Quadratic Functions

and Inequalities
Chapter Overview and Pacing

PACING (days)
Regular Block
LESSON OBJECTIVES Basic/ Basic/
Average Advanced Average Advanced
Graphing Quadratic Functions (pp. 286–293) 1 1 0.5 0.5
• Graph quadratic functions.
• Find and interpret the maximum and minimum values of a quadratic function.
Solving Quadratic Equations by Graphing (pp. 294–300) 1 2 0.5 1
• Solve quadratic equations by graphing. (with 6-2
• Estimate solutions of quadratic equations by graphing. Follow-Up)
Follow-Up: Modeling Real-World Data
Solving Quadratic Equations by Factoring (pp. 301–305) 1 1 0.5 0.5
• Solve quadratic equations by factoring.
• Write a quadratic equation with given roots.
Completing the Square (pp. 306–312) 2 1 1 0.5
• Solve quadratic equations by using the Square Root Property.
• Solve quadratic equations by completing the square.
The Quadratic Formula and the Discriminant (pp. 313–319) 1 1 0.5 0.5
• Solve quadratic equations by using the Quadratic Formula.
• Use the discriminant to determine the number and type of roots of a quadratic equation.
Analyzing Graphs of Quadratic Functions (pp. 320–328) 2 1 1.5 0.5
Preview: Families of Parabolas (with 6-6 (with 6-6
• Analyze quadratic functions of the form y  a (x  h )2  k. Preview) Preview)
• Write a quadratic function in the form y  a (x  h )2  k.
Graphing and Solving Quadratic Inequalities (pp. 329–335) 1 1 0.5 0.5
• Graph quadratic inequalities in two variables.
• Solve quadratic inequalities in one variable.
Study Guide and Practice Test (pp. 336–341) 1 1 0.5 0.5
Standardized Test Practice (pp. 342–343)
Chapter Assessment 1 1 0.5 0.5
TOTAL 11 10 6 5

Pacing suggestions for the entire year can be found on pages T20–T21.

284A Chapter 6 Quadratic Functions and Inequalities

 Timesaving Tools
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All-In-One Planner
and Resource Center
Chapter Resource Manager See pages T12–T13.

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319–320 321–322 323 324 369 SC 11 6-2 6-2 (Follow-Up: graphing calculator)

325–326 327–328 329 330 6-3 6-3 grid paper

331–332 333–334 335 336 369, 371 6-4 6-4 algebra tiles

337–338 339–340 341 342 GCS 38 6-5 6-5 11, 12 posterboard

343–344 345–346 347 348 370 6-6 6-6 (Preview: graphing calculator)
graphing calculator, index cards

349–350 351–352 353 354 370 GCS 37, 6-7 6-7

SC 12

355–368,
372–374

*Key to Abbreviations: GCS  Graphing Calculator and Speadsheet Masters,

SC  School-to-Career Masters,
SM  Science and Mathematics Lab Manual

Chapter 6 Quadratic Functions and Inequalities 284B

 Mathematical Connections
and Background
Continuity of Instruction Graphing Quadratic Functions
When graphing a quadratic function, use all the
available information to produce the most accurate pos-
sible graph. This includes a table of values containing the
vertex and the y-intercept. Be sure the points on the
Prior Knowledge graph are connected with a smooth curve and that the
In the previous chapter students explored graph has a U-shape and not a V-shape at the vertex of
how to factor quadratic expressions, how to the graph. Unlike the letter U, however, the graph
should become progressively wider and have arrows
work with radicals, and how to perform
indicating that the graph continues to infinity.
operations on complex numbers. They have
also solved linear equations and inequalities,
Solving Quadratic Equations
and they are familiar with using formulas.
by Graphing
It is important to distinguish between finding
solutions or roots of an equation and finding zeros of its
related function. An equation of the form ax2  bx  c  0
has a related function, f(x)  ax2  bx  c. The zeros of
f(x) are the x-coordinates of the points where the graph
crosses the x-axis. These x values are the solutions of the
This Chapter related quadratic equation. Without the use of a graph-
Students solve quadratic equations by graph- ing calculator, this method of solving quadratic equa-
ing, by factoring, by completing the square, tions will usually provide only an estimate of solutions.
and by using the Quadratic Formula. They Solutions that appear to be integers should be verified
by substituting them into the original equation.
explore how values of a quadratic equation
are reflected in the parabola that represents
Solving Quadratic Equations
it, and use equations and graphs to explore
quadratic equations that have 0, 1, or 2 roots. by Factoring
They relate the value of the discriminant to When solving a quadratic equation by factoring,
it is important to review factoring techniques. These
the number of roots and to whether the roots
include the techniques for factoring general trinomials,
are rational, irrational, or complex. perfect square trinomials, and a difference of squares.
Also remember to look for a greatest common factor
(GCF) that might be factored out or the possibility of fac-
toring by grouping. Before factoring, the equation should
be rewritten so that one side of the equation is 0. If the
GCF of the terms of the polynomial being factored is a
variable or the product of a number and a variable, such
Future Connections as 3x, one solution to the equation is 0.
Students will continue to explore roots and
zeros of equations, examining higher-order Complete the Square
polynomial equations in Chapter 7. Students Completing the square is a technique most often
will learn to recognize other types of equa- used to solve quadratic equations that are not factorable.
tions that can be solved using the Quadratic To use this technique it is desirable to rewrite the equation
Formula. They will explore systems of quad- so that it is equal to a constant. Then divide the coefficient
of the linear term by 2 and square the result. Add this
ratic inequalities in Chapter 8. value to both sides of the equation. One side of the
equation will now be a perfect square trinomial that can
be rewritten as the square of a binomial. To help isolate the
variable on one side of the equation, take the square root

284C Chapter 6 Quadratic Functions and Inequalities

of both sides, remembering that the square root of the essence this would be an overall addition of 5  5 or 0,
constant on one side of the equation will result in two which does not affect the equality of the statement.
values, one positive and one negative, represented by a For example, y  3x is equivalent to the statement
 sign. Finally, isolate the variable using the Addition, y  3x  5  5. Be especially careful when adding a
Subtraction, Multiplication, and/or Division Properties value inside a set of parentheses, since the use of
of Equality. Simplify any solution that involves the parentheses often involves multiplication. For example
symbol  by simplifying two separate expressions, one y  3(x) is not equivalent to y  3(x  1)  1, but
using the plus sign and the other using the minus sign. instead to y  3(x  1)  3(1).
When using the technique of completing the square,
be sure that the coefficient of the quadratic term is 1.
If it is not 1, divide both sides of the equation by that
Graphing and Solving
coefficient. You can then solve the equation by com- Quadratic Inequalities
pleting the square. One way of solving a linear inequality not dis-
cussed in Chapter 1 is to first solve its related linear
The Quadratic Formula and equation and then test values on either side of this
value in the original inequality. For example, to solve
the Discriminant 3x  2  4, you would solve the equation 3x  2 
While the technique of completing the square 4 and find that x  2. Testing a value less than 2
can be used to solve any quadratic equation, imple- and a value greater than 2 in the inequality reveals
menting this technique can lead to operations involving that the solution to the inequality is the set of values
unwieldy fractions. The Quadratic Formula can also greater than 2. The approach to solving a quadratic
be used to solve any quadratic equation. Using this inequality algebraically is similar. The difference lies
formula, the variable is isolated in the very first step in the fact that many quadratic inequalities have not
and the other steps involve simplifying the solutions one but two solutions. This means that your number
by simplifying radicals and fractions. The discrimi- line is divided in three possible solution sets. Testing
nant is simply the portion of the Quadratic Formula a value from each interval on the number line reveals
that appears underneath the radical, b2  4ac. This which solution set or sets are correct. The solution set
value alone will determine the number and type of of a quadratic inequality will often be a compound
roots (solutions) of the equation. This is because the inequality, so you will want to review this topic from
square root of this value could result in a rational Chapter 1.
number, such as 6, an irrational number, such as 2,
the value 0, or an imaginary number, such as 3i. By
finding and examining just the value of the discrimi-
nant, you can tell very quickly what type of solutions a www.algebra2.com/key_concepts
quadratic equation will have. This can serve as a
check when solving the equation. Additional mathematical information and teaching notes
are available in Glencoe’s Algebra 2 Key Concepts:
Analyzing Graphs of Mathematical Background and Teaching Notes,
which is available at www.algebra2.com/key_concepts.
Quadratic Functions The lessons appropriate for this chapter are as follows.
To write a quadratic equation in the form
• Solving Quadratic Equations by Graphing (Lesson 31)
y  a(x  h)2  k, called vertex form, it is important to
remember that an equation is a statement of equality. • Solving Equations by Factoring (Lesson 27)
When an equation is rewritten in a different form, this • Solving Quadratic Equations by Completing the
equality must be maintained. In previous lessons, if a Square (Lesson 39)
value was added to one side of an equation, it was also • Solving Quadratic Equations by Using the Quadratic
added to the other side of the equation in order to main- Formula (Lesson 32)
tain equality. Another way to maintain equality is to • Graphing Technology: Parent and Family Graphs
add a value to one side and then subtract that same (Lesson 29)
value from that side. For example, if an addition of 5 is • Graphing Quadratic Functions (Lesson 28)
shown on the right side of an equation, a subtraction • More on Axis of Symmetry and Vertices (Lesson 30)
of 5 would also be shown on the right side. So in

Chapter 6 Quadratic Functions and Inequalities 284D

 and Assessment

Type Student Edition Teacher Resources Technology/Internet

Ongoing Prerequisite Skills, pp. 285, 293, 5-Minute Check Transparencies Alge2PASS: Tutorial Plus
INTERVENTION

299, 305, 312, 319, 328 Quizzes, CRM pp. 369–370 www.algebra2.com/self_check_quiz
Practice Quiz 1, p. 305 Mid-Chapter Test, CRM p. 371 www.algebra2.com/extra_examples
Practice Quiz 2, p. 328 Study Guide and Intervention, CRM pp. 313–314,
319–320, 325–326, 331–332, 337–338, 343–344,
349–350
Mixed pp. 293, 299, 305, 312, 319, Cumulative Review, CRM p. 372
Review 328, 335
Error Find the Error, pp. 303, 310, 325 Find the Error, TWE pp. 303, 310, 325
Analysis Common Misconceptions, Unlocking Misconceptions, TWE pp. 288, 295
pp. 289, 308 Tips for New Teachers, TWE pp. 288, 305, 312, 323
Standardized pp. 292, 293, 299, 302, 303, TWE p. 302 Standardized Test Practice
Test Practice 305, 312, 319, 327, 335, 341, Standardized Test Practice, CRM pp. 373–374 CD-ROM
342–343 www.algebra2.com/
standardized_test
Open-Ended Writing in Math, pp. 292, 299, Modeling: TWE pp. 299, 319
Assessment 305, 312, 319, 327, 334 Speaking: TWE pp. 293, 305, 335
Open Ended, pp. 290, 297, 303, Writing: TWE pp. 312, 328
ASSESSMENT

317, 325, 332 Open-Ended Assessment, CRM p. 367

Chapter Study Guide, pp. 336–340 Multiple-Choice Tests (Forms 1, 2A, 2B), TestCheck and Worksheet Builder
Assessment Practice Test, p. 341 CRM pp. 355–360 (see below)
Free-Response Tests (Forms 2C, 2D, 3), MindJogger Videoquizzes
CRM pp. 361–366 www.algebra2.com/
Vocabulary Test/Review, CRM p. 368 vocabulary_review
www.algebra2.com/chapter_test

Key to Abbreviations: TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters

Additional Intervention Resources TestCheck and Worksheet Builder

The Princeton Review’s Cracking the SAT & PSAT This networkable software has three modules for intervention
The Princeton Review’s Cracking the ACT and assessment flexibility:
• Worksheet Builder to make worksheet and tests
ALEKS • Student Module to take tests on screen (optional)
• Management System to keep student records (optional)
Special banks are included for SAT, ACT, TIMSS, NAEP, and
End-of-Course tests.

284E Chapter 6 Quadratic Functions and Inequalities

 Reading and Writing
in Mathematics
Intervention Technology Glencoe Algebra 2 provides numerous opportunities to
Alge2PASS: Tutorial Plus CD-ROM offers a incorporate reading and writing into the mathematics
complete, self-paced algebra curriculum. classroom.

Algebra 2 Alge2PASS Lesson Student Edition

Lesson
• Foldables Study Organizer, p. 285
6-1 10 Graphing Quadratic Equations • Concept Check questions require students to verbalize
6-5 11 Solving Quadratic Equations Using the and write about what they have learned in the lesson.
Quadratic Formula (pp. 290, 297, 303, 310, 317, 325, 332, 336)
6-5 12 Solving Word Problems Using Quadratic • Writing in Math questions in every lesson, pp. 292, 299,
Equations 305, 312, 319, 327, 334
• Reading Study Tip, pp. 306, 313, 316
ALEKS is an online mathematics learning system that • WebQuest, p. 328
adapts assessment and tutoring to the student’s needs.
Subscribe at www.k12aleks.com. Teacher Wraparound Edition
• Foldables Study Organizer, pp. 285, 336
• Study Notebook suggestions, pp. 290, 297, 303, 310,
Intervention at Home 317, 325, 332
• Modeling activities, pp. 299, 319
Log on for student study help. • Speaking activities, pp. 293, 305, 335
• For each lesson in the Student Edition, there are Extra • Writing activities, pp. 312, 328
Examples and Self-Check Quizzes. • Differentiated Instruction, (Verbal/Linguistic), p. 296
www.algebra2.com/extra_examples • ELL Resources, pp. 284, 292, 296, 298, 304, 311,
www.algebra2.com/self_check_quiz 318, 327, 334, 336
• For chapter review, there is vocabulary review, test
practice, and standardized test practice. Additional Resources
www.algebra2.com/vocabulary_review
• Vocabulary Builder worksheets require students to
www.algebra2.com/chapter_test
define and give examples for key vocabulary terms as
www.algebra2.com/standardized_test
they progress through the chapter. (Chapter 6 Resource
Masters, pp. vii-viii)
• Reading to Learn Mathematics master for each lesson
For more information on Intervention and (Chapter 6 Resource Masters, pp. 317, 323, 329, 335,
Assessment, see pp. T8–T11. 341, 347, 353)
• Vocabulary PuzzleMaker software creates crossword,
jumble, and word search puzzles using vocabulary lists
that you can customize.
• Teaching Mathematics with Foldables provides
suggestions for promoting cognition and language.
• Reading and Writing in the Mathematics Classroom
• WebQuest and Project Resources

For more information on Reading and Writing in

Mathematics, see pp. T6–T7.
Chapter 6 Quadratic Functions and Inequalities 284F
 Notes Quadratic Functions
and Inequalities
Have students read over the list
of objectives and make a list of
any words with which they are
not familiar. • Lesson 6-1 Graph quadratic functions.
Key Vocabulary
• Lessons 6-2 through 6-5 Solve quadratic equations. • root (p. 294)
• Lesson 6-3 Write quadratic equations and • zero (p. 294)
functions. • completing the square (p. 307)
• Lesson 6-6 Analyze graphs of quadratic functions. • Quadratic Formula (p. 313)
• Lesson 6-7 Graph and solve quadratic inequalities. • discriminant (p. 316)
Point out to students that this is
only one of many reasons why
each objective is important.
Others are provided in the
introduction to each lesson. Quadratic functions can be used to model real-world
phenomena like the motion of a falling object. They
can also be used to model the shape of architectural
structures such as the supporting cables of a
suspension bridge. You will learn to calculate
the value of the discriminant of a quadratic
equation in order to describe the position
of the supporting cables of the
Golden Gate Bridge in
Lesson 6-5.

NCTM Local
Lesson Standards Objectives
6-1 2, 3, 6, 8, 9, 10
6-2 1, 2, 6, 8, 9, 10
6-2 2, 5, 6, 8, 10
Follow-Up
6-3 1, 2, 3, 6, 7, 8,
9
6-4 1, 2, 3, 6, 7, 8,
9, 10
6-5 1, 2, 6, 8, 9
284 Chapter 6 Quadratic Functions and Inequalities
6-6 2, 8, 10
Preview
6-6 2, 6, 7, 8, 9, 10
6-7 2, 3, 6, 8, 9, 10
Vocabulary Builder ELL

The Key Vocabulary list introduces students to some of the main vocabulary terms
Key to NCTM Standards: included in this chapter. For a more thorough vocabulary list with pronunciations of
1=Number & Operations, 2=Algebra, new words, give students the Vocabulary Builder worksheets found on pages vii and
3=Geometry, 4=Measurement, viii of the Chapter 6 Resource Masters. Encourage them to complete the definition
5=Data Analysis & Probability, 6=Problem of each term as they progress through the chapter. You may suggest that they add
Solving, 7=Reasoning & Proof, these sheets to their study notebooks for future reference when studying for the
8=Communication, 9=Connections, Chapter 6 test.
10=Representation
284 Chapter 6 Quadratic Functions and Inequalities
Prerequisite Skills To be successful in this chapter, you’ll need to master This section provides a review of
these skills and be able to apply them in problem-solving situations. Review the basic concepts needed before
these skills before beginning Chapter 6.
beginning Chapter 6. Page
For Lessons 6-1 and 6-2 Graph Functions
references are included for
additional student help.
Graph each equation by making a table of values. (For review, see Lesson 2-1.)
1. y  2x  3 2. y  x  5 3. y  x2  4 4. y  x2  2x  1 Prerequisite Skills in the Getting
1–4. See pp. 343A-343F. Ready for the Next Lesson section
For Lessons 6-1, 6-2, and 6-5 Multiply Polynomials
at the end of each exercise set
Find each product. (For review, see Lesson 5-2.)
review a skill needed in the next
5. (x  4)(7x  12) 6. (x  5)2 7. (3x  1)2 8. (3x  4)(2x  9) lesson.
7x 2  16x  48 x2  10x  25 9x 2  6x  1 6x 2  35x  36
For Lessons 6-3 and 6-4 Factor Polynomials
For Prerequisite
Factor completely. If the polynomial is not factorable, write prime. (For review, see Lesson 5-4.)
Lesson Skill
9. x2  11x  30 10. x2  13x  36 11. x2  x  56 12. x2  5x  14
6-2 Evaluating Functions (p. 293)
13. x2 x2 14. x2  10x  25 15. x2  22x  121 16. x2  9
prime (x  5) 2 (x  11) 2 (x  3)(x  3) 6-3 Factoring Trinomials (p. 299)
For Lessons 6-4 and 6-5 Simplify Radical Expressions 6-4 Simplifying Radicals (p. 305)
Simplify. (For review, see Lessons 5-6 and 5-9.) 6-5 Evaluating Expressions (p. 312)
 15
17. 225  43
18. 48  65
19. 180 20. 68 
 217 6-6 Perfect Square Trinomials
 5i
21. 25  4i 2
22. 32 23. 270  24. 15
 3i 30 
 i 15 (p. 319)
9. (x  6)(x  5) 10. (x  4)(x  9) 11. (x  8)(x  7) 12. (x  2)(x  7) 6-7 Inequalities (p. 328)

Make this Foldable to record information about quadratic

functions and inequalities. Begin with one sheet of 11"  17"
paper.
Fold and Cut Refold and Label

Fold in half lengthwise. Refold along lengthwise

Then fold in fourths crosswise. fold and staple uncut section
Cut along the middle fold at top. Label the section with
from the edge to the last a lesson number and close
crease as shown. to form a booklet.

6-1 6-2 6-3 6-6 6-7

Vo
c
5

ab
6-

Reading and Writing As you read and study the chapter, fill the journal with
notes, diagrams, and examples for each lesson.

Chapter 6 Quadratic Functions and Inequalities 285

TM

Sequencing Information and Progression of Knowledge

After students make their Foldable, have them label a section for
each lesson in Chapter 6 and a section for vocabulary. As students
For more information progress through the lessons, have them summarize key concepts
about Foldables, see and note the order in which they are presented. Ask students to
Teaching Mathematics write about why the concepts were presented in this sequence. If
with Foldables. they cannot see the logic in the sequence, have them reorder the
key concepts and justify their reasoning.

Chapter 6 Quadratic Functions and Inequalities 285

 Lesson Graphing Quadratic Functions
Notes

• Graph quadratic functions.

1 Focus • Find and interpret the maximum and

minimum values of a quadratic function.
C06-15p
Vocabulary
5-Minute Check can income from a rock
• quadratic function
Transparency 6-1 Use as a • quadratic term
concert be maximized? Rock Concert Income
quiz or a review of Chapter 5. • linear term Rock music managers handle publicity

Income (thousands of dollars)

P (x )
• constant term and other business issues for the artists 80
Mathematical Background notes • parabola they manage. One group’s manager has
are available for this lesson on • axis of symmetry found that based on past concerts, the 60

p. 284C. • vertex predicted income for a performance is

• maximum value P(x)  50x2  4000x  7500, where x is 40
• minimum value the price per ticket in dollars. The graph
Building on Prior of this quadratic function is shown 20

Knowledge at the right. Notice that at first the

income increases as the price per ticket
0 20 40 60 80 x
In Chapter 5, students wrote and increases, but as the price continues
Ticket Price (dollars)
to increase, the income declines.
solved various equations and
inequalities. In this lesson, they
will relate quadratic equations to
GRAPH QUADRATIC FUNCTIONS A quadratic function is described by an
their graphs. equation of the following form.

can income from a rock linear term

concert be maximized?
Ask students: f(x)  ax2  bx  c, where a  0
• How is the income represented
quadratic term constant term
in the given function? by P(x)
• What is significant about the The graph of any quadratic function is called a parabola . One way to graph a
value of P(x) when x  40 (the quadratic function is to graph ordered pairs that satisfy the function.
ticket price of $40)? The value of
P(x) is greatest when x  40, or Example 1 Graph a Quadratic Function
the income is at its maximum Graph f(x)  2x2  8x  9 by making a table of values.
value when x  40. First, choose integer values for x. Then, evaluate the function for each x value.
Graph the resulting coordinate pairs and connect the points with a smooth curve.
f (x )
x 2x 2 ⴚ 8x ⴙ 9 f (x) (x, f (x))
0 2(0)2  8(0)  9 9 (0, 9)
1 2(1)2  8(1)  9 3 (1, 3)
2 2(2)2  8(2)  9 1 (2, 1)
3 2(3)2  8(3)  9 3 (3, 3)
4 2(4)2  8(4)  9 9 (4, 9)
f (x )  2x 2  8x  9
O x

286 Chapter 6 Quadratic Functions and Inequalities

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 6 Resource Masters Teaching Algebra With Manipulatives 5-Minute Check Transparency 6-1
• Study Guide and Intervention, pp. 313–314 Masters, p. 243 Answer Key Transparencies
• Skills Practice, p. 315
• Practice, p. 316 Technology
• Reading to Learn Mathematics, p. 317 Alge2PASS: Tutorial Plus, Lesson 10
• Enrichment, p. 318 Interactive Chalkboard
 TEACHING TIP All parabolas have an axis of symmetry . If you were to fold a
Tell students they will
derive the equation for
parabola along its axis of symmetry, the portions of the parabola
on either side of this line would match.
y
2 Teach
the axis of symmetry in The point at which the axis of symmetry intersects a parabola
Lesson 6-6, Exercise 53, is called the vertex . The y-intercept of a quadratic function, GRAPH QUADRATIC
after they have learned the equation of the axis of symmetry, and the x-coordinate FUNCTIONS
about a technique called of the vertex are related to the equation of the function as
completing the square. shown below.
In-Class Examples Power
Point®

1 Graph f(x)  x2  3x  1 by
O x making a table of values.
x 3 2 1 0 1
Graph of a Quadratic Function f(x) 1 3 3 1 3
• Words Consider the graph of y  ax2  bx  c, where a  0.
f (x )
• The y-intercept is a(0)2  b(0)  c or c.
b
• The equation of the axis of symmetry is x  .
2a
b
• The x-coordinate of the vertex is .
2a
• Model y O x

b
axis of symmetry: x  
2a
y – intercept: c f (x)  x 2  3x  1
O x

vertex
Teaching Tip Make sure
students understand that the
graph shows values for all the
points that satisfy the function,
Knowing the location of the axis of symmetry, y-intercept, and vertex can help even when the x value is not an
you graph a quadratic function. integer. For example, the vertex
is (1.5, 3.25).
Example 2 Axis of Symmetry, y-Intercept, and Vertex
Consider the quadratic function f(x)  x2  9  8x.
2 Consider the quadratic
function f(x)  2  4x  x2.
a. Find the y-intercept, the equation of the axis of symmetry, and the
x-coordinate of the vertex. a. Find the y-intercept, the equa-
Begin by rearranging the terms of the function so that the quadratic term is first, tion of the axis of symmetry,
the linear term is second, and the constant term is last. Then identify a, b, and c. and the x-coordinate of the
f(x)  ax2  bx  c vertex. 2; x  2; 2
f(x)  x2  9  8x → f(x)  1x2  8x  9
b. Make a table of values that
includes the vertex.
So, a  1, b  8, and c  9.
x 0 1 2 3 4
The y-intercept is 9. You can find the equation of the axis of symmetry using
a and b. f(x) 2 1 2 1 2
b
x   Equation of the axis of symmetry c. Use this information to graph
2a
8
x   a  1, b  8
the function.
2(1)
f (x )
x  4 Simplify.

The equation of the axis of symmetry is x  4. Therefore, the x-coordinate of

the vertex is 4. (4, 2)

www.algebra2.com/extra_examples Lesson 6-1 Graphing Quadratic Functions 287 O (3, –1) x

(2, –2)

x2

Lesson 6-1 Graphing Quadratic Functions 287

 Study Tip b. Make a table of values that includes the vertex.
Choose some values for x that are less than 4 and some that are greater than
Intervention Symmetry 4. This ensures that points on each side of the axis of symmetry are graphed.
New When discuss- Sometimes it is convenient
to use symmetry to help
ing Example 2, find other points on the x x2 ⴙ 8x ⴙ 9 f (x) (x, f (x))
make sure graph of a parabola. Each 6 (6)2  8(6)  9 3 (6, 3)
point on a parabola has a
students realize mirror image located the
5 (5)2  8(5)  9 6 (5, 6)
that f(x) and y can be used same distance from the 4 (4)2  8(4)  9 7 (4, 7) Vertex
interchangeably, and also that axis of symmetry on
3 (3)2  8(3)  9 6 (3, 6)
the other side of the
the maximum or minimum parabola. 2 (2)2  8(2)  9 3 (2, 3)
value of the function is given
by the y-coordinate of the f (x )
3 3
c. Use this information to graph the function.
vertex of the parabola.
2 2 Graph the vertex and y-intercept. Then graph f (x ) (0, 9)
the points from your table connecting them 8
and the y-intercept with a smooth curve. As a x  4
O x
check, draw the axis of symmetry, x  4, 4
MAXIMUM AND as a dashed line. The graph of the function
MINIMUM VALUES should be symmetrical about this line. 12 8 4 O 4x
4
In-Class Example Power
Point® (4, 7) 8

3 Consider the function

f(x)  x2  2x  3.
a. Determine whether the
function has a maximum or a MAXIMUM AND MINIMUM VALUES The y-coordinate of the vertex of
a quadratic function is the maximum value or minimum value obtained by the
minimum value. maximum function.
b. State the maximum or mini-
mum value of the function. 4
Maximum and Minimum Value
Teaching Tip Remind students • Words The graph of f(x)  ax2  bx  c, where a  0,
to use the equation for the axis • opens up and has a minimum value when a  0, and
of symmetry to determine the
• opens down and has a maximum value when a  0.
x-coordinate of the vertex and
then to find the y-coordinate of • Models a is positive. a is negative.
the vertex.

Example 3 Maximum or Minimum Value

Consider the function f(x) ⴝ x2 ⴚ 4x ⴙ 9.
Interactive
a. Determine whether the function has a maximum or a minimum value.
Chalkboard
PowerPoint®
For this function, a  1, b  4, and c  9. Since a  0, the graph opens up and
Presentations the function has a minimum value.
288 Chapter 6 Quadratic Functions and Inequalities
This CD-ROM is a customizable
Microsoft® PowerPoint®
presentation that includes:
• Step-by-step, dynamic solutions of Unlocking Misconceptions
each In-Class Example from the
Minimum and Maximum Values Make sure students understand
Teacher Wraparound Edition
that a parabola which opens upward is the graph of a function with a
• Additional, Your Turn exercises for minimum value and that a parabola which opens downward is the
each example graph of a function with a maximum value. Compare these parabolas to
• The 5-Minute Check Transparencies valleys (where the altitude of the valley floor is a minimum) and hills
• Hot links to Glencoe Online (where the peak of the hill is the maximum altitude).
Study Tools

288 Chapter 6 Quadratic Functions and Inequalities

Study Tip b. State the maximum or minimum value of the f (x )
In-Class Example Power
function. Point®
Common 12
Misconception The minimum value of the function is the
The terms minimum point y-coordinate of the vertex. 4 ECONOMICS A souvenir shop
8
and minimum value are 4 sells about 200 coffee mugs
The x-coordinate of the vertex is  or 2.
not interchangeable. The 2(1) 4 each month for $6 each. The
minimum point on the f (x )  x 2  4x  9
Find the y-coordinate of the vertex by evaluating shop owner estimates that for
graph of a quadratic the function for x  2. 4 O 4 8 x
function is the set of each $0.50 increase in the
coordinates that describe f(x)  x2  4x  9 Original function price, he will sell about 10
the location of the vertex.
The minimum value of a f(2)  (2)2  4(2)  9 or 5 x  2 fewer coffee mugs per month.
function is the y-coordinate
Therefore, the minimum value of the function is 5. a. How much should the owner
of the minimum point. It is
the smallest value obtained charge for each mug in order
when f(x) is evaluated for to maximize the monthly
all values of x.
When quadratic functions are used to model real-world situations, their income from their sales? $8
maximum or minimum values can have real-world meaning. b. What is the maximum
monthly income the owner
Example 4 Find a Maximum Value can expect to make from
FUND-RAISING Four hundred people came to last year’s winter play at these items? $1280
Sunnybrook High School. The ticket price was $5. This year, the Drama Club
is hoping to earn enough money to take a trip to a Broadway play. They
estimate that for each $0.50 increase in the price, 10 fewer people will attend
their play.
a. How much should the tickets cost in order to maximize the income from this
year’s play?

Words The income is the number of tickets multiplied by the price per
ticket.
Variables Let x  the number of $0.50 price increases.
Then 5  0.50x  the price per ticket and
400  10x  the number of tickets sold.
Let I(x) = income as a function of x.
The the number multiplied the price
income is of tickets by per ticket.







Equation I(x)  (400  10x) (5  0.50x)

 400(5)  400(0.50x)  10x(5)  10x(0.50x)
Fund-Raising  2000  200x  50x  5x2 Multiply.

The London Marathon,  2000  150x  5x2 Simplify.

which has been run  5x2  150x  2000 Rewrite in ax2  bx  c form.
through the streets of
London, England, annually
I(x) is a quadratic function with a  5, b  150, and c  2000. Since
since 1981, has historically
raised more money than a  0, the function has a maximum value at the vertex of the graph.
any other charity sports Use the formula to find the x-coordinate of the vertex.
event. In 2000, this event b
x-coordinate of the vertex   Formula for the x-coordinate of the vertex
raised an estimated £20 2a
million ($31.6 million U.S. 150
  a  5, b  150
dollars). 2(5)
Source: Guinness World Records  15 Simplify.

This means the Drama Club should make 15 price increases of $0.50 to
maximize their income. Thus, the ticket price should be 5  0.50(15) or $12.50.
(continued on the next page)
Lesson 6-1 Graphing Quadratic Functions 289

Differentiated Instruction
Auditory/Musical Ask students to suggest the kinds of musical sounds
they might associate with a parabola that has a maximum value, and
have them contrast this to a sound that might be associated with a
parabola that has a minimum value. For example, an orchestra piece that
rises to a crescendo and then gradually returns to the previous volume
might be seen as being related to a parabola with a maximum value.

Lesson 6-1 Graphing Quadratic Functions 289

 b. What is the maximum income the Drama Club can expect to make?
3 Practice/Apply To determine maximum income, find the maximum value of the function by
evaluating I(x) for x  15.

