chapter 6 resource masters - glencoe/mcgraw-hill v glencoe algebra 2 assessment options the...
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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 3
f (x) 1 3 3 9
x �1 0 �2 1
f (x) 3 2 2 �1
x �3 �2 �1 �4
f (x) �1 0 3 0
11�4
3�2
11�4
3�2
3�2
3�2
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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 6
f (x) 8 0 �1 0 8
x �2 0 2 4 6
f (x) 16 4 0 4 16
x �2 �1 0 1 2
f (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 2
f (x) 5 1 0.5 1 5
x �6 �4 �2 0 2
f (x) 0 12 16 12 0
x 0 2 4 6 8
f (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 incomethe 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
vertex
axis 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
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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
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f(x) � 3x2 � 4x � 3
x
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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
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x
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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�
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© 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 � 0
34. NUMBER THEORY Find two consecutive even positive integers whose product is 624.24, 26
35. NUMBER THEORY Find two consecutive odd positive integers whose product is 323.17, 19
36. GEOMETRY The length of a rectangle is 2 feet more than its width. Find thedimensions of the rectangle if its area is 63 square feet. 7 ft by 9 ft
37. PHOTOGRAPHY The length and width of a 6-inch by 8-inch photograph are reduced bythe 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.
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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�
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1�
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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
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1�
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© 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%
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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 Property
x � 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���
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�b �b2 � 4�ac����2a
�b �b2 ��4ac����
2a
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 338 Glencoe Algebra 2
Roots and the Discriminant
DiscriminantThe 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�
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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
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© 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
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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 complex
21 2 real, irrational
0 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 � �ac
�.
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
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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
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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
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x
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8
4
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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
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© 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
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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 � h
c. In which direction does the graph open if a � 0? If a � 0? up; down
d. 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 iv
c. 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.
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© 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
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ExampleExample
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© 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 Methodcompleting 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} {x1 � 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} {xx � �7 or x � �2} {x2 � x � 5}
10. 2x2 � 5x� 3 0 11. 4x2 � 23x � 15 � 0 12. �6x2 � 11x � 2 � 0
�x�3 � x � � �xx � or x � 5� �xx � �2 or x � �13. 2x2 � 11x � 12 � 0 14. x2 � 4x � 5 � 0 15. 3x2 � 16x � 5 � 0
�xx � or x � 4� �x � x � 5�1�
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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} {xx � �7 or x 5}
9. x2 � 18x � 81 0 10. x2 36{xx � 9} {x�6 � x � 6}
11. x2 � 7x � 0 12. x2 � 7x � 6 � 0{xx � 0 or x � 7} {x�6 � x � �1}
13. x2 � x � 12 � 0 14. x2 � 9x � 18 0{xx � �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{xx � �3 or x � 0} {xx � �2 or x � 1}
19. �x2 � 64 �16x 20. 9x2 � 12x � 9 � 0all reals
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© 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
{xx � �4 or x � 5} {x2 � x � 8}
10. x2 � 14x � 49 � 0 11. x2 � 5x � 14 12. �x2 � 15 � 8x
all reals {xx � �2 or x � 7} {x�5 � x � �3}
13. �x2 � 5x � 7 0 14. 9x2 � 36x � 36 0 15. 9x 12x2
all reals {xx � �2} �xx � 0 or x �16. 4x2 � 4x � 1 � 0 17. 5x2 � 10 � 27x 18. 9x2 � 31x � 12 0
�xx � � � �xx � 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
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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 � 6
b. 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? true
e. 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. {x x � 0 or x � 4}
b. �x2 � 4x 0 iii ii. {x 0 � x � 4}
c. �x2 � 4x � 0 iv iii. {x x 0 or x � 4}
d. �x2 � 4x � 0 i iv. {x 0 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.
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© 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
Chapter 6 Test, Form 1
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 355 Glencoe Algebra 2
Ass
essm
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Write the letter for the correct answer in the blank at the right of each question.
1. Find the y-intercept for f(x) � �(x � 1)2.A. 1 B. �1 C. x D. 0 1.
2. What is the equation of the axis of symmetry of y � �3(x � 6)2 � 12?A. x � 2 B. x � �6 C. x � 6 D. x � �18 2.
3. Find the minimum value of f(x) � x2 � 6x.A. 3 B. �6 C. �9 D. 27 3.
4. The graph of f(x) � �2x2 � x opens _____ and has a _____ value.A. down; maximum B. down; minimumC. up; maximum D. up; minimum 4.
5. The related graph of a quadratic equation is shown at the right.Use the graph to determine the solutions of the equation.A. �2, 3 B. �3, 2C. 0, �6 D. 0, 2 5.
6. The quadratic function f(x) � x2 has _____.A. no zeros B. exactly one zeroC. exactly two zeros D. more than two zeros 6.
For Questions 7 and 8, solve each equation by factoring.
7. x2 � 3x � 10 � 0A. {�5, 2} B. (�2, 5) C. {�2, 5} D. {�10, 1} 7.
8. 2x2 � 6x � 0A. {�3, 0} B. {0, 3} C. {0, 6} D. {�3, 3} 8.
9. Which quadratic equation has roots �2 and 3?A. x2 � x � 6 � 0 B. x2 � x � 6 � 0C. x2 � 6x � 1 � 0 D. x2 � x � 6 � 0 9.
10. To solve x2 � 8x � 16 � 25 by using the Square Root Property, you would first rewrite the equation as _____.A. (x � 4)2 � 25 B. x2 � 8x � 9 � 0C. (x � 4)2 � 5 D. x2 � 8x � 9 10.
11. Find the value of c that makes x2 � 10x � c a perfect square.A. 100 B. 25 C. 10 D. 50 11.
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© Glencoe/McGraw-Hill 356 Glencoe Algebra 2
Chapter 6 Test, Form 1 (continued)
12. The quadratic equation x2 � 6x � 1 is to be solved by completing the square.Which equation would be the first step in that solution?A. x2 � 6x � 1 � 0 B. x2 � 6x � 36 � 1 � 36C. x(x � 6) � 1 D. x2 � 6x � 9 � 1 � 9 12.
13. Find the exact solutions to x2 � 3x � 1 � 0 by using the Quadratic Formula.
A. ��3 �
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For Questions 14 and 15, use the value of the discriminant to determine the number and type of roots for each equation.
14. x2 � 3x � 7 � 0A. 2 complex roots B. 2 real, irrational rootsC. 2 real, rational roots D. 1 real, rational root 14.
15. x2 � 4x � 4A. 2 real, rational roots B. 2 real, irrational rootsC. 1 real, rational root D. no real roots 15.
16. What is the vertex of y � 2(x � 3)2 � 6?A. (�3, �6) B. (3, �6) C. (�3, 6) D. (3, 6) 16.
17. What is the equation of the axis of symmetry of y � �3(x � 6)2 � 1?A. x � 2 B. x � �6 C. x � �3 D. x � 6 17.
18. Which quadratic function has its vertex at (2, 3) and passes through (1, 0)?A. y � 2(x � 2)2 � 3 B. y � �3(x � 2)2 � 3C. y � �3(x � 2)2 � 3 D. y � 2(x � 2)2 � 3 18.
19. Which quadratic inequality is graphed at the right?A. y � (x � 1)2 � 4B. y � �(x � 1)2 � 4C. y � �(x � 1)2 � 4D. y � �(x � 1)2 � 4 19.
20. Solve (x � 4)(x � 2) � 0.A. {x � x � �2 or x � 4} B. {x � �4 � x � 2}C. {x � �2 � x � 4} D. {x � x � �2 or x � 4} 20.
Bonus Find the x-intercepts and the y-intercept of the graph B:of y � 2(x � 4)2 � 18.
NAME DATE PERIOD
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Chapter 6 Test, Form 2A
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 357 Glencoe Algebra 2
Ass
essm
ent
Write the letter for the correct answer in the blank at the right of each question.
1. Identify the y-intercept and the axis of symmetry for the graph of f(x) � 10x2 � 40x � 42.A. 42; x � 4 B. 0; x � �4 C. 42; x � �2 D. �42; x � 2 1.
2. Identify the quadratic function graphed at the right.A. f(x) � �x2 � 2xB. f(x) � �x2 � 2xC. f(x) � x2 � 2xD. f(x) � �(x � 2)2 2.
3. Determine whether f(x) � 4x2 � 16x � 6 has a maximum or a minimum value and find that value.A. minimum; �10 B. minimum; 2 C. maximum; �10 D. maximum; 2 3.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
4. �x2 � 4xA. 4, 0 B. �4, 0C. between �4 and 4 D. �2, 4 4.
5. x2 � 2x � 5A. between �4 and �3; B. between �2 and �1;
between 1 and 2 between 3 and 4C. no real solutions D. �1, �6 5.
For Questions 6 and 7, solve each equation by factoring.
6. x2 � 3x � 18A. {6} B. {�6, 3} C. {�9, 2} D. {�3, 6} 6.
7. 3x2 � 20 � 7x
A. {�10, 2} B. ��5, �43�� C. ��4, �
53�� D. ��20, �
13�� 7.
8. Which quadratic equation has roots �2 and �15�?
A. x2 � 4x � 4 � 0 B. 5x2 � 9x � 2 � 0C. 5x2 � 9x � 2 � 0 D. 5x2 � 11x � 2 � 0 8.
9. To solve 9x2 � 12x � 4 � 49 by using the Square Root Property, you would first rewrite the equation as _____.A. 9x2 � 12x � 45 � 0 B. (3x � 2)2 � �49C. (3x � 2)2 � 7 D. (3x � 2)2 � 49 9.
10. Find the value of c that makes x2 � 9x � c a perfect square.
A. �841� B. �
92� C. ��
841� D. 81 10.
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© Glencoe/McGraw-Hill 358 Glencoe Algebra 2
Chapter 6 Test, Form 2A (continued)
11. The quadratic equation x2 � 8x � �20 is to be solved by completing the square. Which equation would be a step in that solution?A. (x � 4)2 � 4 B. x � 4 � �2iC. x2 � 8x � 20 � 0 D. x2 � 8x � 16 � �20 11.
12. Find the exact solutions to 3x2 � 5x � 1 by using the Quadratic Formula.
A. ��5 �6
�13�� B. �
5 �3�13�� C. �
5 �6�37�� D. �
5 �6�13�� 12.
For Questions 13 and 14, use the value of the discriminant to determine the number and type of roots for each equation.
13. 2x2 � 7x � 9 � 0A. 2 real, rational B. 2 real, irrationalC. 2 complex D. 1 real, rational 13.
14. x2 � 20 � 12x � 16A. 1 real, irrational B. 2 real, rationalC. no real D. 1 real, rational 14.
15. Identify the vertex, axis of symmetry, and direction of opening for
y � �12�(x � 8)2 � 2.
A. (�8, 2); x � �8; up B. (�8, �2); x � �8; downC. (8, �2); x � 8; up D. (8, 2); x � 8; up 15.
16. Which quadratic function has its vertex at (�2, 7) and opens down?A. y � �3(x � 2)2 � 7 B. y � (x � 2)2 � 7C. y � �12(x � 2)2 � 7 D. y � �2(x � 2)2 � 7 16.
17. Write y � x2 � 4x � 1 in vertex form.A. y � (x � 2)2 � 5 B. y � (x � 2)2 � 5C. y � (x � 2)2 � 1 D. y � (x � 2)2 � 3 17.
18. Write an equation for the parabola whose vertex is at (�8, 4) and passes through (�6, �2).
A. y � ��32�(x � 8)2 � 4 B. y � ��
14�(x � 8)2 � 4
C. y � �32�(x � 6)2 � 2 D. y � ��
32�(x � 8)2 � 4 18.
19. Which quadratic inequality is graphed at the right?A. y � (x � 2)(x � 3) B. y � (x � 2)(x � 3)C. y � (x � 2)(x � 3) D. y (x � 2)(x � 3) 19.
20. Solve x2 � 2x � 24.A. {x � �4 � x � 6} B. {x � �6 � x � 4}C. {x � x � �6 or x � 4} D. {x � x � �4 or x � 6} 20.
Bonus Write a quadratic equation with roots ��i�
43�
�. B:
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Chapter 6 Test, Form 2B
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© Glencoe/McGraw-Hill 359 Glencoe Algebra 2
Ass
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Write the letter for the correct answer in the blank at the right of each question.
1. Identify the y-intercept and the axis of symmetry for the graph of f(x) � �3x2 � 6x � 12.A. 2; x � �12 B. 12; x � 1 C. �2; x � 0 D. �12; x � �1 1.
2. Identify the quadratic function graphed at the right.A. f(x) � x2 � 4xB. f(x) � �x2 � 4xC. f(x) � �x2 � 4xD. f(x) � �(x � 4)2 2.
3. Determine whether f(x) � �5x2 � 10x � 6 has a maximum or a minimum value and find that value.A. minimum; �1 B. maximum; 11 C. maximum; �1 D. minimum; 11 3.
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
4. x2 � 4xA. �4, 0 B. between �4 and 4C. 2, �4 D. 0, 4 4.
5. x2 � 2x � �2A. between �3 and �2; B. between �1 and 0;
between 0 and 1 between 2 and 3C. no real solutions D. �1, 1 5.
For Questions 6 and 7, solve each equation by factoring.
6. x2 � 3x � 28A. {�4, 7} B. {�14, 2} C. {�7, 4} D. {�2, 14} 6.
7. 5x2 � 4 � 19x
A. ��2, �25�� B. ���
25�, 2� C. ���
15�, 4� D. ��4, �
15�� 7.
8. Which quadratic equation has roots 7 and ��23�?
A. 2x2 � 11x � 21 � 0 B. 3x2 � 19x � 14 � 0C. 3x2 � 23x � 14 � 0 D. 2x2 � 11x � 21 � 0 8.
9. To solve 4x2 � 28x � 49 � 25 by using the Square Root Property, you would first rewrite the equation as _____.A. (2x � 7)2 � 25 B. (2x � 7)2 � 5C. (2x � 7)2 � �5 D. 4x2 � 28x � 24 � 0 9.
10. Find the value of c that makes x2 � 5x � c a perfect square trinomial.
A. �2156�
B. �54� C. �
245� D. �
52� 10.
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© Glencoe/McGraw-Hill 360 Glencoe Algebra 2
Chapter 6 Test, Form 2B (continued)
11. The quadratic equation x2 � 18x � �106 is to be solved by completing the square. Which equation would be a step in that solution?A. (x � 9)2 � 25 B. x2 � 18x � 106 � 0C. x � 9 � �5i D. x2 � 18x � 81 � �106 11.
12. Find the exact solutions to 2x2 � 5x � 1 by using the Quadratic Formula.
A. ��5 �4
�17�� B. �
5 �4�17�� C. �
5 �4�33�� D. �
5 �2�17�� 12.
For Questions 13 and 14, use the value of the discriminant to determine the number and type of roots for each equation.
13. 3x2 � x � 12 � 0A. 2 complex roots B. 1 real, rational rootC. 2 real, rational roots D. 2 real, irrational roots 13.
14. x2 � 10 � 3x � 3A. 2 complex roots B. 2 real, irrational rootsC. 1 real, rational root D. 2 real, rational roots 14.
15. Identify the vertex, axis of symmetry, and direction of opening for y � �8(x � 2)2.A. (�8, �2); x � �8 up B. (�2, 0); x � �2; downC. (2, 0); x � 2; down D. (�2, �8); x � �2; down 15.
16. Which quadratic function has its vertex at (�3, 5) and opens down?A. y � (x � 3)2 � 5 B. y � (x � 3)2 � 5C. y � �(x � 3)2 � 5 D. y � �(x � 3)2 � 5 16.
17. Write y � x2 � 18x � 52 in vertex form.A. y � (x � 9)2 � 113 B. y � (x � 9)2 � 29C. y � (x � 9)2 � 52 D. y � (x � 9)2 � 29 17.
