Quadratic Functions - Shakopee Public Schools

Quadratic Functions
5A Quadratic Functions
and Complex Numbers
Lab
Explore Parameter Changes
5-1 Using Transformations to
Graph Quadratic Functions
5-2 Properties of Quadratic
Functions in Standard Form
Lab
Explore Graphs and Factors
5-3 Solving Quadratic Equations
by Graphing and Factoring
5-4 Completing the Square
5-5 Complex Numbers and Roots
5-6 The Quadratic Formula
5B Applying Quadratic
Functions
5-7
Solving Quadratic Inequalities
5-8 Curve Fitting with Quadratic
Models
5-9 Operations with Complex
Numbers
• Make connections among representations of quadratic functions.
• Use various methods to solve quadratic
equations and apply them to realworld problems.
Planetary Pass
How far could you throw a football if
you were on Mars or Saturn? You can
find the answer by using quadratic
functions.
KEYWORD: MB7 ChProj
310
Chapter 5
Vocabulary
Match each term on the left with a definition on the right.
A. a change in a function rule and its graph
1. linear equation
2. solution set
3. transformation
4. x-intercept
B. the x-coordinate of the point where a graph crosses
the x-axis
C. the group of values that make an equation or
inequality true
D. a letter or symbol that represents a number
E. an equation whose graph is a line
Squares and Square Roots
Simplify each expression.
5. 3.2 2
6.
(_25 )
2
7. √
121
8.
1
_
√
16
Simplify Radical Expressions
Simplify each expression.
9. √
72
10. 2( √
144 - 4)
11. √
33 · √
75
√
54
12. _
√
3
Multiply Binomials
Multiply.
13. (x - 2)(x - 6)
14. (x + 9)(x - 9)
15. (x + 2)(x + 7)
16. (2x - 3)(5x + 1)
Solve Multi-Step Equations
Solve each equation.
17. 2x + 10 = - 32
2 (x - 1) = 11
19. _
3
18. 2x - (1 - x) = 2
20. 2(x + 5) - 5x = 1
Graph Linear Functions
Graph each function.
21. y = -x
23. y = -3x + 6
22. y = 2x - 1
1x + 2
24. y = _
3
Quadratic Functions
311
Previously, you
• graphed and transformed
•
•
•
linear functions.
solved linear equations and
inequalities.
fit data using linear models.
used and performed
operations with real numbers.
You will study
• graphing and transforming
•
•
•
quadratic functions.
solving quadratic equations
and inequalities.
fitting data to quadratic
models.
using and performing
operations with imaginary
and other complex numbers.
You can use the skills
in this chapter
• in advanced math classes,
•
•
312
Chapter 5
including Precalculus.
in other classes, such as
Chemistry, Physics, and
Economics.
outside of school to identify
patterns and model data.
Key
Vocabulary/Vocabulario
absolute value of a
complex number
valor absoluto de un
número complejo
complex conjugate
conjugado complejo
complex number
número complejo
imaginary number
número imaginario
maximum value
valor máximo
minimum value
valor mínimo
parabola
parábola
quadratic function
función cuadrática
vertex form
forma en vértice
zero of a function
cero de una función
Thinking About Vocabulary
To become familiar with some of the
vocabulary terms in the chapter, consider
the following. You may refer to the chapter,
the glossary, or a dictionary if you like.
1. Quadratic is from the Latin quadrum,
which means “square.” A quadratic
function always contains a square
of the variable, such as x 2. What is a
quadrilateral, and how does it relate
to a square? What are some other words
that use the root quad-, and what do
they mean?
2. The word conjugate can mean “joined
together, especially in pairs.” Name
some mathematical relationships that
involve pairs.
3. What might the terms maximum value
or minimum value of a function refer to?
4. The word vertex can mean “highest
point.” What might the vertex form of
a quadratic function indicate about the
function’s graph?
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Try This
Describe two representations you could use to solve each problem.