I(x)  5x2  150x  2000 Income function

Study Notebook I(15)  5(15)2  150(15)  2000 x  15

 3125 Use a calculator.
Have students—
• add the definitions/examples of Thus, the maximum income the Drama Club can expect is $3125.
the vocabulary terms to their CHECK Graph this function on a graphing
Vocabulary Builder worksheets for calculator, and use the CALC
menu to confirm this solution.
Chapter 6.
KEYSTROKES: 2nd [CALC] 4
• add the information in the Key
0 ENTER 25 ENTER ENTER
Concept box on p. 287 about the
graph of a quadratic function to At the bottom of the display are the
[5, 50] scl: 5 by [100, 4000] scl: 500
coordinates of the maximum point on
their notebook. the graph of y  5x2  150x  2000.
• make labeled sketches similar to The y value of these coordinates is the
maximum value of the function, or 3125. ⻫
those on p. 288 illustrating the
maximum and minimum values of
the graphs of quadratic functions.
• include any other item(s) that they
find helpful in mastering the skills Concept Check 1. OPEN ENDED Give an example of a quadratic function. Identify its quadratic
in this lesson. 1. Sample answer: term, linear term, and constant term.
f (x)  3x 2  5x ⴚ 6; 2. Identify the vertex and the equation of the axis of symmetry for each function
3 x 2, 5x, ⴚ6 graphed below. a. (2, 1); x  2 b. (3, 2); x  3
About the Exercises… a. f (x ) b. f (x )

Organization by Objective O x
• Graph Quadratic
Functions: 14–31, 44
• Maximum and Minimum
Values: 32–43, 45–53 O x

Odd/Even Assignments
Exercises 14–31 and 32–43 are 3. State whether the graph of each quadratic function opens up or down. Then state
structured so that students whether the function has a maximum or minimum value.
practice the same concepts a. f(x)  3x2  4x  5 up; min. b. f(x)  2x2  9 down; max.
whether they are assigned c. f(x)  5x2  8x  2 down; max. d. f(x)  6x2  5x up; min.
odd or even problems.
Alert! Exercises 58–63 require a Guided Practice Complete parts a–c for each quadratic function.
graphing calculator. GUIDED PRACTICE KEY a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate
Exercises Examples
of the vertex.
Assignment Guide 4–9 1, 2
b. Make a table of values that includes the vertex.
Basic: 15–27 odd, 33–43 odd, 10–12 3 c. Use this information to graph the function. 4–9. See margin.
44–47, 54–57, 64–78 13 4 4. f(x)  4x2 5. f(x)  x2  2x
Average: 15–43 odd, 46–50, 6. f(x)  x2  4x  1 7. f(x)  x2  8x  3
53–57, 64–78 (optional: 58–63) 8. f(x)  2x2  4x  1 9. f(x)  3x2  10x
Advanced: 14–42 even, 46–74
290 Chapter 6 Quadratic Functions and Inequalities
(optional: 75–78)

Answers 4c. f (x) 5a. 0; x  1; 1 5c. f (x)

4a. 0; x  0; 0 5b. x f(x)

O (0, 0)
4b. x f(x) x 3 3
1 4 f (x)  4x 2 2 0
0 0 1 1
O x
1 4 0 0 f (x)  x 2  2x
(1, 1)
1 3

290 Chapter 6 Quadratic Functions and Inequalities

 Determine whether each function has a maximum or a minimum value. Then find Answers
the maximum or minimum value of each function.
10. f(x)  x2  7 11. f(x)  x2  x  6 12. f(x)  4x2  12x  9 8a. 1; x  1; 1
25
max.; 7 min.;  min.; 0 8b. x f(x)
4
Application 13. NEWSPAPERS Due to increased production costs,
the Daily News must increase its subscription 1 7
rate. According to a recent survey, the number of
subscriptions will decrease by about 1250 for each Subscription Rate 0 1
25¢ increase in the subscription rate. What weekly $7.50/wk 1 1
subscription rate will maximize the newspaper's
income from subscriptions? $8.75 Current Circulation 2 1
50,000 3 7
8c. f (x)
★ indicates increased difficulty
Practice and Apply
Homework Help Complete parts a–c for each quadratic function.
For See a. Find the y-intercept, the equation of the axis of symmetry, and the f (x)  2x 2  4x  1
Exercises Examples
14–19 1
x-coordinate of the vertex.
O (1, 1) x
20–31 2 b. Make a table of values that includes the vertex.
32–43, 54 3
44–53 4
c. Use this information to graph the function. 14–31. See pp. 343A–343F.
14. f(x)  2x2 15. f(x)  5x2
Extra Practice 5 5
See page 839. 16. f(x)  x2 4 17. f(x)  x2  9 9a. 0; x    ;  
3 3
18. f(x)  2x2  4 19. f(x)  3x2  1 9b. x f(x)
20. f(x)  x2  4x  4 21. f(x)  x2  9x  9
3 3
22. f(x)  x2  4x  5 23. f(x)  x2  12x  36
2 8
24. f(x)  3x2  6x  1 25. f(x)  2x2  8x  3
5 25
26. f(x)  3x2  4x 27. f(x)  2x2  5x 3 3
★ 28. f(x)  0.5x2  1 ★ 29. f(x)  0.25x2  3x 1 7
★ 30. f(x)  12x2  3x  92 ★ 31. f(x)  x2  23x  89 0 0
9c. f (x)
Determine whether each function has a maximum or a minimum value. Then find 4
the maximum or minimum value of each function.
32. f(x)  3x2 min.; 0 33. f(x)  x2  9 max.; 9 4 2 O 2 x
34. f(x)  x2  8x  2 min.; 14 35. f(x)  x2  6x  2 min.; 11 4
36. f(x)  4x  x2  1 max.; 5 37. f(x)  3  x2  6x max.; 12
9 7 8
38. f(x)  2x  2x2  5 min.;  39. f(x)  x  2x2  1 max.;  f (x)  3x 2  10x
Architecture 2 8 ( 53 ,  253)
40. f(x)  7  3x2  12x max.; 5 41. f(x)  20x  5x2  9 min.; 11 12
The Exchange House in
1 3 1
London, England, is 42. f(x)  x2  2x  3 max.; 5 43. f(x)  x2  5x  2 min.; 10
supported by two interior 2 4 3
and two exterior steel
arches. V-shaped braces ARCHITECTURE For Exercises 44 and 45, use the following information.
add stability to the The shape of each arch supporting the Exchange House can be modeled
structure. by h(x)  0.025x2  2x, where h(x) represents the height of the arch and
Source: Council on Tall Buildings x represents the horizontal distance from one end of the base in meters.
and Urban Habitat
44. Write the equation of the axis of symmetry, and find the coordinates of the
vertex of the graph of h(x). x  40; (40, 40)
45. According to this model, what is the maximum height of the arch? 40 m
www.algebra2.com/self_check_quiz Lesson 6-1 Graphing Quadratic Functions 291

6a. 1; x  2; 2 6c. f (x) 7a. 3; x  4; 4 7c. f (x)

6b. x f(x) 2
f (x)  x  4x  1 7b. x f(x)
(2, 3) 10 8 4 O x
0 1 6 9 4
1 2 5 12
8
2 3 O x 4 13
2
3 2 3 12 f (x)  x  8x  3 12
(4, 13)
4 1 2 9

Lesson 6-1 Graphing Quadratic Functions 291

 Study Guide
NAME ______________________________________________ DATE

andIntervention
Intervention,
____________ PERIOD _____
PHYSICS For Exercises 46 and 47, use the following information.
6-1 Study Guide and
p. 313 (shown) and p. 314 An object is fired straight up from the top of a 200-foot tower at a velocity of
Graphing Quadratic Functions
Graph Quadratic Functions
80 feet per second. The height h(t) of the object t seconds after firing is given by
Quadratic Function A function defined by an equation of the form f (x)  ax 2  bx  c, where a  0 h(t)  16t2  80t  200.
b
A parabola with these characteristics: y intercept: c ; axis of symmetry: x   ;
Graph of a Quadratic
Function x-coordinate of vertex: 
b
2a
2a
46. Find the maximum height reached by the object and the time that the height
Example
is reached. 300 ft, 2.5 s
Find the y-intercept, the equation of the axis of symmetry, and the
x-coordinate of the vertex for the graph of f(x)  x2  3x  5. Use this information

Lesson 6-1
to graph the function. 47. Interpret the meaning of the y-intercept in the context of this problem.
a  1, b  3, and c  5, so the y-intercept is 5. The equation of the axis of symmetry is
(3)
x 3
or 
2(1)
3
. The x-coordinate of the vertex is 
2
.
2
The y-intercept is the initial height of the object.
3
Next make a table of values for x near  .
2
x ft
x
0
x 2  3x  5
02  3(0)  5
f(x)
5
(x, f(x))
(0, 5)
f (x )
CONSTRUCTION For Exercises 48–50, use the following
1 12 3(1)  5 3 (1, 3) information. 49. 60 ft by 30 ft
 23   32   32 , 4 
3 2 11 11
 3  5 
2
2 22  3(2)  5
4
3 (2, 3)
Steve has 120 feet of fence to make a rectangular kennel for his
O x
3 32  3(3)  5 5 (3, 5) dogs. He will use his house as one side.
Exercises 48. Write an algebraic expression for the kennel’s length. 120 – 2x
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 49. What dimensions produce a kennel with the greatest area?
of the vertex.

50. Find the maximum area of the kennel. 1800 ft2

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 x ft
8, x  3, 3 2, x  1, 1 3, x  1, 1
x 3 2 1 4 x 1 0 2 1 x 1 0 2 3
f (x) 1 0 3 0 f (x) 3 2 2 1 f (x) 1 3 3 9

12
f (x ) f (x ) f (x ) TOURISM For Exercises 51 and 52, use the following information.
(–1, 3) 4 12
8
–8 –4 O 4 8x
8
A tour bus in the historic district of Savannah, Georgia, serves 300 customers a day.
4
–4 4
(1, 1)
The charge is $8 per person. The owner estimates that the company would lose
–8 –4 O 4 x
(–3, –1)
–4
–8
–4 O 4 8 x
20 passengers a day for each $1 fare increase.
Gl NAME
/M G ______________________________________________
Hill 313 DATE ____________
GlPERIOD
Al _____
b 2
51. What charge would give the most income for the company? $11.50
Skills
6-1 Practice,
Practice (Average)
p. 315 and 52. If the company raised their fare to this price, how much daily income should
Practice, p. 316Functions
(shown)
Graphing Quadratic they expect to bring in? $2645
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.

★ 53. GEOMETRY A rectangle is inscribed in an isosceles

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  1

15; x  4; 4 12; x  2; 2 1; x  0.5; 0.5 Tourism triangle as shown. Find the dimensions of the inscribed
x 0 2 4 6 8 x 6 4 2 0 2 x 1 0 0.5 1 2 Known as the Hostess City rectangle with maximum area. (Hint: Use similar
f (x) 15 3 1 3 15 f (x) 0 12 16 12 0 f (x) 5 1 0.5 1 5 8 in.
of the South, Savannah,
f (x ) f (x ) f (x ) triangles.) 5 in. by 4 in.
16

12
(–2, 16)
16 Georgia, is a popular
8
12
tourist destination. One of
8
4 the first planned cities in 10 in.
4
O 2 4 6
(4, –1)
8x
–6 –4 –2 O 2x O
(0.5, 0.5)
x
the Americas, Savannah’s 54. CRITICAL THINKING Write an expression for
Historic District is based on the minimum value of a function of the form
Determine whether each function has a maximum or a minimum value. Then find a grid-like pattern of streets y  ax2  c, where a  0. Explain your reasoning.
the maximum or minimum value of each function.
and alleys surrounding Then use this function to find the minimum value
4. f(x)  x2  2x  8 5. f(x)  x2  6x  14 6. v(x)  x2  14x  57
min.; 9 min.; 5 max.; 8 open spaces called squares. of y  8.6x2  12.5. c; See margin for explanation; 12.5.
7. f(x)  2x2  4x  6 8. f(x)  x2  4x  1
2
9. f(x)  x2  8x  24
Source: savannah-online.com
3
min.; 8 max.; 3 max.; 0

10. GRAVITATION From 4 feet above a swimming pool, Susan throws a ball upward with a
velocity of 32 feet per second. The height h(t) of the ball t seconds after Susan throws it
is given by h(t)  16t2  32t  4. Find the maximum height reached by the ball and
55. WRITING IN MATH Answer the question that was posed at the beginning of the
the time that this height is reached. 20 ft; 1 s
lesson. See margin.
11. HEALTH CLUBS Last year, the SportsTime Athletic Club charged $20 to participate in
an aerobics class. Seventy people attended the classes. The club wants to increase the
class price this year. They expect to lose one customer for each $1 increase in the price.
How can income from a rock concert be maximized?
a. What price should the club charge to maximize the income from the aerobics classes?
$45 Include the following in your answer:
b. What is the maximum income the SportsTime Athletic Club can expect to make?
$2025 • an explanation of why income increases and then declines as the ticket price
Gl NAME
/M G ______________________________________________
Hill 316 DATE ____________
Gl PERIOD
Al _____
b 2
increases, and
Reading
6-1 Readingto
to Learn
Learn Mathematics
Mathematics, p. 317 ELL • an explanation of how to algebraically and graphically determine what ticket
Graphing Quadratic Functions price should be charged to achieve maximum income.
Pre-Activity How can income from a rock concert be maximized?
Read the introduction to Lesson 6-1 at the top of page 286 in your textbook.
• Based on the graph in your textbook, for what ticket price is the income
the greatest? $40
• Use the graph to estimate the maximum income. about $72,000
Standardized 56. The graph of which of the following equations is symmetrical about the
Reading the Lesson
1. a. For the quadratic function f(x)  2x2  5x  3, 2x2 is the quadratic term, Test Practice y-axis? C
y  x2  3x  1 y  x2  x
5x is the linear term, and 3 is the constant term.
3 1
A B
b. For the quadratic function f(x)  4  x  3x2, a  ,b , and
c 4 . C y  6x2  9 D y  3x2  3x  1
2. Consider the quadratic function f(x)  ax2  bx  c, where a  0.
a. The graph of this function is a parabola .
292 Chapter 6 Quadratic Functions and Inequalities
b. The y-intercept is c .
b
x  
c. The axis of symmetry is the line 2a .

d. If a  0, then the graph opens upward and the function has a NAME ______________________________________________ DATE ____________ PERIOD _____

minimum value.
Enrichment,
6-1 Enrichment p. 318
Answer
e. If a  0, then the graph opens downward and the function has a
maximum value.
Finding the Axis of Symmetry of a Parabola 54. The x-coordinate of the vertex of
3. Refer to the graph at the right as you complete the f (x )
As you know, if f(x)  ax2  bx  c is a quadratic function, the values of x 0
y  ax 2  c is   or 0, so the
following sentences. (–2, 4)
b   
b2  4ac b   
b2  4ac
parabola that make f(x) equal to zero are  and  .
a. The curve is called a

b. The line x  2 is called the axis of symmetry .

. 2a 2a
2a
b
The average of these two number values is .
y-coordinate of the vertex, the
O x
(0, –1) 2a f(x) b
c. The point (2, 4) is called the vertex . x = – ––
2a
The function f(x) has its maximum or minimum
b
d. Because the graph contains the point (0, 1), 1 is
the y-intercept .
value when x  . Since the axis of symmetry
2a
of the graph of f (x) passes through the point where O
f(x ) = ax 2 + bx + c
x
minimum of the function, is
Helping You Remember
4. How can you remember the way to use the x2 term of a quadratic function to tell
the maximum or minimum occurs, the axis of
symmetry has the equation x  .
b
2a
b
(– –– b
2a , f ( 2a ((
– ––
a(0)2  c or c.
whether the function has a maximum or a minimum value? Sample answer:
Remember that the graph of f(x)  x 2 (with a  0) is a U-shaped curve
that opens up and has a minimum. The graph of g(x)  x 2 (with a  0) Example Find the vertex and axis of symmetry for f(x)  5x 2  10x  7.
is just the opposite. It opens down and has a maximum.
b
Use x  .
2a
10

292 Chapter 6 Quadratic Functions and Inequalities

 57. Which of the following tables represents a quadratic relationship between the
two variables x and y? C
A x 1 2 3 4 5 B x 1 2 3 4 5
4 Assess
y 3 3 3 3 3 y 5 4 3 2 1 Open-Ended Assessment
C D
Speaking Ask students to explain
x 1 2 3 4 5 x 1 2 3 4 5
how to tell by examining a quad-
y 6 3 2 3 6 y 4 3 4 3 4
ratic function whether its graph
will have a maximum or mini-
Graphing MAXIMA AND MINIMA You can use the MINIMUM or MAXIMUM feature on a mum value. Then ask them to
Calculator graphing calculator to find the minimum or maximum value of a quadratic function.
This involves defining an interval that includes the vertex of the parabola. A lower
give an example of what such a
bound is an x value left of the vertex, and an upper bound is an x value right of value might mean in a real-world
the vertex. problem.
Step 1 Graph the function so that the vertex of the parabola is visible.
Step 2 Select 3:minimum or 4:maximum from the CALC menu. Getting Ready for
Step 3 Using the arrow keys, locate a left bound and press ENTER . Lesson 6-2
Step 4 Locate a right bound and press ENTER twice. The cursor appears on the PREREQUISITE SKILL Lesson 6-2
maximum or minimum value of the function, and the coordinates are presents solving quadratic equa-
displayed. tions by graphing. Finding points
Find the coordinates of the maximum or minimum value of each quadratic
on the graph of the function
function to the nearest hundredth. involves evaluating quadratic
58. f(x)  3x2  7x  2 ⴚ2.08 59. f(x)  5x2  8x 3.2 functions. Exercises 75–78 should
60. f(x)  2x2  3x  2 0.88 61. f(x)  6x2  9x 3.38
be used to determine your
62. f(x)  7x2  4x  1 0.43 63. f(x)  4x2  5x 1.56
students’ familiarity with
evaluating functions.

Maintain Your Skills Answer

Mixed Review Simplify. (Lesson 5-9) 74. y
64. i14 1 65. (4  3i)  (5  6i) 66. (7  2i)(1  i) 9  5i
y  3x
1  3i
Solve each equation. (Lesson 5-8) (1, 3) yx4
67. 5   68.  69.  2 4
n  12  n
3
b  2  0 23 x  5  6  4 13

Perform the indicated operations. (Lesson 4-2)

O x
70. [4 1 3]  [6 5 8] [10 ⴚ4 5] 71. [2 5 7]  [3 8 1] [5 ⴚ13 8]
ⴚ24
 14 
3 6 0

0 12
72. 4 72 4
5 11
9
 ⴚ288 20 ⴚ44
ⴚ16 36 
73. 2 7 1
3
4 2
ⴚ
3
ⴚ8

74. Graph the system of equations y  3x and y  x  4. State the solution. Is the
system of equations consistent and independent, consistent and dependent, or
inconsistent? (Lesson 3-1) See margin for graph; (ⴚ1, 3); consistent and
independent.
Getting Ready for PREREQUISITE SKILL Evaluate each function for the given value.
the Next Lesson (To review evaluating functions, see Lesson 2-1.)
75. f(x)  x2  2x  3, x  2 5 76. f(x)  x2  4x  5, x  3 8
2
77. f(x)  3x2  7x, x  2 2 78. f(x)  x2  2x  1, x  3 1
3
Lesson 6-1 Graphing Quadratic Functions 293

Answer
55. If a quadratic function can be used to model ticket price • You can locate the vertex of the parabola on the graph of the
versus profit, then by finding the x-coordinate of the vertex of function. It occurs when x  40. Algebraically, this is found
the parabola you can determine the price per ticket that by calculating
should be charged to achieve maximum profit. Answers b 4000
x    which, for this case, is x   or 40. Thus
should include the following. 2a 2(50)
• If the price of a ticket is too low, then you won’t make the ticket price should be set at $40 each to achieve
enough money to cover your costs, but if the ticket price is maximum profit.
too high fewer people will buy them.
Lesson 6-1 Graphing Quadratic Functions 293
 Lesson Solving Quadratic Equations
Notes by Graphing
• Solve quadratic equations by graphing.

1 Focus • Estimate solutions of quadratic equations by graphing.

Vocabulary does a quadratic function model a free-fall ride?

5-Minute Check • quadratic equation As you speed to the top of a free-fall ride, you are pressed against your seat so
Transparency 6-2 Use as a • root that you feel like you’re being pushed downward. Then as you free-fall, you fall
quiz or a review of Lesson 6-1. • zero at the same rate as your seat. Without the force of your seat pressing on you,
you feel weightless. The height above the ground (in feet) of an object in free-fall
Mathematical Background notes can be determined by the quadratic function h(t)  16t2  h0, where t is the
time in seconds and the initial height is h0 feet.
are available for this lesson on
p. 284C.

does a quadratic SOLVE QUADRATIC EQUATIONS When a quadratic function is set equal to
a value, the result is a quadratic equation. A quadratic equation can be written in
function model a the form ax2  bx  c  0, where a  0.
free-fall ride?
Study Tip The solutions of a quadratic equation are called the f (x )
Ask students: roots of the equation. One method for finding the roots of
Reading Math
• The acceleration of a free-falling In general, equations have a quadratic equation is to find the zeros of the related
object due to Earth’s gravity is roots, functions have zeros, quadratic function. The zeros of the function are the
(1, 0)
and graphs of functions x-intercepts of its graph. These are the solutions of the
32 ft/sec2. It is given as a neg- have x-intercepts. related equation because f(x)  0 at those points. The zeros O (3, 0) x
ative value because the acceler- of the function graphed at the right are 1 and 3.
ation is downward, toward
Earth’s surface. How is this
Example 1 Two Real Solutions
fact represented in the height
function? The coefficient 16 is Solve x2  6x  8  0 by graphing.
the one half of the acceleration due Graph the related quadratic function f(x)  x2  6x  8. The equation of the axis
to gravity in a downward direction. 6
of symmetry is x   or 3. Make a table using x values around 3. Then,
2(1)
• How far has a person fallen graph each point.
f (x )
1 second after beginning a free x 5 4 3 2 1
fall? after 2 seconds? after f (x ) 3 0 1 0 3
3 seconds? 16 ft; 64 ft; 144 ft
From the table and the graph, we can see that the zeros of O x
the function are 4 and 2. Therefore, the solutions of
the equation are 4 and 2. f (x )  x 2  6x  8

CHECK Check the solutions by substituting each

solution into the equation to see if it is satisfied.
x2  6x  8  0 x2  6x  8  0
? ?
(4)2  6(4)  8  0 (2)2  6(2)  8  0
00⻫ 00⻫

The graph of the related function in Example 1 had two zeros; therefore, the
quadratic equation had two real solutions. This is one of the three possible outcomes
when solving a quadratic equation.
294 Chapter 6 Quadratic Functions and Inequalities

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 6 Resource Masters School-to-Career Masters, p. 11 5-Minute Check Transparency 6-2
• Study Guide and Intervention, pp. 319–320 Answer Key Transparencies
• Skills Practice, p. 321
• Practice, p. 322 Technology
• Reading to Learn Mathematics, p. 323 Interactive Chalkboard
• Enrichment, p. 324
• Assessment, p. 369
 Solutions of a Quadratic Equation
• Words A quadratic equation can have one real solution, two real solutions, or
no real solution.
2 Teach
Study Tip • Models One Real Solution Two Real Solutions No Real Solution SOLVE QUADRATIC
One Real Solution f (x ) f (x ) f (x ) EQUATIONS
When a quadratic
equation has one real
O
In-Class Examples Power
Point®
solution, it really has two
O x x O x
solutions that are the
same number. 1 Solve x2  3x  4  0 by
graphing. 1 and 4
f (x )
Example 2 One Real Solution
O x
Solve 8x  x2  16 by graphing.
Write the equation in ax2  bx  c  0 form.
8x  x2  16 → x2  8x  16  0 Subtract 16 from each side.
Graph the related quadratic function f (x )
f(x)  x2  8x  16. f (x )  x 2  8x 16
f (x )  x 2  3x  4
x 2 3 4 5 6 O x
f (x ) 4 1 0 1 4
2 Solve x2  4x  4 by
Notice that the graph has only one x-intercept, 4. graphing. 2
Thus, the equation’s only solution is 4.
f (x )

Example 3 No Real Solution

NUMBER THEORY Find two real numbers whose sum is 6 and whose product
is 10 or show that no such numbers exist.
O x
Explore Let x  one of the numbers. Then 6  x  the other number.
f (x )  x 2  4x  4
Plan Since the product of the two numbers is 10, you know that
x(6  x)  10.
x(6  x)  10 Original equation Teaching Tip In Example 3,
6x  x2  10 Distributive Property inform students that while this
equation does not have any real
x2  6x  10  0 Subtract 10 from each side.
solutions, it does have a solution
Solve You can solve x2
 6x  10  0 by graphing the related function in the set of complex numbers,
f(x)  x2  6x  10. the topic of Lesson 5-9. Such
f (x )
equations will be discussed
x 1 2 3 4 5
again in Lessons 6-4 and 6-5.
f (5
x ) 5 2 1 2 5
f (x )  x 2  6x 10

x
3 NUMBER THEORY Find two
Notice that the graph has no x-intercepts. O
This means that the original equation has no
real numbers whose sum is 4
real solution. Thus, it is not possible for two and whose product is 5 or
numbers to have a sum of 6 and a product show that no such numbers
of 10. exist.
Examine Try finding the product of several pairs of f (x )
numbers whose sum is 6. Is the product of
each pair less than 10 as the graph suggests?

www.algebra2.com/extra_examples Lesson 6-2 Solving Quadratic Equations by Graphing 295

x
O
Unlocking Misconceptions f (x )  x2  4x  5

Equations and Functions Some students may notice that the equation The graph of the related function
derived in Example 2, x2  8x  16  0, is equivalent to the equation does not intersect the x-axis.
x2  8x  16  0. Either equation can be produced from the other by
Therefore, no such numbers exist.
multiplying each side by 1. These two equations have the same solution,
4. However, stress that the related functions, f(x)  x2  8x  16 and
f(x)  x2  8x  16 are not equivalent. This can be seen by looking at
their graphs, which open in opposite directions.

Lesson 6-2 Solving Quadratic Equations by Graphing 295

ESTIMATE SOLUTIONS ESTIMATE SOLUTIONS Often exact roots cannot be found by graphing. In
this case, you can estimate solutions by stating the consecutive integers between
In-Class Examples Power
Point®
which the roots are located.

4 Solve x2  6x  3  0 by Example 4 Estimate Roots

graphing. If exact roots cannot Solve x2  4x  1  0 by graphing. If exact roots cannot be found, state the
be found, state the consecutive consecutive integers between which the roots are located.
integers between which the The equation of the axis of symmetry of the related
Study Tip 4
roots are located. One solution function is x   or 2.
2(1) f (x )
is between 0 and 1, and the other Location of Roots f (x )  x 2  4x 1
is between 5 and 6. Notice in the table of x 0 1 2 3 4
values that the value of
f (x ) f (x ) 1 2 3 2 1
the function changes
x from negative to positive O x
O between the x values of The x-intercepts of the graph are between 0 and 1 and
0 and 1, and 3 and 4. between 3 and 4. So, one solution is between 0 and 1, and
the other is between 3 and 4.

For many applications, an exact answer is not required, and approximate

f (x )  x 2  6x  3 solutions are adequate. Another way to estimate the solutions of a quadratic
equation is by using a graphing calculator.

5 ROYAL GORGE BRIDGE The Example 5 Write and Solve an Equation

highest bridge in the United
EXTREME SPORTS On March 12, 1999, Adrian
States is the Royal Gorge Nicholas broke the world record for the longest Jumps from
plane at
Bridge in Colorado. The deck human flight. He flew 10 miles from his drop 35,000 ft
of the bridge is 1053 feet above point in 4 minutes 55 seconds using a specially
the river below. Suppose a designed, aerodynamic suit. Using the information
at the right and ignoring air resistance, how long
marble is dropped over the would Mr. Nicholas have been in free-fall had
Free-fall

railing from a height of 3 feet he not used this special suit? Use the formula
Opens
above the bridge deck. How h(t)  16t2  h0, where the time t is in seconds parachute
long will it take the marble to and the initial height h0 is in feet. at 500 ft

reach the surface of the water, We need to find t when h0  35,000 and
assuming there is no air h(t)  500. Solve 500  16t2  35,000.
resistance? Use the formula 500  16t2  35,000 Original equation
h(t)  16t2  h0, where t is 0  16t2  34,500 Subtract 500 from each side.
the time in seconds and h0 is
Graph the related function y  16t2  34,500
the initial height above the using a graphing calculator. Adjust your
water in feet. about 8 s window so that the x-intercepts of the graph
are visible.

Use the ZERO feature, 2nd [CALC], to find

the positive zero of the function, since time
cannot be negative. Use the arrow keys to
locate a left bound for the zero and press ENTER .
Then, locate a right bound and press ENTER twice.
The positive zero of the function is approximately
46.4. Mr. Nicholas would have been in free-fall for
[60, 60] scl: 5 by
about 46 seconds.
[40000, 40000] scl: 5000

296 Chapter 6 Quadratic Functions and Inequalities

Differentiated Instruction ELL

Verbal/Linguistic Have students discuss with a partner or in a small
group the methods for multiplying and dividing monomial expressions
with exponents, and also numbers written in scientific notation. Ask
them to work together to develop a list of common errors for such
problems, and to suggest ways to correct and avoid these errors.

296 Chapter 6 Quadratic Functions and Inequalities

 Concept Check 1. Define each term and explain how they are related. See margin.
3 Practice/Apply
a. solution b. root c. zero of a function d. x-intercept
2. OPEN ENDED Give an example of a quadratic function and state its related
quadratic equation. Sample answer: f (x)  3x 2  2x  1; 3x 2  2x  1  0 Study Notebook
3. Explain how you can estimate the solutions of a quadratic equation by Have students—
examining the graph of its related function. See margin.
• add the definitions/examples of
Guided Practice Use the related graph of each equation to determine its solutions. the vocabulary terms to their
GUIDED PRACTICE KEY 4. x2  3x  4  0 4, 1 5. 2x2  2x  4  0 2, 1 6. x2  8x  16  0 4 Vocabulary Builder worksheets for
Exercises Examples f (x ) f (x ) f (x ) Chapter 6.
4–6 1–3 • add the graphic representations of
O x
7–12 1–4
13 3 O 1
x
the three possible types of real
solutions of a quadratic equation
O x shown in the Key Concept box on
f (x )  x 2  3x  4 f (x )  2x 2  2x  4 f (x )  x 2  8x  16
p. 295.
• include any other item(s) that they
Solve each equation by graphing. If exact roots cannot be found, state the find helpful in mastering the skills
consecutive integers between which the roots are located.
in this lesson.
12. between 1 and 0; 7. x2  7x  0 7, 0 8. x2  2x  24  0 4, 6 9. x2  3x  28 7, 4
between 1 and 2 10. 25  x2  10x  0 5 11. 4x2  7x  15  0 12. 2x2  2x  3  0
between 2 and 1; 3
Application 13. NUMBER THEORY Use a quadratic equation to find two real numbers whose
sum is 5 and whose product is 14, or show that no such numbers exist. 2, 7

★ indicates increased difficulty About the Exercises…

Practice and Apply Organization by Objective
• Solve Quadratic Equations:
Homework Help Use the related graph of each equation to determine its solutions. 14–21, 24–31, 36–41
For See 1
Exercises Examples 14. x2  6x  0 0, 6 15. x2  6x  9  0 3 16. 2x2  x  6  0 2,1 • Estimate Solutions: 22, 23,
2
14–19 1–3 f (x ) f (x ) f (x ) 32–35, 42–46
4
20–37 1–4 f (x )  2x 2  x  6
38–41 3 Odd/Even Assignments
42–46 5 O 2 4 6 8x Exercises 14–41 are structured
4
Extra Practice so that students practice the
O x same concepts whether they
See page 840. 8
f (x )  x 2  6x  9 are assigned odd or even
O x
12 f (x )  x 2  6x
problems.
1
17. 0.5x2  0 0 18. 2x2  5x  3  0 , 3 19. 3x2  1  0 no real Alert! Exercises 51–56 require a
2 solutions
f (x ) f (x ) f (x )
graphing calculator.
O x
f (x )  0.5 x 2
O O x Assignment Guide
x f (x )  3 x 2  1
Basic: 15–45 odd, 47–50, 57–72
Average: 15–45 odd, 47–50,
f (x )  2 x 2  5 x  3 57–72 (optional: 51–56)
Advanced: 14–46 even, 47–66
www.algebra2.com/self_check_quiz Lesson 6-2 Solving Quadratic Equations by Graphing 297 (optional: 67–72)

Answers
1a. The solution is the value that satisfies 1d. An x-intercept is the point at which a 3. The x-intercepts of the related function
an equation. graph crosses the x-axis. The are the solutions to the equation. You
1b. A root is a solution of an equation. solutions, or roots, of a quadratic can estimate the solutions by stating
equation are the zeros of the related the consecutive integers between
1c. A zero is the x value of a function that
quadratic function. You can find the which the x-intercepts are located.
makes the function equal to 0.
zeros of a quadratic function by finding
the x-intercepts of its graph.

