Upload
others
View
72
Download
7
Embed Size (px)
Citation preview
Chapter 6Resource 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 6 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: 0-07-828009-5 Algebra 2Chapter 6 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 6-1Study Guide and Intervention . . . . . . . . 313–314Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 315Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 316Reading to Learn Mathematics . . . . . . . . . . 317Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 318
Lesson 6-2Study Guide and Intervention . . . . . . . . 319–320Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 321Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 322Reading to Learn Mathematics . . . . . . . . . . 323Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 324
Lesson 6-3Study Guide and Intervention . . . . . . . . 325–326Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 327Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 328Reading to Learn Mathematics . . . . . . . . . . 329Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 330
Lesson 6-4Study Guide and Intervention . . . . . . . . 331–332Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 333Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 334Reading to Learn Mathematics . . . . . . . . . . 335Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 336
Lesson 6-5Study Guide and Intervention . . . . . . . . 337–338Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 339Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 340Reading to Learn Mathematics . . . . . . . . . . 341Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 342
Lesson 6-6Study Guide and Intervention . . . . . . . . 343–344Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 345Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 346Reading to Learn Mathematics . . . . . . . . . . 347Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 348
Lesson 6-7Study Guide and Intervention . . . . . . . . 349–350Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 351Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 352Reading to Learn Mathematics . . . . . . . . . . 353Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 354
Chapter 6 AssessmentChapter 6 Test, Form 1 . . . . . . . . . . . . 355–356Chapter 6 Test, Form 2A . . . . . . . . . . . 357–358Chapter 6 Test, Form 2B . . . . . . . . . . . 359–360Chapter 6 Test, Form 2C . . . . . . . . . . . 361–362Chapter 6 Test, Form 2D . . . . . . . . . . . 363–364Chapter 6 Test, Form 3 . . . . . . . . . . . . 365–366Chapter 6 Open-Ended Assessment . . . . . . 367Chapter 6 Vocabulary Test/Review . . . . . . . 368Chapter 6 Quizzes 1 & 2 . . . . . . . . . . . . . . . 369Chapter 6 Quizzes 3 & 4 . . . . . . . . . . . . . . . 370Chapter 6 Mid-Chapter Test . . . . . . . . . . . . 371Chapter 6 Cumulative Review . . . . . . . . . . . 372Chapter 6 Standardized Test Practice . . 373–374
Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A32
© Glencoe/McGraw-Hill iv Glencoe Algebra 2
Teacher’s Guide to Using theChapter 6 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 6 Resource Masters includes the core materials neededfor Chapter 6. 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 6-1.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 6Resource 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 342–343. 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 _____
66
© 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 6.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to your AlgebraStudy Notebook to review vocabulary at the end of the chapter.
Vocabulary Term Found on Page Definition/Description/Example
axis of symmetry
completing the square
constant term
discriminant
dihs·KRIH·muh·nuhnt
linear term
maximum value
minimum value
parabola
puh·RA·buh·luh
quadratic equation
kwah·DRA·tihk
Quadratic Formula
(continued on the next page)
© Glencoe/McGraw-Hill viii Glencoe Algebra 2
Vocabulary Term Found on Page Definition/Description/Example
quadratic function
quadratic inequality
quadratic term
roots
Square Root Property
vertex
vertex form
Zero Product Property
zeros
Reading to Learn MathematicsVocabulary Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
66
Study Guide and InterventionGraphing Quadratic Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
6-16-1
© Glencoe/McGraw-Hill 313 Glencoe Algebra 2
Less
on
6-1
Graph Quadratic Functions
Quadratic Function A function defined by an equation of the form f (x) ! ax2 " bx " c, where a # 0
Graph of a Quadratic A parabola with these characteristics: y intercept: c ; axis of symmetry: x ! ;Function x-coordinate of vertex:
Find the y-intercept, the equation of the axis of symmetry, and thex-coordinate of the vertex for the graph of f(x) ! x2 " 3x # 5. Use this informationto graph the function.
a ! 1, b ! $3, and c ! 5, so the y-intercept is 5. The equation of the axis of symmetry is
x ! or . The x-coordinate of the vertex is .
Next make a table of values for x near .
x x2 " 3x # 5 f(x ) (x, f(x ))
0 02 $ 3(0) " 5 5 (0, 5)
1 12 $3(1) " 5 3 (1, 3)
! "2$ 3! " " 5 ! , "
2 22 $ 3(2) " 5 3 (2, 3)
3 32 $ 3(3) " 5 5 (3, 5)
For Exercises 1–3, complete parts a–c for each quadratic function.a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate
of the vertex.b. Make a table of values that includes the vertex.c. Use this information to graph the function.1. f(x) ! x2 " 6x " 8 2. f(x) ! $x2 $2x " 2 3. f(x) ! 2x2 $ 4x " 3
8, x ! "3, "3 2, x ! "1, "1 3, x ! 1, 1
x
f(x)
O
12
8
4
4 8–4
x
f(x)
O
4
–4
–8
4 8–8 –4
x
(x)
O 4–4
4
8
–8
12
–4
x 1 0 2 3f (x) 1 3 3 9
x "1 0 "2 1f (x) 3 2 2 "1
x "3 "2 "1 "4f (x) "1 0 3 0
11%4
3%2
11%4
3%2
3%2
3%2
x
f(x)
O
3%2
3%2
3%2
$($3)%2(1)
$b%2a
$b%2a
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 314 Glencoe Algebra 2
Maximum and Minimum Values The y-coordinate of the vertex of a quadraticfunction is the maximum or minimum value of the function.
Maximum or Minimum Value The graph of f(x ) ! ax2 " bx " c, where a # 0, opens up and has a minimumof a Quadratic Function when a & 0. The graph opens down and has a maximum when a ' 0.
Determine whether each function has a maximum or minimumvalue. Then find the maximum or minimum value of each function.
Study Guide and Intervention (continued)
Graphing Quadratic Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
6-16-1
ExampleExample
a. f(x) ! 3x2 " 6x # 7For this function, a ! 3 and b ! $6.Since a & 0, the graph opens up, and thefunction has a minimum value.The minimum value is the y-coordinateof the vertex. The x-coordinate of the vertex is ! $ ! 1.
Evaluate the function at x ! 1 to find theminimum value.f(1) ! 3(1)2 $ 6(1) " 7 ! 4, so theminimum value of the function is 4.
$6%2(3)
$b%2a
b. f(x) ! 100 " 2x " x2
For this function, a ! $1 and b ! $2.Since a ' 0, the graph opens down, andthe function has a maximum value.The maximum value is the y-coordinate ofthe vertex. The x-coordinate of the vertex is ! $ ! $1.
Evaluate the function at x ! $1 to findthe maximum value.f($1) ! 100 $ 2($1) $ ($1)2 ! 101, sothe minimum value of the function is 101.
$2%2($1)
$b%2a
ExercisesExercises
Determine whether each function has a maximum or minimum value. Then findthe maximum or minimum value of each function.
1. f(x) ! 2x2 $ x " 10 2. f(x) ! x2 " 4x $ 7 3. f(x) ! 3x2 $ 3x " 1
min., 9 min., "11 min.,
4. f(x) ! 16 " 4x $x2 5. f(x) ! x2 $ 7x " 11 6. f(x) ! $x2 " 6x $ 4
max., 20 min., " max., 5
7. f(x) ! x2 " 5x " 2 8. f(x) ! 20 " 6x $ x2 9. f(x) ! 4x2 " x " 3
min., " max., 29 min., 2
10. f(x) ! $x2 $ 4x " 10 11. f(x) ! x2 $ 10x " 5 12. f(x) ! $6x2 " 12x " 21
max., 14 min., "20 max., 27
13. f(x) ! 25x2 " 100x " 350 14. f(x) ! 0.5x2 " 0.3x $ 1.4 15. f(x) ! " $ 8
min., 250 min., "1.445 max., "7 31$
x%4
$x2%2
15$
17$
5$
1$
7$
Skills PracticeGraphing Quadratic Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
6-16-1
© Glencoe/McGraw-Hill 315 Glencoe Algebra 2
Less
on
6-1
For each quadratic function, find the y-intercept, the equation of the axis ofsymmetry, and the x-coordinate of the vertex.
1. f(x) ! 3x2 2. f(x) ! x2 " 1 3. f(x) ! $x2 " 6x $ 150; x ! 0; 0 1; x ! 0; 0 "15; x ! 3; 3
4. f(x) ! 2x2 $ 11 5. f(x) ! x2 $ 10x " 5 6. f(x) ! $2x2 " 8x " 7"11; x ! 0; 0 5; x ! 5; 5 7; x ! 2; 2
Complete parts a–c for each quadratic function.a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate
of the vertex.b. Make a table of values that includes the vertex.c. Use this information to graph the function.
7. f(x) ! $2x2 8. f(x) ! x2 $ 4x " 4 9. f(x) ! x2 $ 6x " 80; x ! 0; 0 4; x ! 2; 2 8; x ! 3; 3
Determine whether each function has a maximum or a minimum value. Then findthe maximum or minimum value of each function.
10. f(x) ! 6x2 11. f(x) ! $8x2 12. f(x) ! x2 " 2xmin.; 0 max.; 0 min.; "1
13. f(x) ! x2 " 2x " 15 14. f(x) ! $x2 " 4x $ 1 15. f(x) ! x2 " 2x $ 3min.; 14 max.; 3 min.; "4
16. f(x) ! $2x2 " 4x $ 3 17. f(x) ! 3x2 " 12x " 3 18. f(x) ! 2x2 " 4x " 1max.; "1 min.; "9 min.; "1
x
f(x)
Ox
f(x)
O
16
12
8
4
2–2 4 6
x
f(x)
O
x 0 2 3 4 6f (x) 8 0 "1 0 8
x "2 0 2 4 6f (x) 16 4 0 4 16
x "2 "1 0 1 2f (x) "8 "2 0 "2 "8
© Glencoe/McGraw-Hill 316 Glencoe Algebra 2
Complete parts a–c for each quadratic function.a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate
of the vertex.b. Make a table of values that includes the vertex.c. Use this information to graph the function.
1. f(x) ! x2 $ 8x " 15 2. f(x) ! $x2 $ 4x " 12 3. f(x) ! 2x2 $ 2x " 115; x ! 4; 4 12; x ! "2; "2 1; x ! 0.5; 0.5
Determine whether each function has a maximum or a minimum value. Then findthe maximum or minimum value of each function.
4. f(x) ! x2 " 2x $ 8 5. f(x) ! x2 $ 6x " 14 6. v(x) ! $x2 " 14x $ 57min.; "9 min.; 5 max.; "8
7. f(x) ! 2x2 " 4x $ 6 8. f(x) ! $x2 " 4x $ 1 9. f(x) ! $%23%x2 " 8x $ 24
min.; "8 max.; 3 max.; 0
10. GRAVITATION From 4 feet above a swimming pool, Susan throws a ball upward with avelocity of 32 feet per second. The height h(t) of the ball t seconds after Susan throws itis given by h(t) ! $16t2 " 32t " 4. Find the maximum height reached by the ball andthe time that this height is reached. 20 ft; 1 s
11. HEALTH CLUBS Last year, the SportsTime Athletic Club charged $20 to participate inan aerobics class. Seventy people attended the classes. The club wants to increase theclass price this year. They expect to lose one customer for each $1 increase in the price.
a. What price should the club charge to maximize the income from the aerobics classes?$45
b. What is the maximum income the SportsTime Athletic Club can expect to make?$2025
16
12
8
4
x
f(x)
O 2–2–4–6x
f(x)
O
16
12
8
4
2 4 6 8
x "1 0 0.5 1 2f (x) 5 1 0.5 1 5
x "6 "4 "2 0 2f (x) 0 12 16 12 0
x 0 2 4 6 8f (x) 15 3 "1 3 15
Practice (Average)
Graphing Quadratic Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
6-16-1
Reading to Learn MathematicsGraphing Quadratic Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
6-16-1
© Glencoe/McGraw-Hill 317 Glencoe Algebra 2
Less
on
6-1
Pre-Activity How can income from a rock concert be maximized?
Read the introduction to Lesson 6-1 at the top of page 286 in your textbook.• Based on the graph in your textbook, for what ticket price is the income
the greatest? $40• Use the graph to estimate the maximum income. about $72,000
Reading the Lesson1. a. For the quadratic function f(x) ! 2x2 " 5x " 3, 2x2 is the term,
5x is the term, and 3 is the term.
b. For the quadratic function f(x) ! $4 " x $ 3x2, a ! , b ! , and
c ! .
2. Consider the quadratic function f(x) ! ax2 " bx " c, where a # 0.
a. The graph of this function is a .
b. The y-intercept is .
c. The axis of symmetry is the line .
d. If a & 0, then the graph opens and the function has a
value.
e. If a ' 0, then the graph opens and the function has a
value.
3. Refer to the graph at the right as you complete the following sentences.
a. The curve is called a .
b. The line x ! $2 is called the .
c. The point ($2, 4) is called the .
d. Because the graph contains the point (0, $1), $1 is
the .
Helping You Remember4. How can you remember the way to use the x2 term of a quadratic function to tell
whether the function has a maximum or a minimum value? Sample answer:Remember that the graph of f(x) ! x2 (with a % 0) is a U-shaped curvethat opens up and has a minimum. The graph of g(x) ! "x2 (with a & 0)is just the opposite. It opens down and has a maximum.
y-intercept
vertexaxis of symmetry
parabola
x
f(x)
O(0, –1)
(–2, 4)
maximumdownward
minimumupward
x ! "$2ba$
c
parabola
"41"3
constantlinearquadratic
© Glencoe/McGraw-Hill 318 Glencoe Algebra 2
Finding the Axis of Symmetry of a ParabolaAs you know, if f(x) ! ax2 " bx " c is a quadratic function, the values of x
that make f(x) equal to zero are and .
The average of these two number values is $%2ba%.
The function f(x) has its maximum or minimum
value when x ! $%2ba%. Since the axis of symmetry
of the graph of f (x) passes through the point where the maximum or minimum occurs, the axis of
symmetry has the equation x ! $%2ba%.
Find the vertex and axis of symmetry for f(x) ! 5x2 # 10x " 7.
Use x ! $%2ba%.
x ! $%21(05)% ! $1 The x-coordinate of the vertex is $1.
Substitute x ! $1 in f(x) ! 5x2 " 10x $ 7.f($1) ! 5($1)2 " 10($1) $ 7 ! $12The vertex is ($1,$12).The axis of symmetry is x ! $%2
ba%, or x ! $1.
Find the vertex and axis of symmetry for the graph of each function using x ! "$2
ba$.
1. f(x) ! x2 $ 4x $ 8 2. g(x) ! $4x2 $ 8x " 3
3. y ! $x2 " 8x " 3 4. f(x) ! 2x2 " 6x " 5
5. A(x) ! x2 " 12x " 36 6. k(x) ! $2x2 " 2x $ 6
O
f(x)
x
– –, f( ( (( b––2a b––2a
b––2ax = –
f(x) = ax2 + bx + c
$b $ #b2 $ 4$ac$%%%2a
$b " #b2 $ 4$ac$%%%2a
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
6-16-1
ExampleExample
Study Guide and InterventionSolving Quadratic Equations by Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
6-26-2
© Glencoe/McGraw-Hill 319 Glencoe Algebra 2
Less
on
6-2
Solve Quadratic Equations
Quadratic Equation A quadratic equation has the form ax2 " bx " c ! 0, where a # 0.
Roots of a Quadratic Equation solution(s) of the equation, or the zero(s) of the related quadratic function
The zeros of a quadratic function are the x-intercepts of its graph. Therefore, finding the x-intercepts is one way of solving the related quadratic equation.
Solve x2 # x " 6 ! 0 by graphing.
Graph the related function f(x) ! x2 " x $ 6.
The x-coordinate of the vertex is ! $ , and the equation of the
axis of symmetry is x ! $ .
Make a table of values using x-values around $ .
x $1 $ 0 1 2
f(x) $6 $6 $6 $4 0
From the table and the graph, we can see that the zeros of the function are 2 and $3.
Solve each equation by graphing.
1. x2 " 2x $ 8 ! 0 2, "4 2. x2 $ 4x $ 5 ! 0 5, "1 3. x2 $ 5x " 4 ! 0 1, 4
4. x2 $ 10x " 21 ! 0 5. x2 " 4x " 6 ! 0 6. 4x2 " 4x " 1 ! 0
3, 7 no real solutions " 1$
x
f(x)
Ox
f(x)
O
x
f(x)
O
x
f(x)
O
x
f(x)
Ox
f(x)
O
1%4
1%2
1%2
1%2
1%2
$b%2a x
f(x)
O
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 320 Glencoe Algebra 2
Estimate Solutions Often, you may not be able to find exact solutions to quadraticequations by graphing. But you can use the graph to estimate solutions.
Solve x2 " 2x " 2 ! 0 by graphing. If exact roots cannot be found,state the consecutive integers between which the roots are located.
The equation of the axis of symmetry of the related function is
x ! $ ! 1, so the vertex has x-coordinate 1. Make a table of values.
x $1 0 1 2 3
f (x) 1 $2 $3 $2 1
The x-intercepts of the graph are between 2 and 3 and between 0 and$1. So one solution is between 2 and 3, and the other solution isbetween 0 and $1.
Solve the equations by graphing. If exact roots cannot be found, state theconsecutive integers between which the roots are located.
1. x2 $ 4x " 2 ! 0 2. x2 " 6x " 6 ! 0 3. x2 " 4x " 2! 0
between 0 and 1; between "2 and "1; between "1 and 0;between 3 and 4 between "5 and "4 between "4 and "3
4. $x2 " 2x " 4 ! 0 5. 2x2 $ 12x " 17 ! 0 6. $ x2 " x " ! 0
between 3 and 4; between 2 and 3; between "2 and "1;between "2 and "1 between 3 and 4 between 3 and 4
x
f(x)
O
x
f(x)
Ox
f(x)
O
5%2
1%2
x
f(x)
Ox
f(x)
Ox
f(x)
O
$2%2(1)
x
f(x)
O
Study Guide and Intervention (continued)
Solving Quadratic Equations by Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
6-26-2
ExampleExample
ExercisesExercises
Skills PracticeSolving Quadratic Equations By Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
6-26-2
© Glencoe/McGraw-Hill 321 Glencoe Algebra 2
Less
on
6-2
Use the related graph of each equation to determine its solutions.
1. x2 " 2x $ 3 ! 0 2. $x2 $ 6x $ 9 ! 0 3. 3x2 " 4x " 3 ! 0
"3, 1 "3 no real solutions
Solve each equation by graphing. If exact roots cannot be found, state theconsecutive integers between which the roots are located.
4. x2 $ 6x " 5 ! 0 5. $x2 " 2x $ 4 ! 0 6. x2 $ 6x " 4 ! 01, 5 no real solutions between 0 and 1;
between 5 and 6
Use a quadratic equation to find two real numbers that satisfy each situation, orshow that no such numbers exist.
7. Their sum is $4, and their product is 0. 8. Their sum is 0, and their product is $36.
"x2 " 4x ! 0; 0, "4 "x2 # 36 ! 0; "6, 6
x
f(x)
O 6–6 12–12
36
24
12
x
f(x)
O
x
f(x)
O
x
f(x)
O
x
f(x)
O
x
f(x)
O
f(x) ! 3x2 # 4x # 3
x
f(x)
O
f(x) ! "x2 " 6x " 9
x
f(x)
O
f(x) ! x2 # 2x " 3
© Glencoe/McGraw-Hill 322 Glencoe Algebra 2
Use the related graph of each equation to determine its solutions.
1. $3x2 " 3 ! 0 2. 3x2 " x " 3 ! 0 3. x2 $ 3x " 2 ! 0
"1, 1 no real solutions 1, 2Solve each equation by graphing. If exact roots cannot be found, state theconsecutive integers between which the roots are located.
4. $2x2 $ 6x " 5 ! 0 5. x2 " 10x " 24 ! 0 6. 2x2 $ x $ 6 ! 0between 0 and 1; "6, "4 between "2 and "1, between "4 and "3 2
Use a quadratic equation to find two real numbers that satisfy each situation, orshow that no such numbers exist.
7. Their sum is 1, and their product is $6. 8. Their sum is 5, and their product is 8.
For Exercises 9 and 10, use the formula h(t) ! v0t " 16t2, where h(t) is the heightof an object in feet, v0 is the object’s initial velocity in feet per second, and t is thetime in seconds.
9. BASEBALL Marta throws a baseball with an initial upward velocity of 60 feet per second.Ignoring Marta’s height, how long after she releases the ball will it hit the ground? 3.75 s
10. VOLCANOES A volcanic eruption blasts a boulder upward with an initial velocity of240 feet per second. How long will it take the boulder to hit the ground if it lands at thesame elevation from which it was ejected? 15 s
"x2 # 5x " 8 ! 0;no such realnumbers exist
"x2 # x # 6 ! 0;3, "2
x
f(x)
O
x
f(x)
O
x
f(x)
O–4 –2–6
12
8
4
x
f(x)
O
f(x) ! x2 " 3x # 2
x
f(x)
O
f(x) ! 3x2 # x # 3
x
f(x)
O
f(x) ! "3x2 # 3
Practice (Average)
Solving Quadratic Equations By Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
6-26-2
Reading to Learn MathematicsSolving Quadratic Equations by Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
6-26-2
© Glencoe/McGraw-Hill 323 Glencoe Algebra 2
Less
on
6-2
Pre-Activity How does a quadratic function model a free-fall ride?
Read the introduction to Lesson 6-2 at the top of page 294 in your textbook.
Write a quadratic function that describes the height of a ball t seconds afterit is dropped from a height of 125 feet. h(t) ! "16t 2 # 125
Reading the Lesson
1. The graph of the quadratic function f(x) ! $x2 " x " 6 is shown at the right. Use the graph to find the solutions of thequadratic equation $x2 " x " 6 ! 0. "2 and 3
2. Sketch a graph to illustrate each situation.
a. A parabola that opens b. A parabola that opens c. A parabola that opensdownward and represents a upward and represents a downward and quadratic function with two quadratic function with represents a real zeros, both of which are exactly one real zero. The quadratic function negative numbers. zero is a positive number. with no real zeros.
Helping You Remember
3. Think of a memory aid that can help you recall what is meant by the zeros of a quadraticfunction.
Sample answer: The basic facts about a subject are sometimes calledthe ABCs. In the case of zeros, the ABCs are the XYZs, because thezeros are the x-values that make the y-values equal to zero.
x
y
Ox
y
Ox
y
O
x
y
O
© Glencoe/McGraw-Hill 324 Glencoe Algebra 2
Graphing Absolute Value Equations You can solve absolute value equations in much the same way you solved quadratic equations. Graph the related absolute value function for each equation using a graphing calculator. Then use the ZERO feature in the CALC menu to find its real solutions, if any. Recall that solutions are points where the graph intersects the x-axis.
For each equation, make a sketch of the related graph and find the solutions rounded to the nearest hundredth.
1. | x " 5| ! 0 2. |4x $ 3| " 5 ! 0 3. | x $ 7| ! 0
5 No solutions 7
4. | x " 3| $ 8 ! 0 5. $| x " 3| " 6 ! 0 6. | x $ 2| $ 3 ! 0
"11, 5 "9, 3 "1, 5
7. |3x " 4| ! 2 8. | x " 12| ! 10 9. | x | $ 3 ! 0
"2, "$23$ "22, "2 "3, 3
10. Explain how solving absolute value equations algebraically and finding zeros of absolute value functions graphically are related.Sample answer: values of x when solving algebraically are the x-intercepts (or zeros) of the function when graphed.
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
6-26-2
Study Guide and InterventionSolving Quadratic Equations by Factoring
NAME ______________________________________________ DATE ____________ PERIOD _____
6-36-3
© Glencoe/McGraw-Hill 325 Glencoe Algebra 2
Less
on
6-3
Solve Equations by Factoring When you use factoring to solve a quadratic equation,you use the following property.
Zero Product Property For any real numbers a and b, if ab ! 0, then either a ! 0 or b !0, or both a and b ! 0.
Solve each equation by factoring.ExampleExamplea. 3x2 ! 15x
3x2 ! 15x Original equation
3x2 $ 15x ! 0 Subtract 15x from both sides.
3x(x $ 5) ! 0 Factor the binomial.
3x ! 0 or x $ 5 ! 0 Zero Product Property
x ! 0 or x ! 5 Solve each equation.
The solution set is {0, 5}.
b. 4x2 " 5x ! 214x2 $ 5x ! 21 Original equation
4x2 $ 5x $ 21 ! 0 Subtract 21 from both sides.
(4x " 7)(x $ 3) ! 0 Factor the trinomial.
4x " 7 ! 0 or x $ 3 ! 0 Zero Product Property
x ! $ or x ! 3 Solve each equation.
The solution set is %$ , 3&.7%4
7%4
ExercisesExercises
Solve each equation by factoring.
1. 6x2 $ 2x ! 0 2. x2 ! 7x 3. 20x2 ! $25x
!0, " {0, 7} !0, " "4. 6x2 ! 7x 5. 6x2 $ 27x ! 0 6. 12x2 $ 8x ! 0
!0, " !0, " !0, "7. x2 " x $ 30 ! 0 8. 2x2 $ x $ 3 ! 0 9. x2 " 14x " 33 ! 0
{5, "6} ! , "1" {"11, "3}
10. 4x2 " 27x $ 7 ! 0 11. 3x2 " 29x $ 10 ! 0 12. 6x2 $ 5x $ 4 ! 0
! , "7" !"10, " !" , "13. 12x2 $ 8x " 1 ! 0 14. 5x2 " 28x $ 12 ! 0 15. 2x2 $ 250x " 5000 ! 0
! , " ! , "6" {100, 25}
16. 2x2 $ 11x $ 40 ! 0 17. 2x2 " 21x $ 11 ! 0 18. 3x2 " 2x $ 21 ! 0
!8, " " !"11, " ! , "3"19. 8x2 $ 14x " 3 ! 0 20. 6x2 " 11x $ 2 ! 0 21. 5x2 " 17x $ 12 ! 0
! , " !"2, " ! , "4"22. 12x2 " 25x " 12 ! 0 23. 12x2 " 18x " 6 ! 0 24. 7x2 $ 36x " 5 ! 0
!" , " " !" , "1" ! , 5"1$
1$
3$
4$
3$
1$
1$
3$
7$
1$
5$
2$
1$
1$
4$
1$
1$
1$
3$
2$
9$
7$
5$
1$
© Glencoe/McGraw-Hill 326 Glencoe Algebra 2
Write Quadratic Equations To write a quadratic equation with roots p and q, let(x $ p)(x $ q) ! 0. Then multiply using FOIL.