18. Write an equation for the parabola whose vertex is at (�5, 7) and passes through (�3, �1).
A. y � ��111�
(x � 5)2 � 7 B. y � �2(x � 5)2 � 7
C. y � ��12�(x � 5)2 � 7 D. y � ��
12�(x � 5)2 � 7 18.
19. Which quadratic inequality is graphed at the right?A. y � (x � 3)(x � 1) B. y � (x � 3)(x � 1)C. y � (x � 3)(x � 1) D. y (x � 3)(x � 1) 19.
20. Solve 2x � 3 � x2.A. {x � �1 � x � 3} B. {x � �3 � x � 1}C. {x � x � �1 or x � 3} D. {x � x � �3 or x � 1} 20.
Bonus Write a quadratic equation with roots ��i�
32�
�. B:
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Chapter 6 Test, Form 2C
© Glencoe/McGraw-Hill 361 Glencoe Algebra 2
1. Graph f(x) � �5x2 � 10x, labeling the y-intercept, vertex, 1.and axis of symmetry.
2. Determine whether f(x) � �3x2 � 6x � 1 has a maximum 2.or a minimum value and find that value.
For Questions 3 and 4, solve each equation by graphing.If exact roots cannot be found, state the consecutive integers between which the roots are located.
3. x2 � 6x � 83.
4. x2 � x � 5 � 0 4.
5. Solve 5x2 � 13x � 6 by factoring. 5.
6. GEOMETRY The length of a rectangle is 7 inches longer 6.than its width. If the area of the rectangle is 144 square inches, what are its dimensions?
7. Write a quadratic equation with �6 and �34� as its roots. 7.
Write the equation in the form ax2 � bx � c � 0, where a, b, and c are integers.
Solve each equation by using the Square Root Property.
8. x2 � 6x � 9 � 25 8.
9. 4x2 � 20x � 25 � 7 9.
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© Glencoe/McGraw-Hill 362 Glencoe Algebra 2
Chapter 6 Test, Form 2C (continued)
For Questions 10 and 11, solve each equation by completing the square.
10. x2 � 4x � 9 � 0 10.
11. 2x2 � 3x � 2 � 0 11.
12. Find the exact solutions to 5x2 � 3x � 2 by using the Quadratic Formula. 12.
For Questions 13 and 14, find the value of the discriminant for each quadratic equation. Then describe the number and type of roots for the equation.
13. 9x2 � 12x � 4 � 0 13.
14. 4x2 � 1 � 9x � 2 14.
15. Identify the vertex, axis of symmetry, and direction of 15.
opening for y � ��23�(x � 5)2 � 7.
16. Write an equation for the parabola with vertex at (2, �1) 16.and y-intercept 5.
17. Write y � x2 � 6x � 8 in vertex form. 17.
18. PHYSICS The height h (in feet) of a certain rocket 18.t seconds after it leaves the ground is modeled by h(t) � �16t2 � 48t � 15. Write the function in vertex form and find the maximum height reached by the rocket.
19. Graph y x2 � 6x � 9. 19.
20. Solve 2x2 � 5x � 3 � 0 algebraically. 20.
Bonus Write a quadratic equation with roots ���37�
�. Write the B:
equation in the form ax2 � bx � c � 0, where a, b, and c are integers.
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Chapter 6 Test, Form 2D
© Glencoe/McGraw-Hill 363 Glencoe Algebra 2
1. Graph f(x) � x2 � 4x � 3, labeling the y-intercept, vertex, 1.and axis of symmetry.
2. Determine whether f(x) � 5x2 � 20x � 3 has a maximum or 2.a minimum value and find that value.
For Questions 3 and 4, solve each equation by graphing.If exact roots cannot be found, state the consecutive integers between which the roots are located.
3. x2 � 2x � 3 � 0 3.
4. 2x2 � 2x � 3 � 0 4.
5. Solve 3x2 � x � 4 by factoring. 5.
6. GEOMETRY The length of a rectangle is 10 inches longer 6.than its width. If the area of the rectangle is 144 square inches, what are its dimensions?
7. Write a quadratic equation with �4 and �32� as its roots. 7.
Write the equation in the form ax2 � bx � c � 0, where a, b, and c are integers.
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© Glencoe/McGraw-Hill 364 Glencoe Algebra 2
Chapter 6 Test, Form 2D (continued)
Solve each equation by using the Square Root Property.
8. x2 � 14x � 49 � 16 8.
9. 9x2 � 12x � 4 � 6 9.
Solve each equation by completing the square.
10. x2 � 8x � 14 � 0 10.
11. 3x2 � x � 2 � 0 11.
12. Find the exact solutions to 2x2 � 9x � 5 by using the 12.Quadratic Formula.
For Questions 13 and 14, find the value of the discriminant for each quadratic equation. Then describe the number and type of roots for the equation.
13. 25x2 � 20x � 4 � 0 13.
14. 2x2 � 10x � 9 � 2x 14.
15. Identify the vertex, axis of symmetry, and direction of 15.opening for y � �(x � 6)2 � 5.
16. Write an equation for the parabola with vertex at (�4, 2) 16.and y-intercept �2.
17. Write y � x2 � 4x � 8 in vertex form. 17.
18. PHYSICS The height h (in feet) of a certain rocket 18.t seconds after it leaves the ground is modeled by h(t) � �16t2 � 64t � 12. Write the function in vertex form and find the maximum height reached by the rocket.
19. Graph y � x2 � 4x � 4.19.
20. Solve 2x2 � 7x � 15 0 algebraically. 20.
Bonus Write a quadratic equation with roots ���45�
�. B:
Write the equation in the form ax2 � bx � c � 0,where a, b, and c are integers.
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Chapter 6 Test, Form 3
© Glencoe/McGraw-Hill 365 Glencoe Algebra 2
1. Graph f(x) � 3 � 3x2 � 2x, labeling the y-intercept, vertex, 1.and axis of symmetry.
2. Determine whether f(x) � 1 � �35�x � �
34�x2
has a maximum or a minimum value and find that value.
3. BUSINESS Khalid charges $10 for a one-year subscription to his on-line newsletter. Khalid currently has 600 subscribers and he estimates that for each $1 decrease in the subscription price, he would gain 100 new subscribers. What subscription price will maximize Khalid’s 2.income? If he charges this price, how much income should Khalid expect? 3.
For Questions 4–6, solve each equation by graphing. 4.If exact roots cannot be found, state the consecutive integers between which the roots are located.
4. 0.5x2 � 9 � 4.5x
5. �23�x � 3 � �
13�x2 5.
6. 4x(x � 3) � �9 6.
7. Solve 18x2 � 15 � 39x by factoring. 7.
8. Write a quadratic equation with ��23� and 1.75 as its roots. 8.
Write the equation in the form ax2 � bx � c � 0, where a, b, and c are integers.
9. If the roots of an equation are �5 and 3, what is the 9.equation of the axis of symmetry?
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2
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2
2
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© Glencoe/McGraw-Hill 366 Glencoe Algebra 2
Chapter 6 Test, Form 3 (continued)
10. Solve 4x2 � 2x � 0.25 � 1.44 by using the 10.Square Root Property.
For Questions 11 and 12, solve each equation by completing the square.
11. 2x2 � �52�x � 2 � 0 11.
12. x2 � 2.5x � 3 � 0.5 12.
13. Find the exact solutions to �14�x2 � 3x � 1 � 0 by using the 13.
Quadratic Formula.
14. Find the value of the discriminant for 14.3x(0.2x � 0.4) � 1 � 0.9. Then describe the number and type of roots for the equation.
15. Find all values of k such that x2 � kx � 1 � 0 has two 15.complex roots.
16. Write an equation of the parabola with equation 16.
y � ��35��x � �
12��
2� �
52�, translated 4 units left and 2 units up.
Then identify the vertex, axis of symmetry, and direction of opening of your function.
17. PHYSICS The height h (in feet) of a certain aircraft 17.t seconds after it leaves the ground is modeled by h(t) � �9.1t2 � 591.5t � 20,388.125. Write the function in vertex form and find the maximum height reached by the aircraft.
18. Write an equation for the parabola that has the same 18.
vertex as y � �13�x2 � 6x � �
823� and x-intercept 1.
19. Graph y �(x2 � 2x) � 5.25. 19.
20. Solve �x � �72��(x � 1)2 � 0. 20.
Bonus Write a quadratic equation with roots ��3 �42i�5��. B:
Write the equation in the form ax2 � bx � c � 0,where a, b, and c are integers.
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Chapter 6 Open-Ended Assessment
© Glencoe/McGraw-Hill 367 Glencoe Algebra 2
Demonstrate your knowledge by giving a clear, concise solutionto each problem. Be sure to include all relevant drawings andjustify your answers. You may show your solution in more thanone way or investigate beyond the requirements of the problem.
1. Mr. Moseley asked the students in his Algebra class to work ingroups to solve (x � 3)2 � 25, stating that each student in thefirst group to solve the equation correctly would earn five bonuspoints on the next quiz. Mi-Ling’s group solved the equationusing the Square Root Property. Emilia’s group used theQuadratic Formula to find the solutions. In which group wouldyou prefer to be? Explain your reasoning.
2. The next day, Mr. Moseley had his students work in pairs toreview for their chapter exam. He asked each student to write apractice problem for his or her partner. Len wrote the followingproblem for his partner, Jocelyn: Write an equation for theparabola whose vertex is (�3, �4), that passes through (�1, 0),and that opens downward.a. Jocelyn had trouble solving Len’s problem. Explain why.b. How would you change Len’s problem?c. Make the change you suggested in part b and complete the
problem.
3. a. Write a quadratic function in vertex form whose maximumvalue is 8.
b. Write a quadratic function that transforms the graph of yourfunction from part a so that it is shifted horizontally. Explainthe change you made and describe the transformation thatresults from this change.
4. When asked to write f(x) � 2x2 � 12x � 5 in vertex form, Josephwrote:
f(x) � 2x2 � 12x � 5Step 1 f(x) � 2(x2 � 6x) � 5Step 2 f(x) � 2(x2 � 6x � 9) � 5 � 9Step 3 f(x) � 2(x � 3)2 � 4Is Joseph’s answer correct? Explain your reasoning.
5. The graph of y � x2 � 4x � 4 is shown. Susan used this graph to solve three quadratic inequalities. Her three solutions are given below. Replace each ● with an inequality symbol (, �, �, �) so that each solution is correct. Explain your reasoning for each.a. The solution of x2 � 4x � 4 ● 0 is
{x � x �2 or x � �2}.b. The solution of x2 � 4x � 4 ● 0 is .c. The solution of x2 � 4x � 4 ● 0 is all real numbers.
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© Glencoe/McGraw-Hill 368 Glencoe Algebra 2
Chapter 6 Vocabulary Test/Review
Write whether each sentence is true or false. If false, replace the underlined word or words to make a true sentence.
1. The Square Root Property is used when a quadratic 1.equation is solved by factoring.
2. In f(x) � 3x2 � 2x � 5, the linear term is 5. 2.
3. 2x2 � 3x � 4 � 0 is an example of a quadratic equation. 3.
4. The solutions of a quadratic equation are called its zeros. 4.
5. The quadratic function y � 2(x � 3)2 � 1 is written in 5.vertex form.
6. If a parabola opens upward, the y-coordinate of the vertex 6.is the maximum value.
7. In f(x) � �x2 � 2x � 1, the constant term is �x2. 7.
8. It is necessary to identify the values of a, b, and c in order 8.to solve a quadratic equation by completing the square.
9. The highest or lowest point on a parabola is called the 9.vertex.
10. In the Quadratic Formula, the expression b2 � 4ac is 10.called the quadratic term.
In your own words—Define each term.
11. parabola
12. axis of symmetry
axis of symmetrycompleting the squareconstant termdiscriminantlinear term
maximum valueminimum valueparabolaquadratic equationQuadratic Formula
quadratic functionquadratic inequalityquadratic termrootsSquare Root Property
vertexvertex formZero Product Propertyzeros
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Chapter 6 Quiz (Lessons 6–1 and 6–2)
66
© Glencoe/McGraw-Hill 369 Glencoe Algebra 2
For Questions 1 and 2, consider f(x) � x2 � 2x � 3.
1. Find the y-intercept, the equation of the axis of symmetry, 1.and the x-coordinate of the vertex.
2. Graph the function, labeling the y-intercept, vertex, and 2.axis of symmetry.
3. Determine whether f(x) � 2x2 � 8x � 9 has a maximum or 3.a minimum value and find that value.
Solve each equation. If exact roots cannot be found, state the consecutive integers between which the roots are located.
4. x2 � 2x � 3 4.
5. x2 � 4x � 7 � 0 5.
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Chapter 6 Quiz (Lessons 6–3 and 6–4)
For Questions 1 and 2, solve each equation by factoring.1. 3x2 � 10 � 13x 2. x2 � 4x � 45
3. STANDARDIZED TEST PRACTICE What is the integer solution of 6x2 � 9 � 21x?
A. �3 B. 3 C. �12� D. 2
Write a quadratic equation with the given roots. Write the equation in the form ax2 � bx � c � 0, where a, b,and c are integers.
4. �6 and 2 5. �23� and �4
Solve each equation by using the Square Root Property.6. x2 � 8x � 16 � 36 7. x2 � 2x � 1 � 45
8. 25x2 � 20x � 4 � 3
Solve each equation by completing the square.9. x2 � 10x � 11 10. x2 � 4x � 29 � 11
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1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
© Glencoe/McGraw-Hill 370 Glencoe Algebra 2
1. Solve x2 � 4x � 1 by using the Quadratic Formula. 1.Find exact solutions.
2. Find the value of the discriminant for 3x2 � 6x � 11. Then 2.describe the number and type of roots for the equation.
3. Graph y � �(x � 2)2 � 1. Show and label the vertex and 3.axis of symmetry.
4. Write y � �3x2 � 12x � 6 in vertex form. 4.
5. Write an equation for the parabola whose vertex is at 5.(�5, 0) and passes through (0, 50).
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Chapter 6 Quiz (Lesson 6–7)
1. Graph y � ��13�(x � 2)2 � 3. 1.
2. Use the graph of its related function to write the 2.solutions of �x2 � 6x � 5 � 0.
3. Solve 0 � x2 � 4x � 3 by graphing. 3.
4. Solve 4x2 � 1 � 4x algebraically. 4.
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Chapter 6 Quiz (Lessons 6–5 and 6–6)
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Chapter 6 Mid-Chapter Test (Lessons 6–1 through 6–4)
© Glencoe/McGraw-Hill 371 Glencoe Algebra 2
Write the letter for the correct answer in the blank at the right of each question.
1. Which function is graphed?A. f(x) � x2 � 2x � 3B. f(x) � x2 � 2x � 3C. f(x) � x2 � x � 3D. f(x) � (x � 3)2 1.
2. By the Zero Product Property, if (2x � 1)(x � 5) � 0, then _____.
A. x � 1 or x � 5 B. x � �12� or x � 5
C. x � ��12� or x � �5 D. x � �1 or x � �5 2.
3. Write a quadratic equation with 7 and �25� as its roots.
Write the equation in the form ax2 � bx � c � 0, where a, b, and c are integers.A. y � 5x2 � 37x � 14 B. y � 2x2 � 9x � 35C. y � 5x2 � 37x � 14 D. y � 2x2 � 9x � 35 3.
4. The quadratic equation x2 � 4x � 16 is to be solved by completing the square. Which equation would be a step in that solution?A. (x � 2)2 � �20 B. x2 � 4x � 16 � 0C. (x � 2)2 � 20 D. (x � 2)2 � 4 4.
5. Solve x2 � 6x � �6. If exact roots cannot be found, state the consecutive integers between which the roots are located.A. �2, �3 B. between �4 and �3; between �2 and �1C. �3 D. between �5 and �4; between �2 and �1 5.