Lesson 6-2 Solving Quadratic Equations by Graphing 297

 Study Guide
NAME ______________________________________________ DATE

andIntervention
Intervention,
____________ PERIOD _____
Solve each equation by graphing. If exact roots cannot be found, state the
6-2 Study Guide and
p. 319 consecutive integers between which the roots are located.
Solving(shown) and by
Quadratic Equations p.Graphing
320
Solve Quadratic Equations 20. x2  3x  0 0, 3 21. x2  4x  0 0, 4
22. between 5 and 22. x2  4x  4  0 23. x2  2x  1  0
Quadratic Equation A quadratic equation has the form ax 2  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.
4; between 0 and 1 24. x2  x  20 4, 5 25. x2  9x  18 3, 6
23. between 1 and
26. 14x   49  0 7 27. 12x  x2  36 6
Example Solve x2  x  6  0 by graphing.
x2
Graph the related function f(x)  x2  x  6. f (x ) 0; between 2 and 3 1 1 1
 3x  9 1, 3 29. 4x2  8x  5  , 2
b 1
The x-coordinate of the vertex is     , and the equation of the
1
2a 2 O x
32. between 4 and 28. 2x2
axis of symmetry is x    .
2 2 1 12 2
Make a table of values using x-values around   .
1
2
3; between 0 and 1 30. 2x  5x  12 4, 1
2 31. 2x  x  15 2, 3
2
x 1 
1
0 1 2
33. between 0 and 1; 2 2
32. x2  3x  2  0 33. x2  4x  2  0
2

Lesson 6-2
1
6 6  6 4
f(x)
4
0
between 3 and 4
From the table and the graph, we can see that the zeros of the function are 2 and 3.
34. between 1 and 34. 2x2  3x  3  0 35. 0.5x2  3  0
Exercises
Solve each equation by graphing. 0; between 2 and 3 36. x2  2x  5  0 no real solutions 37. x2  4x  6  0 no real solutions
1. x2  2x  8  0 2, 4 2. x2  4x  5  0 5, 1 3. x2  5x  4  0 1, 4 35. between 3 and
2; between 2 and 3
f (x ) f (x ) f (x )

O x O x

NUMBER THEORY Use a quadratic equation to find two real numbers that satisfy
O x each situation, or show that no such numbers exist.
38. Their sum is 17, and their product is 72. 8, 9
4. x2  10x  21  0 5. x2  4x  6  0 6. 4x2  4x  1  0
f (x ) f (x ) f (x )
39. Their sum is 7, and their product is 14. See pp. 343A–343F.
O x
40. Their sum is 9, and their product is 24. See pp. 343A–343F.
O x
41. Their sum is 12, and their product is 28. 2, 14
O x
1
3, 7 no real solutions 
2

For Exercises 42–44, use the formula h(t)  v0t  16t2 where h(t) is the height of
Gl NAME
/M G ______________________________________________
Hill 319 DATE ____________
GlPERIOD
Al _____
b 2

Skills
6-2 Practice,
Practice (Average)
p. 321 and
Practice, p. 322
Solving Quadratic (shown)
Equations By Graphing
an object in feet, v0 is the object’s initial velocity in feet per second, and t is the
Use the related graph of each equation to determine its solutions. time in seconds.
1. 3x2  3  0 2. 3x2  x  3  0 3. x2  3x  2  0
f (x )  3x 2  3
f (x ) f (x ) f (x ) 42. ARCHERY An arrow is shot upward with a velocity of 64 feet per second.
Ignoring the height of the archer, how long after the arrow is released does it hit
O x
the ground? 4 s
f (x )  3x 2  x  3 f (x )  x 2  3x  2
O x O x

1, 1 no real solutions 1, 2 43. TENNIS A tennis ball is hit upward with a velocity of 48 feet per second.
Solve each equation by graphing. If exact roots cannot be found, state the
consecutive integers between which the roots are located.
Ignoring the height of the tennis player, how long does it take for the ball to fall to
4. 2x2  6x  5  0
between 0 and 1;
5. x2  10x  24  0
6, 4
6. 2x2  x  6  0
between 2 and 1,
the ground? 3 s
between 4 and 3 2
12
f (x )  2x 2  6x  5
f (x ) f (x )

O
f (x )

x
44. BOATING A boat in distress launches a flare straight up with a velocity of
8

4
190 feet per second. Ignoring the height of the boat, how many seconds will it
–6 –4 –2 O x take for the flare to hit the water? about 12 s
O x f (x )  2x 2  x  6
f (x )  x 2  10x  24

Use a quadratic equation to find two real numbers that satisfy each situation, or
show 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.
45. LAW ENFORCEMENT Police officers can use the length of skid marks to help
f (x ) x 2  x  6  0; f (x ) x 2  5x  8  0;
f (x )   x 2  x  6
3, 2 O x
f (x )  x 2  5x  8 no such real
numbers exist
determine the speed of a vehicle before the brakes were applied. If the skid
s2
marks are on dry concrete, the formula   d can be used. In the formula,
O x
24
s represents the speed in miles per hour, and d represents the length of the skid
For Exercises 9 and 10, use the formula h(t)  v0t  16t 2, where h(t) is the height
of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the
time in seconds.
marks in feet. If the length of the skid marks on dry concrete are 50 feet, how
9. BASEBALL Marta throws a baseball with an initial upward velocity of 60 feet per second. Empire State fast was the car traveling? about 35 mph
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 of
240 feet per second. How long will it take the boulder to hit the ground if it lands at the
Building
same elevation from which it was ejected? 15 s
Located on the 86th floor,
Gl NAME
/M G ______________________________________________
Hill 322 DATE ____________
Gl PERIOD
Al _____
b 2 46. EMPIRE STATE BUILDING Suppose you could conduct an experiment by
Reading 1050 feet (320 meters)
6-2 Readingto
to Learn
Learn Mathematics ELL dropping a small object from the Observatory of the Empire State Building. How
Mathematics, p. 323 above the streets of New
Solving Quadratic Equations by Graphing
York City, the Observatory long would it take for the object to reach the ground, assuming there is no air
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. offers panoramic views resistance? Use the information at the left and the formula h(t)  16t2  h0,
Write a quadratic function that describes the height of a ball t seconds after from within a glass- where t is the time in seconds and the initial height h0 is in feet. about 8 s
it is dropped from a height of 125 feet. h(t)  16t 2  125
enclosed pavilion and
Reading the Lesson
from the surrounding
1. The graph of the quadratic function f(x)  x2  x  6 y
open-air promenade. 47. CRITICAL THINKING A quadratic function has values f(4)  11, f(2)  9,
is shown at the right. Use the graph to find the solutions of the
quadratic equation x2  x  6  0. 2 and 3 Source: www.esbnyc.com and f(0)  5. Between which two x values must f(x) have a zero? Explain your
reasoning. 4 and 2; See margin for explanation.
O x
298 Chapter 6 Quadratic Functions and Inequalities
2. Sketch a graph to illustrate each situation.

a. A parabola that opens b. A parabola that opens c. A parabola that opens

downward 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 NAME ______________________________________________ DATE ____________ PERIOD _____
negative numbers.
y
zero is a positive number.
y
with no real zeros.
y Enrichment,
6-2 Enrichment p. 324 Answer
O x O x O x
Graphing Absolute Value Equations 47. The value of the function changes from
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
negative to positive, therefore the
Helping You Remember CALC menu to find its real solutions, if any. Recall that solutions are points
where the graph intersects the x-axis. value of the function is zero between
3. Think of a memory aid that can help you recall what is meant by the zeros of a quadratic
function.
Sample answer: The basic facts about a subject are sometimes called the
For each equation, make a sketch of the related graph and find the these two numbers.
solutions rounded to the nearest hundredth.
ABCs. In the case of zeros, the ABCs are the XYZs, because the zeros
are the x-values that make the y-values equal to zero. 1. | x  5 |  0 2. | 4x  3 |  5  0 3. | x  7 |  0
5 No solutions 7

298 Chapter 6 Quadratic Functions and Inequalities

 48. WRITING IN MATH Answer the question that was posed at the beginning of
the lesson. See pp. 343A–343F.
How does a quadratic function model a free-fall ride?
4 Assess
Include the following in your answer:
Open-Ended Assessment
• a graph showing the height at any given time of a free-fall ride that lifts riders
to a height of 185 feet, and Modeling Have students draw
• an explanation of how to use this graph to estimate how long the riders would parabolas in various positions and
be in free-fall if the ride were allowed to hit the ground before stopping. label them to show how many
real roots they have and approxi-
Standardized 49. If one of the roots of the equation x2  kx  12  0 is 4, what is the value of k? A mately where those roots occur.
Test Practice A 1 B 0 C 1 D 3

50. For what value of x does f(x)  x2  5x  6 reach its minimum value? B Getting Ready for
A 3 B
5

2
C 2 D 5 Lesson 6-3
PREREQUISITE SKILL Lesson 6-3
Extending SOLVE ABSOLUTE VALUE EQUATIONS BY GRAPHING Similar to quadratic presents solving quadratic equa-
the Lesson equations, you can solve absolute value equations by graphing. Graph the related tions by factoring. Frequently this
absolute value function for each equation using a graphing calculator. Then use the involves factoring a trinomial
ZERO feature, 2nd [CALC], to find its real solutions, if any, rounded to the nearest
hundredth.
expression on one side of an
equation. Exercises 67–72 should
51. x  1  0 1 52. x  3  0 3
be used to determine your stu-
53. x  4  1  0 3, 5 54. x  4  5  0 9, 1
dents’ familiarity with factoring
55. 23x  8  0 1.33 56. 2x  3  1  0 no real solutions
trinomials.
Maintain Your Skills
Assessment Options
Mixed Review Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of Quiz (Lessons 6-1 and 6-2) is
the vertex for each quadratic function. Then graph the function by making a table
of values. (Lesson 6-1) 57–59. See margin for graphs. available on p. 369 of the Chapter 6
57. f(x)  x2  6x  4 58. f(x)  4x2  8x  1
1
59. f(x)  x2  3x  4 Resource Masters.
4
4; x  3; 3 1; x  1; 1 4; x  6; 6
Simplify. (Lesson 5-9)
2i 1 3 4 10 2 1i 1 5 Answers
60.    i 61.    i 62.    i
3i 5 5 5  i 13 13 3  2i 13 13
57. f (x)
Evaluate the determinant of each matrix. (Lesson 4-3)

   
2 1 6 6 5 2
63. 
6 4
3 2
24  64. 5 0 3 8 65. 3 0 6 ⴚ60
O x
3 2 11 1 4 2

66. COMMUNITY SERVICE A drug awareness program is

being presented at a theater that seats 300 people. f (x)  x 2  6x  4
Proceeds will be donated to a local drug information (3, 5)
center. If every two adults must bring at least one
student, what is the maximum amount of money that
can be raised? (Lesson 3-4) $500
58. f (x)
68. (x  10)(x  10)
69. (x  7)(x  4)
Getting Ready for PREREQUISITE SKILL Factor completely.
(1, 3) f (x)  4x 2  8x  1
the Next Lesson (To review factoring trinomials, see Lesson 5-4.)

71. (3x  2)(x  2) 67. x2  5x x (x  5) 68. x2  100 69. x2  11x  28

72. 2(3x  2)(x  3) 70. x2  18x  81 (x  9)2 71. 3x2  8x  4 72. 6x2  14x  12
O x
Lesson 6-2 Solving Quadratic Equations by Graphing 299

59.
f (x)  1 x 2  3x  4
4
f (x)
8

4
O
12 8 4 x
4
(6, 5)

Lesson 6-2 Solving Quadratic Equations by Graphing 299

Graphing
Calculator
A Follow-Up of Lesson 6-2
Investigation
A Follow-Up of Lesson 6-2

Getting Started Modeling Real-World Data

You can use a TI-83 Plus to model data points whose curve of best fit is quadratic.

Know Your Calculator When FALLING WATER Water is allowed to drain from a hole made in a 2-liter bottle.
students use the procedure in The table shows the level of the water y measured in centimeters from the bottom
Step 2 to copy the regression of the bottle after x seconds. Find and graph a linear regression equation and a
quadratic regression equation. Determine which equation is a better fit for the data.
equation from Step 1 to the Y=
list, the coefficients will have Time (s) 0 20 40 60 80 100 120 140 160 180 200 220
several more digits than the Water level (cm) 42.6 40.7 38.9 37.2 35.8 34.3 33.3 32.3 31.5 30.8 30.4 30.1
coefficients displayed on the
home screen. The coefficients on Find a linear regression equation. Find a quadratic regression equation.
the home screen are rounded • Enter the times in L1 and the water levels in L2. • Find the quadratic regression equation. Then
versions of those in the Y= list. Then find a linear regression equation. copy the equation to the Y= list and graph.
KEYSTROKES: Review lists and finding a linear KEYSTROKES: STAT 5 ENTER
Scientific Notation In Step 1,
regression equation on page 87. 5
the value of the coefficient a is VARS ENTER GRAPH
displayed as 2.1035215E4. • Graph a scatter plot and the regression equation.
Point out that this is how the KEYSTROKES: Review graphing a regression

calculator displays the scientific equation on page 87.

notation 2.1035215
104.

Teach [0, 260] scl: 1 by [25, 45] scl: 5

• Make sure students have cleared The graph of the linear regression equation
appears to pass through just two data points.
the L1 and L2 lists before entering [0, 260] scl: 1 by [25, 45] scl: 5 However, the graph of the quadratic
new data. Also have them enter regression equation fits the data very well.
the WINDOW dimensions shown.
• For Step 1, point out that you Exercises 1–4. See margin. Average Braking Distance on
can use the same keystrokes For Exercises 1– 4, use the graph of the braking Dry Pavement
shown in Step 2, substituting 4 distances for dry pavement.
for the first 5, to select LinReg. 1. Find and graph a linear regression equation and 300 284
a quadratic regression equation for the data.
Distance (ft)

• If an error message appears in Determine which equation is a better fit for the data. 188
200
Step 2, have students clear the 2. Use the CALC menu with each regression equation 134
160

Y= list before trying Step 2 again. to estimate the braking distance at speeds of 100 90
100 and 150 miles per hour. 40
• If students need to review enter- 18
3. How do the estimates found in Exercise 2 compare? 0
ing data or selecting statistical 20 30 45 55 60 65 80
4. How might choosing a regression equation that
plots, refer them to p. 87. does not fit the data well affect predictions made Speed (mph)

• Have students complete by using the equation? Source: Missouri Department of Revenue

Exercises 1–4. www.algebra2.com/other_calculator_keystrokes

Assess 300 Chapter 6 Quadratic Functions and Inequalities

Ask students:
Answers
• What does it mean when the
points on a scatter plot appear 1. See pp. 343A–343F.
to lie along a curved path? 2. linear: (100, 345), (150, 562); quadratic: (100, 440),
The equation that best models the (150, 990)
situation may be quadratic, and is 3. The quadratic estimates are much greater.
probably not linear.
4. Sample answer: Choosing a model that does not fit
the data well may cause inaccurate predictions
when the data are very large or small.
300 Chapter 6 Quadratic Functions and Inequalities
0284-343F Alg 2 Ch06-828000 11/22/02 10:38 PM Page 301

Solving Quadratic Equations Lesson

by Factoring Notes
• Solve quadratic equations by factoring.
• Write a quadratic equation with given roots.

is the Zero Product Property used in geometry?

1 Focus
The length of a rectangle is 5 inches more than its x⫹5 5-Minute Check
width, and the area of the rectangle is 24 square inches. Transparency 6-3 Use as a
To find the dimensions of the rectangle you need to quiz or review of Lesson 6-2.
solve the equation x(x ⫹ 5) ⫽ 24 or x2 ⫹ 5x ⫽ 24. x

Mathematical Background notes

are available for this lesson on
p. 284C.
SOLVE EQUATIONS BY FACTORING In the last lesson, you learned to solve
a quadratic equation like the one above by graphing. Another way to solve this Building on Prior
equation is by factoring. Consider the following products.
Knowledge
7(0) ⫽ 0 0(⫺2) ⫽ 0 In Lesson 6-2, students solved
(6 ⫺ 6)(0) ⫽ 0 ⫺4(⫺5 ⫹ 5) ⫽ 0 quadratic equations by graphing.
In this lesson, they use factoring
Notice that in each case, at least one of the factors is zero. These examples illustrate
the Zero Product Property.
as a method for finding the roots
of a quadratic equation.
Zero Product Property is the Zero Product Prop-
• Words For any real numbers a and b, if ab ⫽ 0, then either a ⫽ 0, b ⫽ 0, or erty used in geometry?
both a and b equal zero.
Ask students:
• Example If (x ⫹ 5)(x ⫺ 7) ⫽ 0, then x ⫹ 5 ⫽ 0 and/or x ⫺ 7 ⫽ 0.
• What is the product of the
length and the width of the
rectangle? 24 in2
Example 1 Two Roots
• What is the difference between
Solve each equation by factoring.
the length and the width of the
a. x2 ⫽ 6x rectangle? 5 in.
x2 ⫽ 6x Original equation
x2 ⫺ 6x ⫽ 0 Subtract 6x from each side.
x(x ⫺ 6) ⫽ 0 Factor the binomial.

x⫽0 or x ⫺ 6 ⫽ 0 Zero Product Property

x ⫽ 6 Solve the second equation.

The solution set is {0, 6}.

CHECK Substitute 0 and 6 for x in the original equation.

x2 ⫽ 6x x2 ⫽ 6x
(0)2 ⱨ 6(0) (6)2 ⱨ 6(6)
0⫽0⻫ 36 ⫽ 36 ⻫
Lesson 6-3 Solving Quadratic Equations by Factoring 301

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 6 Resource Masters 5-Minute Check Transparency 6-3
• Study Guide and Intervention, pp. 325–326 Real-World Transparency 6
• Skills Practice, p. 327 Answer Key Transparencies
• Practice, p. 328
• Reading to Learn Mathematics, p. 329 Technology
• Enrichment, p. 330 Interactive Chalkboard

Lesson x-x Lesson Title 301

 b. 2x2  7x  15
2 Teach 2x2
2x2  7x  15
 7x  15  0
Original equation
Subtract 15 from each side.

SOLVE EQUATIONS BY (2x  3)(x  5)  0 Factor the trinomial.

FACTORING 2x  3  0 or x50 Zero Product Property
2x  3 x  5 Solve each equation.
In-Class Examples Power
Point®
3
x  
2
The solution set is 5,  . Check each solution.
Teaching Tip In Example 1a, 3
2
some students may suggest solv-
ing the equation by dividing both
sides by x. Point out that this Study Tip Example 2 Double Root
cannot be done because the
value of x could be zero, and Double Roots Solve x2  16x  64  0 by factoring.
The application of the
division by zero is undefined. x2  16x  64  0 Original equation
Zero Product Property
produced two identical (x  8)(x  8)  0 Factor.
1 Solve each equation by equations, x  8  0, x80 or x  8  0 Zero Product Property
factoring. both of which have a
root of 8. For this reason, x8 x  8 Solve each equation.
a. x2  4x {0, 4} 8 is called the double The solution set is {8}.
root of the equation.
b. 3x2  5x  2  
3
1
,2 CHECK The graph of the related function,
f(x)  x2  16x  64, intersects the x-axis only
2 Solve x2  6x  9 by once. Since the zero of the function is 8, the
factoring. {3} solution of the related equation is 8.

Teaching Tip Point out that the

term repeated root is sometimes Standardized Example 3 Greatest Common Factor
used as a substitute for the term Test Practice Multiple-Choice Test Item
double root.

3 What is the positive solution What is the positive solution of the equation 3x2  3x  60  0?
of the equation A 4 B 2 C 5 D 10
2x2  8x  42  0? D
Read the Test Item
A 3 B5 You are asked to find the positive solution of the given quadratic equation. This
C6 D7 implies that the equation also has a solution that is not positive. Since a quadratic
equation can either have one, two, or no solutions, we should expect to find two
Teaching Tip Ask students why solutions to this equation.
dividing each side of the equation
in this example results in an Solve the Test Item
equivalent equation, without the Solve this equation by factoring. But before trying to factor 3x2  3x  60 into two
binomials, look for a greatest common factor. Notice that each term is divisible by 3.
possibility of losing a root. (The
right side of the equation is 0, Test-Taking Tip 3x2  3x  60  0 Original equation
not f(x) or y, and dividing by 2 Because the problem asked 3(x2  x  20)  0 Factor.
means that you can be sure that for a positive solution, x2  x  20  0 Divide each side by 3.
choice A could have been
you are not dividing by zero.) eliminated even before the (x  4)(x  5)  0 Factor.
expression was factored. x40 or x  5  0 Zero Product Property
x  4 x5 Solve each equation.

WRITE QUADRATIC Both solutions, 4 and 5, are listed among the answer choices. Since the question
asked for the positive solution, the answer is C.
EQUATIONS
In-Class Example Power
Point®
302 Chapter 6 Quadratic Functions and Inequalities

4 Write a quadratic equation

2 Example 3 Point out to students that by reading
with   and 6 as its roots.
3
Standardized
the question carefully and noting exactly what is
Write the equation in the form Test Practice asked for (the positive solution), they can quickly
ax2  bx  c  0, where a, b, eliminate answer choice A because it is negative.
and c are integers. Sample Choice A is an attractive (though incorrect) choice because it is indeed a
answer: 3x 2  16x  12  0 solution of the equation, just not the positive one.

302 Chapter 6 Quadratic Functions and Inequalities

 WRITE QUADRATIC EQUATIONS You have seen that a quadratic equation
of the form (x  p)(x  q)  0 has roots p and q. You can use this pattern to find a
quadratic equation for a given pair of roots. 3 Practice/Apply
Study Tip Example 4 Write an Equation Given Roots
1
Writing an Write a quadratic equation with  and ⴚ5 as its roots. Write the equation in the
Equation 2 Study Notebook
The pattern form ax2 ⴙ bx ⴙ c ⴝ 0, where a, b, and c are integers.
(x  p)(x  q)  0 (x  p)(x  q)  0 Write the pattern.
Have students—
produces one equation • add the definitions/examples of
with roots p and q.
In fact, there are an
x  12[x  (5)]  0 1
Replace p with  and q with 5.
2 the vocabulary terms to their
infinite number of
equations that have
 1

x   (x  5)  0
2
Simplify. Vocabulary Builder worksheets for
these same roots. 9 5
x  x    0
2 Use FOIL.
Chapter 6.
2 2
• include any other item(s) that they
2x2  9x  5  0 Multiply each side by 2 so that b and c are integers.
find helpful in mastering the skills
1
A quadratic equation with roots  and 5 and integral coefficients is in this lesson.
2
2x2  9x  5  0. You can check this result by graphing the related function.

About the Exercises…

Concept Check 1. Write the meaning of the Zero Product Property. Organization by Objective
1. Sample answer: If • Solve Equations by
2. OPEN ENDED Choose two integers. Then, write an equation with those
the product of two roots in the form ax2  bx  c  0, where a, b, and c are integers. Factoring: 14–33, 42–26
factors is zero, then at • Write Quadratic Equations:
least one of the 3. FIND THE ERROR Lina and Kristin are solving x2  2x  8. 34–41
factors must be zero.
2. Sample answer: Kristin Odd/Even Assignments
Lina
roots 6 and 5; Exercises 14–41 are structured
x 2  x  30  0 x2 + 2x = 8
x 2 + 2x = 8 so that students practice the
3. Kristin; the Zero x2 + 2x – 8 = 0
x(x + 2) = 8 same concepts whether they
Product Property (x + 4)(x – 2) = 0
applies only when x = 8 or x + 2 = 8 are assigned odd or even
x + 4 = 0 or x – 2= 0
one side of the x= 6 problems.
x = –4 x=2
equation is 0.
Who is correct? Explain your reasoning. Assignment Guide
Basic: 15–31 odd, 35–43 odd,
Guided Practice Solve each equation by factoring.
47–68
GUIDED PRACTICE KEY 4. x2  11x  0 {0, 11} 5. x2  6x  16  0 {8, 2}
6. x2  49 {7, 7} 7. x2  9  6x {3}
Average: 15–43 odd, 47–68
Exercises Examples
4–9 1, 2 8. 4x2
3
4 
 13x  12 , 4 9. 5x2  5x  60  0 {3, 4} Advanced: 14–44 even, 45–62
10–12 4 (optional: 63–68)
13 3 Write a quadratic equation with the given roots. Write the equation in the form
ax2 ⴙ bx ⴙ c ⴝ 0, where a, b, and c are integers. All: Practice Quiz 1 (1–5)
1 4 3 1
10. 4, 7 11. ,  12. , 
2 3 5 3
x 2 ⴚ 3x ⴚ 28 ⴝ 0 6x 2 ⴚ 11x ⴙ 4 ⴝ 0 15x 2 ⴙ 14x ⴙ 3 ⴝ 0
Standardized 13. Which of the following is the sum of the solutions of x2  2x  8  0? D
Test Practice A 6 B 4 C 2 D 2 FIND THE ERROR
To show that
www.algebra2.com/extra_examples Lesson 6-3 Solving Quadratic Equations by Factoring 303 neither factor on the
left side of Lina’s second
equation needs to be 8, ask
students to name several pairs of
Differentiated Instruction numbers whose product is 8.
Visual/Spatial Provide each student with a sheet of grid paper. Have Possible pairs are 2 and 4, 0.5 and
students begin by first drawing a coordinate grid with two points on the 16, and 1 and 8.
x-axis plotted as the roots of a quadratic equation. Then ask students to
draw several different parabolas that might be the graphs of different
equations having those two points as their solutions. Point out that this
demonstrates that the steps demonstrated in Example 4 yield just one
of the possible equations having the given roots.

Lesson 6-3 Solving Quadratic Equations by Factoring 303

 ★ indicates increased difficulty
NAME ______________________________________________ DATE ____________ PERIOD _____

Study
6-3 Study Guide andIntervention
Guide and Intervention, Practice and Apply
p. 325
Solving(shown) and by
Quadratic Equations p.Factoring
326
Solve Equations by Factoring
you use the following property.
When you use factoring to solve a quadratic equation,
Homework Help Solve each equation by factoring.
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. For See 14. x2  5x  24  0 {8, 3} 15. x2  3x  28  0 {4,7}
Exercises Examples
Example
a. 3x2  15x
Solve each equation by factoring.
b. 4x2  5x  21 14–33, 1, 2 16. x2  25 {5, 5} 17. x2  81 {9, 9}
3x2  15x Original equation
3x  15x  0
2 Subtract 15x from both sides.
4x2  5x  21 Original equation
4x2  5x  21  0 Subtract 21 from both sides.
42–46 18. x2  3x  18 {6, 3} 19. x2  4x  21 {3, 7}
5
  3

3x(x  5)  0 (4x  7)(x  3)  0 Factor the trinomial. 34–41 4
20. 3x2  5x 0,  21. 4x2  3x 0, 
Factor the binomial.

3x  0 or x  5  0 Zero Product Property 4x  7  0 or x  3  0 Zero Product Property

x  0 or x  5 Solve each equation. 7
x    or x  3 Solve each equation.
51–52 3 3 4
22. x2  36  12x {6} 23. x2  64  16x {8}
4
The solution set is {0, 5}.
7
The solution set is   , 3 .  Extra Practice 1
 
1

4
Exercises
See page 840. 24. 4x2  7x  2 2,  25. 4x2  17x  4 , 4
Solve each equation by factoring. 4 4
1. 6x2  2x  0 2. x2  7x 3. 20x2  25x 26. 4x  8x  3
2 27. 6x  6  13x
2
26. , 
1 3
0, 31 0, 54 8 2
 
3 9

{0, 7}
2 2 28. 9x2  30x  16 ,  29. 16x2  48x  27 , 
4. 6x2  7x 5. 6x2  27x  0 6. 12x2  8x  0
3 3 4 4

Lesson 6-3
0, 76 0, 92 0, 32
7. x2  x  30  0 8. 2x2 x30 9. x2  14x  33  0
2
 3
27. , 
3 2 30. 2x2  12x  16  0 {2, 4} 31. 3x2  6x  9  0 {3, 1}
{5, 6} 23 , 1 {11, 3}

10. 4x2  27x  7  0 11. 3x2  29x  10  0 12. 6x2  5x  4  0 ★ 32. Find the roots of x(x  6)(x  5)  0. 0, 6, 5
14 , 7 10, 31 12 , 34
13. 12x2  8x  1  0 14. 5x2  28x  12  0 15. 2x2  250x  5000  0 ★ 33. Solve x3  9x by factoring. 0, 3, 3
16 , 12 25 , 6 {100, 25}

16. 2x2  11x  40  0 17. 2x2  21x  11  0 18. 3x2  2x  21  0

8, 25 11, 21 73 , 3 Write a quadratic equation with the given roots. Write the equation in the form
19. 8x2  14x  3  0 20. 6x2  11x  2  0 21. 5x2  17x  12  0 ax2 ⴙ bx ⴙ c ⴝ 0, where a, b, and c are integers.
 3 1
,   1
2,   3
 , 4
34. x 2  9x  20  0 35. 2, 7 36. 4, 5 37. 6, 8
2 4 6 5
22. 12x2  25x  12  0 23. 12x2  18x  6  0 24. 7x2  36x  5  0 34. 4, 5
43 , 34 12 , 1 17 , 5
35. x 2  5x  14  0 1
38. , 3
1
39. , 5
2 3
40. ,  41. , 
3 4
Gl NAME
/M G ______________________________________________
Hill 325 DATE ____________
GlPERIOD
Al _____
b 2 2 3 3 4 2 5
Skills
6-3 Practice,
Practice (Average)
p. 327 and 36. x 2  x  20  0 2x 2  7x  3  0 3x 2  16x  5  0 12x 2  x  6  0
Practice, p. 328
Solving Quadratic (shown)
Equations by Factoring 37. x 2  14x  48  0
Solve each equation by factoring.
42. DIVING To avoid hitting any rocks below, a cliff diver
1. x2  4x  12  0 {6, 2} 2. x2  16x  64  0 {8} 3. x2  20x  100  0 {10} 41. 10x 2  23x  jumps up and out. The equation h  16t2  4t  26
describes her height h in feet t seconds after jumping.
4. x2  6x  8  0 {2, 4} 5. x2  3x  2  0 {2, 1} 6. x2  9x  14  0 {2, 7}
12  0 Find the time at which she returns to a height of
7. x2  4x  0 {0, 4} 8. 7x2  4x 0,   4
9. x2  25  10x {5}
26 feet. 1 s
7
10. 10x2  9x 0,   9
11. x2  2x  99 {9, 11} 26 ft h 26 ft
12. x2  12x  36 {6}
10
13. 5x2  35x  60  0 {3, 4}
4
14. 36x2  25  ,   56 5
6 15. 2x2  8x  90  0 {9, 5} 43. NUMBER THEORY Find two consecutive even integers
16. 3x2  2x  1  0  , 1  31 
17. 6x2  9x 0, 
3
2 whose product is 224. 14, 16
18. 3x2  24x  45  0 {5, 3} 19. 15x2  19x  6  0   ,    3
5
2
3
20. 3x2  8x  4 2,   2
3  32
21. 6x2  5x  6  ,  
2
3 44. PHOTOGRAPHY A rectangular photograph is 8 centimeters wide and
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.
12 centimeters long. The photograph is enlarged by increasing the length
22. 7, 2 23. 0, 3 24. 5, 8 and width by an equal amount in order to double its area. What are the
x 2  9x  14  0 x 2  3x  0 x 2  3x  40  0
dimensions of the new photograph? 12 cm by 16 cm
25. 7, 8 26. 6, 3 27. 3, 4
x 2  15x  56  0 x 2  9x  18  0 x 2  x  12  0

28. 1, 
1
2
1
29.  , 2
3
30. 0,  
7
2
FORESTRY For Exercises 45 and 46, use the following information.
2x 2  3x  1  0 3x 2  7x  2  0 2x 2  7x  0 Lumber companies need to be able to estimate the number of board feet that a given
1
31.  , 3
3
32. 4, 
1
3
33.   ,  
2
3
4
5 log will yield. One of the most commonly used formulas for estimating board feet is
3x 2  8x  3  0 3x 2  13x  4  0 15x 2  22x  8  0 L
34. NUMBER THEORY Find two consecutive even positive integers whose product is 624. the Doyle Log Rule, B  (D2  8D  16), where B is the number of board feet, D is
24, 26 16
35. NUMBER THEORY Find two consecutive odd positive integers whose product is 323.
17, 19
the diameter in inches, and L is the length of the log in feet.
36. GEOMETRY The length of a rectangle is 2 feet more than its width. Find the
dimensions of the rectangle if its area is 63 square feet. 7 ft by 9 ft 45. Rewrite Doyle's formula for logs that are 16 feet long. B  D 2  8D  16
37. PHOTOGRAPHY The length and width of a 6-inch by 8-inch photograph are reduced by
the same amount to make a new photograph whose area is half that of the original. By
how many inches will the dimensions of the photograph have to be reduced? 2 in.
★ 46. Find the root(s) of the quadratic equation you wrote in Exercise 45. What do
Gl NAME
/M G ______________________________________________
Hill 328 DATE ____________
Gl PERIOD
Al _____
b 2
the root(s) tell you about the kinds of logs for which Doyle’s rule makes
Reading
Readingto
to Learn sense? See margin.
6-3 Learn Mathematics
Mathematics, p. 329 ELL Forestry
Solving Quadratic Equations by Factoring
A board foot is a measure
Pre-Activity How is the Zero Product Property used in geometry?
of lumber volume. One 47. CRITICAL THINKING For a quadratic equation of the form (x  p)(x  q)  0,
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? piece of lumber 1 foot long show that the axis of symmetry of the related quadratic function is located
It represents the area of the rectangle, since the area is the
product of the width and length. by 1 foot wide by 1 inch halfway between the x-intercepts p and q. See margin.
thick measures one board
Reading the Lesson foot. CRITICAL THINKING Find a value of k that makes each statement true.
1. The solution of a quadratic equation by factoring is shown below. Give the reason for
Source: www.wood-worker.com 1
each step of the solution.
48. 3 is a root of 2x2  kx  21  0. 1 49.  is a root of 2x2  11x  k. 6
x2  10x  21 Original equation
2
x2  10x  21  0 Add 21 to each side.
(x  3)(x  7)  0 Factor the trinomial. 304 Chapter 6 Quadratic Functions and Inequalities
x  3  0 or x  7  0 Zero Product Property
x3 x7 Solve each equation.
The solution set is {3, 7} .

