oin
ts i
s gr
aph
ed a
s or
dere
d pa
irs
in a
coo
rdin
ate
plan
e,th
e gr
aph
is
call
ed a
sca
tter
plo
t.A
sca
tter
plo
t ca
n b
e u
sed
to d
eter
min
e if
th
ere
isa
rela
tion
ship
am
ong
the
data
.
BA
SEB
ALL
Th
e ta
ble
bel
ow s
how
s th
e n
um
ber
of
hom
e ru
ns
and
run
s b
atte
d i
n f
or v
ario
us
bas
ebal
l p
laye
rs w
ho
won
th
e M
ost
Val
uab
le P
laye
rA
war
d d
uri
ng
the
1990
s.M
ake
a sc
atte
r p
lot
of t
he
dat
a.
Sour
ce:N
ew Y
ork
Times
Alm
anac
Mak
e a
scat
ter
plo
t fo
r th
e d
ata
in e
ach
tab
le b
elow
.
1.FU
EL E
FFIC
IEN
CY
Th
e ta
ble
belo
w s
how
s th
e av
erag
e fu
el e
ffic
ien
cy i
n m
iles
per
gal
lon
of
new
car
s m
anu
fact
ure
d du
rin
g th
e ye
ars
list
ed.
Sour
ce:N
ew Y
ork
Times
Alm
anac
2.C
ON
GR
ESS
Th
e ta
ble
belo
w s
how
s th
e n
um
ber
of
wom
en s
ervi
ng
in t
he
Un
ited
Sta
tes
Con
gres
s du
rin
g th
e ye
ars
1987
�19
99.
Sour
ce:W
all S
treet
Jour
nal A
lman
ac
Co
ng
ress
ion
al S
essi
on
Nu
mb
er o
f Wo
men
100
25
101
31
102
33
103
55
104
58
105
62
Sess
ion
of
Co
ng
ress
Number of Women
100
102
104
70 56 42 28 14 0
Wo
men
in
Co
ng
ress
Year
Fu
el E
ffic
ien
cy (
mp
g)
1960
15.5
1970
14.1
1980
22.6
1990
26.9
Year
Miles per Gallon
1960
1970
1980
1990
36 30 24 18 12 6 0Avera
ge F
uel
Eff
icie
ncy
Ho
me
Ru
ns
MV
P H
Rs
an
d R
BIs
Runs Batted In
126
024
3618
3042
48
150
125
100 75 50 25
Ho
me
Ru
ns
Ru
ns
Bat
ted
In
3311
4
3911
6
4013
0
2861
4112
8
4714
4
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill82
Gle
ncoe
Alg
ebra
2
Pred
icti
on
Eq
uat
ion
sA
lin
e of
fit
is a
lin
e th
at c
lose
ly a
ppro
xim
ates
a s
et o
f da
tagr
aph
ed i
n a
sca
tter
plo
t.T
he
equ
atio
n o
f a
lin
e of
fit
is
call
ed a
pre
dic
tion
eq
uat
ion
beca
use
it
can
be
use
d to
pre
dict
val
ues
not
giv
en i
n t
he
data
set
.
To
fin
d a
pred
icti
on e
quat
ion
for
a s
et o
f da
ta,s
elec
t tw
o po
ints
th
at s
eem
to
repr
esen
t th
eda
ta w
ell.
Th
en t
o w
rite
th
e pr
edic
tion
equ
atio
n,u
se w
hat
you
kn
ow a
bou
t w
riti
ng
a li
nea
req
uat
ion
wh
en g
iven
tw
o po
ints
on
th
e li
ne.
STO
RA
GE
CO
STS
Acc
ord
ing
to a
cer
tain
pre
dic
tion
eq
uat
ion
,th
eco
st o
f 20
0 sq
uar
e fe
et o
f st
orag
e sp
ace
is $
60.T
he
cost
of
325
squ
are
feet
of
stor
age
spac
e is
$16
0.
a.F
ind
th
e sl
ope
of t
he
pre
dic
tion
eq
uat
ion
.Wh
at d
oes
it r
epre
sen
t?S
ince
th
e co
st d
epen
ds u
pon
th
e sq
uar
e fo
otag
e,le
t x
repr
esen
t th
e am
oun
t of
sto
rage
spac
e in
squ
are
feet
an
d y
repr
esen
t th
e co
st i
n d
olla
rs.T
he
slop
e ca
n b
e fo
un
d u
sin
g th
e
form
ula
m�
.So,
m�
��
0.8
Th
e sl
ope
of t
he
pred
icti
on e
quat
ion
is
0.8.
Th
is m
ean
s th
at t
he
pric
e of
sto
rage
in
crea
ses
80¢
for
each
on
e-sq
uar
e-fo
ot i
ncr
ease
in
sto
rage
spa
ce.
b.
Fin
d a
pre
dic
tion
eq
uat
ion
.U
sin
g th
e sl
ope
and
one
of t
he
poin
ts o
n t
he
lin
e,yo
u c
an u
se t
he
poin
t-sl
ope
form
to
fin
da
pred
icti
on e
quat
ion
.
y�
y 1�
m(x
�x 1
)P
oint
-slo
pe f
orm
y�
60 �
0.8(
x�
200)
(x1,
y1)
�(2
00,
60),
m�
0.8
y�
60 �
0.8x
�16
0D
istr
ibut
ive
Pro
pert
y
y�
0.8x
�10
0A
dd 6
0 to
bot
h si
des.
A p
redi
ctio
n e
quat
ion
is
y�
0.8x
�10
0.
SALA
RIE
ST
he
tab
le b
elow
sh
ows
the
year
s of
exp
erie
nce
for
eig
ht
tech
nic
ian
s at
Lew
is T
ech
omat
ic a
nd
th
e h
ourl
y ra
te o
f p
ay e
ach
tec
hn
icia
n e
arn
s.U
se t
he
dat
afo
r E
xerc
ises
1 a
nd
2.
Exp
erie
nce
(ye
ars)
94
31
106
128
Ho
url
y R
ate
of
Pay
$17
$10
$10
$7$1
9$1
2$2
0$1
5
1.D
raw
a s
catt
er p
lot
to s
how
how
yea
rs o
f ex
peri
ence
are
re
late
d to
hou
rly
rate
of
pay.
Dra
w a
lin
e of
fit
.S
ee g
rap
h.
2.W
rite
a p
redi
ctio
n e
quat
ion
to
show
how
yea
rs o
f ex
peri
ence
(x)
are
rela
ted
to h
ourl
y ra
te o
f pa
y (y
).S
amp
le a
nsw
eru
sin
g (
1,7)
an
d (
9,17
):y
�1.
25x
�5.