Write a quadratic equation with the given roots. Write the equationin the form ax2 # bx # c ! 0.
Study Guide and Intervention (continued)
Solving Quadratic Equations by Factoring
NAME ______________________________________________ DATE ____________ PERIOD _____
6-36-3
ExampleExample
a. 3, "5(x $ p)(x $ q) ! 0 Write the pattern.
(x $ 3)[x $ ($5)] ! 0 Replace p with 3, q with $5.
(x $ 3)(x " 5) ! 0 Simplify.
x2 " 2x $ 15 ! 0 Use FOIL.
The equation x2 " 2x $ 15 ! 0 has roots 3 and $5.
b. " ,
(x $ p)(x $ q) ! 0
'x $ !$ "(!x $ " ! 0
!x " "!x $ " ! 0
( ! 0
! 24 ( 0
24x2 " 13x $ 7 ! 0
The equation 24x2 " 13x $ 7 ! 0 has
roots $ and .1%3
7%8
24 ( (8x " 7)(3x $ 1)%%%24
(3x $ 1)%3
(8x " 7)%8
1%3
7%8
1%3
7%8
1$3
7$8
ExercisesExercises
Write a quadratic equation with the given roots. Write the equation in the formax2 # bx # c ! 0.
1. 3, $4 2. $8, $2 3. 1, 9x2 # x " 12 ! 0 x2 # 10x # 16 ! 0 x2 " 10x # 9 ! 0
4. $5 5. 10, 7 6. $2, 15x2 # 10x # 25 ! 0 x2 " 17x # 70 ! 0 x2 " 13x " 30 ! 0
7. $ , 5 8. 2, 9. $7,
3x2 " 14x " 5 ! 0 3x2 " 8x # 4 ! 0 4x2 # 25x " 21 ! 0
10. 3, 11. $ , $1 12. 9,
5x2 " 17x # 6 ! 0 9x2 # 13x # 4 ! 0 6x2 " 55x # 9 ! 0
13. , $ 14. , $ 15. ,
9x2 " 4 ! 0 8x2 " 6x " 5 ! 0 35x2 " 22x # 3 ! 0
16. $ , 17. , 18. ,
16x2 " 42x " 49 8x2 " 10x # 3 ! 0 48x2 " 14x # 1 ! 0
1%6
1%8
3%4
1%2
7%2
7%8
1%5
3%7
1%2
5%4
2%3
2%3
1%6
4%9
2%5
3%4
2%3
1%3
Skills PracticeSolving Quadratic Equations by Factoring
NAME ______________________________________________ DATE ____________ PERIOD _____
6-36-3
© Glencoe/McGraw-Hill 327 Glencoe Algebra 2
Less
on
6-3
Solve each equation by factoring.
1. x2 ! 64 {"8, 8} 2. x2 $ 100 ! 0 {10, "10}
3. x2 $ 3x " 2 ! 0 {1, 2} 4. x2 $ 4x " 3 ! 0 {1, 3}
5. x2 " 2x $ 3 ! 0 {1, "3} 6. x2 $ 3x $ 10 ! 0 {5, "2}
7. x2 $ 6x " 5 ! 0 {1, 5} 8. x2 $ 9x ! 0 {0, 9}
9. $x2 " 6x ! 0 {0, 6} 10. x2 " 6x " 8 ! 0 {"2, "4}
11. x2 ! $5x {0, "5} 12. x2 $ 14x " 49 ! 0 {7}
13. x2 " 6 ! 5x {2, 3} 14. x2 " 18x ! $81 {"9}
15. x2 $ 4x ! 21 {"3, 7} 16. 2x2 " 5x $ 3 ! 0 ! , "3"
17. 4x2 " 5x $ 6 ! 0 ! , "2" 18. 3x2 $ 13x $ 10 ! 0 !" , 5"
Write a quadratic equation with the given roots. Write the equation in the formax2 # bx # c ! 0, where a, b, and c are integers.
19. 1, 4 x2 " 5x # 4 ! 0 20. 6, $9 x2 # 3x " 54 ! 0
21. $2, $5 x2 # 7x # 10 ! 0 22. 0, 7 x2 " 7x ! 0
23. $ , $3 3x2 #10x # 3 ! 0 24. $ , 8x2 " 2x " 3 ! 0
25. Find two consecutive integers whose product is 272. 16, 17
3%4
1%2
1%3
2$
3$
1$
© Glencoe/McGraw-Hill 328 Glencoe Algebra 2
Solve each equation by factoring.
1. x2 $ 4x $ 12 ! 0 {6, "2} 2. x2 $ 16x " 64 ! 0 {8} 3. x2 $ 20x " 100 ! 0 {10}
4. x2 $ 6x " 8 ! 0 {2, 4} 5. x2 " 3x " 2 ! 0 {"2, "1} 6. x2 $ 9x " 14 ! 0 {2, 7}
7. x2 $ 4x ! 0 {0, 4} 8. 7x2 ! 4x !0, " 9. x2 " 25 ! 10x {5}
10. 10x2 ! 9x !0, " 11. x2 ! 2x " 99 {"9, 11}
12. x2 " 12x ! $36 {"6} 13. 5x2 $ 35x " 60 ! 0 {3, 4}
14. 36x2 ! 25 ! , " " 15. 2x2 $ 8x $ 90 ! 0 {9, "5}
16. 3x2 " 2x $ 1 ! 0 ! , "1" 17. 6x2 ! 9x !0, "18. 3x2 " 24x " 45 ! 0 {"5, "3} 19. 15x2 " 19x " 6 ! 0 !" , " "20. 3x2 $ 8x ! $4 !2, " 21. 6x2 ! 5x " 6 ! , " "Write a quadratic equation with the given roots. Write the equation in the formax2 # bx # c ! 0, where a, b, and c are integers.
22. 7, 2 23. 0, 3 24. $5, 8x2 " 9x # 14 ! 0 x2 " 3x ! 0 x2 " 3x " 40 ! 0
25. $7, $8 26. $6, $3 27. 3, $4x2 # 15x # 56 ! 0 x2 # 9x # 18 ! 0 x2 # x " 12 ! 0
28. 1, 29. , 2 30. 0, $
2x2 " 3x # 1 ! 0 3x2 " 7x # 2 ! 0 2x2 # 7x ! 0
31. , $3 32. 4, 33. $ , $
3x2 # 8x " 3 ! 0 3x2 " 13x # 4 ! 0 15x2 # 22x # 8 ! 034. NUMBER THEORY Find two consecutive even positive integers whose product is 624.
24, 2635. NUMBER THEORY Find two consecutive odd positive integers whose product is 323.
17, 1936. GEOMETRY The length of a rectangle is 2 feet more than its width. Find the
dimensions of the rectangle if its area is 63 square feet. 7 ft by 9 ft37. PHOTOGRAPHY The length and width of a 6-inch by 8-inch photograph are reduced by
the same amount to make a new photograph whose area is half that of the original. Byhow many inches will the dimensions of the photograph have to be reduced? 2 in.
4%5
2%3
1%3
1%3
7%2
1%3
1%2
2$
3$
2$
2$
3$
3$
1$
5$
5$
9$
4$
Practice (Average)
Solving Quadratic Equations by Factoring
NAME ______________________________________________ DATE ____________ PERIOD _____
6-36-3
Reading to Learn MathematicsSolving Quadratic Equations by Factoring
NAME ______________________________________________ DATE ____________ PERIOD _____
6-36-3
© Glencoe/McGraw-Hill 329 Glencoe Algebra 2
Less
on
6-3
Pre-Activity How is the Zero Product Property used in geometry?
Read the introduction to Lesson 6-3 at the top of page 301 in your textbook.
What does the expression x(x " 5) mean in this situation?
It represents the area of the rectangle, since the area is theproduct of the width and length.
Reading the Lesson
1. The solution of a quadratic equation by factoring is shown below. Give the reason foreach step of the solution.
x2 $ 10x ! $21 Original equation
x2 $ 10x " 21 ! 0 Add 21 to each side.(x $ 3)(x $ 7) ! 0 Factor the trinomial.x $ 3 ! 0 or x $ 7 ! 0 Zero Product Property
x ! 3 x ! 7 Solve each equation.The solution set is .
2. On an algebra quiz, students were asked to write a quadratic equation with $7 and 5 asits roots. The work that three students in the class wrote on their papers is shown below.
Marla Rosa Larry(x $7)(x " 5) ! 0 (x " 7)(x $ 5) ! 0 (x " 7)(x $ 5) ! 0x2 $ 2x $ 35 ! 0 x2 " 2x $ 35 ! 0 x2 $ 2x $ 35 ! 0
Who is correct? RosaExplain the errors in the other two students’ work.
Sample answer: Marla used the wrong factors. Larry used the correctfactors but multiplied them incorrectly.
Helping You Remember
3. A good way to remember a concept is to represent it in more than one way. Describe analgebraic way and a graphical way to recognize a quadratic equation that has a doubleroot.
Sample answer: Algebraic: Write the equation in the standard form ax2 # bx # c ! 0 and examine the trinomial. If it is a perfect squaretrinomial, the quadratic function has a double root. Graphical: Graph therelated quadratic function. If the parabola has exactly one x-intercept,then the equation has a double root.
{3, 7}
© Glencoe/McGraw-Hill 330 Glencoe Algebra 2
Euler’s Formula for Prime NumbersMany mathematicians have searched for a formula that would generate prime numbers. One such formula was proposed by Euler and uses a quadratic polynomial, x2 " x " 41.
Find the values of x2 # x # 41 for the given values of x. State whether each value of the polynomial is or is not a prime number.
1. x ! 0 2. x ! 1 3. x ! 2
4. x ! 3 5. x ! 4 6. x ! 5
7. x ! 6 8. x ! 17 9. x ! 28
10. x ! 29 11. x ! 30 12. x ! 35
13. Does the formula produce all prime numbers greater than 40? Give examples in your answer.
14. Euler’s formula produces primes for many values of x, but it does not work for all of them. Find the first value of x for which the formula fails.(Hint: Try multiples of ten.)
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
6-36-3
Study Guide and InterventionCompleting the Square
NAME ______________________________________________ DATE ____________ PERIOD _____
6-46-4
© Glencoe/McGraw-Hill 331 Glencoe Algebra 2
Less
on
6-4
Square Root Property Use the following property to solve a quadratic equation that isin the form “perfect square trinomial ! constant.”
Square Root Property For any real number x if x2 ! n, then x ! )n.
Solve each equation by using the Square Root Property.ExampleExamplea. x2 " 8x # 16 ! 25
x2 $ 8x " 16 ! 25(x $ 4)2 ! 25
x $ 4 ! #25$ or x $ 4 ! $#25$x ! 5 " 4 ! 9 or x ! $5 " 4 ! $1
The solution set is {9, $1}.
b. 4x2 " 20x # 25 ! 324x2 $ 20x " 25 ! 32
(2x $ 5)2 ! 322x $ 5 ! #32$ or 2x $ 5 ! $#32$2x $ 5 ! 4#2$ or 2x $ 5 ! $4#2$
x !
The solution set is % &.5 ) 4#2$%%2
5 ) 4#2$%%2
ExercisesExercises
Solve each equation by using the Square Root Property.
1. x2 $ 18x " 81 ! 49 2. x2 " 20x " 100 ! 64 3. 4x2 " 4x " 1 ! 16
{2, 16} {"2, "18} ! , " "
4. 36x2 " 12x " 1 ! 18 5. 9x2 $ 12x " 4 ! 4 6. 25x2 " 40x " 16 ! 28
! " !0, " ! "
7. 4x2 $ 28x " 49 ! 64 8. 16x2 " 24x " 9 ! 81 9. 100x2 $ 60x " 9 ! 121
! , " " ! , "3" {"0.8, 1.4}
10. 25x2 " 20x " 4 ! 75 11. 36x2 " 48x " 16 ! 12 12. 25x2 $ 30x " 9 ! 96
! " ! " ! "3 ' 4#6$$$"2 ' #3$$$"2 ' 5#3$$$
3$
1$
15$
"4 ' 2#7$$$4$"1 ' 3#2$$$
5$
3$
© Glencoe/McGraw-Hill 332 Glencoe Algebra 2
Complete the Square To complete the square for a quadratic expression of the form x2 " bx, follow these steps.
1. Find . ➞ 2. Square . ➞ 3. Add ! "2to x2 " bx.b
%2b%2
b%2
Study Guide and Intervention (continued)
Completing the Square
NAME ______________________________________________ DATE ____________ PERIOD _____
6-46-4
Find the value ofc that makes x2 # 22x # c aperfect square trinomial. Thenwrite the trinomial as thesquare of a binomial.
Step 1 b ! 22; ! 11
Step 2 112 ! 121Step 3 c ! 121
The trinomial is x2 " 22x " 121,which can be written as (x " 11)2.
b%2
Solve 2x2 " 8x " 24 ! 0 bycompleting the square.
2x2 $ 8x $ 24 ! 0 Original equation
! Divide each side by 2.
x2 $ 4x $ 12 ! 0 x2 $ 4x $ 12 is not a perfect square.x2 $ 4x ! 12 Add 12 to each side.
x2 $ 4x " 4 ! 12 " 4 Since !$ "2
! 4, add 4 to each side.
(x $ 2)2 ! 16 Factor the square.x $ 2 ! )4 Square Root Property
x ! 6 or x ! $ 2 Solve each equation.
The solution set is {6, $2}.
4%2
0%2
2x2 $ 8x $ 24%%2
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find the value of c that makes each trinomial a perfect square. Then write thetrinomial as a perfect square.
1. x2 $ 10x " c 2. x2 " 60x " c 3. x2 $ 3x " c
25; (x " 5)2 900; (x # 30)2 ; %x " &2
4. x2 " 3.2x " c 5. x2 " x " c 6. x2 $ 2.5x " c
2.56; (x # 1.6)2 ; %x # &2 1.5625; (x " 1.25)2
Solve each equation by completing the square.
7. y2 $ 4y $ 5 ! 0 8. x2 $ 8x $ 65 ! 0 9. s2 $ 10s " 21 ! 0"1, 5 "5, 13 3, 7
10. 2x2 $ 3x " 1 ! 0 11. 2x2 $ 13x $ 7 ! 0 12. 25x2 " 40x $ 9 ! 0
1, " , 7 , "
13. x2 " 4x " 1 ! 0 14. y2 " 12y " 4 ! 0 15. t2 " 3t $ 8 ! 0
"2 ' #3$ "6 ' 4#2$ "3 ' #41$$$2
9$
1$
1$
1$
1$
1$
1%2
3$
9$
Skills PracticeCompleting the Square
NAME ______________________________________________ DATE ____________ PERIOD _____
6-46-4
© Glencoe/McGraw-Hill 333 Glencoe Algebra 2
Less
on
6-4
Solve each equation by using the Square Root Property.
1. x2 $ 8x " 16 ! 1 3, 5 2. x2 " 4x " 4 ! 1 "1, "3
3. x2 " 12x " 36 ! 25 "1, "11 4. 4x2 $ 4x " 1 ! 9 "1, 2
5. x2 " 4x " 4 ! 2 "2 ' #2$ 6. x2 $ 2x " 1 ! 5 1 ' #5$
7. x2 $ 6x " 9 ! 7 3 ' #7$ 8. x2 " 16x " 64 ! 15 "8 ' #15$
Find the value of c that makes each trinomial a perfect square. Then write thetrinomial as a perfect square.
9. x2 " 10x " c 25; (x # 5)2 10. x2 $ 14x " c 49; (x " 7)2
11. x2 " 24x " c 144; (x # 12)2 12. x2 " 5x " c ; %x # &2
13. x2 $ 9x " c ; %x " &2 14. x2 $ x " c ; %x " &2
Solve each equation by completing the square.
15. x2 $ 13x " 36 ! 0 4, 9 16. x2 " 3x ! 0 0, "3
17. x2 " x $ 6 ! 0 2, "3 18. x2 $ 4x $ 13 ! 0 2 ' #17$
19. 2x2 " 7x $ 4 ! 0 "4, 20. 3x2 " 2x $ 1 ! 0 , "1
21. x2 " 3x $ 6 ! 0 22. x2 $ x $ 3 ! 0
23. x2 ! $11 'i #11$ 24. x2 $ 2x " 4 ! 0 1 ' i #3$
1 ' #13$$$2"3 ' #33$$$2
1$
1$
1$
1$
9$
81$
5$
25$
© Glencoe/McGraw-Hill 334 Glencoe Algebra 2
Solve each equation by using the Square Root Property.
1. x2 " 8x " 16 ! 1 2. x2 " 6x " 9 ! 1 3. x2 " 10x " 25 ! 16
"5, "3 "4, "2 "9, "1
4. x2 $ 14x " 49 ! 9 5. 4x2 " 12x " 9 ! 4 6. x2 $ 8x " 16 ! 8
4, 10 " , " 4 ' 2#2$
7. x2 $ 6x " 9 ! 5 8. x2 $ 2x " 1 ! 2 9. 9x2 $ 6x " 1 ! 2
3 ' #5$ 1 ' #2$
Find the value of c that makes each trinomial a perfect square. Then write thetrinomial as a perfect square.
10. x2 " 12x " c 11. x2 $ 20x " c 12. x2 " 11x " c
36; (x # 6)2 100; (x " 10)2 ; %x # &2
13. x2 " 0.8x " c 14. x2 $ 2.2x " c 15. x2 $ 0.36x " c
0.16; (x # 0.4)2 1.21; (x " 1.1)2 0.0324; (x " 0.18)2
16. x2 " x " c 17. x2 $ x " c 18. x2 $ x " c
; %x # &2 ; %x " &2 ; %x " &2
Solve each equation by completing the square.
19. x2 " 6x " 8 ! 0 "4, "2 20. 3x2 " x $ 2 ! 0 , "1 21. 3x2 $ 5x " 2 ! 0 1,
22. x2 " 18 ! 9x 23. x2 $ 14x " 19 ! 0 24. x2 " 16x $ 7 ! 06, 3 7 ' #30$ "8 ' #71$
25. 2x2 " 8x $ 3 ! 0 26. x2 " x $ 5 ! 0 27. 2x2 $ 10x " 5 ! 0
28. x2 " 3x " 6 ! 0 29. 2x2 " 5x " 6 ! 0 30. 7x2 " 6x " 2 ! 0
31. GEOMETRY When the dimensions of a cube are reduced by 4 inches on each side, thesurface area of the new cube is 864 square inches. What were the dimensions of theoriginal cube? 16 in. by 16 in. by 16 in.
32. INVESTMENTS The amount of money A in an account in which P dollars is invested for2 years is given by the formula A ! P(1 " r)2, where r is the interest rate compoundedannually. If an investment of $800 in the account grows to $882 in two years, at whatinterest rate was it invested? 5%
"3 ' i#5$$$7"5 ' i#23$$$4
"3 ' i #15$$$2
5 ' #15$$$2"1 ' #21$$$2
"4 ' #22$$$2
2$
2$
5$
25$
1$
1$
5$
25$
5%3
1%4
5%6
11$
121$
1 ' #2$$3
5$
1$
Practice (Average)
Completing the Square
NAME ______________________________________________ DATE ____________ PERIOD _____
6-46-4
Reading to Learn MathematicsCompleting the Square
NAME ______________________________________________ DATE ____________ PERIOD _____
6-46-4
© Glencoe/McGraw-Hill 335 Glencoe Algebra 2
Less
on
6-4
Pre-Activity How can you find the time it takes an accelerating race car toreach the finish line?
Read the introduction to Lesson 6-4 at the top of page 306 in your textbook.
Explain what it means to say that the driver accelerates at a constant rateof 8 feet per second square.
If the driver is traveling at a certain speed at a particularmoment, then one second later, the driver is traveling 8 feetper second faster.
Reading the Lesson
1. Give the reason for each step in the following solution of an equation by using theSquare Root Property.
x2 $ 12x " 36 ! 81 Original equation
(x $ 6)2 ! 81 Factor the perfect square trinomial.x $ 6 ! )#81$ Square Root Propertyx $ 6 ! )9 81 ! 9
x $ 6 ! 9 or x $ 6 ! $9 Rewrite as two equations.x ! 15 x ! $3 Solve each equation.
2. Explain how to find the constant that must be added to make a binomial into a perfectsquare trinomial.
Sample answer: Find half of the coefficient of the linear term and squareit.
3. a. What is the first step in solving the equation 3x2 " 6x ! 5 by completing the square?Divide the equation by 3.
b. What is the first step in solving the equation x2 " 5x $ 12 ! 0 by completing thesquare? Add 12 to each side.
Helping You Remember
4. How can you use the rules for squaring a binomial to help you remember the procedurefor changing a binomial into a perfect square trinomial?One of the rules for squaring a binomial is (x # y)2 ! x2 # 2xy # y2. Incompleting the square, you are starting with x2 # bx and need to find y2. This shows you that b ! 2y, so y ! . That is why you must take half of the coefficient and square it to get the constant that must be added tocomplete the square.
b$
© Glencoe/McGraw-Hill 336 Glencoe Algebra 2
The Golden Quadratic EquationsA golden rectangle has the property that its length can be written as a " b, where a is the width of the
rectangle and %a "a
b% ! %
ab%. Any golden rectangle can be
divided into a square and a smaller golden rectangle,as shown.
The proportion used to define golden rectangles can be used to derive two quadratic equations. These aresometimes called golden quadratic equations.
Solve each problem.
1. In the proportion for the golden rectangle, let a equal 1. Write the resulting quadratic equation and solve for b.
2. In the proportion, let b equal 1. Write the resulting quadratic equation and solve for a.
3. Describe the difference between the two golden quadratic equations you found in exercises 1 and 2.
4. Show that the positive solutions of the two equations in exercises 1 and 2 are reciprocals.
5. Use the Pythagorean Theorem to find a radical expression for the diagonal of a golden rectangle when a ! 1.
6. Find a radical expression for the diagonal of a golden rectangle when b ! 1.
a
a
a
b
b
a
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
6-46-4
Study Guide and InterventionThe Quadratic Formula and the Discriminant
NAME ______________________________________________ DATE ____________ PERIOD _____
6-56-5
© Glencoe/McGraw-Hill 337 Glencoe Algebra 2
Less
on
6-5
Quadratic Formula The Quadratic Formula can be used to solve any quadraticequation once it is written in the form ax2 " bx " c ! 0.
Quadratic Formula The solutions of ax 2 " bx " c ! 0, with a # 0, are given by x ! .
Solve x2 " 5x ! 14 by using the Quadratic Formula.
Rewrite the equation as x2 $ 5x $ 14 ! 0.
x ! Quadratic Formula
! Replace a with 1, b with $5, and c with $14.
! Simplify.
!
! 7 or $2
The solutions are $2 and 7.
Solve each equation by using the Quadratic Formula.
1. x2 " 2x $ 35 ! 0 2. x2 " 10x " 24 ! 0 3. x2 $ 11x " 24 ! 0
5, "7 "4, "6 3, 8
4. 4x2 " 19x $ 5 ! 0 5. 14x2 " 9x " 1 ! 0 6. 2x2 $ x $ 15 ! 0
, "5 " , " 3, "
7. 3x2 " 5x ! 2 8. 2y2 " y $ 15 ! 0 9. 3x2 $ 16x " 16 ! 0
"2, , "3 4,
10. 8x2 " 6x $ 9 ! 0 11. r2 $ " ! 0 12. x2 $ 10x $ 50 ! 0
" , , 5 ' 5#3$
13. x2 " 6x $ 23 ! 0 14. 4x2 $ 12x $ 63 ! 0 15. x2 $ 6x " 21 ! 0
"3 ' 4#2$ 3 ' 2i#3$3 ' 6#2$$$
1$
2$
3$
3$
2%25
3r%5
4$
5$
1$
5$
1$
1$
1$
5 ) 9%2
5 ) #81$%%2
$($5) ) #($5)2$$ 4(1$)($14$)$%%%%2(1)
$b ) #b2 $ 4$ac$%%%2a
$b ) #b2 $$4ac$%%%2a
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 338 Glencoe Algebra 2
Roots and the Discriminant
Discriminant The expression under the radical sign, b2 $ 4ac, in the Quadratic Formula is called the discriminant.
Roots of a Quadratic Equation
Discriminant Type and Number of Roots
b2 $ 4ac & 0 and a perfect square 2 rational roots
b2 $ 4ac & 0, but not a perfect square 2 irrational roots
b2 $ 4ac ! 0 1 rational root
b2 $ 4ac ' 0 2 complex roots
Find the value of the discriminant for each equation. Then describethe number and types of roots for the equation.
Study Guide and Intervention (continued)
The Quadratic Formula and the Discriminant
NAME ______________________________________________ DATE ____________ PERIOD _____
6-56-5
ExampleExample
a. 2x2 # 5x # 3The discriminant is b2 $ 4ac ! 52 $ 4(2)(3) or 1.The discriminant is a perfect square, sothe equation has 2 rational roots.
b. 3x2 " 2x # 5The discriminant is b2 $ 4ac ! ($2)2 $ 4(3)(5) or $56.The discriminant is negative, so theequation has 2 complex roots.
ExercisesExercises
For Exercises 1$12, complete parts a$c for each quadratic equation.a. Find the value of the discriminant.b. Describe the number and type of roots.c. Find the exact solutions by using the Quadratic Formula.
1. p2 " 12p ! $4 128; 2. 9x2 $ 6x " 1 ! 0 0; 3. 2x2 $ 7x $ 4 ! 0 81; two irrational roots; one rational root; 2 rational roots; " ,4"6 ' 4#2$
4. x2 " 4x $ 4 ! 0 32; 5. 5x2 $ 36x " 7 ! 0 1156; 6. 4x2 $ 4x " 11 ! 0
2 irrational roots; 2 rational roots; "160; 2 complexroots; "2 ' 2#2$ , 7
7. x2 $ 7x " 6 ! 0 25; 8. m2 $ 8m ! $14 8; 9. 25x2 $ 40x ! $16 0; 2 rational roots; 2 irrational roots; 1 rational root; 1, 6 4 ' #2$
10. 4x2 " 20x " 29 ! 0 "64; 11. 6x2 " 26x " 8 ! 0 484; 12. 4x2 $ 4x $ 11 ! 0 192; 2 complex roots; 2 rational roots; 2 irrational roots;
4$
1 ' i #10$$$1$
1$
1$
Skills PracticeThe Quadratic Formula and the Discriminant
NAME ______________________________________________ DATE ____________ PERIOD _____
6-56-5
© Glencoe/McGraw-Hill 339 Glencoe Algebra 2
Less
on
6-5
Complete parts a$c for each quadratic equation.a. Find the value of the discriminant.b. Describe the number and type of roots.c. Find the exact solutions by using the Quadratic Formula.