2. On an algebra quiz, students were asked to write a quadratic equation with 7 and 5 as NAME ______________________________________________ DATE ____________ PERIOD _____
its roots. The work that three students in the class wrote on their papers is shown below.
Marla Rosa Larry Enrichment,
6-3 Enrichment p. 330
Answer
(x 7)(x  5)  0 (x  7)(x  5)  0 (x  7)(x  5)  0
x2  2x  35  0
Who is correct? Rosa
x2  2x  35  0 x2  2x  35  0
Euler’s Formula for Prime Numbers 46. 4; The logs must have a diameter
Explain the errors in the other two students’ work.
Sample answer: Marla used the wrong factors. Larry used the correct
Many mathematicians have searched for a formula that would generate prime
numbers. One such formula was proposed by Euler and uses a quadratic greater than 4 in. for the rule to
polynomial, x2  x  41.
factors but multiplied them incorrectly.
produce positive board feet values.
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.
Helping You Remember
1. x  0 2. x  1 3. x  2
3. A good way to remember a concept is to represent it in more than one way. Describe an
algebraic way and a graphical way to recognize a quadratic equation that has a double 41, prime 43, prime 47, prime
root.
Sample answer: Algebraic: Write the equation in the standard form
ax 2  bx  c  0 and examine the trinomial. If it is a perfect square
trinomial, the quadratic function has a double root. Graphical: Graph the 4. x  3 5. x  4 6. x  5
related quadratic function. If the parabola has exactly one x-intercept,
then the equation has a double root. 53, prime 61, prime 71, prime

304 Chapter 6 Quadratic Functions and Inequalities

 50. WRITING IN MATH Answer the question that was posed at the beginning of the
lesson. See pp. 343A–343F.
How is the Zero Product Property used in geometry?
4 Assess
Include the following in your answer:
Open-Ended Assessment
• an explanation of how to find the dimensions of the rectangle using the Zero
Product Property, and Speaking Ask students to give a
• why the equation x(x  5)  24 is not solved by using x  24 and x  5  24. verbal explanation of the Zero
Product Property. They should
1 1
Standardized 51. Which quadratic equation has roots  and ? D
2 3 discuss why it is true and how it
Test Practice A 5x2  5x  2  0 B 5x2  5x  1  0 is used in finding the roots of a
C 6x2  5x  1  0 D 6x2  5x  1  0 quadratic equation, demonstrating
the technique using an example.
52. If the roots of a quadratic equation are 6 and 3, what is the equation of the axis
of symmetry? B
3 1
A x1 B x   C x   D x  2
2 2 Intervention
New Suggest that
Maintain Your Skills students who
have difficulty
Mixed Review Solve each equation by graphing. If exact roots cannot be found, state the
consecutive integers between which the roots are located. (Lesson 6-2) understanding
53. 5, 1 53. f(x)  x2  4x  5 54. f(x)  4x2  4x  1 55. f(x)  3x2  10x  4 the Zero Product Property try
1
54.  to find two nonzero numbers
2 56. Determine whether f(x)  3x2  12x  7 has a maximum or a minimum value. whose product is zero. Students
55. between 1 and 0; Then find the maximum or minimum value. (Lesson 6-1) min.; 19
between 3 and 4 should quickly determine that
57. 32  23
Simplify. (Lesson 5-6) at least one of the numbers
57. 3 6  2   3 59. 5  8
  48
58. 108
3 2
must be zero in order for their
58. 53
Solve each system of equations. (Lesson 3-2) 
33  202 product to be zero.
60. 4a  3b  4 61. 2r  s  1 62. 3x  2y  3
3a  2b  4 (ⴚ4, ⴚ4) r  s  8 (3, ⴚ5)
1

3x  y  3 , 2
3  Getting Ready for
Getting Ready for PREREQUISITE SKILL Simplify. (To review simplifying radicals, see Lesson 5-5.) Lesson 6-4
the Next Lesson 63. 8 22  25
64. 20  33
65. 27 PREREQUISITE SKILL Lesson 6-4
66.  
50 5i2  2i 3
67. 12  4i 3
68. 48 presents solving quadratic equa-
tions by completing the square.
The process involves evaluating
P ractice Quiz 1 Lessons 6-1 through 6-3 radicals. Exercises 63–68 should
be used to determine your
1. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of students’ familiarity with
the vertex for f(x)  3x2  12x  4. Then graph the function by making a table of
values. (Lesson 6-1) 4, x  2; 2; See margin for graph.
simplifying radicals.

2. Determine whether f(x)  3  x2  5x has a maximum or minimum value. Then Assessment Options
find this maximum or minimum value. (Lesson 6-1) max.; 37 or 91
4 4 Practice Quiz 1 The quiz
3. Solve 2x2  11x  12  0 by graphing. If exact roots cannot be found, state the provides students with a brief
1
consecutive integers between which the roots are located. (Lesson 6-2) 1, 4 review of the concepts and skills
2

4. Solve 2x  9x  5  0 by factoring. (Lesson 6-3) 5, 2
2 5, 11

2
in Lessons 6-1 through 6-3.
1 Lesson numbers are given to the
5. Write a quadratic equation with roots 4 and  . Write the equation in the form
3 right of exercises or instruction
ax2  bx  c  0, where a, b, and c are integers. (Lesson 6-3) 3x 2  11x  4  0 lines so students can review
concepts not yet mastered.
www.algebra2.com/self_check_quiz Lesson 6-3 Solving Quadratic Equations by Factoring 305

Answer (Practice Quiz 1)

47. y  (x  p)(x  q) The axis of symmetry is the 1. f (x)
y  x 2  px  qx  pq average of the x-intercepts.
4
y  x 2  (p  q)x  pq Therefore the axis of symmetry f (x)  3x 2  12x  4
a  1, b  (p  q), c  pq is located halfway between the
O 4 8 12 x
b x-intercepts.
axis of symmetry: x    4
2a
(p  q)
x    8
(2, 8)
2(1)
pq
x
2
Lesson 6-3 Solving Quadratic Equations by Factoring 305
 Lesson Completing the Square
Notes

• Solve quadratic equations by using the Square Root Property.

1 Focus • Solve quadratic equations by completing the square.

Vocabulary can you find the time it takes an accelerating

5-Minute Check • completing the square race car to reach the finish line?
Transparency 6-4 Use as a Under a yellow caution flag, race car
quiz or review of Lesson 6-3. drivers slow to a speed of 60 miles per
hour. When the green flag is waved,
Mathematical Background notes the drivers can increase their speed.
are available for this lesson on Suppose the driver of one car is 500 feet
p. 284C. from the finish line. If the driver accelerates
at a constant rate of 8 feet per second
squared, the equation t2  22t  121  246
Building on Prior represents the time t it takes the driver
Knowledge to reach this line. To solve this equation,
you can use the Square Root Property.
In Lesson 6-3, students solved
quadratic equations by factoring.
In this lesson they use two other SQUARE ROOT PROPERTY You have solved equations like x2  25  0 by
methods to solve equations: the factoring. You can also use the Square Root Property to solve such an equation.
Square Root Property and This method is useful with equations like the one above that describes the race car’s
speed. In this case, the quadratic equation contains a perfect square trinomial set
completing the square. equal to a constant.
can you find the time it
takes an accelerating Study Tip Square Root Property
race car to reach the finish line? Reading Math For any real number n, if x2  n, then x  
n.
Ask students: n is read plus or
minus the square root
• The number 121 is a perfect of n.
square. What is the square root Example 1 Equation with Rational Roots
of 121? 11 Solve x2  10x  25  49 by using the Square Root Property.
• How does the number 11 relate x2  10x  25  49 Original equation
to the coefficient of the x term? (x  5)2  49 Factor the perfect square trinomial.
TEACHING TIP
The coefficient of x is twice 11. 
x  5  49 Square Root Property
Have students solve
Example 1 by factoring x  5  7 7
49
and compare the results.
x  5  7 Add 5 to each side.
Point out that Example 2
cannot be solved by x  5  7 or x  5  7 Write as two equations.
factoring. x2 x  12 Solve each equation.

The solution set is {2, 12}. You can check this result by using factoring to solve the
original equation.

Roots that are irrational numbers may be written as exact answers in radical form
or as approximate answers in decimal form when a calculator is used.
306 Chapter 6 Quadratic Functions and Inequalities

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 6 Resource Masters Teaching Algebra With Manipulatives 5-Minute Check Transparency 6-4
• Study Guide and Intervention, pp. 331–332 Masters, pp. 244, 245–246 Answer Key Transparencies
• Skills Practice, p. 333
• Practice, p. 334 Technology
• Reading to Learn Mathematics, p. 335 Interactive Chalkboard
• Enrichment, p. 336
• Assessment, pp. 369, 371
 Example 2 Equation with Irrational Roots
Solve x2  6x  9  32 by using the Square Root Property. 2 Teach
x2  6x  9  32 Original equation
(x  3)2  32 Factor the perfect square trinomial. SQUARE ROOT PROPERTY

x  3  32 Square Root Property
In-Class Examples Power
x  3  42   42
Add 3 to each side; 32 Point®

x  3  42 or x  3  42 Write as two equations. Teaching Tip In Example 1,

x
8.7 x
2.7 Use a calculator. point out that both constants in
The exact solutions of this equation are 3  42 and 3  42 . The approximate the equation, 25 and 49, are
solutions are 2.7 and 8.7. Check these results by finding and graphing the related perfect squares.
quadratic function.
x2  6x  9  32 Original equation
1 Solve x2  14x  49  64 by
x2  6x  23  0 Subtract 32 from each side.
using the Square Root
Property. {15, 1}
y x2  6x  23 Related quadratic function

CHECK Use the ZERO function of a graphing

Teaching Tip Point out that the
constant on the right side of the
calculator. The approximate zeros of the
related function are 2.7 and 8.7. equation given in Example 2, is
not a perfect square. Stress that
this occurrence means the roots
will be irrational numbers involv-
ing radicals. Also emphasize the
use of the  symbol in the step
where the Square Root Property
is utilized.
COMPLETE THE SQUARE The Square Root Property can only be used to
solve quadratic equations when the side containing the quadratic expression is a
perfect square. However, few quadratic expressions are perfect squares. To make a 2 Solve x2  10x  25  12 by
quadratic expression a perfect square, a method called completing the square may using the Square Root
be used. Property. {5 23}
In a perfect square trinomial, there is a relationship between the coefficient of the
linear term and the constant term. Consider the pattern for squaring a sum.
(x  7)2  x2  2(7)x  72 Square of a sum pattern
 x2  14x  49 Simplify. COMPLETE THE SQUARE
Teaching Tip When discussing the
 
14 2

2
72 Notice that 49 is 72 and 7 is one-half of 14. steps for completing the square,
emphasize that the coefficient of
You can use this pattern of coefficients to complete the square of a quadratic the quadratic term must be 1.
expression.

Completing the Square

• Words To complete the square for any quadratic expression of the form
x2 ⴙ bx, follow the steps below.
Step 1 Find one half of b, the coefficient of x.
Step 2 Square the result in Step 1.
Step 3 Add the result of Step 2 to x2  bx.

• Symbols x2  bx  b22  x  b22

www.algebra2.com/extra_examples Lesson 6-4 Completing the Square 307

Lesson 6-4 Completing the Square 307

In-Class Examples Power Example 3 Complete the Square
Point®
Find the value of c that makes x2  12x  c a perfect square. Then write the
3 Find the value of c that makes trinomial as a perfect square.
x2  16x  c a perfect square. Step 1 Find one half of 12.
12
  6
2
Then write the trinomial as a Step 2 Square the result of Step 1. 62  36
perfect square. 64; (x  8)2
Step 3 Add the result of Step 2 to x2  12x. x2  12x  36
4 Solve  4x  12  0 by
x2 The trinomial x2  12x  36 can be written as (x  6)2.
completing the square. {6, 2}
Teaching Tip Remind students You can solve any quadratic equation by completing the square. Because you are
solving an equation, add the value you use to complete the square to each side.
to think carefully about the
difference between multiplying
a quantity by a factor of 2 and
squaring a quantity. Completing the Square
Use algebra tiles to complete the square for the equation x2  2x  3  0.
Represent x2  2x  3  0 Add 3 to each side of the
on an equation mat. mat. Remove the zero pairs.

2 2 1 1
x x x x x x
  1

1 1 1 1 1 1 1 1 1

x 2  2x  3  0 x 2  2x  3  3  03

Begin to arrange the x2 To complete the square, add

and x tiles into a square. 1 yellow 1 tile to each side.
The completed equation is
x 2  2x  1  4 or (x  1)2  4.

2 1 1 2 1 1
x x x x
 1  1 1
x x 1

x 2  2x  3 x 2  2x  1  31

Model
Use algebra tiles to complete the square for each equation.
1. x2  2x  4  0 (x  1)2  5 2. x2  4x  1  0 (x  2)2  3
3. x  6x  5 (x  3)  4
2 2 4. x2  2x  1 (x  1)2  0

Study Tip Example 4 Solve an Equation by Completing the Square

Common Solve x2  8x  20  0 by completing the square.
Misconception x2  8x  20  0 Notice that x2  8x  20 is not a perfect square.
When solving equations
by completing the square, x  8x  20
2 Rewrite so the left side is of the form x2  bx.
8 2
b 2
don’t forget to add  to x  8x  16  20  16
2 Since   16, add 16 to each side.
2 2
each side of the equation. (x  4)  36
2 Write the left side as a perfect square by factoring.

308 Chapter 6 Quadratic Functions and Inequalities

Algebra Activity
Materials: algebra tiles, equation mat
• Ask students why the choice was made to add 3 unit tiles to each side of
the equation mat in Step 2. Sample answer: In order to simplify the work
arranging the tiles into a square in Step 3.
• Remind students that an x tile is x units long and 1 unit wide. Stress that the
width is the same as the length of each side of a unit tile.

308 Chapter 6 Quadratic Functions and Inequalities

 x  4  6 Square Root Property
In-Class Examples Power
Point®
x  4  6 Add 4 to each side.
x  4  6 or x  4  6 Write as two equations.
Teaching Tip Some students
x2 x  10 The solution set is {10, 2}. may notice that the left side of
You can check this result by using factoring to solve the original equation. the equation in Example 5 can
be factored into the product of
two binomials: (2x  3)(x  1).
When the coefficient of the quadratic term is not 1, you must first divide the Then the Zero Product Property
equation by that coefficient before completing the square. can be used to obtain the same
solutions. This is a good time to
Example 5 Equation with a ⴝ 1 point out that more than one
method of solution is often
Solve 2x2 ⴚ 5x  3  0 by completing the square.
possible when solving a
2x2  5x  3  0 Notice that 2x2  5x  3 is not a perfect square.
quadratic equation.
5 3
x2  x     0 Divide by the coefficient of quadratic term, 2.
2 2
5 3
5 Solve 3x2  2x  1  0 by
3
x  x  
2 Subtract  from each side.
2 completing the square.
2 2
5 25 3
x  x      
2
2 16 2
25
16 5 2 25
 25
Since   2  , add  to each side.
2 16 16  13 , 1
 
5 2 1 Write the left side as a perfect square by factoring.
x    
4 16 Simplify the right side. 6 Solve x2  2x  3  0 by com-
5
x    
1
Square Root Property pleting the square. {1 i 2}
4 4
5
x     
1 5
Add  to each side.
4
Teaching Tip Ask students to
4 4
name three possible ways to
5 1 5 1
x     or x     Write as two equations. check the solutions to an equa-
4 4 4 4
tion that they have solved by
The solution set is 1,  .
3 3
x   x1 completing the square. graphing
2 2
the related function, factoring the
equation, substituting the solu-
Not all solutions of quadratic equations are real numbers. In some cases, the tions into the original equation
solutions are complex numbers of the form a  bi, where b  0.

Example 6 Equation with Complex Solutions

Solve x2  4x  11  0 by completing the square.
x2  4x  11  0 Notice that x2  4x  11 is not a perfect square.
x2  4x  11 Rewrite so the left side is of the form x2  bx.
4 2
x2  4x  4  11  4 Since   4, add 4 to each side.
2
(x  2)2  7 Write the left side as a perfect square by factoring.

x  2   7 Square Root Property
x  2   i7 i
1
x  2  i7 Subtract 2 from each side.

The solution set is {2  i7, 2  i7 }. Notice

that these are imaginary solutions.

CHECK A graph of the related function shows that

the equation has no real solutions since the
graph has no x-intercepts. Imaginary
solutions must be checked algebraically by
substituting them in the original equation.

Lesson 6-4 Completing the Square 309

Differentiated Instruction
Kinesthetic Have students work with algebra tiles to help them write five
equations that can be solved by completing the square. Provide each stu-
dent with one x2 tile, several x tiles, and several unit tiles. Have students
begin by creating a square arrangement of their tiles and then work back-
wards through the steps shown in the Algebra Activity on p. 308 to find a
quadratic equation. After students have written their five equations, ask
them to trade their equations with another student and then use their
algebra tiles to find the solutions of the equations they receive.
Lesson 6-4 Completing the Square 309
 3 Practice/Apply Concept Check 1. Explain what it means to complete the square.
1. Completing the 2. Determine whether the value of c that makes ax2  bx  c a perfect square
square allows you to trinomial is sometimes, always, or never negative. Explain your reasoning.
rewrite one side of a 3. FIND THE ERROR Rashid and Tia are solving 2x2  8x  10  0 by completing
Study Notebook quadratic equation in the square.
Have students— the form of a perfect
square. Once in this Rashid T ia
• add the definitions/examples of form, the equation is
the vocabulary terms to their solved by using the 2x2 – 8x + 10 = 0 2x 2 – 8x + 10 = 0
Square Root Property. 2x2 – 8x = –10 x 2 – 4x = 0 – 5
Vocabulary Builder worksheets for 2
2. Never; see margin 2x2 – 8x + 16 = –10 + 16 x – 4x + 4 = –5 + 4
Chapter 6. for explanation. (x – 4)2 = 6 (x – 2) 2 = –1
• include any other item(s) that they x – 4 =–+ ∑6 x – 2 = +– i
find helpful in mastering the skills x = 4–+ ∑6 x = 2 +– i
in this lesson.
Who is correct? Explain your reasoning. Tia; see margin for explanation.

Guided Practice Solve each equation by using the Square Root Property.
4 2
GUIDED PRACTICE KEY 4. x2  14x  49  9 {10, 4} 5. 9x2  24x  16  2  3
Exercises Examples
Find the value of c that makes each trinomial a perfect square. Then write the
About the Exercises… 4, 5, 1, 2 trinomial as a perfect square.
Organization by Objective 12, 13
6, 7 3
6. x2  12x  c 36; (x  6)2
9

3 2
7. x2  3x  c ; x  
4 2 
• Square Root Property: 14–23 8–11 4–6
• Complete the Square: 24–51, Solve each equation by completing the square.
53 8. x2  3x  18  0 {6, 3} 9. x2  8x  11  0 4 5 { }
Odd/Even Assignments
10. x2 { 
 2x  6  0 1 i 5 } 

11. 2x2  3x  3  0 3 33

4
Exercises 14–47 are structured Application ASTRONOMY For Exercises 12 and 13, use the following information.
so that students practice the 1
The height h of an object t seconds after it is dropped is given by h  gt2  h0,
same concepts whether they 2
are assigned odd or even where h0 is the initial height and g is the acceleration due to gravity. The acceleration
due to gravity near Earth’s surface is 9.8 m/s2, while on Jupiter it is 23.1 m/s2. Suppose
problems. 7 5
Alert! Exercise 51 involves
18.   2 an object is dropped from an initial height of 100 meters from the surface of each planet.
12. On which planet should the object reach the ground first? Jupiter

5 11
research on the Internet or
other reference materials.
19.  3 13. Find the time it takes for the object to reach the ground on each planet to the
nearest tenth of a second. Earth: 4.5 s, Jupiter: 2.9 s
★ indicates increased difficulty
Assignment Guide Practice and Apply
Basic: 15–19 odd, 23–47 odd,
Homework Help Solve each equation by using the Square Root Property.
52, 54–72 For See
Exercises Examples
14. x2  4x  4  25 {3, 7} 15. x2  10x  25  49 {2, 12}
Average: 15–47 odd, 52–72 14–23, 48 1, 2 16. x2 { 
 8x  16  7 4 7 } {
17. x2  6x  9  8 3 22 }
Advanced: 14–48 even, 49–52, 24–31 3
18. 4x2  28x  49  5 19. 9x2  30x  25  11
32–47, 4–6
54–68 (optional: 69–72) 49–50, 53 ★ 20. x2  x  4  16 4, 4
1 9 5 1
★ 21. x2  1.4x  0.49  0.81 {1.6, 0.2}
Extra Practice 22. MOVIE SCREENS The area A in square feet of a projected picture on a movie
See page 840. screen is given by A  0.16d2, where d is the distance from the projector to
FIND THE ERROR the screen in feet. At what distance will the projected picture have an area
of 100 square feet? 25 ft
Point out that,
while it is possible to 310 Chapter 6 Quadratic Functions and Inequalities

complete the square when the

coefficient of the x2 term is some-
thing other than 1, it is much Answers
easier to first divide each side by b
2. The value of c that makes ax 2  bx  c a perfect square trinomial is the square of  and
the coefficient and students will 2
the square of a number can never be negative.
also be less likely to make an
error like the one made by Rashid 3. Before completing the square, you must first check to see that the coefficient of the
shown here. quadratic term is 1. If it is not, you must first divide the equation by that coefficient.

310 Chapter 6 Quadratic Functions and Inequalities

 23. ENGINEERING In an engineering test, a rocket sled is propelled into a Study
NAME ______________________________________________ DATE

Guide andIntervention
Intervention,
____________ PERIOD _____

6-4 Study Guide and

target. The sled’s distance d in meters from the target is given by the formula p. 331 (shown) and p. 332
Completing the Square
d  1.5t2  120, where t is the number of seconds after rocket ignition. How Square Root Property Use the following property to solve a quadratic equation that is
many seconds have passed since rocket ignition when the sled is 10 meters from in the form “perfect square trinomial  constant.”

For any real number x if x 2  n, then x  n.

the target? about 8.56 s Square Root Property

Example Solve each equation by using the Square Root Property.

a. x2  8x  16  25 b. 4x2  20x  25  32
Find the value of c that makes each trinomial a perfect square. Then write the x2  8x  16  25 4x2  20x  25  32
(x  4)2  25 (2x  5)2  32
trinomial as a perfect square. x  4   25 or x  4  25
 2x  5  32
 or 2x  5  32

x  5  4  9 or x  5  4  1 2x  5  42 or 2x  5  42

24. x2  16x  c 64; (x  8)2 25. x2  18x  c 81; (x  9)2 The solution set is {9, 1}. 5  42
x  
2


225
26. x2  15x  c ; x  
4 
15 2
2  49
27. x2  7x  c ; x  
4
7 2
2    5  242
The solution set is  .

28. x2  0.6x  c 0.09; (x  0.3)2 29. x2  2.4x  c 1.44; (x  1.2)2 Exercises

8
3
16
9 
30. x2  x  c ; x  
4 2
3  5 25 5 2
31. x2  x  c ; x  
2 16 4   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} 32 , 25

Engineering Solve each equation by completing the square.
Reverse ballistic testing— 32. x2  8x  15  0 {3, 5} 33. x2  2x  120  0 {12, 10} 4. 36x2  12x  1  18 5. 9x2  12x  4  4 6. 25x2  40x  16  28

accelerating a target on a  
1 32

0, 34  
4 27


{
34. x2  2x  6  0 1 7 } 
35. x2  4x  1  0 2 3 { } 6 5

Lesson 6-4
sled to impact a stationary
test item at the end of the 36. x2  4x  5  0 {2 i } 37. x2  6x  13  0 {3 2i} 7. 4x2  28x  49  64 8. 16x2  24x  9  81 9. 100x2  60x  9  121
track—was pioneered at  2 23 , 3
5
 1

15 1
,  {0.8, 1.4}
the Sandia National 38. 2x2  3x  5  0 , 1 39. 2x2  3x  1  0 , 1 2

2 2
Laboratories’ Rocket 40. 3x2  5x  1  0 41. 3x2  4x  2  0 10. 25x2  20x  4  75 11. 36x2  48x  16  12 12. 25x2  30x  9  96
Sled Track Facility in
42. 2x2  7x  12  0 43. 3x2  5x  4  0  
2 53

 
2 3

 
3 46


Albuquerque, New Mexico. 5 3 5

This facility provides a 44. x2  1.4x  1.2 {2, 0.6} 45. x2  4.7x  2.8 {0.7, 4} Gl NAME
/M G ______________________________________________
Hill 331 DATE ____________
GlPERIOD
Al _____
b 2

10,000-foot track for Skills Practice, p. 333 and

2 26 1
  3 23 3
 
6-4 Practice
46. x2  x    0  3 47. x2  x    0  2
(Average)
testing items at very high 3 9 3 2 16 4 Practice,
Completingp.
the334
Square(shown)
speeds. Solve each equation by using the Square Root Property.

1. x2  8x  16  1 2. x2  6x  9  1 3. x2  10x  25  16
Source: www.sandia.gov 48. FRAMING A picture has a square frame that 5, 3 4, 2 9, 1
is 2 inches wide. The area of the picture is one-third s 4. x2  14x  49  9 5. 4x2  12x  9  4 6. x2  8x  16  8
1 5
 ,  4 22

of the total area of the picture and frame. What 4, 10
2 2

 6 
5 13
7. x2  6x  9  5 8. x2  2x  1  2 9. 9x2  6x  1  2
are the dimensions of the picture to the nearest 1 2

40.  quarter of an inch? 51 in. by 51 in.
3 5
 1 2
 
3

2 in. Find the value of c that makes each trinomial a perfect square. Then write the
2 2 trinomial as a perfect square.

41.  
2 10  10. x2  12x  c
36; (x  6)2
11. x2  20x  c
100; (x  10)2
12. x2  11x  c
121
; x   
11 2

3 4 2
13. x2  0.8x  c 14. x2  2.2x  c 15. x2  0.36x  c

42.   
7 i 47
2 in. 0.16; (x  0.4)2 1.21; (x  1.1)2 0.0324; (x  0.18)2
5 1 5
4 16. x2   x  c
6
17. x2   x  c
4
18. x2   x  c
3
25
; x  
5 2
 1

1 2
; x    25
5 2
; x   
43.   
5 i 23

144 12 64 8 36 6
GOLDEN RECTANGLE For Exercises 49–51, use the A E B Solve each equation by completing the square.
6 following information. 19. x2  6x  8  0 4, 2 20. 3x2  x  2  0  , 1
2
3
21. 3x2  5x  2  0 1, 
2
3
A golden rectangle is one that can be divided into a 22. x2  18  9x 23. x2  14x  19  0 24. x2  16x  7  0
6, 3 7 30
 8 71

square and a second rectangle that is geometrically 1
25. 2x2  8x  3  0 26. x2  x  5  0 27. 2x2  10x  5  0
similar to the original rectangle. The ratio of the 4 22

2
 1 21

2
 5 15

2


length of the longer side to the shorter side of a 1 x1 28. x2  3x  6  0 29. 2x2  5x  6  0 30. 7x2  6x  2  0
3 i 15
  5 i 23
  3 i 5
 
golden rectangle is called the golden ratio. D x F C 2 4 7

x 1 31. GEOMETRY When the dimensions of a cube are reduced by 4 inches on each side, the
49. ,  49. Find the ratio of the length of the longer side to surface area of the new cube is 864 square inches. What were the dimensions of the
1 x1 original cube? 16 in. by 16 in. by 16 in.

the length of the shorter side for rectangle ABCD 32. INVESTMENTS The amount of money A in an account in which P dollars is invested for
2 years is given by the formula A  P(1  r)2, where r is the interest rate compounded
and for rectangle EBCF. annually. If an investment of $800 in the account grows to $882 in two years, at what


1  5
interest rate was it invested? 5%

50.  50. Find the exact value of the golden ratio by setting the two ratios in Exercise 49 Gl
Reading
NAME
/M G ______________________________________________
Hill 334 DATE ____________
Gl PERIOD
Al _____
b 2
2 6-4 Readingto
to Learn
Learn Mathematics
equal and solving for x. (Hint: The golden ratio is a positive value.) Mathematics, p. 335 ELL
Completing the Square
Pre-Activity How can you find the time it takes an accelerating race car to
51. RESEARCH Use the Internet or other reference to find examples of the golden reach the finish line?
Read the introduction to Lesson 6-4 at the top of page 306 in your textbook.
rectangle in architecture. What applications does the reciprocal of the golden Explain what it means to say that the driver accelerates at a constant rate
ratio have in music? See margin. of 8 feet per second square.
If the driver is traveling at a certain speed at a particular
moment, then one second later, the driver is traveling 8 feet
per second faster.

52. CRITICAL THINKING Find all values of n such that x2  bx    n has
b 2
Reading the Lesson
2
a. one real root. n  0 b. two real roots. n  0
1. Give the reason for each step in the following solution of an equation by using the
c. two imaginary roots. Square Root Property.

n0 x2  12x  36  81 Original equation

www.algebra2.com/self_check_quiz Lesson 6-4 Completing the Square 311 (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.
NAME ______________________________________________ DATE ____________ PERIOD _____ x  15 x  3 Solve each equation.
51. Sample answers: The golden rectangle is
Enrichment,
6-4 Enrichment p. 336 2. Explain how to find the constant that must be added to make a binomial into a perfect

found in much of ancient Greek architecture, square trinomial.

Sample answer: Find half of the coefficient of the linear term and square it.

such as the Parthenon, as well as in modern The Golden Quadratic Equations

A golden rectangle has the property that its length a b
3. a. What is the first step in solving the equation 3x2  6x  5 by completing the square?
Divide the equation by 3.

architecture, such as in the windows of the can be written as a  b, where a is the width of the
ab a
rectangle and   . Any golden rectangle can be
b. What is the first step in solving the equation x2  5x  12  0 by completing the
Lesson 6-4

a b square? Add 12 to each side.

United Nations building. Many songs have divided into a square and a smaller golden rectangle,
as shown.
a a

Helping You Remember

their climax at a point occurring 61.8% of The proportion used to define golden rectangles can be
used to derive two quadratic equations. These are
sometimes called golden quadratic equations.
a b
4. How can you use the rules for squaring a binomial to help you remember the procedure
for changing a binomial into a perfect square trinomial?
One of the rules for squaring a binomial is (x  y) 2  x 2  2xy  y 2. In
the way through the piece, with 0.618 being Solve each problem.
completing the square, you are starting with x 2  bx and need to find y 2.
b
This shows you that b  2y, so y  
about the reciprocal of the golden ratio. The 1. In the proportion for the golden rectangle, let a equal 1. Write the resulting
quadratic equation and solve for b.
. That is why you must take half of
2
the coefficient and square it to get the constant that must be added to
b10
reciprocal of the golden ratio is also used in b2
1  5
b   
complete the square.

2
the design of some violins. 2 In the proportion let b equal 1 Write the resulting quadratic equation and

Lesson 6-4 Completing the Square 311

 ★ 53. KENNEL A kennel owner has 164 feet of fencing
4 Assess with which to enclose a rectangular region. He wants
to subdivide this region into three smaller rectangles
of equal length, as shown. If the total area to be w

Open-Ended Assessment enclosed is 576 square feet, find the dimensions of

the entire enclosed region. (Hint: Write an expression ᐉ ᐉ ᐉ
Writing Have students write a for ᐉ in terms of w.) 18 ft by 32 ft or 64 ft by 9 ft
summary of the various
techniques for solving quadratic 54. WRITING IN MATH Answer the question that was posed at the beginning
of the lesson. See margin.
equations using the Square Root
How can you find the time it takes an accelerating race car to reach the
Property. Students should finish line?
provide written examples of each Include the following in your answer:
technique.
• an explanation of why t2  22t  121  246 cannot be solved by factoring,
and
• a description of the steps you would take to solve the equation
Intervention t2  22t  121  246.
New Suggest stu-
Standardized 55. What is the absolute value of the product of the two solutions for x in
dents write a Test Practice x2  2x  2  0? D
summary of A 1 B 0 C 1 D 2
the various
methods that can be used to 56. For which value of c will the roots of x2  4x  c  0 be real and equal? D
solve quadratic equations. A 1 B 2 C 3 D 4 E 5
Ask them which method they
prefer to use, and why they
like that method the best.
Maintain Your Skills
Mixed Review Write a quadratic equation with the given root(s). Write the equation in the form
Getting Ready for ax2  bx  c  0, where a, b, and c are integers. (Lesson 6-3)
Lesson 6-5 57. 2, 1 58. 3, 9
1
59. 6, 
3
60.  , 
1
3
3
4
PREREQUISITE SKILL Lesson 6-5 x 2  3x  2  0 x 2  6x  27  0 3x 2  19x  6  0 12x 2  13x  3  0
Solve each equation by graphing. If exact roots cannot be found, state the
presents the Quadratic Formula. consecutive integers between which the roots are located. (Lesson 6-2)
The first step in evaluating the 1
61. between 4 and 61. 3x2  4  8x 62. x2  48  14x 6, 8 63. 2x2  11x  12 4, 1
formula is to evaluate the expres- 2
3; between 0 and 1 3

sion under the radical sign. Use 64. Write the seventh root of 5 cubed using exponents. (Lesson 5-7) 5 7
Exercises 69–72 to determine
Solve each system of equations by using inverse matrices. (Lesson 4-8)
your students’ familiarity with
65. 5x  3y  5 66. 6x  5y  8
evaluating expressions.
7x  5y  11 (2, 5)
43
 6
3x  y  7 , 
21 7 
Assessment Options CHEMISTRY For Exercises 67 and 68, use the following information.
Quiz (Lessons 6-3 and 6-4) is For hydrogen to be a liquid, its temperature must be within 2°C of 257°C. (Lesson 1-4)
available on p. 369 of the Chapter 6 67. Write an equation to determine the greatest and least temperatures for this
substance. x  (257)  2
Resource Masters.
68. Solve the equation. greatest: 255°C; least: 259°C
Mid-Chapter Test (Lessons 6-1
through 6-4) is available on Getting Ready for PREREQUISITE SKILL Evaluate b2  4ac for the given values of a, b, and c.
p. 371 of the Chapter 6 Resource the Next Lesson (To review evaluating expressions, see Lesson 1-1.)
Masters. 69. a  1, b  7, c  3 37 70. a  1, b  2, c  5 16
71. a  2, b  9, c  5 121 72. a  4, b  12, c  9 0
312 Chapter 6 Quadratic Functions and Inequalities

Answer
54. To find the distance traveled by • Since the expression t 2  22t  125 is prime, the solutions of t 2  22t  121  246
the accelerating race car in the cannot be obtained by factoring.
given situation, you must solve • Rewrite t 2  22t  121 as (t  11)2. Solve (t  11)2  246 by applying the Square Root
the equation t 2  22t  121  Property. Then, subtract 11 from each side. Using a calculator, the two solutions are
246 or t 2  22t  125  0. about 4.7 or 26.7. Since time cannot be negative, the driver takes about 4.7 seconds
Answers should include the to reach the finish line.
following.