75
Exp
erie
nce
(ye
ars)
Hourly Pay ($)
20
610
48
1214
24 20 16 12 8 4
Tech
nic
ian
Sala
ries
100
� 125
160
�60
��
325
�20
0y 2
�y 1
� x 2�
x 1
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Mo
del
ing
Rea
l-W
orl
d D
ata:
Usi
ng
Sca
tter
Plo
ts
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-5
2-5
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A15 Glencoe Algebra 2
An
swer
s
Answers (Lesson 2-5)
Skil
ls P
ract
ice
Mo
del
ing
Rea
l-W
orl
d D
ata:
Usi
ng
Sca
tter
Plo
ts
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-5
2-5
©G
lenc
oe/M
cGra
w-H
ill83
Gle
ncoe
Alg
ebra
2
Lesson 2-5
For
Exe
rcis
es 1
–3,c
omp
lete
par
ts a
–c f
or e
ach
set
of
dat
a.
a.D
raw
a s
catt
er p
lot.
b.
Use
tw
o or
der
ed p
airs
to
wri
te a
pre
dic
tion
eq
uat
ion
.c.
Use
you
r p
red
icti
on e
qu
atio
n t
o p
red
ict
the
mis
sin
g va
lue.
1.1a
.
1b.
Sam
ple
an
swer
usi
ng
(1,
1) a
nd
(8,
15):
y�
2x�
11c
.S
amp
le a
nsw
er:
19
2.2a
.
2b.
Sam
ple
an
swer
usi
ng
(5,
9) a
nd
(40
,44)
:y
�x
�4
2c.
Sam
ple
an
swer
:54
3.3a
.
3b.
Sam
ple
an
swer
usi
ng
(2,
16)
and
(7,
34):
y�
3.6x
�8.
83c
.S
amp
le a
nsw
er:
19.6
13
57
24
68
36 30 24 18 12 6 0x
yx
y
116
216
3?
422
530
734
836
515
2535
1020
3040
40 32 24 16 8 0x
yx
y
59
1017
2022
2530
3538
4044
50?
13
57
24
68
15 12 9 6 3 0x
yx
y
11
35
47
611
712
815
10?
©G
lenc
oe/M
cGra
w-H
ill84
Gle
ncoe
Alg
ebra
2
For
Exe
rcis
es 1
–3,c
omp
lete
par
ts a
–c f
or e
ach
set
of
dat
a.a.
Dra
w a
sca
tter
plo
t.b
.U
se t
wo
ord
ered
pai
rs t
o w
rite
a p
red
icti
on e
qu
atio
n.
c.U
se y
our
pre
dic
tion
eq
uat
ion
to
pre
dic
t th
e m
issi
ng
valu
e.
1.FU
EL E
CO
NO
MY
Th
e ta
ble
give
s th
e ap
prox
imat
e w
eigh
ts i
n t
ons
and
esti
mat
es
for
over
all
fuel
eco
nom
y in
mil
es p
er g
allo
n
for
seve
ral
cars
.1b
.Sam
ple
an
swer
usi
ng
(1.
4,24
) an
d
(2.4
,15)
:y
��
9x�
36.6
1c.S
amp
le a
nsw
er:
18.6
mi/g
al
2.A
LTIT
UD
EIn
mos
t ca
ses,
tem
pera
ture
dec
reas
es w
ith
in
crea
sin
g al
titu
de.A
s A
nch
ara
driv
es i
nto
th
e m
oun
tain
s,h
er c
ar t
her
mom
eter
reg
iste
rs t
he
tem
pera
ture
s (°
F)
show
nin
th
e ta
ble
at t
he
give
n a
ltit
ude
s (f
eet)
.
2b.S
amp
le a
nsw
er u
sin
g (
7500
,61)
an
d
(970
0,50
):y
��
0.00
5x�
98.5
2c.S
amp
le a
nsw
er:
38.5
°F
3.H
EALT
HA
lton
has
a t
read
mil
l th
at u
ses
the
tim
e on
th
e tr
eadm
ill
and
the
spee
d of
w
alki
ng
or r
un
nin
g to
est
imat
e th
e n
um
ber
of C
alor
ies
he
burn
s du
rin
g a
wor
kou
t.T
he
tabl
e gi
ves
wor
kou
t ti
mes
an
d C
alor
ies
burn
ed f
or s
ever
al w
orko
uts
.
3b.S
amp
le a
nsw
er u
sin
g (
24,2
80)
and
(48,
440)
:y
�6.
67x
�11
9.92
3c.S
amp
le a
nsw
er:
abo
ut
520
calo
ries
Tim
e (m
in)
Calories Burned
010
2030
4050
555
1525
3545
500
400
300
200
100
Bu
rnin
g C
alo
ries
Tim
e (m
in)
1824
3040
4248
5260
Cal
ori
es B
urn
ed26
028
032
038
040
044
047
5?
Alt
itu
de
(ft)
Temperature (�F)
07,
000
8,00
09,
000
10,0
00
65 60 55 50 45
Tem
pera
ture
Vers
us
Alt
itu
de
Alt
itu
de
(ft)
7500
8200
8600
9200
9700
10,4
0012
,000
Tem
per
atu
re (
�F)
6158
5653
5046
?
Wei
gh
t (t
on
s)
Fuel Economy (mi/gal)
00.
51.
01.
52.
02.
5
30 25 20 15 10 5Fuel
Eco
no
my V
ers
us
Weig
ht
Wei
gh
t (t
on
s)1.
31.
41.
51.
82
2.1
2.4
Mile
s p
er G
allo
n29
2423
21?
1715
Pra
ctic
e (
Ave
rag
e)
Mo
del
ing
Rea
l-W
orl
d D
ata:
Usi
ng
Sca
tter
Plo
ts
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-5
2-5
© Glencoe/McGraw-Hill A16 Glencoe Algebra 2
Answers (Lesson 2-5)
Readin
g t
o L
earn
Math
em
ati
csM
od
elin
g R
eal-
Wo
rld
Dat
a:U
sin
g S
catt
er P
lots
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-5
2-5
©G
lenc
oe/M
cGra
w-H
ill85
Gle
ncoe
Alg
ebra
2
Lesson 2-5
Pre-
Act
ivit
yH
ow c
an a
lin
ear
equ
atio
n m
odel
th
e n
um
ber
of
Cal
orie
s yo
u b
urn
exer
cisi
ng?
Rea
d th
e in
trod
uct
ion
to
Les
son
2-5
at
the
top
of p
age
81 i
n y
our
text
book
.
•If
a w
oman
ru
ns
5.5
mil
es p
er h
our,
abou
t h
ow m
any
Cal
orie
s w
ill
she
burn
in
an
hou
r?S
amp
le a
nsw
er:
572
Cal
ori
es
•If
a m
an r
un
s 7.
5 m
iles
per
hou
r,ab
out
how
man
y C
alor
ies
wil
l h
e bu
rnin
hal
f an
hou
r?S
amp
le a
nsw
er:
397
Cal
ori
es
Rea
din
g t
he
Less
on
1.S
upp
ose
that
a s
et o
f da
ta c
an b
e m
odel
ed b
y a
lin
ear
equ
atio
n.E
xpla
in t
he
diff
eren
cebe
twee
n a
sca
tter
plo
t of
th
e da
ta a
nd
a gr
aph
of
the
lin
ear
equ
atio
n t
hat
mod
els
that
data
.S
amp
le a
nsw
er:T
he
scat
ter
plo
t is
a d
iscr
ete
gra
ph
.It
is m
ade
up
just
of
the
ind
ivid
ual
po
ints
th
at r
epre
sen
t th
e d
ata
po
ints
.Th
e lin
ear
equ
atio
nh
as a
co
nti
nu
ou
s g
rap
h t
hat
is t
he
line
that
bes
t fi
ts t
he
dat
a p
oin
ts.