1. x2 $ 8x " 16 ! 0 2. x2 $ 11x $ 26 ! 0
0; 1 rational root; 4 225; 2 rational roots; "2, 13
3. 3x2 $ 2x ! 0 4. 20x2 " 7x $ 3 ! 0
4; 2 rational roots; 0, 289; 2 rational roots; " ,
5. 5x2 $ 6 ! 0 6. x2 $ 6 ! 0
120; 2 irrational roots; ' 24; 2 irrational roots; '#6$
7. x2 " 8x " 13 ! 0 8. 5x2 $ x $ 1 ! 0
12; 2 irrational roots; "4 ' #3$ 21; 2 irrational roots;
9. x2 $ 2x $ 17 ! 0 10. x2 " 49 ! 0
72; 2 irrational roots; 1 ' 3#2$ "196; 2 complex roots; '7i
11. x2 $ x " 1 ! 0 12. 2x2 $ 3x ! $2
"3; 2 complex roots; "7; 2 complex roots;
Solve each equation by using the method of your choice. Find exact solutions.
13. x2 ! 64 '8 14. x2 $ 30 ! 0 '#30$
15. x2 $ x ! 30 "5, 6 16. 16x2 $ 24x $ 27 ! 0 , "
17. x2 $ 4x $ 11 ! 0 2 ' #15$ 18. x2 $ 8x $ 17 ! 0 4 ' #33$
19. x2 " 25 ! 0 '5i 20. 3x2 " 36 ! 0 '2i #3$
21. 2x2 " 10x " 11 ! 0 22. 2x2 $ 7x " 4 ! 0
23. 8x2 " 1 ! 4x 24. 2x2 " 2x " 3 ! 0
25. PARACHUTING Ignoring wind resistance, the distance d(t) in feet that a parachutistfalls in t seconds can be estimated using the formula d(t) ! 16t2. If a parachutist jumpsfrom an airplane and falls for 1100 feet before opening her parachute, how many secondspass before she opens the parachute? about 8.3 s
"1 ' i#5$$$21 ' i$4
7 ' #17$$$4"5 ' #3$$$2
3$
9$
3 ' i #7$$$41 ' i #3$$$2
1 ' #21$$$10
#30$$5
1$
3$
2$
© Glencoe/McGraw-Hill 340 Glencoe Algebra 2
Complete parts a$c for each quadratic equation.a. Find the value of the discriminant.b. Describe the number and type of roots.c. Find the exact solutions by using the Quadratic Formula.
1. x2 $ 16x " 64 ! 0 2. x2 ! 3x 3. 9x2 $ 24x " 16 ! 0
0; 1 rational; 8 9; 2 rational; 0, 3 0; 1 rational;
4. x2 $ 3x ! 40 5. 3x2 " 9x $ 2 ! 0 105; 6. 2x2 " 7x ! 0
169; 2 rational; "5, 8 2 irrational; 49; 2 rational; 0, "
7. 5x2 $ 2x " 4 ! 0 "76; 8. 12x2 $ x $ 6 ! 0 289; 9. 7x2 " 6x " 2 ! 0 "20; 2 complex; 2 rational; , " 2 complex;
10. 12x2 " 2x $ 4 ! 0 196; 11. 6x2 $ 2x $ 1 ! 0 28; 12. x2 " 3x " 6 ! 0 "15; 2 rational; , " 2 irrational; 2 complex;
13. 4x2 $ 3x2 $ 6 ! 0 105; 14. 16x2 $ 8x " 1 ! 0 15. 2x2 $ 5x $ 6 ! 0 73; 2 irrational; 0; 1 rational; 2 irrational;
Solve each equation by using the method of your choice. Find exact solutions.
16. 7x2 $ 5x ! 0 0, 17. 4x2 $ 9 ! 0 '
18. 3x2 " 8x ! 3 , "3 19. x2 $ 21 ! 4x "3, 7
20. 3x2 $ 13x " 4 ! 0 , 4 21. 15x2 " 22x ! $8 " , "
22. x2 $ 6x " 3 ! 0 3 ' #6$ 23. x2 $ 14x " 53 ! 0 7 ' 2i
24. 3x2 ! $54 '3i #2$ 25. 25x2 $ 20x $ 6 ! 0
26. 4x2 $ 4x " 17 ! 0 27. 8x $ 1 ! 4x2
28. x2 ! 4x $ 15 2 ' i #11$ 29. 4x2 $ 12x " 7 ! 0
30. GRAVITATION The height h(t) in feet of an object t seconds after it is propelled straight upfrom the ground with an initial velocity of 60 feet per second is modeled by the equationh(t) ! $16t2 " 60t. At what times will the object be at a height of 56 feet? 1.75 s, 2 s
31. STOPPING DISTANCE The formula d ! 0.05s2 " 1.1s estimates the minimum stoppingdistance d in feet for a car traveling s miles per hour. If a car stops in 200 feet, what is thefastest it could have been traveling when the driver applied the brakes? about 53.2 mi/h
3 ' #2$$2
2 ' #3$$21 ' 4i$
2 ' #10$$$5
4$
2$
1$
1$
3$
5$
5 ' #73$$$41$3 ' #105$$$8
"3 ' i #15$$$1 ' #7$$6
2$
1$
"3 ' i #5$$$72$
3$1 ' i #19$$$5
7$"9 ' #105$$$6
4$
Practice (Average)
The Quadratic Formula and the Discriminant
NAME ______________________________________________ DATE ____________ PERIOD _____
6-56-5
Reading to Learn MathematicsThe Quadratic Formula and the Discriminant
NAME ______________________________________________ DATE ____________ PERIOD _____
6-56-5
© Glencoe/McGraw-Hill 341 Glencoe Algebra 2
Less
on
6-5
Pre-Activity How is blood pressure related to age?
Read the introduction to Lesson 6-5 at the top of page 313 in your textbook.
Describe how you would calculate your normal blood pressure using one ofthe formulas in your textbook.
Sample answer: Substitute your age for A in the appropriateformula (for females or males) and evaluate the expression.
Reading the Lesson
1. a. Write the Quadratic Formula. x !
b. Identify the values of a, b, and c that you would use to solve 2x2 $ 5x ! $7, but donot actually solve the equation.
a ! b ! c !
2. Suppose that you are solving four quadratic equations with rational coefficients andhave found the value of the discriminant for each equation. In each case, give thenumber of roots and describe the type of roots that the equation will have.
Value of Discriminant Number of Roots Type of Roots
64 2 real, rational$8 2 complex21 2 real, irrational0 1 real, rational
Helping You Remember
3. How can looking at the Quadratic Formula help you remember the relationshipsbetween the value of the discriminant and the number of roots of a quadratic equationand whether the roots are real or complex?Sample answer: The discriminant is the expression under the radical inthe Quadratic Formula. Look at the Quadratic Formula and consider whathappens when you take the principal square root of b2 " 4ac and apply' in front of the result. If b2 " 4ac is positive, its principal square rootwill be a positive number and applying ' will give two different realsolutions, which may be rational or irrational. If b2 " 4ac ! 0, itsprincipal square root is 0, so applying ' in the Quadratic Formula willonly lead to one solution, which will be rational (assuming a, b, and c areintegers). If b2 " 4ac is negative, since the square roots of negativenumbers are not real numbers, you will get two complex roots,corresponding to the # and " in the ' symbol.
7"52
"b ' #b2 "4$ac$$$2a
© Glencoe/McGraw-Hill 342 Glencoe Algebra 2
Sum and Product of Roots Sometimes you may know the roots of a quadratic equation without knowing the equationitself. Using your knowledge of factoring to solve an equation, you can work backward tofind the quadratic equation. The rule for finding the sum and product of roots is as follows:
Sum and Product of RootsIf the roots of ax2 " bx " c ! 0, with a ≠ 0, are s1 and s2, then s1 " s2 ! $%
ba% and s1 ( s2 ! %a
c%.
A road with an initial gradient, or slope, of 3% can be represented bythe formula y ! ax2 # 0. 03x # c, where y is the elevation and x is the distance alongthe curve. Suppose the elevation of the road is 1105 feet at points 200 feet and 1000feet along the curve. You can find the equation of the transition curve. Equationsof transition curves are used by civil engineers to design smooth and safe roads.
The roots are x ! 3 and x ! $8.
3 " ($8) ! $5 Add the roots.3($8) ! $24 Multiply the roots.
Equation: x2 " 5x $ 24 ! 0
Write a quadratic equation that has the given roots.
1. 6, $9 2. 5, $1 3. 6, 6
x2 # 3x " 54 ! 0 x2 " 4x " 5 ! 0 x2 " 12x # 36 ! 0
4. 4 ) #3$ 6. $%25%, %
27% 6.
x2 " 8x # 13 ! 0 35x2 # 4x " 4 ! 0 49x2 " 42x # 205 ! 0
Find k such that the number given is a root of the equation.
7. 7; 2x2 " kx $ 21 ! 0 8. $2; x2 $ 13x " k ! 0 "11 "30
$2 ) 3#5$%%7
x
y
O
(–5–2, –301–4)
10
–10
–20
–30
2 4–2–4–6–8
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
6-56-5
ExampleExample
Study Guide and InterventionAnalyzing Graphs of Quadratic Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
6-66-6
© Glencoe/McGraw-Hill 343 Glencoe Algebra 2
Less
on
6-6
Analyze Quadratic Functions
The graph of y ! a (x $ h)2 " k has the following characteristics:• Vertex: (h, k )
Vertex Form • Axis of symmetry: x ! hof a Quadratic • Opens up if a & 0Function • Opens down if a ' 0
• Narrower than the graph of y ! x2 if a & 1• Wider than the graph of y ! x2 if a ' 1
Identify the vertex, axis of symmetry, and direction of opening ofeach graph.
a. y ! 2(x # 4)2 " 11The vertex is at (h, k) or ($4, $11), and the axis of symmetry is x ! $4. The graph opensup, and is narrower than the graph of y ! x2.
a. y ! " (x " 2)2 # 10
The vertex is at (h, k) or (2, 10), and the axis of symmetry is x ! 2. The graph opensdown, and is wider than the graph of y ! x2.
Each quadratic function is given in vertex form. Identify the vertex, axis ofsymmetry, and direction of opening of the graph.
1. y ! (x $ 2)2 " 16 2. y ! 4(x " 3)2 $ 7 3. y ! (x $ 5)2 " 3
(2, 16); x ! 2; up ("3, "7); x ! "3; up (5, 3); x ! 5; up
4. y ! $7(x " 1)2 $ 9 5. y ! (x $ 4)2 $ 12 6. y ! 6(x " 6)2 " 6
("1, "9); x ! "1; down (4, "12); x ! 4; up ("6, 6); x ! "6; up
7. y ! (x $ 9)2 " 12 8. y ! 8(x $ 3)2 $ 2 9. y ! $3(x $ 1)2 $ 2
(9, 12); x ! 9; up (3, "2); x ! 3; up (1, "2); x ! 1; down
10. y ! $ (x " 5)2 " 12 11. y ! (x $ 7)2 " 22 12. y ! 16(x $ 4)2 " 1
("5, 12); x ! "5; down (7, 22); x ! 7; up (4, 1); x ! 4; up
13. y ! 3(x $ 1.2)2 " 2.7 14. y ! $0.4(x $ 0.6)2 $ 0.2 15. y ! 1.2(x " 0.8)2 " 6.5
(1.2, 2.7); x ! 1.2; up (0.6, "0.2); x ! 0.6; ("0.8, 6.5); x ! "0.8;down up
4%3
5%2
2%5
1%5
1%2
1$4
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 344 Glencoe Algebra 2
Write Quadratic Functions in Vertex Form A quadratic function is easier tograph when it is in vertex form. You can write a quadratic function of the form y ! ax2 " bx " c in vertex from by completing the square.
Write y ! 2x2 " 12x # 25 in vertex form. Then graph the function.
y ! 2x2 $ 12x " 25y ! 2(x2 $ 6x) " 25y ! 2(x2 $ 6x " 9) " 25 $ 18y ! 2(x $ 3)2 " 7
The vertex form of the equation is y ! 2(x $ 3)2 " 7.
Write each quadratic function in vertex form. Then graph the function.
1. y ! x2 $ 10x " 32 2. y ! x2 " 6x 3. y ! x2 $ 8x " 6y ! (x " 5)2 # 7 y ! (x # 3)2 " 9 y ! (x " 4)2 " 10
4. y ! $4x2 " 16x $ 11 5. y ! 3x2 $ 12x " 5 6. y ! 5x2 $ 10x " 9y ! "4(x " 2)2 # 5 y ! 3(x " 2)2 " 7 y ! 5(x" 1)2 # 4
x
y
O
x
y
O
x
y
O
x
y
O 4–4 8
8
4
–4
–8
–12
x
y
O
x
y
O
x
y
O
Study Guide and Intervention (continued)
Analyzing Graphs of Quadratic Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
6-66-6
ExampleExample
ExercisesExercises
Skills PracticeAnalyzing Graphs of Quadratic Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
6-66-6
© Glencoe/McGraw-Hill 345 Glencoe Algebra 2
Less
on
6-6
Write each quadratic function in vertex form, if not already in that form. Thenidentify the vertex, axis of symmetry, and direction of opening.
1. y ! (x $ 2)2 2. y ! $x2 " 4 3. y ! x2 $ 6y ! (x " 2)2 # 0; y ! "(x " 0)2 # 4; y ! (x " 0)2 " 6;(2, 0); x ! 2; up (0, 4); x ! 0; down (0, "6); x ! 0; up
4. y ! $3(x " 5)2 5. y ! $5x2 " 9 6. y ! (x $ 2)2 $ 18y ! "3(x # 5)2 # 0; y ! "5(x " 0)2 # 9; y ! (x " 2)2 " 18; ("5, 0); x ! "5; down (0, 9); x ! 0; down (2, "18); x ! 2; up
7. y ! x2 $ 2x $ 5 8. y ! x2 " 6x " 2 9. y ! $3x2 " 24xy ! (x " 1)2 " 6; y ! (x # 3)2 " 7; y ! "3(x " 4)2 # 48; (1, "6); x ! 1; up ("3, "7); x ! "3; up (4, 48); x ! 4; down
Graph each function.
10. y ! (x $ 3)2 $ 1 11. y ! (x " 1)2 " 2 12. y ! $(x $ 4)2 $ 4
13. y ! $ (x " 2)2 14. y ! $3x2 " 4 15. y ! x2 " 6x " 4
Write an equation for the parabola with the given vertex that passes through thegiven point.
16. vertex: (4, $36) 17. vertex: (3, $1) 18. vertex: ($2, 2)point: (0, $20) point: (2, 0) point: ($1, 3)y ! (x " 4)2 " 36 y ! (x " 3)2 " 1 y ! (x # 2)2 # 2
x
y
Ox
y
O
x
y
O
1%2
x
y
O
x
y
Ox
y
O
© Glencoe/McGraw-Hill 346 Glencoe Algebra 2
Write each quadratic function in vertex form, if not already in that form. Thenidentify the vertex, axis of symmetry, and direction of opening.
1. y ! $6(x " 2)2 $ 1 2. y ! 2x2 " 2 3. y ! $4x2 " 8xy ! "6(x # 2)2 " 1; y ! 2(x # 0)2 # 2; y ! "4(x " 1)2 # 4;("2, "1); x ! "2; down (0, 2); x ! 0; up (1, 4); x ! 1; down
4. y ! x2 " 10x " 20 5. y ! 2x2 " 12x " 18 6. y ! 3x2 $ 6x " 5y ! (x # 5)2 " 5; y ! 2(x # 3)2; ("3, 0); y ! 3(x " 1)2 # 2; ("5, "5); x ! "5; up x ! "3; up (1, 2); x ! 1; up
7. y ! $2x2 $ 16x $ 32 8. y ! $3x2 " 18x $ 21 9. y ! 2x2 " 16x " 29y ! "2(x # 4)2; y ! "3(x " 3)2 # 6; y ! 2(x # 4)2 " 3; ("4, 0); x ! "4; down (3, 6); x ! 3; down ("4, "3); x ! "4; up
Graph each function.
10. y ! (x " 3)2 $ 1 11. y ! $x2 " 6x $ 5 12. y ! 2x2 $ 2x " 1
Write an equation for the parabola with the given vertex that passes through thegiven point.
13. vertex: (1, 3) 14. vertex: ($3, 0) 15. vertex: (10, $4)point: ($2, $15) point: (3, 18) point: (5, 6)y ! "2(x " 1)2 # 3 y ! (x # 3)2 y ! (x " 10)2 " 4
16. Write an equation for a parabola with vertex at (4, 4) and x-intercept 6.y ! "(x " 4)2 # 4
17. Write an equation for a parabola with vertex at ($3, $1) and y-intercept 2.y ! (x # 3)2 " 1
18. BASEBALL The height h of a baseball t seconds after being hit is given by h(t) ! $16t2 " 80t " 3. What is the maximum height that the baseball reaches, andwhen does this occur? 103 ft; 2.5 s
19. SCULPTURE A modern sculpture in a park contains a parabolic arc thatstarts at the ground and reaches a maximum height of 10 feet after ahorizontal distance of 4 feet. Write a quadratic function in vertex formthat describes the shape of the outside of the arc, where y is the heightof a point on the arc and x is its horizontal distance from the left-handstarting point of the arc. y ! " (x " 4)2 # 105
$
10 ft
4 ft
1$
2$
1$
x
y
O
x
y
O
Practice (Average)
Analyzing Graphs of Quadratic Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
6-66-6
Reading to Learn MathematicsAnalyzing Graphs of Quadratic Equations
NAME ______________________________________________ DATE ____________ PERIOD _____
6-66-6
© Glencoe/McGraw-Hill 347 Glencoe Algebra 2
Less
on
6-6
Pre-Activity How can the graph of y ! x2 be used to graph any quadraticfunction?
Read the introduction to Lesson 6-6 at the top of page 322 in your textbook.
• What does adding a positive number to x2 do to the graph of y ! x2?It moves the graph up.
• What does subtracting a positive number to x before squaring do to thegraph of y ! x2? It moves the graph to the right.
Reading the Lesson
1. Complete the following information about the graph of y ! a(x $ h)2 " k.
a. What are the coordinates of the vertex? (h, k)b. What is the equation of the axis of symmetry? x ! hc. In which direction does the graph open if a & 0? If a ' 0? up; downd. What do you know about the graph if a ' 1?
It is wider than the graph of y ! x2.If a & 1? It is narrower than the graph of y ! x2.
2. Match each graph with the description of the constants in the equation in vertex form.
a. a & 0, h & 0, k ' 0 iii b. a ' 0, h ' 0, k ' 0 ivc. a ' 0, h ' 0, k & 0 ii d. a & 0, h ! 0, k ' 0 i
i. ii. iii. iv.
Helping You Remember
3. When graphing quadratic functions such as y ! (x " 4)2 and y ! (x $ 5)2, many studentshave trouble remembering which represents a translation of the graph of y ! x2 to the leftand which represents a translation to the right. What is an easy way to remember this?Sample answer: In functions like y ! (x # 4)2, the plus sign puts thegraph “ahead” so that the vertex comes “sooner” than the origin and thetranslation is to the left. In functions like y ! (x " 5)2, the minus puts thegraph “behind” so that the vertex comes “later” than the origin and thetranslation is to the right.
x
y
Ox
y
Ox
y
Ox
y
O
© Glencoe/McGraw-Hill 348 Glencoe Algebra 2
Patterns with Differences and Sums of SquaresSome whole numbers can be written as the difference of two squares andsome cannot. Formulas can be developed to describe the sets of numbersalgebraically.
If possible, write each number as the difference of two squares.Look for patterns.
1. 0 02 " 02 2. 1 12 " 02 3. 2 cannot 4. 3 22 " 12
5. 4 22 " 02 6. 5 32 " 22 7. 6 cannot 8. 7 42 " 32
9. 8 32 " 12 10. 9 32 " 02 11. 10 cannot 12. 11 62 " 52
13. 12 42 " 22 14. 13 72 " 62 15. 14 cannot 16. 15 42 " 12
Even numbers can be written as 2n, where n is one of the numbers 0, 1, 2, 3, and so on. Odd numbers can be written 2n # 1. Use these expressions for these problems.
17. Show that any odd number can be written as the difference of two squares.2n # 1 ! (n # 1)2 " n2
18. Show that the even numbers can be divided into two sets: those that can be written in the form 4n and those that can be written in the form 2 " 4n.Find 4n for n ! 0, 1, 2, and so on. You get {0, 4, 8, 12, …}. For 2 # 4n,you get {2, 6, 10, 12, …}. Together these sets include all even numbers.
19. Describe the even numbers that cannot be written as the difference of two squares. 2 # 4n, for n ! 0, 1, 2, 3, …
20. Show that the other even numbers can be written as the difference of two squares. 4n ! (n # 1)2 " (n " 1)2
Every whole number can be written as the sum of squares. It is never necessary to use more than four squares. Show that this is true for the whole numbers from 0 through 15 by writing each one as the sum of the least number of squares.
21. 0 02 22. 1 12 23. 2 12 # 12
24. 3 12 # 12 # 12 25. 4 22 26. 5 12 # 22
27. 6 12 # 12 # 22 28. 7 12 # 12 # 12 # 22 29. 8 22 # 22
30. 9 32 31. 10 12 # 32 32. 11 12 # 12 # 32
33. 12 12 # 12 # 12 # 32 34. 13 22 # 32 35. 14 12 # 22 # 32
36. 15 12 # 12 # 22 # 32
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
6-66-6
Study Guide and InterventionGraphing and Solving Quadratic Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
6-76-7
© Glencoe/McGraw-Hill 349 Glencoe Algebra 2
Less
on
6-7
Graph Quadratic Inequalities To graph a quadratic inequality in two variables, usethe following steps:
1. Graph the related quadratic equation, y ! ax2 " bx " c.Use a dashed line for ' or &; use a solid line for * or +.
2. Test a point inside the parabola.If it satisfies the inequality, shade the region inside the parabola;otherwise, shade the region outside the parabola.
Graph the inequality y % x2 # 6x # 7.
First graph the equation y ! x2 " 6x " 7. By completing the square, you get the vertex form of the equation y ! (x " 3)2 $ 2,so the vertex is ($3, $2). Make a table of values around x ! $3,and graph. Since the inequality includes &, use a dashed line.Test the point ($3, 0), which is inside the parabola. Since ($3)2 " 6($3) " 7 ! $2, and 0 & $2, ($3, 0) satisfies the inequality. Therefore, shade the region inside the parabola.
Graph each inequality.
1. y & x2 $ 8x " 17 2. y * x2 " 6x " 4 3. y + x2 " 2x " 2
4. y ' $x2 " 4x $ 6 5. y + 2x2 " 4x 6. y & $2x2 $ 4x " 2
x
y
O
x
y
O
x
y
O
x
y
O
x
y
O
x
y
O
x
y
O
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 350 Glencoe Algebra 2
Solve Quadratic Inequalities Quadratic inequalities in one variable can be solvedgraphically or algebraically.
To solve ax2 " bx " c ' 0:First graph y ! ax2 " bx " c. The solution consists of the x-values
Graphical Methodfor which the graph is below the x-axis.To solve ax2 " bx " c & 0:First graph y ! ax2 " bx " c. The solution consists the x-values for which the graph is above the x-axis.
Find the roots of the related quadratic equation by factoring,
Algebraic Method completing the square, or using the Quadratic Formula.2 roots divide the number line into 3 intervals.Test a value in each interval to see which intervals are solutions.
If the inequality involves * or +, the roots of the related equation are included in thesolution set.
Solve the inequality x2 " x " 6 ( 0.
First find the roots of the related equation x2 $ x $ 6 ! 0. Theequation factors as (x $ 3)(x " 2) ! 0, so the roots are 3 and $2.The graph opens up with x-intercepts 3 and $2, so it must be on or below the x-axis for $2 * x * 3. Therefore the solution set is {x$2 * x * 3}.
Solve each inequality.
1. x2 " 2x ' 0 2. x2 $ 16 ' 0 3. 0 ' 6x $ x2 $ 5
{x⏐"2 & x & 0} {x⏐"4 & x & 4} {x⏐1 & x & 5}
4. c2 * 4 5. 2m2 $ m ' 1 6. y2 ' $8
{c⏐"2 ( c ( 2} !m⏐" & m & 1" )
7. x2 $ 4x $ 12 ' 0 8. x2 " 9x " 14 & 0 9. $x2 " 7x $ 10 + 0
{x⏐"2 & x & 6} {x⏐x & "7 or x % "2} {x⏐2 ( x ( 5}
10. 2x2 " 5x$ 3 * 0 11. 4x2 $ 23x " 15 & 0 12. $6x2 $ 11x " 2 ' 0
!x⏐"3 ( x ( " !x⏐x & or x % 5" !x⏐x & "2 or x % "13. 2x2 $ 11x " 12 + 0 14. x2 $ 4x " 5 ' 0 15. 3x2 $ 16x " 5 ' 0
!x⏐x & or x % 4" ) !x⏐ & x & 5"1$
3$
1$
3$
1$
1$
x
y
O
Study Guide and Intervention (continued)
Graphing and Solving Quadratic Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
6-76-7
ExampleExample
ExercisesExercises
Skills PracticeGraphing and Solving Quadratic Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
6-76-7
© Glencoe/McGraw-Hill 351 Glencoe Algebra 2
Less
on
6-7
Graph each inequality.
1. y + x2 $ 4x " 4 2. y * x2 $ 4 3. y & x2 " 2x $ 5
Use the graph of its related function to write the solutions of each inequality.
4. x2 $ 6x " 9 * 0 5. $x2 $ 4x " 32 + 0 6. x2 " x $ 20 & 0
3 "8 ( x ( 4 x & "5 or x % 4
Solve each inequality algebraically.