312 Chapter 6 Quadratic Functions and Inequalities

 The Quadratic Formula Lesson
and the Discriminant Notes
• Solve quadratic equations by using the Quadratic Formula.
• Use the discriminant to determine the number and type of roots of a quadratic
equation. 1 Focus
Vocabulary is blood pressure related to age?
• Quadratic Formula 5-Minute Check
• discriminant As people age, their arteries lose their elasticity, Transparency 6-5 Use as a
which causes blood pressure to increase. For quiz or review of Lesson 6-4.
healthy women, average systolic blood pressure
is estimated by P  0.01A2  0.05A  107,
where P is the average blood pressure in
Mathematical Background notes
millimeters of mercury (mm Hg) and A is are available for this lesson on
the person’s age. For healthy men, average p. 284D.
systolic blood pressure is estimated by
P  0.006A2 – 0.02A  120.
Building on Prior
Knowledge
QUADRATIC FORMULA You have seen that exact solutions to some quadratic In Lesson 6-4, students solved
equations can be found by graphing, by factoring, or by using the Square Root quadratic equations by completing
Property. While completing the square can be used to solve any quadratic equation, the square. In this lesson, students
the process can be tedious if the equation contains fractions or decimals. Fortunately,
a formula exists that can be used to solve any quadratic equation of the form generalize this procedure as they
ax2  bx  c  0. This formula can be derived by solving the general form of a complete the square for the
quadratic equation. general quadratic equation to
ax2  bx  c  0 General quadratic equation derive the Quadratic Formula.
b c
x2  x    0 Divide each side by a.
a a is blood pressure
b c c related to age?
x2  x   Subtract  from each side.
a
a a
b b2 c b2 Ask students:
x2  x  2    2 Complete the square.
a 4a a 4a
• As the value of A increases in
b2  4ac
 b 2
2a 
x    
4a2
Factor the left side. Simplify the right side. these equations, what happens
b   
b2  4ac to the value of P? It increases.
x     
2a Square Root Property
2a • Which way do the parabolas
b   
b2  4ac
b open that are the graphs of
x    
2a Subtract  from each side.
2a
2a
these equations? upward
b  
b  4ac
2
x   2a Simplify.

This equation is known as the Quadratic Formula .

Study Tip
Reading Math Quadratic Formula
The Quadratic Formula is
The solutions of a quadratic equation of the form ax2  bx  c  0, where a  0,
read x equals the opposite
are given by the following formula.
of b, plus or minus the
square root of b squared b b 
 4ac 2
minus 4ac, all divided
x   2a
by 2a.

Lesson 6-5 The Quadratic Formula and the Discriminant 313

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 6 Resource Masters Graphing Calculator and 5-Minute Check Transparency 6-5
• Study Guide and Intervention, pp. 337–338 Spreadsheet Masters, p. 38 Answer Key Transparencies
• Skills Practice, p. 339
• Practice, p. 340 Technology
• Reading to Learn Mathematics, p. 341 Alge2PASS: Tutorial Plus, Lessons 11, 12
• Enrichment, p. 342 Interactive Chalkboard

Lesson x-x Lesson Title 313

 Example 1 Two Rational Roots
2 Teach Solve x2  12x  28 by using the Quadratic Formula.
First, write the equation in the form ax2  bx  c  0 and identify a, b, and c.
QUADRATIC FORMULA ax2  bx  c  0

In-Class Examples Power

Point® x2  12x  28 1x2  12x  28  0

1 Solve x2  8x  33 by using Then, substitute these values into the Quadratic Formula.
the Quadratic Formula. 3, 11 b  
b  4ac
2
x   2a Quadratic Formula
Teaching Tip Encourage stu- (12)  
(12)  
 4(1)(28)
2
dents to write down the values x   2(1) Replace a with 1, b with 12, and c with 28.
of a, b, and c from the standard 12  
144 
112
form of the quadratic equation x   2 Simplify.
Study Tip
before they begin substituting 
12  256
into the formula. Quadratic x   2 Simplify.
Formula
12  16
2 Solve x2  34x  289  0 by Although factoring may be x     16
256
an easier method to solve 2
using the Quadratic Formula. the equations in Examples 12  16
x   or x  
12  16
Write as two equations.
17 1 and 2, the Quadratic 2 2
Formula can be used to
solve any quadratic
 14  2 Simplify.
equation.
The solutions are 2 and 14. Check by substituting each of these values into the
original equation.

When the value of the radicand in the Quadratic Formula is 0, the quadratic
equation has exactly one rational root.

Example 2 One Rational Root

Solve x2  22x  121  0 by using the Quadratic Formula.
Identify a, b, and c. Then, substitute these values into the Quadratic Formula.

b  
b  4ac
2
x   2a Quadratic Formula

(22)   (22)2 
 
4(1)(121)
x   2(1) Replace a with 1, b with 22, and c with 121.

22  0
x  2 Simplify.

22
x   or 11 0  0
2
The solution is 11.

CHECK A graph of the related function shows that

there is one solution at x  11.

[15, 5] scl: 1 by [5, 15] scl: 1

314 Chapter 6 Quadratic Functions and Inequalities

Teacher to Teacher
Lori Haldorson & Cathy Hokkanen Blaine H.S., Blaine, MN
"To help students memorize the Quadratic Formula, we sing it
to the tune of ‘Pop Goes the Weasel’."

314 Chapter 6 Quadratic Functions and Inequalities

 You can express irrational roots exactly by writing them in radical form.
In-Class Examples Power
Point®
Example 3 Irrational Roots
Teaching Tip Some students
Solve 2x2  4x  5  0 by using the Quadratic Formula. may have commented that the
b  
b  4ac
2 equations in Examples 1 and 2
x   2a
Quadratic Formula could have been solved by factor-
ing. Before beginning Example 3,
(4)  
(4) 2 
4(2)(5)
x   2(2) Replace a with 2, b with 4, and c with 5. stress that many quadratic equa-
tions cannot easily be solved by

4  56 factoring. State that the quadratic
x   4 Simplify.
equation presented in Example 3

4  214 2  
14 is such an equation. Emphasize
x   4 or 
2 
56   
4 • 14 or 214
that the Quadratic Formula

2  14 
2  14 provides a way to find the roots
The exact solutions are  and . The approximate solutions
2 2 for any quadratic equation.
are 2.9 and 0.9.

CHECK Check these results by graphing

3 Solve x2  6x  2  0 by
the related quadratic function, y  2x  4x  5
2
using the Quadratic Formula.
y  2x2  4x  5. Using the ZERO 3 7, or approximately 0.4
function of a graphing calculator, and 5.6
the approximate zeros of the related
function are 2.9 and 0.9. Teaching Tip Remind students
[10, 10] scl: 1 by [10, 10] scl: 1 that conjugate pairs are two
complex numbers of the form
a  bi and a  bi.
When using the Quadratic Formula, if the radical contains a negative value, the
solutions will be complex. Complex solutions always appear in conjugate pairs. 4 Solve x2  13  6x by using
the Quadratic Formula.
Example 4 Complex Roots 3 2i
Solve x2 – 4x  13 by using the Quadratic Formula.
Study Tip b  
b  4ac
2
x   2a Quadratic Formula
Concept Check
Using the
Quadratic (4)  
x  
(4)2 
 
 4(1)(13)
Replace a with 1, b with 4, and c with 13.
Real Roots or Imaginary Roots
Formula 2(1) Ask students to look back at the
Remember that to

4  36 first four examples in this lesson
correctly identify a, b, and x   2 Simplify.
c for use in the Quadratic and see how they might predict
Formula, the equation 4  6i whether the roots will be real or
must be written in the x   36  36(1
 ) or 6i
2
form ax2  bx  c  0. imaginary. If the values of a and c
x  2  3i Simplify.
have the same sign and 4 times their
product is greater than the square of
The solutions are the complex numbers 2  3i and 2  3i. the value of b, then the roots will be
imaginary.
A graph of the related function shows that the
solutions are complex, but it cannot help you find
them.

[15, 5] scl: 1 by [2, 18] scl: 1

www.algebra2.com/extra_examples Lesson 6-5 The Quadratic Formula and the Discriminant 315

Lesson 6-5 The Quadratic Formula and the Discriminant 315

ROOTS AND THE CHECK To check complex solutions, you must substitute them into the original
equation. The check for 2  3i is shown below.
DISCRIMINANT
x2  4x  13 Original equation

In-Class Example Power (2  3i)2  4(2  3i) ⱨ 13 x  2  3i

Point®
4  12i  9i2  8  12i ⱨ 13 Sum of a square; Distributive Property
5 Find the value of the dis- 4  9i2 ⱨ 13 Simplify.
criminant for each quadratic 4  9  13 ⻫ i 2  1
equation. Then describe the
number and type of roots for
the equation. Study Tip ROOTS AND THE DISCRIMINANT In Examples 1, 2, 3, and 4, observe the
Reading Math relationship between the value of the expression under the radical and the roots of
a. x2  6x  9  0 Remember that the the quadratic equation. The expression b2  4ac is called the discriminant.
0; one rational root solutions of an equation
b  
b2  4ac discriminant
are called roots. x  
b. x2  3x  5  0 2a
11; two complex roots The value of the discriminant can be used to determine the number and type of
c. x2  8x  4  0 roots of a quadratic equation.
80; two irrational roots
d. x2  11x  10  0
Discriminant
81; two rational roots Consider ax2 ⴙ bx ⴙ c ⴝ 0.
Type and Example of Graph
Value of Discriminant
Number of Roots of Related Function
b2  4ac  0; y
2 real,
b2  4ac is
rational roots
a perfect square.
O x
b2  4ac  0;
b2  4ac is not 2 real,
a perfect square. irrational roots

b2  4ac  0 1 real,
rational root
O x

b2  4ac  0 2 complex roots

O x

Study Tip Example 5 Describe Roots

Using the Find the value of the discriminant for each quadratic equation. Then describe
Discriminant the number and type of roots for the equation.
The discriminant can help
you check the solutions of
a. 9x2  12x  4  0 b. 2x2  16x  33  0
a quadratic equation. Your a  9, b  12, c  4 a  2, b  16, c  33
solutions must match in
b2  4ac  (12)2  4(9)(4) b2  4ac  (16)2  4(2)(33)
number and in type to
those determined by the  144  144  256  264
discriminant.
0  8
The discriminant is 0, so The discriminant is negative, so
there is one rational root. there are two complex roots.
316 Chapter 6 Quadratic Functions and Inequalities

Differentiated Instruction
Logical Have students use their classification skills to create a classroom
poster listing the four different types of roots that can result when solving
a quadratic equation. Each listing should include a sample equation that
results in that type of roots and an explanation of how the value of the
discriminant is indicative of the root type. Graphs like those shown on
p. 316 can be added to the poster.

316 Chapter 6 Quadratic Functions and Inequalities

 c. 5x2  8x  1  0 d. 7x  15x2  4  0
a  5, b  8, c  1
b2  4ac  (8)2  4(5)(1)
a  15, b  7, c  4
b2  4ac  (7)2  4(15)(4)
3 Practice/Apply
 64  20  49  240
 44  289 or 172
The discriminant is 44, The discriminant is 289, Study Notebook
which is not a perfect which is a perfect square.
square. Therefore, there Therefore, there are Have students—
are two irrational roots. two rational roots. • add the definitions/examples of
the vocabulary terms to their
You have studied a variety of methods for solving quadratic equations. The table Vocabulary Builder worksheets for
below summarizes these methods.
Chapter 6.
• copy the information provided in
Solving Quadratic Equations
the Concept Summary on p. 317
Method Can be Used When to Use
into their notebook.
Use only if an exact answer is not
required. Best used to check the
• include any other item(s) that they
Graphing sometimes
reasonableness of solutions found find helpful in mastering the skills
algebraically.
in this lesson.
Use if the constant term is 0 or if
Factoring sometimes the factors are easily determined.
Example x2  3x  0

Use for equations in which a perfect

Square Root Property sometimes square is equal to a constant.
Example (x  13)2  9 About the Exercises…
Useful for equations of the form Organization by Objective
Completing the
Square
always x2  bx  c  0, where b is even. • Quadratic Formula: 14–39,
Example x2  14x  9  0
42–44
Useful when other methods fail or • Roots and the Discriminant:
Quadratic Formula always are too tedious. 14–27, 40, 41
Example 3.4x2  2.5x  7.9  0
Odd/Even Assignments
Exercises 14–39 are structured
so that students practice the
same concepts whether they
Concept Check 1. OPEN ENDED Sketch the graph of a quadratic equation whose discriminant is are assigned odd or even
a. positive. b. negative. c. zero. a–c. See margin. problems.
2. The square root of a 2. Explain why the roots of a quadratic equation are complex if the value of the Assignment Guide
negative number is a discriminant is less than 0.
complex number. 3. Describe the relationship that must exist between a, b, and c in the equation Basic: 15–25 odd, 29–39 odd,
ax2  bx  c  0 in order for the equation to have exactly one solution. 40, 41, 45–66
b2 ⴚ 4ac must equal 0. Average: 15–39 odd, 40–43,
Guided Practice Complete parts a–c for each quadratic equation. 45–66
a. Find the value of the discriminant. 4–7. See margin.
GUIDED PRACTICE KEY Advanced: 14–38 even, 42–60
Exercises Examples b. Describe the number and type of roots.
(optional: 61–66)
4–7 1–5 c. Find the exact solutions by using the Quadratic Formula.
8–11 1–4 4. 8x2  18x  5  0 5. 2x2  4x  1  0
12, 13 1–4
6. 4x2  4x  1  0 7. x2  3x  8  5 Answers
Lesson 6-5 The Quadratic Formula and the Discriminant 317 4a. 484 4b. 2 rational
1 5
4c.  ,  
4 2
1a. Sample answer: 1b. Sample answer: 1c. Sample answer: 5a. 8 5b. 2 irrational
y y y
2 2
5c. 
2
6a. 0 6b. one rational
1
6c.  
2
O x O x O x 7a. 3 7b. 2 complex
3 i 3
7c. 
2

Lesson 6-5 The Quadratic Formula and the Discriminant 317

 Study
NAME ______________________________________________ DATE

Guide andIntervention
Intervention,
____________ PERIOD _____
Solve each equation using the method of your choice. Find exact solutions.
6-5 Study Guide and
p. 337 (shown) and p.Discriminant
338 8. x2  8x  0 0, 8 9. x2  5x  6  0 3, 2
5 ⴞ i 2
The Quadratic Formula and the
Quadratic Formula The Quadratic Formula can be used to solve any quadratic
equation once it is written in the form ax2  bx  c  0. 10. x2 
 2x  2  0 1 ⴞ 3 11. 4x2  20x  25  2  2
b   
b 2  4ac
Quadratic Formula The solutions of ax 2  bx  c  0, with a  0, are given by x   .
2a

Example Solve x2  5x  14 by using the Quadratic Formula.

Application PHYSICS For Exercises 12 and 13, use the following information.
Rewrite the equation as x2  5x  14  0. The height h(t) in feet of an object t seconds after it is propelled straight up from the
x  
b   
 4ac
b2
2a
Quadratic Formula ground with an initial velocity of 85 feet per second is modeled by h(t)  16t2  85t.
(5)  
(5)2  )
 4(1)(14
 

5 81

2(1)
Replace a with 1, b with 5, and c with 14.
12. at about 0.7 s and 12. When will the object be at a height of 50 feet?
 
59
2
Simplify.
again at about 4.6 s 13. Will the object ever reach a height of 120 feet? Explain your reasoning.

2
 7 or 2 No; see margin for explanation.
★ indicates increased difficulty
The solutions are 2 and 7.

Exercises
Solve each equation by using the Quadratic Formula. Practice and Apply
1. x2  2x  35  0 2. x2  10x  24  0 3. x2  11x  24  0
5, 7 4, 6 3, 8
Homework Help Complete parts a–c for each quadratic equation.
4. 4x2  19x  5  0 5. 14x2  9x  1  0 6. 2x2  x  15  0 For See a. Find the value of the discriminant. 14–27. See margin.
1
 , 5
4
1
 1
, 
2 7
5
3,  
2
Exercises Examples
7. 3x2  5x  2 8. 2y2  y  15  0 9. 3x2  16x  16  0
14–27 1–5 b. Describe the number and type of roots.
1 5 4 28–39, 1–4
2, 
3
 , 3
2
4, 
3 c. Find the exact solutions by using the Quadratic Formula.
42–44
10. 8x2  6x  9  0 11. r2      0
3r
5
2
25
12. x2  10x  50  0 40–41 5 14. x2  3x  3  0 15. x2  16x  4  0
3 3
 , 2 1
,  5 53

Lesson 6-5

2 4 5 5
16. x2  2x  5  0 17. x2  x  6  0
13. x2  6x  23  0 14. 4x2  12x  63  0 15. x2  6x  21  0 Extra Practice
3 42
 3 62

 3 2i 3
 See page 841. 18. 12x2  5x  2  0 19. 3x2  5x  2  0
2

Gl NAME
/M G ______________________________________________
Hill 337 DATE ____________
GlPERIOD
Al _____
b 2 20. x2  4x  3  4 21. 2x  5  x2
Skills
6-5 Practice,
Practice p. 339 and
Practice,
(Average)
p. Formula
340 (shown) 22. 9x2  6x  4  5 23. 25  4x2  20x
The Quadratic and the Discriminant
Complete parts ac for each quadratic equation. 24. 4x2  7  9x 25. 3x  6  6x2
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. ★ 26. 34x2  13x  1  0 ★ 27. 0.4x2  x  0.3  0
 16x  64  0  3x  24x  16  0
1. x2
0; 1 rational; 8
2. x2
9; 2 rational; 0, 3
3. 9x2
4
0; 1 rational; 
28–39. See pp. 343A–343F.
4. x2  3x  40 5. 3x2  9x  2  0 105; 6. 2x2  7x  0
3
Solve each equation by using the method of your choice. Find exact solutions.
9 105

28. x2  30x  64  0 29. 7x2  3  0 30. x2  4x  7  0
7
169; 2 rational; 5, 8 2 irrational;  49; 2 rational; 0,  
6 2
7. 5x2  2x  4  0 76; 8. 12x2  x  6  0 289; 9. 7x2  6x  2  0 20;
1 i 19
2 complex;  
5
3
2 rational;  2
, 
4 3
3 i 5
2 complex;  
7
31. 2x2  6x  3  0 32. 4x2 80 33. 4x2  81  36x
10. 12x2  2x  4  0 196;
1
2 rational;  2
, 
11. 6x2  2x  1  0 28;
1 7
2 irrational;  
12. x2  3x  6  0 15;
2 complex; 
3 i 15
 34. 4(x  3)2  28 35. 3x2  10x  7 36. x2  9  8x
2 3 6 2
13. 4x2  3x2  6  0 105; 14. 16x2  8x  1  0 15. 2x2  5x  6  0 73; 37. 10x2  3x  0 38. 2x2  12x  7  5 39. 21  (x  2)2  5
3 105
2 irrational;   1
0; 1 rational;  5 73
2 irrational;  
8 4 4

Solve each equation by using the method of your choice. Find exact solutions. BRIDGES For Exercises 40 and 41, use the following information.
16. 7x2  5x  0 0, 
5
7
17. 4x2  9  0 
3
2
The supporting cables of the Golden Gate Bridge approximate the shape of a
18. 3x2  8x  3  , 3
1
3
19. x2  21  4x 3, 7 parabola. The parabola can be modeled by the quadratic function y  0.00012x2  6,
20. 3x2  13x  4  0  , 4
1
3
21. 15x2  22x  8   ,  
2
3
4
5
where x represents the distance from the axis of symmetry and y represents the
22. x2  6x  3  0 3 6
 23. x2  14x  53  0 7 2i height of the cables. The related quadratic equation is 0.00012x2  6  0.
24. 3x2  54 3i 2
 2 10
25. 25x2  20x  6  0  
5
Bridges 40. Calculate the value of the discriminant.  0.00288
1 4i
26. 4x2  4x  17  0 
2
2 3
27. 8x  1  4x2  
2
The Golden Gate, located
28. x2  4x  15 2 i 11
 29. 4x2 3 2
 12x  7  0   in San Francisco, California, 41. What does the discriminant tell you about the supporting cables of the Golden
2
30. GRAVITATION The height h(t) in feet of an object t seconds after it is propelled straight up is the tallest bridge in the Gate Bridge? See pp. 343A–343F.
from the ground with an initial velocity of 60 feet per second is modeled by the equation
h(t)  16t2  60t. At what times will the object be at a height of 56 feet? 1.75 s, 2 s world, with its towers
31. STOPPING DISTANCE The formula d  0.05s2  1.1s estimates the minimum stopping
distance d in feet for a car traveling s miles per hour. If a car stops in 200 feet, what is the
extending 746 feet above FOOTBALL For Exercises 42 and 43, use the following information.
the water and the floor The average NFL salary A(t) (in thousands of dollars) from 1975 to 2000 can be
of the bridge extending estimated using the function A(t)  2.3t2  12.4t  73.7, where t is the number of
Gl NAME
/M G ______________________________________________
Hill 340 DATE ____________
Gl PERIOD
Al _____
b 2
Reading
6-5 Readingto
to Learn
Learn Mathematics
Mathematics, p. 341 ELL 220 feet above water. years since 1975. 42. D: 0
t
25, R: 73.7
A(t)
1201.2
The Quadratic Formula and the Discriminant
Source:
Pre-Activity How is blood pressure related to age?
www.goldengatebridge.org 42. Determine a domain and range for which this function makes sense.
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 of
the formulas in your textbook.
43. According to this model, in what year did the average salary first exceed
Sample answer: Substitute your age for A in the appropriate
formula (for females or males) and evaluate the expression.
1 million dollars? 1998
Reading the Lesson
b   
b 2 4ac
Online Research Data Update What is the current average NFL salary?
1. a. Write the Quadratic Formula. x  
2a How does this average compare with the average given by the function used
b. Identify the values of a, b, and c that you would use to solve 2x2  5x  7, but do
not actually solve the equation. in Exercises 42 and 43? Visit www.algebra2.com/data_update to learn more.
a 2 b 5 c 7

2. Suppose that you are solving four quadratic equations with rational coefficients and 318 Chapter 6 Quadratic Functions and Inequalities
have found the value of the discriminant for each equation. In each case, give the
number 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
NAME ______________________________________________ DATE ____________ PERIOD _____
8

21
2
2
complex
real, irrational Enrichment,
6-5 Enrichment p. 342
Answer
13. The discriminant of 16t 2  85t  120
0 1 real, rational
Sum and Product of Roots
Helping You Remember
3. How can looking at the Quadratic Formula help you remember the relationships
Sometimes you may know the roots of a quadratic equation without knowing the equation
itself. Using your knowledge of factoring to solve an equation, you can work backward to is 455, indicating that the equation
find the quadratic equation. The rule for finding the sum and product of roots is as follows:
between the value of the discriminant and the number of roots of a quadratic equation
and whether the roots are real or complex?
Sum and Product of Roots
If the roots of ax 2  bx  c  0, with a ≠ 0, are s1 and s2,
b c
has no real solutions.
Sample answer: The discriminant is the expression under the radical in then s1  s2   and s1 s2  .
a a
the Quadratic Formula. Look at the Quadratic Formula and consider what
happens when you take the principal square root of b2  4ac and apply Example
in front of the result. If b2  4ac is positive, its principal square root A road with an initial gradient, or slope, of 3% can be represented by
will be a positive number and applying will give two different real the formula y  ax2  0. 03x  c, where y is the elevation and x is the distance along
solutions, which may be rational or irrational. If b2  4ac  0, its the curve. Suppose the elevation of the road is 1105 feet at points 200 feet and 1000
feet along the curve. You can find the equation of the transition curve. Equations
principal square root is 0, so applying in the Quadratic Formula will
of transition curves are used by civil engineers to design smooth and safe roads.
only lead to one solution, which will be rational (assuming a, b, and c are
integers). If b 2  4ac is negative, since the square roots of negative The roots are x  3 and x  8. y
numbers are not real numbers, you will get two complex roots, 3  (8)  5 Add the roots. 10
corresponding to the  and  in the symbol. 3(8)  24 Multiply the roots.
–8 –6 –4 –2 O 2 4 x
Equation: x2  5x  24  0
–10

318 Chapter 6 Quadratic Functions and Inequalities

 44. HIGHWAY SAFETY Highway safety engineers can use the formula
d  0.05s2  1.1s to estimate the minimum stopping distance d in feet for a
vehicle traveling s miles per hour. If a car is able to stop after 125 feet, what
is the fastest it could have been traveling when the driver first applied the
4 Assess
brakes? about 40.2 mph Open-Ended Assessment
45. CRITICAL THINKING Find all values of k such that x2  kx  9  0 has Modeling Ask students to sketch
a. one real root. k  6 b. two real roots. c. no real roots. graphs of parabolas that illustrate
k  6 or k  6 6  k  6 each of the four types of roots for
46. WRITING IN MATH Answer the question that was posed at the beginning of quadratic equations. Have them
the lesson. See pp. 343A–343F. label each sketch with the type of
How is blood pressure related to age? value the discriminant of the cor-
Include the following in your answer: responding quadratic equation
• an expression giving the average systolic blood pressure for a person of your would have.
age, and
• an example showing how you could determine A in either formula given a
specific value of P. Getting Ready for
Lesson 6-6
Standardized 47. If 2x2  5x  9  0, then x could equal which of the following? D
PREREQUISITE SKILL Lesson 6-6
Test Practice A 1.12 B 1.54 C 2.63 D 3.71
presents the analysis of the graphs
48. Which best describes the nature of the roots of the equation x2  3x  4  0? C of quadratic functions. To graph a
A real and equal B real and unequal
quadratic function, it is helpful if
C complex D real and complex
the function is written in vertex
form, which often requires
students to complete the square.
Recognition of perfect square
Maintain Your Skills trinomials is an important part of
completing the square. Exercises
Mixed Review Solve each equation by using the Square Root Property. (Lesson 6-4) 61–66 should be used to deter-
49. x2  18x  81  25 50. x2  8x  16  7 51. 4x2  4x  1  8 mine your students’ familiarity
14, 4 
4 ⴞ 7 1 ⴞ 22 with perfect square trinomials.

Solve each equation by factoring. (Lesson 6-3) 2
2
52. 4x2  8x  0 2, 0 53. x2  5x  14 2, 7 54. 3x2  10  17x , 5
3
Simplify. (Lesson 5-5) Answers
55. a 56. 
100p12 57. 
3
8b20 a4b10 q2 10p 6q 64b6c6 4b 2c 2
22a. 0
58. ANIMALS The fastest-recorded physical action of any living thing is the wing 22b. one rational
beat of the common midge. This tiny insect normally beats its wings at a rate of
1
133,000 times per minute. At this rate, how many times would the midge beat 22c. 
its wings in an hour? Write your answer in scientific notation. (Lesson 5-1) 3
7.98  106 23a. 0
Solve each system of inequalities. (Lesson 3-3) 59–60. See pp. 343A–343F.
23b. one rational
59. x  y 9 60. x  1
xy 3 y 1 5
23c.  
yx4 y x 2
24a. 31
Getting Ready for PREREQUISITE SKILL State whether each trinomial is a perfect square. If it is,
24b. 2 complex
the Next Lesson factor it. (To review perfect square trinomials, see Lesson 5-4.)
61. x2  5x  10 no 62. x2  14x  49 yes; (x  7)2 9 i 31

24c. 
63. 4x2  12x  9 yes; (2x  3)2 64. 25x2  20x  4 yes; (5x  2)2 8
65. 9x2  12x  16 no 66. 36x2  60x  25 yes; (6x  5)2 25a. 135
www.algebra2.com/self_check_quiz Lesson 6-5 The Quadratic Formula and the Discriminant 319
25b. 2 complex
1 i 15

25c. 
4
Answers 28
26a. 
9
14a. 21 16a. 16 18a. 121 20a. 20 26b. 2 irrational
14b. 2 irrational 16b. 2 complex 18b. 2 rational 20b. 2 irrational 2 4 7
1 2 26c. 
3 21
14c.   16c. 1 2i 18c.   ,  20c. 2 5 9
2 4 3
17a. 23 21a. 24 27a. 1.48
15a. 240 19a. 49
17b. 2 complex 21b. 2 irrational 27b. 2 irrational
15b. 2 irrational 19b. 2 rational
1 i 23
 21c. 1 6 1 20.37

17c.  1 27c. 
15c. 8 215  2 19c. 2,  0.8
3
Lesson 6-5 The Quadratic Formula and the Discriminant 319
Graphing
Calculator A Preview of Lesson 6-6
Investigation
A Preview of Lesson 6-6

Getting Started Families of Parabolas

The general form of a quadratic equation is y  a(x  h)2  k. Changing the values
of a, h, and k results in a different parabola in the family of quadratic functions. You
Know Your Calculator Students can use a TI-83 Plus graphing calculator to analyze the effects that result from
can use the calculator to confirm changing each of these parameters.
the location of the vertex of each
parabola. A good way to do this
is to change the window settings 3Example 1
for the x-axis to [9.4, 9.4]. Then Graph each set of equations on the same screen in the standard viewing
use the Trace feature and sym- window. Describe any similarities and differences among the graphs.
metry properties of parabolas to y ⴝ x2, y ⴝ x2 ⴙ 3, y ⴝ x2 ⴚ 5
check that the graph is symmetric The graphs have the same shape, and all open up. The vertex of
with respect to the vertical line each graph is on the y-axis. However, the graphs have different
vertical positions. y  x2  3
through the point that appears to y  x2
be the vertex. y  x2  5

Teach
Example 1 shows how changing the value of k in the equation y  a(x  h)2  k
• Ask students to describe the translates the parabola along the y-axis. If k  0, the parabola is translated k units up,
and if k  0, it is translated k units down.
three constants (a, h, and k) in
the general form of a quadratic How do you think changing the value of h will affect the graph of y  x2?
equation y  a(x  h)2  k.
Sample answer: a: coefficient of
the squared quantity involving the 3Example 2
variable x; h: value subtracted from Graph each set of equations on the same screen in the standard viewing
x in the quantity being squared and window. Describe any similarities and differences among the graphs.
then multiplied by a; k: value y ⴝ x2, y ⴝ (x ⴙ 3)2, y ⴝ (x ⴚ 5)2
added at the end These three graphs all open up and have the same shape. The
• Before discussing the examples, vertex of each graph is on the x-axis. However, the graphs
have different horizontal positions.
have students make a conjecture y  (x  3)2
about the effect of the value of y  x2 y  (x  5)2
each of the constants a, h, and k
on the graph of the parabola.
• After completing the discussion
of Example 3, have students Example 2 shows how changing the value of h in the equation y  a(x  h)2  k
translates the graph horizontally. If h  0, the graph translates to the right h units.
compare the conjectures they If h  0, the graph translates to the left h units.
made at the beginning of the
investigation to the knowledge www.algebra2.com/other_calculator_keystrokes
they gained during the
discussions.
• Have students complete 320 Chapter 6 Quadratic Functions and Inequalities
Exercises 1–15.

320 Chapter 6 Quadratic Functions and Inequalities

How does the value a affect the graph of y  x2?