6. Solve x2 � 4x � 3 � 0 by graphing. 6.
7. Determine whether f(x) � �12�x2 � x � 9 7.
has a maximum or a minimum value and find that value.
For Questions 8 and 9, solve each equation by factoring.
8. x2 � 7x � 18 9. 4x2 � x 9.
10. Solve 9x2 � 6x � 1 � 5 by using the Square Root Property. 10.
y
xO
Part II
Part I
NAME DATE PERIOD
SCORE 66
Ass
essm
ent
xO
f(x )
8.
© Glencoe/McGraw-Hill 372 Glencoe Algebra 2
Chapter 6 Cumulative Review (Chapters 1–6)
1. Find the value of 12 � 36 � 4 � (5 � 7)2. (Lesson 1-1) 1.
2. Find the slope of the line that is parallel to the line with 2.equation 3x � 4y � 10. (Lesson 2-3)
3. Describe the system 2x � 3y � 21 and y � 5 � �23�x as 3.
consistent and independent, consistent and dependent, or inconsistent. (Lesson 3-1)
4. Find the coordinates of the vertices of the figure formed 4.by the system of inequalities. (Lesson 3-3)
x � �2 x � y � 6y � �2 x � y � �2
5. Find the value of � �. (Lesson 4-5) 5.
6. Solve � � � � � by using inverse matrices. 6.
(Lesson 4-8)
7. Use synthetic division to find 7.
(2x4 � 5x3 � x2 � 10x � 4) � (x � 3). (Lesson 5-3)
8. Use a calculator to approximate �4
983� to three 8.decimal places. (Lesson 5-5)
9. Solve �x � 2� � 1 � 8. (Lesson 5-8) 9.
10. PHYSICS An object is thrown straight up from the top of 10.a 100-foot platform at a velocity of 48 feet per second. The
height h(t) of the object t seconds after being thrown is given by h(t) � �16t2 � 48t � 100. Find the maximum height reached by the object and the time it takes to achieve this height. (Lesson 6-1)
11. Solve x2 � 2x � 3 by graphing. (Lesson 6-2)
11.
12. Solve 4x2 � 4x � 24 by factoring. (Lesson 6-3) 12.
13. Find the value of the discriminant for 7x2 � 5x � 1 � 0. 13.Then describe the number and type of roots for the equation. (Lesson 6-5)
14. Write y � x2 � 7x � 5 in vertex form. (Lesson 6-6) 14.
y
xO
11�13
ab
�13
4�2
124
5�6
NAME DATE PERIOD
66
Standardized Test Practice (Chapters 1–6)
© Glencoe/McGraw-Hill 373 Glencoe Algebra 2
1. If �ab� � �
32�, then 8a equals which of the following?
A. 16b B. 12b C. �32b� D. �
83�b 1.
2. 20% of 3 yards is how many fifths of 9 feet?E. 1 F. 6 G. 10 H. 15 2.
3. If u � v and t � 0, which of the following are true?I. ut � vt II. u � t � v � t III. u � t � v � tA. I only B. III onlyC. I and II only D. I, II, and III 3.
4. Which of the following is the greatest?
E. �23� F. �
79� G. �
1105�
H. �181�
4.
5. If 2a � 3b represents the perimeter of a rectangle and a � 2b represents its width, the length is ______.
A. 7b B. b C. �72b� D. 14b 5.
6. In the figure, what is the area of the shaded region?E. 30 F. 36G. 54 H. 27 6.
7. Mr. Salazár rented a car for d days. The rental agency charged x dollars per day plus c cents per mile for the model he selected.When Mr. Salazár returned the car, he paid a total of T dollars.In terms of d, x, c, and T, how many miles did he drive?
A. T � (xd � c) B. T � �xcd� C. �xd
T� c� D. �
T �c
xd� 7.
8. If P(3, 2) and Q(7, 10) are the endpoints of the diameter of a circle, what is the area of the circle?E. 2�5� F. 80 G. 4�5� H. 20 8.
9. If (x � y)2 � 100 and xy � 20, what is the value of x2 � y2?A. 120 B. 140 C. 80 D. 60 9.
10. The tenth term in the sequence 7, 12, 19, 28, … is ______.E. 124 F. 103 G. 57 H. 147 10. HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
NAME DATE PERIOD
66
Ass
essm
ent
6
3
15
Part 1: Multiple Choice
Instructions: Fill in the appropriate oval for the best answer.
© Glencoe/McGraw-Hill 374 Glencoe Algebra 2
Standardized Test Practice (continued)
11. If t2 � 6t � �9, what is the value of �t � �12��
2? 11. 12.
12. All four walls of a rectangular room that is 14 feet wide, 20 feet long, and 8 feet high, are to be painted. What is the minimum cost of paint if one gallon covers at most 130 square feet and the paint costs $22 per gallon?
13. The bar graph shows the distribution of votes among the 13. 14.candidates for senior class president. If 220 seniors voted in all, how many students voted for either Theo or Pam?
14. Find the median of x, 2x � 1, �2x
� � 13, 45, and
x � 22 if the mean of this set of numbers is 83.
Column A Column B
15. 15.
16. 1 a c 16.
17. x2 � 25 17.y2 � 16
yx
DCBA
�ac
��ac
�
DCBA
zx
DCBA
y˚ y˚ z˚x˚
Part 3: Quantitative Comparison
Instructions: Compare the quantities in columns A and B. Shade in if the quantity in column A is greater; if the quantity in column B is greater; if the quantities are equal; or if the relationship cannot be determined from the information given.
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
Part 2: Grid In
Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.
NAME DATE PERIOD
66
NAME DATE PERIOD
JoeyAnaPamTheo
20
30
10
0
40
Perc
ent
of
vote
s re
ceiv
ed 50
Candidates
A
D
C
B
Standardized Test PracticeStudent Record Sheet (Use with pages 342–343 of the Student Edition.)
© Glencoe/McGraw-Hill A1 Glencoe Algebra 2
NAME DATE PERIOD
66
An
swer
s
Select the best answer from the choices given and fill in the corresponding oval.
1 4 7 9
2 5 8 10
3 6
Solve the problem and write your answer in the blank.
For Questions 14–20, also enter your answer by writing each number or symbol ina box. Then fill in the corresponding oval for that number or symbol.
11 15 17 19
12
13
14 16 18 20
Select the best answer from the choices given and fill in the corresponding oval.
21 23 25 27
22 24 26 28 DCBADCBADCBADCBA
DCBADCBADCBADCBA
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
DCBADCBA
DCBADCBADCBADCBA
DCBADCBADCBADCBA
Part 2 Short Response/Grid InPart 2 Short Response/Grid In
Part 1 Multiple ChoicePart 1 Multiple Choice
Part 3 Quantitative ComparisonPart 3 Quantitative Comparison
© Glencoe/McGraw-Hill A2 Glencoe Algebra 2
Answers (Lesson 6-1)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Gra
ph
ing
Qu
adra
tic
Fu
nct
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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
Qu
adra
tic
Fu
nct
ion
Afu
nctio
n de
fined
by
an e
quat
ion
of t
he f
orm
f(x
) �
ax2
�bx
�c,
whe
re a
�0
Gra
ph
of
a Q
uad
rati
cA
par
abo
law
ith t
hese
cha
ract
eris
tics:
yin
terc
ept:
c;
axis
of
sym
met
ry:
x�
;F
un
ctio
nx-
coor
dina
te o
f ve
rtex
:
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-in
terc
ept
is 5
.Th
e eq
uat
ion
of
the
axis
of
sym
met
ry i
s
x�
or
.Th
e x-
coor
din
ate
of t
he
vert
ex i
s .
Nex
t m
ake
a ta
ble
of v
alu
es f
or x
nea
r .
xx
2�
3x�
5f(
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.
Mak
e a
tab
le o
f va
lues
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
�6x
�8
2.f(
x) �
�x2
�2x
�2
3.f(
x) �
2x2
�4x
�3
8,x
��
3,�
32,
x�
�1,
�1
3,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
3
f(x
)1
33
9
x�
10
�2
1
f(x
)3
22
�1
x�
3�
2�
1�
4
f(x
)�
10
30
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-co
ordi
nat
e of
th
e ve
rtex
of
a qu
adra
tic
fun
ctio
n i
s th
e m
axim
um
or
min
imu
m v
alu
e of
th
e fu
nct
ion
.
Max
imu
m o
r M
inim
um
Val
ue
The
gra
ph o
f f(
x)
�ax
2�
bx�
c, w
here
a�
0, o
pens
up
and
has
a m
inim
umo
f a
Qu
adra
tic
Fu
nct
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 G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Gra
ph
ing
Qu
adra
tic
Fu
nct
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-1
6-1
Exam
ple
Exam
ple
a.f(
x) �
3x2
�6x
�7
For
th
is f
un
ctio
n,a
�3
and
b�
�6.
Sin
ce a
�0,
the
grap
h o
pen
s u
p,an
d th
efu
nct
ion
has
a m
inim
um
val
ue.
Th
e m
inim
um
val
ue
is t
he
y-co
ordi
nat
eof
th
e ve
rtex
.Th
e x-
coor
din
ate
of t
he
vert
ex i
s �
��
1.
Eva
luat
e th
e fu
nct
ion
at
x�
1 to
fin
d th
em
inim
um
val
ue.
f(1)
�3(
1)2
�6(
1) �
7 �
4,so
th
em
inim
um
val
ue
of t
he
fun
ctio
n i
s 4.
�6
� 2(3)
�b
� 2a
b.f
(x)
�10
0 �
2x�
x2
For
th
is f
un
ctio
n,a
��
1 an
d b
��
2.S
ince
a�
0,th
e gr
aph
ope
ns
dow
n,a
nd
the
fun
ctio
n h
as a
max
imu
m v
alu
e.T
he
max
imu
m v
alu
e is
th
e y-
coor
din
ate
ofth
e ve
rtex
.Th
e x-
coor
din
ate
of t
he
vert
ex
is
��
��
1.
Eva
luat
e th
e fu
nct
ion
at
x�
�1
to f
ind
the
max
imu
m v
alu
e.f(
�1)
�10
0 �
2(�
1) �
(�1)
2�
101,
soth
e m
inim
um
val
ue
of t
he
fun
ctio
n i
s 10
1.
�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�
1011
.f(x
) �
x2�
10x
�5
12.f
(x)
��
6x2
�12
x�
21
max
.,14
min
.,�
20m
ax.,
27
13.f
(x)
�25
x2�
100x
�35
014
.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
ph
ing
Qu
adra
tic
Fu
nct
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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;0
5;x
�5;
57;
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
.M
ake
a ta
ble
of
valu
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.
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)
�6x
211
.f(x
) �
�8x
212
.f(x
) �
x2�
2xm
in.;
0m
ax.;
0m
in.;
�1
13.f
(x)
�x2
�2x
�15
14.f
(x)
��
x2�
4x�
115
.f(x
) �
x2�
2x�
3m
in.;
14m
ax.;
3m
in.;
�4
16.f
(x)
��
2x2
�4x
�3
17.f
(x)
�3x
2�
12x
�3
18.f
(x)
�2x
2�
4x�
1m
ax.;
�1
min
.;�
9m
in.;
�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
)8
0�
10
8
x�
20
24
6
f(x
)16
40
416
x�
2�
10
12
f(x
)�
8�
20
�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
.M
ake
a ta
ble
of
valu
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�
8x�
152.
f(x)
��
x2�
4x�
123.
f(x)
�2x
2�
2x�
115
;x
�4;
412
;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
,Su
san
th
row
s a
ball
upw
ard
wit
h a
velo
city
of
32 f
eet
per
seco
nd.
Th
e h
eigh
t h
(t)
of t
he
ball
tse
con
ds a
fter
Su
san
th
row
s it
is g
iven
by
h(t
) �
�16
t2�
32t
�4.
Fin
d th
e m
axim
um
hei
ght
reac
hed
by
the
ball
an
dth
e ti
me
that
th
is h
eigh
t is
rea
ched
.20
ft;
1 s
11.H
EALT
H C
LUB
SL
ast
year
,th
e S
port
sTim
e A
thle
tic
Clu
b ch
arge
d $2
0 to
par
tici
pate
in
an a
erob
ics
clas
s.S
even
ty p
eopl
e at
ten
ded
the
clas
ses.
Th
e cl
ub
wan
ts t
o in
crea
se t
he
clas
s pr
ice
this
yea
r.T
hey
exp
ect
to l
ose
one
cust
omer
for
eac
h $
1 in
crea
se i
n t
he
pric
e.
a.W
hat
pri
ce s
hou
ld t
he
clu
b ch
arge
to
max
imiz
e th
e in
com
e fr
om t
he
aero
bics
cla
sses
?$4
5b
.W
hat
is
the
max
imu
m i
nco
me
the
Spo
rtsT
ime
Ath
leti
c C
lub
can
exp
ect
to m
ake?
$202
5
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
)5
10.
51
5
x�
6�
4�
20
2
f(x
)0
1216
120
x0
24
68
f(x
)15
3�
13
15
Pra
ctic
e (
Ave
rag
e)
Gra
ph
ing
Qu
adra
tic
Fu
nct
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-1
6-1
© Glencoe/McGraw-Hill A4 Glencoe Algebra 2
Answers (Lesson 6-1)
Readin
g t
o L
earn
Math
em
ati
csG
rap
hin
g Q
uad
rati
c F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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
uct
ion
to
Les
son
6-1
at
the
top
of p
age
286
in y
our
text
book
.
•B
ased
on
th
e gr
aph
in
you
r te
xtbo
ok,f
or w
hat
tic
ket
pric
e is
th
e in
com
eth
e gr
eate
st?
$40
•U
se t
he
grap
h t
o es
tim
ate
the
max
imu
m i
nco
me.
abo
ut
$72,
000
Rea
din
g t
he
Less
on
1.a.
For
th
e qu
adra
tic
fun
ctio
n f
(x)
�2x
2�
5x�
3,2x
2is
th
e te
rm,
5xis
th
e te
rm,a
nd
3 is
th
e te
rm.
b.
For
th
e qu
adra
tic
fun
ctio
n f
(x)
��
4 �
x�
3x2 ,
a�
,b�
,an
d
c�
.
2.C
onsi
der
the
quad
rati
c fu
nct
ion
f(x
) �
ax2
�bx
�c,
wh
ere
a�
0.
a.T
he
grap
h o
f th
is f
un
ctio
n i
s a
.
b.
Th
e y-
inte
rcep
t is
.
c.T
he
axis
of
sym
met
ry i
s th
e li
ne
.
d.
If a
�0,
then
th
e gr
aph
ope
ns
and
the
fun
ctio
n h
as a
valu
e.
e.If
a�
0,th
en t
he
grap
h o
pen
s an
d th
e fu
nct
ion
has
a
valu
e.
3.R
efer
to
the
grap
h a
t th
e ri
ght
as y
ou c
ompl
ete
the
foll
owin
g se
nte
nce
s.
a.T
he
curv
e is
cal
led
a .
b.
Th
e li
ne
x�
�2
is c
alle
d th
e .
c.T
he
poin
t (�
2,4)
is
call
ed t
he
.
d.
Bec
ause
th
e gr
aph
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 u
se t
he
x2te
rm o
f a
quad
rati
c fu
nct
ion
to
tell
wh
eth
er t
he
fun
ctio
n h
as a
max
imu
m o
r a
min
imu
m v
alu
e?S
amp
le a
nsw
er:
Rem
emb
er t
hat
th
e g
rap
h o
f f(
x) �
x2
(wit
h a
�0)
is a
U-s
hap
ed c
urv
eth
at o
pen
s u
p a
nd
has
a m
inim
um
.Th
e g
rap
h o
f g
(x)
��
x2
(wit
h a
�0)
is ju
st t
he
op
po
site
.It
op
ens
do
wn
an
d h
as a
max
imu
m.
y-in
terc
ept
vert
ex
axis
of
sym
met
ry
par
abo
la
x
f(x)
O ( 0, –
1)
( –2,
4)
max
imu
md
ow
nw
ard
min
imu
mu
pw
ard
x�
�� 2b a�
c
par
abo
la
�4
1�
3
con
stan
tlin
ear
qu
adra
tic
©G
lenc
oe/M
cGra
w-H
ill31
8G
lenc
oe A
lgeb
ra 2
Fin
din
g t
he
Axi
s o
f S
ymm
etry
of
a P
arab
ola
As
you
kn
ow,i
f f(
x) �
ax2
�bx
�c
is a
qu
adra
tic
fun
ctio
n,t
he
valu
es o
f x
that
mak
e f(
x) e
qual
to
zero
are
an
d .