1. A triangle with coordinates A(3, 5), B(2, 2), and C(3, -2) is translated 3 units
left and 2 units up. Give the coordinates of the image.
2. A bottle of juice from a vending machine costs $1.50. Hiroshi buys a bottle by
inserting 8 coins in quarters and dimes. If Hiroshi receives 5 cents in change,
how many quarters did he use? how many dimes?
3. What is the slope of the line that passes through the point (6, 9) and has a
y-intercept of 3?
Quadratic Functions
313
5-1
Explore Parameter Changes
You can use a graphing calculator to explore how changes in the parameters
of a quadratic function affect its graph. Recall from Lesson 1-9 that the
quadratic parent function is f (x) = x 2 and that its graph is a parabola.
Use with Lesson 5-1
KEYWORD: MB7 Lab5
Activity
Describe what happens when you change the value of k in the quadratic function g(x) = x 2 + k.
1 Choose three values for k. Use 0, -5 (a negative value),
and 4 (a positive value). Press
, and enter X2 for
Y1, X2 - 5 for Y2, and X2 + 4 for Y3.
2 Change the style of the graphs of Y1 and Y2 so that
you can tell which graph represents which function.
To do this, move the cursor to the graph style
indicator next to Y1. Press
to cycle through
the options. For Y1, which represents the parent
function, choose the thick line.
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3 Next, change the line style for Y2 to the dotted line.
4 Graph the functions in the square window by
pressing
and choosing 5 : ZSquare.
Notice that the graphs are identical except that the graph
of Y2 is shifted 5 units down and the graph of Y3 has been
shifted 4 units up from the graph of Y1.
You can conclude that the parameter k in the function
g(x) = x 2 + k has the effect of translating the parent
function f (x) = x 2 k units up if k is positive and ⎜k⎟ units
down if k is negative.
Try This
Use your graphing calculator to compare the graph of each function to the graph
of f (x) = x 2. Describe how the graphs differ.
1. g(x) = (x - 4 )2
2. g(x) = (x + 3)2
3. g(x) = -x 2
4. Make a Conjecture Use your graphing calculator to determine what happens
2
when you change the value of h in the quadratic function g(x) = (x - h ) . Check
both positive and negative values of h.
5. Make a Conjecture Use your graphing calculator to determine what happens
when you change the value of a in the quadratic function g(x) = ax 2. Check values
of a that are greater than 1 and values of a that are between 0 and 1.
314
Chapter 5 Quadratic Functions
5-1
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Using Transformations to Graph
Quadratic Functions
A2.3.3
andgraphs
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IA-2.8
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functions
2.1a Analyze,
Recognize
parent
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polynomial,
exponential,
radical,
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and logarithmic
2
of
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y = predict
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functions
and
effects
Why learn this?
You can use transformations of quadratic
functions to analyze changes in braking distance.
(See Example 5.)
Objectives
Transform quadratic
functions.
Describe the effects
of changes in the
coefficients of
2
y = a(x - h) + k.
Vocabulary
quadratic function
parabola
vertex of a parabola
vertex form
In Chapters 2 and 3, you studied linear functions
of the form f (x) = mx + b. A quadratic function
is a function that can be written in the form
2
f (x) = a(x - h) + k (a ≠ 0). In a quadratic
function, the variable is always squared. The table
shows the linear and quadratic parent functions.
Linear and Quadratic Parent Functions
ALGEBRA
NUMBERS
GRAPH
Linear Parent Function
f(x) = x
. . . methods and tools
which may include
graphing calculators.
x
-2
-1
0
1
2
f(x) = x
-2
-1
0
1
2
x
-2
-1
0
1
2
4
1
0
1
4
Quadratic Parent Function
f(x) = x 2
f(x) = x
2
Notice that the graph of the parent function f (x) = x 2 is a U-shaped curve
called a parabola . As with other functions, you can graph a quadratic function
by plotting points with coordinates that make the equation true.
EXAMPLE
1
Graphing Quadratic Functions Using a Table
Graph f (x) = x 2 - 6x + 8 by using a table.
