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
1 2 3 4 5 6 7 8 9 10 066 11 10 09 08 07 06 05 04 03 02
Glencoe/McGraw-Hill
© Glencoe/McGraw-Hill iii Glencoe Algebra 2
Contents
Vocabulary Builder . . . . . . . . . . . . . . . . vii
Lesson 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 ! {y⏐y # "1}; yes R ! all reals; yes R ! all reals; yes
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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}; yes7. {(!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) "1412. 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 function
because 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 mapping A 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 mapping A 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
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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 ! yesvariable 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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© 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 equation14x # 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 yes3. 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. yes7. 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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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
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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.
y ! "3x $ 9 y ! x y ! x $ 1 4&9
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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 formy ! 2 " 2(x ! 8) (x1, y1) " (8, 2), m " 2y ! 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 formy ! 5 " 3(x ! (!1)) (x1, y1) " (!1, 5), m " 3y ! 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 " 919. RESERVOIRS The surface of Grand Lake is at an elevation of 648 feet. During the
current 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
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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& $ &6
y& ! 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 formy ! 60 " 0.8(x ! 200) (x1, y1) " (200, 60), m " 0.8y ! 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 $ 36.61c. 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 30Source: 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) ! 3⏐x⏐ " 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 {y⏐y # 0} {y⏐y ) 1}
x
y
O
x
y
O
x
y
O
x%3
x
f (x)
O
x
f (x)
O
x 3⏐x⏐ " 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 integers6. g(x) " 2x 7. f(x) " x # 1
D ! all reals, D ! all reals, R ! {y⏐y # 1}R ! nonnegative reals
8. f(x) " 'x if x ' 0 9. h(x) " '3 if x ' !12 if x ( 0 x # 1 if x > 1
D ! all reals, D ! {x⏐x ( "1 or x * 1},R ! {y⏐y ( 0 or y ! 2} R ! {y⏐y * 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) " !2x 4. f(x) " x # 1
D ! all reals, D ! all reals,R ! nonpositive reals R ! nonnegative reals
5. f(x) " 'x # 2 if x ) ! 2 6. h(x) " '4 ! x if x * 03x 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
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x%2
x
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1%2
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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 ) 3⏐x " 1⏐.
First graph the equation y " 3x ! 1.Since the inequality is ), the graph of the boundary is solid.Test (0, 0).0 )? 30 ! 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 ! 2x * 3
4. y ' !x ! 3 5. x # y ( 4 6. x # 1 # 2y ' 0
7. 2 ! x # y * !1 8. y ' 3x ! 3 9. y ) 1 ! x # 4
x
y
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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
y
Ox
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Ox
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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 * !3x # 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
x
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1%2
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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
© Glencoe/McGraw-Hill A2 Glencoe Algebra 2
Answers (Lesson 2-1)
Stu
dy
Gu
ide
and I
nte
rven
tion
Rel
atio
ns a
nd F
unct
ions
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-1
2-1
©G
lenc
oe/M
cGra
w-Hi
ll57
Gle
ncoe
Alg
ebra
2
Lesson 2-1
Gra
ph
Rel
atio
ns
A r
elat
ion
can
be r
epre
sent
ed a
s a
set
of o
rder
ed p
airs
or
as a
neq
uati
on;t
he r
elat
ion
is t
hen
the
set
of a
ll or
dere
d pa
irs
(x,y
) th
at m
ake
the
equa
tion
tru
e.T
he d
omai
nof
a r
elat
ion
is t
he s
et o
f al
l fir
st c
oord
inat
es o
f th
e or
dere
d pa
irs,
and
the
ran
geis
the
set
of
all s
econ
d co
ordi
nate
s.A
fu
nct
ion
is a
rel
atio
n in
whi
ch e
ach
elem
ent
of t
he d
omai
n is
pai
red
wit
h ex
actl
y on
eel
emen
t of
the
ran
ge.Y
ou c
an t
ell i
f a
rela
tion
is a
fun
ctio
n by
gra
phin
g,th
en u
sing
the
vert
ical
lin
e te
st.I
f a
vert
ical
line
inte
rsec
ts t
he g
raph
at
mor
e th
an o
ne p
oint
,the
rela
tion
is n
ot a
fun
ctio
n.
Gra
ph
th
e eq
uat
ion
y!
2x"
3 an
d f
ind
th
e d
omai
n a
nd
ran
ge.D
oes
the
equ
atio
n r
epre
sen
t a
fun
ctio
n?
Mak
e a
tabl
e of
val
ues
to f
ind
orde
red
pair
s th
at
sati
sfy
the
equa
tion
.The
n gr
aph
the
orde
red
pair
s.
The
dom
ain
and
rang
e ar
e bo
th a
ll re
al n
umbe
rs.T
hegr
aph
pass
es t
he v
erti
cal l
ine
test
,so
it is
fun
ctio
n.
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
.
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};y
es4.
y"
x2!
15.
y"
x!
46.
y"
3x#
2
D !
all r
eals
,D
!al
l rea
ls,
D !
all r
eals
,R
!{y
⏐y#
"1}
;yes
R !
all r
eals
;yes
R !
all r
eals
;yesx
y
O
x
y
O
x
y
O
x
y
O
x
y
O
x
y
O
x
y
O
xy
!1
!5
0!
3
1!
1
21
33
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-Hi
ll58
Gle
ncoe
Alg
ebra
2
Equ
atio
ns
of
Fun
ctio
ns
and
Rel
atio
ns
Equ
atio
ns t
hat
repr
esen
t fu
ncti
ons
are
ofte
n w
ritt
en in
fu
nct
ion
al n
otat
ion
.For
exa
mpl
e,y
"10
!8x
can
be w
ritt
en a
s f(
x) "
10 !
8x.T
his
nota
tion
em
phas
izes
the
fac
t th
at t
he v
alue
s of
y,t
he d
epen
den
tva
riab
le,d
epen
d on
the
val
ues
of x
,the
in
dep
end
ent
vari
able
.
To e
valu
ate
a fu
ncti
on,o
r fi
nd a
fun
ctio
nal v
alue
,mea
ns t
o su
bsti
tute
a g
iven
val
ue in
the
dom
ain
into
the
equ
atio
n to
fin
d th
e co
rres
pond
ing
elem
ent
in t
he r
ange
.
Giv
en t
he
fun
ctio
n f
(x)
!x2
$2x
,fin
d e
ach
val
ue.
a.f(
3)
f(x)
"x2
#2x
Orig
inal
func
tion
f(3)
"32
#2(
3)Su
bstit
ute.
"15
Sim
plify
.
b.f(
5a)
f(x)
"x2
#2x
Orig
inal
func
tion
f(5a
) "(5
a)2
#2(
5a)
Subs
titut
e.
"25
a2#
10a
Sim
plify
.
Fin
d e
ach
val
ue
if f
(x)
!"
2x$
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
) an
d f(
5).
f(2)
!7,
f(5)
!49
17.C
ompa
re t
he v
alue
s of
f(2
) $
f(5)
and
f(2
$5)
.f(
2) %
f(5)
!34
3,f(
2 %
5) !
199
2 & 31 % 3
b2& 10
b % 21 & 2a1 & 4
2 & x
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Rel
atio
ns a
nd F
unct
ions
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-1
2-1
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A3 Glencoe Algebra 2
An
swer
s
Answers (Lesson 2-1)
Skil
ls P
ract
ice
Rel
atio
ns a
nd F
unct
ions
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-1
2-1
©G
lenc
oe/M
cGra
w-Hi
ll59
Gle
ncoe
Alg
ebra
2
Lesson 2-1
Det
erm
ine
wh
eth
er e
ach
rel
atio
n i
s a
fun
ctio
n.W
rite
yes
or n
o.
1.ye
s2.
no
3.ye
s4.
no
Gra
ph
eac
h r
elat
ion
or
equ
atio
n a
nd
fin
d t
he
dom
ain
an
d r
ange
.Th
en d
eter
min
ew
het
her
th
e re
lati
on o
r eq
uat
ion
is
a fu
nct
ion
.
5.{(
2,!
3),(
2,4)
,(2,
!1)
}6.
{(2,
6),(
6,2)
}
D !
{2},
R !
{"3,
"1,
4};n
oD
!{2
,6},
R !
{2,6
};ye
s7.
{(!
3,4)
,(!
2,4)
,(!
1,!
1),(
3,!
1)}
8.x
"!
2
D !
{"3,
"2,
"1,
3},
D !
{"2}
,R !
all r
eals
;no
R !
{"1,
4};y
es
Fin
d e
ach
val
ue
if f
(x)
!2x
"1
and
g(x
) !
2 "
x2.
9.f(
0)"
110
.f(1
2)23
11.g
(4)
"14
12.f
(!2)
"5
13.g
(!1)
114
.f(d
)2d
"1
x
y
O
( –2,
4)
( –3,
4)
( –1,
–1)
( 3, –
1)x
y
O
( 2, 6
) ( 6, 2
)
x
y
O
( 2, 4
)
( 2, –
1)
( 2, –
3)
x
y
O
x
y
O
xy
12
24
36
D 3
R 1 5
D 100
200
300
R 50
100
150
©G
lenc
oe/M
cGra
w-Hi
ll60
Gle
ncoe
Alg
ebra
2
Det
erm
ine
wh
eth
er e
ach
rel
atio
n i
s a
fun
ctio
n.W
rite
yes
or n
o.
1.no
2.ye
s
3.ye
s4.
no
Gra
ph
eac
h r
elat
ion
or
equ
atio
n a
nd
fin
d t
he
dom
ain
an
d r
ange
.Th
en d
eter
min
ew
het
her
th
e re
lati
on o
r eq
uat
ion
is
a fu
nct
ion
.
5.{(
!4,
!1)
,(4,
0),(
0,3)
,(2,
0)}
6.y
"2x
!1
D !
{"4,
0,2,
4},
D !
all r
eals
,R !
all r
eals
;yes
R !
{"1,
0,3}
;yes
Fin
d e
ach
val
ue
if f
(x)
!an
d g
(x)
!"
2x$
3.
7.f(
3)1
8.f(
!4)
"9.
g !"2
10.f
(!2)
unde
fined
11.g
(!6)
1512
.f(m
!2)
13.M
USI
CT
he o
rder
ed p
airs
(1,
16),
(2,1
6),(
3,32
),(4
,32)
,and
(5,
48)
repr
esen
t th
e co
st o
fbu
ying
var
ious
num
bers
of
CD
s th
roug
h a
mus
ic c
lub.
Iden
tify
the
dom
ain
and
rang
e of
the
rela
tion
.Is
the
rela
tion
a f
unct
ion?
D !
{1,2
,3,4
,5},
R !
{16,
32,4
8};y
es
14.C
OM
PUTI
NG
If a
com
pute
r ca
n do
one
cal
cula
tion
in 0
.000
0000
015
seco
nd,t
hen
the
func
tion
T(n
) "0.