3Example 3
Assess
Graph each set of equations on the same screen in the standard viewing
window. Describe any similarities and differences among the graphs.
Ask students:
a. y ⴝ x2, y ⴝ ⴚx2 • In the general form of a quad-
The graphs have the same vertex and the same shape. ratic equation, which constant
However, the graph of y  x2 opens up and the graph y  x2 would you change to move the
of y  x2 opens down.
graph left or right? h
y  x 2
• Which constant would you
change to move the graph up
y  4x 2
or down? k
1
b. y ⴝ x2, y ⴝ 4x2, y ⴝ x2 • Which constant would you
4
The graphs have the same vertex, (0, 0), but each has a change to make the graph
different shape. The graph of y  4x2 is narrower than wider or narrower? a
1
the graph of y  x2. The graph of y  x2 is wider
4 y  14 x 2 y  x2
than the graph of y  x2.
Answers
[10, 10] scl: 1 by [5, 15] scl: 1
1. Changing the value of h moves the
Changing the value of a in the equation y  a(x  h)2  k can affect the direction of graph to the left and the right. If
the opening and the shape of the graph. If a  0, the graph opens up, and if a  0, h  0, the graph translates to the
the graph opens down or is reflected over the x-axis. If a  1, the graph is narrower right, and if h  0, it translates to
than the graph of y  x2. If a  1, the graph is wider than the graph of y  x2.
Thus, a change in the absolute value of a results in a dilation of the graph of y  x2.
the left. In y  x 2, the vertex is at
(0, 0) and in y  (x  2)2, the
Exercises 1–3. See margin. vertex is at (2, 0). The graph has
Consider y ⴝ a(x ⴚ h)2 ⴚ k. been translated to the right.
1. How does changing the value of h affect the graph? Give an example.
2. Changing the value of k moves the
2. How does changing the value of k affect the graph? Give an example.
graph up and down. If k  0, the
3. How does using a instead of a affect the graph? Give an example.
graph translates upward, and if
Examine each pair of equations and predict the similarities and differences k  0, it translates downward. In
in their graphs. Use a graphing calculator to confirm your predictions. Write
y  x 2, the vertex is at (0, 0) and
a sentence or two comparing the two graphs. 4–15. See pp. 343A–343F.
4. y  x2, y  x2  2.5 5. y  x2, y  x2  9
in y  x2  3, the vertex is at
6. y  x2, y  3x2 7. y  x2, y  6x2
(0, 3). The graph has been
1 1
translated downward.
8. y  x2, y  (x  3)2 9. y  x2, y  x2  2
3 3 3. Using a instead of a reflects the
10. y  x2, y  (x  7)2 11. y  x2, y  3(x  4)2  7 graph over the x-axis. The graph
1
12. y  x2, y  x2  1 13. y  (x  3)2  2, y  (x  3)2  5 of y  x2 opens upward, while the
4
graph of y  x2 opens downward.
14. y  3(x  2)2  1, 15. y  4(x  2)2  3,
1
y  6(x  2)2 1 y  (x  2)2  1
4

Graphing Calculator Investigation Families of Parabolas 321

Graphing Calculator Investigation Families of Parabolas 321

 Lesson Analyzing Graphs of
Notes Quadratic Functions
• Analyze quadratic functions of the form y  a(x  h)2  k.

1 Focus • Write a quadratic function in the form y  a(x  h)2  k.

can the graph of y ⴝ x2 be used to graph any quadratic

Vocabulary
5-Minute Check function?
• vertex form
Transparency 6-6 Use as a A family of graphs is a group of graphs that y  x2  2
quiz or review of Lesson 6-5. displays one or more similar characteristics. y
The graph of y  x2 is called the parent graph of
Mathematical Background notes the family of quadratic functions. Study the
are available for this lesson on graphs of y  x2, y  x2  2, and y  (x  3)2.
Notice that adding a constant to x2 moves the y  (x  3)2
p. 284D. graph up. Subtracting a constant from x before y  x2
squaring it moves the graph to the right. O x
can the graph of y  x2
be used to graph any
quadratic function?
Ask students: ANALYZE QUADRATIC Axis of
• For the function y  x2, what FUNCTIONS Notice that each Equation Vertex
Symmetry
function above can be written in the form
value of x makes y equal 0? 0 y  (x  h)2  k, where (h, k) is the vertex
y  x 2 or
(0, 0) x0
What value of x makes y equal y  (x  0)2  0
of the parabola and x  h is its axis of
0 if the function is y  (x  3)2 ? symmetry. This is often referred to as the y  x  2 or
2
(0, 2) x0
vertex form of a quadratic function. y  (x  0)2  2
3
In Chapter 4, you learned that a y  (x  3)2 or
• Compare the graph of (3, 0) x3
translation slides a figure on the coordinate y  (x  3)2  0
y  x2  2 with the graph of plane without changing its shape or size.
y  (x  2)2. What difference As the values of h and k change, the graph
does adding the 2 within the of y  a(x  h)2  k is the graph of y  x2 translated
parentheses make? Sample • h units left if h is negative or h units right if h is positive, and
answer: Adding the 2 inside the • k units up if k is positive or k units down if k is negative.
parentheses moves the graph
2 units to the left rather than Example 1 Graph a Quadratic Function in Vertex Form
2 units up when compared to the Analyze y  (x  2)2  1. Then draw its graph.
graph of y  x 2. This function can be rewritten as y  [x  (2)]2  1. Then h  2 and k  1.
The vertex is at (h, k) or (2, 1), and the axis of symmetry is x  2. The graph has
the same shape as the graph of y  x2, but is translated 2 units left and 1 unit up.
Now use this information to draw the graph. y

Step 1 Plot the vertex, (2, 1).

(4, 5) (0, 5)
Step 2 Draw the axis of symmetry, x  2. 2
y  (x  2)  1
Step 3 Find and plot two points on one side of the (3, 2) (1, 2)
axis of symmetry, such as (1, 2) and (0, 5). (2, 1)
O x
Step 4 Use symmetry to complete the graph.

322 Chapter 6 Quadratic Functions and Inequalities

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 6 Resource Masters Teaching Algebra With Manipulatives 5-Minute Check Transparency 6-6
• Study Guide and Intervention, pp. 343–344 Masters, pp. 247–248 Answer Key Transparencies
• Skills Practice, p. 345
• Practice, p. 346 Technology
• Reading to Learn Mathematics, p. 347 Interactive Chalkboard
• Enrichment, p. 348
• Assessment, p. 370
 How does the value of a in the general form y a b
y  a(x  h)2  k affect a parabola? Compare the graphs of
the following functions to the parent function, y  x2.
1
y  x2
2 Teach
a. y  2x2 b. y  x2
2
1
O x ANALYZE QUADRATIC
c. y  2x2 d. y  x2
2 FUNCTIONS
All of the graphs have the vertex (0, 0) and axis of
symmetry x  0.
c d In-Class Example Power
Point®

1
Notice that the graphs of y  2x2 and y  x2 are dilations of the graph of y  x2.
2
1 Analyze y  (x  3)2  2.
The graph of y  2x2 is narrower than the graph of y  x2, while the graph of Then draw its graph. The ver-
1 tex of the graph is at (3, 2) and the
y  x2 is wider. The graphs of y  2x2 and y  2x2 are reflections of each other
2
1 1 axis of symmetry is x  3. The
over the x-axis, as are the graphs of y  x2 and y  x2. graph has the same shape as the
2 2
Changing the value of a in the equation y  a(x  h)2  k can affect the direction graph of y  x 2, but is translated
of the opening and the shape of the graph. 3 units right and 2 units up.
• If a  0, the graph opens up. f (x )
Study Tip • If a  0, the graph opens down. (1, 6)
(5, 6)
Reading Math • If a  1, the graph is narrower than the graph of y  x2.
a  1 means that a
is a rational number • If a  1, the graph is wider than the graph of y  x2.
between 0 and 1, such
(2, 3) (4, 3)
2
as , or a rational number
(3, 2)
5 f (x )  (x  3)2  1
between ⴚ1 and 0, such
as ⴚ0.3. O x
Quadratic Functions in Vertex Form
The vertex form of a quadratic function is y  a(x  h)2  k.
Teaching Tip To help students
h and k k remember that as |a| increases,
Vertex and Axis of the graph gets narrower and not
Vertical Translation
Symmetry k 0
wider, discuss the fact that a
y xh y greater multiplier for the quan-
tity (x  h)2 will make the corres-
ponding y value greater as well.
y  x 2,
k0 Point out that greater values of
O x y result in a steeper (and thus
O
k0
x narrower) graph.
(h, k )

h a Intervention
Horizontal Translation Direction of Opening and
New Encourage
y  x 2, Shape of Parabola students to ask
h0 y a  1
questions about
y y
any aspects
a0 they may find confusing that
are covered in the Concept
h0 h0
O x
O a 1 x
Summary chart on this page.
y x 2, 
O x a0
a1 Ask them to write and use
their own summary on an
index card. Explain to students
that a thorough understanding
www.algebra2.com/extra_examples Lesson 6-6 Analyzing Graphs of Quadratic Functions 323
of these concepts will save
them time since this knowl-
edge will enable them to sketch
approximate graphs quickly.

Lesson 6-6 Analyzing Graphs of Quadratic Functions 323

WRITE QUADRATIC WRITE QUADRATIC FUNCTIONS IN VERTEX FORM Given a function
of the form y  ax2  bx  c, you can complete the square to write the function in
FUNCTIONS IN VERTEX vertex form.
FORM
In-Class Examples Power
Point®
Example 2 Write y  x2  bx  c in Vertex Form
Write y  x2  8x  5 in vertex form. Then analyze the function.
2 Write y  x2  2x  4 in ver-
y  x2  8x  5 Notice that x2  8x  5 is not a perfect square.
tex form. Then analyze the 8
Complete the square by adding  or 16.
2
function. y  (x  1)2  3; y  (x2  8x  16)  5  16 2
Balance this addition by subtracting 16.
vertex: (1, 3); axis of symmetry:
y  (x  4)2  21 Write x2  8x  16 as a perfect square.
x  1; opens up; The graph has
the same shape as the graph of
This function can be rewritten as y  [x  (4)]2  (21). Written in this way,
y  x 2, but it is translated 1 unit you can see that h  4 and k  21.
left and 3 units up.
The vertex is at (4, 21), and the axis of symmetry is x  4. Since a  1, the
3 Write y  2x2  4x  2 in graph opens up and has the same shape as the graph of y  x2, but it is translated
vertex form. Then analyze 4 units left and 21 units down.
Study Tip
and graph the function.
Check CHECK You can check the vertex and axis of symmetry using the formula
y  2(x  1)2  4; vertex: As an additional check, b
(1, 4); axis of symmetry: graph the function in x   . In the original equation, a  1 and b  8, so the axis of
2a
8
x  1; opens down; The graph Example 2 to verify the symmetry is x   or 4. Thus, the x-coordinate of the vertex is
2(1)
location of its vertex and
is narrower than the graph of axis of symmetry. 4, and the y-coordinate of the vertex is y  (4)2  8(4)  5 or 21.
y  x 2, and it is translated 1 unit
left and 4 units up.
f (x )
When writing a quadratic function in which the coefficient of the quadratic
f (x )  2(x  1)2  4 term is not 1 in vertex form, the first step is to factor out that coefficient from
the quadratic and linear terms. Then you can complete the square and write in
vertex form.

Example 3 Write y  ax2  bx  c in Vertex Form, a ⴝ 1

O x
Write y  3x2  6x  1 in vertex form. Then analyze and graph the function.
y  3x2  6x  1 Original equation
y  3(x2  2x)  1 Group ax2  bx and factor, dividing by a.
Complete the square by adding 1 inside the parentheses.
y  3(x  2x  1)  1  (3)(1) Notice that this is an overall addition of 3(1). Balance
this addition by subtracting 3(1).
y  3(x  1)2  2 Write x2  2x  1 as a perfect square.

The vertex form of this function is y  3(x  1)2  2.

So, h  1 and k  2.
y
The vertex is at (1, 2), and the axis of symmetry is
x  1. Since a  3, the graph opens downward and is
narrower than the graph of y  x2. It is also translated y  3(x  1)2  2
(1.5, 1.25)
1 unit right and 2 units up.
O x
Now graph the function. Two points on the graph to (2, 1)
the right of x  1 are (1.5, 1.25) and (2, 1). Use
symmetry to complete the graph.

324 Chapter 6 Quadratic Functions and Inequalities

Differentiated Instruction
Naturalist Have students observe or research some natural events that
can be modeled by parabolas, such as the fountain’s water stream dis-
cussed in Exercise 14 on p. 326. Students should report their observa-
tions and findings to the class. If students are able to determine a quadratic
function that models the event, they should present the function and
explain how the characteristics of the equation can be used to analyze
its graph.

324 Chapter 6 Quadratic Functions and Inequalities

 If the vertex and one other point on the graph of a parabola are known, you can
write the equation of the parabola in vertex form. In-Class Example Power
Point®

Example 4 Write an Equation Given Points 4 Write an equation for the

parabola whose vertex is at
Write an equation for the parabola whose vertex is at (1, 4) and passes
through (2, 1). (1, 2) and passes through
The vertex of the parabola is at (1, 4), so h  1 and k  4. Since (2, 1) is a point (3, 4). y  1(x  1)2  2
2
on the graph of the parabola, let x  2 and y  1. Substitute these values into the
vertex form of the equation and solve for a.
y  a(x  h)2  k Vertex form
1  a[2  (1)]2  4 Substitute 1 for y, 2 for x, 1 for h, and 4 for k.
1  a(9)  4 Simplify.
3 Practice/Apply
3  9a Subtract 4 from each side.
1
  a Divide each side by 9.
3
The equation of the parabola in vertex form
Study Notebook
y   13 ( x  1)2 
y 4
1
is y  (x  1)2  4. (1, 4 ) Have students—
3
• add the definitions/examples of
1
CHECK A graph of y  (x  1)2  4 (2, 1) the vocabulary terms to their
3
verifies that the parabola passes Vocabulary Builder worksheets for
O x
through the point at (2, 1).
Chapter 6.
• write a summary in their own words
of everything you can tell about
the graph of a parabola when the
function is written in vertex form.
Concept Check 1. Write a quadratic equation that transforms the graph of y  2(x  1)2  3 so that • include any other item(s) that they
1d. y  2(x  2)2  3 it is: find helpful in mastering the skills
1e. Sample answer: a. 2 units up. y  2(x  1)2  5 b. 3 units down. y  2(x  1)2 in this lesson.
y  4(x  1)2  3 c. 2 units to the left. y  2(x  3)2  3 d. 3 units to the right.
1f. Sample answer:
y  (x  1)2  3 e. narrower. f. wider.
3. Sample answer: g. opening in the opposite direction. y  2(x  1)2  3
y  2(x  2)2  1
2. Explain how you can find an equation of a parabola using its vertex and one
other point on its graph. See margin. FIND THE ERROR
3. OPEN ENDED Write the equation of a parabola with a vertex of (2, 1). Students may
find that it will help
4. Jenny; when 4. FIND THE ERROR Jenny and Ruben are writing y  x2  2x  5 in vertex form.
completing the square them avoid errors of this type if
is used to write a Jenny Ruben they specifically write the addition
quadratic function in and subtraction in a separate step
vertex form, the y = x 2 – 2x + 5 y = x2 – 2x + 5
before placing parentheses around
quantity added is then y = (x 2 – 2x + 1) + 5 – 1 y = (x2 – 2x + 1) + 5 + 1
the perfect square trinomial, as in
subtracted from the y = (x – 1) 2 + 4 y = (x – 1) 2 + 6
same side of the y  x2  2x  1  1  5.
equation to maintain Who is correct? Explain your reasoning.
equality.
Answers
Guided Practice Write each quadratic function in vertex form, if not already in that form. Then
2. Substitute the x-coordinate of the
identify the vertex, axis of symmetry, and direction of opening. 5–7. See margin.
5. y  5(x  3)2  1 6. y  x2  8x  3 7. y  3x2  18x  11
vertex for h and the y-coordinate
of the vertex for k in the equation
Lesson 6-6 Analyzing Graphs of Quadratic Functions 325
y  a(x  h)2  k. Then substitute
the x-coordinate of the other point
for x and the y-coordinate for y
into this equation and solve for a.
Replace a with this value in the
equation you wrote with h and k.
5. (3, 1); x  3; up
6. y  (x  4)2  19, (4, 19);
x  4; up
7. y  3(x  3)2  38; (3, 38);
x  3; down

Lesson 6-6 Analyzing Graphs of Quadratic Functions 325

 GUIDED PRACTICE KEY Graph each function. 8–10. See margin.
1
About the Exercises… Exercises Examples 8. y  3(x  3)2 9. y  (x  1)2  3 10. y  2x2  16x  31
3
5–7 2
Organization by Objective 8–10 1, 3 Write an equation for the parabola with the given vertex that passes through the
• Analyze Quadratic 11–14 4 given point.
Functions: 15, 16, 27–30, 37, 11. vertex: (2, 0) 12. vertex: (3, 6) 13. vertex: (2, 3)
38, 47, 51, 52 point: (1, 4) point: (5, 2) point: (4, 5)
1
• Write Quadratic Functions y  4(x  2)2 y  (x  3)2  6 y   (x  2)2  3
2
in Vertex Form: 19–26, Application 14. FOUNTAINS The height of a fountain’s water 1 ft
31–36, 39–46, 48–50 stream can be modeled by a quadratic
function. Suppose the water from a jet
Odd/Even Assignments reaches a maximum height of 8 feet at a
distance 1 foot away from the jet. If the 8 ft
Exercises 15–46 are structured water lands 3 feet away from the jet, find a
so that students practice the quadratic function that models the height h(d)
same concepts whether they of the water at any given distance d feet from
are assigned odd or even the jet. h(d)  2d 2  4d  6
3 ft
problems.

Assignment Guide ★ indicates increased difficulty

Basic: 15–33 odd, 39–45 odd,
53–71
Practice and Apply
Average: 15–47 odd, 48–50, Homework Help Write each quadratic function in vertex form, if not already in that form. Then
53–71 For See identify the vertex, axis of symmetry, and direction of opening. 19–26. See margin.
Exercises Examples 1
15. y  2(x  3)2 (3, 0); x  3; down 16. y  (x  1)2  2 (1, 2); x  1; up
Advanced: 16–46 even, 48–67 15–26 2 3
(optional: 68–71)
27–38, 47, 1, 3 17. y  5x2  6 (0, 6); x  0; up 18. y  8x2  3 (0, 3); x  0; down
48, 50–52
39–46, 49 4 19. y  x2  4x  8 20. y  x2  6x  1
All: Practice Quiz 2 (1–10)
21. y  3x2  12x 22. y  4x2  24x
Extra Practice
See page 841. 23. y  4x2  8x  3 24. y  2x2  20x  35
25. y  3x2  3x  1 26. y  4x2  12x  11
38. Sample answer:
the graphs have Graph each function. 27–36. See pp. 343A–343F.
the same shape, 27. y  4(x  3)2  1 28. y  (x  5)2  3
Answers but the graph of 1 1
y  2(x  4)2  1 29. y  (x  2)2  4 30. y  (x  3)2  5
4 2
8. y is 1 unit to the left
and 5 units 31. y  x2  6x  2 32. y  x2  8x  18
below the graph of 33. y  4x2  16x  11 34. y  5x2  40x  80
y  2(x  5)2  4. ★ 35. y  1x2  5x  27 ★ 36. y  13x2  4x  15
2 2
★ 37. Write one sentence that compares the graphs of y  0.2(x  3)2  1 and
y  3(x  3) 2 y  0.4(x  3)2  1. Sample answer: the graph of y  0.4(x  3)2  1 is
narrower than the graph of y  0.2(x  3)2  1.
O x ★ 38. Compare the graphs of y  2(x  5)2  4 and y  2(x  4)2  1.
Write an equation for the parabola with the given vertex that passes through the
You can use a quadratic given point.
9. y function to model the 39. vertex: (6, 1) y  9(x – 6)2  1 40. vertex: (4, 3) y  3(x  4)2  3
world population. Visit point: (5, 10) point: (3, 6)
www.algebra2.com/ 2
webquest to continue 41. vertex: (3, 0) y  (x  3)2 42. vertex: (5, 4) y  3(x  5)2  4
3
work on your WebQuest point: (6, 6) point: (3, 8)
1 2 5
project. 43. vertex: (0, 5) y  x  5 44. vertex: (3, 2) y  (x  3)2  2
3 2
1 point: (3, 8) point: (1, 8)
y  3 (x  1)2  3
326 Chapter 6 Quadratic Functions and Inequalities
O x

10. y 19. y  (x  2)2  12; (2, 12); x  2; 24. y  2(x  5)2  15; (5, 15); x  5;
y   2x 2  16x  31 down down
20. y  (x  3)2  8; (3, 8); x  3; up
O x 
25. y  3 x  
2
1 2 7 1
7
 ; ,  ;
4 2 4 
21. y  3(x  2)2  12; (2, 12); x  2;
1
down x    ; up
2
22. y  4(x  3)2  36; (3, 36);
x  3; up 
26. y  4 x  
2
3 2 3
2  3

 20;  , 20 ; x   ; up
2
23. y  4(x  1)2  7; (1, 7); x  1; up

326 Chapter 6 Quadratic Functions and Inequalities

 45. Write an equation for a parabola whose vertex is at the origin and passes Study
NAME ______________________________________________ DATE

Guide andIntervention
Intervention,
____________ PERIOD _____

through (2, 8). y ⴝ ⴚ2x2 6-6 Study Guide and

p. 343 (shown) and p.Functions
344
Analyzing Graphs of Quadratic
Analyze Quadratic Functions

Lesson 6-6
The graph of y  a(x  h)2  k has the following characteristics:
46. Write an equation for a parabola with vertex at (3, 4) and y-intercept 8. Vertex Form
• Vertex: (h, k)
• Axis of symmetry: x  h

4 of a Quadratic • Opens up if a  0

y ⴝ (x ⴙ 3)2 ⴚ 4 Function • Opens down if a  0

• Narrower than the graph of y  x 2 if a  1
3 • Wider than the graph of y  x 2 if a  1

47. AEROSPACE NASA’s KC135A aircraft flies in parabolic arcs to simulate

Example Identify the vertex, axis of symmetry, and direction of opening of
the weightlessness experienced by astronauts in space. The height h of each graph.

the aircraft (in feet) t seconds after it begins its parabolic flight can be a. y  2(x  4)2  11
The vertex is at (h, k) or (4, 11), and the axis of symmetry is x  4. The graph opens
modeled by the equation h(t)  9.09(t  32.5)2  34,000. What is the up, and is narrower than the graph of y  x2.
1
a. y    (x  2)2  10
maximum height of the aircraft during this maneuver and when does 4
The vertex is at (h, k) or (2, 10), and the axis of symmetry is x  2. The graph opens
it occur? 34,000 feet; 32.5 s after the aircraft begins its parabolic flight down, and is wider than the graph of y  x2.

Exercises
Each quadratic function is given in vertex form. Identify the vertex, axis of
symmetry, and direction of opening of the graph.

Aerospace DIVING For Exercises 48–50, use the following information. 1. y  (x  2)2  16 2. y  4(x  3)2  7
1
3. y   (x  5)2  3
2
The distance of a diver above the water d(t) (in feet) t seconds after diving off a (2, 16); x  2; up (3, 7); x  3; up (5, 3); x  5; up
The KC135A has the
nickname “Vomit Comet.” platform is modeled by the equation d(t)  16t2  8t  30. 4. y  7(x  1)2  9
1
5. y   (x  4)2  12 6. y  6(x  6)2  6
5

It starts its ascent at 48. Find the time it will take for the diver to hit the water. about 1.6 s (1, 9); x  1; down (4, 12); x  4; up (6, 6); x  6; up

24,000 feet. As it 2
7. y   (x  9)2  12 8. y  8(x  3)2  2 9. y  3(x  1)2  2
49. Write an equation that models the diver’s distance above the water if the 5
approaches maximum (9, 12); x  9; up (3, 2); x  3; up (1, 2); x  1; down

height, the engines are platform were 20 feet higher. d(t)  16t 2  8t  50 5 4
10. y    (x  5)2  12 11. y   (x  7)2  22 12. y  16(x  4)2  1
2 3
stopped, and the aircraft is 50. Find the time it would take for the diver to hit the water from this new (5, 12); x  5; down (7, 22); x  7; up (4, 1); x  4; up
allowed to free-fall at a height. about 2.0 s 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
determined angle. Zero (1.2, 2.7); x  1.2; up (0.6, 0.2); x  0.6; (0.8, 6.5); x  0.8;
gravity is achieved for down up

25 seconds as the plane LAWN CARE For Exercises 51 and 52, use the following information. Gl NAME
/M G ______________________________________________
Hill 343 DATE ____________
GlPERIOD
Al _____
b 2

reaches the top of its flight Skills

6-6 Practice,
Practice p. 345 and
The path of water from a sprinkler can be modeled by a quadratic function. Practice,
(Average)
p. 346 (shown)
and begins its descent. The three functions below model paths for three different angles of the water. Analyzing Graphs of Quadratic Functions
Write each quadratic function in vertex form, if not already in that form. Then
Source: NASA identify the vertex, axis of symmetry, and direction of opening.
Angle A: y  0.28(x  3.09)2  3.27 1. y  6(x  2)2  1 2. 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;
Angle B: y  0.14(x  3.57)2  2.39 (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  5

Angle C: y  0.09(x  3.22)2  1.53 y  (x  5)2  5;

(5, 5); x  5; up
y  2(x  3)2; (3, 0);
x  3; up
y  3(x  1)2  2;
(1, 2); x  1; up
7. y  2x2  16x  32 8. y  3x2  18x  21 9. y  2x2  16x  29
51. Angle A; the graph 51. Which sprinkler angle will send water the highest? Explain your reasoning. y  2(x  4)2;
(4, 0); x  4; down
y  3(x  3)2  6;
(3, 6); x  3; down
y  2(x  4)2  3;
(4, 3); x  4; up
of the equation for 52. Which sprinkler angle will send water the farthest? Explain your reasoning. Graph each function.

angle A is higher than 10. y  (x  3)2  1

y
11. y  x2  6x  5
y
12. y  2x2  2x  1
y

the other two since

3.27 is greater than 53. CRITICAL THINKING Given y   bx  c with a  0, derive the equation for
ax2 O x

2.39 or 1.53. the axis of symmetry by completing the square and rewriting the equation O x O x

in the form y  a(x  h)2  k. See pp. 343A–343F.

52. Angle B; the vertex Write an equation for the parabola with the given vertex that passes through the

of the equation for given point.

13. vertex: (1, 3) 14. vertex: (3, 0) 15. vertex: (10, 4)
angle B is farther to 54. WRITING IN MATH Answer the question that was posed at the beginning of the
point: (2, 15)
y  2(x  1)2  3
point: (3, 18)
1
y (x  3)2
point: (5, 6)
2
y (x  10)2  4
2 5
the right than the lesson. See pp. 343A–343F. 16. Write an equation for a parabola with vertex at (4, 4) and x-intercept 6.
y  (x  4)2 4
other two since 3.57 17. Write an equation for a parabola with vertex at (3, 1) and y-intercept 2.

is greater than 3.09 How can the graph y ⴝ x2 be used to graph any quadratic function? 1
y (x  3)2  1
3
18. BASEBALL The height h of a baseball t seconds after being hit is given by
or 3.22. Include the following in your answer: h(t)  16t2  80t  3. What is the maximum height that the baseball reaches, and
when does this occur? 103 ft; 2.5 s

• a description of the effects produced by changing a, h, and k in the equation 19. SCULPTURE A modern sculpture in a park contains a parabolic arc that
starts at the ground and reaches a maximum height of 10 feet after a
y  a(x  h)2  k, and horizontal distance of 4 feet. Write a quadratic function in vertex form
that describes the shape of the outside of the arc, where y is the height
10 ft
of a point on the arc and x is its horizontal distance from the left-hand
• a comparison of the graph of y  x2 and the graph of y  a(x  h)2  k using starting point of the arc. 5 2y    (x  4)  10
8 4 ft

values of your own choosing for a, h, and k. Gl NAME

/M G ______________________________________________
Hill 346 DATE ____________
Gl PERIOD
Al _____
b 2
Reading
6-6 Readingto
to Learn
Learn Mathematics
Mathematics, p. 347 ELL
Analyzing Graphs of Quadratic Equations
Standardized 55. If f(x)  x2  5x and f(n)  4, then which of the following could be n? D Pre-Activity How can the graph of y  x2 be used to graph any quadratic

Lesson 6-6
function?

Test Practice A 5 B 4 C 1 D 1 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 the

56. The vertex of the graph of y  2(x   3 is located at which of the following
graph of y  x2? It moves the graph to the right.
6)2
points? B Reading the Lesson
1. Complete the following information about the graph of y  a(x  h)2  k.
A (2, 3) B (6, 3) C (6, 3) D (2, 3) a. What are the coordinates of the vertex? (h, k)

b. What is the equation of the axis of symmetry? x  h

www.algebra2.com/self_check_quiz Lesson 6-6 Analyzing Graphs of Quadratic Functions 327 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  x 2.
If a  1? It is narrower than the graph of y  x 2.

NAME ______________________________________________ DATE ____________ PERIOD _____ 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
Enrichment,
6-6 Enrichment p. 348 c. a  0, h  0, k  0 ii d. a  0, h  0, k  0 i

i. y ii. y iii. y iv. y

Patterns with Differences and Sums of Squares
O x
Some whole numbers can be written as the difference of two squares and O x O x O x
some cannot. Formulas can be developed to describe the sets of numbers
algebraically.

If possible, write each number as the difference of two squares. Helping You Remember
Look for patterns.
3. When graphing quadratic functions such as y  (x  4)2 and y  (x  5)2, many students
1. 0 02  02 2. 1 12  02 3. 2 cannot 4. 3 22  12 have trouble remembering which represents a translation of the graph of y  x2 to the left
and which represents a translation to the right. What is an easy way to remember this?
5. 4 22  02 6. 5 32  22 7. 6 cannot 8. 7 42  32
Sample answer: In functions like y  (x  4)2, the plus sign puts the
graph “ahead” so that the vertex comes “sooner” than the origin and the
9. 8 32  12 10. 9 32  02 11. 10 cannot 12. 11 62  52
translation is to the left. In functions like y  (x  5)2, the minus puts the
13. 12 42  22 14. 13 72  62 15. 14 cannot 16. 15 42  12
graph “behind” so that the vertex comes “later” than the origin and the
translation is to the right.
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.

Lesson 6-6 Analyzing Graphs of Quadratic Functions 327

 Maintain Your Skills
4 Assess Mixed Review Find the value of the discriminant for each quadratic equation. Then describe the
number and type of roots for the equation. (Lesson 6-5)
Open-Ended Assessment 57. 3x2  6x  2  0 58. 4x2  7x  11 59. 2x2  5x  6  0
Writing Give students the 12; 2 irrational 225; 2 rational 23; 2 complex
equation of a specific parabola Solve each equation by completing the square. (Lesson 6-4)
and ask them to put it in vertex 60. x2  10x  17  0 61. x2  6x  18  0 62. 4x2  8x  9
{5 22 } {3 3i }
form, analyze it, and sketch the
graph. Instruct them to show all
Find each quotient. (Lesson 5-3) 63–66. See margin. 
2

2 13

63. (2t3  2t  3)  (t  1) 64. (t3  3t  2)  (t  2)
the steps in their procedures, with
65. (n4  8n3  54n  105)  (n  5) 66. (y4  3y3  y  1)  (y  3)
notes and explanations for each
step, similar to the notes in the 67. EDUCATION The graph shows
right column of the Examples. the number of U.S. students in
study-abroad programs. USA TODAY Snapshots®
(Lesson 2-5)
Getting Ready for More Americans study abroad
67a. Sample answer a. Write a prediction equation The number of U.S. college students in study-abroad
Lesson 6-7 using (1994, 76,302)
and (1997, 99,448):
from the data given. programs rose 11.4% in the year ending June 1997 (latest
available) to about 1% of students. Annual numbers:
b. Use your equation to
PREREQUISITE SKILL Lesson 6-7 y  7715x  predict the number of
Note: Includes any
student getting
credit at a U.S.
presents quadratic inequalities. 15,307,408 students in these programs school for
study abroad

Use Exercises 68–71 to determine in 2005.

your students’ familiarity with Sample answer: 161,167
302
Getting Ready for PREREQUISITE SKILL Determine 76, ,403
inequalities. 84 242
the Next Lesson whether the given value satisfies 4 89, ,448
199 99
the inequality. 199
5
Assessment Options (To review inequalities, see Lesson 1-6.) 199
6
7
68. 2x2  3  0; x  5 yes 199
Practice Quiz 2 The quiz
provides students with a brief 69. 4x2  2x  3  0; x  1 no Source:
Institute of
International Education

review of the concepts and skills 70. 4x2  4x  1 10; x  2 yes By Anne R. Carey and Marcy E. Mullins, USA TODAY

in Lessons 6-4 through 6-6. 71. 6x2  3x  8; x  0 no

Lesson numbers are given to the
right of exercises or instruction
lines so students can review
concepts not yet mastered.
P ractice Quiz 2 Lessons 6-4 through 6-6
Solve each equation by completing the square. (Lesson 6-4)
Quiz (Lessons 6-5 and 6-6) is 1 3i
available on p. 370 of the Chapter 6 }
1. x2  14x  37  0 {7 23 2. 2x2  2x  5  0 
2 
Resource Masters. Find the value of the discriminant for each quadratic equation. Then describe the
number and type of roots for the equation. (Lesson 6-5)
3. 5x2  3x  1  0 11; 2 complex 4. 3x2  4x  7  0 100; 2 rational
Answers
Solve each equation by using the Quadratic Formula. (Lesson 6-5)
3
63. 2t 2  2t   5. x2  9x  11  0 9 55 6. 3x2  4x  4 2 2i 2
t1   2 
3 
64. t 2  2t  1 7. Write an equation for a parabola with vertex at (2, 5) that passes through (1, 1).
2
65. n 3  3n 2  15n  21 (Lesson 6-6) y  (x  2)2  5
3
4 Write each equation in vertex form. Then identify the vertex, axis of symmetry,
66. y 3  1  
y3 and direction of opening. (Lesson 6-6) 8–10. See margin.
8. y  x2  8x  18 9. y  x2  12x  36 10. y  2x2  12x  13

Answers (Practice Quiz 2)

328 Chapter 6 Quadratic Functions and Inequalities
8. y  (x  4)2  2; (4, 2),
x  4; up
9. y  (x  6)2; (6, 0), x  6; down Online Lesson Plans
10. y  2(x  3)2  5; (3, 5),
x  3; up USA TODAY Education’s Online site offers resources and
interactive features connected to each day’s newspaper.
Experience TODAY, USA TODAY’s daily lesson plan, is
available on the site and delivered daily to subscribers.
This plan provides instruction for integrating USA TODAY
graphics and key editorial features into your mathematics
classroom. Log on to www.education.usatoday.com.