2.S
upp
ose
that
tu
itio
n a
t a
stat
e co
lleg
e w
as $
3500
per
yea
r in
199
5 an
d h
as b
een
incr
easi
ng
at a
rat
e of
$22
5 pe
r ye
ar.
a.W
rite
a p
redi
ctio
n e
quat
ion
th
at e
xpre
sses
th
is i
nfo
rmat
ion
.y
�35
00 �
225x
b.
Exp
lain
th
e m
ean
ing
of e
ach
var
iabl
e in
you
r pr
edic
tion
equ
atio
n.
xre
pre
sen
ts t
he
nu
mb
er o
f ye
ar s
ince
199
5 an
d y
rep
rese
nts
th
etu
itio
n in
th
at y
ear.
3.U
se t
his
mod
el t
o pr
edic
t th
e tu
itio
n a
t th
is c
olle
ge i
n 2
007.
$620
0
Hel
pin
g Y
ou
Rem
emb
er
4.L
ook
up
the
wor
d sc
atte
rin
a d
icti
onar
y.H
ow c
an i
ts d
efin
itio
n h
elp
you
to
rem
embe
rth
e m
ean
ing
of t
he
diff
eren
ce b
etw
een
a s
catt
er p
lot
and
the
grap
h o
f a
lin
ear
equ
atio
n?
Sam
ple
an
swer
:To
sca
tter
mea
ns
to b
reak
up
an
d g
o in
man
y d
irec
tio
ns.
Th
e p
oin
ts o
n a
sca
tter
plo
t ar
e b
roke
n u
p.I
n a
sca
tter
plo
t,th
e p
oin
tsar
e sc
atte
red
or
bro
ken
up
.In
th
e g
rap
h o
f a
linea
r eq
uat
ion
,th
e p
oin
tsar
e co
nn
ecte
d t
o f
orm
a c
on
tin
uo
us
line.
©G
lenc
oe/M
cGra
w-H
ill86
Gle
ncoe
Alg
ebra
2
Med
ian
-Fit
Lin
es
A m
edia
n-f
it li
ne
is a
par
ticu
lar
type
of
lin
e of
fit
.Fol
low
th
e st
eps
belo
w t
o fi
nd
the
equ
atio
n o
f th
e m
edia
n-f
it l
ine
for
the
data
.
Ap
pro
xim
ate
Per
cen
tag
e o
f Vio
len
t C
rim
es C
om
mit
ted
by
Juve
nile
s T
hat
Vic
tim
s R
epo
rted
to
Law
En
forc
emen
t
Year
1980
1982
1984
1986
1988
1990
1992
1994
1996
Off
end
ers
3635
3332
3130
2929
30
Sour
ce: U
.S. B
urea
u of
Justi
ce S
tatis
tics
1.D
ivid
e th
e da
ta i
nto
th
ree
appr
oxim
atel
y eq
ual
gro
ups
.Th
ere
shou
ld a
lway
s be
th
e sa
me
nu
mbe
r of
poi
nts
in
th
e fi
rst
and
thir
d gr
oups
.In
th
is c
ase,
ther
e w
ill
be t
hre
e da
ta p
oin
ts i
n e
ach
gro
up.
Gro
up
1G
rou
p 2
Gro
up
3Ye
arO
ffen
der
sYe
arO
ffen
der
sYe
arO
ffen
der
s
2.F
ind
x 1,x
2,an
d x 3
,th
e m
edia
ns
of t
he
xva
lues
in
gro
ups
1,2
,an
d 3,
resp
ecti
vely
.Fin
d y 1
,y2,
and
y 3,t
he
med
ian
s of
th
e y
valu
es i
n g
rou
ps
1,2,
and
3,re
spec
tive
ly.
1982
,198
8,19
94;
35,3
1,29
3.F
ind
an e
quat
ion
of
the
lin
e th
rou
gh (
x 1,y
1) a
nd
(x3,
y 3).
y�
�0.
5x�
1026
4.F
ind
Y,t
he
y-co
ordi
nat
e of
th
e po
int
on t
he
lin
e in
Exe
rcis
e 2
wit
h a
n
x-co
ordi
nat
e of
x2.
32
5.T
he
med
ian
-fit
lin
e is
par
alle
l to
th
e li
ne
in E
xerc
ise
2,bu
t is
on
e-th
ird
clos
er t
o (x
2,y 2
).T
his
mea
ns
it p
asse
s th
rou
gh �x
2,�2 3� Y
��1 3�
y 2�.F
ind
this
or
dere
d pa
ir. a
bo
ut
(198
8,31
.67)
6.W
rite
an
equ
atio
n o
f th
e m
edia
n-f
it l
ine.
y�
�0.
5x�
1025
.67
7.U
se t
he
med
ian
-fit
lin
e to
pre
dict
th
e pe
rcen
tage
of
juve
nil
e vi
olen
t cr
ime
offe
nde
rs i
n 2
010
and
2020
.20
10:
abo
ut
21%
;20
20:
abo
ut1
6%
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-5
2-5
© Glencoe/McGraw-Hill A17 Glencoe Algebra 2
An
swer
s
Answers (Lesson 2-6)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Sp
ecia
l Fu
nct
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-6
2-6
©G
lenc
oe/M
cGra
w-H
ill87
Gle
ncoe
Alg
ebra
2
Lesson 2-6
Step
Fu
nct
ion
s, C
on
stan
t Fu
nct
ion
s, a
nd
th
e Id
enti
ty F
un
ctio
nT
he
char
tbe
low
lis
ts s
ome
spec
ial
fun
ctio
ns
you
sh
ould
be
fam
ilia
r w
ith
.
Fu
nct
ion
Wri
tten
as
Gra
ph
Con
stan
tf(
x) �
cho
rizon
tal l
ine
Iden
tity
f(x)
�x
line
thro
ugh
the
orig
in w
ith s
lope
1
Gre
ates
t In
tege
r F
unct
ion
f(x)
��x
�on
e-un
it ho
rizon
tal s
egm
ents
, w
ith r
ight
end
poin
ts m
issi
ng,
arra
nged
lik
e st
eps
Th
e gr
eate
st i
nte
ger
fun
ctio
n i
s an
exa
mpl
e of
a s
tep
fu
nct
ion
,a f
un
ctio
n w
ith
a g
raph
th
atco
nsi
sts
of h
oriz
onta
l se
gmen
ts.
Iden
tify
eac
h f
un
ctio
n a
s a
con
stan
t fu
nct
ion
,th
e id
enti
ty f
un
ctio
n,
or a
ste
p f
un
ctio
n.
a.b
.
a co
nst
ant
fun
ctio
na
step
fu
nct
ion
Iden
tify
eac
h f
un
ctio
n a
s a
con
stan
t fu
nct
ion
,th
e id
enti
ty f
un
ctio
n,a
gre
ates
tin
tege
r fu
nct
ion
,or
a st
ep f
un
ctio
n.