0000
0000
15n
give
s th
e ti
me
requ
ired
for
the
com
pute
r to
do
nca
lcul
atio
ns.H
ow lo
ng w
ould
it t
ake
the
com
pute
r to
do
5 bi
llion
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 (A
vera
ge)
Rel
atio
ns a
nd F
unct
ions
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
__PE
RIO
D__
___
2-1
2-1
© Glencoe/McGraw-Hill A4 Glencoe Algebra 2
Answers (Lesson 2-1)
Rea
din
g t
o L
earn
Math
emati
csR
elat
ions
and
Fun
ctio
ns
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-1
2-1
©G
lenc
oe/M
cGra
w-Hi
ll61
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
biol
ogy?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 2-
1 at
the
top
of
page
56
in y
our
text
book
.
•R
efer
to
the
tabl
e.W
hat
does
the
ord
ered
pai
r (8
,20)
tel
l you
?Fo
r ade
er,t
he a
vera
ge lo
ngev
ity is
8 y
ears
and
the
max
imum
long
evity
is 2
0 ye
ars.
•Su
ppos
e th
at t
his
tabl
e is
ext
ende
d to
incl
ude
mor
e an
imal
s.Is
it p
ossi
ble
to h
ave
an o
rder
ed p
air
for
the
data
in w
hich
the
fir
st n
umbe
r is
larg
erth
an t
he s
econ
d?Sa
mpl
e an
swer
:No,
the
max
imum
long
evity
mus
t alw
ays
be g
reat
er th
an th
e av
erag
e lo
ngev
ity.
Rea
din
g t
he
Less
on
1.a.
Exp
lain
the
dif
fere
nce
betw
een
a re
lati
on a
nd a
fun
ctio
n.Sa
mpl
e an
swer
:Are
latio
n is
any
set
of o
rder
ed p
airs
.A fu
nctio
n is
a s
peci
al k
ind
ofre
latio
n in
whi
ch e
ach
elem
ent o
f the
dom
ain
is p
aire
d w
ith e
xact
lyon
e el
emen
t in
the
rang
e.b.
Exp
lain
the
dif
fere
nce
betw
een
dom
ain
and
rang
e.Sa
mpl
e an
swer
:The
dom
ain
of a
rela
tion
is th
e se
t of a
ll fir
st c
oord
inat
es o
f the
ord
ered
pai
rs.T
hera
nge
is th
e se
t of a
ll se
cond
coo
rdin
ates
.
2.a.
Wri
te t
he d
omai
n an
d ra
nge
of t
he r
elat
ion
show
n in
the
gra
ph.
D:{"
3,"
2,"
1,0,
3};R
:{"
5,"
4,0,
1,2,
4}b.
Is t
his
rela
tion
a f
unct
ion?
Exp
lain
.Sa
mpl
e an
swer
:No,
it is
not
a fu
nctio
nbe
caus
e on
e of
the
elem
ents
of t
he d
omai
n,3,
is p
aire
d w
ith tw
oel
emen
ts o
f the
rang
e.
Hel
pin
g Y
ou
Rem
emb
er
3.L
ook
up t
he w
ords
dep
ende
ntan
d in
depe
nden
tin
a d
icti
onar
y.H
ow c
an t
he m
eani
ng o
fth
ese
wor
ds h
elp
you
dist
ingu
ish
betw
een
inde
pend
ent
and
depe
nden
t va
riab
les
in a
func
tion
?Sa
mpl
e an
swer
:The
var
iabl
e w
hose
val
ues
depe
nd o
n,or
are
dete
rmin
ed b
y,th
e va
lues
of t
he o
ther
var
iabl
e is
the
depe
nden
t var
iabl
e.
( 0, 4
)
( 3, 1
)
( 3, –
4)( –
1, –
5)
( –2,
0)
( –3,
2)
x
y
O
©G
lenc
oe/M
cGra
w-Hi
ll62
Gle
ncoe
Alg
ebra
2
Map
ping
sT
here
are
thr
ee s
peci
al w
ays
in w
hich
one
set
can
be
map
ped
to a
noth
er.A
set
can
be m
appe
d in
toan
othe
r se
t,on
toan
othe
r se
t,or
can
hav
e a
one-
to-o
neco
rres
pond
ence
wit
h an
othe
r se
t.
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
,ont
oin
to,o
nto
into
,ont
o,in
to,o
nto
one-
to-o
ne
5.6.
7.8.
into
into
,ont
oin
to,o
nto
into
,ont
o,on
e-to
-one
9.C
an a
set
be
map
ped
onto
a se
t w
ith
few
er e
lem
ents
tha
n it
has
?ye
s
10.C
an a
set
be
map
ped
into
a se
t th
at h
as m
ore
elem
ents
tha
n it
has
?ye
s
11.I
f a
map
ping
fro
m s
et A
into
set
Bis
a o
ne-t
o-on
e co
rres
pond
ence
,wha
t ca
n yo
u co
nclu
de a
bout
the
num
ber
of e
lem
ents
in A
and
B?
The
sets
hav
e th
e sa
me
num
ber o
f ele
men
ts.
–2 9 12 5
1 4 –7 0
–2 9 12 5
1 4 –7 0
–315 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
ping
Am
appi
ng fr
om s
et A
to s
et B
wher
e ev
ery
elem
ent o
f Ais
map
ped
to o
ne o
r mor
e el
emen
ts o
f set
B,b
ut n
ever
to a
n el
emen
t not
in B
.
Ont
o m
appi
ngA
map
ping
from
set
Ato
set
Bwh
ere
each
ele
men
t of s
et B
has
at le
ast o
ne e
lem
ent o
f se
t Am
appe
d to
it.
One
-to-o
ne
Am
appi
ng fr
om s
et A
onto
set
Bwh
ere
each
ele
men
t of s
et A
is m
appe
d to
exa
ctly
one
corr
espo
nden
ceel
emen
t of s
et B
and
diffe
rent
ele
men
ts o
f Aar
e ne
ver m
appe
d to
the
sam
e el
emen
t of B
.
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-1
2-1
© Glencoe/McGraw-Hill A5 Glencoe Algebra 2
An
swer
s
Answers (Lesson 2-2)
Stu
dy
Gu
ide
and I
nte
rven
tion
Line
ar E
quat
ions
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-2
2-2
©G
lenc
oe/M
cGra
w-Hi
ll63
Gle
ncoe
Alg
ebra
2
Lesson 2-2
Iden
tify
Lin
ear
Equ
atio
ns
and
Fu
nct
ion
sA
lin
ear
equ
atio
nha
s no
ope
rati
ons
othe
r th
an a
ddit
ion,
subt
ract
ion,
and
mul
tipl
icat
ion
of a
var
iabl
e by
a c
onst
ant.
The
vari
able
s m
ay n
ot b
e m
ulti
plie
d to
geth
er o
r ap
pear
in a
den
omin
ator
.A li
near
equ
atio
n do
esno
t co
ntai
n va
riab
les
wit
h ex
pone
nts
othe
r th
an 1
.The
gra
ph o
f a
linea
r eq
uati
on is
a li
ne.
A l
inea
r fu
nct
ion
is a
fun
ctio
n w
hose
ord
ered
pai
rs s
atis
fy a
line
ar e
quat
ion.
Any
line
arfu
ncti
on c
an b
e w
ritt
en in
the
for
m f
(x) "
mx
#b,
whe
re m
and
bar
e re
al n
umbe
rs.
If a
n eq
uati
on is
line
ar,y
ou n
eed
only
tw
o po
ints
tha
t sa
tisf
y th
e eq
uati
on in
ord
er t
o gr
aph
the
equa
tion
.One
way
is t
o fi
nd t
he x
-int
erce
pt a
nd t
he y
-int
erce
pt a
nd c
onne
ct t
hese
tw
opo
ints
wit
h a
line. Is
f(x
) !
0.2
"a
lin
ear
fun
ctio
n?
Exp
lain
.
Yes;
it is
a li
near
fun
ctio
n be
caus
e it
can
be w
ritt
en in
the
for
mf(
x) "
!x
#0.
2. Is 2
x$
xy"
3y!
0 a
lin
ear
fun
ctio
n?
Exp
lain
.
No;
it is
not
a li
near
fun
ctio
n be
caus
eth
e va
riab
les
xan
d y
are
mul
tipl
ied
toge
ther
in t
he 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
.
The
x-i
nter
cept
is t
he v
alue
of x
whe
n y
"0.
4x!
5y"
20O
rigin
al e
quat
ion
4x!
5(0)
"20
Subs
titut
e 0
for y
.
x"
5Si
mpl
ify.
So t
he x
-int
erce
pt is
5.
Sim
ilarl
y,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
quat
ion
or
fun
ctio
n i
s li
nea
r.W
rite
yes
or n
o.If
no,
exp
lain
.
1.6y
!x
"7
yes
2.9x
"No
;the
3.
f(x)
"2
!ye
sva
riabl
e y
appe
ars
in th
e de
nom
inat
or.
Fin
d t
he
x-in
terc
ept
and
th
e y-
inte
rcep
t of
th
e gr
aph
of
each
equ
atio
n.T
hen
gra
ph
the
equ
atio
n.
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
-int:
2x-
int:
"3;
y-in
t:1.
5
x
y
Ox
y
Ox
y
O
x % 1118 % y
©G
lenc
oe/M
cGra
w-Hi
ll64
Gle
ncoe
Alg
ebra
2
Stan
dar
d F
orm
The
sta
nd
ard
for
mof
a li
near
equ
atio
n is
Ax
#B
y"
C,w
here
A
,B,a
nd C
are
inte
gers
who
se g
reat
est
com
mon
fac
tor
is 1
.
Wri
te e
ach
equ
atio
n i
n s
tan
dar
d f
orm
.Id
enti
fy A
,B,a
nd
C.
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Line
ar E
quat
ions
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-2
2-2
Exam
ple
Exam
ple
a.y
!8x
"5
y"
8x!
5O
rigin
al e
quat
ion
!8x
#y
"!
5Su
btra
ct 8
xfro
m e
ach
side.
8x!
y"
5M
ultip
ly ea
ch s
ide
by !
1.
So A
"8,
B"
!1,
and
C"
5.
b.14
x!
"7y
$21
14x
"!
7y#
21O
rigin
al e
quat
ion
14x
#7y
"21
Add
7yto
eac
h sid
e.2x
#y
"3
Divid
e ea
ch s
ide
by 7
.
So A
"2,
B"
1,an
d C
"3.
Exer
cises
Exer
cises
Wri
te e
ach
equ
atio
n i
n s
tan
dar
d f
orm
.Id
enti
fy A
,B,a
nd
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
!3
B!
"9,
C!
"60
B
!"
3,C
!5
7.0.
4x#
3y"
108.
x"
4y!
79.
2y"
3x#
62x
$15
y!
50;A
!2,
x"
4y!
"7;
A!
1,3x
"2y
!"