7. x2 $ 3x $ 10 ' 0 8. x2 " 2x $ 35 + 0{x⏐"2 & x & 5} {x⏐x ( "7 or x * 5}
9. x2 $ 18x " 81 * 0 10. x2 * 36{x⏐x ! 9} {x⏐"6 & x & 6}
11. x2 $ 7x & 0 12. x2 " 7x " 6 ' 0{x⏐x & 0 or x % 7} {x⏐"6 & x & "1}
13. x2 " x $ 12 & 0 14. x2 " 9x " 18 * 0{x⏐x & "4 or x % 3} {x⏐"6 ( x ( "3}
15. x2 $ 10x " 25 + 0 16. $x2 $ 2x " 15 + 0all reals {x⏐"5 ( x ( 3}
17. x2 " 3x & 0 18. 2x2 " 2x & 4{x⏐x & "3 or x % 0} {x⏐x & "2 or x % 1}
19. $x2 $ 64 * $16x 20. 9x2 " 12x " 9 ' 0all reals )
x
y
O 2
5
x
y
O 2
6
x
y
O
x
y
O
x
y
O
x
y
O
© Glencoe/McGraw-Hill 352 Glencoe Algebra 2
Graph each inequality.
1. y * x2 " 4 2. y & x2 " 6x " 6 3. y ' 2x2 $ 4x $ 2
Use the graph of its related function to write the solutions of each inequality.
4. x2 $ 8x & 0 5. $x2 $ 2x " 3 + 0 6. x2 $ 9x " 14 * 0
x & 0 or x % 8 "3 ( x ( 1 2 ( x ( 7
Solve each inequality algebraically.
7. x2 $ x $ 20 & 0 8. x2 $ 10x " 16 ' 0 9. x2 " 4x " 5 * 0
{x⏐x & "4 or x % 5} {x⏐2 & x & 8} )
10. x2 " 14x " 49 + 0 11. x2 $ 5x & 14 12. $x2 $ 15 + 8x
all reals {x⏐x & "2 or x % 7} {x⏐"5 ( x ( "3}
13. $x2 " 5x $ 7 * 0 14. 9x2 " 36x " 36 * 0 15. 9x * 12x2
all reals {x⏐x ! "2} !x⏐x ( 0 or x * "16. 4x2 " 4x " 1 & 0 17. 5x2 " 10 + 27x 18. 9x2 " 31x " 12 * 0
!x⏐x + " " !x⏐x ( or x * 5" !x⏐"3 ( x ( " "19. FENCING Vanessa has 180 feet of fencing that she intends to use to build a rectangular
play area for her dog. She wants the play area to enclose at least 1800 square feet. Whatare the possible widths of the play area? 30 ft to 60 ft
20. BUSINESS A bicycle maker sold 300 bicycles last year at a profit of $300 each. The makerwants to increase the profit margin this year, but predicts that each $20 increase inprofit will reduce the number of bicycles sold by 10. How many $20 increases in profit canthe maker add in and expect to make a total profit of at least $100,000? from 5 to 10
4$
2$
1$
3$
x
y
O
x
y
Ox
y
O 2 4 6
6
–6
–12
8
x
y
O
x
y
O
Practice (Average)
Graphing and Solving Quadratic Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
6-76-7
Reading to Learn MathematicsGraphing and Solving Quadratic Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
6-76-7
© Glencoe/McGraw-Hill 353 Glencoe Algebra 2
Less
on
6-7
Pre-Activity How can you find the time a trampolinist spends above a certainheight?
Read the introduction to Lesson 6-7 at the top of page 329 in your textbook.
• How far above the ground is the trampoline surface? 3.75 feet• Using the quadratic function given in the introduction, write a quadratic
inequality that describes the times at which the trampolinist is morethan 20 feet above the ground. "16t 2 # 42t # 3.75 % 20
Reading the Lesson
1. Answer the following questions about how you would graph the inequality y + x2 " x $ 6.
a. What is the related quadratic equation? y ! x2 # x " 6b. Should the parabola be solid or dashed? How do you know?
solid; The inequality symbol is *.c. The point (0, 2) is inside the parabola. To use this as a test point, substitute
for x and for y in the quadratic inequality.
d. Is the statement 2 + 02 " 0 $ 6 true or false? truee. Should the region inside or outside the parabola be shaded? inside
2. The graph of y ! $x2 " 4x is shown at the right. Match each of the following related inequalities with its solution set.
a. $x2 " 4x & 0 ii i. {xx ' 0 or x & 4}
b. $x2 " 4x * 0 iii ii. {x0 ' x ' 4}
c. $x2 " 4x + 0 iv iii. {xx * 0 or x + 4}
d. $x2 " 4x ' 0 i iv. {x0 * x * 4}
Helping You Remember
3. A quadratic inequality in two variables may have the form y & ax2 " bx " c,y ' ax2 " bx " c, y + ax2 " bx " c, or y * ax2 " bx " c. Describe a way to rememberwhich region to shade by looking at the inequality symbol and without using a test point.Sample answer: If the symbol is % or *, shade the region above theparabola. If the symbol is & or (, shade the region below the parabola.
x
y
O(0, 0) (4, 0)
(2, 4)
20
© Glencoe/McGraw-Hill 354 Glencoe Algebra 2
Graphing Absolute Value Inequalities You can solve absolute value inequalities by graphing in much the same manner you graphed quadratic inequalities. Graph the related absolute function for each inequality by using a graphing calculator. For & and +, identify the x-values, if any, for which the graph lies below the x-axis. For ' and *, identify the x values, if any, for which the graph lies above the x-axis.
For each inequality, make a sketch of the related graph and find the solutions rounded to the nearest hundredth.
1. |x $ 3| & 0 2. |x| $ 6 ' 0 3. $|x " 4| " 8 ' 0
"6 & x & 6 "12 & x & 4
4. 2|x " 6| $ 2 + 0 5. |3x $ 3| + 0 6. |x $ 7| ' 5
x ( "7 or x * "5 all real numbers 2 & x & 12
7. |7x $ 1| & 13 8. |x $ 3.6| * 4.2 9. |2x " 5| * 7
x & "1.71 or x % 2 "0.6 ( x ( 7.8 "6 ( x ( 1
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
6-76-7
© Glencoe/McGraw-Hill A2 Glencoe Algebra 2
Answers (Lesson 6-1)
Stu
dy
Gu
ide
and I
nte
rven
tion
Gra
phin
g Q
uadr
atic
Fun
ctio
ns
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-1
6-1
©G
lenc
oe/M
cGra
w-H
ill31
3G
lenc
oe A
lgeb
ra 2
Lesson 6-1
Gra
ph
Qu
adra
tic
Fun
ctio
ns
Qua
drat
ic F
unct
ion
Afu
nctio
n de
fined
by
an e
quat
ion
of th
e fo
rm f
(x) !
ax2
"bx
"c,
whe
re a
#0
Gra
ph o
f a Q
uadr
atic
Apa
rabo
law
ith th
ese
char
acte
ristic
s: y
inte
rcep
t: c;
axis
of s
ymm
etry
: x!
;Fu
nctio
nx-
coor
dina
te o
f ver
tex:
Fin
d t
he
y-in
terc
ept,
the
equ
atio
n o
f th
e ax
is o
f sy
mm
etry
,an
d t
he
x-co
ord
inat
e of
th
e ve
rtex
for
th
e gr
aph
of
f(x)
!x2
"3x
#5.
Use
th
is i
nfo
rmat
ion
to g
rap
h t
he
fun
ctio
n.
a!
1,b
!$
3,an
d c
!5,
so t
he y
-int
erce
pt is
5.T
he e
quat
ion
of t
he a
xis
of s
ymm
etry
is
x!
or
.The
x-c
oord
inat
e of
the
ver
tex
is
.
Nex
t m
ake
a ta
ble
of v
alue
s fo
r x
near
.
xx2
"3x
#5
f(x)
(x,f
(x))
002
$3(
0) "
55
(0, 5
)
112
$3(
1) "
53
(1, 3
)
!"2
$3 !
""5
!,
"2
22$
3(2)
"5
3(2
, 3)
332
$3(
3) "
55
(3, 5
)
For
Exe
rcis
es 1
–3,c
omp
lete
par
ts a
–c f
or e
ach
qu
adra
tic
fun
ctio
n.
a.F
ind
th
e y-
inte
rcep
t,th
e eq
uat
ion
of
the
axis
of
sym
met
ry,a
nd
th
e x-
coor
din
ate
of t
he
vert
ex.
b.M
ake
a ta
ble
of v
alu
es t
hat
in
clu
des
th
e ve
rtex
.c.
Use
th
is i
nfo
rmat
ion
to
grap
h t
he
fun
ctio
n.
1.f(
x) !
x2"
6x"
82.
f(x)
!$
x2$
2x"
23.
f(x)
!2x
2$
4x"
38,
x!
"3,
"3
2,x
!"
1,"
13,
x!
1,1 ( 1
, 1)
x
f(x)
O12 8 4
48
–4
( –1,
3)
x
f(x)
O4 –4 –8
48
–8–4
( –3,
–1)
x
f(x)
O4
–4
48
–8
12 –4
x1
02
3f(
x)1
33
9x
"1
0"
21
f(x)
32
2"
1x
"3
"2
"1
"4
f(x)
"1
03
0
11 % 43 % 2
11 % 43 % 2
3 % 23 % 2
x
f (x)
O
3 % 2
3 % 23 % 2
$($
3)%
2(1)
$b
% 2a
$b
% 2a
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill31
4G
lenc
oe A
lgeb
ra 2
Max
imu
m a
nd
Min
imu
m V
alu
esT
he y
-coo
rdin
ate
of t
he v
erte
x of
a q
uadr
atic
func
tion
is t
he m
axim
um o
r m
inim
um v
alue
of
the
func
tion
.
Max
imum
or M
inim
um V
alue
Th
e gr
aph
of f(
x) !
ax2
"bx
"c,
whe
re a
#0,
ope
ns u
p an
d ha
s a
min
imum
of a
Qua
drat
ic F
unct
ion
whe
n a
&0.
The
gra
ph o
pens
dow
n an
d ha
s a
max
imum
whe
n a
'0.
Det
erm
ine
wh
eth
er e
ach
fu
nct
ion
has
a m
axim
um
or
min
imu
mva
lue.
Th
en f
ind
th
e m
axim
um
or
min
imu
m v
alu
e of
eac
h f
un
ctio
n.
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Gra
phin
g Q
uadr
atic
Fun
ctio
ns
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-1
6-1
Exam
ple
Exam
ple
a.f(
x) !
3x2
"6x
#7
For
this
fun
ctio
n,a
!3
and
b!
$6.
Sinc
e a
&0,
the
grap
h op
ens
up,a
nd t
hefu
ncti
on h
as a
min
imum
val
ue.
The
min
imum
val
ue is
the
y-c
oord
inat
eof
the
ver
tex.
The
x-c
oord
inat
e of
the
ve
rtex
is
!$
!1.
Eva
luat
e th
e fu
ncti
on a
t x
!1
to f
ind
the
min
imum
val
ue.
f(1)
!3(
1)2
$6(
1) "
7 !
4,so
the
min
imum
val
ue o
f th
e fu
ncti
on is
4.
$6
% 2(3)
$b
% 2a
b.f(
x) !
100
"2x
"x2
For
this
fun
ctio
n,a
!$
1 an
d b
!$
2.Si
nce
a'
0,th
e gr
aph
open
s do
wn,
and
the
func
tion
has
a m
axim
um v
alue
.T
he m
axim
um v
alue
is t
he y
-coo
rdin
ate
ofth
e ve
rtex
.The
x-c
oord
inat
e of
the
ver
tex
is
!$
!$
1.
Eva
luat
e th
e fu
ncti
on a
t x
!$
1 to
fin
dth
e m
axim
um v
alue
.f(
$1)
!10
0 $
2($
1) $
($1)
2!
101,
soth
e m
inim
um v
alue
of
the
func
tion
is 1
01.
$2
% 2($
1)$
b% 2a
Exer
cises
Exer
cises
Det
erm
ine
wh
eth
er e
ach
fu
nct
ion
has
a m
axim
um
or
min
imu
m v
alu
e.T
hen
fin
dth
e m
axim
um
or
min
imu
m v
alu
e of
eac
h f
un
ctio
n.
1.f(
x) !
2x2
$x
"10
2.f(
x) !
x2"
4x$
73.
f(x)
!3x
2$
3x"
1
min
.,9
min
.,"
11m
in.,
4.f(
x) !
16 "
4x$
x25.
f(x)
!x2
$7x
"11
6.f(
x) !
$x2
"6x
$4
max
.,20
min
.,"
max
.,5
7.f(
x) !
x2"
5x"
28.
f(x)
!20
"6x
$x2
9.f(
x) !
4x2
"x
"3
min
.,"
max
.,29
min
.,2
10.f
(x) !
$x2
$4x
"10
11.f
(x) !
x2$
10x
"5
12.f
(x) !
$6x
2"
12x
"21
max
.,14
min
.,"
20m
ax.,
27
13.f
(x) !
25x2
"10
0x"
350
14.f
(x) !
0.5x
2"
0.3x
$1.
415
.f(x
) !"
$8
min
.,25
0m
in.,
"1.
445
max
.,"
731 $ 32
x % 4$
x2%
215 $ 1617 $ 4
5 $ 4
1 $ 47 $ 8
© Glencoe/McGraw-Hill A3 Glencoe Algebra 2
An
swer
s
Answers (Lesson 6-1)
Skil
ls P
ract
ice
Gra
phin
g Q
uadr
atic
Fun
ctio
ns
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-1
6-1
©G
lenc
oe/M
cGra
w-H
ill31
5G
lenc
oe A
lgeb
ra 2
Lesson 6-1
For
eac
h q
uad
rati
c fu
nct
ion
,fin
d t
he
y-in
terc
ept,
the
equ
atio
n o
f th
e ax
is o
fsy
mm
etry
,an
d t
he
x-co
ord
inat
e of
th
e ve
rtex
.
1.f(
x) !
3x2
2.f(
x) !
x2"
13.
f(x)
!$
x2"
6x$
150;
x!
0;0
1;x
!0;
0"
15;x
!3;
3
4.f(
x) !
2x2
$11
5.f(
x) !
x2$
10x
"5
6.f(
x) !
$2x
2"
8x"
7"
11;x
!0;
05;
x!
5;5
7;x
!2;
2
Com
ple
te p
arts
a–c
for
eac
h q
uad
rati
c fu
nct
ion
.a.
Fin
d t
he
y-in
terc
ept,
the
equ
atio
n o
f th
e ax
is o
f sy
mm
etry
,an
d t
he
x-co
ord
inat
eof
th
e ve
rtex
.b.
Mak
e a
tabl
e of
val
ues
th
at i
ncl
ud
es t
he
vert
ex.
c.U
se t
his
in
form
atio
n t
o gr
aph
th
e fu
nct
ion
.
7.f(
x) !
$2x
28.
f(x)
!x2
$4x
"4
9.f(
x) !
x2$
6x"
80;
x!
0;0
4;x
!2;
28;
x!
3;3
Det
erm
ine
wh
eth
er e
ach
fu
nct
ion
has
a m
axim
um
or
a m
inim
um
val
ue.
Th
en f
ind
the
max
imu
m o
r m
inim
um
val
ue
of e
ach
fu
nct
ion
.
10.f
(x) !
6x2
11.f
(x) !
$8x
212
.f(x
) !x2
"2x
min
.;0
max
.;0
min
.;"
1
13.f
(x) !
x2"
2x"
1514
.f(x
) !$
x2"
4x$
115
.f(x
) !x2
"2x
$3
min
.;14
max
.;3
min
.;"
4
16.f
(x) !
$2x
2"
4x$
317
.f(x
) !3x
2"
12x
"3
18.f
(x) !
2x2
"4x
"1
max
.;"
1m
in.;
"9
min
.;"
1( 3, –
1)x
f (x)
O( 2
, 0)
x
f(x)
O16 12 8 4
2–2
46
( 0, 0
)x
f(x)
O
x0
23
46
f(x)
80
"1
08
x"
20
24
6f(
x)16
40
416
x"
2"
10
12
f(x)
"8
"2
0"
2"
8
©G
lenc
oe/M
cGra
w-H
ill31
6G
lenc
oe A
lgeb
ra 2
Com
ple
te p
arts
a–c
for
eac
h q
uad
rati
c fu
nct
ion
.a.
Fin
d t
he
y-in
terc
ept,
the
equ
atio
n o
f th
e ax
is o
f sy
mm
etry
,an
d t
he
x-co
ord
inat
eof
th
e ve
rtex
.b.
Mak
e a
tabl
e of
val
ues
th
at i
ncl
ud
es t
he
vert
ex.
c.U
se t
his
in
form
atio
n t
o gr
aph
th
e fu
nct
ion
.
1.f(
x) !
x2$
8x"
152.
f(x)
!$
x2$
4x"
123.
f(x)
!2x
2$
2x"
115
;x!
4;4
12;x
!"
2;"
21;
x!
0.5;
0.5
Det
erm
ine
wh
eth
er e
ach
fu
nct
ion
has
a m
axim
um
or
a m
inim
um
val
ue.
Th
en f
ind
the
max
imu
m o
r m
inim
um
val
ue
of e
ach
fu
nct
ion
.
4.f(
x) !
x2"
2x$
85.
f(x)
!x2
$6x
"14
6.v(
x) !
$x2
"14
x$
57m
in.;
"9
min
.;5
max
.;"
8
7.f(
x) !
2x2
"4x
$6
8.f(
x) !
$x2
"4x
$1
9.f(
x) !
$%2 3% x
2"
8x$
24m
in.;
"8
max
.;3
max
.;0
10.G
RA
VIT
ATI
ON
Fro
m 4
fee
t ab
ove
a sw
imm
ing
pool
,Sus
an t
hrow
s a
ball
upw
ard
wit
h a
velo
city
of
32 f
eet
per
seco
nd.T
he h
eigh
t h(
t) o
f th
e ba
ll t
seco
nds
afte
r Su
san
thro
ws
itis
giv
en b
y h(
t) !
$16
t2"
32t
"4.
Fin
d th
e m
axim
um h
eigh
t re
ache
d by
the
bal
l and
the
tim
e th
at t
his
heig
ht is
rea
ched
.20
ft;1
s
11.H
EALT
H C
LUB
SL
ast
year
,the
Spo
rtsT
ime
Ath
leti
c C
lub
char
ged
$20
to p
arti
cipa
te in
an a
erob
ics
clas
s.Se
vent
y pe
ople
att
ende
d th
e cl
asse
s.T
he c
lub
wan
ts t
o in
crea
se t
hecl
ass
pric
e th
is y
ear.
The
y ex
pect
to
lose
one
cus
tom
er f
or e
ach
$1 in
crea
se in
the
pri
ce.
a.W
hat
pric
e sh
ould
the
clu
b ch
arge
to
max
imiz
e th
e in
com
e fr
om t
he a
erob
ics
clas
ses?
$45
b.W
hat
is t
he m
axim
um in
com
e th
e Sp
orts
Tim
e A
thle
tic
Clu
b ca
n ex
pect
to
mak
e?$2
025
f(x)
( 0.5
, 0.5
)x
O
16 12 8 4
( –2,
16)
x
f (x)
O2
–2–4
–6( 4
, –1)
x
f (x)
O16 12 8 4
24
68
x"
10
0.5
12
f(x)
51
0.5
15
x"
6"
4"
20
2f(
x)0
1216
120
x0
24
68
f(x)
153
"1
315
Pra
ctic
e (A
vera
ge)
Gra
phin
g Q
uadr
atic
Fun
ctio
ns
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-1
6-1
© Glencoe/McGraw-Hill A4 Glencoe Algebra 2
Answers (Lesson 6-1)
Rea
din
g t
o L
earn
Math
emati
csG
raph
ing
Qua
drat
ic F
unct
ions
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-1
6-1
©G
lenc
oe/M
cGra
w-H
ill31
7G
lenc
oe A
lgeb
ra 2
Lesson 6-1
Pre-
Act
ivit
yH
ow c
an i
nco
me
from
a r
ock
con
cert
be
max
imiz
ed?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 6-
1 at
the
top
of
page
286
in y
our
text
book
.•
Bas
ed o
n th
e gr
aph
in y
our
text
book
,for
wha
t ti
cket
pri
ce is
the
inco
me
the
grea
test
?$4
0•
Use
the
gra
ph t
o es
tim
ate
the
max
imum
inco
me.
abou
t $72
,000
Rea
din
g t
he
Less
on
1.a.
For
the
quad
rati
c fu
ncti
on f
(x) !
2x2
"5x
"3,
2x2
is t
he
term
,
5xis
the
te
rm,a
nd 3
is t
he
term
.
b.Fo
r th
e qu
adra
tic
func
tion
f(x
) !$
4 "
x$
3x2 ,
a!
,b!
,and
c!
.
2.C
onsi
der
the
quad
rati
c fu
ncti
on f
(x) !
ax2
"bx
"c,
whe
re a
#0.
a.T
he g
raph
of
this
fun
ctio
n is
a
.
b.T
he y
-int
erce
pt is
.
c.T
he a
xis
of s
ymm
etry
is t
he li
ne
.
d.If
a&
0,th
en t
he g
raph
ope
ns
and
the
func
tion
has
a
valu
e.
e.If
a'
0,th
en t
he g
raph
ope
ns
and
the
func
tion
has
a
valu
e.
3.R
efer
to
the
grap
h at
the
rig
ht a
s yo
u co
mpl
ete
the
follo
win
g se
nten
ces.
a.T
he c
urve
is c
alle
d a
.
b.T
he li
ne x
!$
2 is
cal
led
the
.
c.T
he p
oint
($2,
4) is
cal
led
the
.
d.B
ecau
se t
he g
raph
con
tain
s th
e po
int
(0,$
1),$
1 is
the
.
Hel
pin
g Y
ou
Rem
emb
er4.
How
can
you
rem
embe
r th
e w
ay t
o us
e th
e x2
term
of
a qu
adra
tic
func
tion
to
tell
whe
ther
the
fun
ctio
n ha
s a
max
imum
or
a m
inim
um v
alue
?Sa
mpl
e an
swer
:R
emem
ber t
hat t
he g
raph
of f
(x) !
x2(w
ith a
%0)
is a
U-s
hape
d cu
rve
that
ope
ns u
p an
d ha
s a
min
imum
.The
gra
ph o
f g(x
) !"
x2(w
ith a
&0)
is ju
st th
e op
posi
te.I
t ope
ns d
own
and
has
a m
axim
um.
y-in
terc
ept
vert
exax
is o
f sym
met
rypa
rabo
la
x
f(x)
O ( 0, –
1)
( –2,
4)
max
imum
dow
nwar
d
min
imum
upw
ard
x!
"$ 2b a$
c
para
bola
"4
1"
3co
nsta
ntlin
ear
quad
ratic
©G
lenc
oe/M
cGra
w-H
ill31
8G
lenc
oe A
lgeb
ra 2
Find
ing
the
Axi
s of
Sym
met
ry o
f a P
arab
ola
As
you
know
,if f
(x) !
ax2
"bx
"c
is a
qua
drat
ic f
unct
ion,
the
valu
es o
f x
that
mak
e f(
x) e
qual
to
zero
are
an
d .
The
ave
rage
of
thes
e tw
o nu
mbe
r va
lues
is $
% 2b a%.
The
fun
ctio
n f(
x) h
as it
s m
axim
um o
r m
inim
um
valu
e w
hen
x!
$% 2b a%
.Sin
ce t
he a
xis
of s
ymm
etry
of t
he g
raph
of f
(x)
pass
es t
hrou
gh t
he p
oint
whe
re
the
max
imum
or
min
imum
occ
urs,
the
axis
of
sym
met
ry h
as t
he e
quat
ion
x!
$% 2b a%
.
Fin
d t
he
vert
ex a
nd
axi
s of
sym
met
ry f
or f
(x)
!5x
2#
10x
"7.
Use
x!
$% 2b a%
.
x!
$% 21 (0 5)%
!$
1T
he x
-coo
rdin
ate
of t
he v
erte
x is
$1.
Subs
titu
te x
!$
1 in
f(x
) !5x
2"
10x
$7.
f($
1) !
5($
1)2
"10
($1)
$7
!$
12T
he v
erte
x is
($1,
$12
).T
he a
xis
of s
ymm
etry
is x
!$
% 2b a%,o
r x
!$
1.
Fin
d t
he
vert
ex a
nd
axi
s of
sym
met
ry f
or t
he
grap
h o
f ea
ch f
un
ctio
n
usi
ng
x!
"$ 2b a$
.
1.f(
x) !
x2$
4x$
8(2
,"12
);x
!2
2.g(
x) !
$4x
2$
8x"
3("
1,7)
;x!
"1
3.y
!$
x2"
8x"
3(4
,19)
;x!
44.
f(x)
!2x
2"
6x"
5%"
$3 2$ ,$1 2$ &;
x!
"$3 2$
5.A
(x) !
x2"
12x
"36
("6,
0);x
!"
66.
k(x)
!$
2x2
"2x
$6
%$1 2$ ,"
5$1 2$ &;x
!$1 2$
O
f(x)
x
––
,f(
(
((
b –– 2a b –– 2a
b –– 2ax
= –
f(x) =
ax2 +
bx
+ c
$b
$#
b2$
4$
ac$%
%%
2a$
b"
#b2
$4
$ac$
%%
%2a
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-1
6-1
Exam
ple
Exam
ple
© Glencoe/McGraw-Hill A5 Glencoe Algebra 2
An
swer
s
Answers (Lesson 6-2)
Stu
dy
Gu
ide
and I
nte
rven
tion
Solv
ing
Qua
drat
ic E
quat
ions
by
Gra
phin
g
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-2
6-2
©G
lenc
oe/M
cGra
w-H
ill31
9G
lenc
oe A
lgeb
ra 2
Lesson 6-2
Solv
e Q
uad
rati
c Eq
uat
ion
s
Qua
drat
ic E
quat
ion
Aqu
adra
tic e
quat
ion
has
the
form
ax2
"bx
"c
!0,
whe
re a
#0.
Roo
ts o
f a Q
uadr
atic
Equ
atio
nso
lutio
n(s)
of t
he e
quat
ion,
or t
he z
ero(
s) o
f the
rela
ted
quad
ratic
func
tion
The
zer
os o
f a
quad
rati
c fu
ncti
on a
re t
he x
-int
erce
pts
of it
s gr
aph.