Make a table. Plot enough ordered pairs to see both sides of the curve.
x
f(x) = x 2 - 6x + 8
(x, f(x))
1
f(1) = 1 2 - 6(1) + 8 = 3
2
f(2) = 2 2 - 6(2) + 8 = 0
3
f(3) = 3 2 - 6(3) + 8 = -1
4
f(4) = 4 2 - 6(4) + 8 = 0
5
f(5) = 5 2 - 6(5) + 8 = 3
(1, 3)
(2, 0)
(3, -1)
(4, 0)
(5, 3)
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5-1 Using Transformations to Graph Quadratic Functions
315
1. Graph g(x) = -x 2 + 6x - 8 by using a table.
You can also graph quadratic functions by applying transformations to the
parent function f (x) = x 2. Transforming quadratic functions is similar to
transforming linear functions (Lesson 2-6).
Translations of Quadratic Functions
Horizontal Translations
Vertical Translations
Horizontal Shift of ⎪h⎥ Units
Vertical Shift of ⎪k⎥ Units
f(x) = x 2
f (x - h ) = (x - h )
EXAMPLE
2
f(x) = x 2
2
f(x) + k = x 2 + k
Moves left for
h<0
Moves down for
k<0
Moves right for
h>0
Moves up for k > 0
Translating Quadratic Functions
Using the graph of f (x) = x 2 as a guide, describe the transformations, and
then graph each function.
A g(x) = (x + 3) 2 + 1
B g(x) = (x - 2) 2 - 1
Identify h and k.
Identify h and k.
g(x) = (x -(-3)) + 1
2
h
k
Because h = -3, the graph
is translated 3 units left.
Because k = 1, the graph
is translated 1 unit up.
Therefore, g is f translated
3 units left and 1 unit up.
}
h
k
Because h = 2, the graph
is translated 2 units right.
Because k = -1, the graph
is translated 1 unit down.
Therefore, g is f translated
2 units right and 1 unit down.
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Using the graph of f (x) = x 2 as a guide, describe the
transformations, and then graph each function.
2a. g(x) = x 2 - 5
2b. g(x) = (x + 3) 2 - 2
Recall that functions can also be reflected, stretched, or compressed.
316
Chapter 5 Quadratic Functions
Reflections, Stretches, and Compressions of Quadratic Functions
Reflections
Reflection Across x-axis
Reflection Across y-axis
Input values
change.
Output values
change.
f(x) = x 2
f(x) = x 2
-f(x) = -(x 2)
= -x 2
f(-x) = (-x) 2 = x 2
The function
f(x) = x 2 is its own
reflection across
the y-axis.
The function is
flipped across the
x-axis.
Stretches and Compressions
Horizontal Stretch/Compression by a
Factor of b
Vertical Stretch/Compression by a
Factor of a
Output values
change.
Input values
change.
f(x) = x 2
f 1x = 1x
b
b
f(x) = x 2
(_ ) (_ )
EXAMPLE
2
a · f(x) = ax 2
b > 1 stretches away from the y-axis.
a > 1 stretches away from the x-axis.
0 < b < 1 compresses toward the y-axis.
0 < a < 1 compresses toward the x-axis.
3
Reflecting, Stretching, and Compressing Quadratic Functions
Using the graph of f (x) = x 2 as a guide, describe the transformations, and
then graph each function.
2
A g (x) = -4x 2
B g(x) = 1 x
2
Because a is negative, g is a
Because b = 2, g is a
reflection of f across the x-axis.
horizontal stretch of
Because a = 4, g is a vertical
f by a factor of 2.
stretch of f by a factor of 4.
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Using the graph of f (x) = x 2 as a guide, describe the
transformations, and then graph each function.
1 x2
3b. g(x) = -_
3a. g (x) = (2x) 2
2
5-1 Using Transformations to Graph Quadratic Functions
317
If a parabola opens upward, it has a lowest point. If a parabola opens downward,
it has a highest point. This lowest or highest point is the vertex of a parabola .