1.2.
3.
a co
nst
ant
fun
ctio
na
step
fu
nct
ion
the
iden
tity
fu
nct
ionx
f (x)
Ox
f (x)
Ox
f (x)
O
x
f (x)
Ox
f (x)
O
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill88
Gle
ncoe
Alg
ebra
2
Ab
solu
te V
alu
e an
d P
iece
wis
e Fu
nct
ion
sA
not
her
spe
cial
fu
nct
ion
is
the
abso
lute
val
ue
fun
ctio
n,w
hic
h i
s al
so c
alle
d a
pie
cew
ise
fun
ctio
n.
Ab
solu
te V
alu
e F
un
ctio
nf(
x)
�x
tw
o ra
ys th
at a
re m
irror
imag
es o
f eac
h ot
her
and
mee
t at a
poi
nt, t
he v
erte
x
To
grap
h a
spe
cial
fu
nct
ion
,use
its
def
init
ion
an
d yo
ur
know
ledg
e of
th
e pa
ren
t gr
aph
.Fin
dse
vera
l or
dere
d pa
irs,
if n
eces
sary
.
Gra
ph
f(x
) �
3x
�4.
Fin
d se
vera
l or
dere
d pa
irs.
Gra
ph t
he
poin
ts a
nd
con
nec
t th
em.Y
ou w
ould
exp
ect
the
grap
h t
o lo
oksi
mil
ar t
o it
s pa
ren
t fu
nct
ion
,f(x
) �
x.
Gra
ph
f(x
) �
�2xif
x�
2x
�1
if x
�2.
Fir
st,g
raph
th
e li
nea
r fu
nct
ion
f(x
) �
2xfo
r x
�2.
Sin
ce 2
doe
s n
otsa
tisf
y th
is i
neq
ual
ity,
stop
wit
h a
cir
cle
at (
2,4)
.Nex
t,gr
aph
th
eli
nea
r fu
nct
ion
f(x
) �
x�
1 fo
r x
�2.
Sin
ce 2
doe
s sa
tisf
y th
isin
equ
alit
y,be
gin
wit
h a
dot
at
(2,1
).
Gra
ph
eac
h f
un
ctio
n.I
den
tify
th
e d
omai
n a
nd
ran
ge.
1.g(
x) �
��2.
h(x
) �
2x
�1
3.h
(x)
�
do
mai
n:
all r
eal
do
mai
n:
all r
eal
do
mai
n:
all r
eal
nu
mb
ers;
ran
ge:
nu
mb
ers;
ran
ge:
nu
mb
ers;
ran
ge:
all i
nte
ger
s{y
y�
0}{y
y
1}
x
y
O
x
y
O
x
y
Ox � 3
x
f (x)
O
x
f (x)
O
x3
x�
4
0�
4
1�
1
22
�1
�1
�2
2
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Sp
ecia
l Fu
nct
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-6
2-6
Exer
cises
Exer
cises
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
if x
0
2x�
6 if
0 �
x�
21
if x
�2
x � 3
© Glencoe/McGraw-Hill A18 Glencoe Algebra 2
Answers (Lesson 2-6)
Skil
ls P
ract
ice
Sp
ecia
l Fu
nct
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-6
2-6
©G
lenc
oe/M
cGra
w-H
ill89
Gle
ncoe
Alg
ebra
2
Lesson 2-6
Iden
tify
eac
h f
un
ctio
n a
s S
for
ste
p,C
for
con
stan
t,A
for
ab
solu
te v
alu
e,or
P f
orp
iece
wis
e.
1.2.
3.
SC
A
Gra
ph
eac
h f
un
ctio
n.I
den
tify
th
e d
omai
n a
nd
ran
ge.
4.f(
x) �
�x�
1�5.
f(x)
��x
�3�
D �
all r
eals
,R �
all i
nte
ger
sD
�al
l rea
ls,R
�al
l in
teg
ers
6.g(
x) �
2x
7.f(
x) �
x
�1
D �
all r
eals
,D
�al
l rea
ls,R
�{y
y�
1}R
�n
on
neg
ativ
e re
als
8.f(
x) �
�xif
x�
09.
h(x
) �
�3 if
x�
�1
2 if
x�
0x
�1
if x
> 1
D �
all r
eals
,D
�{x
x�
�1
or
x
1},
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y
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x
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O
x
f (x)
O
x
f (x)
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g (x)
O
x
f (x)
O
x
f (x)
O
x
y
O
x
y
Ox
y
O
©G
lenc
oe/M
cGra
w-H
ill90
Gle
ncoe
Alg
ebra
2
Gra
ph
eac
h f
un
ctio
n.I
den
tify
th
e d
omai
n a
nd
ran
ge.
1.f(
x) �
�0.5
x�2.
f(x)
��x
��
2
D �
all r
eals
,R �
all i
nte
ger
sD
�al
l rea
ls,R
�al
l in
teg
ers
3.g(
x) �
�2
x4.
f(x)
�x
�1
D �
all r
eals
,D
�al
l rea
ls,
R �
no
np
osi
tive
rea
ls
R �
no
nn
egat
ive
real
s
5.f(
x) �
�x�
2 if
x
�2
6.h
(x)
��4
�x
if x
0
3xif
x
�2
�2x
�2
if x
�0
D �
all r
eals
,R �
all r
eals
D �
all n
on
zero
rea
ls,R
�al
l rea
ls7.
BU
SIN
ESS
A S
titc
h i
n T
ime
char
ges
8.
BU
SIN
ESS
A w
hol
esal
er c
har
ges
a st
ore
$3.0
0
$40
per
hou
r or
an
y fr
acti
on t
her
eof
per
poun
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r le
ss t
han
20 p
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can
dy a
ndfo
r la
bor.
Dra
w a
gra
ph o
f th
e st
ep
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r po
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d fo
r 20
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raw
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nct
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th
at r
epre
sen
ts t
his
sit
uat
ion
.gr
aph
of
the
fun
ctio
n t
hat
rep
rese
nts
th
issi
tuat
ion
.
Ho
urs
Total Cost ($)
10
35
24
67
280
240
200
160
120 80 40
Lab
or
Co
sts
Pou
nd
s
Cost ($)
50
1525
1020
3035
105 90 75 60 45 30 15
Can
dy C
ost
s
x
h (x)
O
f(x)
xO
x
f (x)
O
x
g (x)
O
x
f (x)
Ox
f (x)
OPra
ctic
e (
Ave
rag
e)
Sp
ecia
l Fu
nct
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-6
2-6
© Glencoe/McGraw-Hill A19 Glencoe Algebra 2
An
swer
s
Answers (Lesson 2-6)
Readin
g t
o L
earn
Math
em
ati
csS
pec
ial F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-6
2-6
©G
lenc
oe/M
cGra
w-H
ill91
Gle
ncoe
Alg
ebra
2
Lesson 2-6
Pre-
Act
ivit
yH
ow d
o st
ep f
un
ctio
ns
app
ly t
o p
osta
ge r
ates
?