The
refo
re,f
indi
ng t
he
x-in
terc
epts
is o
ne w
ay o
f so
lvin
g th
e re
late
d qu
adra
tic
equa
tion
.
Sol
ve x
2#
x "
6 !
0 by
gra
ph
ing.
Gra
ph t
he r
elat
ed f
unct
ion
f(x)
!x2
"x
$6.
The
x-c
oord
inat
e of
the
ver
tex
is
!$
,and
the
equ
atio
n of
the
axis
of
sym
met
ry is
x!
$.
Mak
e a
tabl
e of
val
ues
usin
g x-
valu
es a
roun
d $
.
x$
1$
01
2
f(x)
$6
$6
$6
$4
0
Fro
m t
he t
able
and
the
gra
ph,w
e ca
n se
e th
at t
he z
eros
of
the
func
tion
are
2 a
nd $
3.
Sol
ve e
ach
equ
atio
n b
y gr
aph
ing.
1.x2
"2x
$8
!0
2,"
42.
x2$
4x$
5 !
05,
"1
3.x2
$5x
"4
!0
1,4
4.x2
$10
x"
21 !
05.
x2"
4x"
6 !
06.
4x2
"4x
"1
!0
3,7
no re
al s
olut
ions
"1 $ 2
x
f(x)
Ox
f (x)
O
x
f(x)
O
x
f(x)
O
x
f(x)
Ox
f(x)
O
1 % 41 % 2
1 % 2
1 % 2
1 % 2$
b% 2a
x
f (x)
O
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill32
0G
lenc
oe A
lgeb
ra 2
Esti
mat
e So
luti
on
sO
ften
,you
may
not
be
able
to
find
exa
ct s
olut
ions
to
quad
rati
ceq
uati
ons
by g
raph
ing.
But
you
can
use
the
gra
ph t
o es
tim
ate
solu
tion
s.
Sol
ve x
2"
2x"
2 !
0 by
gra
ph
ing.
If e
xact
roo
ts c
ann
ot b
e fo
un
d,
stat
e th
e co
nse
cuti
ve i
nte
gers
bet
wee
n w
hic
h t
he
root
s ar
e lo
cate
d.
The
equ
atio
n of
the
axi
s of
sym
met
ry o
f th
e re
late
d fu
ncti
on is
x!
$!
1,so
the
ver
tex
has
x-co
ordi
nate
1.M
ake
a ta
ble
of v
alue
s.
x$
10
12
3
f(x)
1$
2$
3$
21
The
x-i
nter
cept
s of
the
gra
ph a
re b
etw
een
2 an
d 3
and
betw
een
0 an
d$
1.So
one
sol
utio
n is
bet
wee
n 2
and
3,an
d th
e ot
her
solu
tion
isbe
twee
n 0
and
$1.
Sol
ve t
he
equ
atio
ns
by g
rap
hin
g.If
exa
ct r
oots
can
not
be
fou
nd
,sta
te t
he
con
secu
tive
in
tege
rs b
etw
een
wh
ich
th
e ro
ots
are
loca
ted
.
1.x2
$4x
"2
!0
2.x2
"6x
"6
!0
3.x2
"4x
"2!
0
betw
een
0 an
d 1;
betw
een
"2
and
"1;
betw
een
"1
and
0;be
twee
n 3
and
4be
twee
n "
5 an
d "
4be
twee
n "
4 an
d "
3
4.$
x2"
2x"
4 !
05.
2x2
$12
x"
17 !
06.
$x2
"x
"!
0
betw
een
3 an
d 4;
betw
een
2 an
d 3;
betw
een
"2
and
"1;
betw
een
"2
and
"1
betw
een
3 an
d 4
betw
een
3 an
d 4 x
f(x)
O
x
f(x)
Ox
f(x)
O
5 % 21 % 2
x
f(x)
Ox
f(x)
Ox
f(x)
O
$2
% 2(1)
x
f (x)
O
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Solv
ing
Qua
drat
ic E
quat
ions
by
Gra
phin
g
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-2
6-2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A6 Glencoe Algebra 2
Answers (Lesson 6-2)
Skil
ls P
ract
ice
Solv
ing
Qua
drat
ic E
quat
ions
By
Gra
phin
g
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-2
6-2
©G
lenc
oe/M
cGra
w-H
ill32
1G
lenc
oe A
lgeb
ra 2
Lesson 6-2
Use
th
e re
late
d g
rap
h o
f ea
ch e
quat
ion
to
det
erm
ine
its
solu
tion
s.
1.x2
"2x
$3
!0
2.$
x2$
6x$
9 !
03.
3x2
"4x
"3
!0
"3,
1"
3no
real
sol
utio
ns
Sol
ve e
ach
equ
atio
n b
y gr
aph
ing.
If e
xact
roo
ts c
ann
ot b
e fo
un
d,s
tate
th
eco
nse
cuti
ve i
nte
gers
bet
wee
n w
hic
h t
he
root
s ar
e lo
cate
d.
4.x2
$6x
"5
!0
5.$
x2"
2x$
4 !
06.
x2$
6x"
4 !
01,
5no
real
sol
utio
nsbe
twee
n 0
and
1;be
twee
n 5
and
6
Use
a q
uad
rati
c eq
uat
ion
to
fin
d t
wo
real
nu
mbe
rs t
hat
sat
isfy
eac
h s
itu
atio
n,o
rsh
ow t
hat
no
such
nu
mbe
rs e
xist
.
7.T
heir
sum
is $
4,an
d th
eir
prod
uct
is 0
.8.
The
ir s
um is
0,a
nd t
heir
pro
duct
is $
36.
"x2
"4x
!0;
0,"
4"
x2#
36 !
0;"
6,6
f(x) !
"x2 #
36
x
f (x)
O6
–612
–12
36 24 12
f(x) !
"x2 "
4x
x
f (x)
O
f(x) !
x2 " 6
x # 4
x
f (x)
Of(x
) ! "
x2 # 2
x " 4x
f (x)
O
f(x) !
x2 " 6
x # 5
x
f (x)
O
x
f (x) O
f(x) !
3x2 #
4x #
3
x
f(x)
O
f(x) !
"x2 "
6x "
9
x
f (x)
O
f(x) !
x2 # 2
x " 3
©G
lenc
oe/M
cGra
w-H
ill32
2G
lenc
oe A
lgeb
ra 2
Use
th
e re
late
d g
rap
h o
f ea
ch e
quat
ion
to
det
erm
ine
its
solu
tion
s.
1.$
3x2
"3
!0
2.3x
2"
x"
3 !
03.
x2$
3x"
2 !
0
"1,
1no
real
sol
utio
ns1,
2S
olve
eac
h e
quat
ion
by
grap
hin
g.If
exa
ct r
oots
can
not
be
fou
nd
,sta
te t
he
con
secu
tive
in
tege
rs b
etw
een
wh
ich
th
e ro
ots
are
loca
ted
.
4.$
2x2
$6x
"5
!0
5.x2
"10
x"
24 !
06.
2x2
$x
$6
!0
betw
een
0 an
d 1;
"6,
"4
betw
een
"2
and
"1,
betw
een
"4
and
"3
2
Use
a q
uad
rati
c eq
uat
ion
to
fin
d t
wo
real
nu
mbe
rs t
hat
sat
isfy
eac
h s
itu
atio
n,o
rsh
ow t
hat
no
such
nu
mbe
rs e
xist
.
7.T
heir
sum
is 1
,and
the
ir p
rodu
ct is
$6.
8.T
heir
sum
is 5
,and
the
ir p
rodu
ct is
8.
For
Exe
rcis
es 9
an
d 1
0,u
se t
he
form
ula
h(t
) !
v 0t
"16
t2,w
her
e h
(t)
is t
he
hei
ght
of a
n o
bjec
t in
fee
t,v 0
is t
he
obje
ct’s
in
itia
l ve
loci
ty i
n f
eet
per
sec
ond
,an
d t
is t
he
tim
e in
sec
ond
s.
9.B
ASE
BA
LLM
arta
thr
ows
a ba
seba
ll w
ith
an in
itia
l upw
ard
velo
city
of 6
0 fe
et p
er s
econ
d.Ig
nori
ng M
arta
’s he
ight
,how
long
aft
er s
he r
elea
ses
the
ball
will
it h
it t
he g
roun
d?3.
75 s
10.V
OLC
AN
OES
A v
olca
nic
erup
tion
bla
sts
a bo
ulde
r up
war
d w
ith
an in
itia
l vel
ocit
y of
240
feet
per
sec
ond.
How
long
will
it t
ake
the
boul
der
to h
it t
he g
roun
d if
it la
nds
at t
hesa
me
elev
atio
n fr
om w
hich
it w
as e
ject
ed?
15 s
"x2
#5x
"8
!0;
no s
uch
real
num
bers
exi
stx
f (x)
Of(x
) ! "
x2 # 5
x " 8
"x2
#x
#6
!0;
3,"
2f(x
) ! "
x2 # x
# 6 x
f (x)
O
x
f (x)
O
f(x) !
2x2 "
x "
6f(x
) ! x2 #
10x
# 2
4x
f (x)
O
f(x) !
"2x
2 " 6
x # 5
x
f (x)
O–4
–2–6
12 8 4
x
f (x)
O
f(x) !
x2 " 3
x # 2
x
f(x) O
f(x) !
3x2 #
x #
3
x
f (x)
O
f(x) !
"3x
2 # 3
Pra
ctic
e (A
vera
ge)
Solv
ing
Qua
drat
ic E
quat
ions
By
Gra
phin
g
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-2
6-2
© Glencoe/McGraw-Hill A7 Glencoe Algebra 2
An
swer
s
Answers (Lesson 6-2)
Rea
din
g t
o L
earn
Math
emati
csSo
lvin
g Q
uadr
atic
Equ
atio
ns b
y G
raph
ing
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-2
6-2
©G
lenc
oe/M
cGra
w-H
ill32
3G
lenc
oe A
lgeb
ra 2
Lesson 6-2
Pre-
Act
ivit
yH
ow d
oes
a qu
adra
tic
fun
ctio
n m
odel
a f
ree-
fall
rid
e?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 6-
2 at
the
top
of
page
294
in y
our
text
book
.
Wri
te a
qua
drat
ic f
unct
ion
that
des
crib
es t
he h
eigh
t of
a b
all t
seco
nds
afte
rit
is d
ropp
ed f
rom
a h
eigh
t of
125
fee
t.h(
t) !
"16
t2#
125
Rea
din
g t
he
Less
on
1.T
he g
raph
of
the
quad
rati
c fu
ncti
on f
(x) !
$x2
"x
"6
is s
how
n at
the
rig
ht.U
se t
he g
raph
to
find
the
sol
utio
ns o
f th
equ
adra
tic
equa
tion
$x2
"x
"6
!0.
"2
and
3
2.Sk
etch
a g
raph
to
illus
trat
e ea
ch s
itua
tion
.
a.A
par
abol
a th
at o
pens
b.
A p
arab
ola
that
ope
ns
c.A
par
abol
a th
at o
pens
dow
nwar
d an
d re
pres
ents
a
upw
ard
and
repr
esen
ts a
do
wnw
ard
and
qu
adra
tic
func
tion
wit
h tw
o qu
adra
tic
func
tion
wit
h
repr
esen
ts a
re
al z
eros
,bot
h of
whi
ch a
reex
actl
y on
e re
al z
ero.
The
qu
adra
tic
func
tion
ne
gati
ve n
umbe
rs.
zero
is a
pos
itiv
e nu
mbe
r.w
ith
no r
eal z
eros
.
Hel
pin
g Y
ou
Rem
emb
er
3.T
hink
of
a m
emor
y ai
d th
at c
an h
elp
you
reca
ll w
hat
is m
eant
by
the
zero
sof
a q
uadr
atic
func
tion
.
Sam
ple
answ
er:T
he b
asic
fact
s ab
out a
sub
ject
are
som
etim
es c
alle
d th
eA
BC
s.In
the
case
of z
eros
,the
AB
Cs
are
the
XYZs
,bec
ause
the
zero
sar
e th
e x-
valu
es th
at m
ake
the
y-va
lues
equ
al to
zer
o.
x
y
Ox
y
Ox
y
O
x
y
O
©G
lenc
oe/M
cGra
w-H
ill32
4G
lenc
oe A
lgeb
ra 2
Gra
phin
g A
bsol
ute
Valu
e Eq
uatio
ns
You
can
solv
e ab
solu
te v
alue
equ
atio
ns in
muc
h th
e sa
me
way
you
sol
ved
quad
rati
c eq
uati
ons.
Gra
ph t
he r
elat
ed a
bsol
ute
valu
e fu
ncti
on f
or e
ach
equa
tion
usi
ng a
gra
phin
g ca
lcul
ator
.The
n us
e th
e ZE
ROfe
atur
e in
the
CA
LCm
enu
to f
ind
its
real
sol
utio
ns,i
f an
y.R
ecal
l tha
t so
luti
ons
are
poin
ts
whe
re t
he g
raph
inte
rsec
ts t
he x
-axi
s.
For
eac
h e
quat
ion
,mak
e a
sket
ch o
f th
e re
late
d g
rap
h a
nd
fin
d t
he
solu
tion
s ro
un
ded
to
the
nea
rest
hu
nd
red
th.
1.|x
"5|
!0
2.|4
x$
3| "
5 !
03.
|x$
7| !
0
"5
No
solu
tions
7
4.|x
"3|
$8
!0
5.$
|x"
3| "
6 !
06.
|x$
2| $
3 !
0
"11
,5"
9,3
"1,
5
7.|3
x "
4| !
28.
|x "
12| !
109.
|x|$
3 !
0
"2,
"$2 3$
"22
,"2
"3,
3
10.E
xpla
in h
ow s
olvi
ng a
bsol
ute
valu
e eq
uati
ons
alge
brai
cally
and
fin
ding
ze
ros
of a
bsol
ute
valu
e fu
ncti
ons
grap
hica
lly a
re r
elat
ed.
Sam
ple
answ
er:v
alue
s of
xw
hen
solv
ing
alge
brai
cally
are
the
x-in
terc
epts
(or z
eros
) of t
he fu
nctio
n w
hen
grap
hed.
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-2
6-2
© Glencoe/McGraw-Hill A8 Glencoe Algebra 2
Answers (Lesson 6-3)
Stu
dy
Gu
ide
and I
nte
rven
tion
Solv
ing
Qua
drat
ic E
quat
ions
by
Fact
orin
g
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-3
6-3
©G
lenc
oe/M
cGra
w-H
ill32
5G
lenc
oe A
lgeb
ra 2
Lesson 6-3
Solv
e Eq
uat
ion
s b
y Fa
cto
rin
gW
hen
you
use
fact
orin
g to
sol
ve a
qua
drat
ic e
quat
ion,
you
use
the
follo
win
g pr
oper
ty.
Zero
Pro
duct
Pro
pert
yFo
r any
real
num
bers
aan
d b,
if a
b!
0, th
en e
ither
a!
0 or
b!
0, o
r bot
h a
and
b!
0.
Sol
ve e
ach
equ
atio
n b
y fa
ctor
ing.
Exam
ple
Exam
ple
a.3x
2!
15x
3x2
!15
xO
rigin
al e
quat
ion
3x2
$15
x!
0Su
btra
ct 1
5xfro
m b
oth
side
s.
3x(x
$5)
!0
Fact
or th
e bi
nom
ial.
3x !
0or
x$
5 !
0Ze
ro P
rodu
ct P
rope
rty
x!
0or
x!
5So
lve
each
equ
atio
n.
The
sol
utio
n se
t is
{0,
5}.
b.4x
2"
5x!
214x
2$
5x!
21O
rigin
al e
quat
ion
4x2
$5x
$21
!0
Subt
ract
21
from
bot
h si
des.
(4x
"7)
(x$
3)!
0Fa
ctor
the
trino
mia
l.
4x"
7 !
0or
x$
3 !
0Ze
ro P
rodu
ct P
rope
rty
x!
$or
x
!3
Solv
e ea
ch e
quat
ion.
The
sol
utio
n se
t is
%$,3
&.7 % 4
7 % 4
Exer
cises
Exer
cises
Sol
ve e
ach
equ
atio
n b
y fa
ctor
ing.
1.6x
2$
2x!
02.
x2!
7x3.
20x2
!$
25x
!0,"
{0,7
}!0,
""
4.6x
2!
7x5.
6x2
$27
x!
06.
12x2
$8x
!0
!0,"
!0,"
!0,"
7.x2
"x
$30
!0
8.2x
2$
x$
3 !
09.
x2"
14x
"33
!0
{5,"
6}!
,"1 "
{"11
,"3}
10.4
x2"
27x
$7
!0
11.3
x2"
29x
$10
!0
12.6
x2$
5x$
4 !
0
!,"
7 "!"
10,
"!"
,"
13.1
2x2
$8x
"1
!0
14.5
x2"
28x
$12
!0
15.2
x2$
250x
"50
00 !
0
!,
"!
,"6 "
{100
,25}
16.2
x2$
11x
$40
!0
17.2
x2"
21x
$11
!0
18.3
x2"
2x$
21 !
0
!8,"
"!"
11,
"!
,"3 "
19.8
x2$
14x
"3
!0
20.6
x2"
11x
$2
!0
21.5
x2"
17x
$12
!0
!,
"!"
2,"
!,"
4 "22
.12x
2"
25x
"12
!0
23.1
2x2
"18
x"
6 !
024
.7x2
$36
x"
5 !
0
!","
"!"
,"1 "
!,5
"1 $ 7
1 $ 23 $ 4
4 $ 3
3 $ 51 $ 6
1 $ 43 $ 2
7 $ 31 $ 2
5 $ 2
2 $ 51 $ 2
1 $ 6
4 $ 31 $ 2
1 $ 31 $ 4
3 $ 2
2 $ 39 $ 2
7 $ 6
5 $ 41 $ 3
©G
lenc
oe/M
cGra
w-H
ill32
6G
lenc
oe A
lgeb
ra 2
Wri
te Q
uad
rati
c Eq
uat
ion
sTo
wri
te a
qua
drat
ic e
quat
ion
wit
h ro
ots
pan
d q,
let
(x$
p)(x
$q)
!0.
The
n m
ulti
ply
usin
g F
OIL
.
Wri
te a
qu
adra
tic
equ
atio
n w
ith
th
e gi
ven
roo
ts.W
rite
th
e eq
uat
ion
in t
he
form
ax2
#bx
#c
!0.
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Solv
ing
Qua
drat
ic E
quat
ions
by
Fact
orin
g
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-3
6-3
Exam
ple
Exam
ple
a.3,
"5 (x
$p)
(x$
q) !
0W
rite
the
patte
rn.
(x$
3)[x
$($
5)]
!0
Rep
lace
pw
ith 3
, qw
ith $
5.
(x$
3)(x
"5)
!0
Sim
plify
.
x2"
2x$
15 !
0U
se F
OIL
.
The
equ
atio
n x2
"2x
$15
!0
has
root
s 3
and
$5.
b."
,
(x$
p)(x
$q)
!0
'x$
!$"(!x
$"!
0
!x"
"!x$
"!0
(!
0
!24
(0
24x2
"13
x$
7 !
0
The
equ
atio
n 24
x2"
13x
$7
!0
has
root
s $
and
.1 % 3
7 % 8
24 (
(8x
"7)
(3x
$1)
%%
%24
(3x
$1)
%3
(8x
"7)
%8
1 % 37 % 8
1 % 37 % 8
1 $ 37 $ 8
Exer
cises
Exer
cises
Wri
te a
qu
adra
tic
equ
atio
n w
ith
th
e gi
ven
roo
ts.W
rite
th
e eq
uat
ion
in
th
e fo
rma
x2#
bx#
c!
0.
1.3,
$4
2.$
8,$
23.
1,9
x2#
x"
12 !
0x2
#10
x#
16 !
0x2
"10
x#
9 !
04.
$5
5.10
,76.
$2,
15x2
#10
x#
25 !
0x2
"17
x#
70 !
0x2
"13
x"
30 !
0
7.$
,58.
2,9.
$7,
3x2
"14
x"
5 !
03x
2"
8x#
4 !
04x
2#
25x
"21
!0
10.3
,11
.$,$
112
.9,
5x2
"17
x#
6 !
09x
2#
13x
#4
!0
6x2
"55
x#
9 !
0
13.
,$14
.,$
15.
,
9x2
"4
!0
8x2
"6x
"5
!0
35x2
"22
x#
3 !
0
16.$
,17
.,
18.
,
16x2
"42
x"
498x
2"
10x
#3
!0
48x2
"14
x#
1 !
0
1 % 61 % 8
3 % 41 % 2
7 % 27 % 8
1 % 53 % 7
1 % 25 % 4
2 % 32 % 3
1 % 64 % 9
2 % 5
3 % 42 % 3
1 % 3
© Glencoe/McGraw-Hill A9 Glencoe Algebra 2
An
swer
s
Answers (Lesson 6-3)
Skil
ls P
ract
ice
Solv
ing
Qua
drat
ic E
quat
ions
by
Fact
orin
g
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-3
6-3
©G
lenc
oe/M
cGra
w-H
ill32
7G
lenc
oe A
lgeb
ra 2
Lesson 6-3
Sol
ve e
ach
equ
atio
n b
y fa
ctor
ing.
1.x2
!64
{"8,
8}2.
x2$
100
!0
{10,
"10
}
3.x2
$3x
"2
!0
{1,2
}4.
x2$
4x"
3 !
0{1
,3}
5.x2
"2x
$3
!0
{1,"
3}6.
x2$
3x$
10 !
0{5
,"2}
7.x2
$6x
"5
!0
{1,5
}8.
x2$
9x!
0{0
,9}
9.$
x2"
6x!
0{0
,6}
10.x
2"
6x"
8 !
0{"
2,"
4}
11.x
2!
$5x
{0,"
5}12
.x2
$14
x"
49 !
0{7
}
13.x
2"
6 !
5x{2
,3}
14.x
2"
18x
!$
81{"
9}
15.x
2$
4x!
21{"
3,7}
16.2
x2"
5x$
3 !
0!
,"3 "
17.4
x2"
5x$
6 !
0!
,"2 "
18.3
x2$
13x
$10
!0
!",5
"
Wri
te a
qu
adra
tic
equ
atio
n w
ith
th
e gi
ven
roo
ts.W
rite
th
e eq
uat
ion
in
th
e fo
rma
x2#
bx#
c!
0,w
her
e a
,b,a
nd
car
e in
tege
rs.
19.1
,4x2
"5x
#4
!0
20.6
,$9
x2#
3x"
54 !
0
21.$
2,$
5x2
#7x
#10
!0
22.0
,7x2
"7x
!0
23.$
,$3
3x2
#10
x#
3 !
024
.$,
8x2
"2x
"3
!0
25.F
ind
two
cons
ecut
ive
inte
gers
who
se p
rodu
ct is
272
.16
,17
3 % 41 % 2
1 % 3
2 $ 33 $ 4
1 $ 2
©G
lenc
oe/M
cGra
w-H
ill32
8G
lenc
oe A
lgeb
ra 2
Sol
ve e
ach
equ
atio
n b
y fa
ctor
ing.
1.x2
$4x
$12
!0
{6,"
2}2.
x2$
16x
"64
!0
{8}
3.x2
$20
x"
100
!0
{10}
4.x2
$6x
"8
!0
{2,4
}5.
x2"
3x"
2 !
0{"
2,"
1}6.
x2$
9x"
14 !
0{2
,7}
7.x2
$4x
!0
{0,4
}8.
7x2
!4x
!0,"
9.x2
"25
!10
x{5
}
10.1
0x2
!9x
!0,"
11.x
2!
2x"
99{"
9,11
}
12.x
2"
12x
!$
36{"
6}13
.5x2
$35
x"
60 !
0{3
,4}
14.3
6x2
!25
!,"
"15
.2x2
$8x
$90
!0
{9,"
5}
16.3
x2"
2x$
1 !
0!
,"1 "
17.6
x2!
9x!0,
"18
.3x2
"24
x"
45 !
0{"
5,"
3}19
.15x
2"
19x
"6
!0
!","
"20
.3x2
$8x
!$
4!2,
"21
.6x2
!5x
"6
!,"
"W
rite
a q
uad
rati
c eq
uat
ion
wit
h t
he
give
n r
oots
.Wri
te t
he
equ
atio
n i
n t
he
form
ax2
#bx
#c
!0,
wh
ere
a,b
,an
d c
are
inte
gers
.
22.7
,223
.0,3
24
.$5,
8x2
"9x
#14
!0
x2"
3x!
0x2
"3x
"40
!0
25.$
7,$
826
.$6,
$3
27.3
,$4
x2#
15x
#56
!0
x2#
9x#
18 !
0x2
#x
"12
!0
28.1
,29
.,2
30.0
,$
2x2
"3x
#1
!0
3x2
"7x
#2
!0
2x2
#7x
!0
31.
,$3
32.4
,33
.$,$
3x2
#8x
"3
!0
3x2
"13
x#
4 !
015
x2#
22x
#8
!0
34.N
UM
BER
TH
EORY
Fin
d tw
o co
nsec
utiv
e ev
en p
osit
ive
inte
gers
who
se p
rodu
ct is
624
.24
,26
35.N
UM
BER
TH
EORY
Fin
d tw
o co
nsec
utiv
e od
d po
siti
ve in
tege
rs w
hose
pro
duct
is 3
23.
17,1
936
.GEO
MET
RYT
he le
ngth
of
a re
ctan
gle
is 2
fee
t m
ore
than
its
wid
th.F
ind
the
dim
ensi
ons
of t
he r
ecta
ngle
if it
s ar
ea is
63
squa
re f
eet.
7 ft
by 9
ft37
.PH
OTO
GR
APH
YT
he le
ngth
and
wid
th o
f a
6-in
ch b
y 8-
inch
pho
togr
aph
are
redu
ced
byth
e sa
me
amou
nt t
o m
ake
a ne
w p
hoto
grap
h w
hose
are
a is
hal
f th
at o
f th
e or
igin
al.B
yho
w m
any
inch
es w
ill t
he d
imen
sion
s of
the
pho
togr
aph
have
to
be r
educ
ed?
2 in
.