The parent function f (x) = x 2 has its vertex at the origin. You can identify the
vertex of other quadratic functions by analyzing the function in vertex form.
2
The vertex form of a quadratic function is f (x) = a(x - h) + k, where a, h,
and k are constants.
Vertex Form of a Quadratic Function
a indicates a reflection
across the x-axis and/or
a vertical stretch or
compression.
h indicates
a horizontal
translation.
k indicates a
vertical translation.
Because the vertex is translated h horizontal units and k vertical units from the
origin, the vertex of the parabola is at (h, k ).
EXAMPLE
4
Writing Transformed Quadratic Functions
Use the description to write the quadratic function in vertex form.
The parent function f (x) = x 2 is reflected across the x-axis, vertically
stretched by a factor of 6, and translated 3 units left to create g.
Step 1 Identify how each transformation affects the constants in
vertex form.
reflection across x-axis:
ProjectbyTitle
vertical stretch
6:
When the quadratic
parent function
f(x) = x 2 is written
in vertex form,
y = a(x - h) 2 + k,
a = 1, h = 0, and
k = 0.
a is negative ⎫
⎬ a = -6
⎭
⎜a⎟ = 6
Algebra
II 2007 Student
translation
left 3 units:
h = -3Edition
Spec Number
Step 2 Write the transformed function.
A207Se2 c05l01002
g(x) = a(x - h) + k
Vertex form of a quadratic function
Created By2
= -6
) + 0 Substitute -6 for a, -3 for h, and 0 for k.
(x -(-3)Corporation
Krosscore
Creation
(x + 3) 2 Date
= -6
Simplify.
01/26/2005
Check Graph both functions on a graphing calculator. Enter f as Y1 and
g as Y2. The graph indicates the identified transformations.
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Use the description to write the quadratic function in
vertex form.
4a. The parent function f (x) = x 2 is vertically compressed by a
factor of __13 and translated 2 units right and 4 units down to
create g.
4b. The parent function f (x) = x 2 is reflected across the x-axis and
translated 5 units left and 1 unit up to create g.
318
Chapter 5 Quadratic Functions
EXAMPLE
5
Automotive Application
The minimum braking distance d in feet for a vehicle on dry concrete is
approximated by the function d(v) = 0.045v 2, where v is the vehicle’s
speed in miles per hour. If the vehicle’s tires are in poor condition, the
braking-distance function is d p(v) = 0.068v 2. What kind of transformation
describes this change, and what does the transformation mean?
Examine both functions in vertex form.
d(v) = 0.045(v - 0) 2 + 0
d p(v) = 0.068(v - 0) 2 + 0
The value of a has increased from 0.045 to 0.068. The increase indicates a
vertical stretch.
Find the stretch factor by comparing the new a-value to the old a-value:
a from d p(v) _
__
= 0.068 ≈ 1.5
0.045
a from d(v)
The function d p represents a vertical stretch of d by a factor of approximately
1.5. Because the value of each function approximates braking distance, a
vehicle with tires in poor condition takes about 1.5 times as many feet to
stop as a vehicle with good tires does.
Check Graph both functions on a graphing
calculator. The graph of d p appears to be
vertically stretched compared with the
graph of d.
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Use the information above to answer the following.
5. The minimum braking distance d n in feet for a vehicle with
2
new tires at optimal inflation is d n(v) = 0.039v , where v is the
vehicle’s speed in miles per hour. What kind of transformation
describes this change from d(v) = 0.045v 2, and what does this
transformation mean?
THINK AND DISCUSS
1. Explain how the values of a, h, and k in the vertex form of a quadratic
function affect the function’s graph.
2. Explain how to determine which of two quadratic functions expressed
in vertex form has a narrower graph.
3. GET ORGANIZED Copy
and complete the graphic
organizer. In each
row, write an equation
that represents the
indicated transformation
of the quadratic parent
function, and show
its graph.