6;A
!3,
B!
15,C
!50
B!
"4,
C!
"7
B!
"2,
C!
"6
10.
x#
y!
2 "
011
.4y
#4x
#12
"0
12.3
x"
!18
6x$
5y!
30;A
!6,
x$
y!
"3;
A!
1,x
!"
6;A
!1,
B!
5,C
!30
B!
1,C
!"
3B
!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!
8,10
x"
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!
500;
A!
1,B
!"
12,C
!"
24B
!"
3,C
!5
B!
2,C
!50
0
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
Line
ar E
quat
ions
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-2
2-2
©G
lenc
oe/M
cGra
w-Hi
ll65
Gle
ncoe
Alg
ebra
2
Lesson 2-2
Sta
te w
het
her
eac
h e
quat
ion
or
fun
ctio
n i
s li
nea
r.W
rite
yes
or n
o.If
no,
exp
lain
you
r re
ason
ing.
1.y
"3x
2.y
"!
2 #
5x
yes
yes
3.2x
#y
"10
4.f(
x) "
4x2
yes
No;t
he e
xpon
ent o
f xis
not
1.
5.!
#y
"15
6.x
"y
#8
No;x
is in
a d
enom
inat
or.
yes
7.g(
x) "
88.
h(x)
"#
x$#3
yes
No;x
is in
side
a s
quar
e ro
ot.
Wri
te e
ach
equ
atio
n i
n s
tan
dar
d f
orm
.Id
enti
fy A
,B,a
nd
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
equ
atio
n.T
hen
gra
ph
the
equ
atio
n.
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-Hi
ll66
Gle
ncoe
Alg
ebra
2
Sta
te w
het
her
eac
h e
quat
ion
or
fun
ctio
n i
s li
nea
r.W
rite
yes
or n
o.If
no,
exp
lain
you
r re
ason
ing.
1.h(
x) "
23ye
s2.
y"
xye
s
3.y
"No
;xis
a d
enom
inat
or.
4.9
!5x
y"
2No
;xan
d y
are
mul
tiplie
d.
Wri
te e
ach
equ
atio
n i
n s
tan
dar
d f
orm
.Id
enti
fy A
,B,a
nd
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,
8,21
Fin
d t
he
x-in
terc
ept
and
th
e y-
inte
rcep
t of
th
e gr
aph
of
each
equ
atio
n.T
hen
gra
ph
the
equ
atio
n.
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
The
equ
atio
n y
"2.
54x
give
s th
e le
ngth
in c
enti
met
ers
corr
espo
ndin
g to
ale
ngth
xin
inch
es.W
hat
is t
he le
ngth
in c
enti
met
ers
of a
1-f
oot
rule
r?30
.48
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 lo
ng-d
ista
nce
calli
ng p
lan,
the
mon
thly
cos
t C
in d
olla
rs is
giv
en b
y th
e lin
ear
func
tion
C(t
) "6
#0.
05t,
whe
re t
is t
he n
umbe
r of
min
utes
tal
ked.
14.W
hat
is t
he t
otal
cos
t of
tal
king
8 h
ours
? of
tal
king
20
hour
s?$3
0;$6
6
15.W
hat
is t
he e
ffec
tive
cos
t pe
r m
inut
e (t
he t
otal
cos
t di
vide
d by
the
num
ber
of m
inut
esta
lked
) of
tal
king
8 h
ours
? of
tal
king
20
hour
s?$0
.062
5;$0
.055
( 1, 0
)
( 0, 3
)
x
y
O
x
y
(–2,
0)
( 0, –
4)
O
( 7, 0
)
( 0, 2
)
x
y
O( –
2, 0
)
( 0, 4
)
x
y
O
3 % 42 % 7
3 % 8
5 % x
2 % 3
Pra
ctic
e (A
vera
ge)
Line
ar E
quat
ions
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-2
2-2
© Glencoe/McGraw-Hill A7 Glencoe Algebra 2
An
swer
s
Answers (Lesson 2-2)
Rea
din
g t
o L
earn
Math
emati
csLi
near
Equ
atio
ns
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-2
2-2
©G
lenc
oe/M
cGra
w-Hi
ll67
Gle
ncoe
Alg
ebra
2
Lesson 2-2
Pre-
Act
ivit
yH
ow d
o li
nea
r eq
uat
ion
s re
late
to
tim
e sp
ent
stu
dyi
ng?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 2-
2 at
the
top
of
page
63
in y
our
text
book
.
•If
Lol
ita
spen
ds 2
hour
s st
udyi
ng m
ath,
how
man
y ho
urs
will
she
hav
e
to s
tudy
che
mis
try?
1ho
urs
•Su
ppos
e th
at L
olit
a de
cide
s to
sta
y up
one
hou
r la
ter
so t
hat
she
now
has
5 ho
urs
to s
tudy
and
do
hom
ewor
k.W
rite
a li
near
equ
atio
n th
at d
escr
ibes
this
sit
uati
on.
x$
y!
5
Rea
din
g t
he
Less
on
1.W
rite
yes
or n
oto
tel
l whe
ther
eac
h lin
ear
equa
tion
is in
sta
ndar
d fo
rm.I
f it
is n
ot,
expl
ain
why
it is
not
.
a.!
x#
2y"
5No
;Ais
neg
ativ
e.
b.9x
!12
y"
!5
yes
c.5x
!7y
"3
yes
d.2x
!y
"1
No;B
is n
ot a
n in
tege
r.
e.0x
#0y
"0
No;A
and
Bar
e bo
th 0
.
f.2x
#4y
"8
No;T
he g
reat
est c
omm
on fa
ctor
of 2
,4,a
nd 8
is 2
,not
1.
2.H
ow c
an y
ou u
se t
he s
tand
ard
form
of
a lin
ear
equa
tion
to
tell
whe
ther
the
gra
ph is
aho
rizo
ntal
line
or
a ve
rtic
al li
ne?
If A
!0,
then
the
grap
h is
a h
oriz
onta
l lin
e.If
B!
0,th
en th
e gr
aph
is a
ver
tical
line
.
Hel
pin
g Y
ou
Rem
emb
er
3.O
ne w
ay t
o re
mem
ber
som
ethi
ng is
to
expl
ain
it t
o an
othe
r pe
rson
.Sup
pose
tha
t yo
u ar
e st
udyi
ng t
his
less
on w
ith
a fr
iend
who
thi
nks
that
she
sho
uld
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 he
r ho
w t
ore
mem
ber
the
corr
ect
way
to
find
inte
rcep
ts o
f a
line?
Sam
ple
answ
er:T
he
x-in
terc
ept i
s th
e x-
coor
dina
te o
f a p
oint
on
the
x-ax
is.E
very
poi
nt o
n th
e x-
axis
has
y-c
oord
inat
e 0,
so le
t y!
0 to
find
an
x-in
terc
ept.
The
y-in
terc
ept i
s th
e y-
coor
dina
te o
f a p
oint
on
the
y-ax
is.E
very
poi
nt o
n th
e y-
axis
has
x-c
oord
inat
e 0,
so le
t x!
0 to
find
a y
-inte
rcep
t.
4 % 7
1 & 2
1 % 2
©G
lenc
oe/M
cGra
w-Hi
ll68
Gle
ncoe
Alg
ebra
2
Gre
ates
t Com
mon
Fac
tor
Supp
ose
we
are
give
n a
linea
r eq
uati
on a
x#
by"
cw
here
a,b
,and
car
e no
nzer
oin
tege
rs,a
nd w
e w
ant
to k
now
if t
here
exi
st in
tege
rs x
and
yth
at s
atis
fy t
heeq
uati
on.W
e co
uld
try
gues
sing
a fe
w 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
lidea
nA
lgor
ithm
,we
can
dete
rmin
e no
t on
ly if
suc
h in
tege
rs x
and
yex
ist,
but
also
find
th
em.T
he fo
llow
ing
exam
ple
show
s ho
w t
his
algo
rith
m w
orks
.
Fin
d i
nte
gers
xan
d y
that
sat
isfy
588
x$
432y
!72
.
Div
ide
the
grea
ter
of t
he t
wo
coef
fici
ents
by
the
less
er t
o ge
t a
quot
ient
and
rem
aind
er.T
hen,
repe
at t
he p
roce
ss b
y di
vidi
ng t
he d
ivis
or b
y th
e re
mai
nder
unti
l you
get
a r
emai
nder
of
0.T
he p
roce
ss c
an b
e w
ritt
en a
s fo
llow
s.
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
)
The
last
non
zero
rem
aind
er is
the
GC
F o
f th
e tw
o co
effi
cien
ts.I
f th
e co
nsta
ntte
rm 7
2 is
div
isib
le b
y th
e G
CF,
then
inte
gers
xan
d y
do e
xist
tha
t sa
tisf
y th
eeq
uati
on.T
o fi
nd x
and
y,w
ork
back
war
d in
the
fol
low
ing
man
ner.
72"
6 $
12"
6 $
[120
!36
(3)]
Subs
titut
e fo
r 12
usin
g (4
)"
6(12
0) !
18(3
6)"
6(12
0) !
18[1
56 !
120(
1)]
Subs
titut
e fo
r 36
usin
g (3
)"
!18
(156
) #
24(1
20)
"!
18(1
56)
#24
[432
!15
6(2)
]Su
bstit
ute
for 1
20 u
sing
(2)
"24
(432
) !
66(1
56)
"24
(432
) !
66[5
88 !
432(
1)]
Subs
titut
e fo
r 156
usin
g (1
)"
588(
!66
) #
432(
90)
Thu
s,x
"!
66 a
nd y
"90
.
Fin
d i
nte
gers
xan
d y
,if
they
exi
st,t
hat
sat
isfy
eac
h e
quat
ion
.
1.27
x#
65y
"3
2.45
x#
144y
"36
x!
"36
and
y!
15x
!"
12 a
nd y
!4
3.90
x#
117y
"10
4.12
3x#
36y
"15
no in
tegr
al s
olut
ions
exi
stx
!25
and
y!
"85
5.10
32x
#10
01y
"1
6.31
25x
#30
87y
"1
x!
"22
6 an
d y
!23
3x
!"
1381
and
y!
1398
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-2
2-2
Exam
ple
Exam
ple
© Glencoe/McGraw-Hill A8 Glencoe Algebra 2
Answers (Lesson 2-3)
Stu
dy
Gu
ide
and I
nte
rven
tion
Slop
e
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-3
2-3
©G
lenc
oe/M
cGra
w-Hi
ll69
Gle
ncoe
Alg
ebra
2
Lesson 2-3
Slo
pe
Slop
e m
of a
Lin
eFo
r poi
nts
(x1,
y 1) a
nd (x
2, y 2
), wh
ere
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"
Slop
e fo
rmul
a
"(x
1, y 1
) "(2
, !1)
, (x 2
, y2)
"(!
4, 5
)
""
!1
Sim
plify
.