4 % 52 % 3
1 % 31 % 3
7 % 21 % 3
1 % 2
2 $ 33 $ 2
2 $ 3
2 $ 33 $ 5
3 $ 21 $ 3
5 $ 65 $ 6
9 $ 10
4 $ 7
Pra
ctic
e (A
vera
ge)
Solv
ing
Qua
drat
ic E
quat
ions
by
Fact
orin
g
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-3
6-3
© Glencoe/McGraw-Hill A10 Glencoe Algebra 2
Answers (Lesson 6-3)
Rea
din
g t
o L
earn
Math
emati
csSo
lvin
g Q
uadr
atic
Equ
atio
ns b
y Fa
ctor
ing
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-3
6-3
©G
lenc
oe/M
cGra
w-H
ill32
9G
lenc
oe A
lgeb
ra 2
Lesson 6-3
Pre-
Act
ivit
yH
ow i
s th
e Z
ero
Pro
du
ct P
rop
erty
use
d i
n g
eom
etry
?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 6-
3 at
the
top
of
page
301
in y
our
text
book
.
Wha
t do
es t
he e
xpre
ssio
n x(
x"
5) m
ean
in t
his
situ
atio
n?
It re
pres
ents
the
area
of t
he re
ctan
gle,
sinc
e th
e ar
ea is
the
prod
uct o
f the
wid
th a
nd le
ngth
.
Rea
din
g t
he
Less
on
1.T
he s
olut
ion
of a
qua
drat
ic e
quat
ion
by f
acto
ring
is s
how
n be
low
.Giv
e th
e re
ason
for
each
ste
p of
the
sol
utio
n.
x2$
10x
!$
21O
rigin
al e
quat
ion
x2$
10x
"21
!0
Add
21
to e
ach
side
.(x
$3)
(x$
7) !
0Fa
ctor
the
trin
omia
l.x
$3
!0
or x
$7
!0
Zero
Pro
duct
Pro
pert
yx
!3
x!
7So
lve
each
equ
atio
n.T
he s
olut
ion
set
is
.
2.O
n an
alg
ebra
qui
z,st
uden
ts w
ere
aske
d to
wri
te a
qua
drat
ic e
quat
ion
wit
h $
7 an
d 5
asit
s ro
ots.
The
wor
k th
at t
hree
stu
dent
s in
the
cla
ss w
rote
on
thei
r pa
pers
is s
how
n be
low
.
Mar
laR
osa
Lar
ry(x
$7)
(x"
5) !
0(x
"7)
(x$
5) !
0(x
"7)
(x$
5) !
0x2
$2x
$35
!0
x2"
2x$
35 !
0x2
$2x
$35
!0
Who
is c
orre
ct?
Ros
aE
xpla
in t
he e
rror
s in
the
oth
er t
wo
stud
ents
’ wor
k.
Sam
ple
answ
er:M
arla
use
d th
e w
rong
fact
ors.
Larr
y us
ed th
e co
rrec
tfa
ctor
s bu
t mul
tiplie
d th
em in
corr
ectly
.
Hel
pin
g Y
ou
Rem
emb
er
3.A
goo
d w
ay t
o re
mem
ber
a co
ncep
t is
to
repr
esen
t it
in m
ore
than
one
way
.Des
crib
e an
alge
brai
c w
ay a
nd a
gra
phic
al w
ay t
o re
cogn
ize
a qu
adra
tic
equa
tion
tha
t ha
s a
doub
lero
ot.
Sam
ple
answ
er:A
lgeb
raic
:Writ
e th
e eq
uatio
n in
the
stan
dard
form
ax
2#
bx#
c!
0 an
d ex
amin
e th
e tr
inom
ial.
If it
is a
per
fect
squ
are
trin
omia
l,th
e qu
adra
tic fu
nctio
n ha
s a
doub
le ro
ot.G
raph
ical
:Gra
ph th
ere
late
d qu
adra
tic fu
nctio
n.If
the
para
bola
has
exa
ctly
one
x-in
terc
ept,
then
the
equa
tion
has
a do
uble
root
.
{3,7
}
©G
lenc
oe/M
cGra
w-H
ill33
0G
lenc
oe A
lgeb
ra 2
Eule
r’s F
orm
ula
for P
rime
Num
bers
Man
y m
athe
mat
icia
ns h
ave
sear
ched
for
a f
orm
ula
that
wou
ld g
ener
ate
prim
e nu
mbe
rs.O
ne s
uch
form
ula
was
pro
pose
d by
Eul
er a
nd u
ses
a qu
adra
tic
poly
nom
ial,
x2"
x"
41.
Fin
d t
he
valu
es o
f x2
#x
#41
for
th
e gi
ven
val
ues
of
x.S
tate
wh
eth
er
each
val
ue
of t
he
pol
ynom
ial
is o
r is
not
a p
rim
e n
um
ber.
1.x
!0
2.x
!1
3.x
!2
41,p
rime
43,p
rime
47,p
rime
4.x
!3
5.x
!4
6.x
!5
53,p
rime
61,p
rime
71,p
rime
7.x
!6
8.x
!17
9.x
!28
83,p
rime
347,
prim
e85
3,pr
ime
10.x
!29
11.x
!30
12.x
!35
911,
prim
e97
1,pr
ime
1301
,prim
e
13.D
oes
the
form
ula
prod
uce
all p
rim
e nu
mbe
rs g
reat
er t
han
40?
Giv
e ex
ampl
es
in y
our
answ
er.
No.
Am
ong
the
prim
es o
mitt
ed a
re 5
9,67
,73,
79,8
9,10
1,10
3,10
7,10
9,an
d 12
7.
14.E
uler
’s f
orm
ula
prod
uces
pri
mes
for
man
y va
lues
of x
,but
it d
oes
not
wor
k fo
r al
l of
them
.Fin
d th
e fi
rst
valu
e of
xfo
r w
hich
the
for
mul
a fa
ils.
(Hin
t:T
ry m
ulti
ples
of
ten.
)
x!
40 g
ives
168
1,w
hich
equ
als
412 .
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-3
6-3
© Glencoe/McGraw-Hill A11 Glencoe Algebra 2
An
swer
s
Answers (Lesson 6-4)
Stu
dy
Gu
ide
and I
nte
rven
tion
Com
plet
ing
the
Squa
re
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-4
6-4
©G
lenc
oe/M
cGra
w-H
ill33
1G
lenc
oe A
lgeb
ra 2
Lesson 6-4
Squ
are
Ro
ot
Pro
per
tyU
se t
he f
ollo
win
g pr
oper
ty t
o so
lve
a qu
adra
tic
equa
tion
tha
t is
in t
he f
orm
“pe
rfec
t sq
uare
tri
nom
ial !
cons
tant
.”
Squa
re R
oot P
rope
rty
For a
ny re
al n
umbe
r xif
x2!
n, th
en x
!)
n.
Sol
ve e
ach
equ
atio
n b
y u
sin
g th
e S
quar
e R
oot
Pro
per
ty.
Exam
ple
Exam
ple
a.x2
"8x
#16
!25
x2$
8x"
16 !
25(x
$4)
2!
25x
$4
!#
25$or
x$
4 !
$#
25$x
!5
"4
!9
or
x!
$5
"4
!$
1
The
sol
utio
n se
t is
{9,
$1}
.
b.4x
2"
20x
#25
!32
4x2
$20
x"
25!
32(2
x$
5)2
!32
2x$
5 !
#32$
or 2
x$
5 !
$#
32$2x
$5
!4#
2$or
2x
$5
!$
4#2$
x!
The
sol
utio
n se
t is
%&.
5 )
4#2$
%% 2
5 )
4#2$
%% 2
Exer
cises
Exer
cises
Sol
ve e
ach
equ
atio
n b
y u
sin
g th
e S
quar
e R
oot
Pro
per
ty.
1.x2
$18
x"
81 !
492.
x2"
20x
"10
0 !
643.
4x2
"4x
"1
!16
{2,1
6}{"
2,"
18}
!,"
"
4.36
x2"
12x
"1
!18
5.9x
2$
12x
"4
!4
6.25
x2"
40x
"16
!28
!"
!0,"
!"
7.4x
2$
28x
"49
!64
8.16
x2"
24x
"9
!81
9.10
0x2
$60
x"
9 !
121
!,"
"!,
"3 "
{"0.
8,1.
4}
10.2
5x2
"20
x"
4 !
7511
.36x
2"
48x
"16
!12
12.2
5x2
$30
x"
9 !
96
!"
!"
!"
3 '
4#6$
$$ 5
"2
'#
3$$
$ 3"
2 '
5#3$
$$ 5
3 $ 21 $ 2
15 $ 2
"4
'2 #
7$$
$ 54 $ 3
"1
'3#
2$$
$ 6
5 $ 23 $ 2
©G
lenc
oe/M
cGra
w-H
ill33
2G
lenc
oe A
lgeb
ra 2
Co
mp
lete
th
e Sq
uar
eTo
com
plet
e th
e sq
uare
for
a q
uadr
atic
exp
ress
ion
of t
he f
orm
x2
"bx
,fo
llow
the
se s
teps
.
1.F
ind
.➞
2.Sq
uare
.➞
3.A
dd !
"2to
x2
"bx
.b % 2
b % 2b % 2
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Com
plet
ing
the
Squa
re
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-4
6-4
Fin
d t
he
valu
e of
cth
at m
akes
x2
#22
x#
ca
per
fect
squ
are
trin
omia
l.T
hen
wri
te t
he
trin
omia
l as
th
esq
uar
e of
a b
inom
ial.
Step
1b
!22
;!
11
Step
211
2!
121
Step
3c
!12
1
The
tri
nom
ial i
s x2
"22
x"
121,
whi
ch c
an b
e w
ritt
en a
s (x
"11
)2.
b % 2
Sol
ve 2
x2"
8x"
24 !
0 by
com
ple
tin
g th
e sq
uar
e.
2x2
$8x
$24
!0
Orig
inal
equ
atio
n
!D
ivid
e ea
ch s
ide
by 2
.
x2$
4x$
12 !
0x2
$4x
$12
is n
ot a
per
fect
squ
are.
x2$
4x!
12Ad
d 12
to e
ach
side
.
x2$
4x"
4 !
12 "
4Si
nce !$
"2!
4, a
dd 4
to e
ach
side
.
(x$
2)2
!16
Fact
or th
e sq
uare
.x
$2
!)
4Sq
uare
Roo
t Pro
perty
x!
6 or
x!
$2
Solv
e ea
ch e
quat
ion.
The
sol
utio
n se
t is
{6,
$2}
.
4 % 2
0 % 22x
2$
8x$
24%
% 2
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Fin
d t
he
valu
e of
cth
at m
akes
eac
h t
rin
omia
l a
per
fect
squ
are.
Th
en w
rite
th
etr
inom
ial
as a
per
fect
squ
are.
1.x2
$10
x"
c2.
x2"
60x
"c
3.x2
$3x
"c
25;(
x"
5)2
900;
(x#
30)2
; %x"
&2
4.x2
"3.
2x"
c5.
x2"
x"
c6.
x2$
2.5x
"c
2.56
;(x
# 1
.6)2
; %x#
&21.
5625
;(x
"1.
25)2
Sol
ve e
ach
equ
atio
n b
y co
mp
leti
ng
the
squ
are.
7.y2
$4y
$5
!0
8.x2
$8x
$65
!0
9.s2
$10
s"
21 !
0"
1,5
"5,
133,
7
10.2
x2$
3x"
1 !
011
.2x2
$13
x$
7 !
012
.25x
2"
40x
$9
!0
1,"
,7,"
13.x
2"
4x"
1 !
014
.y2
"12
y"
4 !
015
.t2
"3t
$8
!0
"2
'#
3$"
6 '
4#2$
"3
'#
41 $$
$ 29 $ 51 $ 5
1 $ 21 $ 2
1 $ 41 $ 16
1 % 2
3 $ 29 $ 4
© Glencoe/McGraw-Hill A12 Glencoe Algebra 2
Answers (Lesson 6-4)
Skil
ls P
ract
ice
Com
plet
ing
the
Squa
re
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-4
6-4
©G
lenc
oe/M
cGra
w-H
ill33
3G
lenc
oe A
lgeb
ra 2
Lesson 6-4
Sol
ve e
ach
equ
atio
n b
y u
sin
g th
e S
quar
e R
oot
Pro
per
ty.
1.x2
$8x
"16
!1
3,5
2.x2
"4x
"4
!1
"1,
"3
3.x2
"12
x"
36 !
25"
1,"
114.
4x2
$4x
"1
!9
"1,
2
5.x2
"4x
"4
!2
"2
'#
2$6.
x2$
2x"
1 !
51
'#
5$
7.x2
$6x
"9
!7
3 '
#7$
8.x2
"16
x"
64 !
15"
8 '
#15$
Fin
d t
he
valu
e of
cth
at m
akes
eac
h t
rin
omia
l a
per
fect
squ
are.
Th
en w
rite
th
etr
inom
ial
as a
per
fect
squ
are.
9.x2
"10
x"
c25
;(x
#5)
210
.x2
$14
x"
c49
;(x
"7)
2
11.x
2"
24x
"c
144;
(x#
12)2
12.x
2"
5x"
c; %x
#&2
13.x
2$
9x"
c; %x
"&2
14.x
2$
x"
c; %x
"&2
Sol
ve e
ach
equ
atio
n b
y co
mp
leti
ng
the
squ
are.
15.x
2$
13x
"36
!0
4,9
16.x
2"
3x!
00,
"3
17.x
2"
x$
6 !
02,
"3
18.x
2$
4x$
13 !
02
'#
17$
19.2
x2"
7x$
4 !
0"
4,20
.3x2
"2x
$1
!0
,"1
21.x
2"
3x$
6 !
022
.x2
$x
$3
!0
23.x
2!
$11
'i#
11$24
.x2
$2x
"4
!0
1 '
i#3$
1 '
#13$
$$ 2
"3
'#
33$$
$ 2
1 $ 31 $ 2
1 $ 21 $ 4
9 $ 281 $ 4
5 $ 225 $ 4
©G
lenc
oe/M
cGra
w-H
ill33
4G
lenc
oe A
lgeb
ra 2
Sol
ve e
ach
equ
atio
n b
y u
sin
g th
e S
quar
e R
oot
Pro
per
ty.
1.x2
"8x
"16
!1
2.x2
"6x
"9
!1
3.x2
"10
x"
25 !
16
"5,
"3
"4,
"2
"9,
"1
4.x2
$14
x"
49 !
95.
4x2
"12
x"
9 !
46.
x2$
8x"
16 !
8
4,10
","
4 '
2#2$
7.x2
$6x
"9
!5
8.x2
$2x
"1
!2
9.9x
2$
6x"
1 !
2
3 '
#5$
1 '
#2$
Fin
d t
he
valu
e of
cth
at m
akes
eac
h t
rin
omia
l a
per
fect
squ
are.
Th
en w
rite
th
etr
inom
ial
as a
per
fect
squ
are.
10.x
2"
12x
"c
11.x
2$
20x
"c
12.x
2"
11x
"c
36;(
x#
6)2
100;
(x"
10)2
; %x#
&2
13.x
2"
0.8x
"c
14.x
2$
2.2x
"c
15.x
2$
0.36
x"
c
0.16
;(x
#0.
4)2
1.21
;(x
"1.
1)2
0.03
24;(
x"
0.18
)2
16.x
2"
x"
c17
.x2
$x
"c
18.x
2$
x"
c
; %x#
&2; %x
"&2
; %x"
&2
Sol
ve e
ach
equ
atio
n b
y co
mp
leti
ng
the
squ
are.
19.x
2"
6x"
8 !
0"
4,"
220
.3x2
"x
$2
!0
,"1
21.3
x2$
5x"
2 !
01,
22.x
2"
18 !
9x23
.x2
$14
x"
19 !
024
.x2
"16
x$
7 !
06,
37
'#
30$"
8 '
#71$
25.2
x2"
8x$
3 !
026
.x2
"x
$5
!0
27.2
x2$
10x
"5
!0
28.x
2"
3x"
6 !
029
.2x2
"5x
"6
!0
30.7
x2"
6x"
2 !
0
31.G
EOM
ETRY
Whe
n th
e di
men
sion
s of
a c
ube
are
redu
ced
by 4
inch
es o
n ea
ch s
ide,
the
surf
ace
area
of
the
new
cub
e is
864
squ
are
inch
es.W
hat
wer
e th
e di
men
sion
s of
the
orig
inal
cub
e?16
in.b
y 16
in.b
y 16
in.
32.I
NV
ESTM
ENTS
The
am
ount
of
mon
ey A
in a
n ac
coun
t in
whi
ch P
dolla
rs is
inve
sted
for
2 ye
ars
is g
iven
by
the
form
ula
A!
P(1
"r)
2 ,w
here
ris
the
inte
rest
rat
e co
mpo
unde
dan
nual
ly.I
f an
inve
stm
ent
of $
800
in t
he a
ccou
nt g
row
s to
$88
2 in
tw
o ye
ars,
at w
hat
inte
rest
rat
e w
as it
inve
sted
?5%
"3
'i#
5$$
$ 7"
5 '
i#23$
$$ 4
"3
'i#
15$$
$ 2
5 '
#15$
$$ 2
"1
'#
21$$
$ 2"
4 '
#22$
$$ 2
2 $ 32 $ 3
5 $ 625 $ 36
1 $ 81 $ 64
5 $ 1225 $ 14
4
5 % 31 % 4
5 % 6
11 $ 212
1$
41 '
#2$
$3
5 $ 21 $ 2
Pra
ctic
e (A
vera
ge)
Com
plet
ing
the
Squa
re
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-4
6-4
© Glencoe/McGraw-Hill A13 Glencoe Algebra 2
An
swer
s
Answers (Lesson 6-4)
Rea
din
g t
o L
earn
Math
emati
csC
ompl
etin
g th
e Sq
uare
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-4
6-4
©G
lenc
oe/M
cGra
w-H
ill33
5G
lenc
oe A
lgeb
ra 2
Lesson 6-4
Pre-
Act
ivit
yH
ow c
an y
ou f
ind
th
e ti
me
it t
akes
an
acc
eler
atin
g ra
ce c
ar t
ore
ach
th
e fi
nis
h l
ine?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 6-
4 at
the
top
of
page
306
in y
our
text
book
.
Exp
lain
wha
t it
mea
ns t
o sa
y th
at t
he d
rive
r ac
cele
rate
s at
a c
onst
ant
rate
of 8
fee
t pe
r se
cond
squ
are.
If th
e dr
iver
is tr
avel
ing
at a
cer
tain
spe
ed a
t a p
artic
ular
mom
ent,
then
one
sec
ond
late
r,th
e dr
iver
is tr
avel
ing
8 fe
etpe
r sec
ond
fast
er.
Rea
din
g t
he
Less
on
1.G
ive
the
reas
on f
or e
ach
step
in t
he f
ollo
win
g so
luti
on o
f an
equ
atio
n by
usi
ng t
heSq
uare
Roo
t P
rope
rty.
x2$
12x
"36
!81
Orig
inal
equ
atio
n
(x$
6)2
!81
Fact
or th
e pe
rfect
squ
are
trin
omia
l.x
$6
!)
#81$
Squa
re R
oot P
rope
rty
x$
6 !
)9
81 !
9x
$6
!9
or x
$6
!$
9R
ewrit
e as
two
equa
tions
.x
!15
x
!$
3So
lve
each
equ
atio
n.
2.E
xpla
in h
ow t
o fi
nd t
he c
onst
ant
that
mus
t be
add
ed t
o m
ake
a bi
nom
ial i
nto
a pe
rfec
tsq
uare
tri
nom
ial.
Sam
ple
answ
er:F
ind
half
of th
e co
effic
ient
of t
he li
near
term
and
squ
are
it.
3.a.
Wha
t is
the
fir
st s
tep
in s
olvi
ng t
he e
quat
ion
3x2
"6x
!5
by c
ompl
etin
g th
e sq
uare
?D
ivid
e th
e eq
uatio
n by
3.
b.W
hat
is t
he f
irst
ste
p in
sol
ving
the
equ
atio
n x2
"5x
$12
!0
by c
ompl
etin
g th
esq
uare
?A
dd 1
2 to
eac
h si
de.
Hel
pin
g Y
ou
Rem
emb
er
4.H
ow c
an y
ou u
se t
he r
ules
for
squ
arin
g a
bino
mia
l to
help
you
rem
embe
r th
e pr
oced
ure
for
chan
ging
a b
inom
ial i
nto
a pe
rfec
t sq
uare
tri
nom
ial?
One
of t
he ru
les
for s
quar
ing
a bi
nom
ial i
s (x
#y)
2!
x2#
2xy
#y2
.In
com
plet
ing
the
squa
re,y
ou a
re s
tart
ing
with
x2
#bx
and
need
to fi
nd y
2 .Th
is s
how
s yo
u th
at b
!2y
,so
y!
.Tha
t is
why
you
mus
t tak
e ha
lf of
th
e co
effic
ient
and
squ
are
it to
get
the
cons
tant
that
mus
t be
adde
d to
com
plet
e th
e sq
uare
.
b $ 2
©G
lenc
oe/M
cGra
w-H
ill33
6G
lenc
oe A
lgeb
ra 2
The
Gol
den
Qua
drat
ic E
quat
ions
A g
old
en r
ecta
ngl
eha
s th
e pr
oper
ty t
hat
its
leng
th
can
be w
ritt
en a
s a
"b,
whe
re a
is t
he w
idth
of
the
rect
angl
e an
d %a
" ab
%!
%a b% .A
ny g
olde
n re
ctan
gle
can
be
divi
ded
into
a s
quar
e an
d a
smal
ler
gold
en r
ecta
ngle
,as
sho
wn.
The
pro
port
ion
used
to
defi
ne g
olde
n re
ctan
gles
can
be
used
to
deri
ve t
wo
quad
rati
c eq
uati
ons.
The
se a
reso
met
imes
calle
d go
lden
qua
drat
ic e
quat
ions
.
Sol
ve e
ach
pro
blem
.
1.In
the
pro
port
ion
for
the
gold
en r
ecta
ngle
,let
aeq
ual 1
.Wri
te t
he r
esul
ting
qu
adra
tic
equa
tion
and
sol
ve f
or b
.
b2#
b"
1!
0 b
!
2.In
the
pro
port
ion,
let
beq
ual 1
.Wri
te t
he r
esul
ting
qua
drat
ic e
quat
ion
and
solv
e fo
r a.
a2"
a"
1!
0a
!
3.D
escr
ibe
the
diff
eren
ce b
etw
een
the
two
gold
en q
uadr
atic
equ
atio
ns y
ou
foun
d in
exe
rcis
es 1
and
2.
The
sign
s of
the
first
-deg
ree
term
s ar
e op
posi
te.
4.Sh
ow t
hat
the
posi
tive
sol
utio
ns o
f th
e tw
o eq
uati
ons
in e
xerc
ises
1 a
nd 2
ar
e re
cipr
ocal
s.
'('
(!!
$"1 4#
5$
!1
5.U
se t
he P
ytha
gore
an T
heor
em t
o fi
nd a
rad
ical
exp
ress
ion
for
the
diag
onal
of
a g
olde
n re
ctan
gle
whe
n a
!1.
d!
6.F
ind
a ra
dica
l exp
ress
ion
for
the
diag
onal
of
a go
lden
rec
tang
le w
hen
b!
1.
d!
#10
#2
$#
5$$
$$ 2
#10
"2
$#
5$$
$$ 2
"( 1
2 )#
( #5$)2
$$ 4
1 #
#5$
$2
"1
##
5$$
$ 2
1 #
#5$
$2
"1
##
5$$
$ 2
a
a a
b b
a
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-4
6-4
© Glencoe/McGraw-Hill A14 Glencoe Algebra 2
Answers (Lesson 6-5)
Stu
dy
Gu
ide
and I
nte
rven
tion
The
Qua
drat
ic F
orm
ula
and
the
Dis
crim
inan
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-5
6-5
©G
lenc
oe/M
cGra
w-H
ill33
7G
lenc
oe A
lgeb
ra 2
Lesson 6-5
Qu
adra
tic
Form
ula
The
Qu
adra
tic
For
mu
laca
n be
use
d to
sol
ve a
nyqu
adra
tic
equa
tion
onc
e it
is w
ritt
en in
the
for
m a
x2"
bx"
c!
0.
Qua
drat
ic F
orm
ula
The
solu
tions
of a
x2"
bx"
c!
0, w
ith a
#0,
are
giv
en b
y x
!.
Sol
ve x
2"
5x!
14 b
y u
sin
g th
e Q
uad
rati
c F
orm
ula
.
Rew
rite
the
equ
atio
n as
x2
$5x
$14
!0.
x!
Qua
drat
ic F
orm
ula
!R
epla
ce a
with
1, b
with
$5,
and
cw
ith $
14.
!Si
mpl
ify.
! !7
or $
2
The
sol
utio
ns a
re $
2 an
d 7.
Sol
ve e
ach
equ
atio
n b
y u
sin
g th
e Q
uad
rati
c F
orm
ula
.
1.x2
"2x
$35
!0
2.x2
"10
x"
24 !
03.
x2$
11x
"24
!0
5,"
7"
4,"
63,
8
4.4x
2"
19x
$5
!0
5.14
x2"
9x"
1 !
06.
2x2
$x
$15
!0
,"5
","
3,"
7.3x
2"
5x!
28.
2y2
"y
$15
!0
9.3x
2$
16x
"16
!0
"2,
,"3
4,
10.8
x2"
6x$
9 !
011
.r2
$"
!0
12.x
2$
10x
$50
!0
",
,5
'5#
3$
13.x
2"
6x$
23 !
014
.4x2
$12
x$
63 !
015
.x2
$6x
"21
!0
"3
'4#
2$3
'2i
#3$
3 '
6 #2 $
$$ 21 $ 5
2 $ 53 $ 4
3 $ 2
2 % 253r % 5
4 $ 35 $ 2
1 $ 3
5 $ 21 $ 7
1 $ 21 $ 45
)9
%2
5 )
#81$
%% 2
$($
5) )
#($
5)2
$$
4(1
$)(
$14
$)$
%%
%%
2(1)
$b
)#
b2$
4$
ac$%
%%
2a
$b
)#
b2$
$4a
c$
%%
%2a
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill33
8G
lenc
oe A
lgeb
ra 2
Ro
ots
an
d t
he
Dis
crim
inan
t
Dis
crim
inan
tTh
e ex
pres
sion
und
er th
e ra
dica
l sig
n, b
2$
4ac,
in th
e Q
uadr
atic
For
mul
a is
cal
led
the
disc
rimin
ant.