Th
e av
erag
e of
th
ese
two
nu
mbe
r va
lues
is
�� 2b a�
.
Th
e fu
nct
ion
f(x
) h
as i
ts m
axim
um
or
min
imu
m
valu
e w
hen
x�
�� 2b a�
.Sin
ce t
he
axis
of
sym
met
ry
of t
he
grap
h o
f f(
x) p
asse
s th
rou
gh t
he
poin
t w
her
e th
e m
axim
um
or
min
imu
m o
ccu
rs,t
he
axis
of
sym
met
ry h
as t
he
equ
atio
n 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-co
ordi
nat
e of
th
e ve
rtex
is
�1.
Su
bsti
tute
x�
�1
in f
(x)
�5x
2�
10x
�7.
f(�
1) �
5(�
1)2
�10
(�1)
�7
��
12T
he
vert
ex i
s (�
1,�
12).
Th
e ax
is o
f sy
mm
etry
is
x�
�� 2b a�
,or
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
�4
4.f(
x) �
2x2
�6x
�5
���3 2� ,
�1 2� �;x
��
�3 2�
5.A
(x)
�x2
�12
x�
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
) =
ax
2 +
bx
+ c
�b
��
b2�
4�
ac��
��
2a�
b�
�b2
�4
�ac�
��
�2a
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-1
6-1
Exam
ple
Exam
ple
© Glencoe/McGraw-Hill A5 Glencoe Algebra 2
An
swer
s
Answers (Lesson 6-2)
Stu
dy G
uid
e a
nd I
nte
rven
tion
So
lvin
g Q
uad
rati
c E
qu
atio
ns
by G
rap
hin
g
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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
Qu
adra
tic
Eq
uat
ion
Aqu
adra
tic e
quat
ion
has
the
form
ax
2�
bx�
c�
0, w
here
a�
0.
Ro
ots
of
a Q
uad
rati
c E
qu
atio
nso
lutio
n(s)
of
the
equa
tion,
or
the
zero
(s)
of t
he r
elat
ed q
uadr
atic
fun
ctio
n
Th
e ze
ros
of a
qu
adra
tic
fun
ctio
n a
re t
he
x-in
terc
epts
of
its
grap
h.T
her
efor
e,fi
ndi
ng
the
x-in
terc
epts
is
one
way
of
solv
ing
the
rela
ted
quad
rati
c eq
uat
ion
.
Sol
ve x
2�
x �
6 �
0 b
y gr
aph
ing.
Gra
ph t
he
rela
ted
fun
ctio
n f
(x)
�x2
�x
�6.
Th
e x-
coor
din
ate
of t
he
vert
ex i
s �
�,a
nd
the
equ
atio
n o
f th
e
axis
of
sym
met
ry i
s x
��
.
Mak
e a
tabl
e of
val
ues
usi
ng
x-va
lues
aro
un
d �
.
x�
1�
01
2
f(x
)�
6�
6�
6�
40
Fro
m t
he
tabl
e an
d th
e gr
aph
,we
can
see
th
at t
he
zero
s of
th
e fu
nct
ion
are
2 a
nd
�3.
Sol
ve e
ach
eq
uat
ion
by
grap
hin
g.
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
rea
l so
luti
on
s�
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
fin
d ex
act
solu
tion
s to
qu
adra
tic
equ
atio
ns
by g
raph
ing.
Bu
t yo
u c
an u
se t
he
grap
h t
o es
tim
ate
solu
tion
s.
Sol
ve x
2�
2x�
2 �
0 b
y gr
aph
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.
Th
e eq
uat
ion
of
the
axis
of
sym
met
ry o
f th
e re
late
d fu
nct
ion
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
�2
1
Th
e x-
inte
rcep
ts o
f th
e gr
aph
are
bet
wee
n 2
an
d 3
and
betw
een
0 a
nd
�1.
So
one
solu
tion
is
betw
een
2 a
nd
3,an
d th
e ot
her
sol
uti
on i
sbe
twee
n 0
an
d �
1.
Sol
ve t
he
equ
atio
ns
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
.
1.x2
�4x
�2
�0
2.x2
�6x
�6
�0
3.x2
�4x
�2�
0
bet
wee
n 0
an
d 1
;b
etw
een
�2
and
�1;
bet
wee
n �
1 an
d 0
;b
etw
een
3 a
nd
4b
etw
een
�5
and
�4
bet
wee
n �
4 an
d �
3
4.�
x2�
2x�
4 �
05.
2x2
�12
x�
17 �
06.
�x2
�x
��
0
bet
wee
n 3
an
d 4
;b
etw
een
2 a
nd
3;
bet
wee
n �
2 an
d �
1;b
etw
een
�2
and
�1
bet
wee
n 3
an
d 4
bet
wee
n 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 G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
So
lvin
g Q
uad
rati
c E
qu
atio
ns
by G
rap
hin
g
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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
So
lvin
g Q
uad
rati
c E
qu
atio
ns
By
Gra
ph
ing
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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
qu
atio
n t
o d
eter
min
e it
s so
luti
ons.
1.x2
�2x
�3
�0
2.�
x2�
6x�
9 �
03.
3x2
�4x
�3
�0
�3,
1�
3n
o r
eal s
olu
tio
ns
Sol
ve e
ach
eq
uat
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.x2
�6x
�5
�0
5.�
x2�
2x�
4 �
06.
x2�
6x�
4 �
01,
5n
o r
eal s
olu
tio
ns
bet
wee
n 0
an
d 1
;b
etw
een
5 a
nd
6
Use
a q
uad
rati
c eq
uat
ion
to
fin
d t
wo
real
nu
mb
ers
that
sat
isfy
eac
h s
itu
atio
n,o
rsh
ow t
hat
no
such
nu
mb
ers
exis
t.
7.T
hei
r su
m i
s �
4,an
d th
eir
prod
uct
is
0.8.
Th
eir
sum
is
0,an
d th
eir
prod
uct
is
�36
.
�x
2�
4x�
0;0,
�4
�x
2�
36 �
0;�
6,6
f(x) �
�x2
� 3
6
x
f (x)
O6
–612
–12
36 24 12
f(x) �
�x2
� 4
x
x
f (x)
O
f(x) �
x2
� 6
x �
4
x
f (x)
Of(x
) � �
x2 �
2x
� 4x
f (x)
O
f(x) �
x2
� 6
x �
5
x
f (x)
O
x
f (x) O
f(x) �
3x2
� 4
x �
3
x
f(x)
O
f(x) �
�x2
� 6
x �
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
qu
atio
n t
o d
eter
min
e it
s so
luti
ons.
1.�
3x2
�3
�0
2.3x
2�
x�
3 �
03.
x2�
3x�
2 �
0
�1,
1n
o r
eal s
olu
tio
ns
1,2
Sol
ve e
ach
eq
uat
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
bet
wee
n 0
an
d 1
;�
6,�
4b
etw
een
�2
and
�1,
bet
wee
n �
4 an
d �
32
Use
a q
uad
rati
c eq
uat
ion
to
fin
d t
wo
real
nu
mb
ers
that
sat
isfy
eac
h s
itu
atio
n,o
rsh
ow t
hat
no
such
nu
mb
ers
exis
t.
7.T
hei
r su
m i
s 1,
and
thei
r pr
odu
ct i
s �
6.8.
Th
eir
sum
is
5,an
d th
eir
prod
uct
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
bje
ct i
n f
eet,
v 0is
th
e ob
ject
’s i
nit
ial
velo
city
in
fee
t p
er s
econ
d,a
nd
tis
th
eti
me
in s
econ
ds.
9.B
ASE
BA
LLM
arta
thr
ows
a ba
seba
ll w
ith
an in
itia
l upw
ard
velo
city
of
60 f
eet
per
seco
nd.
Igno
ring
Mar
ta’s
hei
ght,
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
eru
ptio
n b
last
s a
bou
lder
upw
ard
wit
h a
n i
nit
ial
velo
city
of
240
feet
per
sec
ond.
How
lon
g w
ill
it t
ake
the
bou
lder
to
hit
th
e gr
oun
d if
it
lan
ds a
t th
esa
me
elev
atio
n f
rom
wh
ich
it
was
eje
cted
?15
s
�x2
�5x
�8
�0;
no
su
ch r
eal
nu
mb
ers
exis
tx
f (x)
Of(x
) � �
x2 �
5x
� 8
�x
2�
x�
6 �
0;3,
�2
f(x) �
�x2
� x
� 6 x
f (x)
O
x
f (x)
O
f(x) �
2x2
� x
� 6
f(x) �
x2
� 1
0x �
24
x
f (x)
O
f(x) �
�2x
2 �
6x
� 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 (
Ave
rag
e)
So
lvin
g Q
uad
rati
c E
qu
atio
ns
By
Gra
ph
ing
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-2
6-2
© Glencoe/McGraw-Hill A7 Glencoe Algebra 2
An
swer
s
Answers (Lesson 6-2)
Readin
g t
o L
earn
Math
em
ati
csS
olv
ing
Qu
adra
tic
Eq
uat
ion
s by
Gra
ph
ing
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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 q
uad
rati
c fu
nct
ion
mod
el a
fre
e-fa
ll r
ide?
Rea
d th
e in
trod
uct
ion
to
Les
son
6-2
at
the
top
of p
age
294
in y
our
text
book
.
Wri
te a
qu
adra
tic
fun
ctio
n t
hat
des
crib
es t
he
hei
ght
of a
bal
l t
seco
nds
aft
erit
is
drop
ped
from
a h
eigh
t of
125
fee
t.h
(t)
��
16t2
�12
5
Rea
din
g t
he
Less
on
1.T
he
grap
h o
f th
e qu
adra
tic
fun
ctio
n f
(x)
��
x2�
x�
6 is
sh
own
at
the
righ
t.U
se t
he
grap
h t
o fi
nd
the
solu
tion
s of
th
equ
adra
tic
equ
atio
n �
x2�
x�
6 �
0.�
2 an
d 3
2.S
ketc
h a
gra
ph t
o il
lust
rate
eac
h s
itu
atio
n.
a.A
par
abol
a th
at o
pen
s b
.A
par
abol
a th
at o
pen
s c.
A p
arab
ola
that
ope
ns
dow
nw
ard
and
repr
esen
ts a
u
pwar
d an
d re
pres
ents
a
dow
nw
ard
and
qu
adra
tic
fun
ctio
n w
ith
tw
o qu
adra
tic
fun
ctio
n w
ith
re
pres
ents
a
re
al z
eros
,bot
h o
f w
hic
h a
reex
actl
y on
e re
al z
ero.
Th
e
quad
rati
c fu
nct
ion
n
egat
ive
nu
mbe
rs.
zero
is
a po
siti
ve n
um
ber.
wit
h n
o re
al z
eros
.
Hel
pin
g Y
ou
Rem
emb
er
3.T
hin
k of
a m
emor
y ai
d th
at c
an h
elp
you
rec
all
wh
at i
s m
ean
t by
th
e ze
ros
of a
qu
adra
tic
fun
ctio
n.
Sam
ple
an
swer
:Th
e b
asic
fac
ts a
bo
ut
a su
bje
ct a
re s
om
etim
es c
alle
d t
he
AB
Cs.
In t
he
case
of
zero
s,th
e A
BC
s ar
e th
e X
YZ
s,b
ecau
se t
he
zero
sar
e th
e x-
valu
es t
hat
mak
e th
e y-
valu
es e
qu
al t
o z
ero
.
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
ph
ing
Ab
solu
te V
alu
e E
qu
atio
ns
You
can
sol
ve a
bsol
ute
val
ue
equ
atio
ns
in m
uch
th
e sa
me
way
you
sol
ved
quad
rati
c eq
uat
ion
s.G
raph
th
e re
late
d ab
solu
te v
alu
e fu
nct
ion
for
eac
h
equ
atio
n u
sin
g a
grap
hin
g ca
lcu
lato
r.T
hen
use
th
e ZE
ROfe
atu
re i
n t
he
CALC
men
u t
o fi
nd
its
real
sol
uti
ons,
if a
ny.
Rec
all
that
sol
uti
ons
are
poin
ts
wh
ere
the
grap
h i
nte
rsec
ts t
he
x-ax
is.
For
eac
h e
qu
atio
n,m
ake
a sk
etch
of
the
rela
ted
gra
ph
an
d f
ind
th
e so
luti
ons
rou
nd
ed t
o th
e n
eare
st h
un
dre
dth
.
1.|x
�5|
�0
2.|4
x�
3| �
5 �
03.
|x�
7| �
0
�5
No
so
luti
on
s7
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
abso
lute
val
ue
equ
atio
ns
alge
brai
call
y an
d fi
ndi
ng
zero
s of
abs
olu
te v
alu
e fu
nct
ion
s gr
aph
ical
ly a
re r
elat
ed.
Sam
ple
an
swer
:va
lues
of
xw
hen
so
lvin
g a
lgeb
raic
ally
are
th
e x-
inte
rcep
ts (
or
zero
s) o
f th
e fu
nct
ion
wh
en g
rap
hed
.
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-2
6-2
© Glencoe/McGraw-Hill A8 Glencoe Algebra 2
Answers (Lesson 6-3)
Stu
dy G
uid
e a
nd I
nte
rven
tion
So
lvin
g Q
uad
rati
c E
qu
atio
ns
by F
acto
rin
g
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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
fac
tori
ng
to s
olve
a q
uad
rati
c eq
uat
ion
,yo
u u
se t
he
foll
owin
g pr
oper
ty.
Zer
o P
rod
uct
Pro
per
tyF
or a
ny r
eal n
umbe
rs a
and
b, if
ab
�0,
the
n ei
ther
a�
0 or
b�
0, o
r bo
th a
and
b�
0.
Sol
ve e
ach
eq
uat
ion
by
fact
orin
g.Ex
ampl
eEx
ampl
ea.
3x2
�15
x3x
2�
15x
Orig
inal
equ
atio
n
3x2
�15
x�
0S
ubtr
act
15x
from
bot
h si
des.
3x(x
�5)
�0
Fac
tor
the
bino
mia
l.
3x �
0or
x�
5 �
0Z
ero
Pro
duct
Pro
pert
y
x�
0or
x�
5S
olve
eac
h eq
uatio
n.
Th
e so
luti
on s
et i
s {0
,5}.
b.4
x2�
5x�
214x
2�
5x�
21O
rigin
al e
quat
ion
4x2
�5x
�21
�0
Sub
trac
t 21
fro
m b
oth
side
s.
(4x
�7)
(x�
3)�
0F
acto
r th
e tr
inom
ial.
4x�
7 �
0or
x�
3 �
0Z
ero
Pro
duct
Pro
pert
y
x�
�or
x
�3
Sol
ve e
ach
equa
tion.
Th
e so
luti
on s
et i
s ��
,3�.
7 � 4
7 � 4
Exer
cises
Exer
cises
Sol
ve e
ach
eq
uat
ion
by
fact
orin
g.
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
sT
o w
rite
a q
uad
rati
c eq
uat
ion
wit
h r
oots
pan
d q,
let
(x�
p)(x
�q)
�0.
Th
en m
ult
iply
usi
ng
FO
IL.