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5-1 Using Transformations to Graph Quadratic Functions
319
5-1
Exercises
KEYWORD: MB7 5-1
KEYWORD: MB7 Parent
GUIDED PRACTICE
1. Vocabulary The highest or lowest point on the graph of a quadratic function is the
? . (vertex or parabola)
−−−
SEE EXAMPLE
1
2. f (x) = -2x 2 - 4
p. 315
SEE EXAMPLE
Graph each function by using a table.
2
p. 316
3
p. 317
8. g(x) = 3x 2
11. h(x) = -(5x) 2
SEE EXAMPLE 4
4. h(x) = x 2 + 2x
Using the graph of f(x) = x 2 as a guide, describe the transformations, and then
graph each function.
5. d(x) = (x - 4) 2
SEE EXAMPLE
3. g(x) = -x 2 + 3x - 2
6. g(x) = (x - 3) 2 + 2
( )
1x 2
9. h(x) = _
8
12. g(x) = 4.2x 2
7. h(x) = (x + 1) 2 - 3
10. p(x) = 0.25x 2
2 x2
13. d(x) = -_
3
Use the description to write each quadratic function in vertex form.
14. The parent function f (x) = x 2 is vertically stretched by a factor of 2 and translated
3 units left to create g.
p. 318
15. The parent function f (x) = x 2 is reflected across the x-axis and translated 6 units
down to create h.
SEE EXAMPLE
5
p. 319
16. Physics The safe working load L in pounds for a natural rope can be estimated
by L(r) = 5920r 2, where r is the radius of the rope in inches. For an old rope, the
function L o(r) = 4150r 2 is used to estimate its safe working load. What kind of
transformation describes this change, and what does this transformation mean?
PRACTICE AND PROBLEM SOLVING
Independent Practice
For
See
Exercises Example
17–19
20–25
26–28
29–30
31
1
2
3
4
5
Extra Practice
Skills Practice p. S12
Application Practice p. S36
Graph each function by using a table.
17. f (x) = -x 2 + 4
18. g(x) = x 2 - 2x + 1
19. h(x) = 2x 2 + 4x - 1
Using the graph of f(x) = x 2 as a guide, describe the transformations, and then
graph each function.
20. g(x) = x 2 - 2
21. h(x) = (x + 5) 2
22. j(x) = (x - 1) 2
23. g(x) = (x + 4) 2 - 3
4 x2
26. g(x) = _
7
24. h(x) = (x + 2) 2 + 2
25. j(x) = (x - 4) 2 - 9
1x 2
28. j(x) = _
3
27. h(x) = -20x 2
( )
Use the description to write each quadratic function in vertex form.
29. The parent function f (x) = x 2 is reflected across the x-axis, vertically compressed by
a factor of __12 , and translated 1 unit right to create g.
30. The parent function f (x) = x 2 is vertically stretched by a factor of 2.5 and translated
2 units left and 1 unit up to create h.
31. Consumer Economics The average gas mileage m in miles per gallon for a
compact car is modeled by m(s) = -0.015(s - 47) 2 + 33, where s is the car’s
speed in miles per hour. The average gas mileage for an SUV is modeled by
m u(s) = -0.015(s - 47) 2 + 15. What kind of transformation describes this change,
and what does this transformation mean?
320
Chapter 5 Quadratic Functions
Using f (x) = x 2 as a guide, describe the transformations
for each function.
33. p(x) = -(x - 4) 2
34. g(x) = 8(x + 2) 2
35. h(x) = 4x 2 - 2
1 x2 + 2
36. p(x) = _
4
1x 2
38. h(x) = - _
3
37. g(x) = (3x) 2 + 1
( )
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32. Pets Keille is building a rectangular pen for a pet rabbit. She can buy
wire fencing in a roll of 40 ft or a roll of 80 ft. The graph shows the area
of pens she can build with each type of roll.
a. Describe the function for an 80 ft roll of fencing
as a transformation of the function for a 40 ft roll
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of fencing.
b. Is the largest pen Keille can build with an 80 ft roll
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A. a(x) = 4(x + 8) 2 - 3
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42. Geometry The area A of the circle in the figure can be
represented by A(r) = πr 2, where r is the radius.
a. Write a function B in terms of r that represents the
area of the shaded portion of the figure.
b. Describe B as a transformation of A.
c. What are the reasonable domain and range for each
function? Explain.