Ro
ots
of
a Q
uad
rati
c Eq
uat
ion
Dis
crim
inan
tTy
pe a
nd N
umbe
r of R
oots
b2$
4ac
&0
and
a pe
rfect
squ
are
2 ra
tiona
l roo
ts
b2$
4ac
&0,
but
not
a pe
rfect
squ
are
2 irr
atio
nal r
oots
b2$
4ac
!0
1 ra
tiona
l roo
t
b2$
4ac
'0
2 co
mpl
ex ro
ots
Fin
d t
he
valu
e of
th
e d
iscr
imin
ant
for
each
equ
atio
n.T
hen
des
crib
eth
e n
um
ber
and
typ
es o
f ro
ots
for
the
equ
atio
n.
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
The
Qua
drat
ic F
orm
ula
and
the
Dis
crim
inan
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-5
6-5
Exam
ple
Exam
ple
a.2x
2#
5x#
3T
he d
iscr
imin
ant
is
b2$
4ac
!52
$4(
2)(3
) or
1.
The
dis
crim
inan
t is
a p
erfe
ct s
quar
e,so
the
equa
tion
has
2 r
atio
nal r
oots
.
b.3x
2"
2x#
5T
he d
iscr
imin
ant
is
b2$
4ac
!($
2)2
$4(
3)(5
) or
$56
.T
he d
iscr
imin
ant
is n
egat
ive,
so t
heeq
uati
on h
as 2
com
plex
roo
ts.
Exer
cises
Exer
cises
For
Exe
rcis
es 1
$12
,com
ple
te p
arts
a$
c fo
r ea
ch q
uad
rati
c eq
uat
ion
.a.
Fin
d t
he
valu
e of
th
e d
iscr
imin
ant.
b.D
escr
ibe
the
nu
mbe
r an
d t
ype
of r
oots
.c.
Fin
d t
he
exac
t so
luti
ons
by u
sin
g th
e Q
uad
rati
c F
orm
ula
.
1.p2
"12
p!
$4
128;
2.9x
2$
6x"
1 !
00;
3.2x
2$
7x$
4 !
081
;tw
o irr
atio
nal
root
s;on
e ra
tiona
l roo
t;2
ratio
nal r
oots
;",4
"6
'4 #
2$
4.x2
"4x
$4
!0
32;
5.5x
2$
36x
"7
!0
1156
;6.
4x2
$4x
"11
!0
2 irr
atio
nal r
oots
;2
ratio
nal r
oots
;"
160;
2 co
mpl
ex ro
ots;
"2
'2 #
2$,7
7.x2
$7x
"6
!0
25;
8.m
2$
8m!
$14
8;9.
25x2
$40
x!
$16
0;2
ratio
nal r
oots
;2
irrat
iona
l roo
ts;
1 ra
tiona
l roo
t;1,
64
'#
2$
10.4
x2"
20x
"29
!0
"64
;11
.6x2
"26
x"
8 !
048
4;12
.4x2
$4x
$11
!0
192;
2 co
mpl
ex ro
ots;
2 ra
tiona
l roo
ts;
2 irr
atio
nal r
oots
;4 $ 5
1 '
i#10$
$$ 2
1 $ 5
1 $ 21 $ 3
© Glencoe/McGraw-Hill A15 Glencoe Algebra 2
An
swer
s
Answers (Lesson 6-5)
Skil
ls P
ract
ice
The
Qua
drat
ic F
orm
ula
and
the
Dis
crim
inan
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
__PE
RIO
D__
___
6-5
6-5
©G
lenc
oe/M
cGra
w-H
ill33
9G
lenc
oe A
lgeb
ra 2
Lesson 6-5
Com
ple
te p
arts
a$
c fo
r ea
ch q
uad
rati
c eq
uat
ion
.a.
Fin
d t
he
valu
e of
th
e d
iscr
imin
ant.
b.D
escr
ibe
the
nu
mbe
r an
d t
ype
of r
oots
.c.
Fin
d t
he
exac
t so
luti
ons
by u
sin
g th
e Q
uad
rati
c F
orm
ula
.
1.x2
$8x
"16
!0
2.x2
$11
x$
26 !
0
0;1
ratio
nal r
oot;
422
5;2
ratio
nal r
oots
;"2,
13
3.3x
2$
2x!
04.
20x2
"7x
$3
!0
4;2
ratio
nal r
oots
;0,
289;
2 ra
tiona
l roo
ts;"
,
5.5x
2$
6 !
06.
x2$
6 !
0
120;
2 irr
atio
nal r
oots
;'24
;2 ir
ratio
nal r
oots
;'#
6$
7.x2
"8x
"13
!0
8.5x
2$
x$
1 !
0
12;2
irra
tiona
l roo
ts;"
4 '
#3$
21;2
irra
tiona
l roo
ts;
9.x2
$2x
$17
!0
10.x
2"
49 !
0
72;2
irra
tiona
l roo
ts;1
'3#
2$"
196;
2 co
mpl
ex ro
ots;
'7i
11.x
2$
x"
1 !
012
.2x2
$3x
!$
2
"3;
2 co
mpl
ex ro
ots;
"7;
2 co
mpl
ex ro
ots;
Sol
ve e
ach
equ
atio
n b
y u
sin
g th
e m
eth
od o
f yo
ur
choi
ce.F
ind
exa
ct s
olu
tion
s.
13.x
2!
64'
814
.x2
$30
!0
'#
30$
15.x
2$
x!
30"
5,6
16.1
6x2
$24
x$
27 !
0,"
17.x
2$
4x$
11 !
02
' #
15$18
.x2
$8x
$17
!0
4 '
#33$
19.x
2"
25 !
0'
5i20
.3x2
"36
!0
'2i
#3$
21.2
x2"
10x
"11
!0
22.2
x2$
7x"
4 !
0
23.8
x2"
1 !
4x24
.2x2
"2x
"3
!0
25.P
AR
AC
HU
TIN
GIg
nori
ng w
ind
resi
stan
ce,t
he d
ista
nce
d(t)
in f
eet
that
a p
arac
huti
stfa
lls in
tse
cond
s ca
n be
est
imat
ed u
sing
the
for
mul
a d(
t) !
16t2
.If
a pa
rach
utis
t ju
mps
from
an
airp
lane
and
fal
ls f
or 1
100
feet
bef
ore
open
ing
her
para
chut
e,ho
w m
any
seco
nds
pass
bef
ore
she
open
s th
e pa
rach
ute?
abou
t 8.3
s
"1
'i#
5$$
$ 21
'i
$4
7 '
#17$
$$ 4
"5
'#
3$$
$ 2
3 $ 49 $ 4
3 '
i#7$
$$ 4
1 '
i#3$
$$ 2
1 '
#21$
$$ 10
#30$
$5
1 $ 43 $ 5
2 $ 3
©G
lenc
oe/M
cGra
w-H
ill34
0G
lenc
oe A
lgeb
ra 2
Com
ple
te p
arts
a$
c fo
r ea
ch q
uad
rati
c eq
uat
ion
.a.
Fin
d t
he
valu
e of
th
e d
iscr
imin
ant.
b.D
escr
ibe
the
nu
mbe
r an
d t
ype
of r
oots
.c.
Fin
d t
he
exac
t so
luti
ons
by u
sin
g th
e Q
uad
rati
c F
orm
ula
.
1.x2
$16
x"
64 !
02.
x2!
3x3.
9x2
$24
x"
16 !
0
0;1
ratio
nal;
89;
2 ra
tiona
l;0,
30;
1 ra
tiona
l;
4.x2
$3x
!40
5.3x
2"
9x$
2 !
010
5;6.
2x2
"7x
!0
169;
2 ra
tiona
l;"
5,8
2 irr
atio
nal;
49;2
ratio
nal;
0,"
7.5x
2$
2x"
4 !
0"
76;
8.12
x2$
x$
6 !
028
9;9.
7x2
"6x
"2
!0
"20
;2
com
plex
;2
ratio
nal;
,"2
com
plex
;
10.1
2x2
"2x
$4
!0
196;
11.6
x2$
2x$
1 !
028
;12
.x2
"3x
"6
!0
"15
;2
ratio
nal;
,"2
irrat
iona
l;2
com
plex
;
13.4
x2$
3x2
$6
!0
105;
14.1
6x2
$8x
"1
!0
15.2
x2$
5x$
6 !
073
;2
irrat
iona
l;0;
1 ra
tiona
l;2
irrat
iona
l;
Sol
ve e
ach
equ
atio
n b
y u
sin
g th
e m
eth
od o
f yo
ur
choi
ce.F
ind
exa
ct s
olu
tion
s.
16.7
x2$
5x!
00,
17.4
x2$
9 !
0'
18.3
x2"
8x!
3,"
319
.x2
$21
!4x
"3,
7
20.3
x2$
13x
"4
!0
,421
.15x
2"
22x
!$
8"
,"
22.x
2$
6x"
3 !
03
'#
6$23
.x2
$14
x"
53 !
07
'2i
24.3
x2!
$54
'3i
#2$
25.2
5x2
$20
x$
6 !
0
26.4
x2$
4x"
17 !
027
.8x
$1
!4x
2
28.x
2!
4x$
152
'i#
11$29
.4x2
$12
x"
7 !
0
30. G
RA
VIT
ATI
ON
The
hei
ght
h(t)
in fe
et o
f an
obje
ct t
seco
nds
afte
r it
is p
rope
lled
stra
ight
up
from
the
gro
und
wit
h an
init
ial v
eloc
ity
of 6
0 fe
et p
er s
econ
d is
mod
eled
by
the
equa
tion
h(t)
!$
16t2
"60
t.A
t w
hat
tim
es w
ill t
he o
bjec
t be
at
a he
ight
of
56 f
eet?
1.75
s,2
s31
.STO
PPIN
G D
ISTA
NC
ET
he f
orm
ula
d!
0.05
s2"
1.1s
esti
mat
es t
he m
inim
um s
topp
ing
dist
ance
din
feet
for
a ca
r tr
avel
ing
sm
iles
per
hour
.If a
car
sto
ps in
200
feet
,wha
t is
the
fast
est
it c
ould
hav
e be
en t
rave
ling
whe
n th
e dr
iver
app
lied
the
brak
es?
abou
t 53.
2 m
i/h
3 '
#2$
$2
2 '
#3$
$2
1 '
4i$
2
2 '
#10$
$$ 54 $ 5
2 $ 31 $ 3
1 $ 3
3 $ 25 $ 7
5 '
#73$
$$ 4
1 $ 43
'#
105
$$
$ 8
"3
'i #
15$$
$ 21
'#
7$$
62 $ 3
1 $ 2
"3
'i#
5$$
$ 72 $ 3
3 $ 41
'i#
19$$
$ 5
7 $ 2"
9 '
#10
5$
$$ 6
4 $ 3
Pra
ctic
e (A
vera
ge)
The
Qua
drat
ic F
orm
ula
and
the
Dis
crim
inan
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-5
6-5
© Glencoe/McGraw-Hill A16 Glencoe Algebra 2
Answers (Lesson 6-5)
Rea
din
g t
o L
earn
Math
emati
csTh
e Q
uadr
atic
For
mul
a an
d th
e D
iscr
imin
ant
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-5
6-5
©G
lenc
oe/M
cGra
w-H
ill34
1G
lenc
oe A
lgeb
ra 2
Lesson 6-5
Pre-
Act
ivit
yH
ow i
s bl
ood
pre
ssu
re r
elat
ed t
o ag
e?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 6-
5 at
the
top
of
page
313
in y
our
text
book
.
Des
crib
e ho
w y
ou w
ould
cal
cula
te y
our
norm
al b
lood
pre
ssur
e us
ing
one
ofth
e fo
rmul
as in
you
r te
xtbo
ok.
Sam
ple
answ
er:S
ubst
itute
you
r age
for A
in th
e ap
prop
riate
form
ula
(for f
emal
es o
r mal
es) a
nd e
valu
ate
the
expr
essi
on.
Rea
din
g t
he
Less
on
1.a.
Wri
te t
he Q
uadr
atic
For
mul
a.x
!
b.Id
enti
fy t
he v
alue
s of
a,b
,and
cth
at y
ou w
ould
use
to
solv
e 2x
2$
5x!
$7,
but
dono
t ac
tual
ly s
olve
the
equ
atio
n.
a!
b!
c!
2.Su
ppos
e th
at y
ou a
re s
olvi
ng f
our
quad
rati
c eq
uati
ons
wit
h ra
tion
al c
oeff
icie
nts
and
have
fou
nd t
he v
alue
of
the
disc
rim
inan
t fo
r ea
ch e
quat
ion.
In e
ach
case
,giv
e th
enu
mbe
r of
roo
ts a
nd d
escr
ibe
the
type
of
root
s th
at t
he e
quat
ion
will
hav
e.
Valu
e of
Dis
crim
inan
tN
umbe
r of R
oots
Type
of R
oots
642
real
,rat
iona
l$
82
com
plex
212
real
,irr
atio
nal
01
real
,rat
iona
l
Hel
pin
g Y
ou
Rem
emb
er
3.H
ow c
an lo
okin
g at
the
Qua
drat
ic F
orm
ula
help
you
rem
embe
r th
e re
lati
onsh
ips
betw
een
the
valu
e of
the
dis
crim
inan
t an
d th
e nu
mbe
r of
roo
ts o
f a
quad
rati
c eq
uati
onan
d w
heth
er t
he r
oots
are
rea
l or
com
plex
?Sa
mpl
e an
swer
:The
dis
crim
inan
t is
the
expr
essi
on u
nder
the
radi
cal i
nth
e Q
uadr
atic
For
mul
a.Lo
ok a
t the
Qua
drat
ic F
orm
ula
and
cons
ider
wha
tha
ppen
s w
hen
you
take
the
prin
cipa
l squ
are
root
of b
2"
4ac
and
appl
y'
in fr
ont o
f the
resu
lt.If
b2"
4ac
is p
ositi
ve,i
ts p
rinci
pal s
quar
e ro
otw
ill b
e a
posi
tive
num
ber a
nd a
pply
ing
'w
ill g
ive
two
diffe
rent
real
solu
tions
,whi
ch m
ay b
e ra
tiona
l or i
rrat
iona
l.If
b2"
4ac
!0,
itspr
inci
pal s
quar
e ro
ot is
0,s
o ap
plyi
ng '
in th
e Q
uadr
atic
For
mul
a w
illon
ly le
ad to
one
sol
utio
n,w
hich
will
be
ratio
nal (
assu
min
g a,
b,an
d c
are
inte
gers
).If
b2"
4ac
is n
egat
ive,
sinc
e th
e sq
uare
root
s of
neg
ativ
enu
mbe
rs a
re n
ot re
al n
umbe
rs,y
ou w
ill g
et tw
o co
mpl
ex ro
ots,
corr
espo
ndin
g to
the
#an
d "
in th
e '
sym
bol.7
"5
2
"b
'#
b2"
4$
ac $$
$ 2a
©G
lenc
oe/M
cGra
w-H
ill34
2G
lenc
oe A
lgeb
ra 2
Sum
and
Pro
duct
of R
oots
So
met
imes
you
may
kno
w t
he r
oots
of
a qu
adra
tic
equa
tion
wit
hout
kno
win
g th
e eq
uati
onit
self.
Usi
ng y
our
know
ledg
e of
fac
tori
ng t
o so
lve
an e
quat
ion,
you
can
wor
k ba
ckw
ard
tofi
nd t
he q
uadr
atic
equ
atio
n.T
he r
ule
for
find
ing
the
sum
and
pro
duct
of
root
s is
as
follo
ws:
Sum
and
Pro
duct
of R
oots
If th
e ro
ots
of a
x2"
bx"
c!
0, w
ith a
≠0,
are
s1
and
s 2,
then
s1
"s 2
!$
%b a%an
d s 1
(s 2
!% ac % .
A r
oad
wit
h a
n i
nit
ial g
rad
ien
t,or
slo
pe,
of 3
% c
an b
e re
pre
sen
ted
by
the
form
ula
y!
ax2
#0.
03x
#c,
wh
ere
yis
th
e el
evat
ion
an
d x
is t
he
dis
tan
ce a
lon
gth
e cu
rve.
Su
pp
ose
the
elev
atio
n o
f th
e ro
ad i
s 11
05 f
eet
at p
oin
ts 2
00 f
eet
and
100
0fe
et a
lon
g th
e cu
rve.
You
can
fin
d t
he
equ
atio
n o
f th
e tr
ansi
tion
cu
rve.
Equ
atio
ns
of t
ran
siti
on c
urv
es a
re u
sed
by
civi
l en
gin
eers
to
des
ign
sm
ooth
an
d s
afe
road
s.
The
roo
ts a
re x
!3
and
x!
$8.
3 "
($8)
!$
5Ad
d th
e ro
ots.
3($
8) !
$24
Mul
tiply
the
root
s.
Equ
atio
n:x2
"5x
$24
!0
Wri
te a
qu
adra
tic
equ
atio
n t
hat
has
th
e gi
ven
roo
ts.
1.6,
$9
2.5,
$1
3.6,
6
x2#
3x"
54 !
0x2
"4x
"5
!0
x2"
12x
#36
!0
4.4
)#
3$6.
$%2 5% ,
%2 7%6.
x2"
8x#
13 !
035
x2#
4x"
4 !
049
x2"
42x
#20
5 !
0
Fin
d k
such
th
at t
he
nu
mbe
r gi
ven
is
a ro
ot o
f th
e eq
uat
ion
.
7.7;
2x2
"kx
$21
!0
8.$
2;x2
$13
x"
k!
0 "
11"
30
$2
)3#
5$%
% 7
x
y
O
(–5 – 2, –3
01 – 4)
10 –10
–20
–30
24
–2–4
–6–8
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-5
6-5
Exam
ple
Exam
ple
© Glencoe/McGraw-Hill A17 Glencoe Algebra 2
An
swer
s
Answers (Lesson 6-6)
Stu
dy
Gu
ide
and I
nte
rven
tion
Ana
lyzi
ng G
raph
s of
Qua
drat
ic F
unct
ions
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-6
6-6
©G
lenc
oe/M
cGra
w-H
ill34
3G
lenc
oe A
lgeb
ra 2
Lesson 6-6
An
alyz
e Q
uad
rati
c Fu
nct
ion
s
The
grap
h of
y!
a(x
$h)
2"
kha
s th
e fo
llow
ing
char
acte
ristic
s:•
Verte
x: (h
, k)
Vert
ex F
orm
•Ax
is o
f sym
met
ry: x
!h
of a
Qua
drat
ic•
Ope
ns u
p if
a&
0Fu
nctio
n•
Ope
ns d
own
if a
'0
•N
arro
wer
than
the
grap
h of
y!
x2if a
&
1•
Wid
er th
an th
e gr
aph
of y
!x2
if a
'
1
Iden
tify
th
e ve
rtex
,axi
s of
sym
met
ry,a
nd
dir
ecti
on o
f op
enin
g of
each
gra
ph
.
a.y
!2(
x#
4)2
"11
The
ver
tex
is a
t (h
,k)
or ($
4,$
11),
and
the
axis
of
sym
met
ry is
x!
$4.
The
gra
ph o
pens
up,a
nd is
nar
row
er t
han
the
grap
h of
y !
x2.
a.y
!"
(x"
2)2
#10
The
ver
tex
is a
t (h
,k)
or (
2,10
),an
d th
e ax
is o
f sy
mm
etry
is x
!2.
The
gra
ph o
pens
dow
n,an
d is
wid
er t
han
the
grap
h of
y !
x2.
Eac
h q
uad
rati
c fu
nct
ion
is
give
n i
n v
erte
x fo
rm.I
den
tify
th
e ve
rtex
,axi
s of
sym
met
ry,a
nd
dir
ecti
on o
f op
enin
g of
th
e gr
aph
.
1.y
!(x
$2)
2"
162.
y!
4(x
"3)
2$
73.
y!
(x$
5)2
"3
(2,1
6);x
!2;
up("
3,"
7);x
!"
3;up
(5,3
);x
!5;
up
4.y
!$
7(x
"1)
2$
95.
y!
(x$
4)2
$12
6.y
!6(
x"
6)2
"6
("1,
"9)
;x!
"1;
dow
n(4
,"12
);x
!4;
up("
6,6)
;x!
"6;
up
7.y
!(x
$9)
2"
128.
y!
8(x
$3)
2$
29.
y!
$3(
x$
1)2
$2
(9,1
2);x
!9;
up(3
,"2)
;x!
3;up
(1,"
2);x
!1;
dow
n
10.y
!$
(x"
5)2
"12
11.y
!(x
$7)
2"
2212
.y!
16(x
$4)
2"
1
("5,
12);
x!
"5;
dow
n(7
,22)
;x!
7;up
(4,1
);x
!4;
up
13.y
!3(
x$
1.2)
2"
2.7
14.y
!$
0.4(
x$
0.6)
2$
0.2
15.y
!1.
2(x
"0.
8)2
"6.
5
(1.2
,2.7
);x
!1.
2;up
(0.6
,"0.
2);x
!0.
6;("
0.8,
6.5)
;x!
"0.
8;do
wn
up
4 % 35 % 2
2 % 5
1 % 5
1 % 2
1 $ 4
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill34
4G
lenc
oe A
lgeb
ra 2
Wri
te Q
uad
rati
c Fu
nct
ion
s in
Ver
tex
Form
A q
uadr
atic
fun
ctio
n is
eas
ier
togr
aph
whe
n it
is in
ver
tex
form
.You
can
wri
te a
qua
drat
ic f
unct
ion
of t
he f
orm
y
!ax
2"
bx"
cin
ver
tex
from
by
com
plet
ing
the
squa
re.
Wri
te y
!2x
2"
12x
#25
in
ver
tex
form
.Th
en g
rap
h t
he
fun
ctio
n.
y!
2x2
$12
x"
25y
!2(
x2$
6x) "
25y
!2(
x2$
6x"
9) "
25 $
18y
!2(
x$
3)2
"7
The
ver
tex
form
of
the
equa
tion
is y
!2(
x$
3)2
"7.
Wri
te e
ach
qu
adra
tic
fun
ctio
n i
n v
erte
x fo
rm.T
hen
gra
ph
th
e fu
nct
ion
.
1.y
!x2
$10
x "
322.
y !
x2"
6x3.
y!
x2$
8x"
6y
!(x
"5)
2#
7y
!(x
#3)
2"
9y
!(x
"4)
2"
10
4.y
!$
4x2
"16
x$
115.
y!
3x2
$12
x"
56.
y!
5x2
$10
x"
9y
!"
4(x
"2)
2#
5y
!3(
x"
2)2
"7
y!
5(x"
1)2
#4 x
y
O
x
y
O
x
y
O
x
y
O4
–48
8 4 –4 –8 –12
x
y
O
x
y
O
x
y
O
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Ana
lyzi
ng G
raph
s of
Qua
drat
ic F
unct
ions
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-6
6-6
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A18 Glencoe Algebra 2
Answers (Lesson 6-6)
Skil
ls P
ract
ice
Ana
lyzi
ng G
raph
s of
Qua
drat
ic F
unct
ions
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-6
6-6
©G
lenc
oe/M
cGra
w-H
ill34
5G
lenc
oe A
lgeb
ra 2
Lesson 6-6
Wri
te e
ach
qu
adra
tic
fun
ctio
n i
n v
erte
x fo
rm,i
f n
ot a
lrea
dy
in t
hat
for
m.T
hen
iden
tify
th
e ve
rtex
,axi
s of
sym
met
ry,a
nd
dir
ecti
on o
f op
enin
g.
1.y
!(x
$2)
22.
y!
$x2
"4
3.y
!x2
$6
y!
(x"
2)2
#0;
y!
"(x
"0)
2#
4;y
!(x
"0)
2"
6;(2
,0);
x!
2;up
(0,4
);x
!0;
dow
n(0
,"6)
;x!
0;up
4.y
!$
3(x
"5)
25.
y!
$5x
2"
96.
y!
(x$
2)2
$18
y!
"3(
x#
5)2
#0;
y!
"5(
x"
0)2
#9;
y!
(x"
2)2
"18
;("
5,0)
;x!
"5;
dow
n(0
,9);
x!
0;do
wn
(2,"
18);
x!
2;up
7.y
!x2
$2x
$5
8.y
!x2
"6x
"2
9.y
!$
3x2
"24
xy
!(x
"1)
2"
6;y
!(x
#3)
2"
7;y
!"
3(x
"4)
2#
48;
(1,"
6);x
!1;
up("
3,"
7);x
!"
3;up
(4,4
8);x
!4;
dow
n
Gra
ph
eac
h f
un
ctio
n.
10.y
!(x
$3)
2$
111
.y!
(x"
1)2
"2
12.y
!$
(x$
4)2
$4
13.y
!$
(x"
2)2
14.y
!$
3x2
"4
15.y
!x2
"6x
"4
Wri
te a
n e
quat
ion
for
th
e p
arab
ola
wit
h t
he
give
n v
erte
x th
at p
asse
s th
rou
gh t
he
give
n p
oin
t.
16.v
erte
x:(4
,$36
)17
.ver
tex:
(3,$
1)18
.ver
tex:
($2,
2)po
int:
(0,$
20)
poin
t:(2
,0)
poin
t:($
1,3)
y!
(x"
4)2
"36
y!
(x"
3)2
"1
y!
(x#
2)2
#2x
y
Ox
y
O
x
y
O
1 % 2
x
y
O
x
y
Ox
y
O
©G
lenc
oe/M
cGra
w-H
ill34
6G
lenc
oe A
lgeb
ra 2
Wri
te e
ach
qu
adra
tic
fun
ctio
n i
n v
erte
x fo
rm,i
f n
ot a
lrea
dy
in t
hat
for
m.T
hen
iden
tify
th
e ve
rtex
,axi
s of
sym
met
ry,a
nd
dir
ecti
on o
f op
enin
g.
1.y
!$
6(x
"2)
2$
12.
y!
2x2
"2
3.y
!$
4x2
"8x
y!
"6(
x#
2)2
"1;
y!
2(x
#0)
2#
2;y
!"
4(x
"1)
2#
4;("
2,"
1);x
!"
2;do
wn
(0,2
);x
!0;
up(1
,4);
x!