The
slo
pe o
f th
e 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 o
rder
ed
pair
(!
1,!
3).T
hen,
acco
rdin
g to
the
sl
ope,
go u
p 4
unit
san
d ri
ght
5 un
its.
Plo
t th
e ne
w p
oint
(4,1
).C
onne
ct t
hepo
ints
and
dra
w
the
line.
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)
and
(!
4,3)
"5.
(5,1
0) a
nd (
!1,
!2)
26.
(!1,
!4)
and
(!13
,2)"
7.(7
,!2)
and
(3,
3)"
8.(!
5,9)
and
(5,
5)"
9.(4
,!2)
and
(!
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
hrou
gh (
0,2)
pass
es t
hrou
gh (
1,4)
pass
es t
hrou
gh (!
2,!
5)
13.s
lope
"1
14.s
lope
"!
15.s
lope
"
pass
es t
hrou
gh (!
4,6)
pass
es t
hrou
gh (
!3,
0)pa
sses
thr
ough
(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-Hi
ll70
Gle
ncoe
Alg
ebra
2
Para
llel a
nd
Per
pen
dic
ula
r Li
nes
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Slop
e
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-3
2-3
In a
pla
ne,n
onve
rtic
al li
nes
wit
h th
esa
me
slop
e ar
e p
aral
lel.
All
vert
ical
lines
are
par
alle
l.
x
y
O
slop
e !
m
slop
e !
m
In a
pla
ne,t
wo
obliq
ue li
nes
are
per
pen
dic
ula
rif
and
only
if t
he p
rodu
ct o
f th
eir
slop
es is
!1.
Any
vert
ical
line
is p
erpe
ndic
ular
to
any
hori
zont
al li
ne.
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
lines
.
The
slo
pe o
f th
e fi
rst
line
is
"1.
The
slo
pe o
f th
e se
cond
line
is
"!
.
The
slo
pes
are
not
equa
l and
the
pro
duct
of
the
slop
es is
not
!1,
so t
he li
nes
are
neit
her
para
llel n
or p
erpe
ndic
ular
.
Are
th
e li
nes
par
alle
l,p
erp
end
icu
lar,
or n
eith
er?
1.th
e lin
e pa
ssin
g th
roug
h (4
,3)
and
(1.!
3) a
nd t
he li
ne p
assi
ng t
hrou
gh (
1,2)
and
(!1,
3)pe
rpen
dicu
lar
2.th
e lin
e pa
ssin
g th
roug
h (2
,8)
and
(!2,
2) a
nd t
he li
ne p
assi
ng t
hrou
gh (
0,9)
and
(6,
0)ne
ither
3.th
e lin
e pa
ssin
g th
roug
h (3
,9)
and
(!2,
!1)
and
the
gra
ph o
f y"
2xpa
ralle
l
4.th
e lin
e w
ith
x-in
terc
ept
!2
and
y-in
terc
ept
5 an
d th
e lin
e w
ith
x-in
terc
ept
2 an
d y-
inte
rcep
t !
5pa
ralle
l
5.th
e lin
e w
ith
x-in
terc
ept
1 an
d y-
inte
rcep
t 3
and
the
line
wit
h x-
inte
rcep
t 3
and
y-in
terc
ept
1ne
ither
6.th
e lin
e pa
ssin
g th
roug
h (!
2,!
3) a
nd (
2,5)
and
the
gra
ph o
f x#
2y"
10pe
rpen
dicu
lar
7.th
e lin
e pa
ssin
g th
roug
h (!
4,!
8) a
nd (
6,!
4) a
nd t
he g
raph
of
2x!
5y"
5pa
ralle
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
Slop
e
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-3
2-3
©G
lenc
oe/M
cGra
w-Hi
ll71
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)
unde
fined
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
roug
h (0
,1),
perp
endi
cula
r to
15.p
asse
s th
roug
h (0
,!5)
,par
alle
l to
the
a lin
e w
hose
slo
pe is
gr
aph
of y
"1
16.H
IKIN
GN
aom
i lef
t fr
om a
n el
evat
ion
of 7
400
feet
at
7:00
A.M
.and
hik
ed t
o an
ele
vati
onof
980
0 fe
et b
y 11
:00
A.M
.Wha
t w
as h
er r
ate
of c
hang
e in
alt
itud
e?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-Hi
ll72
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)un
defin
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
roug
h (3
,0),
perp
endi
cula
r12
.pas
ses
thro
ugh
(!3,
!1)
,par
alle
l to
a lin
e
to a
line
who
se s
lope
is
who
se s
lope
is !
1
DEP
REC
IATI
ON
For
Exe
rcis
es 1
3–15
,use
th
e fo
llow
ing
info
rmat
ion
.A
mac
hine
tha
t or
igin
ally
cos
t $1
5,60
0 ha
s a
valu
e of
$75
00 a
t th
e en
d of
3 y
ears
.The
sam
em
achi
ne h
as a
val
ue o
f $2
800
at t
he e
nd o
f 8
year
s.
13.F
ind
the
aver
age
rate
of
chan
ge in
val
ue (
depr
ecia
tion
) of
the
mac
hine
bet
wee
n it
spu
rcha
se a
nd t
he e
nd o
f 3
year
s."
$270
0 pe
r yea
r14
.Fin
d th
e av
erag
e ra
te o
f ch
ange
in v
alue
of
the
mac
hine
bet
wee
n th
e en
d of
3 y
ears
and
the
end
of 8
yea
rs.
"$9
40 p
er y
ear
15.I
nter
pret
the
sig
n of
you
r an
swer
s.It
is n
egat
ive
beca
use
the
valu
e is
dec
reas
ing.
( –3,
–1)
x
y
O
(3, 0
)x
y
O
3 % 2
( 2, –
3)
x
y
O( 0
, 2)
x
y
O
4 % 5
(2, 1
)
x
y
O
(0, –
3)
x
y
O
3 % 4
7 & 25 & 35 & 17
5 & 4
Pra
ctic
e (A
vera
ge)
Slop
e
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-3
2-3
© Glencoe/McGraw-Hill A10 Glencoe Algebra 2
Answers (Lesson 2-3)
Rea
din
g t
o L
earn
Math
emati
csSl
ope
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-3
2-3
©G
lenc
oe/M
cGra
w-Hi
ll73
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
ucti
on t
o L
esso
n 2-
3 at
the
top
of
page
68
in y
our
text
book
.
•W
hat
is t
he g
rade
of
a ro
ad t
hat
rise
s 40
fee
t ov
er a
hor
izon
tal d
ista
nce
of 1
000
feet
?4%
•W
hat
is t
he g
rade
of
a ro
ad t
hat
rise
s 52
5 m
eter
s ov
er a
hor
izon
tal
dist
ance
of
10 k
ilom
eter
s? (
1 ki
lom
eter
"10
00 m
eter
s)5.
25%
Rea
din
g t
he
Less
on
1.D
escr
ibe
each
typ
e of
slo
pe a
nd in
clud
e a
sket
ch.
Type
of S
lope
Desc
riptio
n of
Gra
phSk
etch
Posit
iveTh
e lin
e ris
es to
the
right
.
Zero
The
line
is h
oriz
onta
l.
Nega
tive
The
line
falls
to th
e rig
ht.
Unde
fined
The
line
is v
ertic
al.
2.a.
How
are
the
slo
pes
of t
wo
nonv
erti
cal p
aral
lel l
ines
rel
ated
?Th
ey a
re e
qual
.b.
How
are
the
slo
pes
of t
wo
obliq
ue p
erpe
ndic
ular
line
s re
late
d?Th
eir p
rodu
ct is
"1.
Hel
pin
g Y
ou
Rem
emb
er
3.L
ook
up t
he t
erm
s gr
ade,
pitc
h,sl
ant,
and
slop
e.H
ow c
an e
very
day
mea
ning
s of
the
sew
ords
hel
p yo
u re
mem
ber
the
defi
niti
on o
f sl
ope?
Sam
ple
answ
er:A
ll th
ese
wor
dsca
n be
use
d w
hen
you
desc
ribe
how
muc
h a
thin
g sl
ants
upw
ard
ordo
wnw
ard.
You
can
desc
ribe
this
num
eric
ally
by
com
parin
g ris
e to
run.
x
y
O
x
y
O
x
y
O
x
y
O
©G
lenc
oe/M
cGra
w-Hi
ll74
Gle
ncoe
Alg
ebra
2
Aer
ial S
urve
yors
and
Are
aM
any
land
reg
ions
hav
e ir
regu
lar
shap
es.A
eria
l sur
veyo
rs
supp
ly a
eria
l map
pers
wit
h lis
ts o
f co
ordi
nate
s an
d el
evat
ions
fo
r th
e ar
eas
that
nee
d to
be
phot
ogra
phed
fro
m t
he a
ir.T
hese
m
aps
prov
ide
info
rmat
ion
abou
t th
e ho
rizo
ntal
and
ver
tica
l fe
atur
es o
f th
e la
nd.
Ste
p 1
Lis
t th
e or
dere
d pa
irs
for
the
vert
ices
in
coun
terc
lock
wis
e or
der,
repe
atin
g th
e fi
rst
orde
red
pair
at
the
bott
om o
f th
e lis
t.
Ste
p 2
Fin
d D
,the
sum
of
the
dow
nwar
d di
agon
al p
rodu
cts
(fro
m le
ft t
o ri
ght)
.D
"(5
$5)
#(2
$1)
#(2
$3)
#(6
$7)
"25
#2
#6
#42
or
75
Ste
p 3
Fin
d U
,the
sum
of
the
upw
ard
diag
onal
pro
duct
s (f
rom
left
to
righ
t).
U"
(2 $
7) #
(2 $
5) #
(6 $
1) #
(5 $
3)"
14 #
10 #
6 #
15 o
r 45
Ste
p 4
Use
the
for
mul
a A
"%1 2% (
D!
U)
to f
ind
the
area
.
A"
%1 2% (75
!45
)
"%1 2% (
30)
or 1
5
The
are
a is
15
squa
re u
nits
.Cou
nt t
he n
umbe
r of
squ
are
unit
s en
clos
ed b
y th
e po
lygo
n.D
oes
this
res
ult
seem
rea
sona
ble?
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
quar
e u
nit
s.
1.2.
3.
20 u
nits
214
uni
ts2
34 u
nits
2
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
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-3
2-3
© Glencoe/McGraw-Hill A11 Glencoe Algebra 2
An
swer
s
Answers (Lesson 2-4)
Stu
dy
Gu
ide
and I
nte
rven
tion
Writ
ing
Line
ar E
quat
ions
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-4
2-4
©G
lenc
oe/M
cGra
w-Hi
ll75
Gle
ncoe
Alg
ebra
2
Lesson 2-4
Form
s o
f Eq
uat
ion
s
Slop
e-In
terc
ept F
orm
of
a L
inea
r Equ
atio
ny
"m
x#
b, w
here
mis
the
slope
and
bis
the
y-in
terc
ept
Poin
t-Slo
pe F
orm
y
!y 1
"m
(x!
x 1),
wher
e (x
1, y 1
) are
the
coor
dina
tes
of a
poi
nt o
n th
e lin
e an
d of
a L
inea
r Equ
atio
nm
is th
e slo
pe o
f the
line
Wri
te a
n e
quat
ion
in
slop
e-in
terc
ept
form
for
th
e li
ne
that
has
slo
pe
"2
and
pas
ses
thro
ugh
th
ep
oin
t (3
,7).