Rea
d th
e in
trod
uct
ion
to
Les
son
2-6
at
the
top
of p
age
89 i
n y
our
text
book
.
•W
hat
is
the
cost
of
mai
lin
g a
lett
er t
hat
wei
ghs
0.5
oun
ce?
$0.3
4 o
r 34
cen
ts•
Giv
e th
ree
diff
eren
t w
eigh
ts o
f le
tter
s th
at w
ould
eac
h c
ost
55 c
ents
to
mai
l.A
nsw
ers
will
var
y.S
amp
le a
nsw
er:
1.1
ou
nce
s,1.
9 o
un
ces,
2.0
ou
nce
s
Rea
din
g t
he
Less
on
1.F
ind
the
valu
e of
eac
h e
xpre
ssio
n.
a.�
3�
��3�
�
b.
6.2
�
�6.2
��
c.�
4.01
�
��4.
01�
�
2.T
ell
how
th
e n
ame
of e
ach
kin
d of
fu
nct
ion
can
hel
p yo
u r
emem
ber
wh
at t
he
grap
h
look
s li
ke.
a.co
nst
ant
fun
ctio
nS
amp
le a
nsw
er:
So
met
hin
g is
co
nst
ant
if it
do
es n
ot
chan
ge.
Th
e y-
valu
es o
f a
con
stan
t fu
nct
ion
do
no
t ch
ang
e,so
th
eg
rap
h is
a h
ori
zon
tal l
ine.
b.
abso
lute
val
ue
fun
ctio
nS
amp
le a
nsw
er:T
he
abso
lute
val
ue
of
a n
um
ber
tells
yo
u h
ow
far
it is
fro
m 0
on
th
e n
um
ber
lin
e.It
mak
es n
o d
iffe
ren
cew
het
her
yo
u g
o t
o t
he
left
or
rig
ht
so lo
ng
as
you
go
th
e sa
me
dis
tan
ce e
ach
tim
e.
c.st
ep f
un
ctio
nS
amp
le a
nsw
er:
A s
tep
fu
nct
ion
’s g
rap
h lo
oks
like
ste
ps
that
go
up
or
do
wn
.
d.
iden
tity
fu
nct
ion
Sam
ple
an
swer
:Th
e x-
an
d y
-val
ues
are
alw
ays
iden
tica
lly t
he
sam
e fo
r an
y p
oin
t o
n t
he
gra
ph
.So
th
e g
rap
h is
a li
ne
thro
ug
h t
he
ori
gin
th
at h
as s
lop
e 1.
Hel
pin
g Y
ou
Rem
emb
er
3.M
any
stu
den
ts f
ind
the
grea
test
in
tege
r fu
nct
ion
con
fusi
ng.
Exp
lain
how
you
can
use
an
um
ber
lin
e to
fin
d th
e va
lue
of t
his
fu
nct
ion
for
an
y re
al n
um
ber.
An
swer
s w
ill v
ary.
Sam
ple
an
swer
:D
raw
a n
um
ber
lin
e th
at s
ho
ws
the
inte
ger
s.To
fin
d t
he
valu
e o
f th
e g
reat
est
inte
ger
fu
nct
ion
fo
r an
y re
al n
um
ber
,pla
ce t
hat
nu
mb
er o
n t
he
nu
mb
er li
ne.
If it
is a
n in
teg
er,t
he
valu
e o
f th
e fu
nct
ion
isth
e n
um
ber
itse
lf.I
f n
ot,
mov
e to
th
e in
teg
er d
irec
tly
to t
he
left
of
the
nu
mb
er y
ou
ch
ose
.Th
is in
teg
er w
ill g
ive
the
valu
e yo
u n
eed
.
�5
4.01
66.
2
�3
3
©G
lenc
oe/M
cGra
w-H
ill92
Gle
ncoe
Alg
ebra
2
Gre
ates
t In
teg
er F
un
ctio
ns
Use
th
e gr
eate
st i
nte
ger
fun
ctio
n �
x�to
exp
lore
som
e u
nu
sual
gra
phs.
It w
ill
be h
elpf
ul
to m
ake
a ch
art
of v
alu
es f
or e
ach
fu
nct
ion
s an
d to
use
a c
olor
ed
pen
or
pen
cil.
Gra
ph
eac
h f
un
ctio
n.
1.y
�2x
��x
�2.
y�
�� �x x� ��
3.y
��� �0 0. .5 5x x
� �
1 1� ��
4.y
�� �x x��
x
y
O1
–1–2
–3–4
23
4
4 3 2 1 –1 –2 –3 –4
x
y
O1
–1–2
–3–4
23
4
4 3 2 1 –1 –2 –3 –4
x
y
O1
–1–2
–3–4
23
4
4 3 2 1 –1 –2 –3 –4
x
y
O1
–1–2
–3–4
23
4
4 3 2 1 –1 –2 –3 –4
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-6
2-6
© Glencoe/McGraw-Hill A20 Glencoe Algebra 2
Answers (Lesson 2-7)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Gra
ph
ing
Ineq
ual
itie
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-7
2-7
©G
lenc
oe/M
cGra
w-H
ill93
Gle
ncoe
Alg
ebra
2
Lesson 2-7
Gra
ph
Lin
ear
Ineq
ual
itie
s.A
lin
ear
ineq
ual
ity,
like
y�
2x�
1,re
sem
bles
a l
inea
req
uat
ion
,bu
t w
ith
an
in
equ
alit
y si
gn i
nst
ead
of a
n e
qual
s si
gn.T
he
grap
h o
f th
e re
late
dli
nea
r eq
uat
ion
sep
arat
es t
he
coor
din
ate
plan
e in
to t
wo
hal
f-pl
anes
.Th
e li
ne
is t
he
bou
nda
ry o
f ea
ch h
alf-
plan
e.
To
grap
h a
lin
ear
ineq
ual
ity,
foll
ow t
hes
e st
eps.
1.G
raph
th
e bo
un
dary
,th
at i
s,th
e re
late
d li
nea
r eq
uat
ion
.If
the
ineq
ual
ity
sym
bol
is
or
�,t
he
bou
nda
ry i
s so
lid.
If t
he
ineq
ual
ity
sym
bol
is �
or
,th
e bo
un
dary
is
dash
ed.
2.C
hoo
se a
poi
nt
not
on
th
e bo
un
dary
an
d te
st i
t in
th
e in
equ
alit
y.(0
,0)
is a
goo
d po
int
toch
oose
if
the
bou
nda
ry d
oes
not
pas
s th
rou
gh t
he
orig
in.
3.If
a t
rue
ineq
ual
ity
resu
lts,
shad
e th
e h
alf-
plan
e co
nta
inin
g yo
ur
test
poi
nt.
If a
fal
sein
equ
alit
y re
sult
s,sh
ade
the
oth
er h
alf-
plan
e.
Gra
ph
x�
2y�
4.
Th
e bo
un
dary
is
the
grap
h o
f x
�2y
�4.