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 G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
So
lvin
g Q
uad
rati
c E
qu
atio
ns
by F
acto
rin
g
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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
, q
with
�5.
(x�
3)(x
�5)
�0
Sim
plify
.
x2�
2x�
15 �
0U
se F
OIL
.
Th
e eq
uat
ion
x2
�2x
�15
�0
has
roo
ts
3 an
d �
5.
b.�
,
(x�
p)(x
�q)
�0
�x�
�����
x�
��0
�x�
��x�
��0
��
0
�24
�0
24x2
�13
x�
7 �
0
Th
e eq
uat
ion
24x
2�
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
�0
x2
�10
x�
16 �
0x
2�
10x
�9
�0
4.�
55.
10,7
6.�
2,15
x2
�10
x�
25 �
0x
2�
17x
�70
�0
x2
�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
35x
2�
22x
�3
�0
16.�
,17
.,
18.
,
16x
2�
42x
�49
8x2
�10
x�
3 �
048
x2
�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
So
lvin
g Q
uad
rati
c E
qu
atio
ns
by F
acto
rin
g
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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
eq
uat
ion
by
fact
orin
g.
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
,4x
2�
5x�
4 �
020
.6,�
9x
2�
3x�
54 �
0
21.�
2,�
5x
2�
7x�
10 �
022
.0,7
x2
�7x
�0
23.�
,�3
3x2
�10
x�
3 �
024
.�,
8x2
�2x
�3
�0
25.F
ind
two
con
secu
tive
in
tege
rs w
hos
e pr
odu
ct i
s 27
2.16
,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
eq
uat
ion
by
fact
orin
g.
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,
8x
2�
9x�
14 �
0x
2�
3x�
0x
2�
3x�
40 �
0
25.�
7,�
826
.�6,
�3
27.3
,�4
x2
�15
x�
56 �
0x
2�
9x�
18 �
0x
2�
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
�22
x�
8 �
0
34.N
UM
BER
TH
EORY
Fin
d tw
o co
nse
cuti
ve e
ven
pos
itiv
e in
tege
rs w
hos
e pr
odu
ct i
s 62
4.24
,26
35.N
UM
BER
TH
EORY
Fin
d tw
o co
nse
cuti
ve o
dd p
osit
ive
inte
gers
wh
ose
prod
uct
is
323.
17,1
936
.GEO
MET
RYT
he
len
gth
of
a re
ctan
gle
is 2
fee
t m
ore
than
its
wid
th.F
ind
the
dim
ensi
ons
of t
he
rect
angl
e if
its
are
a is
63
squ
are
feet
.7
ft b
y 9
ft
37.P
HO
TOG
RA
PHY
Th
e le
ngt
h a
nd
wid
th o
f a
6-in
ch b
y 8-
inch
ph
otog
raph
are
red
uce
d by
the
sam
e am
oun
t to
mak
e a
new
ph
otog
raph
wh
ose
area
is
hal
f th
at o
f th
e or
igin
al.B
yh
ow m
any
inch
es w
ill
the
dim
ensi
ons
of t
he
phot
ogra
ph h
ave
to b
e re
duce
d?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 (
Ave
rag
e)
So
lvin
g Q
uad
rati
c E
qu
atio
ns
by F
acto
rin
g
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-3
6-3
© Glencoe/McGraw-Hill A10 Glencoe Algebra 2
Answers (Lesson 6-3)
Readin
g t
o L
earn
Math
em
ati
csS
olv
ing
Qu
adra
tic
Eq
uat
ion
s by
Fac
tori
ng
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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
uct
ion
to
Les
son
6-3
at
the
top
of p
age
301
in y
our
text
book
.
Wh
at d
oes
the
expr
essi
on x
(x�
5) m
ean
in
th
is s
itu
atio
n?
It r
epre
sen
ts t
he
area
of
the
rect
ang
le,s
ince
th
e ar
ea is
th
ep
rod
uct
of
the
wid
th a
nd
len
gth
.
Rea
din
g t
he
Less
on
1.T
he
solu
tion
of
a qu
adra
tic
equ
atio
n b
y fa
ctor
ing
is s
how
n b
elow
.Giv
e th
e re
ason
for
each
ste
p of
th
e so
luti
on.
x2�
10x
��
21O
rigin
al e
quat
ion
x2�
10x
�21
�0
Ad
d 2
1 to
eac
h s
ide.
(x�
3)(x
�7)
�0
Fact
or
the
trin
om
ial.
x�
3 �
0 or
x �
7 �
0Z
ero
Pro
du
ct P
rop
erty
x�
3 x
�7
So
lve
each
eq
uat
ion
.
Th
e so
luti
on s
et i
s .
2.O
n a
n a
lgeb
ra q
uiz
,stu
den
ts w
ere
aske
d to
wri
te a
qu
adra
tic
equ
atio
n w
ith
�7
and
5 as
its
root
s.T
he
wor
k th
at t
hre
e st
ude
nts
in
th
e cl
ass
wro
te o
n t
hei
r pa
pers
is
show
n b
elow
.
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
Wh
o is
cor
rect
?R
osa
Exp
lain
th
e er
rors
in
th
e ot
her
tw
o st
ude
nts
’ wor
k.
Sam
ple
an
swer
:M
arla
use
d t
he
wro
ng
fac
tors
.Lar
ry u
sed
th
e co
rrec
tfa
cto
rs b
ut
mu
ltip
lied
th
em in
corr
ectl
y.
Hel
pin
g Y
ou
Rem
emb
er
3.A
goo
d w
ay t
o re
mem
ber
a co
nce
pt i
s to
rep
rese
nt
it i
n m
ore
than
on
e w
ay.D
escr
ibe
anal
gebr
aic
way
an
d a
grap
hic
al w
ay t
o re
cogn
ize
a qu
adra
tic
equ
atio
n t
hat
has
a d
oubl
ero
ot.
Sam
ple
an
swer
:A
lgeb
raic
:Wri
te t
he
equ
atio
n in
th
e st
and
ard
fo
rm
ax2
�b
x�
c�
0 an
d e
xam
ine
the
trin
om
ial.
If it
is a
per
fect
sq
uar
etr
ino
mia
l,th
e q
uad
rati
c fu
nct
ion
has
a d
ou
ble
ro
ot.
Gra
ph
ical
:G
rap
h t
he
rela
ted
qu
adra
tic
fun
ctio
n.I
f th
e p
arab
ola
has
exa
ctly
on
e x-
inte
rcep
t,th
en t
he
equ
atio
n h
as a
do
ub
le r
oo
t.
{3,7
}
©G
lenc
oe/M
cGra
w-H
ill33
0G
lenc
oe A
lgeb
ra 2
Eu
ler’
s F
orm
ula
fo
r P
rim
e N
um
ber
sM
any
mat
hem
atic
ian
s h
ave
sear
ched
for
a f
orm
ula
th
at w
ould
gen
erat
e pr
ime
nu
mbe
rs.O
ne
such
for
mu
la w
as p
ropo
sed
by E
ule
r an
d 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
rim
e43
,pri
me
47,p
rim
e
4.x
�3
5.x
�4
6.x
�5
53,p
rim
e61
,pri
me
71,p
rim
e
7.x
�6
8.x
�17
9.x
�28
83,p
rim
e34
7,p
rim
e85
3,p
rim
e
10.x
�29
11.
x�
3012
.x�
35
911,
pri
me
971,
pri
me
1301
,pri
me
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
on
g t
he
pri
mes
om
itte
d a
re 5
9,67
,73,
79,8
9,10
1,10
3,10
7,10
9,an
d 1
27.
14.E
ule
r’s
form
ula
pro
duce
s pr
imes
for
man
y va
lues
of
x,bu
t it
doe
s n
ot w
ork
for
all
of t
hem
.Fin
d th
e fi
rst
valu
e of
xfo
r w
hic
h t
he
form
ula
fai
ls.
(Hin
t:T
ry m
ult
iple
s of
ten
.)
x�
40 g
ives
168
1,w
hic
h e
qu
als
412 .
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-3
6-3
© Glencoe/McGraw-Hill A11 Glencoe Algebra 2
An
swer
s
Answers (Lesson 6-4)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Co
mp
leti
ng
th
e S
qu
are
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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
foll
owin
g pr
oper
ty t
o so
lve
a qu
adra
tic
equ
atio
n t
hat
is
in t
he
form
“pe
rfec
t sq
uar
e tr
inom
ial
�co
nst
ant.
”
Sq
uar
e R
oo
t P
rop
erty
For
any
rea
l num
ber
xif
x2
�n,
the
n x
�
n.
Sol
ve e
ach
eq
uat
ion
by
usi
ng
the
Sq
uar
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
Th
e so
luti
on s
et i
s {9
,�1}
.
b.4
x2�
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�
Th
e so
luti
on s
et i
s �
�.5
4�
2��
� 2
5
4 �2�
�� 2
Exer
cises
Exer
cises
Sol
ve e
ach
eq
uat
ion
by
usi
ng
the
Sq
uar
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
eT
o co
mpl
ete
the
squ
are
for
a qu
adra
tic
expr
essi
on o
f th
e fo
rm
x2�
bx,
foll
ow t
hes
e st
eps.
1.F
ind
.➞
2.S
quar
e .
➞3.
Add
��2
to x
2�
bx.
b � 2b � 2
b � 2
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Co
mp
leti
ng
th
e S
qu
are
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-4
6-4
Fin
d t
he
valu
e of
cth
at m
akes
x2
�22
x�
ca
per
fect
sq
uar
e tr
inom
ial.
Th
enw
rite
th
e tr
inom
ial
as t
he
squ
are
of a
bin
omia
l.
Ste
p 1
b�
22;
�11
Ste
p 2
112
�12
1S
tep
3c
�12
1
The
tri
nom
ial
is x
2�
22x
�12
1,w
hic
h c
an b
e w
ritt
en a
s (x
�11
)2.
b � 2
Sol
ve 2
x2�
8x�
24 �
0 b
yco
mp
leti
ng
the
squ
are.
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�
12A
dd 1
2 to
eac
h si
de.
x2�
4x�
4 �
12 �
4S
ince
���2
�4,
add
4 t
o ea
ch s
ide.
(x�
2)2
�16
Fac
tor
the
squa
re.
x�
2 �
4
Squ
are
Roo
t P
rope
rty
x�
6 or
x�
�2
Sol
ve e
ach
equa
tion.
Th
e so
luti
on s
et i
s {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
sq
uar
e.T
hen
wri
te t
he
trin
omia
l as
a p
erfe
ct s
qu
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
��2
1.56
25;
(x�
1.25
)2
Sol
ve e
ach
eq
uat
ion
by
com
ple
tin
g th
e sq
uar
e.
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
Co
mp
leti
ng
th
e S
qu
are
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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
eq
uat
ion
by
usi
ng
the
Sq
uar
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
sq
uar
e.T
hen
wri
te t
he
trin
omia
l as
a p
erfe
ct s
qu
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
eq
uat
ion
by
com
ple
tin
g th
e sq
uar
e.
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
eq
uat
ion
by
usi
ng
the
Sq
uar
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
sq
uar
e.T
hen
wri
te t
he
trin
omia
l as
a p
erfe
ct s
qu
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
eq
uat
ion
by
com
ple
tin
g th
e sq
uar
e.
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
Wh
en t
he
dim
ensi
ons
of a
cu
be a
re r
edu
ced
by 4
in
ches
on
eac
h s
ide,
the
surf
ace
area
of
the
new
cu
be i
s 86
4 sq
uar
e in
ches
.Wh
at w
ere
the
dim
ensi
ons
of t
he
orig
inal
cu
be?
16 in
.by
16 in
.by
16 in
.
32.I
NV
ESTM
ENTS
Th
e am
oun
t of
mon
ey A
in a
n a
ccou
nt
in w
hic
h P
doll
ars
is i
nve
sted
for
2 ye
ars
is g
iven
by
the
form
ula
A�
P(1
�r)
2 ,w
her
e r
is t
he
inte
rest
rat
e co
mpo
un
ded
ann
ual
ly.I
f an
in
vest
men
t of
$80
0 in
th
e ac
cou
nt
grow
s to
$88
2 in
tw
o ye
ars,
at w
hat
inte
rest
rat
e w
as i
t in
vest
ed?
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 (
Ave
rag
e)
Co
mp
leti
ng
th
e S
qu
are
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-4
6-4
© Glencoe/McGraw-Hill A13 Glencoe Algebra 2
An
swer
s
Answers (Lesson 6-4)
Readin
g t
o L
earn
Math
em
ati
csC
om
ple
tin
g t
he
Sq
uar
e
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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
uct
ion
to
Les
son
6-4
at
the
top
of p
age
306
in y
our
text
book
.
Exp
lain
wh
at i
t m
ean
s to
say
th
at t
he
driv
er a
ccel
erat
es a
t a
con
stan
t ra
teof
8 f
eet
per
seco
nd
squ
are.
If t
he
dri
ver
is t
rave
ling
at
a ce
rtai
n s
pee
d a
t a
par
ticu
lar
mo
men
t,th
en o
ne
seco
nd
late
r,th
e d
rive
r is
tra
velin
g 8
fee
tp
er s
eco
nd
fas
ter.
Rea
din
g t
he
Less
on
1.G
ive
the
reas
on f
or e
ach
ste
p in
th
e fo
llow
ing
solu
tion
of
an e
quat
ion
by
usi
ng
the
Squ
are
Roo
t P
rope
rty.
x2�
12x
�36
�81
Orig
inal
equ
atio
n
(x�
6)2
�81
Fact
or
the
per
fect
sq
uar
e tr
ino
mia
l.
x�
6 �
�
81�S
qu
are
Ro
ot
Pro
per
ty
x�
6 �
9
81 �
9
x�
6 �
9 or
x�
6 �
�9
Rew
rite
as
two
eq
uat
ion
s.
x�
15
x�
�3
So
lve
each
eq
uat
ion
.
2.E
xpla
in h
ow t
o fi
nd
the
con
stan
t th
at m
ust
be
adde
d to
mak
e a
bin
omia
l in
to a
per
fect
squ
are
trin
omia
l.
Sam
ple
an
swer
:Fin
d h
alf
of
the
coef
ficie
nt
of
the
linea
r te
rm a
nd
sq
uar
e it.
3.a.
Wh
at i
s th
e fi
rst
step
in
sol
vin
g th
e eq
uat
ion
3x2
�6x
�5
by c
ompl
etin
g th
e sq
uar
e?D
ivid
e th
e eq
uat
ion
by
3.
b.
Wh
at i
s th
e fi
rst
step
in
sol
vin
g th
e eq
uat
ion
x2
�5x
�12
�0
by c
ompl
etin
g th
esq
uar
e?A
dd
12
to e
ach
sid
e.
Hel
pin
g Y
ou
Rem
emb
er
4.H
ow c
an y
ou u
se t
he
rule
s fo
r sq
uar
ing
a bi
nom
ial
to h
elp
you
rem
embe
r th
e pr
oced
ure
for
chan
gin
g a
bin
omia
l in
to a
per
fect
squ
are
trin
omia
l?
On
e o
f th
e ru
les
for
squ
arin
g a
bin
om
ial i
s (x
�y
)2�
x2
�2x
y�
y2 .
Inco
mp
leti
ng
th
e sq
uar
e,yo
u a
re s
tart
ing
wit
h x
2�
bx
and
nee
d t
o f
ind
y2 .