328 Chapter 6 Quadratic Functions and Inequalities

 Graphing and Solving Lesson
Quadratic Inequalities Notes
• Graph quadratic inequalities in two variables.
• Solve quadratic inequalities in one variable.

can you find the time a trampolinist

1 Focus
Vocabulary
spends above a certain height? 5-Minute Check
• quadratic inequality
Trampolining was first featured as an Olympic Transparency 6-7 Use as a
sport at the 2000 Olympics in Sydney, Australia. quiz or review of Lesson 6-6.
The competitors performed two routines
consisting of 10 different skills. Suppose the Mathematical Background notes
height h(t) in feet of a trampolinist above are available for this lesson on
the ground during one bounce is modeled
by the quadratic function h(t)  16t2  42t p. 284D.
 3.75. We can solve a quadratic inequality to
determine how long this performer is more than
a certain distance above the ground.
Building on Prior
Knowledge
In Lesson 6-6, students analyzed
Study Tip GRAPH QUADRATIC INEQUALITIES You can graph quadratic and graphed equations. In this
inequalities in two variables using the same techniques you used to graph lesson, students use the same
Look Back
For review of graphing linear inequalities in two variables. y y techniques to graph and solve
linear inequalities, see
Lesson 2-7.
Step 1 Graph the related quadratic equation, inequalities.
y  ax2  bx  c. Decide if the parabola x x
O O
should be solid or dashed. can you find the time a
or   or 
trampolinist spends
y above a certain height?
Step 2 Test a point (x1, y1) inside the parabola. (x 1, y 1)
TEACHING TIP Ask students:
Check to see if this point is a solution of
Remind students that the inequality.
(0, 0) is a good point to
O x • What is a trampoline and how
use as a test point. ?
y1  a(x1)2  b(x1)  c does a trampolinist use it?
y y
Ask a student who is familiar with
Step 3 If (x1, y1) is a solution, shade the region this sport to explain it to those who
inside the parabola. If (x1, y1) is not a
solution, shade the region outside the may not have seen it.
O x O x
parabola. • Which way does the parabola
(x1, y1) is (x1, y1) is not
a solution. a solution.
for the given quadratic
equation open? downward
Example 1 Graph a Quadratic Inequality
Graph y  x2  6x  7. y
y  x 2  6x  7

Step 1 Graph the related quadratic equation,

y  x2  6x  7. O x
Since the inequality symbol is , the parabola
should be dashed.
(continued on the next page)
Lesson 6-7 Graphing and Solving Quadratic Inequalities 329

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 6 Resource Masters Graphing Calculator and 5-Minute Check Transparency 6-7
• Study Guide and Intervention, pp. 349–350 Spreadsheet Masters, p. 37 Answer Key Transparencies
• Skills Practice, p. 351 School-to-Career Masters, p. 12
• Practice, p. 352 Technology
• Reading to Learn Mathematics, p. 353 Interactive Chalkboard
• Enrichment, p. 354
• Assessment, p. 370

Lesson x-x Lesson Title 329

 Step 2 Test a point inside the parabola, such as (3, 0). y

2 Teach y  x2  6x  7
?
0  (3)2  6(3)  7
y  x 2  6x  7

(3, 0) O x
GRAPH QUADRATIC ?
0  9  18  7
INEQUALITIES ?
0 2 ⫻
In-Class Example Power
Point®
So, (3, 0) is not a solution of the inequality.
Step 3 Shade the region outside the parabola.
1 Graph y  x2  3x  2.
y SOLVE QUADRATIC INEQUALITIES To solve a quadratic inequality in one
variable, you can use the graph of the related quadratic function.
To solve ax2  bx  c  0, graph y  ax2  bx  c. Identify the x values for which
the graph lies below the x-axis.
a 0 a 0

O x
y  x 2  3x  2 x1 x2
x1 x2

Teaching Tip For In-Class {x | x 1  x  x 2}

Example 1, encourage students
to write the calculations as in For , include the x-intercepts in the solution.
Step 2 of Example 1 when test- To solve ax2  bx  c  0, graph y  ax2  bx  c. Identify the x values for which
ing a point inside the parabola. the graph lies above the x-axis.
Emphasize that they must write
a 0 a 0
a question mark over each
inequality sign after substituting
the coordinates of the point for x1 x2
the variables in the inequality. x1 x2

{x | x  x 1 or x  x 2}

SOLVE QUADRATIC For  , include the x-intercepts in the solution.

INEQUALITIES
Example 2 Solve ax2  bx  c  0
In-Class Example Power
Point®
Study Tip Solve x2  2x  3  0 by graphing.
Solving Quadratic The solution consists of the x values for which the graph of the related quadratic
2 Solve x2  4x  3  0 by Inequalities by function lies above the x-axis. Begin by finding the roots of the related equation.
graphing. Graphing
A precise graph of the x2  2x  3  0 Related equation
y related quadratic function (x  3)(x  1)  0 Factor.
is not necessary since the
zeros of the function were
x  3  0 or x  1  0 Zero Product Property
y
found algebraically. x  3 x  1 Solve each equation.
Sketch the graph of a parabola that has
x-intercepts at 3 and 1. The graph should
O x
O x open up since a  0.

y  x 2  4x  3 The graph lies above the x-axis to the left of

x  3 and to the right of x  1. Therefore, the y  x 2  2x  3
solution set is {xx  3 or x  1}.
{x | x  1 or x  3}
330 Chapter 6 Quadratic Functions and Inequalities
Teaching Tip Suggest that
students try solving first by
factoring, but if that does not
quickly yield a solution, then
use the Quadratic Formula.

330 Chapter 6 Quadratic Functions and Inequalities

 Example 3 Solve ax2  bx  c
0 In-Class Examples Power
Point®
Solve 0  3x2  7x  1 by graphing.
This inequality can be rewritten as 3x2  7x  1 0. The solution consists of the 3 Solve 0 2x2  6x  1 by
x values for which the graph of the related quadratic function lies on and below the graphing.
x-axis. Begin by finding the roots of the related equation. y
3x2  7x  1  0 Related equation y  2x 2  6x  1
b  b 
 4ac
2
x   2a Use the Quadratic Formula.

(7)   (7)2 
 
 4(3)(1)
x   2(3) Replace a with 3, b with 7, and c with 1.


7  61 
7  61
x   or x   Simplify and write as two equations.
6 6
O x
x
2.47 x
0.14 Simplify.

Sketch the graph of a parabola that has y

{x | 3.16
x
0.16}
x-intercepts of 2.47 and 0.14. The graph should
open up since a  0.
Teaching Tip When discussing
The graph lies on and below the x-axis at y  3x 2  7x  1 Example 4, be aware that some
x  0.14 and x  2.47 and between these two x students may not be familiar with
O
values. Therefore, the solution set of the all of the aspects of the game of
inequality is approximately {x0.14 x 2.47}.
football. Ask students who are
familiar with the terms in this
CHECK Test one value of x less than 0.14, one between 0.14 and 2.47, and one example and the margin note to
greater than 2.47 in the original inequality. explain them to the class.
Test x  1. Test x  0. Test x  3.
0  3x2  7x  1 0  3x2  7x  1 0  3x2  7x  1
4 SPORTS The height of a ball
Football ? 3(1)2  7(1)  1 ? 3(0)2  7(0)  1 ? 3(3)2  7(3)  1 above the ground after it is
A long hang time allows 0 0 0
the kicking team time to 09⫻ 0  1 ⻫ 05⫻
thrown upwards at 40 feet per
provide good coverage second can be modeled by the
on a punt return. The function h(x)  40x  16x2,
suggested hang time Real-world problems that involve vertical motion can often be solved by using a
for high school and college
where the height h(x) is given
quadratic inequality.
punters is 4.5–4.6 seconds. in feet and the time x is in
Source: www.takeaknee.com seconds. At what time in its
Example 4 Write an Inequality
flight is the ball within 15 feet
FOOTBALL The height of a punted football can be modeled by the function
H(x)  4.9x2  20x  1, where the height H(x) is given in meters and the time x of the ground? The ball is within
is in seconds. At what time in its flight is the ball within 5 meters of the ground? 15 feet of the ground for the first
The function H(x) describes the height of the football. Therefore, you want to find 0.46 second of its flight and again
the values of x for which H(x) 5. after 2.04 seconds until the ball
H(x) 5 Original inequality hits the ground at 2.5 seconds.
4.9x2  20x  1 5 H(x)  4.9x2  20x  1
4.9x2  20x  4 0 Subtract 5 from each side.

Graph the related function y  4.9x2  20x  4

using a graphing calculator. The zeros of the
function are about 0.21 and 3.87, and the graph lies
below the x-axis when x  0.21 or x  3.87.
Thus, the ball is within 5 meters of the ground for
the first 0.21 second of its flight and again after
3.87 seconds until the ball hits the ground at
4.13 seconds. [1.5, 5] scl: 1 by [5, 20] scl: 5

www.algebra2.com/extra_examples Lesson 6-7 Graphing and Solving Quadratic Inequalities 331

Differentiated Instruction
Intrapersonal Have students think about how the graph of a quadratic
inequality helps them understand what the inequality means. Ask them
to explore which is more meaningful to them (and therefore easier for
them to grasp), the quadratic inequality itself or the graph of the
inequality. Ask them to give an explanation of their choice.

Lesson 6-7 Graphing and Solving Quadratic Inequalities 331

 You can also solve quadratic inequalities algebraically.
In-Class Example Power
Point®
Example 5 Solve a Quadratic Inequality
Teaching Tip An alternative
way to solve the inequality in Solve x2  x  6 algebraically.
Example 5 is to first subtract 6 First solve the related quadratic equation x2  x  6.
from both sides of the inequality, Study Tip x2  x  6 Related quadratic equation
obtaining x2  x  6  0. After Solving Quadratic x2  x  6  0 Subtract 6 from each side.
factoring the left side as Inequalities (x  3)(x  2)  0 Factor.
(x  3)(x  2), point out that Algebraically
for the product (x  3)(x  2) As with linear inequalities, x30 or x  2  0 Zero Product Property
the solution set of a
to be greater than 0, either both x  3 x  2 Solve each equation.
quadratic inequality can
binomials must be positive or be all real numbers or Plot –3 and 2 on a number line. Use circles since these values are not solutions of
both must be negative. This fact the empty set, . The the original inequality. Notice that the number line is now separated into three
can be used to test the three solution is all real intervals.
numbers when all three
intervals of the number line. test points satisfy the x  3 3  x  2 x 2
inequality. It is the empty
5 Solve x2  x 2 set when none of the
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
algebraically. {x | 2
x
1} tests points satisfy the
inequality. Test a value in each interval to see if it satisfies the original inequality.

x  3 3  x  2 x2

3 Practice/Apply Test x  4.

x2  x  6
Test x  0.
x2  x  6
Test x  4.
x2  x  6
? ? ?
(4)2  (4)  6 02 06 42  4  6
12  6 ⻫ 0  6⫻ 20  6 ⻫
Study Notebook The solution set is {xx  3 or x  2}. This is shown on the number line below.
Have students—
• complete the definitions/examples 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7

for the remaining terms on their

Vocabulary Builder worksheets for
Chapter 6.
• write the basic steps shown in
Example 1 for solving a quadratic Concept Check 1. Determine which inequality, y  (x  3)2  1 y

inequality by graphing. or y (x  3)2  1, describes the graph at the

right. y  (x  3)2  1
• include any other item(s) that they
y  (x  3)2  1
2. OPEN ENDED List three points you might
find helpful in mastering the skills test to find the solution of (x  3)(x  5)  0. O x
in this lesson. Sample answer: one number less than 3,
one number between 3 and 5, and one
number greater than 5
3. Examine the graph of y  x2  4x  5 at the y
right. a. x  1, 5 b. x
1 or x  5 4
a. What are the solutions of 0  x2  4x  5? O x
b. What are the solutions of x2  4x  5  0? 2 2 4 6

c. What are the solutions of x2  4x  5 0? 4

1
x
5 8
y  x 2  4x  5

332 Chapter 6 Quadratic Functions and Inequalities

332 Chapter 6 Quadratic Functions and Inequalities

 Guided Practice Graph each inequality. 4–7. See margin.
GUIDED PRACTICE KEY 4. y  x2  10x  25 5. y  x2  16 About the Exercises…
Exercises Examples 6. y  2x2  4x  3 7. y x2  5x  6
Organization by Objective
4–7 1 y • Graph Quadratic
8 2, 3
8. Use the graph of the related function of
x2  6x  5  0, which is shown at the right, y  x 2  6x  5 Inequalities: 14–25
9–12 2, 3, 5
13 4 to write the solutions of the inequality. • Solve Quadratic
x  1 or x  5 Inequalities: 26–48
Solve each inequality algebraically. x
9. x2  6x  7  0 {x1  x  7} O
Odd/Even Assignments
10. x2  x  12  0 {xx  3 or x  4} Exercises 14–41 are structured
11. x2  10x  25 ∅ so that students practice the

x
3 }
12. x2 3 {x3 same concepts whether they
are assigned odd or even
Application 13. BASEBALL A baseball player hits a high problems.
pop-up with an initial upward velocity of
Alert! Exercises 53–58 require a
30 meters per second, 1.4 meters above the 30 m/s
ground. The height h(t) of the ball in meters graphing calculator.
t seconds after being hit is modeled by
h(t)  4.9t2  30t  1.4. How long does a
player on the opposing team have to catch 1.4 m
Assignment Guide
the ball if he catches it 1.7 meters above the Basic: 15–41 odd, 45, 49–52,
ground? about 6.1 s 59–71
★ indicates increased difficulty
Average: 15–45 odd, 49–52,
Practice and Apply 59–71 (optional: 53–58)
Homework Help Graph each inequality. 14–25. See pp. 343A–343F. Advanced: 14–44 even, 46–71
For See 14. y  x2  3x  18 15. y  x2  7x  8 16. y x2  4x  4
Exercises Examples
14–25 1 17. y x2  4x 18. y  x2  36 19. y  x2  6x  5
26–29 2, 3
20. y x2  3x  10 21. y  x2  7x  10 22. y  x2  10x  23
30–42 2, 3, 5
43–48 4 23. y  x2  13x  36 24. y  2x2  3x  5 25. y  2x2  x  3

Extra Practice Use the graph of its related function to write the solutions of each inequality.
See page 841.
26. x2  10x  25  0 5 27. x2  4x  12 0 2
x
6
y y x
x 2 O 2 4 6
O
4
y   x 2  10 x  25
8

12
y  x 2  4 x  12
16

28. x2  9  0 x  3 or x  3 29. x2  10x  21  0 x  7 or

x  3
y y

4 y   x 2  10 x  21
x
4 2 O 2 4
O x
4

8
y  x2  9

www.algebra2.com/self_check_quiz Lesson 6-7 Graphing and Solving Quadratic Inequalities 333

Answers
4. y 5. y 6. y 7. y
12 y  2x 2  4x  3 y  x 2  5x  6
8 12
4
O
8
4 2 2 4x
4
8 4
12 O x
O x 2 O 2 4 6x
y  x 2  10x  25 20 y  x 2  16

Lesson 6-7 Graphing and Solving Quadratic Inequalities 333

 Study Guide
NAME ______________________________________________ DATE

andIntervention
Intervention,
____________ PERIOD _____
Solve each inequality algebraically.
6-7 Study Guide and
p. 349 (shown)
Graphing and Solvingand p. Inequalities
Quadratic 350 30. x2 ⫺ 3x ⫺ 18  0 {xx  3 or x  6} 31. x2  3x ⫺ 28  0 {x7  x  4}
Graph Quadratic Inequalities
the following steps:
To graph a quadratic inequality in two variables, use
32. x2 ⫺ 4x 5 {x1
x
5} 33. x2  2x  24 {xx
6 or x  4}
⫺ x  12  0 {x4
x
3} 35. ⫺x2 ⫺ 6x  7 0 {xx
7 or x  1}
1. Graph the related quadratic equation, y  ax2  bx  c.
Use a dashed line for  or ; use a solid line for or . 34. ⫺x2
1

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. 36. 9x2 ⫺ 6x  1 0 xx   37. 4x2  20x  25  0 all reals
3
38. x2  12x  ⫺36 ∅ 39. ⫺x2  14x ⫺ 49  0 {xx  7}

Lesson 6-7
Example Graph the inequality y  x2  6x  7.
First graph the equation y  x2  6x  7. By completing the
41. 16x2  9  24x ∅
y
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,
40. 18x ⫺ x2 81 all reals
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 O x
inequality. Therefore, shade the region inside the parabola.
★ 42. Solve (x ⫺ 1)(x  4)(x ⫺ 3)  0. {x4  x  1 or x  3}
Exercises
Graph each inequality.
43. LANDSCAPING Kinu wants to plant a garden and surround it with decorative
1. y  x2  8x  17 2. y x2  6x  4 3. y  x2  2x  2
y y y stones. She has enough stones to enclose a rectangular garden with a perimeter
of 68 feet, but she wants the garden to cover no more than 240 square feet. What
could the width of her garden be? 0 to 10 ft or 24 to 34 ft
O x

O x O x

4. y  x2  4x  6 5. y  2x2  4x 6. y  2x2  4x  2

More About . . . 44. BUSINESS A mall owner has determined that the relationship between
y y y

O x
monthly rent charged for store space r (in dollars per square foot) and monthly
O x
profit P(r) (in thousands of dollars) can be approximated by the function
O x P(r)  8.1r2  46.9r ⫺ 38.2. Solve each quadratic equation or inequality.
Explain what each answer tells about the relationship between monthly rent and
profit for this mall. a–d. See margin.
Gl NAME
/M G ______________________________________________
Hill 349 DATE ____________
GlPERIOD
Al _____
b 2

Skills
6-7 Practice,
Practice (Average)
p. 351 and a. 8.1r2  46.9r ⫺ 38.2  0 b. 8.1r2  46.9r ⫺ 38.2  0
Practice, p. Solving
Graphing and 352 Quadratic
(shown)Inequalities c. 8.1r2  46.9r ⫺ 38.2  10 d. 8.1r2  46.9r ⫺ 38.2  10
Graph each inequality.
1. y x2 4 2. y  x2  6x  6 3. y  2x2  4x  2
y y y

Landscape 45. GEOMETRY A rectangle is 6 centimeters longer than it is wide. Find the
O x
O x
Architect possible dimensions if the area of the rectangle is more than 216 square
O x
Landscape architects centimeters. The width should be greater than 12 cm and the length shoud be
Use the graph of its related function to write the solutions of each inequality.
design outdoor spaces greater than 18 cm.
4. x2  8x  0 5. x2  2x  3  0 6. x2  9x  14 0 so that they are not only
6
y y y
functional, but beautiful FUND-RAISING For Exercises 46–48, use the following information.
O x

O 2 4 6 8x
and compatible with the The girls’ softball team is sponsoring a fund-raising trip to see a professional
–6
O x
natural environment. baseball game. They charter a 60-passenger bus for $525. In order to make a profit,
–12
they will charge $15 per person if all seats on the bus are sold, but for each empty
x  0 or x  8 3
x
1 2
x
7
Online Research seat, they will increase the price by $1.50 per person.
For information about a
Solve each inequality algebraically.
7. x2  x  20  0 8. x2  10x  16  0 9. x2  4x  5 0 career as a landscape ★ 46. Write a quadratic function giving the softball team’s profit P(n) from this
{xx  4 or x  5} {x2  x  8} architect, visit: fund-raiser as a function of the number of passengers n.
10. x2  14x  49  0
all reals
11. x2  5x  14
{xx  2 or x  7}
12. x2  15  8x
{x5
x
3}
www.algebra2.com/ ★ 47. What is the minimum number of passengers needed in order for the softball
13. x2  5x  7 0 14. 9x2  36x  36 0 15. 9x 12x2
careers team not to lose money? 6
{xx  2} xx
0 or x  43
★ 48. What is the maximum profit the team can make with this fund-raiser, and how
all reals

46. P(n)  n[15 

16. 4x2  4x  1  0 17. 5x2  10  27x 18. 9x2  31x  12 0

xx  12 xx

52 or x  5 x3
x
49 many passengers will it take to achieve this maximum?
19. FENCING Vanessa has 180 feet of fencing that she intends to use to build a rectangular 1.5(60  n)]  525 or $1312.50; 35 passengers
1.5n2  105n  525
play area for her dog. She wants the play area to enclose at least 1800 square feet. What
are 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 maker
wants to increase the profit margin this year, but predicts that each $20 increase in
49. CRITICAL THINKING Graph the intersection of the graphs of y x2  4 and
profit will reduce the number of bicycles sold by 10. How many $20 increases in profit can
the maker add in and expect to make a total profit of at least $100,000? from 5 to 10
y  x2 ⫺ 4. See margin.
Gl NAME
/M G ______________________________________________
Hill 352 DATE ____________
Gl PERIOD
Al _____
b 2
Reading
6-7 Readingto
to Learn
Learn Mathematics
Mathematics, p. 353 ELL 50. WRITING IN MATH Answer the question that was posed at the beginning of
Graphing and Solving Quadratic Inequalities
Pre-Activity How can you find the time a trampolinist spends above a certain
height?
the lesson. See margin.
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
How can you find the time a trampolinist spends above a certain height?
• Using the quadratic function given in the introduction, write a quadratic
inequality that describes the times at which the trampolinist is more Include the following in your answer:
than 20 feet above the ground. 16t 2  42t  3.75  20
• a quadratic inequality that describes the time the performer spends more than
Reading the Lesson 10 feet above the ground, and
1. Answer the following questions about how you would graph the inequality
y  x2  x  6. • two approaches to solving this quadratic inequality.
a. What is the related quadratic equation? y  x 2  x  6

b. Should the parabola be solid or dashed? How do you know? 334 Chapter 6 Quadratic Functions and Inequalities
solid; The inequality symbol is .
c. The point (0, 2) is inside the parabola. To use this as a test point, substitute
0 for x and 2 for y in the quadratic inequality.

d. Is the statement 2  02  0  6 true or false? true

NAME ______________________________________________ DATE ____________ PERIOD _____
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 y

Enrichment,
6-7 Enrichment p. 354
of the following related inequalities with its solution set. (2, 4)

a. x2  4x  0 ii i. {xx  0 or x  4} Graphing Absolute Value Inequalities

(0, 0) (4, 0)
b. x2  4x 0 iii ii. {x0  x  4} O x You can solve absolute value inequalities by graphing in much the same
manner you graphed quadratic inequalities. Graph the related absolute function
c. x2  4x  0 iv iii. {xx 0 or x  4}
for each inequality by using a graphing calculator. For  and , identify the
d. x2  4x  0 i iv. {x0 x 4} 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.

Helping You Remember For each inequality, make a sketch of the related graph and find the
solutions rounded to the nearest hundredth.
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 remember 1. |x  3|  0 2. |x|  6  0 3. |x  4|  8  0
which region to shade by looking at the inequality symbol and without using a test point.
x  3 or x  3 6  x  6 12  x  4
Sample answer: If the symbol is  or , shade the region above the
parabola. If the symbol is  or
, shade the region below the parabola.

334 Chapter 6 Quadratic Functions and Inequalities

 Standardized 51. Which is a reasonable estimate of the area under y
Test Practice the curve from x  0 to x  18? C
A 29 square units
8

4
4 Assess
B 58 square units
C 116 square units
Open-Ended Assessment
O 4 8 12 16 x
D 232 square units Speaking Have students explain
how to test points in the coordi-
52. If (x  1)(x  2) is positive, then A nate plane in order to determine
A x  1 or x  2. B x  1 or x  2. which region represents the solu-
C 1  x  2. D 2  x  1. tion to a quadratic inequality.
Also ask them to explain how to
Extending SOLVE ABSOLUTE VALUE INEQUALITIES BY GRAPHING Similar to quadratic analyze the graph of a quadratic
the Lesson inequalities, you can solve absolute value inequalities by graphing.
equation in order to determine
Graph the related absolute value function for each inequality using a graphing the solution set for a quadratic
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
inequality.
lies above the x-axis.
53. x  2  0 {xall reals, x  2} 54. x  7  0 {x7  x  7} Assessment Options
55. x  3 6  0 {xx  9 or x  3} 56. 2x  3  1  0 {xx
3.5 or Quiz (Lesson 6-7) is available on
x  2.5}
57. 5x  4  2 0 58. 4x  1  3  0 no real solutions
p. 370 of the Chapter 6 Resource
{x1.2
x
0.4} Masters.

Maintain Your Skills

Mixed Review Write each equation in vertex form. Then identify the vertex, axis of symmetry,
and direction of opening. (Lesson 6-6)
59. y  (x  1)2  8; 59. y  x2  2x  9 60. y  2x2  16x  32
1
61. y  x2  6x  18 Answers
2
(1, 8), x  1; up
49. y
60. y  2(x  4)2; Solve each equation using the method of your choice. Find exact solutions.
(4, 0), x  4; down (Lesson 6-5) y  x 2  4
1
61. y  (x  6)2; 62. x2  12x  32  0 63. x2  7  5x
64. 3x2  6x  2  3
2
(6, 0), x  6; up 4, 8 3 ⴞ 26

5 ⴞ i 3 Simplify. (Lesson 5-2) 65. 4a2b2  2a2b  4ab2  12a  7b 3
O x
63. 
2 65. (2a2b  3ab2  5a  6b )  (4a2b2  7ab2  b  7a)
66. (x3  3x2y  4xy2  y3)  (7x3  x2y  9xy2  y3) 6x 3  4x 2y  13xy 2 y  x2  4
1
67. x3y2(x4y  x3y1  x2y2) xy 3  y  
x
68. (5a  3)(1  3a) 15a2  14a  3
50. Answers should include the
Find each product, if possible. (Lesson 4-3) following.

 
3 3 • 16t 2  42t  3.75  10
69. 
6
4
3
7
 3

2 5
6

ⴚ21
ⴚ13
48
22
 70. [2 6 3] 9 0 [ ⴚ54 6]
• One method of solving this
2 4
inequality is to graph the
related quadratic function h(t ) 
71. LAW ENFORCEMENT Thirty-four states classify drivers having at least a
0.1 blood alcohol content (BAC) as intoxicated. An infrared device measures a 16t 2  42t  3.75  10. The
person’s BAC through an analysis of his or her breath. A certain detector measures interval(s) at which the graph is
BAC to within 0.002. If a person’s actual blood alcohol content is 0.08, write and above the x-axis represents the
solve an absolute value equation to describe the range of BACs that might register times when the trampolinist is
on this device. (Lesson 1-6) x  0.08
0.002; 0.078
x
0.082
above 10 feet. A second method
Lesson 6-7 Graphing and Solving Quadratic Inequalities 335 of solving this inequality would
be find the roots of the related
quadratic equation
Answers 16t 2  42t  3.75  10  0
44a. 0.98, 4.81; The owner will break even if he charges $0.98 or $4.81 per square foot. and then test points in the three
intervals determined by these
44b. 0.98  r  4.81; The owner will make a profit if the rent is between $0.98 and $4.81.
roots to see if they satisfy the
44c. 1.34  r  4.45; If rent is set between $1.34 and $4.45 per sq ft, the profit will be inequality. The interval(s) at
greater than $10,000. which the inequality is satisfied
44d. r  1.34 or r  4.45; If rent is set between $0 and $1.34 or above $4.45 per sq ft, the represent the times when the
profit will be less than $10,000. trampolinist is above 10 feet.

Lesson 6-7 Graphing and Solving Quadratic Inequalities 335

 Study Guide Study Guide and Review
and Review
Vocabulary and Concept Check
Vocabulary and
axis of symmetry (p. 287) parabola (p. 286) Square Root Property (p. 306)
Concept Check completing the square (p. 307) quadratic equation (p. 294) vertex (p. 287)
constant term (p. 286) Quadratic Formula (p. 313) vertex form (p. 322)
• This alphabetical list of discriminant (p. 316) quadratic function (p. 286) Zero Product Property (p. 301)
vocabulary terms in Chapter 6 linear term (p. 286) quadratic inequality (p. 329) zeros (p. 294)
includes a page reference maximum value (p. 288) quadratic term (p. 286)
where each term was minimum value (p. 288) roots (p. 294)
introduced.
Choose the letter of the term that best matches each phrase.
• Assessment A vocabulary
1. the graph of any quadratic function f
test/review for Chapter 6 is a. axis of symmetry
2. process used to create a perfect square trinomial b
available on p. 368 of the b. completing the square
3. the line passing through the vertex of a parabola and
Chapter 6 Resource Masters. dividing the parabola into two mirror images a c. discriminant
4. a function described by an equation of the form d. constant term
f(x)  ax2  bx  c, where a  0 h e. linear term
Lesson-by-Lesson 5. the solutions of an equation i f. parabola
Review 6. y  a(x  h)2  k j g. Quadratic Formula
7. in the Quadratic Formula, the expression under the h. quadratic function
For each lesson, radical sign, b2  4ac c i. roots
• the main ideas are b   
b  4ac 2
j. vertex form
8. x   2a g
summarized,
• additional examples review
concepts, and
• practice exercises are provided.
6-1 Graphing Quadratic Functions
Vocabulary See pages Concept Summary y
y  ax 2  bx  c
286–293.
PuzzleMaker The graph of y  ax2  bx  c, a  0, b
y -intercept: c axis of symmetry: x   2a
• opens up, and the function has a O x
ELL The Vocabulary PuzzleMaker minimum value when a  0, and
software improves students’ mathematics • opens down, and the function has a vertex
vocabulary using four puzzle formats— maximum value when a  0.
crossword, scramble, word search using a
word list, and word search using clues. Example Find the maximum or minimum value of f (x )
Students can work on a computer screen f(x)  x2  4x  12. 4 O 4 8x

or from a printed handout. Since a  0, the graph opens down and the function 4
f (x )  x 2  4x  12
has a maximum value. The maximum value of the
8
function is the y-coordinate of the vertex. The
4
MindJogger x-coordinate of the vertex is x   or 2. Find 12
2(1)
Videoquizzes the y-coordinate by evaluating the function for x  2.
f(x)  x2  4x  12 Original function
ELL MindJogger Videoquizzes f(2)  (2)2  4(2)  12 or 8 Replace x with 2.
provide an alternative review of concepts
Therefore, the maximum value of the function is 8.
presented in this chapter. Students work
in teams in a game show format to gain 336 Chapter 6 Quadratic Functions and Inequalities www.algebra2.com/vocabulary_review
points for correct answers. The questions
are presented in three rounds. TM

Round 1 Concepts (5 questions) Discuss with students how they might recognize a key concept that
Round 2 Skills (4 questions) needs to be included in the Foldable. Ask them to include a transition
Round 3 Problem Solving (4 questions) sentence or two in their notes that relates one topic to the next.
For more information Suggest that they use this discussion as they review their Foldable
about Foldables, see to add, delete, or reorganize material in order to make it more
Teaching Mathematics useful to them.
with Foldables.
Encourage students to refer to their Foldables while completing the
Study Guide and Review and to use them in preparing for the
Chapter Test.

336 Chapter 6 Quadratic Functions and Inequalities

0284-343F Alg 2 Ch06-828000 11/22/02 9:14 PM Page 337

Chapter 6 Study Guide and Review Study Guide and Review

Exercises Complete parts a–c for each quadratic function.