Use
th
e sl
ope-
inte
rcep
t fo
rm,y
��
x�
2,to
gra
ph t
he
bou
nda
ry l
ine.
Th
e bo
un
dary
lin
e sh
ould
be
soli
d.
Now
tes
t th
e po
int
(0,0
).
0 �
2(0)
�?
4(x
, y
) �
(0,
0)
0 �
4fa
lse
Sh
ade
the
regi
on t
hat
doe
s n
otco
nta
in (
0,0)
.
Gra
ph
eac
h i
neq
ual
ity.
1.y
�3x
�1
2.y
�x
�5
3.4x
�y
�
1
4.y
��
45.
x�
y
66.
0.5x
�0.
25y
�1.
5
x
y
O
x
y
O
x
y
O
x � 2
x
y
O
x
y
O
x
y
O
1 � 2x
y O
Exer
cises
Exer
cises
Exam
ple
Exam
ple
©G
lenc
oe/M
cGra
w-H
ill94
Gle
ncoe
Alg
ebra
2
Gra
ph
Ab
solu
te V
alu
e In
equ
alit
ies
Gra
phin
g ab
solu
te v
alu
e in
equ
alit
ies
is s
imil
arto
gra
phin
g li
nea
r in
equ
alit
ies.
Th
e gr
aph
of
the
rela
ted
abso
lute
val
ue
equ
atio
n i
s th
ebo
un
dary
.Th
is b
oun
dary
is
grap
hed
as
a so
lid
lin
e if
th
e in
equ
alit
y is
or
�,a
nd
dash
ed i
fth
e in
equ
alit
y is
�or
.C
hoo
se a
tes
t po
int
not
on
th
e bo
un
dary
to
dete
rmin
e w
hic
h r
egio
nto
sh
ade.
Gra
ph
y
3x
�1
.
Fir
st g
raph
th
e eq
uat
ion
y�
3x
�1
.S
ince
th
e in
equ
alit
y is
,t
he
grap
h o
f th
e bo
un
dary
is
soli
d.T
est
(0,0
).0
?3
0 �
1(x
, y)
�(0
, 0)
0 ?
3�
1�
1�
1
0
3tr
ue
Sh
ade
the
regi
on t
hat
con
tain
s (0
,0).
Gra
ph
eac
h i
neq
ual
ity.
1.y
�x
�
12.
y
2x
�1
3.y
�2
x
3
4.y
��
x
�3
5.x
�
y�
46.
x�
1�
2y�
0
7.2
�x
�y
�
18.
y�
3x
�3
9.y
1
�x
�4 x
y
O
x
y
O
x
y
O
x
y
O
x
y
O
x
y
O
x
y
Ox
y
Ox
y
O
x
y
O
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Gra
ph
ing
Ineq
ual
itie
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-7
2-7
Exer
cises
Exer
cises
Exam
ple
Exam
ple
© Glencoe/McGraw-Hill A21 Glencoe Algebra 2
An
swer
s
Answers (Lesson 2-7)
Skil
ls P
ract
ice
Gra
ph
ing
Ineq
ual
itie
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-7
2-7
©G
lenc
oe/M
cGra
w-H
ill95
Gle
ncoe
Alg
ebra
2
Lesson 2-7
Gra
ph
eac
h i
neq
ual
ity.
1.y
�1
2.y
�x
�2
3.x
�y
�4
4.x
�3
�y
5.2
�y
�x
6.y
��
x
7.x
�y
��
28.
9x�
3y�
6 �
09.
y�
1 �
2x
10.y
�7
��
911
.x�
�5
12.y
�x
x
y
Ox
y
Ox
y
O
x
y
Ox
y
Ox
y
O
x
y
Ox
y
O
x
y
O
x
y
O
x
y
Ox
y
O
©G
lenc
oe/M
cGra
w-H
ill96
Gle
ncoe
Alg
ebra
2
Gra
ph
eac
h i
neq
ual
ity.
1.y
��
32.
x�
23.
x�
y�
�4
4.y
��
3x�
55.
y�
x�
36.
y�
1 �
�x
7.x
�3y
�6
8.y
�x
�
19.
y�
�3
x�
1�
2
CO
MPU
TER
SF
or E
xerc
ises
10–
12,u
se t
he
foll
owin
g in
form
atio
n.
A s
choo
l sy
stem
is
buyi
ng
new
com
pute
rs.T
hey
wil
l bu
y de
skto
p co
mpu
ters
cos
tin
g $1
000
per
un
it,a
nd
not
eboo
k co
mpu
ters
cos
tin
g $1
200
per
un
it.T
he
tota
l co
st o
f th
e co
mpu
ters
can
not
exc
eed
$80,
000.
10.W
rite
an
in
equ
alit
y th
at d
escr
ibes
th
is s
itu
atio
n.
1000
d�
1200
n�
80,0
00
11.G
raph
th
e in
equ
alit
y.
12.I
f th
e sc
hoo
l w
ants
to
buy
50 o
f th
e de
skto
p co
mpu
ters
an
d 25
of
the
not
eboo
k co
mpu
ters
,w
ill
they
hav
e en
ough
mon
ey?
yes
Des
kto
ps
Notebooks
100
3050
2040
6070
8090
100
80 70 60 50 40 30 20 10
Co
mp
ute
rs P
urc
hase
d
x
y
O
x
y
Ox
y O
x
y
Ox
y
O
x
y
O
1 � 2
x
y
O
x
y
O
x
y
OPra
ctic
e
Gra
ph
ing
Ineq
ual
itie
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-7
2-7
© Glencoe/McGraw-Hill A22 Glencoe Algebra 2
Answers (Lesson 2-7)
Readin
g t
o L
earn
Math
em
ati
csG
rap
hin
g In
equ
alit
ies
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-7
2-7
©G
lenc
oe/M
cGra
w-H
ill97
Gle
ncoe
Alg
ebra
2
Lesson 2-7
Pre-
Act
ivit
yH
ow d
o in
equ
alit
ies
app
ly t
o fa
nta
sy f
ootb
all?
Rea
d th
e in
trod
uct
ion
to
Les
son
2-7
at
the
top
of p
age
96 i
n y
our
text
book
.
•W
hic
h o
f th
e co
mbi
nat
ion
s of
yar
ds a
nd
tou
chdo
wn
s li
sted
wou
ld D
ana
con
side
r a
good
gam
e?T
he
firs
t o
ne:
168
yard
s an
d
3 to
uch
do
wn
s•
Su
ppos
e th
at i
n o
ne
of t
he
gam
es D
ana
play
s,M
oss
gets
157
rec
eivi
ng
yard
s.W
hat
is
the
smal
lest
nu
mbe
r of
tou
chdo
wn
s h
e m
ust
get
in
ord
erfo
r D
ana
to c
onsi
der
this
a g
ood
gam
e?3
Rea
din
g t
he
Less
on
1.W
hen
gra
phin
g a
lin
ear
ineq
ual
ity
in t
wo
vari
able
s,h
ow d
o yo
u k
now
wh
eth
er t
o m
ake
the
bou
nda
ry a
sol
id l
ine
or a
das
hed
lin
e? If
th
e sy
mb
ol i
s �
or
,t
he
line
isso
lid.