Subs
titu
te f
or m
,x,a
nd y
in t
he
slop
e-in
terc
ept
form
.y
"m
x#
bSl
ope-
inte
rcep
t for
m
7 "
(!2)
(3) #
b(x
, y) "
(3, 7
), m
"!
2
7 "
!6
#b
Sim
plify
.
13 "
bAd
d 6
to b
oth
sides
.
The
y-i
nter
cept
is 1
3.T
he e
quat
ion
in
slop
e-in
terc
ept
form
is y
"!
2x#
13.
Wri
te a
n e
quat
ion
in
slop
e-in
terc
ept
form
for
th
e li
ne
that
has
slo
pe
and
x-i
nte
rcep
t 5.
y"
mx
#b
Slop
e-in
terc
ept f
orm
0 "
!"(5
) #
b(x
, y) "
(5, 0
), m
"
0 "
#b
Sim
plify
.
!"
bSu
btra
ct
from
bot
h sid
es.
The
y-i
nter
cept
is !
.The
slo
pe-i
nter
cept
form
is y
"x
!.
5 % 31 % 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
quat
ion
in
slo
pe-
inte
rcep
t fo
rm f
or t
he
lin
e th
at s
atis
fies
eac
h s
et o
fco
nd
itio
ns.
1.sl
ope
!2,
pass
es t
hrou
gh (!
4,6)
2.sl
ope
,y-i
nter
cept
4
y!
"2x
"2
y!
x$
4
3.sl
ope
1,pa
sses
thr
ough
(2,
5)4.
slop
e !
,pas
ses
thro
ugh
(5,!
7)
y!
x$
3y
!"
x$
6
Wri
te a
n e
quat
ion
in
slo
pe-
inte
rcep
t fo
rm f
or e
ach
gra
ph
.
5.6.
7.
y!
"3x
$9
y!
xy
!x
$14 & 9
1 & 95 & 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-Hi
ll76
Gle
ncoe
Alg
ebra
2
Para
llel a
nd
Per
pen
dic
ula
r Li
nes
Use
the
slo
pe-i
nter
cept
or
poin
t-sl
ope
form
to
find
equa
tion
s of
line
s th
at a
re p
aral
lel o
r pe
rpen
dicu
lar
to a
giv
en li
ne.R
emem
ber
that
par
alle
llin
es h
ave
equa
l slo
pe.T
he s
lope
s of
tw
o pe
rpen
dicu
lar
lines
are
neg
ativ
e re
cipr
ocal
s,th
atis
,the
ir p
rodu
ct is
!1.
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Writ
ing
Line
ar E
quat
ions
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-4
2-4
Wri
te a
n e
quat
ion
of
the
lin
e th
at p
asse
s th
rou
gh (
8,2)
an
d i
sp
erp
end
icu
lar
to t
he
lin
e w
hos
e eq
uat
ion
is
y!
"x
$3.
The
slo
pe o
f th
e gi
ven
line
is !
.Sin
ce t
he
slop
es o
f pe
rpen
dicu
lar
lines
are
neg
ativ
ere
cipr
ocal
s,th
e sl
ope
of t
he p
erpe
ndic
ular
line
is 2
.U
se t
he s
lope
and
the
giv
en p
oint
to
wri
teth
e eq
uati
on.
y!
y1
"m
(x!
x 1)
Poin
t-slo
pe fo
rmy
!2
"2(
x!
8)(x
1, y 1
) "(8
, 2),
m"
2y
!2
"2x
!16
Dist
ribut
ive P
rop.
y"
2x!
14Ad
d 2
to e
ach
side.
An
equa
tion
of
the
line
is y
"2x
!14
.
1 % 2
1 & 2
Wri
te a
n e
quat
ion
of
the
lin
e th
at p
asse
s th
rou
gh (
"1,
5) a
nd
is
par
alle
l to
the
grap
h o
f y
!3x
$1.
The
slo
pe o
f th
e gi
ven
line
is 3
.Sin
ce t
hesl
opes
of
para
llel l
ines
are
equ
al,t
he s
lope
of t
he p
aral
lel l
ine
is a
lso
3.U
se t
he s
lope
and
the
giv
en p
oint
to
wri
teth
e eq
uati
on.
y!
y 1"
m(x
!x 1
)Po
int-s
lope
form
y!
5 "
3(x
!(!
1))
(x1,
y 1) "
(!1,
5),
m"
3y
!5
"3x
#3
Dist
ribut
ive P
rop.
y"
3x#
8Ad
d 5
to e
ach
side.
An
equa
tion
of
the
line
is y
"3x
#8.
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Wri
te a
n e
quat
ion
in
slo
pe-
inte
rcep
t fo
rm f
or t
he
lin
e th
at s
atis
fies
eac
h s
et o
fco
nd
itio
ns.
1.pa
sses
thr
ough
(!4,
2),p
aral
lel t
o th
e lin
e w
hose
equ
atio
n is
y"
x#
5y
!x
$4
2.pa
sses
thr
ough
(3,
1),p
erpe
ndic
ular
to
the
grap
h of
y"
!3x
#2
y!
x
3.pa
sses
thr
ough
(1,
!1)
,par
alle
l to
the
line
that
pas
ses
thro
ugh
(4,1
) an
d (2
,!3)
y!
2x"
3
4.pa
sses
thr
ough
(4,
7),p
erpe
ndic
ular
to
the
line
that
pas
ses
thro
ugh
(3,6
) an
d (3
,15)
y!
7
5.pa
sses
thr
ough
(8,
!6)
,per
pend
icul
ar t
o th
e gr
aph
of 2
x!
y"
4y
!"
x"
2
6.pa
sses
thr
ough
(2,
!2)
,per
pend
icul
ar t
o th
e gr
aph
of x
#5y
"6
y!
5x"
12
7.pa
sses
thr
ough
(6,
1),p
aral
lel t
o th
e lin
e w
ith
x-in
terc
ept
!3
and
y-in
terc
ept
5
y!
x"
9
8.pa
sses
thr
ough
(!2,
1),p
erpe
ndic
ular
to
the
line
y"
4x!
11y
!"
x$
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
Writ
ing
Line
ar E
quat
ions
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-4
2-4
©G
lenc
oe/M
cGra
w-Hi
ll77
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
quat
ion
.
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
quat
ion
in
slo
pe-
inte
rcep
t fo
rm f
or e
ach
gra
ph
.
9.10
.11
.
y!
3x"
1y
!"
1y
!"
2x$
3
Wri
te a
n e
quat
ion
in
slo
pe-
inte
rcep
t fo
rm f
or t
he
lin
e th
at s
atis
fies
eac
h s
et o
fco
nd
itio
ns.
12.s
lope
3,p
asse
s th
roug
h (1
,!3)
13.s
lope
!1,
pass
es t
hrou
gh (
0,0)
y!
3x"
6y
!"
x
14.s
lope
!2,
pass
es t
hrou
gh (
0,!
5)15
.slo
pe 3
,pas
ses
thro
ugh
(2,0
)
y!
"2x
"5
y!
3x"
6
16.p
asse
s th
roug
h (!
1,!
2) a
nd (
!3,
1)17
.pas
ses
thro
ugh
(!2,
!4)
and
(1,
8)
y!
"x
"y
!4x
$4
18.x
-int
erce
pt 2
,y-i
nter
cept
!6
19.x
-int
erce
pt
,y-i
nter
cept
5
y!
3x"
6y
!"
2x$
5
20.p
asse
s th
roug
h (3
,!1)
,per
pend
icul
ar t
o th
e gr
aph
of y
"!
x!
4.y
!3x
"10
1 % 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-Hi
ll78
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
quat
ion
.
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
quat
ion
in
slo
pe-
inte
rcep
t fo
rm f
or e
ach
gra
ph
.
7.8.
9.
y!
2y
!x
"2
y!
"x
$1
Wri
te a
n e
quat
ion
in
slo
pe-
inte
rcep
t fo
rm f
or t
he
lin
e th
at s
atis
fies
eac
h s
et o
fco
nd
itio
ns.
10.s
lope
!5,
pass
es t
hrou
gh (!
3,!
8)11
.slo
pe
,pas
ses
thro
ugh
(10,
!3)
y!
"5x
"23
y!
x"
11
12.s
lope
0,p
asse
s th
roug
h (0
,!10
)13
.slo
pe !
,pas
ses
thro
ugh
(6,!
8)
y!
"10
y!
"x
"4
14.p
asse
s th
roug
h (3
,11)
and
(!6,
5)15
.pas
ses
thro
ugh
(7,!
2) a
nd (
3,!
1)
y!
x$
9y
!"
x"
16.x
-int
erce
pt 3
,y-i
nter
cept
217
.x-i
nter
cept
!5,
y-in
terc
ept
7
y!
"x
$2
y!
x$
7
18.p
asse
s th
roug
h (!
8,!
7),p
erpe
ndic
ular
to
the
grap
h of
y"
4x!
3y
!"
x"
919
.RES
ERV
OIR
ST
he s
urfa
ce o
f G
rand
Lak
e is
at
an e
leva
tion
of
648
feet
.Dur
ing
the
curr
ent
drou
ght,
the
wat
er le
vel i
s dr
oppi
ng a
t a
rate
of
3 in
ches
per
day
.If
this
tre
ndco
ntin
ues,
wri
te a
n eq
uati
on t
hat
give
s th
e el
evat
ion
in f
eet
of t
he s
urfa
ce o
f G
rand
Lak
eaf
ter
xda
ys.
y!
"0.
25x
$64
820
.BU
SIN
ESS
Tony
Mar
coni
’s c
ompa
ny m
anuf
actu
res
CD
-RO
M d
rive
s.T
he c
ompa
ny w
illm
ake
$150
,000
pro
fit
if it
man
ufac
ture
s 10
0,00
0 dr
ives
,and
$1,
750,
000
prof
it if
itm
anuf
actu
res
500,
000
driv
es.T
he r
elat
ions
hip
betw
een
the
num
ber
of d
rive
sm
anuf
actu
red
and
the
prof
it is
line
ar.W
rite
an
equa
tion
tha
t gi
ves
the
prof
it P
whe
n n
driv
es a
re m
anuf
actu
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 (A
vera
ge)
Writ
ing
Line
ar E
quat
ions
NAM
E__
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____
____
____
____
____
____
____
____
____
____
DATE
____
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PERI
OD
____
_
2-4
2-4
© Glencoe/McGraw-Hill A13 Glencoe Algebra 2
An
swer
s
Answers (Lesson 2-4)
Rea
din
g t
o L
earn
Math
emati
csW
ritin
g Li
near
Equ
atio
ns
NAM
E__
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____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-4
2-4
©G
lenc
oe/M
cGra
w-Hi
ll79
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
busi
nes
s?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 2-
4 at
the
top
of
page
75
in y
our
text
book
.