Th
is s
ho
ws
you
th
at b
�2y
,so
y�
.Th
at is
why
yo
u m
ust
tak
e h
alf
of
the
coef
fici
ent
and
sq
uar
e it
to
get
th
e co
nst
ant
that
mu
st b
e ad
ded
to
com
ple
te t
he
squ
are.
b � 2
©G
lenc
oe/M
cGra
w-H
ill33
6G
lenc
oe A
lgeb
ra 2
Th
e G
old
en Q
uad
rati
c E
qu
atio
ns
A g
old
en r
ecta
ngl
eh
as t
he
prop
erty
th
at i
ts l
engt
h
can
be
wri
tten
as
a�
b,w
her
e a
is t
he
wid
th o
f th
e
rect
angl
e an
d �a
� ab
��
�a b� .A
ny
gold
en r
ecta
ngl
e ca
n b
e
divi
ded
into
a s
quar
e an
d a
smal
ler
gold
en r
ecta
ngl
e,as
sh
own
.
Th
e pr
opor
tion
use
d to
def
ine
gold
en r
ecta
ngl
es c
an b
e u
sed
to d
eriv
e tw
o qu
adra
tic
equ
atio
ns.
The
se a
reso
met
imes
call
ed g
old
en q
uad
rati
c eq
uat
ion
s.
Sol
ve e
ach
pro
ble
m.
1.In
th
e pr
opor
tion
for
th
e go
lden
rec
tan
gle,
let
aeq
ual
1.W
rite
th
e re
sult
ing
quad
rati
c eq
uat
ion
an
d so
lve
for
b.
b2
�b
�1
�0
b�
2.In
th
e pr
opor
tion
,let
beq
ual
1.W
rite
th
e re
sult
ing
quad
rati
c eq
uat
ion
an
d so
lve
for
a.
a2
�a
�1
�0
a�
3.D
escr
ibe
the
diff
eren
ce b
etw
een
the
two
gold
en q
uad
rati
c eq
uat
ion
s yo
u
fou
nd
in e
xerc
ises
1 a
nd
2.
Th
e si
gn
s o
f th
e fi
rst-
deg
ree
term
s ar
e o
pp
osi
te.
4.S
how
th
at t
he
posi
tive
sol
uti
ons
of t
he
two
equ
atio
ns
in e
xerc
ises
1 a
nd
2 ar
e re
cipr
ocal
s.
���
���
��1 4�
5�
�1
5.U
se t
he
Pyt
hag
orea
n T
heo
rem
to
fin
d a
radi
cal
expr
essi
on f
or t
he
diag
onal
of
a g
olde
n r
ecta
ngl
e w
hen
a�
1.
d�
6.F
ind
a ra
dica
l ex
pres
sion
for
th
e di
agon
al o
f a
gold
en r
ecta
ngl
e 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
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-4
6-4
© Glencoe/McGraw-Hill A14 Glencoe Algebra 2
Answers (Lesson 6-5)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Th
e Q
uad
rati
c F
orm
ula
an
d t
he
Dis
crim
inan
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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
Th
e Q
uad
rati
c F
orm
ula
can
be
use
d to
sol
ve a
ny
quad
rati
ceq
uat
ion
on
ce i
t is
wri
tten
in
th
e fo
rm a
x2�
bx�
c�
0.
Qu
adra
tic
Fo
rmu
laT
he s
olut
ions
of
ax2
�bx
�c
�0,
with
a�
0, a
re g
iven
by
x�
.
Sol
ve x
2�
5x�
14 b
y u
sin
g th
e Q
uad
rati
c F
orm
ula
.
Rew
rite
th
e eq
uat
ion
as
x2�
5x�
14 �
0.
x�
Qua
drat
ic F
orm
ula
�R
epla
ce a
with
1,
bw
ith �
5, a
nd c
with
�14
.
�S
impl
ify.
� �7
or �
2
Th
e so
luti
ons
are
�2
and
7.
Sol
ve e
ach
eq
uat
ion
by
usi
ng
the
Qu
adra
tic
For
mu
la.
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
��
4ac
��
��
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
tT
he e
xpre
ssio
n un
der
the
radi
cal s
ign,
b2
�4a
c, in
the
Qua
drat
ic F
orm
ula
is c
alle
d th
e d
iscr
imin
ant.
Ro
ots
of
a Q
uad
rati
c Eq
uat
ion
Dis
crim
inan
tTy
pe
and
Nu
mb
er o
f R
oo
ts
b2
�4a
c�
0 an
d a
perf
ect
squa
re2
ratio
nal r
oots
b2
�4a
c�
0, b
ut n
ot
a pe
rfec
t sq
uare
2 irr
atio
nal r
oots
b2
�4a
c�
01
ratio
nal r
oot
b2
�4a
c�
02
com
plex
roo
ts
Fin
d t
he
valu
e of
th
e d
iscr
imin
ant
for
each
eq
uat
ion
.Th
en d
escr
ibe
the
nu
mb
er a
nd
typ
es o
f ro
ots
for
the
equ
atio
n.
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Th
e Q
uad
rati
c F
orm
ula
an
d t
he
Dis
crim
inan
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-5
6-5
Exam
ple
Exam
ple
a.2x
2�
5x�
3T
he
disc
rim
inan
t is
b2
�4a
c�
52�
4(2)
(3)
or 1
.T
he
disc
rim
inan
t is
a p
erfe
ct s
quar
e,so
the
equ
atio
n h
as 2
rat
ion
al r
oots
.
b.3
x2�
2x�
5T
he
disc
rim
inan
t is
b2
�4a
c�
(�2)
2�
4(3)
(5)
or �
56.
Th
e di
scri
min
ant
is n
egat
ive,
so t
he
equ
atio
n 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.
Des
crib
e th
e n
um
ber
an
d t
ype
of r
oots
.c.
Fin
d t
he
exac
t so
luti
ons
by
usi
ng
the
Qu
adra
tic
For
mu
la.
1.p2
�12
p�
�4
128;
2.9x
2�
6x�
1 �
00;
3.2x
2�
7x�
4 �
081
;tw
o ir
rati
on
alro
ots
;o
ne
rati
on
al r
oo
t;2
rati
on
al r
oo
ts;
�,4
�6
�4 �
2�
4.x2
�4x
�4
�0
32;
5.5x
2�
36x
�7
�0
1156
;6.
4x2
�4x
�11
�0
2 ir
rati
on
al r
oo
ts;
2 ra
tio
nal
ro
ots
;�
160;
2 co
mp
lex
roo
ts;
�2
�2 �
2�,7
7.x2
�7x
�6
�0
25;
8.m
2�
8m�
�14
8;9.
25x2
�40
x�
�16
0;2
rati
on
al r
oo
ts;
2 ir
rati
on
al r
oo
ts;
1 ra
tio
nal
ro
ot;
1,6
4 �
�2�
10.4
x2�
20x
�29
�0
�64
;11
.6x2
�26
x�
8 �
048
4;12
.4x2
�4x
�11
�0
192;
2 co
mp
lex
roo
ts;
2 ra
tio
nal
ro
ots
;2
irra
tio
nal
ro
ots
;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
Th
e Q
uad
rati
c F
orm
ula
an
d t
he
Dis
crim
inan
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
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.
Des
crib
e th
e n
um
ber
an
d t
ype
of r
oots
.c.
Fin
d t
he
exac
t so
luti
ons
by
usi
ng
the
Qu
adra
tic
For
mu
la.
1.x2
�8x
�16
�0
2.x2
�11
x�
26 �
0
0;1
rati
on
al r
oo
t;4
225;
2 ra
tio
nal
ro
ots
;�
2,13
3.3x
2�
2x�
04.
20x2
�7x
�3
�0
4;2
rati
on
al r
oo
ts;
0,28
9;2
rati
on
al r
oo
ts;
�,
5.5x
2�
6 �
06.
x2�
6 �
0
120;
2 ir
rati
on
al r
oo
ts;
�24
;2
irra
tio
nal
ro
ots
;�
�6�
7.x2
�8x
�13
�0
8.5x
2�
x�
1 �
0
12;
2 ir
rati
on
al r
oo
ts;
�4
��
3�21
;2
irra
tio
nal
ro
ots
;
9.x2
�2x
�17
�0
10.x
2�
49 �
0
72;
2 ir
rati
on
al r
oo
ts;
1 �
3�2�
�19
6;2
com
ple
x ro
ots
;�
7i
11.x
2�
x�
1 �
012
.2x2
�3x
��
2
�3;
2 co
mp
lex
roo
ts;
�7;
2 co
mp
lex
roo
ts;
Sol
ve e
ach
eq
uat
ion
by
usi
ng
the
met
hod
of
you
r ch
oice
.Fin
d e
xact
sol
uti
ons.
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
nor
ing
win
d re
sist
ance
,th
e di
stan
ce d
(t)
in f
eet
that
a p
arac
hu
tist
fall
s in
tse
con
ds c
an b
e es
tim
ated
usi
ng
the
form
ula
d(t
) �
16t2
.If
a pa
rach
uti
st ju
mps
from
an
air
plan
e an
d fa
lls
for
1100
fee
t be
fore
ope
nin
g h
er p
arac
hu
te,h
ow m
any
seco
nds
pass
bef
ore
she
open
s th
e pa
rach
ute
?ab
ou
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.
Des
crib
e th
e n
um
ber
an
d t
ype
of r
oots
.c.
Fin
d t
he
exac
t so
luti
ons
by
usi
ng
the
Qu
adra
tic
For
mu
la.
1.x2
�16
x�
64 �
02.
x2�
3x3.
9x2
�24
x�
16 �
0
0;1
rati
on
al;
89;
2 ra
tio
nal
;0,
30;
1 ra
tio
nal
;
4.x2
�3x
�40
5.3x
2�
9x�
2 �
010
5;6.
2x2
�7x
�0
169;
2 ra
tio
nal
;�
5,8
2 ir
rati
on
al;
49;
2 ra
tio
nal
;0,
�
7.5x
2�
2x�
4 �
0�
76;
8.12
x2�
x�
6 �
028
9;9.
7x2
�6x
�2
�0
�20
;
2 co
mp
lex;
2 ra
tio
nal
;,�
2 co
mp
lex;
10.1
2x2
�2x
�4
�0
196;
11.6
x2�
2x�
1 �
028
;12
.x2
�3x
�6
�0
�15
;
2 ra
tio
nal
;,�
2 ir
rati
on
al;
2 co
mp
lex;
13.4
x2�
3x2
�6
�0
105;
14.1
6x2
�8x
�1
�0
15.2
x2�
5x�
6 �
073
;
2 ir
rati
on
al;
0;1
rati
on
al;
2 ir
rati
on
al;
Sol
ve e
ach
eq
uat
ion
by
usi
ng
the
met
hod
of
you
r ch
oice
.Fin
d e
xact
sol
uti
ons.
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 f
eet
of a
n ob
ject
tse
cond
s af
ter
it is
pro
pelle
d st
raig
ht u
pfr
om t
he
grou
nd
wit
h a
n i
nit
ial
velo
city
of
60 f
eet
per
seco
nd
is m
odel
ed b
y th
e eq
uat
ion
h(t
) �
�16
t2�
60t.
At
wh
at t
imes
wil
l th
e ob
ject
be
at a
hei
ght
of 5
6 fe
et?
1.75
s,2
s
31.S
TOPP
ING
DIS
TAN
CE
Th
e fo
rmu
la d
�0.
05s2
�1.
1ses
tim
ates
th
e m
inim
um
sto
ppin
gdi
stan
ce d
in f
eet
for
a ca
r tr
avel
ing
sm
iles
per
hou
r.If
a c
ar s
tops
in 2
00 f
eet,
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?
abo
ut
53.2
mi/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 (
Ave
rag
e)
Th
e Q
uad
rati
c F
orm
ula
an
d t
he
Dis
crim
inan
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-5
6-5
© Glencoe/McGraw-Hill A16 Glencoe Algebra 2
Answers (Lesson 6-5)
Readin
g t
o L
earn
Math
em
ati
csT
he
Qu
adra
tic
Fo
rmu
la a
nd
th
e D
iscr
imin
ant
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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 b
lood
pre
ssu
re r
elat
ed t
o ag
e?
Rea
d th
e in
trod
uct
ion
to
Les
son
6-5
at
the
top
of p
age
313
in y
our
text
book
.
Des
crib
e h
ow y
ou w
ould
cal
cula
te y
our
nor
mal
blo
od p
ress
ure
usi
ng
one
ofth
e fo
rmu
las
in y
our
text
book
.
Sam
ple
an
swer
:S
ub
stit
ute
yo
ur
age
for
Ain
th
e ap
pro
pri
ate
form
ula
(fo
r fe
mal
es o
r m
ales
) an
d e
valu
ate
the
exp
ress
ion
.
Rea
din
g t
he
Less
on
1.a.
Wri
te t
he
Qu
adra
tic
For
mu
la.
x�
b.
Iden
tify
th
e va
lues
of
a,b,
and
cth
at y
ou w
ould
use
to
solv
e 2x
2�
5x�
�7,
but
don
ot a
ctu
ally
sol
ve t
he
equ
atio
n.
a�
b�
c�
2.S
upp
ose
that
you
are
sol
vin
g fo
ur
quad
rati
c eq
uat
ion
s w
ith
rat
ion
al c
oeff
icie
nts
an
dh
ave
fou
nd
the
valu
e of
th
e di
scri
min
ant
for
each
equ
atio
n.I
n e
ach
cas
e,gi
ve t
he
nu
mbe
r of
roo
ts a
nd
desc
ribe
th
e ty
pe o
f ro
ots
that
th
e eq
uat
ion
wil
l h
ave.
Val
ue
of
Dis
crim
inan
tN
um
ber
of
Ro
ots
Typ
e o
f R
oo
ts
642
real
,rat
ion
al
�8
2co
mp
lex
212
real
,irr
atio
nal
01
real
,rat
ion
al
Hel
pin
g Y
ou
Rem
emb
er
3.H
ow c
an l
ooki
ng
at t
he
Qu
adra
tic
For
mu
la h
elp
you
rem
embe
r th
e re
lati
onsh
ips
betw
een
th
e va
lue
of t
he
disc
rim
inan
t an
d th
e n
um
ber
of r
oots
of
a qu
adra
tic
equ
atio
nan
d w
het
her
th
e ro
ots
are
real
or
com
plex
?
Sam
ple
an
swer
:Th
e d
iscr
imin
ant
is t
he
exp
ress
ion
un
der
th
e ra
dic
al in
the
Qu
adra
tic
Fo
rmu
la.L
oo
k at
th
e Q
uad
rati
c F
orm
ula
an
d c
on
sid
er w
hat
hap
pen
s w
hen
yo
u t
ake
the
pri
nci
pal
sq
uar
e ro
ot
of
b2
�4a
can
d a
pp
ly�
in f
ron
t o
f th
e re
sult
.If
b2
�4a
cis
po
siti
ve,i
ts p
rin
cip
al s
qu
are
roo
tw
ill b
e a
po
siti
ve n
um
ber
an
d a
pp
lyin
g �
will
giv
e tw
o d
iffe
ren
t re
also
luti
on
s,w
hic
h m
ay b
e ra
tio
nal
or
irra
tio
nal
.If
b2
�4a
c�
0,it
sp
rin
cip
al s
qu
are
roo
t is
0,s
o a
pp
lyin
g �
in t
he
Qu
adra
tic
Fo
rmu
la w
illo
nly
lead
to
on
e so
luti
on
,wh
ich
will
be
rati
on
al (
assu
min
g a
,b,a
nd
car
ein
teg
ers)
.If
b2
�4a
cis
neg
ativ
e,si
nce
th
e sq
uar
e ro
ots
of
neg
ativ
en
um
ber
s ar
e n
ot
real
nu
mb
ers,
you
will
get
tw
o c
om
ple
x ro
ots
,co
rres
po
nd
ing
to
th
e �
and
�in
th
e �
sym
bo
l.7�
52
�b
��
b2
�4
�ac �
��
2a
©G
lenc
oe/M
cGra
w-H
ill34
2G
lenc
oe A
lgeb
ra 2
Su
m a
nd
Pro
du
ct o
f R
oo
ts
Som
etim
es y
ou m
ay k
now
th
e ro
ots
of a
qu
adra
tic
equ
atio
n w
ith
out
know
ing
the
equ
atio
nit
self
.Usi
ng
you
r kn
owle
dge
of f
acto
rin
g to
sol
ve a
n e
quat
ion
,you
can
wor
k ba
ckw
ard
tofi
nd
the
quad
rati
c eq
uat
ion
.Th
e ru
le f
or f
indi
ng
the
sum
an
d pr
odu
ct o
f ro
ots
is a
s fo
llow
s:
Su
m a
nd
Pro
du
ct o
f R
oo
tsIf
the
root
s of
ax2
�bx
�c
�0,
with
a≠
0, a
re s
1an
d 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
grad
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.