Answers
a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate
of the vertex. 10c. f (x)
b. Make a table of values that includes the vertex. f (x)
x 2  2x  15
c. Use this information to graph the function. (See Example 2 on pages 287 and 288.) 8
9. f(x)
x2  6x  20 10. f(x)
x2  2x  15 11. f(x)
x2  8x  7
12. f(x)
2x  12x  9 13. f(x)
x  4x  3
2 2 14. f(x)
3x2  9x  6 2 O 2 4 x
9–14. See margin. 8
Determine whether each function has a maximum or a minimum value. Then
find the maximum or minimum value of each function. 16
(See Example 3 on pages 288 and 289.) (1, 16)
15. f(x)
4x2  3x  5 16. f(x)
3x2  2x  2 17. f(x)
2x2 7
89 5 11a. 7; x
4; 4
min.; ⴚ max.; ⴚ max.; 7
16 3 11b. x f(x)
6-2 Solving Quadratic Equations by Graphing 2 5
See pages
294–299.
Concept Summary 3 8
• The solutions, or roots, of a quadratic equation are the zeros of the related 4 9
quadratic function. You can find the zeros of a quadratic function by finding
the x-intercepts of its graph. 5 8
• A quadratic equation can have one real solution, two real solutions, or no 6 5
real solution.
One Real Solution Two Real Solutions No Real Solution
11c. f (x)
f (x ) f (x ) f (x ) 4

O 4 8 12 x
O x O x O x
4

Example Solve 2x2  5x  2
0 by graphing. 8 f (x)
x 2  8x  7

5 5
The equation of the axis of symmetry is x
 or x
. f (x ) (4, 9)
2(2) 4

x 0
1

5
 2
5
 12a. 9; x
3; 3
2 4 2
f (x) 2 0
9
 0 2 12b. x f(x)
8

1 1 1
The zeros of the related function are  and 2. Therefore, the O x
2
solutions of the equation are  and 2.
1 2
f (x )
2x  5x  2 2 7
2
3 9
Exercises Solve each equation by graphing. If exact roots cannot be found, 4 7
state the consecutive integers between which the roots are located.
(See Examples 1–3 on pages 294 and 295.) 19. 2, 5 5 1
3
18. x2  36
0 6, 6 19. x2  3x  10
0 20. 2x2  x  3
0 1,  12c.
1 2 f (x)
21. x2  40x  80
0 22. 3x2  6x  2
0 23. (x  3)2  5
0 2, 8 (3, 9)
5
21. between 3 and 2; between 38 and 37 8
22. between 2 and 1; between 1 and 0 f (x)
2x 2  12x  9
Chapter 6 Study Guide and Review 337
4

O 4 8 12 x
9a. 20; x
3; 3 9c. f (x) 10a. 15; x
1; 1
9b. x f(x) 10b. x f(x) 4
24
5 15 1 12
16
f (x )
x 2  6x  20 (continued on the next page)
4 12 0 15
(3, 11) 8
3 11 1 16
2 12 2 15
8 4 O 4 8x
1 15 3 12

Chapter 6 Study Guide and Review 337

Study Guide and Review Chapter 6 Study Guide and Review

6-3 Solving Quadratic Equations by Factoring

Answers See pages Concept Summary
301–305.
13a. 3; x  2; 2 • Zero Product Property: For any real numbers a and b, if ab  0, then either
a  0, b  0, or both a and b  0.
13b. x f(x)
4 3
Example Solve x2  9x  20  0 by factoring.
x2  9x  20  0 Original equation
3 0
(x  4)(x  5)  0 Factor the trinomial.
2 1 x  4  0 or x  5  0 Zero Product Property
1 0 x  4 x  5 The solution set is {5, 4}.
0 3
Exercises Solve each equation by factoring. (See Examples 1–3 on pages 301 and 302.)
13c. f (x) 24. x2  4x  32  0 25. 3x2  6x  3  0 {1} 26. 5y2  80 {4, 4}
2
f (x)  x  4x  3 27. 2c2  18c  44  0 28. 25x2  30x  9 
3
5  1
3
3
29. 6x2  7x  3 , 
2 
24. {4, 8} 27. {11, 2}
(2, 1) Write a quadratic equation with the given root(s). Write the equation in the form
O x ax2  bx  c, where a, b, and c are integers. (See Example 4 on page 303.)
1
30. 4, 25 31. 10, 7 32.  , 2 3x 2  7x  2  0
3

x 2  29x  100  0 x 2  3x  70  0

6-4 Completing the Square

3 3 See pages Concept Summary
14a. 6; x  ;  306–312.
2 2 • To complete the square for any quadratic expression x2  bx:
14b. x f(x) Step 1 Find one half of b, the coefficient of x.
3 6 Step 2 Square the result in Step 1.
2 0 Step 3 Add the result of Step 2 to x2  bx. x2  bx    x  
b 2 b 2
2 2
32 34
Example Solve x2  10x  39  0 by completing the square.
1 0 x2  10x  39  0 Notice that x2  10x  39  0 is not a perfect square.

0 6 x2  10x  39 Rewrite so the left side is of the form x2  bx.

10
x2  10x  25  39  25 Since   25, add 25 to each side.
2
2
14c. f (x) (x  5)2  64 Write the left side as a perfect square by factoring.
121
34. ; x  5  8 Square Root Property
4 x  5  8 or x  5  8 Rewrite as two equations.
 11 2
x  
2  x3 x  13 The solution set is {13, 3}.

36. , 5
3 Exercises Find the value of c that makes each trinomial a perfect square. Then
f (x)  3x 2  9x  6 2 write the trinomial as a perfect square. (See Example 3 on page 307.)
O x 37. 3 25
 33. x2  34x  c 34. x2 11x  c
7 49 7 2
35. x2  x  c ; x  
16 4  
( 32 ,  34 ) 289; (x  17)2 2


5 i 7 Solve each equation by completing the square. (See Examples 4–6 on pages 308 and 309.)
38. 
4 36. 2x2  7x  15  0 37. 2n2  12n  22  0 38. 2x2  5x  7  3

338 Chapter 6 Quadratic Functions and Inequalities

338 Chapter 6 Quadratic Functions and Inequalities

 Chapter 6 Study Guide and Review Study Guide and Review

6-5 The Quadratic Formula and the Discriminant Answers

See pages Concept Summary
313–319. b   
b2  4ac 39a. 24
• Quadratic Formula: x   2a where a  0
39b. 2 complex
Solve x2  5x  66  0 by using the Quadratic Formula.
b   
b  4ac
2 39c. 1 i 6
x   2a Quadratic Formula
40a. 104
(5)  
(5)  
 4(1)(66)
2
40b. 2 irrational
  2(1) Replace a with 1, b with 5, and c with 66.

5  17 26
40c. 3 
  Simplify.
2
2
5  17 5  17 41a. 73
x   or x   Write as two equations.
2 2
 11  6 The solution set is {11, 6}. 41b. 2 irrational
7 73

41c. 
Exercises Complete parts a–c for each quadratic equation. 6
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.
(See Examples 1–4 on pages 314–316.) 39–41. See margin.
39. x2  2x  7  0 40. 2x2  12x  5  0 41. 3x2  7x  2  0

6-6 Analyzing Graphs of Quadratic Functions

See pages Concept Summary
322–328.
• As the values of h and k change, the graph of y  (x  h)2  k is the graph
of y  x2 translated
• h units left if h is negative or h units right if h is positive.
• k units up if k is positive or k units down if k is negative.
• Consider the equation y  a(x  h)2  k.
• If a  0, the graph opens up; if a  0 the graph opens down.
• If a  1, the graph is narrower than the graph of y  x2.
• If a  1, the graph is wider than the graph of y  x2.
Example Write the quadratic function y  3x2  42x  142 in vertex form. Then identify the
vertex, axis of symmetry, and direction of opening.
y  3x2  42x  142 Original equation
y  3(x2  14x)  142 Group ax2  bx and factor, dividing by a.
14 2
Complete the square by adding 3 .
y  3(x2  14x  49)  142  3(49) 2
Balance this with a subtraction of 3(49).
y  3(x  7)2  5 Write x2  14x  7 as a perfect square.

So, a  3, h  7, and k  5. The vertex is at (7, 5), and the axis of symmetry is
x  7. Since a is positive, the graph opens up.
Chapter 6 Study Guide and Review 339

Chapter 6 Study Guide and Review 339

 • Extra Practice, see pages 839–841.
Study Guide and Review • Mixed Problem Solving, see page 867.

Exercises Write each equation in vertex form, if not already in that form.
Answers Then identify the vertex, axis of symmetry, and direction of opening.
(See Examples 1 and 3 on pages 322 and 324.) 42–44. See margin.
42. (2, 3); x  2; down 1
42. y  6(x  2)2  3 43. y  5x2  35x  58 44. y  x2  8x
 7 2
  
13 7 13 3
43. y  5 x     ;   ,   ;
2 4 2 4 Graph each function. (See Examples 1–3 on pages 322 and 324.) 45–47. See margin.
7 45. y  (x  2)2  2 46. y  2x2  8x  10 47. y  9x2  18x  6
x    ; up
2
Write an equation for the parabola with the given vertex that passes through
1
44. y    (x  12)2  48; (12, 48); the given point. (See Example 4 on page 325.)
3
48. vertex: (4, 1) 49. vertex: (2, 3) 50. vertex: (3, 5)
x  12; down point: (2, 13) point: (6, 11) point: (0, 14)
45. y
1
y  3(x  4)2  1 y  (x  2)2  3 y  (x  3)2  5
2
6-7 Graphing and Solving Quadratic Inequalities
See pages Concept Summary
329–335.
• Graph quadratic inequalities in two variables as follows.
O x Step 1 Graph the related quadratic equation, y  ax2  bx  c. Decide if the
parabola should be solid or dashed.
y  (x  2)2  2 Step 2 Test a point (x1, y1) inside the parabola. Check to see if this point is a
solution of the inequality.
46. y
Step 3 If (x1, y1) is a solution, shade the region inside the parabola. If (x1, y1) is not a
solution, shade the region outside the parabola.
• To solve a quadratic inequality in one variable, graph the related quadratic function.
Identify the x values for which the graph lies below the x-axis for  and . Identify
the x values for which the graph lies above the x-axis for  and  .

y  2x 2  8x  10 Example Solve x2  3x  10  0 by graphing.

O x Find the roots of the related equation.
0  x2  3x  10 Related equation
0  (x  5)(x  2) Factor. y
47. y y  x 2  3x  10
x  5  0 or x  2  0 Zero Product Property
2
y  9x  18x  6 x  5 x2 Solve each equation. O x
8 4 4
Sketch the graph of the parabola that has x-intercepts at
O x 5 and 2. The graph should open up since a  0. The
graph lies below the x-axis between x  5 and x  2.
Therefore, the solution set is {x5  x  2}.

Exercises Graph each inequality. (See Example 1 on pages 329 and 330.) 51–53. See
margin.
51. y  x2  5x  15 52. y 4x2  36x  17 53. y  x2  7x  11
51. y Solve each inequality. (See Examples 2, 3, and 5 on pages 330–332.) 54–59. See pp. 343A–
25
54. 6x2  5x  4 55. 8x  x2  16 56. 2x2  5x  12 343F.
15 57. 2x2  5x  3 58. 4x2  9 4x 59. 3x2  5  6x

5 y  x 2  5x  15 340 Chapter 6 Quadratic Functions and Inequalities

O 1 3 5 7x
10
52. y 53. y
10 y  x 2  7x  11

4 O 2 6 10 x
20 O x

40

60
y  4x 2  36x  17

340 Chapter 6 Quadratic Functions and Inequalities

 Practice Test Practice Test

Vocabulary and Concepts

Assessment Options
Choose the word or term that best completes each statement.
Vocabulary Test A vocabulary
1. The y-coordinate of the vertex of the graph of y  ax2  bx  c is the (maximum, test/review for Chapter 6 can be
minimum) value obtained by the function when a is positive.
found on p. 368 of the Chapter 6
2. (The Square Root Property, Completing the square) can be used to solve any
quadratic equation. Resource Masters.
Chapter Tests There are six
Skills and Applications Chapter 6 Tests and an Open-
Complete parts a–c for each quadratic function.
Ended Assessment task available
a. Find the y-intercept, the equation of the axis of symmetry, and the
in the Chapter 6 Resource Masters.
x-coordinate of the vertex.
b. Make a table of values that includes the vertex. Chapter 6 Tests
c. Use this information to graph the function. 3–5. See pp. 343A–343F. Form Type Level Pages
3. f(x)  x2  2x  5 4. f(x)  3x2  8x 5. f(x)  2x2  7x  1 1 MC basic 355–356
Determine whether each function has a maximum or a minimum value. 2A MC average 357–358
Then find the maximum or minimum value of each function.
2B MC average 359–360
6. f(x)  x2  6x  9 min.; 0 7. f(x)  3x2  12x  24 min.; 36 8. f(x)  x2  4x max.; 4
2C FR average 361–362
9. Write a quadratic equation with roots 4 and 5. Write the equation in the form
ax2  bx  c  0, where a, b, and c are integers. x 2  x  20  0 2D FR average 363–364
3 FR advanced 365–366
Solve each equation using the method of your choice. Find exact solutions. 10–18. See margin.
10. x2  x  42  0 11. 1.6x2  3.2x  18  0 12. 15x2  16x  7  0 MC = multiple-choice questions
19 FR = free-response questions
13. x2  8x  48  0 14. x2  12x  11  0 15. x2  9x    0
4
16. 3x2  7x  31  0 17. 10x2  3x  1 18. 11x2  174x  221  0 Open-Ended Assessment
19. BALLOONING At a hot-air balloon festival, you throw a weighted marker straight Performance tasks for Chapter 6
down from an altitude of 250 feet toward a bull’s eye below. The initial velocity of can be found on p. 367 of the
the marker when it leaves your hand is 28 feet per second. Find how long it will Chapter 6 Resource Masters. A
take the marker to hit the target by solving the equation 16t2  28t  250  0. about 3.17 s
sample scoring rubric for these
Write each equation in vertex form, if not already in that form. Then identify tasks appears on p. A28.
the vertex, axis of symmetry, and direction of opening. 20–22. See pp. 343A–343F.
20. y  (x  2)2  3 21. y  x2  10x  27 22. y  9x2  54x  8
Graph each inequality. 23–25. See pp. 343A–343F. TestCheck and
23. y x2  6x  7 24. y  2x2  9
1
25. y  x2  3x  1 Worksheet Builder
2
Solve each inequality. 26. {x7  x  5} 27–28. See pp. 343A–343F.
This networkable software has
26. (x  5)(x  7)  0 27. 3x2  16 28. 5x2  x  2  0
three modules for assessment.
• Worksheet Builder to make
29. PETS A rectangular turtle pen is 6 feet long by 4 feet wide. The pen is enlarged
by increasing the length and width by an equal amount in order to double its worksheets and tests.
area. What are the dimensions of the new pen? 8 ft by 6 ft • Student Module to take tests
30. STANDARDIZED TEST PRACTICE Which of the following is the sum of both on-screen.
solutions of the equation x2  8x  48  0? B • Management System to keep
A 16 B 8 C 4 D 12 student records.
www.algebra2.com/chapter_test Chapter 6 Practice Test 341
Answers
9 5
10. 7, 6 11.   , 
2 2
Portfolio Suggestion 7 1
12.   ,  13. 12, 4
5 3
Introduction In this chapter quadratic equations have been graphed and
1 19
solved using many different methods, often following a process that involved 14. 11, 1 15.   , 
2 2
numerous steps.
7 421
 1 1
Ask Students Select an item from this chapter that shows your best work, 16.  17.   , 
6 2 5
including a graph, and place it in your portfolio. Explain why you believe it to be 13
your best work and how you came to choose this particular piece of work. 18. 17, 
11

Chapter 6 Practice Test 341

 Standardized
Test Practice

These two pages contain practice Part 1 Multiple Choice 

16x 2 
64x 
64
6. If x  0, then  x2
 ? B
questions in the various formats
that can be found on the most Record your answers on the answer sheet A 2 B 4
provided by your teacher or on a sheet of
frequently given standardized paper.
C 8 D 16
tests.
1. In a class of 30 students, half are girls and
A practice answer sheet for these 24 ride the bus to school. If 4 of the girls do 7. If x and p are both greater than zero and
not ride the bus to school, how many boys in 4x2p2  xp  33  0, then what is the value
two pages can be found on p. A1 this class ride the bus to school? C of p in terms of x? D
of the Chapter 6 Resource Masters.
A 2 B 11 A 
3 B
11

NAME DATE PERIOD x 4x
Standardized C 13 D 15
6 Standardized Test Practice
Test Practice 3 11
Student Recording Sheet,342–343 of p. A1Edition.) C  D 
Student Record Sheet (Use with pages the Student
4x 4x
Part 1 Multiple Choice
2. In the figure below, the measures of
⬔m  ⬔n  ⬔p  ? D
Select the best answer from the choices given and fill in the corresponding oval.

1 A B C D 4 A B C D 7 A B C D 9 A B C D

2 A B C D 5 A B C D 8 A B C D 10 A B C D A 90 B 180 8. For all positive integers n, n .

 3n
3 A B C D 6 A B C D

C 270 D 360 Which of the following equals 12? C

Part 2 Short Response/Grid In
Solve the problem and write your answer in the blank. A 4 B 8
For Questions 14–20, also enter your answer by writing each number or symbol in
a box. Then fill in the corresponding oval for that number or symbol.
m˚ C 16 D 32
11 15 17 19

12 / / / / / /
. . . . . . . . . . . .
0 0 0 0 0 0 0 0 0
13 1 1 1 1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3 3 3 3 3
4
5
6
4
5
6
4
5
6
4
5
6
4
5
4
5
4
5
4
5
4
5
4
5
4
5
4
5 n˚ 9. Which number is the sum of both solutions of
6 6 6 6 6 6 6 6

p˚ the equation x2  3x  18  0? C
7 7 7 7
Answers

7 7 7 7 7 7 7 7
8 8 8 8 8 8 8 8 8 8 8 8
9 9 9 9 9 9 9 9 9 9 9 9

14 16 18 20
A 6 B 3
3. Of the points (4, 2), (1, 3), (1, 3), (3, 1),
/ / / / / / / /
.

1
.
0
1
.
0
1
.
0
1
.

1
.
0
1
.
0
1
.
0
1
.

1
.
0
1
.
0
1
.
0
1
.

1
.
0
1
.
0
1
.
0
1
C 3 D 6
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
and (2, 1), which three lie on the same side of
the line y – x  0? C
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

(4, 2), (1, 3), (2, 1)

9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

A
Part 3 Quantitative Comparison 10. One of the roots of the polynomial
Select the best answer from the choices given and fill in the corresponding oval. B (4, 2), (1, 3), (3, 1) 5
21 A B C D 23 A B C D 25 A B C D 27 A B C D 6x2  kx  20  0 is . What is the value
2
22 A B C D 24 A B C D 26 A B C D 28 A B C D
C (4, 2), (1, 3), (2, 1) of k? C
4
D (1, 3), (1, 3), (3, 1) A 23 B 
3
C 23 D 7
Additional Practice 4. If k is an integer, then which of the following
must also be integers? B
See pp. 373–374 in the Chapter 6 5k  5 5k  5 5k  k
2
I.  II.  III. 
Resource Masters for additional 5k k1 5k
standardized test practice. A I only B II only
Test-Taking Tip
C I and II D II and III
Questions 8, 11, 13, 16, 21, and 27 Be sure to
use the information that describes the variables in
5. Which of the following is a factor of x2  7x  8? D any standardized test item. For example, if an item
A x2 B x1 says that x  0, check to be sure that your solution
for x is not a negative number.
C x4 D x8
342 Chapter 6 Quadratic Functions and Inequalities

TestCheck and
Log On for Test Practice
Worksheet Builder
The Princeton Review offers
additional test-taking tips and Special banks of standardized test
practice problems at their web site. Visit questions similar to those on the SAT,
www.princetonreview.com or ACT, TIMSS 8, NAEP 8, and Algebra 1
www.review.com End-of-Course tests can be found on
this CD-ROM.

342 Chapter 6 Quadratic Functions and Inequalities

 Aligned and
verified by

Part 2 Short Response/Grid In Part 3 Quantitative Comparison

Record your answers on the answer sheet Compare the quantity in Column A and the
provided by your teacher or on a sheet of quantity in Column B. Then determine
paper. whether:
11. If n is a three-digit number that can be A the quantity in Column A is greater,
expressed as the product of three B the quantity in Column B is greater,
consecutive even integers, what is one
possible value of n? 192, 480, or 960 C the two quantities are equal, or
D the relationship cannot be determined
12. If x and y are different positive integers and
from the information given.
x  y  6, what is one possible value of
3x  5y? 20, 22, 26, or 28
Column A Column B
13. If a circle of radius 12 inches has its radius 21. s0
decreased by 6 inches, by what percent is its
area decreased? 75% s increased by
300% of s
4s C
14. What is the least positive integer k for which
12k is the cube of an integer? 18
22. In 䉭ABC, side 
AB has length 8, and side 
BC
15. If AB  BC in the y has length 4.
figure, what is the C (2, 11)
the length of
y-coordinate of
C
side A 
10 D
point B? 7 B (x, y )

A (2, 3) 23. the perimeter of a the perimeter of a

rectangle with rectangle with area D
O x
area 8 units 10 units

16. In the figure, if O is the center of the circle,

what is the value of x? 35 24. 2350  2349 2349 C

25. t59
x˚ 110˚ t3 7 A
O
26. x2  12x  36  0
x 5 B
17. Let a♦b be defined as the sum of all
integers greater than a and less than b. For 27. pq
example, 6♦10  7  8  9 or 25. What is
the value of (75♦90)  (76♦89)? 165 |p| |q| D

18. If x2  y2  42 and x  y  6, what is the 28.

value of x  y? 7 71˚
x y
19. By what amount does the sum of the roots
exceed the product of the roots of the 54˚
equation (x  7)(x  3)  0? 25
20. If x2  36 and y2  9, what is the greatest the measure of the measure of
A
possible value of (x  y)2? 81 side x side y

www.algebra2.com/standardized_test Chapter 6 Standardized Test Practice 343

Chapter 6 Standardized Test Practice 343

 Page 284, Chapter 6 Getting Started 17a. 9; x  0; 0
1. y 2. y 17b. x f(x) 17c. f (x)

O x 2 5 4

y  2x  3 1 8
4 2 O 2 4x
O x y  x  5 0 9
4
1 8
2 5
(0, 9) f (x )  x 2  9

3. y 4. y 18a. 4; x  0; 0
y  x 2  2x  1
18b. x f(x) 18c. f (x)

2 4

y  x2  4 O x 1 2
0 4 O x
O x 1 2
2 4
f (x )  2x 2  4 (0, 4)

Page 291, Lesson 6-1 19a. 1; x  0; 0

Additional Answers for Chapter 6

14a. 0; x  0; 0 19b. 19c.

x f(x) f (x)
14b. x f(x) 14c. f (x)
2 13
2 8 1 4
1 2 0 1
0 0 1 4 f (x )  3x 2  1 (0, 1)
1 2 2 13
f (x )  2x 2 O x
2 8 (0, 0)
O x
20a. 4; x  2; 2
15a. 0; x  0; 0 20b. 20c.
x f(x) f (x)
15b. x f(x) 15c. f (x)
0 4
O
2 20 (0, 0) x 1 1
f (x )  5x 2
1 5 2 0
0 0 3 1 f (x )  x 2  4x  4
1 5 4 4 O (2, 0) x
2 20

21a. 9; x  4.5; 4.5

16a. 4; x  0; 0 21b. 21c.
x f(x) f (x)
16b. x f(x) 16c. f (x)
3 9 2
2 8 12
4 11 O 4 8 12 x
1 5 8 4.5 11.25 4
0 4 5 11
8 f (x )  x 2  9x  9
1 5 (0, 4) f (x )  x 2  4 6 9
2 8 4 2 O 2 4x
12 (4 12 , 1114 )

343A Chapter 6 Additional Answers

22a. 5; x  2; 2 5 5
27a. 0; x    ;  
22b. 22c. 4 4
x f(x) f (x)
O x 27b. x f(x) 27c. f (x)
0 5
3 3
1 8
f (x )  2x 2  5x
2 2
2 9 O
5 25
3 8   x
4 8
4 5 2
f (x )  x  4x  5 1 3
(2, 9)
0 0
( 54 ,  258)
23a. 36; x  6; 6
23b. 23c. 28a. 1; x  0; 0
x f(x) f (x)
28b. x f(x) 28c. f (x)
8 4 6
2 1
7 1 4 f (x )  0.5x 2  1
1
6 0 1 
f (x )  x 2  12x  36 2
2
5 1 0 1
O
4 4 16 12 8 4 O x 1 
1
x
(6, 0) 2
(0, 1)
2 1
24a. 1; x  1; 1

Additional Answers for Chapter 6

24b. 24c. 29a. 0; x  6; 6
x f(x) f (x)
29b. x f(x) 29c. (6, 9) f (x)
3 8 f (x )  3x 2  6x  1
8
8 8
2 1 4
7 8.75
1 4 O x
6 9
0 1 8 4 O x
5 8.75 4
1 8
(1, 4) 4 8
f (x )  0.25x 2  3x

25a. 3; x  2, 2 9
25b. 25c. 30a.  ; x  3, 3
x f(x) f (x) 2
(2, 5)
0 3
30b. x f(x) 30c. f (x)
2
f (x )  2x  8x  3 5 2
1 3
2 5 4 0.5
3 3 O x 3 0
(3, 0)
4 3 2 0.5
O x
1 2 f (x )  12 x 2  3x  92
2 2
26a. 0; x    ,  
3 3 8 1 1
26b. 26c. 31a.   ; x   ; 
x f(x) f (x) 9 3 3

2 4
( 23 , 43 ) 31b. x f(x) 31c. f (x)

1 x 7
1 O 1 
2 4 9
  8
3 3 0 
f (x )  3x 2  4x 9 2
f (x )  x 2  3 x  9
8
0 0 1
 1 O
1 7 3 x
5
1  (13 , 1)
9
7
2 1
9

Chapter 6 Additional Answers 343B

 Pages 298–299, Lesson 6-2 Page 305, Lesson 6-3
39. Let x be the first number. 50. Answers should include the following.
Then, 7  x is the other number. • Subtract 24 from each side of x 2  5x  24 so that
x(7  x)  14 the equation becomes x 2  5x  24  0. Factor the
x 2  7x  14  0 left side as (x  3)(x  8). Set each factor equal to
y Since the graph of the zero. Solve each equation for x. The solutions to the
2
y  x  7x  14 related function does not equation are 3 and 8. Since length cannot be
intersect the x-axis, this negative, the width of the rectangle is 3 inches, and
O x
equation has no real the length is 3  5 or 8 inches.
solutions. Therefore no • To use the Zero Product Property, one side of the
such numbers exist. equation must equal zero.

Pages 318–319, Lesson 6-5

21
40. Let x be the first number. 28. 2, 32 29. i 
7
Then, 9  x is the other number. 3  15 
30. 2  i 3
 31. 
x(9  x)  24 2
x 2  9x  24  0 9
32. 2
 33. 
y Since the graph of the 2
related function does not 5  46 
34. 3  i 7
 35. 
O x intersect the x-axis, this 3
3
Additional Answers for Chapter 6

2
y  x  9x  24 equation has no real 36. 4  7
 37. 0,  
solutions. Therefore no 10
such numbers exist. 38. 3  22  39. 2, 6
41. This means that the cables do not touch the floor of
the bridge, since the graph does not intersect the
x-axis and the roots are imaginary.
48. Answers should include the following. 46. The person’s age can be substituted for A in the appro-
priate formula, depending upon their gender, and their
• h (t )
180
2
h (t )  16t  185 average blood pressure calculated. See student’s work.
160 • If a woman’s blood pressure is given to be 118, then
140 solve the equation 118  0.01A2  0.05A  107 to
120 find the value of A. Use the Quadratic Formula, sub-
100 stituting 0.01 for a, 0.05 for b, and 11 for c. This gives
80 solutions of about 35.8 or 30.8. Since age cannot
60 be negative, the only valid solution for A is 30.8.
40 59. y 60. y
20 x1
xy9 8
0 1 2 3 4 5 t yx4 6 yx
4
• Locate the positive x-intercept at about 3.4. This 2
represents the time when the height of the ride is 0.
6 4 O 2 4 6 8 x O x
Thus, if the ride were allowed to fall to the ground, it
4 xy  3 y  1
would take about 3.4 seconds.
6

Page 300, Follow-Up of Lesson 6-2

Graphing Calculator Investigation
Page 321, Preview of Lesson 6-6
1. linear: y  4.343x  89.669; Graphing Calculator Investigation
quadratic: y  0.044x2  0.003x  0.218
4. Both graphs have the same
shape, but the graph of
y  x2  2.5 is 2.5 units
above the graph of y  x2.

[0, 85] scl: 5 by [0, 300] scl: 20 [0, 85] scl: 5 by [0, 300] scl: 20
The quadratic equation fits the data better.
343C Chapter 6 Additional Answers
 5. Both graphs have the same 13. The graphs have the same
shape, but the graph of shape, but the graph of
y  x2 opens downward y  (x  3)2  5 is 7 units
while the graph of above the graph of
y  x2  9 opens upward y  (x  3)2  2.
and is 9 units lower than
the graph of y  x2.
14. The graph of
6. The graph of y  3x2 is
y  6(x  2)2  1 is
narrower than the graph of
narrower than the graph of
y  x2.
y  3(x  2)2  1.

15. The graph of

7. The graph of y  6x 2 1
y   (x  2)2  1 is wider
opens downward and is 4
narrower than the graph of than the graph of
y  x 2. y  4(x  2)2  3, and its
vertex is 2 units above the
vertex of y  4(x  2)2  3.

8. The graphs have the same Pages 326–327, Lesson 6-6

Additional Answers for Chapter 6

shape, but the graph of 27. y 28. y
y  (x  3)2 is 3 units to the
left of the graph of y  x2. O x
y  (x  5)2  3

9. The graphs have the same

shape and open downward, O x
y  4(x  3)2  1
but the graph of
1
y   x 2  2 is 2 units 29. y 30.
3 y
above the graph of
1
y   x 2.
3 O x
10. The graphs have the same
shape, but the graph of 1
y  (x  7)2 is 7 units to the y  4 (x  2)2  4
right of the graph of y  x 2. 1
O x y  2 (x  3)2  5

31. y 32. y

11. The graph of

O x
y  3(x  4)2  7 is 4 units
to the left, 7 units below,
and narrower than the
graph of y  x2. y  x 2  8x  18
O x
1 y  x 2  6x  2
12. The graph of y   x 2  1
4
opens downward, is wider 33. y 34. y
y  4x 2  16x  11
than and 1 unit above the y  5x 2  40x  80
1
graph of y   x 2. O x
4

O x

Chapter 6 Additional Answers 343D

 35. y 36. y 18. y 19. y
5
y   1 x 2  5x  27
2 2
8 4 O 4 8x
O x
10
O x
20
O x
y  1 x 2  4x  15 30
3
y  x 2  36 y  x 2  6x  5

53. y  ax 2  bx  c 20. y 21. y  x 2  7x  10

y  x 2  3x  10 14 y
 b
y  a x2  x  c
a  10 20

 b b 2
y  a x2  x    c  a 
a 2a  b 2
2a   6 12

b2

ya x c
b 2
2a 4a 2 4

6 4 2 O 2x 12 8 4 O 4x
b 4
The axis of symmetry is x  h or   .
2a
54. All quadratic equations are a transformation of the par-
ent graph y  x 2. By identifying these transformations 22. y 23. y
y  x 2  13x  36
when a quadratic function is written in vertex form, you 2
y  x  10x  23 6
can redraw the graph of y  x2. Answers should
Additional Answers for Chapter 6

include the following. 2

• In the equation y  a(x  h)2  k, h translated the O x O 2 6 10 x
graph of y  x 2 h units to the right when h is positive 4
and h units to the left when h is negative. The graph of
y  x 2 is translated k units up when k is positive and 8
k units down when k is negative. When a is positive,
the graph opens upward and when a is negative, the 24. 25.
graph opens downward. If the absolute value of a is y y
less than 1, the graph will be narrower than the graph 4
of y  x 2, and if the absolute value of a is greater than
1, the graph will be wider than the graph of y  x 2. 2 O 2 4 x
• Sample answer: y  2(x  2)2  3 is the graph of O x
4
y  x2 translated 2 units left and 3 units down. The y  2x 2  3x  5
graph opens upward, but is narrower that the graph 8
y  2x 2  x  3
of y  x 2.

Page 333, Lesson 6-7

Page 340, Chapter 6 Study Guide and Review
14. 15.
15
y y y  x 2  7x  8 
54. x x    or x  
4
3
1
2
12 55. all reals
5
8 4
5
O 4 8x 8

56. x 4  x  
2
3

57. x x    or x  3
4 1
15
2
y  x 2  3x  18
4 x
58. x  
1  10 1  10

25 O 4 8
x 
2 2
16. y 17. y 59. x x  
3  26  3  26
or x   

3 3

O
x

O x y  x 2  4x
y  x 2  4x  4

343E Chapter 6 Additional Answers

Page 341, Chapter 6 Practice Test 20. (2, 3); x  2; up
3a. (0, 5); x  1; 1 21. y  (x  5)2  2; (5, 2); x  5; up
3b. x f(x) 3c. f (x) 22. y  9(x  3)2  73; (3, 73); x  3; down
1 8 23. y 24. y
( 1, 4) 20
0 5
10 y  2x 2  9
1 4 f (x)  x 2  2x  5
2 5 12 8 4 O x
3 8 x 10
O

y  x 2  6x  7 20
O x
4 4
4a. (0, 0); x   ; 
3 3
25. y
4b. x f(x) 4c. f (x) 4 16 y   12 x 2  3x  1
( ,
3 3 )
0 0
1 5 f (x)  3x 2  8x
4 16
 
3 3
2 4 O x
O x
3 3

Additional Answers for Chapter 6

43 43 
7 7
5a. (0, 1); x    ;  
4 4 
27. x x   or x  
3 3
1  41
 1  41

28. x x   or x  
5b. x f(x) 5c. f (x)
( 74 ,418) 10 10
3 2
f (x)  2x 2  7x  1
2 5
7 41
 
4 8
1 4 O x
0 1

Chapter 6 Additional Answers 343F