•If
the
tot
al c
ost
of p
rodu
cing
a p
rodu
ct is
giv
en b
y th
e eq
uati
on
y"
5400
#1.
37x,
wha
t is
the
fix
ed c
ost?
Wha
t is
the
var
iabl
e co
st
(for
eac
h it
em p
rodu
ced)
?$5
400;
$1.3
7•
Wri
te a
line
ar e
quat
ion
that
des
crib
es t
he f
ollo
win
g si
tuat
ion:
A c
ompa
ny t
hat
man
ufac
ture
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 co
mpu
ter.
y!
228,
750
$85
2x
Rea
din
g t
he
Less
on
1.a.
Wri
te t
he s
lope
-int
erce
pt f
orm
of
the
equa
tion
of
a lin
e.T
hen
expl
ain
the
mea
ning
of
each
of
the
vari
able
s in
the
equ
atio
n.y
!m
x$
b;m
is th
e sl
ope
and
bis
the
y-in
terc
ept.
The
varia
bles
xan
d y
are
the
coor
dina
tes
of a
ny p
oint
on
the
line.
b.W
rite
the
poi
nt-s
lope
for
m o
f th
e eq
uati
on o
f a
line.
The
n ex
plai
n th
e m
eani
ng o
f ea
chof
the
var
iabl
es in
the
equ
atio
n.y
"y 1
!m
(x"
x 1);
mis
the
slop
e.x
and
yar
e th
e co
ordi
nate
s of
any
poi
nt o
n th
e lin
e.x 1
and
y 1ar
e th
e co
ordi
nate
s of
one
spe
cific
poi
nt o
n th
e lin
e.
2.Su
ppos
e th
at y
our
alge
bra
teac
her
asks
you
to
wri
te t
he p
oint
-slo
pe f
orm
of
the
equa
tion
of t
he li
ne t
hrou
gh t
he p
oint
s (!
6,7)
and
(!
3,!
2).Y
ou w
rite
y#
2 "
!3(
x#
3) a
ndyo
ur c
lass
mat
e w
rite
s y
!7
"!
3(x
#6)
.Whi
ch o
f yo
u is
cor
rect
?E
xpla
in. Y
ou a
rebo
th c
orre
ct.E
ither
poi
nt m
ay b
e us
ed a
s (x
1,y 1
) in
the
poin
t-slo
pe fo
rm.
You
used
("3,
"2)
,and
you
r cla
ssm
ate
used
("6,
7).
3.Yo
u ar
e as
ked
to w
rite
an
equa
tion
of
two
lines
tha
t pa
ss t
hrou
gh (
3,!
5),o
ne o
f th
empa
ralle
l to
and
one
of t
hem
per
pend
icul
ar t
o th
e lin
e w
hose
equ
atio
n is
y"
!3x
#4.
The
fir
st s
tep
in f
indi
ng t
hese
equ
atio
ns is
to
find
the
ir s
lope
s.W
hat
is t
he s
lope
of
the
para
llel l
ine?
Wha
t is
the
slo
pe o
f th
e pe
rpen
dicu
lar
line?
"3;
Hel
pin
g Y
ou
Rem
emb
er
4.M
any
stud
ents
hav
e tr
oubl
e re
mem
beri
ng t
he p
oint
-slo
pe f
orm
for
a li
near
equ
atio
n.H
ow c
an y
ou u
se t
he d
efin
itio
n of
slo
pe t
o re
mem
ber
this
for
m?
Sam
ple
answ
er:
Writ
e th
e de
finiti
on o
f slo
pe:m
!.M
ultip
ly b
oth
side
s of
this
equa
tion
by x
2"
x 1.D
rop
the
subs
crip
ts in
y2
and
x 2.T
his
give
s th
epo
int-s
lope
form
of t
he e
quat
ion
of a
line
.
y 2"
y 1& x 2
"x 1
1 & 3
©G
lenc
oe/M
cGra
w-Hi
ll80
Gle
ncoe
Alg
ebra
2
Two-
Inte
rcep
t For
m o
f a L
inea
r Equ
atio
nYo
u ar
e al
read
y fa
mili
ar w
ith
the
slop
e-in
terc
ept
form
of a
line
ar e
quat
ion,
y"
mx
#b.
Lin
ear
equa
tion
s ca
n al
so b
e w
ritt
en in
the
form
% ax %#
% by %"
1 w
ith
x-
inte
rcep
t a
and
y-in
terc
ept
b.T
his
is c
alle
d tw
o-in
terc
ept
form
.
Dra
w t
he
grap
h o
f & "x 3&
$& 6y &
!1.
The
gra
ph c
ross
es t
he x
-axi
s at
!3
and
the
y-ax
is a
t 6.
Gra
ph
(!3,
0) a
nd (0
,6),
then
dra
w a
str
aigh
t lin
e th
roug
h th
em.
Wri
te 3
x$
4y!
12 i
n t
wo-
inte
rcep
t fo
rm.
% 13 2x %#
% 14 2y %"
%1 12 2%Di
vide
by 1
2 to
obt
ain
1 on
the
right
sid
e.
% 4x %#
% 3y %"
1Si
mpl
ify.
The
x-i
nter
cept
is 4
;the
y-i
nter
cept
is 3
.
Use
th
e gi
ven
in
terc
epts
a a
nd
b,t
o w
rite
an
equ
atio
n i
n t
wo-
inte
rcep
t fo
rm.T
hen
dra
w t
he
grap
h.
See
stud
ents
’gra
phs.
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
equ
atio
n i
n t
wo-
inte
rcep
t fo
rm.T
hen
dra
w t
he
grap
h.
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
NAM
E__
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____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-4
2-4
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
©G
lencoe/McG
raw-HillA14
Glencoe Algebra 2
Answers
(Lesson 2-5)
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 formy ! 60 " 0.8(x ! 200) (x1, y1) " (200, 60), m " 0.8y ! 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
©G
lencoe/McG
raw-HillA15
Glencoe Algebra 2
Answers
Answers
(Lesson 2-5)
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 $ 36.61c. 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)
Cal
ori
es B
urn
ed
0 10 20 30 40 50 555 15 25 35 45
500
400
300
200
100
Burning 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
© Glencoe/McGraw-Hill A16 Glencoe Algebra 2
Answers (Lesson 2-5)
Rea
din
g t
o L
earn
Math
emati
csM
odel
ing
Rea
l-Wor
ld D
ata:
Usi
ng S
catte
r Plo
ts
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-5
2-5
©G
lenc
oe/M
cGra
w-Hi
ll85
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 C
alor
ies
you
bu
rnex
erci
sin
g?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 2-
5 at
the
top
of
page
81
in y
our
text
book
.
•If
a w
oman
run
s 5.
5 m
iles
per
hour
,abo
ut h
ow m
any
Cal
orie
s w
ill s
hebu
rn in
an
hour
?Sa
mpl
e an
swer
:572
Cal
orie
s
•If
a m
an r
uns
7.5
mile
s pe
r ho
ur,a
bout
how
man
y C
alor
ies
will
he
burn
in h
alf
an h
our?
Sam
ple
answ
er:3
97 C
alor
ies
Rea
din
g t
he
Less
on
1.Su
ppos
e th
at a
set
of
data
can
be
mod
eled
by
a lin
ear
equa
tion
.Exp
lain
the
dif
fere
nce
betw
een
a sc
atte
r pl
ot o
f th
e da
ta a
nd a
gra
ph o
f th
e lin
ear
equa
tion
tha
t m
odel
s th
atda
ta.
Sam
ple
answ
er:T
he s
catte
r plo
t is
a di
scre
te g
raph
.It i
s m
ade
up ju
st o
fth
e in
divi
dual
poi
nts
that
repr
esen
t the
dat
a po
ints
.The
line
ar e
quat
ion
has
a co
ntin
uous
gra
ph th
at is
the
line
that
bes
t fits
the
data
poi
nts.
2.Su
ppos
e th
at t
uiti
on a
t a
stat
e co
llege
was
$35
00 p
er y
ear
in 1
995
and
has
been
incr
easi
ng a
t a
rate
of
$225
per
yea
r.
a.W
rite
a p
redi
ctio
n eq
uati
on t
hat
expr
esse
s th
is in
form
atio
n.y
!35
00 $
225x
b.E
xpla
in t
he m
eani
ng o
f ea
ch v
aria
ble
in y
our
pred
icti
on e
quat
ion.
xre
pres
ents
the
num
ber o
f yea
r sin
ce 1
995
and
yre
pres
ents
the
tuiti
on in
that
yea
r.
3.U
se t
his
mod
el t
o pr
edic
t th
e tu
itio
n at
thi
s co
llege
in 2
007.
$620
0
Hel
pin
g Y
ou
Rem
emb
er
4.L
ook
up t
he w
ord
scat
ter
in a
dic
tion
ary.
How
can
its
defi
niti
on h
elp
you
to r
emem
ber
the
mea
ning
of
the
diff
eren
ce b
etw
een
a sc
atte
r pl
ot a
nd t
he g
raph
of
a lin
ear
equa
tion
?Sa
mpl
e an
swer
:To
scat
term
eans
to b
reak
up
and
go in
man
y di
rect
ions
.Th
e po
ints
on
a sc
atte
r plo
t are
bro
ken
up.I
n a
scat
ter p
lot,
the
poin
tsar
e sc
atte
red
or b
roke
n up
.In
the
grap
h of
a li
near
equ
atio
n,th
e po
ints
are
conn
ecte
d to
form
a c
ontin
uous
line
.
©G
lenc
oe/M
cGra
w-Hi
ll86
Gle
ncoe
Alg
ebra
2
Med
ian-
Fit L
ines
A
med
ian
-fit
lin
eis
a p
arti
cula
r ty
pe o
f lin
e of
fit
.Fol
low
the
ste
ps b
elow
to
find
the
equ
atio
n of
the
med
ian-
fit
line
for
the
data
.
Appr
oxim
ate
Perc
enta
ge o
f Vio
lent
Crim
es C
omm
itted
by
Juve
nile
s Th
at V
ictim
s Re
port
ed to
Law
Enf
orce
men
t
Year
1980
1982
1984
1986
1988
1990
1992
1994
1996
Offe
nder
s36
3533
3231
3029
2930
Sour
ce: U
.S. B
urea
u of
Justi
ce S
tatist
ics
1.D
ivid
e th
e da
ta in
to t
hree
app
roxi
mat
ely
equa
l gro
ups.