Eq
uat
ion
sof
tra
nsi
tion
cu
rves
are
use
d b
y ci
vil
engi
nee
rs t
o d
esig
n s
moo
th a
nd
saf
e ro
ads.
Th
e ro
ots
are
x�
3 an
d x
��
8.
3 �
(�8)
��
5A
dd t
he r
oots
.
3(�
8) �
�24
Mul
tiply
the
roo
ts.
Equ
atio
n:x
2�
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
�0
x2
�4x
�5
�0
x2
�12
x�
36 �
0
4.4
�
3�6.
��2 5� ,
�2 7�6.
x2
�8x
�13
�0
35x
2�
4x�
4 �
049
x2
�42
x�
205
�0
Fin
d k
such
th
at t
he
nu
mb
er g
iven
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,
–30
1 – 4)
10 –10
–20
–30
24
–2–4
–6–8
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-5
6-5
Exam
ple
Exam
ple
© Glencoe/McGraw-Hill A17 Glencoe Algebra 2
An
swer
s
Answers (Lesson 6-6)
Stu
dy G
uid
e a
nd I
nte
rven
tion
An
alyz
ing
Gra
ph
s o
f Q
uad
rati
c F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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
gra
ph o
f y
�a
(x�
h)2
�k
has
the
follo
win
g ch
arac
teris
tics:
•V
erte
x: (
h, k
)V
erte
x F
orm
•A
xis
of s
ymm
etry
: x
�h
of
a Q
uad
rati
c•
Ope
ns u
p if
a�
0F
un
ctio
n•
Ope
ns d
own
if a
�0
•N
arro
wer
tha
n th
e gr
aph
of y
�x
2if
a
�1
•W
ider
tha
n th
e gr
aph
of y
�x
2if
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
Th
e ve
rtex
is
at (
h,k
) or
(�
4,�
11),
and
the
axis
of
sym
met
ry i
s x
��
4.T
he
grap
h o
pen
su
p,an
d is
nar
row
er t
han
th
e gr
aph
of
y �
x2.
a.y
��
(x�
2)2
�10
Th
e ve
rtex
is
at (
h,k
) or
(2,
10),
and
the
axis
of
sym
met
ry i
s x
�2.
Th
e gr
aph
ope
ns
dow
n,a
nd
is w
ider
th
an t
he
grap
h o
f 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;u
p(�
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;d
ow
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;u
p(3
,�2)
;x
�3;
up
(1,�
2);
x�
1;d
ow
n
10.y
��
(x�
5)2
�12
11.y
�(x
�7)
2�
2212
.y�
16(x
�4)
2�
1
(�5,
12);
x�
�5;
do
wn
(7,2
2);
x�
7;u
p(4
,1);
x�
4;u
p
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;u
p(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
uad
rati
c fu
nct
ion
is
easi
er t
ogr
aph
wh
en i
t is
in
ver
tex
form
.You
can
wri
te a
qu
adra
tic
fun
ctio
n o
f th
e fo
rm
y�
ax2
�bx
�c
in v
erte
x fr
om b
y co
mpl
etin
g th
e sq
uar
e.
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)
�25
y�
2(x2
�6x
�9)
�25
�18
y�
2(x
�3)
2�
7
Th
e ve
rtex
for
m o
f th
e eq
uat
ion
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 G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
An
alyz
ing
Gra
ph
s o
f Q
uad
rati
c F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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
An
alyz
ing
Gra
ph
s o
f Q
uad
rati
c F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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;u
p(0
,4);
x�
0;d
ow
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;d
ow
n(0
,9);
x�
0;d
ow
n(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;u
p(�
3,�
7);
x�
�3;
up
(4,4
8);
x�
4;d
ow
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
qu
atio
n f
or t
he
par
abol
a w
ith
th
e gi
ven
ver
tex
that
pas
ses
thro
ugh
th
egi
ven
poi
nt.
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;u
px
��
3;u
p(1
,2);
x�
1;u
p
7.y
��
2x2
�16
x�
328.
y�
�3x
2�
18x
�21
9.y
�2x
2�
16x
�29
y�
�2(
x�
4)2 ;
y�
�3(
x�
3)2
�6;
y�
2(x
�4)
2�
3;(�
4,0)
;x
��
4;d
ow
n(3
,6);
x�
3;d
ow
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
qu
atio
n f
or t
he
par
abol
a w
ith
th
e gi
ven
ver
tex
that
pas
ses
thro
ugh
th
egi
ven
poi
nt.
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
equ
atio
n f
or a
par
abol
a w
ith
ver
tex
at (
4,4)
an
d x-
inte
rcep
t 6.
y�
�(x
�4)
2�
4
17.W
rite
an
equ
atio
n f
or a
par
abol
a w
ith
ver
tex
at (
�3,
�1)
an
d y-
inte
rcep
t 2.
y�
(x�
3)2
�1
18.B
ASE
BA
LLT
he
hei
ght
hof
a b
aseb
all
tse
con
ds a
fter
bei
ng
hit
is
give
n b
y h
(t)
��
16t2
�80
t�
3.W
hat
is
the
max
imu
m h
eigh
t th
at t
he
base
ball
rea
ches
,an
dw
hen
doe
s th
is o
ccu
r?10
3 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
oun
d an
d re
ach
es a
max
imu
m h
eigh
t of
10
feet
aft
er a
hor
izon
tal
dist
ance
of
4 fe
et.W
rite
a q
uad
rati
c fu
nct
ion
in
ver
tex
form
that
des
crib
es t
he
shap
e of
th
e ou
tsid
e of
th
e ar
c,w
her
e y
is t
he
hei
ght
of a
poi
nt
on t
he
arc
and
xis
its
hor
izon
tal
dist
ance
fro
m t
he
left
-han
dst
arti
ng
poin
t of
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 (
Ave
rag
e)
An
alyz
ing
Gra
ph
s o
f Q
uad
rati
c F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-6
6-6
© Glencoe/McGraw-Hill A19 Glencoe Algebra 2
An
swer
s
Answers (Lesson 6-6)
Readin
g t
o L
earn
Math
em
ati
csA
nal
yzin
g G
rap
hs
of
Qu
adra
tic
Eq
uat
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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
use
d t
o gr
aph
an
y q
uad
rati
cfu
nct
ion
?
Rea
d th
e in
trod
uct
ion
to
Les
son
6-6
at
the
top
of p
age
322
in y
our
text
book
.
•W
hat
doe
s ad
din
g a
posi
tive
nu
mbe
r to
x2
do t
o th
e gr
aph
of
y�
x2?
It m
oves
th
e g
rap
h u
p.
•W
hat
doe
s su
btra
ctin
g a
posi
tive
nu
mbe
r to
xbe
fore
squ
arin
g do
to
the
grap
h o
f y
�x2
?It
mov
es t
he
gra
ph
to
th
e ri
gh
t.
Rea
din
g t
he
Less
on
1.C
ompl
ete
the
foll
owin
g in
form
atio
n a
bou
t th
e gr
aph
of
y�
a(x
�h
)2�
k.
a.W
hat
are
th
e co
ordi
nat
es o
f th
e ve
rtex
?(h
,k)
b.
Wh
at i
s th
e eq
uat
ion
of
the
axis
of
sym
met
ry?
x�
h
c.In
wh
ich
dir
ecti
on d
oes
the
grap
h o
pen
if
a�
0? I
f a
�0?
up
;d
ow
n
d.
Wh
at d
o yo
u k
now
abo
ut
the
grap
h i
f a
�
1?It
is w
ider
th
an t
he
gra
ph
of
y�
x2 .
If
a�
1?It
is n
arro
wer
th
an t
he
gra
ph
of
y�
x2 .
2.M
atch
eac
h g
raph
wit
h t
he
desc
ript
ion
of
the
con
stan
ts i
n t
he
equ
atio
n i
n v
erte
x fo
rm.
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 f
unct
ions
suc
h 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
�x2
to t
he le
ftan
d w
hich
rep
rese
nts
a tr
ansl
atio
n to
the
rig
ht.W
hat
is a
n ea
sy w
ay t
o re
mem
ber
this
?
Sam
ple
an
swer
:In
fu
nct
ion
s lik
e y
�(x
�4)
2 ,th
e p
lus
sig
n p
uts
th
eg
rap
h “
ahea
d”
so t
hat
th
e ve
rtex
co
mes
“so
on
er”
than
th
e o
rig
in a
nd
th
etr
ansl
atio
n is
to
th
e le
ft.I
n f
un
ctio
ns
like
y�
(x�
5)2 ,
the
min
us
pu
ts t
he
gra
ph
“b
ehin
d”
so t
hat
th
e ve
rtex
co
mes
“la
ter”
than
th
e o
rig
in a
nd
th
etr
ansl
atio
n is
to
th
e ri
gh
t.
x
y
Ox
y
Ox
y
Ox
y
O
©G
lenc
oe/M
cGra
w-H
ill34
8G
lenc
oe A
lgeb
ra 2
Pat
tern
s w
ith
Dif
fere
nce
s an
d S
um
s o
f S
qu
ares
Som
e w
hol
e n
um
bers
can
be
wri
tten
as
the
diff
eren
ce o
f tw
o sq
uar
es a
nd
som
e ca
nn
ot.F
orm
ula
s ca
n b
e de
velo
ped
to d
escr
ibe
the
sets
of
nu
mbe
rsal
gebr
aica
lly.
If p
ossi
ble
,wri
te e
ach
nu
mb
er a
s 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
can
no
t4.
322
�12
5.4
22�
026.
532
�22
7.6
can
no
t8.
742
�32
9.8
32�
1210
.932
�02
11.
10ca
nn
ot
12.1
162
�52
13.1
242
�22
14.1
372
�62
15.1
4ca
nn
ot
16.1
542
�12
Eve
n n
um
ber
s ca
n b
e w
ritt
en a
s 2n
,wh
ere
nis
on
e of
th
e n
um
ber
s 0,
1,2,
3,an
d s
o on
.Od
d n
um
ber
s ca
n b
e w
ritt
en 2
n�
1.U
se t
hes
e ex
pre
ssio
ns
for
thes
e p
rob
lem
s.
17.S
how
th
at a
ny
odd
nu
mbe
r ca
n b
e w
ritt
en a
s th
e di
ffer
ence
of
two
squ
ares
.2n
�1
�(n
�1)
2�
n2
18.S
how
th
at t
he
even
nu
mbe
rs c
an b
e di
vide
d in
to t
wo
sets
:th
ose
that
can
be
wri
tten
in
th
e fo
rm 4
nan
d th
ose
that
can
be
wri
tten
in
th
e fo
rm 2
�4n
.F
ind
4n
for
n�
0,1,
2,an
d s
o o
n.Y
ou
get
{0,
4,8,
12,…
}.F
or
2 �
4n,y
ou
get
{2,
6,10
,12,
…}.
Tog
eth
er t
hes
e se
ts in
clu
de
all e
ven
nu
mb
ers.
19.D
escr
ibe
the
even
nu
mbe
rs t
hat
can
not
be
wri
tten
as
the
diff
eren
ce o
f tw
o sq
uar
es.
2 �
4n,f
or
n�
0,1,
2,3,
…
20.S
how
th
at t
he
oth
er e
ven
nu
mbe
rs c
an b
e w
ritt
en a
s th
e di
ffer
ence
of
two
squ
ares
.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
mb
ers
from
0 t
hro
ugh
15
by
wri
tin
g ea
ch o
ne
as t
he
sum
of
the
leas
t n
um
ber
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
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-6
6-6
© Glencoe/McGraw-Hill A20 Glencoe Algebra 2
Answers (Lesson 6-7)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Gra
ph
ing
an
d S
olv
ing
Qu
adra
tic
Ineq
ual
itie
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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
sT
o gr
aph
a q
uad
rati
c in
equ
alit
y in
tw
o va
riab
les,
use
the
foll
owin
g st
eps:
1.G
raph
th
e re
late
d qu
adra
tic
equ
atio
n,y
�ax
2�
bx�
c.U
se a
das
hed
lin
e fo
r �
or �
;use
a s
olid
lin
e fo
r
or �
.
2.T
est
a po
int
insi
de t
he
para
bola
.If
it
sati
sfie
s th
e in
equ
alit
y,sh
ade
the
regi
on i
nsi
de t
he
para
bola
;ot
her
wis
e,sh
ade
the
regi
on o
uts
ide
the
para
bola
.
Gra
ph
th
e in
equ
alit
y y
�x2
�6x
�7.
Fir
st g
raph
th
e eq
uat
ion
y�
x2�
6x�
7.B
y co
mpl
etin
g th
e sq
uar
e,yo
u g
et t
he
vert
ex f
orm
of
the
equ
atio
n y
�(x
�3)
2�
2,so
th
e ve
rtex
is
(�3,
�2)
.Mak
e a
tabl
e of
val
ues
aro
un
d x
��
3,an
d gr
aph
.Sin
ce t
he
ineq
ual
ity
incl
ude
s �
,use
a d
ash
ed l
ine.
Tes
t th
e po
int
(�3,
0),w
hic
h i
s in
side
th
e pa
rabo
la.S
ince
(�
3)2
�6(
�3)
�7
��
2,an
d 0
��
2,(�
3,0)
sat
isfi
es t
he
ineq
ual
ity.
Th
eref
ore,
shad
e th
e re
gion
in
side
th
e pa
rabo
la.
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
Qu
adra
tic
ineq
ual
itie
s in
on
e va
riab
le c
an b
e so
lved
grap
hic
ally
or
alge
brai
call
y.
To s
olve
ax
2�
bx�
c�
0:F
irst
grap
h y
�ax
2�
bx�
c. T
he s
olut
ion
cons
ists
of
the
x-va
lues
Gra
ph
ical
Met
ho
dfo
r w
hich
the
gra
ph is
bel
ow
the
x-ax
is.
To s
olve
ax
2�
bx�
c�
0:F
irst
grap
h y
�ax
2�
bx�
c. T
he s
olut
ion
cons
ists
the
x-v
alue
s fo
r w
hich
the
gra
ph is
ab
ove
the
x-ax
is.
Fin
d th
e ro
ots
of t
he r
elat
ed q
uadr
atic
equ
atio
n by
fac
torin
g,
Alg
ebra
ic M
eth
od
com
plet
ing
the
squa
re,
or u
sing
the
Qua
drat
ic F
orm
ula.
2 ro
ots
divi
de t
he n
umbe
r lin
e in
to 3
inte
rval
s.Te
st a
val
ue in
eac
h in
terv
al t
o se
e w
hich
inte
rval
s ar
e so
lutio
ns.
If t
he
ineq
ual
ity
invo
lves
or
�,t
he
root
s of
th
e re
late
d eq
uat
ion
are
in
clu
ded
in t
he
solu
tion
set
.
Sol
ve t
he
ineq
ual
ity
x2�
x�
6 �
0.
Fir
st f
ind
the
root
s of
th
e re
late
d eq
uat
ion
x2
�x
�6
�0.
Th
eeq
uat
ion
fac
tors
as
(x�
3)(x
�2)
�0,
so t
he
root
s ar
e 3
and
�2.
Th
e gr
aph
ope
ns
up
wit
h x
-in
terc
epts
3 a
nd
�2,
so i
t m
ust
be
on
or b
elow
th
e x-
axis
for
�2
x
3.