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43. Critical Thinking What type of graph would a function of the form
2
f (x) = a(x - h) + k have if a = 0? What type of function would it be?
44. Write About It Describe the graph of f (x) = 999,999(x + 5) 2 + 5 without graphing it.
45. This problem will prepare you for the Multi-Step Test Prep on page 364.
The height h in feet of a baseball on Earth after t seconds can be modeled by
the function h(t) = -16(t - 1.5) 2 + 36, where -16 is a constant in ft/s 2 due to
Earth’s gravity.
a. What if...? The gravity on Mars is only 0.38 times that on Earth. If the same
baseball were thrown on Mars, it would reach a maximum height 59 feet higher
and 2.5 seconds later than on Earth. Describe the transformations that must be
applied to make the function model the height of the baseball on Mars.
b. Write a height function for the baseball thrown on Mars.
5-1 Using Transformations to Graph Quadratic Functions
321
Use the graph for Exercises 46 and 47.
46. Which best describes how the graph of the function
y = -x 2 was transformed to produce the graph shown?
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Translation 2 units right and 2 units up
Translation 2 units right and 2 units down
Translation 2 units left and 2 units up
Translation 2 units left and 2 units down
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47. Which gives the function rule for the parabola shown?
f(x) = (x + 2) 2 - 2
f(x) = (x - 2) 2 - 2
f(x) = -(x - 2) 2 - 2
f(x) = -(x + 2) 2 - 2
48. Which shows the functions below in order from widest to narrowest of their
corresponding graphs?
1 x2
1 x2
m(x) = _
n(x) = 4x 2
p(x) = 6x 2
q(x) = -_
6
2
m, n, p, q
m, q, n, p
q, p, n, m
q, m, n, p
49. Which of the following functions has its vertex below the x-axis?
f(x) = (x - 7) 2
f(x) = -2x 2
f(x) = x 2 - 8
f(x) = -(x + 3) 2
50. Gridded Response What is the y-coordinate of the vertex of the graph of
f(x) = -3(x - 1) 2 + 5?
CHALLENGE AND EXTEND
51. Identify the transformations of the graph of f (x) = -3(x + 3) 2 - 3 that would cause
the graph’s image to have a vertex at (3, 3). Then write the transformed function.
52. Consider the functions f (x) = (2x) 2 - 2 and g(x) = 4x 2 - 2.
a. Describe each function as a transformation of the quadratic parent function.
b. Graph both functions on the coordinate plane.
c. Make a conjecture about the relationship between the two functions.
d. Write the rule for a horizontal compression of the parent function that would
give the same graph as f (x) = 9x 2.
SPIRAL REVIEW
53. Packaging Peanuts are packaged in cylindrical containers. A small container is
7 in. tall and has a radius of 2 in. A large container is 5.5 in. tall and has a radius
twice that of the small container. The price of the large container is three times the
price of the small container. Is this price justified? Explain. (Previous course)
Identify the parent function for g from its function rule. (Lesson 1-9)
54. g(x) = 4x + √3
55. g(x) = 3 √
x+4
Write each function in slope-intercept form. Then graph the function. (Lesson 2-3)
1 y + 4 = -1
56. 2y + 5x = 14
57. x - _
2
322
Chapter 5 Quadratic Functions
5-2
*/
Properties of Quadratic
Functions in Standard Form
A2.3.3
describe, function
and sketch
ofthe
quadratic
functions including
2.3b Analyze,
Graph a quadratic
andgraphs
identify
x- and y-intercepts
and the lines
of
symmetry.
maximum
or minimum value, using various methods and tools . . .
Why learn this?