The
re s
houl
d al
way
s be
the
sam
e nu
mbe
r of
poi
nts
in t
he f
irst
and
thi
rd g
roup
s.In
thi
s ca
se,t
here
w
ill b
e th
ree
data
poi
nts
in e
ach
grou
p.
Gro
up 1
Gro
up 2
Gro
up 3
Year
Offe
nder
sYe
arO
ffend
ers
Year
Offe
nder
s
2.F
ind
x 1,x
2,an
d x 3
,the
med
ians
of
the
xva
lues
in g
roup
s 1,
2,an
d 3,
resp
ecti
vely
.Fin
d y 1
,y2,
and
y 3,t
he m
edia
ns o
f th
e y
valu
es in
gro
ups
1,2,
and
3,re
spec
tive
ly.
1982
,198
8,19
94;3
5,31
,29
3.F
ind
an e
quat
ion
of t
he li
ne t
hrou
gh (x
1,y 1
) an
d (x
3,y 3
).y
!"
0.5x
$10
26
4.F
ind
Y,t
he y
-coo
rdin
ate
of t
he p
oint
on
the
line
in E
xerc
ise
2 w
ith
an
x-co
ordi
nate
of x
2.32
5.T
he m
edia
n-fi
t lin
e is
par
alle
l to
the
line
in E
xerc
ise
2,bu
t is
one
-thi
rd
clos
er t
o (x
2,y 2
).T
his
mea
ns it
pas
ses
thro
ugh
!x 2,%
2 3% Y#
%1 3% y2".
Fin
d th
is
orde
red
pair
. abo
ut (1
988,
31.6
7)
6.W
rite
an
equa
tion
of
the
med
ian-
fit
line.
y!
"0.
5x$
1025
.67
7.U
se t
he m
edia
n-fi
t lin
e to
pre
dict
the
per
cent
age
of ju
veni
le v
iole
nt c
rim
e of
fend
ers
in 2
010
and
2020
.20
10:a
bout
21%
;202
0:ab
out1
6%
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-5
2-5
© Glencoe/McGraw-Hill A17 Glencoe Algebra 2
An
swer
s
Answers (Lesson 2-6)
Stu
dy
Gu
ide
and I
nte
rven
tion
Spec
ial F
unct
ions
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-6
2-6
©G
lenc
oe/M
cGra
w-Hi
ll87
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 c
hart
belo
w li
sts
som
e sp
ecia
l fun
ctio
ns y
ou s
houl
d be
fam
iliar
wit
h.
Func
tion
Writ
ten
asG
raph
Cons
tant
f(x) "
cho
rizon
tal l
ine
Iden
tity
f(x) "
xlin
e th
roug
h th
e or
igin
with
slo
pe 1
Gre
ates
t Int
eger
Fun
ctio
nf(x
) "%x
&on
e-un
it ho
rizon
tal s
egm
ents
, with
righ
t end
poin
ts m
issin
g, a
rrang
ed
like
step
s
The
gre
ates
t in
tege
r fu
ncti
on is
an
exam
ple
of a
ste
p fu
nct
ion
,a fu
ncti
on w
ith
a gr
aph
that
cons
ists
of
hori
zont
al s
egm
ents
.
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
nsta
nt f
unct
ion
a st
ep f
unct
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
nsta
nt fu
nctio
na
step
func
tion
the
iden
tity
func
tionx
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-Hi
ll88
Gle
ncoe
Alg
ebra
2
Ab
solu
te V
alu
e an
d P
iece
wis
e Fu
nct
ion
sA
noth
er s
peci
al f
unct
ion
is t
heab
solu
te v
alu
e fu
nct
ion
,whi
ch is
als
o ca
lled
a p
iece
wis
e fu
nct
ion
.
Abso
lute
Val
ue F
unct
ion
f(x) "
x
two
rays
that
are
mirr
or im
ages
of e
ach
othe
r and
mee
t at a
poi
nt, t
he v
erte
x
To g
raph
a s
peci
al f
unct
ion,
use
its
defi
niti
on a
nd y
our
know
ledg
e of
the
par
ent
grap
h.F
ind
seve
ral o
rder
ed p
airs
,if
nece
ssar
y.
Gra
ph
f(x
) !
3⏐x⏐
"4.
Fin
d se
vera
l ord
ered
pai
rs.G
raph
the
poi
nts
and
conn
ect
them
.You
wou
ld e
xpec
t th
e gr
aph
to lo
oksi
mila
r to
its
pare
nt f
unct
ion,
f(x)
"x
.
Gra
ph
f(x
) !
!2xif
x(
2x
"1
if x
#2.
Fir
st,g
raph
the
line
ar f
unct
ion
f(x)
"2x
for
x'
2.Si
nce
2 do
es n
otsa
tisf
y th
is in
equa
lity,
stop
wit
h a
circ
le a
t (2
,4).
Nex
t,gr
aph
the
linea
r fu
ncti
on f
(x) "
x!
1 fo
r x
(2.
Sinc
e 2
does
sat
isfy
thi
sin
equa
lity,
begi
n w
ith
a do
t 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)
"2
x#
13.
h(x
) "
dom
ain:
all r
eal
dom
ain:
all r
eal
dom
ain:
all r
eal
num
bers
;ran
ge:
num
bers
;ran
ge:
num
bers
;ran
ge:
all i
nteg
ers
{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
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Spec
ial F
unct
ions
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
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
©G
lencoe/McG
raw-HillA18
Glencoe Algebra 2
Answers
(Lesson 2-6)
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 integers6. g(x) " 2x 7. f(x) " x # 1
D ! all reals, D ! all reals, R ! {y⏐y # 1}R ! nonnegative reals
8. f(x) " 'x if x ' 0 9. h(x) " '3 if x ' !12 if x ( 0 x # 1 if x > 1
D ! all reals, D ! {x⏐x ( "1 or x * 1},R ! {y⏐y ( 0 or y ! 2} R ! {y⏐y * 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) " !2x 4. f(x) " x # 1
D ! all reals, D ! all reals,R ! nonpositive reals R ! nonnegative reals
5. f(x) " 'x # 2 if x ) ! 2 6. h(x) " '4 ! x if x * 03x 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
Pounds
Co
st (
$)
50 15 2510 20 30 35
105
90
75
60
45
30
15
Candy Costs
x
h(x)
O
f(x)
xO
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
© Glencoe/McGraw-Hill A19 Glencoe Algebra 2
An
swer
s
Answers (Lesson 2-6)
Rea
din
g t
o L
earn
Math
emati
csSp
ecia
l Fun
ctio
ns
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-6
2-6
©G
lenc
oe/M
cGra
w-Hi
ll91
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
ucti
on t
o L
esso
n 2-
6 at
the
top
of
page
89
in y
our
text
book
.
•W
hat
is t
he c
ost
of m
ailin
g a
lett
er t
hat
wei
ghs
0.5
ounc
e?$0
.34
or 3
4 ce
nts
•G
ive
thre
e di
ffer
ent
wei
ghts
of
lett
ers
that
wou
ld e
ach
cost
55
cent
s to
mai
l.An
swer
s w
ill v
ary.
Sam
ple
answ
er:1
.1 o
unce
s,1.
9 ou
nces
,2.0
oun
ces
Rea
din
g t
he
Less
on
1.F
ind
the
valu
e of
eac
h ex
pres
sion
.
a.!
3"
%!3&
"
b.6
.2
"%6
.2&"
c.!
4.01
"
%!4.
01&"
2.Te
ll ho
w t
he n
ame
of e
ach
kind
of
func
tion
can
hel
p yo
u re
mem
ber
wha
t th
e gr
aph
look
s lik
e.
a.co
nsta
nt f
unct
ion
Sam
ple
answ
er:S
omet
hing
is c
onst
ant i
f it d
oes
not
chan
ge.T
he y
-val
ues
of a
con
stan
t fun
ctio
n do
not
cha
nge,
so th
egr
aph
is a
hor
izon
tal l
ine.
b.ab
solu
te v
alue
fun
ctio
nSa
mpl
e an
swer
:The
abs
olut
e va
lue
of a
num
ber
tells
you
how
far i
t is
from
0 o
n th
e nu
mbe
r lin
e.It
mak
es n
o di
ffere
nce
whe
ther
you
go
to th
e le
ft or
righ
t so
long
as
you
go th
e sa
me
dist
ance
eac
h tim
e.c.
step
fun
ctio
nSa
mpl
e an
swer
:A s
tep
func
tion’
s gr
aph
look
s lik
e st
eps
that
go
up o
r dow
n.
d.id
enti
ty f
unct
ion
Sam
ple
answ
er:T
he x
- and
y-v
alue
s ar
e al
way
sid
entic
ally
the
sam
e fo
r any
poi
nt o
n th
e gr
aph.
So th
e gr
aph
is a
line
thro
ugh
the
orig
in th
at h
as s
lope
1.
Hel
pin
g Y
ou
Rem
emb
er
3.M
any
stud
ents
fin
d th
e gr
eate
st in
tege
r fu
ncti
on c
onfu
sing
.Exp
lain
how
you
can
use
anu
mbe
r lin
e to
fin
d th
e va
lue
of t
his
func
tion
for
any
rea
l num
ber.
Answ
ers
will
var
y.Sa
mpl
e an
swer
:Dra
w a
num
ber l
ine
that
sho
ws
the
inte
gers
.To
find
the
valu
e of
the
grea
test
inte
ger f
unct
ion
for a
ny re
al n
umbe
r,pl
ace
that
num
ber o
n th
e nu
mbe
r lin
e.If
it is
an
inte
ger,
the
valu
e of
the
func
tion
isth
e nu
mbe
r its
elf.
If no
t,m
ove
to th
e in
tege
r dire
ctly
to th
e le
ft of
the
num
ber y
ou c
hose
.Thi
s in
tege
r will
giv
e th
e va
lue
you
need
.
"5
4.01
66.
2
"3
3
©G
lenc
oe/M
cGra
w-Hi
ll92
Gle
ncoe
Alg
ebra
2
Gre
ates
t Int
eger
Fun
ctio
nsU
se t
he g
reat
est
inte
ger
func
tion
%x&
to e
xplo
re s
ome
unus
ual g
raph
s.It
will
be
hel
pful
to
mak
e a
char
t of
val
ues
for
each
fun
ctio
ns a
nd t
o us
e a
colo
red
pen
or p
enci
l.
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
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-6
2-6
© Glencoe/McGraw-Hill A20 Glencoe Algebra 2
Answers (Lesson 2-7)
Stu
dy
Gu
ide
and I
nte
rven
tion
Gra
phin
g In
equa
litie
s
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-7
2-7
©G
lenc
oe/M
cGra
w-Hi
ll93
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 li
near
equa
tion
,but
wit
h an
ineq
ualit
y si
gn in
stea
d of
an
equa
ls s
ign.