Th
eref
ore
the
solu
tion
set
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}
{x1
�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}
{x2
�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
�xx
�o
r x
�4 �
�x
�x
�5 �
1 � 33 � 2
1 � 63 � 4
1 � 2
1 � 2
x
y
O
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Gra
ph
ing
an
d S
olv
ing
Qu
adra
tic
Ineq
ual
itie
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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
ph
ing
an
d S
olv
ing
Qu
adra
tic
Ineq
ual
itie
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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 o
r x
�4
Sol
ve e
ach
in
equ
alit
y al
geb
raic
ally
.
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
{xx
�9}
{x�
6 �
x�
6}
11.x
2�
7x�
012
.x2
�7x
�6
�0
{xx
�0
or
x�
7}{x
�6
�x
��
1}
13.x
2�
x�
12 �
014
.x2
�9x
�18
0
{xx
��
4 o
r 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
{xx
��
3 o
r x
�0}
{xx
��
2 o
r 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 o
r x
�8
�3
�x
�1
2 �
x�
7
Sol
ve e
ach
in
equ
alit
y al
geb
raic
ally
.
7.x2
�x
�20
�0
8.x2
�10
x�
16 �
09.
x2�
4x�
5
0
{xx
��
4 o
r x
�5}
{x2
�x
�8}
10.x
2�
14x
�49
�0
11.x
2�
5x�
1412
.�x2
�15
�8x
all r
eals
{xx
��
2 o
r x
�7}
{x�
5 �
x�
�3}
13.�
x2�
5x�
7
014
.9x2
�36
x�
36
015
.9x
12
x2
all r
eals
{xx
��
2}�x
x�
0 o
r x
�
16.4
x2�
4x�
1 �
017
.5x2
�10
�27
x18
.9x2
�31
x�
12
0
�xx
��
��x
x�
or
x
5 ��x
�3
�x
��
�19
.FEN
CIN
GV
anes
sa h
as 1
80 f
eet
of f
enci
ng
that
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e in
ten
ds t
o u
se t
o bu
ild
a re
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gula
rpl
ay a
rea
for
her
dog
.Sh
e w
ants
th
e pl
ay a
rea
to e
ncl
ose
at l
east
180
0 sq
uar
e fe
et.W
hat
are
the
poss
ible
wid
ths
of t
he
play
are
a?30
ft
to 6
0 ft
20.B
USI
NES
SA
bic
ycle
mak
er s
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300
bicy
cles
last
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r at
a p
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20 i
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it w
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he n
umbe
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ycle
s so
ld b
y 10
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y $2
0 in
crea
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in p
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aker
add
in
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d ex
pect
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at l
east
$10
0,00
0?fr
om
5 t
o 1
0
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 (
Ave
rag
e)
Gra
ph
ing
an
d S
olv
ing
Qu
adra
tic
Ineq
ual
itie
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-7
6-7
© Glencoe/McGraw-Hill A22 Glencoe Algebra 2
Answers (Lesson 6-7)
Readin
g t
o L
earn
Math
em
ati
csG
rap
hin
g a
nd
So
lvin
g Q
uad
rati
c In
equ
alit
ies
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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
uct
ion
to
Les
son
6-7
at
the
top
of p
age
329
in y
our
text
book
.
•H
ow f
ar a
bove
th
e gr
oun
d is
th
e tr
ampo
lin
e su
rfac
e?3.
75 f
eet
•U
sin
g th
e qu
adra
tic
fun
ctio
n g
iven
in
th
e in
trod
uct
ion
,wri
te a
qu
adra
tic
ineq
ual
ity
that
des
crib
es t
he
tim
es a
t w
hic
h t
he
tram
poli
nis
t is
mor
eth
an 2
0 fe
et a
bove
th
e gr
oun
d.�
16t2
�42
t�
3.75
�20
Rea
din
g t
he
Less
on
1.A
nsw
er t
he
foll
owin
g qu
esti
ons
abou
t h
ow y
ou w
ould
gra
ph t
he
ineq
ual
ity
y�
x2�
x�
6.
a.W
hat
is
the
rela
ted
quad
rati
c eq
uat
ion
?y
�x
2�
x�
6
b.
Sh
ould
th
e pa
rabo
la b
e so
lid
or d
ash
ed?
How
do
you
kn
ow?
solid
;Th
e in
equ
alit
y sy
mb
ol i
s
.
c.T
he
poin
t (0
,2)
is i
nsi
de t
he
para
bola
.To
use
th
is a
s a
test
poi
nt,
subs
titu
te
for
xan
d fo
r y
in t
he
quad
rati
c in
equ
alit
y.
d.
Is t
he
stat
emen
t 2
�02
�0
�6
tru
e or
fal
se?
tru
e
e.S
hou
ld t
he
regi
on i
nsi
de o
r ou
tsid
e th
e pa
rabo
la b
e sh
aded
?in
sid
e
2.T
he
grap
h o
f y
��
x2�
4xis
sh
own
at
the
righ
t.M
atch
eac
h
of t
he
foll
owin
g re
late
d in
equ
alit
ies
wit
h i
ts s
olu
tion
set
.
a.�
x2�
4x�
0ii
i.{x
x�
0 or
x�
4}
b.
�x2
�4x
0
iiiii
.{x
0 �
x�
4}
c.�
x2�
4x�
0iv
iii.
{xx
0
or x
�4}
d.
�x2
�4x
�0
iiv
.{x
0
x
4}
Hel
pin
g Y
ou
Rem
emb
er
3.A
qu
adra
tic
ineq
ual
ity
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
ine
qual
ity
sym
bol
and
wit
hout
usi
ng a
tes
t po
int.
Sam
ple
an
swer
:If
th
e sy
mb
ol i
s �
or
,s
had
e th
e re
gio
n a
bov
e th
ep
arab
ola
.If
the
sym
bo
l is
�o
r �
,sh
ade
the
reg
ion
bel
ow
th
e p
arab
ola
.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
ph
ing
Ab
solu
te V
alu
e In
equ
alit
ies
You
can
sol
ve a
bsol
ute
val
ue
ineq
ual
itie
s by
gra
phin
g in
mu
ch t
he
sam
e m
ann
er y
ou g
raph
ed q
uad
rati
c in
equ
alit
ies.
Gra
ph t
he
rela
ted
abso
lute
fu
nct
ion
fo
r ea
ch i
neq
ual
ity
by u
sin
g a
grap
hin
g ca
lcu
lato
r.F
or �
and
�,i
den
tify
th
e x-
valu
es,i
f an
y,fo
r w
hic
h t
he
grap
h l
ies
belo
wth
e x-
axis
.For
�an
d
,ide
nti
fy
the
xva
lues
,if
any,
for
wh
ich
th
e gr
aph
lie
s ab
ove
the
x-ax
is.
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 o
r 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
�5
all r
eal n
um
ber
s2
�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
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-7
6-7
© Glencoe/McGraw-Hill A23 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10. A
D
C
C
D
A
B
A
B
C
1 and 7; 14
C
B
C
B
D
C
A
D
D
B
A
B
B
C
B
B
A
C
B
B
Chapter 6 Assessment Answer KeyForm 1 Form 2APage 355 Page 356 Page 357
An
swer
s
(continued on the next page)
© Glencoe/McGraw-Hill A24 Glencoe Algebra 2
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
A
A
B
D
C
B
A
D
B
C
C
A
B
D
A
C
D
B
C
B
D
C
A
B
A
D
D
C
D
B
Chapter 6 Assessment Answer KeyForm 2A (continued) Form 2BPage 358 Page 359 Page 360
Sample answer: 16x2 � 3 � 0
Sample answer: 9x2 � 2 � 0
© Glencoe/McGraw-Hill A25 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B: 9x2 � 7 � 0
�x � x � ��12
� or x � 3�
y
xO
y � (x � 3)2 � 1
y � �32
�(x � 2)2 � 1
(�5, �7); x � �5; down
33; 2 real, irrational roots
0; 1 real, rational root
�3 �
1i0�31��
��2, �12
��{�2 � �13�}
��5 �2
�7���{�8, 2}
4x2 � 21x � 18 � 0
9 in. by 16 in.
��3, �25
��
y
xO
y
xO
2, 4
maximum; 4
Chapter 6 Assessment Answer KeyForm 2CPage 361 Page 362
An
swer
s
xO
f(x )
(3, 0)
(1, 5)
f (x) � �5x 2 � 10x
x � 1
between �2 and �1;between 1 and 2 h(t ) � �16(t � 1.5)2 �
51; 51 ft
© Glencoe/McGraw-Hill A26 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B: 16x2 � 5 � 0
�x � ��32
� � x � 5�
y
xO
y � (x � 2)2 � 4
y � ��14
�(x � 4)2 � 2
(6, �5); x � 6; down
�8; 2 complex roots
0; 1 real, rational root
�9 �
4�41��
��1, �23
��{4 � �2�}
���2 �3
�6���{3, 11}
2x2 � 5x � 12
8 in. by 18 in.
��1, �43
��
y
xO
between �1 and 0;between 1 and 2
y
xO
1, �3
minimum; �17
Chapter 6 Assessment Answer KeyForm 2DPage 363 Page 364
xO
f(x )
(2, �1)
(0, 3)
f (x) � x 2 � 4x � 3
x � 2
h(t) � �16(t � 2)2 �76; 76 ft
© Glencoe/McGraw-Hill A27 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B: 16x2 � 24x � 29 � 0
�x � x � ��72
� or x � 1�
y
xO
y � ��22090
�(x � 9)2 � �229�
h(t) � �9.1(t � 32.5)2 �
30,000; 30,000 ft
�2 � k � 2
1.2; two real,irrational roots
6 � 4�2�
{�3.5, 1}
��5 �8i�39���
{�0.35, 0.85}
x � �1
12x2 � 13x � 14 � 0
��12
�, �53
��
y
xO
between 1 and 2
y
xO
2
2
between �3 and �2;between 4 and 5
y
xO
2
2
3, 6
$8.00; $6400
minimum; �2225�
xO
f(x )
(0, 3)
f (x) � 3x 2 � 2x � 3
x � � 13
– , 13
83( )
An
swer
s
Chapter 6 Assessment Answer KeyForm 3Page 365 Page 366
y � ��35
��x � �72
�� 2� �
12
�;
���72
�, ��12
��; x � ��72
�;
down
© Glencoe/McGraw-Hill A28 Glencoe Algebra 2
Chapter 6 Assessment Answer KeyPage 367, Open-Ended Assessment
Scoring Rubric
Score General Description Specific Criteria
• Shows thorough understanding of the concepts ofgraphing, analyzing, and finding the maximum andminimum values of quadratic functions; solving quadraticequations; and solving inequalities.
• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Goes beyond requirements of some or all problems.
• Shows an understanding of the concepts of graphing,analyzing, and finding the maximum and minimum valuesof quadratic functions; solving quadratic equations; andsolving inequalities.
• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Satisfies all requirements of problems.
• Shows an understanding of most of the concepts ofgraphing, analyzing, and finding the maximum andminimum values of quadratic functions; solving quadraticequations; and solving inequalities.
• May not use appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are satisfactory.• Satisfies the requirements of most of the problems.
• Final computation is correct.• No written explanations or work is shown to substantiate
the final computation.• Satisfies minimal requirements of some of the problems.
• Shows little or no understanding of most of the conceptsof graphing, analyzing, and finding the maximum andminimum values of quadratic functions; solving quadraticequations; and solving inequalities.
• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Does not satisfy requirements of problems.• No answer may be given.
0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given
1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation
2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem
3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation
4 SuperiorA correct solution that is supported by well-developed, accurateexplanations
Chapter 6 Assessment Answer KeyPage 367, Open-Ended Assessment
Sample Answers
© Glencoe/McGraw-Hill A29 Glencoe Algebra 2
1. Student responses should indicate thatusing the Square Root Property, as Mi-Ling’s group did, would take lesstime than the other two methods sincethe equation is already set up as aperfect square set equal to a constant.To solve using either of the other twomethods, the binomial would need to beexpanded and the constant on the rightbrought to the left side of the equalsign.
2a. Jocelyn had trouble because theproblem is impossible. No suchparabola exists.
2b. Student responses will vary. One of thethree conditions must be omitted ormodified. Sample answer: Delete“...and passes through (�1, 0).”
2c. Answers will vary and depend on theanswer for part b. For example, for thesample answer in part b above, apossible equation is:y � �2(x � 3)2 � 4.
3a. Answer must be of the form y � a(x � h)2 � 8 where h is any realnumber and a � 0.
3b. Answers must be of the form y � a[x � (h � n)]2 � 8 where h and arepresent the same values as in part a.The student choice is for the value ofn. The student should indicate that thegraph will shift to the left n units ifhis or her value of n is negative, butwill shift the graph to the right n unitsif the chosen value of n is positive.
4. Students should indicate that Joseph’sanswer is not correct. In Step 2, whenhe completed the square by inserting�9 inside the parentheses, he actuallyadded 2(9) � 18 to the right side of theequation, so he must subtract 18 fromthe constant on the same side, ratherthan add 9, to keep the statementsequivalent. The correct solution is f(x) � 2(x � 3)2 � 23.
5a. �; The graph is strictly above the x-axis for all values of x other than 2.
5b. �; The graph is never below the x-axis.
5c. �; The graph is always on or above the x-axis.
In addition to the scoring rubric found on page A28, the following sample answers may be used as guidance in evaluating open-ended assessment items.
An
swer
s
© Glencoe/McGraw-Hill A30 Glencoe Algebra 2
Chapter 6 Assessment Answer KeyVocabulary Test/Review Quiz (Lessons 6–1 and 6–2) Quiz (Lessons 6–5 and 6–6)
Page 368 Page 369 Page 370
1. false; Zero ProductProperty
2. false; constant term
3. false; quadraticinequality
4. false; roots
5. true
6. false; minimumvalue
7. false; quadraticterm
8. false; (the)Quadratic Formula
9. true
10. false; discriminant
11. Sample answer: A parabola is asmooth curve thatis the graph of aquadratic function.
12. Sample answer: An axis of symmetryis a line along whichyou can fold a graphand get matchingparts on both sidesof the line.
1.
2.
3.
4.
5.
Quiz (Lessons 6–3 and 6–4)
Page 369
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
1.
2.
3.
4.
5.
Quiz (Lesson 6–7)
Page 370
1.
2.
3.
4. all reals
{x � x � 1 or x � 3}
y
xO
{x � 1 � x � 5}
y
xO
y � 2(x � 5)2
y � �3(x � 2)2 � 6
xO
y
(2, �1)
x � 2
�96; 2 complex roots
2 � �5�
{2 � i�14�}
{�1, 11}
���2 �5
�3���{1 � 3�5�}
{�10, 2}
3x2 � 10x � 8 � 0
x2 � 4x � 12 � 0
B
{�9, 5}
��5, �23
��
between 1 and 2;between �6 and �5
3, �1
minimum; 1
�3; x � �1; �1
xO
f(x )
(�1, �4) (0, �3)
f (x) � x 2 � 2x � 3
x � �1
© Glencoe/McGraw-Hill A31 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.y � �x � �
72
��2� �
249�
�3, 2 complex roots
{�2, 3}
y
xO
�1, 3
136 ft; 1.5 s
51
5.599
(2, �3)
92
(�2, 0), (�2, 8),(0, �2), (8, �2)
inconsistent
��34
�
17
���1 �3
�5����0, �
14
��{�2, 9}
minimum; �9�12
�
y
xO
1, 3
D
C
A
B
B
Chapter 6 Assessment Answer KeyMid-Chapter Test Cumulative ReviewPage 371 Page 372
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2x3 � x2 � 2x �
4 � �x �
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© Glencoe/McGraw-Hill A32 Glencoe Algebra 2
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4 9 / 4
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Chapter 6 Assessment Answer KeyStandardized Test PracticePage 373 Page 374