1;do
wn
4.y
!x2
"10
x"
205.
y!
2x2
"12
x"
186.
y!
3x2
$6x
"5
y!
(x#
5)2
"5;
y!
2(x
#3)
2 ;("
3,0)
;y
!3(
x"
1)2
#2;
("5,
"5)
;x!
"5;
upx
!"
3;up
(1,2
);x
!1;
up7.
y!
$2x
2$
16x
$32
8.y
!$
3x2
"18
x$
219.
y!
2x2
"16
x"
29y
!"
2(x
#4)
2 ;y
!"
3(x
"3)
2#
6;y
!2(
x#
4)2
"3;
("4,
0);x
!"
4;do
wn
(3,6
);x
!3;
dow
n("
4,"
3);x
!"
4;up
Gra
ph
eac
h f
un
ctio
n.
10.y
!(x
"3)
2$
111
.y!
$x2
"6x
$5
12.y
!2x
2$
2x"
1
Wri
te a
n e
quat
ion
for
th
e p
arab
ola
wit
h t
he
give
n v
erte
x th
at p
asse
s th
rou
gh t
he
give
n p
oin
t.
13.v
erte
x:(1
,3)
14.v
erte
x:($
3,0)
15
.ver
tex:
(10,
$4)
poin
t:($
2,$
15)
poin
t:(3
,18)
poin
t:(5
,6)
y!
"2(
x"
1)2
#3
y!
(x#
3)2
y!
(x"
10)2
"4
16.W
rite
an
equa
tion
for
a p
arab
ola
wit
h ve
rtex
at
(4,4
) an
d x-
inte
rcep
t 6.
y!
"(x
"4)
2#
417
.Wri
te a
n eq
uati
on f
or a
par
abol
a w
ith
vert
ex a
t ($
3,$
1) a
nd y
-int
erce
pt 2
.y
!(x
#3)
2"
118
.BA
SEB
ALL
The
hei
ght
hof
a b
aseb
all t
seco
nds
afte
r be
ing
hit
is g
iven
by
h(t)
!$
16t2
"80
t"
3.W
hat
is t
he m
axim
um h
eigh
t th
at t
he b
aseb
all r
each
es,a
ndw
hen
does
thi
s oc
cur?
103
ft;2.
5 s
19.S
CU
LPTU
RE
A m
oder
n sc
ulpt
ure
in a
par
k co
ntai
ns a
par
abol
ic a
rc t
hat
star
ts a
t th
e gr
ound
and
rea
ches
a m
axim
um h
eigh
t of
10
feet
aft
er a
hori
zont
al d
ista
nce
of 4
fee
t.W
rite
a q
uadr
atic
fun
ctio
n in
ver
tex
form
that
des
crib
es t
he s
hape
of
the
outs
ide
of t
he a
rc,w
here
yis
the
hei
ght
of a
poi
nt o
n th
e ar
c an
dx
is it
s ho
rizo
ntal
dis
tanc
e fr
om t
he le
ft-h
and
star
ting
poi
nt o
f th
e ar
c.y
!"
(x"
4)2
#10
5 $ 8
10 ft
4 ft
1 $ 3
2 $ 51 $ 2
x
y O
x
y
O
x
y
O
Pra
ctic
e (A
vera
ge)
Ana
lyzi
ng G
raph
s of
Qua
drat
ic F
unct
ions
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-6
6-6
© Glencoe/McGraw-Hill A19 Glencoe Algebra 2
An
swer
s
Answers (Lesson 6-6)
Rea
din
g t
o L
earn
Math
emati
csA
naly
zing
Gra
phs
of Q
uadr
atic
Equ
atio
ns
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-6
6-6
©G
lenc
oe/M
cGra
w-H
ill34
7G
lenc
oe A
lgeb
ra 2
Lesson 6-6
Pre-
Act
ivit
yH
ow c
an t
he
grap
h o
f y
!x2
be u
sed
to
grap
h a
ny
quad
rati
cfu
nct
ion
?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 6-
6 at
the
top
of
page
322
in y
our
text
book
.
•W
hat
does
add
ing
a po
siti
ve n
umbe
r to
x2
do t
o th
e gr
aph
of y
!x2
?It
mov
es th
e gr
aph
up.
•W
hat
does
sub
trac
ting
a p
osit
ive
num
ber
to x
befo
re s
quar
ing
do t
o th
egr
aph
of y
!x2
?It
mov
es th
e gr
aph
to th
e rig
ht.
Rea
din
g t
he
Less
on
1.C
ompl
ete
the
follo
win
g in
form
atio
n ab
out
the
grap
h of
y!
a(x
$h)
2"
k.
a.W
hat
are
the
coor
dina
tes
of t
he v
erte
x?(h
,k)
b.W
hat
is t
he e
quat
ion
of t
he a
xis
of s
ymm
etry
?x
!h
c.In
whi
ch d
irec
tion
doe
s th
e gr
aph
open
if a
&0?
If
a'
0?up
;dow
nd.
Wha
t do
you
kno
w a
bout
the
gra
ph if
a
'1?
It is
wid
er th
an th
e gr
aph
of y
!x2
.If
a
&1?
It is
nar
row
er th
an th
e gr
aph
of y
!x2
.
2.M
atch
eac
h gr
aph
wit
h th
e de
scri
ptio
n of
the
con
stan
ts in
the
equ
atio
n in
ver
tex
form
.
a.a
&0,
h&
0,k
'0
iiib.
a'
0,h
'0,
k'
0iv
c.a
'0,
h'
0,k
&0
iid.
a&
0,h
!0,
k'
0i
i.ii
.ii
i.iv
.
Hel
pin
g Y
ou
Rem
emb
er
3.W
hen
grap
hing
qua
drat
ic fu
ncti
ons
such
as
y!
(x"
4)2
and
y!
(x$
5)2 ,
man
y st
uden
tsha
ve t
roub
le r
emem
beri
ng w
hich
rep
rese
nts
a tr
ansl
atio
n of
the
gra
ph o
f y!
x2to
the
left
and
whi
ch r
epre
sent
s a
tran
slat
ion
to t
he r
ight
.Wha
t is
an
easy
way
to
rem
embe
r th
is?
Sam
ple
answ
er:I
n fu
nctio
ns li
ke y
!(x
#4)
2 ,th
e pl
us s
ign
puts
the
grap
h “a
head
”so
that
the
vert
ex c
omes
“so
oner
”th
an th
e or
igin
and
the
tran
slat
ion
is to
the
left.
In fu
nctio
ns li
ke y
!(x
"5)
2 ,th
e m
inus
put
s th
egr
aph
“beh
ind”
so th
at th
e ve
rtex
com
es “
late
r”th
an th
e or
igin
and
the
tran
slat
ion
is to
the
right
.
x
y
Ox
y
Ox
y
Ox
y
O
©G
lenc
oe/M
cGra
w-H
ill34
8G
lenc
oe A
lgeb
ra 2
Patte
rns
with
Diff
eren
ces
and
Sum
s of
Squ
ares
Som
e w
hole
num
bers
can
be
wri
tten
as
the
diff
eren
ce o
f tw
o sq
uare
s an
dso
me
cann
ot.F
orm
ulas
can
be
deve
lope
d to
des
crib
e th
e se
ts o
f nu
mbe
rsal
gebr
aica
lly.
If p
ossi
ble,
wri
te e
ach
nu
mbe
r as
th
e d
iffe
ren
ce o
f tw
o sq
uar
es.
Loo
k f
or p
atte
rns.
1.0
02"
022.
112
"02
3.2
cann
ot4.
322
"12
5.4
22"
026.
532
"22
7.6
cann
ot8.
742
"32
9.8
32"
1210
.932
"02
11.1
0ca
nnot
12.1
162
"52
13.1
242
"22
14.1
372
"62
15.1
4ca
nnot
16.1
542
"12
Eve
n n
um
bers
can
be
wri
tten
as
2n,w
her
e n
is o
ne
of t
he
nu
mbe
rs
0,1,
2,3,
and
so
on.O
dd
nu
mbe
rs c
an b
e w
ritt
en 2
n#
1.U
se t
hes
e ex
pre
ssio
ns
for
thes
e p
robl
ems.
17.S
how
tha
t an
y od
d nu
mbe
r ca
n be
wri
tten
as
the
diff
eren
ce o
f tw
o sq
uare
s.2n
#1
!(n
#1)
2"
n2
18.S
how
tha
t th
e ev
en n
umbe
rs c
an b
e di
vide
d in
to t
wo
sets
:tho
se t
hat
can
be w
ritt
en in
the
for
m 4
nan
d th
ose
that
can
be
wri
tten
in t
he f
orm
2 "
4n.
Find
4n
for n
!0,
1,2,
and
so o
n.Yo
u ge
t {0,
4,8,
12,…
}.Fo
r 2 #
4n,y
ouge
t {2,
6,10
,12,
…}.
Toge
ther
thes
e se
ts in
clud
e al
l eve
n nu
mbe
rs.
19.D
escr
ibe
the
even
num
bers
tha
t ca
nnot
be
wri
tten
as
the
diff
eren
ce o
f tw
o sq
uare
s.2
#4n
,for
n!
0,1,
2,3,
…20
.Sho
w t
hat
the
othe
r ev
en n
umbe
rs c
an b
e w
ritt
en a
s th
e di
ffer
ence
of
two
squa
res.
4n!
(n#
1)2
"(n
"1)
2
Eve
ry w
hol
e n
um
ber
can
be
wri
tten
as
the
sum
of
squ
ares
.It
is n
ever
n
eces
sary
to
use
mor
e th
an f
our
squ
ares
.Sh
ow t
hat
th
is i
s tr
ue
for
the
wh
ole
nu
mbe
rs f
rom
0 t
hro
ugh
15
by w
riti
ng
each
on
e as
th
e su
m o
f th
e le
ast
nu
mbe
r of
squ
ares
.
21.0
0222
.112
23.2
12#
12
24.3
12#
12#
1225
.422
26.5
12#
22
27.6
12#
12#
2228
.712
#12
#12
#22
29.8
22#
22
30.9
3231
.10
12#
3232
.11
12#
12#
32
33.1
212
#12
#12
#32
34.1
322
#32
35.1
412
#22
#32
36.1
512
#12
#22
#32
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-6
6-6
© Glencoe/McGraw-Hill A20 Glencoe Algebra 2
Answers (Lesson 6-7)
Stu
dy
Gu
ide
and I
nte
rven
tion
Gra
phin
g an
d So
lvin
g Q
uadr
atic
Ineq
ualit
ies
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-7
6-7
©G
lenc
oe/M
cGra
w-H
ill34
9G
lenc
oe A
lgeb
ra 2
Lesson 6-7
Gra
ph
Qu
adra
tic
Ineq
ual
itie
sTo
gra
ph a
qua
drat
ic in
equa
lity
in t
wo
vari
able
s,us
eth
e fo
llow
ing
step
s:
1.G
raph
the
rel
ated
qua
drat
ic e
quat
ion,
y!
ax2
"bx
"c.
Use
a d
ashe
d lin
e fo
r '
or &
;use
a s
olid
line
for
*or
+.
2.Te
st a
poi
nt in
side
the
par
abol
a.If
it s
atis
fies
the
ineq
ualit
y,sh
ade
the
regi
on in
side
the
par
abol
a;ot
herw
ise,
shad
e th
e re
gion
out
side
the
par
abol
a.
Gra
ph
th
e in
equ
alit
y y
%x2
#6x
#7.
Fir
st g
raph
the
equ
atio
n y
!x2
"6x
"7.
By
com
plet
ing
the
squa
re,y
ou g
et t
he v
erte
x fo
rm o
f th
e eq
uati
on y
!(x
"3)
2$
2,so
the
ver
tex
is ($
3,$
2).M
ake
a ta
ble
of v
alue
s ar
ound
x!
$3,
and
grap
h.Si
nce
the
ineq
ualit
y in
clud
es &
,use
a d
ashe
d lin
e.Te
st t
he p
oint
($3,
0),w
hich
is in
side
the
par
abol
a.Si
nce
($3)
2"
6($
3) "
7 !
$2,
and
0 &
$2,
($3,
0) s
atis
fies
the
in
equa
lity.
The
refo
re,s
hade
the
reg
ion
insi
de t
he p
arab
ola.
Gra
ph
eac
h i
neq
ual
ity.
1.y
&x2
$8x
"17
2.y
*x2
"6x
"4
3.y
+x2
"2x
"2
4.y
'$
x2"
4x$
65.
y+
2x2
"4x
6.y
&$
2x2
$4x
"2
x
y
O
x
y
O
x
y
O
x
y
O
x
y
O
x
y
O
x
y
O
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill35
0G
lenc
oe A
lgeb
ra 2
Solv
e Q
uad
rati
c In
equ
alit
ies
Qua
drat
ic in
equa
litie
s in
one
var
iabl
e ca
n be
sol
ved
grap
hica
lly o
r al
gebr
aica
lly.
To s
olve
ax2
"bx
"c
'0:
Firs
t gra
ph y
!ax
2"
bx"
c. T
he s
olut
ion
cons
ists
of t
he x
-val
ues
Gra
phic
al M
etho
dfo
r whi
ch th
e gr
aph
is b
elow
the
x-ax
is.
To s
olve
ax2
"bx
"c
&0:
Firs
t gra
ph y
!ax
2"
bx"
c. T
he s
olut
ion
cons
ists
the
x-va
lues
fo
r whi
ch th
e gr
aph
is a
bove
the
x-ax
is.
Find
the
root
s of
the
rela
ted
quad
ratic
equ
atio
n by
fact
orin
g,
Alg
ebra
ic M
etho
dco
mpl
etin
g th
e sq
uare
, or u
sing
the
Qua
drat
ic F
orm
ula.
2 ro
ots
divi
de th
e nu
mbe
r lin
e in
to 3
inte
rval
s.Te
st a
val
ue in
eac
h in
terv
al to
see
whi
ch in
terv
als
are
solu
tions
.
If t
he in
equa
lity
invo
lves
*or
+,t
he r
oots
of
the
rela
ted
equa
tion
are
incl
uded
in t
heso
luti
on s
et.
Sol
ve t
he
ineq
ual
ity
x2"
x"
6 (
0.
Fir
st f
ind
the
root
s of
the
rel
ated
equ
atio
n x2
$x
$6
!0.
The
equa
tion
fac
tors
as
(x$
3)(x
"2)
!0,
so t
he r
oots
are
3 a
nd $
2.T
he g
raph
ope
ns u
p w
ith
x-in
terc
epts
3 a
nd $
2,so
it m
ust
be o
n or
bel
ow t
he x
-axi
s fo
r $
2 *
x*
3.T
here
fore
the
sol
utio
n se
t is
{x$
2 *
x*
3}.
Sol
ve e
ach
in
equ
alit
y.
1.x2
"2x
'0
2.x2
$16
'0
3.0
'6x
$x2
$5
{x⏐"
2 &
x&
0}{x
⏐"4
&x
&4}
{x⏐1
&x
&5}
4.c2
*4
5.2m
2$
m'
16.
y2'
$8
{c⏐"
2 (
c (
2}!m
⏐"&
m&
1 ")
7.x2
$4x
$12
'0
8.x2
"9x
"14
&0
9.$
x2"
7x$
10 +
0
{x⏐"
2 &
x&
6}{x
⏐x&
"7
or x
%"
2}{x
⏐2 (
x(
5}
10.2
x2"
5x$
3 *
011
.4x2
$23
x"
15 &
012
.$6x
2$
11x
"2
'0
!x⏐"
3 (
x(
"!x⏐
x&
or x
%5 "
!x⏐x
&"
2 or
x%
"13
.2x2
$11
x"
12 +
014
.x2
$4x
"5
'0
15.3
x2$
16x
"5
'0
!x⏐x
&or
x%
4 ")
!x⏐&
x&
5 "1 $ 3
3 $ 2
1 $ 63 $ 4
1 $ 2
1 $ 2
x
y
O
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Gra
phin
g an
d So
lvin
g Q
uadr
atic
Ineq
ualit
ies
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-7
6-7
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A21 Glencoe Algebra 2
An
swer
s
Answers (Lesson 6-7)
Skil
ls P
ract
ice
Gra
phin
g an
d So
lvin
g Q
uadr
atic
Ineq
ualit
ies
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-7
6-7
©G
lenc
oe/M
cGra
w-H
ill35
1G
lenc
oe A
lgeb
ra 2
Lesson 6-7
Gra
ph
eac
h i
neq
ual
ity.
1.y
+x2
$4x
"4
2.y
*x2
$4
3.y
&x2
"2x
$5
Use
th
e gr
aph
of
its
rela
ted
fu
nct
ion
to
wri
te t
he
solu
tion
s of
eac
h i
neq
ual
ity.
4.x2
$6x
"9
*0
5.$
x2$
4x"
32 +
06.
x2"
x$
20 &
0
3"
8 (
x(
4x
&"
5 or
x%
4
Sol
ve e
ach
in
equ
alit
y al
gebr
aica
lly.
7.x2
$3x
$10
'0
8.x2
"2x
$35
+0
{x⏐"
2 &
x&
5}{x
⏐x(
"7
or x
*5}
9.x2
$18
x"
81 *
010
.x2
*36
{x⏐x
!9}
{x⏐"
6 &
x&
6}
11.x
2$
7x&
012
.x2
"7x
"6
'0
{x⏐x
&0
or x
%7}
{x⏐"
6 &
x&
"1}
13.x
2"
x$
12 &
014
.x2
"9x
"18
*0
{x⏐x
&"
4 or
x%
3}{x
⏐"6
(x
("
3}
15.x
2$
10x
"25
+0
16.$
x2$
2x"
15 +
0al
l rea
ls{x
⏐"5
(x
(3}
17.x
2"
3x&
018
.2x2
"2x
&4
{x⏐x
&"
3 or
x%
0}{x
⏐x&
"2
or x
%1}
19.$
x2$
64 *
$16
x20
.9x2
"12
x"
9 '
0al
l rea
ls)
x
y O2
5
x
y O2
6
x
y O
x
y
O
x
y
O
x
y
O
©G
lenc
oe/M
cGra
w-H
ill35
2G
lenc
oe A
lgeb
ra 2
Gra
ph
eac
h i
neq
ual
ity.
1.y
*x2
"4
2.y
&x2
"6x
"6
3.y
'2x
2$
4x$
2
Use
th
e gr
aph
of
its
rela
ted
fu
nct
ion
to
wri
te t
he
solu
tion
s of
eac
h i
neq
ual
ity.
4.x2
$8x
&0
5.$
x2$
2x"
3 +
06.
x2$
9x"
14 *
0
x&
0 or
x%
8"
3 (
x(
12
(x
(7
Sol
ve e
ach
in
equ
alit
y al
gebr
aica
lly.
7.x2
$x
$20
&0
8.x2
$10
x"
16 '
09.
x2"
4x"
5 *
0
{x⏐x
&"
4 or
x%
5}{x
⏐2 &
x&
8})
10.x
2"
14x
"49
+0
11.x
2$
5x&
1412
.$x2
$15
+8x
all r
eals
{x⏐x
&"
2 or
x%
7}{x
⏐"5
(x
("
3}
13.$
x2"
5x$
7 *
014
.9x2
"36
x"
36 *
015
.9x
*12
x2
all r
eals
{x⏐x
!"
2}!x⏐
x(
0 or
x*
"16
.4x2
"4x
"1
&0
17.5
x2"
10 +
27x
18.9
x2"
31x
"12
*0
!x⏐x
+"
"!x⏐
x(
or x
*5 "
!x⏐"
3 (
x(
""
19.F
ENC
ING
Vane
ssa
has
180
feet
of
fenc
ing
that
she
inte
nds
to u
se t
o bu
ild a
rec
tang
ular
play
are
a fo
r he
r do
g.Sh
e w
ants
the
pla
y ar
ea t
o en
clos
e at
leas
t 18
00 s
quar
e fe
et.W
hat
are
the
poss
ible
wid
ths
of t
he p
lay
area
?30
ft to
60
ft20
.BU
SIN
ESS
A b
icyc
le m
aker
sol
d 30
0 bi
cycl
es la
st y
ear
at a
pro
fit o
f $30
0 ea
ch.T
he m
aker
wan
ts t
o in
crea
se t
he p
rofi
t m
argi
n th
is y
ear,
but
pred
icts
tha
t ea
ch $
20 in
crea
se in
prof
it w
ill r
educ
e th
e nu
mbe
r of
bic
ycle
s so
ld b
y 10
.How
man
y $2
0 in
crea
ses
in p
rofit
can
the
mak
er a
dd in
and
exp
ect
to m
ake
a to
tal p
rofi
t of
at
leas
t $1
00,0
00?
from
5 to
10
4 $ 92 $ 5
1 $ 2
3 $ 4x
y
O
x
y
Ox
y
O2
46
6 –6 –12
8
x
y Ox
y
O
x
y
OPra
ctic
e (A
vera
ge)
Gra
phin
g an
d So
lvin
g Q
uadr
atic
Ineq
ualit
ies
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-7
6-7
© Glencoe/McGraw-Hill A22 Glencoe Algebra 2
Answers (Lesson 6-7)
Rea
din
g t
o L
earn
Math
emati
csG
raph
ing
and
Solv
ing
Qua
drat
ic In
equa
litie
s
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-7
6-7
©G
lenc
oe/M
cGra
w-H
ill35
3G
lenc
oe A
lgeb
ra 2
Lesson 6-7
Pre-
Act
ivit
yH
ow c
an y
ou f
ind
th
e ti
me
a tr
amp
olin
ist
spen
ds
abov
e a
cert
ain
hei
ght?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 6-
7 at
the
top
of
page
329
in y
our
text
book
.
•H
ow f
ar a
bove
the
gro
und
is t
he t
ram
polin
e su
rfac
e?3.
75 fe
et•
Usi
ng t
he q
uadr
atic
fun
ctio
n gi
ven
in t
he in
trod
ucti
on,w
rite
a q
uadr
atic
ineq
ualit
y th
at d
escr
ibes
the
tim
es a
t w
hich
the
tra
mpo
linis
t is
mor
eth
an 2
0 fe
et a
bove
the
gro
und.
"16
t2#
42t#
3.75
%20
Rea
din
g t
he
Less
on
1.A
nsw
er t
he f
ollo
win
g qu
esti
ons
abou
t ho
w y
ou w
ould
gra
ph t
he in
equa
lity
y+
x2"
x$
6.
a.W
hat
is t
he r
elat
ed q
uadr
atic
equ
atio
n?y
!x2
#x
"6
b.Sh
ould
the
par
abol
a be
sol
id o
r da
shed
? H
ow d
o yo
u kn
ow?
solid
;The
ineq
ualit
y sy
mbo
l is
*.
c.T
he p
oint
(0,
2) is
insi
de t
he p
arab
ola.
To u
se t
his
as a
tes
t po
int,
subs
titu
te
for
xan
d fo
r y
in t
he q
uadr
atic
ineq
ualit
y.
d.Is
the
sta
tem
ent
2 +
02"
0 $
6 tr
ue o
r fa
lse?
true
e.Sh
ould
the
reg
ion
insi
de o
r ou
tsid
e th
e pa
rabo
la b
e sh
aded
?in
side
2.T
he g
raph
of y
!$
x2"
4xis
sho
wn
at t
he r
ight
.Mat
ch e
ach
of t
he f
ollo
win
g re
late
d in
equa
litie
s w
ith
its
solu
tion
set
.
a.$
x2"
4x&
0ii
i.{xx
'0
or x
&4}
b.$
x2"
4x*
0iii
ii.{
x0
'x
'4}
c.$
x2"
4x+
0iv
iii.
{xx
*0
or x
+4}
d.$
x2"
4x'
0i
iv.
{x0
*x
*4}
Hel
pin
g Y
ou
Rem
emb
er
3.A
qua
drat
ic in
equa
lity
in t
wo
vari
able
s m
ay h
ave
the
form
y&
ax2
"bx
"c,
y'
ax2
"bx
"c,
y+
ax2
"bx
"c,
or y
*ax
2"
bx"
c.D
escr
ibe
a w
ay t
o re
mem
ber
whi
ch r
egio
n to
sha
de b
y lo
okin
g at
the
ineq
ualit
y sy
mbo
l and
wit
hout
usi
ng a
tes
t po
int.
Sam
ple
answ
er:I
f the
sym
bol i
s %
or *
,sha
de th
e re
gion
abo
ve th
epa
rabo
la.I
f the
sym
bol i
s &
or (
,sha
de th
e re
gion
bel
ow th
e pa
rabo
la.x
y
O( 0
, 0)
( 4, 0
)
( 2, 4
)
20
©G
lenc
oe/M
cGra
w-H
ill35
4G
lenc
oe A
lgeb
ra 2
Gra
phin
g A
bsol
ute
Valu
e In
equa
litie
s Yo
u ca
n so
lve
abso
lute
val
ue in
equa
litie
s by
gra
phin
g in
muc
h th
e sa
me
man
ner
you
grap
hed
quad
rati
c in
equa
litie
s.G
raph
the
rel
ated
abs
olut
e fu
ncti
on
for
each
ineq
ualit
y by
usi
ng a
gra
phin
g ca
lcul
ator
.For
&an
d +
,ide
ntif
y th
e x-
valu
es,i
f an
y,fo
r w
hich
the
gra
ph li
es b
elow
the
x-ax
is.F
or '
and
*,i
dent
ify
the
xva
lues
,if
any,
for
whi
ch t
he g
raph
lies
abo
veth
e x-
axis
.
For
eac
h i
neq
ual
ity,
mak
e a
sket
ch o
f th
e re
late
d g
rap
h a
nd
fin
d t
he
solu
tion
s ro
un
ded
to
the
nea
rest
hu
nd
red
th.
1.|x
$3|
&0
2.|x|
$6
'0
3.$
|x "
4| "
8 '
0
x%
3 or
x&
3"
6 &
x&
6"
12 &
x&
4
4.2|x
"6|
$2
+0
5.|3x
$3|
+0
6.|x
$7|
'5
x(
"7
or x
*"
5al
l rea
l num
bers
2 &
x&
12
7.|7x
$1|
&13
8.|x
$3.
6|*
4.2
9.|2x
"5|
*7
x&
"1.
71 o
r x%
2"
0.6
(x
(7.
8"
6 (
x(
1
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DAT
E__
____
____
__PE
RIO
D__
___
6-7
6-7