The
gra
ph o
f th
e re
late
dlin
ear
equa
tion
sep
arat
es t
he c
oord
inat
e pl
ane
into
tw
o ha
lf-p
lane
s.T
he li
ne is
the
boun
dary
of
each
hal
f-pl
ane.
To g
raph
a li
near
ineq
ualit
y,fo
llow
the
se s
teps
.
1.G
raph
the
bou
ndar
y,th
at is
,the
rel
ated
line
ar e
quat
ion.
If t
he in
equa
lity
sym
bol i
s )
or (
,the
bou
ndar
y is
sol
id.I
f th
e in
equa
lity
sym
bol i
s '
or *
,the
bou
ndar
y is
das
hed.
2.C
hoos
e a
poin
t no
t on
the
bou
ndar
y an
d te
st it
in t
he in
equa
lity.
(0,0
) is
a g
ood
poin
t to
choo
se if
the
bou
ndar
y do
es n
ot p
ass
thro
ugh
the
orig
in.
3.If
a t
rue
ineq
ualit
y re
sult
s,sh
ade
the
half
-pla
ne c
onta
inin
g yo
ur t
est
poin
t.If
a f
alse
ineq
ualit
y re
sult
s,sh
ade
the
othe
r ha
lf-p
lane
.
Gra
ph
x$
2y#
4.
The
bou
ndar
y is
the
gra
ph o
f x#
2y"
4.
Use
the
slo
pe-i
nter
cept
for
m,y
"!
x#
2,to
gra
ph t
he b
ound
ary
line.
The
bou
ndar
y lin
e sh
ould
be
solid
.
Now
tes
t th
e po
int
(0,0
).
0 #
2(0)
(?
4(x
, y) "
(0, 0
)
0 (
4fa
lse
Shad
e th
e re
gion
tha
t do
es n
otco
ntai
n (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-Hi
ll94
Gle
ncoe
Alg
ebra
2
Gra
ph
Ab
solu
te V
alu
e In
equ
alit
ies
Gra
phin
g ab
solu
te v
alue
ineq
ualit
ies
is s
imila
rto
gra
phin
g lin
ear
ineq
ualit
ies.
The
gra
ph o
f th
e re
late
d ab
solu
te v
alue
equ
atio
n is
the
boun
dary
.Thi
s bo
unda
ry is
gra
phed
as
a so
lid li
ne if
the
ineq
ualit
y is
)or
(,a
nd d
ashe
d if
the
ineq
ualit
y is
'or
*.C
hoos
e a
test
poi
nt n
ot o
n th
e bo
unda
ry t
o de
term
ine
whi
ch r
egio
nto
sha
de.
Gra
ph
y)
3⏐x
"1⏐
.
Fir
st g
raph
the
equ
atio
n y
"3
x!
1.
Sinc
e th
e in
equa
lity
is )
,the
gra
ph o
f th
e bo
unda
ry is
sol
id.
Test
(0,
0).
0 )?
30
!1
(x, y
) "(0
, 0)
0 )?
3!
1!
1"
1
0 )
3tru
e
Shad
e th
e re
gion
tha
t co
ntai
ns (
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
!
35.
x
#y
(4
6.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
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Gra
phin
g In
equa
litie
s
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
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
phin
g In
equa
litie
s
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-7
2-7
©G
lenc
oe/M
cGra
w-Hi
ll95
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-Hi
ll96
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 sys
tem
is b
uyin
g ne
w c
ompu
ters
.The
y w
ill
buy
desk
top
com
pute
rs c
osti
ng $
1000
per
uni
t,an
dno
tebo
ok c
ompu
ters
cos
ting
$12
00 p
er u
nit.
The
tot
al
cost
of
the
com
pute
rs c
anno
t ex
ceed
$80
,000
.
10.W
rite
an
ineq
ualit
y th
at d
escr
ibes
thi
s si
tuat
ion.
1000
d$
1200
n)
80,0
00
11.G
raph
the
ineq
ualit
y.
12.I
f th
e sc
hool
wan
ts t
o bu
y 50
of
the
desk
top
com
pute
rs a
nd 2
5 of
the
not
eboo
k co
mpu
ters
,w
ill t
hey
have
eno
ugh
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
has
edx
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 (A
vera
ge)
Gra
phin
g In
equa
litie
s
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-7
2-7
© Glencoe/McGraw-Hill A22 Glencoe Algebra 2
Answers (Lesson 2-7)
Rea
din
g t
o L
earn
Math
emati
csG
raph
ing
Ineq
ualit
ies
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-7
2-7
©G
lenc
oe/M
cGra
w-Hi
ll97
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
ucti
on t
o L
esso
n 2-
7 at
the
top
of
page
96
in y
our
text
book
.
•W
hich
of
the
com
bina
tion
s of
yar
ds a
nd t
ouch
dow
ns li
sted
wou
ld D
ana
cons
ider
a g
ood
gam
e?Th
e fir
st o
ne:1
68 y
ards
and
3
touc
hdow
ns•
Supp
ose
that
in o
ne o
f th
e ga
mes
Dan
a pl
ays,
Mos
s ge
ts 1
57 r
ecei
ving
yard
s.W
hat
is t
he s
mal
lest
num
ber
of t
ouch
dow
ns h
e m
ust
get
in o
rder
for
Dan
a to
con
side
r th
is a
goo
d ga
me?
3
Rea
din
g t
he
Less
on
1.W
hen
grap
hing
a li
near
ineq
ualit
y in
tw
o va
riab
les,
how
do
you
know
whe
ther
to
mak
eth
e bo
unda
ry a
sol
id li
ne o
r a
dash
ed li
ne?
If th
e sy
mbo
l is
#or
),t
he li
ne is
solid
.If
the
sym
bol i
s *
or (
,the
line
is d
ashe
d.
2.H
ow d
o yo
u kn
ow w
hich
sid
e of
the
bou
ndar
y to
sha
de?
Sam
ple
answ
er:I
f the
test
poin
t giv
es a
true
ineq
ualit
y,sh
ade
the
regi
on c
onta
inin
g th
e te
st p
oint
.If
the
test
poi
nt g
ives
a fa
lse
ineq
ualit
y,sh
ade
the
regi
on n
otco
ntai
ning
the
test
poi
nt.
3.M
atch
eac
h in
equa
lity
wit
h it
s gr
aph.
a.y
*2x
!3
iiib.
y'
!2x
#3
ivc.
y(
2x!
3ii
d.y
(!
2x#
3i
i.ii
.ii
i.iv
.
Hel
pin
g Yo
u R
emem
ber
4.D
escr
ibe
som
e w
ays
in w
hich
gra
phin
g an
ineq
ualit
y in
one
var
iabl
e on
a n
umbe
r lin
e is
sim
ilar
to g
raph
ing
an in
equa
lity
in t
wo
vari
able
s in
a c
oord
inat
e pl
ane.
How
can
wha
tyo
u kn
ow a
bout
gra
phin
g in
equa
litie
s on
a n
umbe
r lin
e he
lp y
ou t
o gr
aph
ineq
ualit
ies
ina
coor
dina
te p
lane
?Sa
mpl
e an
swer
:A b
ound
ary
on a
coo
rdin
ate
grap
h is
sim
ilar t
o an
end
poin
t on
a nu
mbe
r lin
e gr
aph.
A da
shed
line
is s
imila
r to
a ci
rcle
on
a nu
mbe
r lin
e:bo
th a
re o
pen
and
mea
n no
t inc
lude
d;th
eyre
pres
ent t
he s
ymbo
ls *
and
(.A
sol
id li
ne is
sim
ilar t
o a
dot o
n a
num
ber l
ine:
both
are
clo
sed
and
mea
n in
clud
ed;t
hey
repr
esen
t the
sym
bols
#an
d )
.
x
y
O
x
y
Ox
y
O
x
y
O
©G
lenc
oe/M
cGra
w-Hi
ll98
Gle
ncoe
Alg
ebra
2
Alg
ebra
ic P
roof
The
fol
low
ing
para
grap
h st
ates
a r
esul
t yo
u m
ight
be
aske
d to
pro
ve in
am
athe
mat
ics
cour
se.P
arts
of
the
para
grap
h ar
e nu
mbe
red.
01L
et n
be a
pos
itiv
e in
tege
r.
02A
lso,
let
n 1"
s(n 1
) be
the
sum
of
the
squa
res
of t
he d
igit
s in
n.
03T
hen
n 2"
s(n 1
) is
the
sum
of
the
squa
res
of t
he d
igit
s of
n1,
and
n 3"
s(n 2
)is
the
sum
of
the
squa
res
of t
he d
igit
s of
n2.
04In
gen
eral
,nk
"s(
n k!
1) is
the
sum
of
the
squa
res
of t
he d
igit
s of
nk
!1.
05C
onsi
der
the
sequ
ence
:n,n
1,n 2
,n3,
…,n
k,…
.
06In
thi
s se
quen
ce e
ithe
r al
l the
ter
ms
from
som
e k
on h
ave
the
valu
e 1,
07or
som
e te
rm,s
ay n
j,ha
s th
e va
lue
4,so
tha
t th
e ei
ght
term
s 4,
16,3
7,58
,89,
145,
42,a
nd 2
0 ke
ep r
epea
ting
fro
m t
hat
poin
t on
.
Use
th
e p
arag
rap
h t
o an
swer
th
ese
ques
tion
s.
1.U
se t
he s
ente
nce
in li
ne 0
1.L
ist
the
firs
t fi
ve v
alue
s of
n.
1,2,
3,4,
5
2.U
se 9
246
for
nan
d gi
ve a
n ex
ampl
e to
sho
w t
he m
eani
ng o
f lin
e 02
.n 1
!s(
9246
) !13
7,be
caus
e 13
7 !
81 #
4 #
16 #
36
3.In
line
02,
whi
ch s
ymbo
l sho
ws
a fu
ncti
on?
Exp
lain
the
func
tion
in a
sen
tenc
e.s(
n);t
he s
um o
f the
squ
ares
of t
he d
igits
of a
num
ber i
s a
func
tion
of th
e nu
mbe
r
4.Fo
r n
"92
46,f
ind
n 2an
d n 3
as d
escr
ibed
in s
ente
nce
03.
n 2!
59,n
3!
106
5.H
ow d
o th
e fi
rst
four
sen
tenc
es r
elat
e to
sen
tenc
e 05
?Th
ey e
xpla
in h
ow to
com
pute
the
term
s of
the
sequ
ence
.
6.U
se n
"31
and
fin
d th
e fi
rst
four
ter
ms
of t
he s
eque
nce.
31,1
0,1,
1
7.W
hich
sen
tenc
e of
the
par
agra
ph is
illu
stra
ted
by n
"31
?se
nten
ce 0
6
8.U
se n
"61
and
fin
d th
e fi
rst
ten
term
s.61
,37,
58,8
9,14
5,42
,20,
4,16
,37
9.W
hich
sen
tenc
e is
illu
stra
ted
by n
"61
?se
nten
ce 0
7
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
2-7
2-7