If t
he
sym
bo
l is
o
r �
,th
e lin
e is
das
hed
.
2.H
ow d
o yo
u k
now
wh
ich
sid
e of
th
e bo
un
dary
to
shad
e?S
amp
le a
nsw
er:
If t
he
test
po
int
giv
es a
tru
e in
equ
alit
y,sh
ade
the
reg
ion
co
nta
inin
g t
he
test
po
int.
Ifth
e te
st p
oin
t g
ives
a f
alse
ineq
ual
ity,
shad
e th
e re
gio
n n
ot
con
tain
ing
the
test
po
int.
3.M
atch
eac
h i
neq
ual
ity
wit
h i
ts g
raph
.
a.y
2x
�3
iiib
.y
��
2x�
3iv
c.y
�2x
�3
iid
.y
��
2x�
3i
i.ii
.ii
i.iv
.
Hel
pin
g Y
ou R
emem
ber
4.D
escr
ibe
som
e w
ays
in w
hic
h g
raph
ing
an i
neq
ual
ity
in o
ne
vari
able
on
a n
um
ber
lin
e is
sim
ilar
to
grap
hin
g an
in
equ
alit
y in
tw
o va
riab
les
in a
coo
rdin
ate
plan
e.H
ow c
an w
hat
you
kn
ow a
bou
t gr
aph
ing
ineq
ual
itie
s on
a n
um
ber
lin
e h
elp
you
to
grap
h i
neq
ual
itie
s in
a co
ordi
nat
e pl
ane?
Sam
ple
an
swer
:A
bo
un
dar
y o
n a
co
ord
inat
e g
rap
h is
sim
ilar
to a
n e
nd
po
int
on
a n
um
ber
lin
e g
rap
h.A
das
hed
lin
e is
sim
ilar
toa
circ
le o
n a
nu
mb
er li
ne:
bo
th a
re o
pen
an
d m
ean
no
t in
clu
ded
;th
eyre
pre
sen
t th
e sy
mb
ols
an
d �
.A s
olid
lin
e is
sim
ilar
to a
do
t o
n a
nu
mb
er li
ne:
bo
th a
re c
lose
d a
nd
mea
n in
clu
ded
;th
ey r
epre
sen
t th
esy
mb
ols
�an
d
.
x
y
O
x
y
Ox
y
O
x
y
O
©G
lenc
oe/M
cGra
w-H
ill98
Gle
ncoe
Alg
ebra
2
Alg
ebra
ic P
roo
fT
he
foll
owin
g pa
ragr
aph
sta
tes
a re
sult
you
mig
ht
be a
sked
to
prov
e in
am
ath
emat
ics
cou
rse.
Par
ts o
f th
e pa
ragr
aph
are
nu
mbe
red.
01L
et n
be a
pos
itiv
e in
tege
r.
02A
lso,
let
n1
�s(
n1)
be
the
sum
of
the
squ
ares
of
the
digi
ts i
n n
.
03T
hen
n2
�s(
n1)
is
the
sum
of
the
squ
ares
of
the
digi
ts o
f n
1,an
d n
3�
s(n
2)is
th
e su
m o
f th
e sq
uar
es o
f th
e di
gits
of
n2.
04In
gen
eral
,nk
�s(
nk
�1)
is
the
sum
of
the
squ
ares
of
the
digi
ts o
f n
k�
1.
05C
onsi
der
the
sequ
ence
:n,n
1,n
2,n
3,…
,nk,
….
06In
th
is s
equ
ence
eit
her
all
th
e te
rms
from
som
e k
on h
ave
the
valu
e 1,
07or
som
e te
rm,s
ay n
j,h
as t
he
valu
e 4,
so t
hat
th
e ei
ght
term
s 4,
16,3
7,58
,89,
145,
42,a
nd
20 k
eep
repe
atin
g fr
om t
hat
poi
nt
on.
Use
th
e p
arag
rap
h t
o an
swer
th
ese
qu
esti
ons.
1.U
se t
he
sen
ten
ce i
n l
ine
01.L
ist
the
firs
t fi
ve v
alu
es o
f n
.1,
2,3,
4,5
2.U
se 9
246
for
nan
d gi
ve a
n e
xam
ple
to s
how
th
e m
ean
ing
of l
ine
02.
n1
�s
(924
6) �
137,
bec
ause
137
�81
�4
�16
�36
3.In
lin
e 02
,whi
ch s
ymbo
l sh
ows
a fu
ncti
on?
Exp
lain
the
fun
ctio
n in
a s
ente
nce.
s(n
);th
e su
m o
f th
e sq
uar
es o
f th
e d
igit
s o
f a
nu
mb
er is
a f
un
ctio
n
of
the
nu
mb
er
4.F
or n
�92
46,f
ind
n2
and
n3
as d
escr
ibed
in
sen
ten
ce 0
3.n
2�
59,n
3�
106
5.H
ow d
o th
e fi
rst
fou
r se
nte
nce
s re
late
to
sen
ten
ce 0
5?T
hey
exp
lain
ho
w t
o c
om
pu
te t
he
term
s o
f th
e se
qu
ence
.
6.U
se n
�31
an
d fi
nd
the
firs
t fo
ur
term
s of
th
e se
quen
ce.
31,1
0,1,
1
7.W
hic
h s
ente
nce
of
the
para
grap
h i
s il
lust
rate
d by
n�
31?
sen
ten
ce 0
6
8.U
se n
�61
an
d fi
nd
the
firs
t te
n t
erm
s.61
,37,
58,8
9,14
5,42
,20,
4,16
,37
9.W
hic
h s
ente
nce
is
illu
stra
ted
by n
�61
?se
nte
nce
07
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
2-7
2-7
© 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:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12. A
B
C
B
B
C
B
A
C
D
B
D
k � 10
D
A
C
B
D
C
D
D
B
A
B
C
D
A
B
B
D
C
C
C
Chapter 2 Assessment Answer KeyForm 1 Form 2APage 99 Page 100 Page 101
An
swer
s
(continued on the next page)
© Glencoe/McGraw-Hill A24 Glencoe Algebra 2
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: k � �16
C
C
B
A
B
D
C
A
D
D
B
C
D
A
D
B
A
C
D
C
k � 5
B
D
C
B
C
B
A
D
Chapter 2 Assessment Answer KeyForm 2A (continued) Form 2BPage 102 Page 103 Page 104
© Glencoe/McGraw-Hill A25 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B: k � �7
p
a80
91011121314151617
5 6 7 8 9 10 11 12 13 14
Poin
ts S
core
d
Field Goals Attempted
y � ��32
�x
y � 2x � 7
y
xO
�14
y
xO
y
xO
xO
f (x )
y
xO
yes
5a2 � 8a
�3
no
yes
D � {�3}; R � {0, 1, 2, 3}; no
y
xO
(�3, 1)
(�3, 0)
(�3, 2)(�3, 3)
Chapter 2 Assessment Answer KeyForm 2CPage 105 Page 106
An
swer
s
No, because a variable appears in the denominator.
5x � 16y � 18; A � 5,B � �16, C � 18
x-intercept is 3; y-intercept is �2
Sample answer using(6, 9) and (10, 15):
p � �32
�a; 30
step function; D � all reals, R � all integers
© Glencoe/McGraw-Hill A26 Glencoe Algebra 2
1.
D � {�4, 0, 2, 4}; R � {0, 4}; yes
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B: k � �3
t
n0
12141618202224262830
6 7 8 9 10 11 12 13 14 15
Tick
ets
Sold
Calls Made
y � ��52
�x
y � �x � 1
y
xO
�10
y
xO
y
xO
xO
f (x )
y
xO
yes
�4a2 � 2a � 3
14
yes
no
y
xO
(�4, 0)
(0, 0)
(4, 0)
(2, 4)
Chapter 2 Assessment Answer KeyForm 2DPage 107 Page 108
No, because the variablesare multiplied together.
2x � 56y � 1; A � 2, B � �56, C � 1x-intercept is �4; y-intercept is 3
Sample answer using (6, 12) and (8, 16):t � 2n; 32
step function; D � all reals, R � all integers
© Glencoe/McGraw-Hill A27 Glencoe Algebra 2
1.
D � {x � x � 1}; R � all reals; no
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
(160,150)
18.
19.
20.
B:k � �
35
�
y � �� x � 2 �
t
d0
20406080
100120140160
60 100 140 180 220
Tim
e (m
in)
Distance (km)
4x � 2y � 1
c
p
0048
1216202428
5 10 15 20 25 30 35
12p � 15c � 360
y � �4x � �23
�
y � ��25
�x � 2
$15.22 per year
5.6
y
xO
y
xO
absolute value function
x-intercept is �27
�;
no y-intercept
25x � 5y � 3;A � 25, B � �5, C � 3
A. yesB. no
3x � 5
�8
no
y
xO
Chapter 2 Assessment Answer KeyForm 3Page 109 Page 110
An
swer
s
Sample answer using
(40, 30) and (200, 150):
t � �34
�d; 120 min; much
lower
f (x) � ��2x if x � �1
0 if �1 � x � 2x if x � 2
© Glencoe/McGraw-Hill A28 Glencoe Algebra 2
Chapter 2 Assessment Answer KeyPage 111, Open-Ended Assessment
Scoring Rubric
Score General Description Specific Criteria
• Shows thorough understanding of the concepts ofrelations and functions, linear equations and inequalities,scatter plots, and prediction equations.
• 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 relations andfunctions, linear equations and inequalities, scatter plots,and prediction equations.
• 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 ofrelations and functions, linear equations and inequalities,scatter plots, and prediction equations.
• 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 conceptsof relations and functions, linear equations andinequalities, scatter plots, and prediction equations.
• 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
Chapter 2 Assessment Answer KeyPage 111, Open-Ended Assessment
Sample Answers
© Glencoe/McGraw-Hill A29 Glencoe Algebra 2
1. Students should describe two situations:If given as a mapping, a set of orderedpairs, or a table, determine whethereach member of the domain is pairedwith exactly one member of the range.If given as a graph, determine whetherthe graph passes the vertical line test.Functions must satisfy both of theseconditions.
2. Sample answer: The speed of a cardecreases as you apply the brakes. Thus,the rate of change of the speed withrespect to time is negative.
3. slope-intercept form: y � �12�x � 5
standard form: x � 2y � �10Sample answer: The slope-interceptform is most useful when graphing sincethe slope and the y-intercept can beeasily determined.
4. Students should indicate that the valuefor 1994 is likely to be more accuratethan the value for 2005 because valuesin the future may vary considerablyfrom the known data.
5. Students should state that all of thegraphs have the same shape, that thegraph of g(x) is the graph of the parentfunction f(x) translated, or shifted, left 2 units, and that the graph of h(x) is thegraph of f(x) translated right 3 units.The graph of y � � x � 500 � is the graphof f(x) translated left 500 units.
6. Alessia needed a test point to determinewhich side of the line to shade. Studentsshould indicate that Alessia made apoor choice since the point (�1, 7) lieson the graph of the boundary line and,therefore, does not provide theinformation she needs to complete thegraph.
7. The graph of the relation is an infiniteset of points represented graphically asa shaded region. Any vertical line willtherefore pass through an infinitenumber of points in the region. Thus,the relation is not a function.
In addition to the scoring rubric found on page A28, the following sample answers may be used as guidance in evaluating open-ended assessment items.
An
swer
s
© Glencoe/McGraw-Hill A30 Glencoe Algebra 2
Chapter 2 Assessment Answer KeyVocabulary Test/Review Quiz (Lessons 2–1 and 2–2) Quiz (Lessons 2–5 and 2–6)
Page 112 Page 113 Page 114
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11. Sample answer: Thevertical line test letsyou use the graphof a relation to tellwhether the relationis a function. Eachvertical line mustintersect the graphin at most onepoint.
12. Sample answer: Alinear function is afunction that can bewritten in the formf(x) � mx � b,where m and b arereal numbers.
1.
2.
3.
4.
5.
Quiz (Lessons 2–3 and 2–4)
Page 113
1.
2.
3.
4.
5.
1.
(5, 25)
2.
3.
D � all reals; R � {y � y � 0}
Quiz (Lesson 2–7)
Page 114
1.2.
3.
4. y
xO
y
xO
y
xO
y � 3x � 1
y
xO
y
x
20
1520253035404550
4 6 8 10 12 14 16 18 20
y � �2x � 11
B
y
xO
undefined
�32
�
5
no
d
j
f
b
c
a
g
h
i
e D � all reals; R � all reals; yes
x-intercept is 4; y-intercept is 3;
y
xO
5x � y � �10; A � 5,B � �1, C � �10
Sample answer using (10, 21) and (20, 41): y � 2x � 1; 61
© Glencoe/McGraw-Hill A31 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
D � {0, 1, 2, 4}; R � {�2, 3, 4}; yes
7.
D � all reals; R � all reals; yes
8.
9.
10.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
y
xO
D � all real numbers;R � {y � y � 8}
about $237,610
��52
�
x-intercept is �83
�;
y-intercept is �2
119
D � {2, 3, 4}; R � {�7, 0}; no
�1�2 0 1 2 4 5 63
{y � �2 � y � 6} or [�2, 6)
�1�2�3�4 0 1 2 43
{x � x � �1} or (��, �1]
{3, 11}
Q, R
1
y � ��13
�x � 1
�18
�
5
y
xO
y
xO
(4, �2)
(1, 3)
(2, 4)
(0, 3)
B
D
C
C
D
Chapter 2 Assessment Answer KeyMid-Chapter Test Cumulative ReviewPage 115 Page 116
An
swer
s
© Glencoe/McGraw-Hill A32 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
3 2
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
7 5
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
2 1
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
1 2
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
Chapter 2 Assessment Answer KeyStandardized Test Practice
Page 117 Page 118