Quadratic functions can be used to find the
maximum power generated by the engine
of a speedboat. (See Example 4.)
Objectives
Define, identify, and
graph quadratic
functions.
Identify and use
maximums and minimums
of quadratic functions to
solve problems.
Vocabulary
axis of symmetry
standard form
minimum value
maximum value
When you transformed quadratic functions
in the previous lesson, you saw that reflecting
the parent function across the y-axis results in
the same function.
f (x) = x 2
g(x) = (-x) 2 = x 2
. . . which may include a
graphing calculator.
This shows that parabolas are symmetric curves.
The axis of symmetry is the line through the
vertex of a parabola that divides the parabola into
two congruent halves.
Axis of Symmetry
Quadratic Functions
WORDS
ALGEBRA
GRAPH
The axis of symmetry is a The quadratic function
2
vertical line through the f(x) = a(x - h) + k has the
vertex of the function’s
axis of symmetry x = h.
graph.
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EXAMPLE
1
Identifying the Axis of Symmetry
Identify the axis of symmetry for the graph of f (x) = 2(x + 2) 2 - 3.
Rewrite the function to find the value of h.
f (x) = 2⎡⎣x - (-2)⎦⎤2 - 3
Because h = -2, the axis of symmetry is the
vertical line x = -2.
Check Analyze the graph on a graphing
calculator. The parabola is symmetric
about the vertical line x = -2.
1. Identify the axis of symmetry for the graph of
f (x) = (x - 3) 2 + 1.
5- 2 Properties of Quadratic Functions in Standard Form
323
Another useful form of writing quadratic functions is the standard form. The
standard form of a quadratic function is f (x) = ax 2 + bx + c, where a ≠ 0.
The coefficients a, b, and c can show properties of the graph of the function. You
can determine these properties by expanding the vertex form.
f (x) = a(x - h) + k
2
f (x) = a(x 2 - 2xh + h 2) + k
Multiply to expand (x - h) .
f (x) = a(x 2) - a(2hx) + a(h 2) + k
Distribute a.
f (x) = ax 2 + (-2ah)x + (ah 2 + k)
Simplify and group like terms.
a=a
2
-2ah = b
ah 2 + k = c
 a in standard form is the same as in vertex form. It indicates
a = a  whether a reflection and/or vertical stretch or compression
has been applied.

b = -2ah
b
b
 Solving for h gives h = ____
= - ___
. Therefore, the axis of
-2a
2a
 symmetry, x = h, for a quadratic function in standard form is
x = -__
b
.

2a
 Notice that the value of c is the same value given by the vertex
 y-intercept.
c = ah 2 + k  form of f when x = 0: f (0) = a(0 - h) 2 + k = ah 2 + k. So c is the
These properties can be generalized to help you graph quadratic functions.
Properties of a Parabola
For f(x) = ax 2 + bx + c, where a, b, and c are real numbers and a ≠ 0, the
parabola has these properties:
The parabola opens upward if a > 0 and
downward if a < 0.
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_
b
The axis of symmetry is the vertical line x = - .
2a
b
b
.
The vertex is the point - , f 2a
2a
The y-intercept is c.
(
EXAMPLE
2
_
( _))
Graphing Quadratic Functions in Standard Form
A Consider the function f (x) = x 2 - 4x + 6.
a. Determine whether the graph opens upward or downward.
Because a is positive, the parabola opens upward.
b. Find the axis of symmetry.
b.
The axis of symmetry is given by x = - _
2a
(-4)
x = - _ = 2 Substitute -4 for b and 1 for a.
2(1)
The axis of symmetry is the line x = 2.
324
Chapter 5 Quadratic Functions
Ý
c. Find the vertex.
The vertex lies on the axis of symmetry, so the x-coordinate is 2.
The y-coordinate is the value of the function at this x-value, or f (2).
f (2) = (2)2 - 4(2) + 6 = 2
The vertex is (2, 2).
d. Find the y-intercept.
Because c = 6, the y-intercept is 6.
e. Graph the function.
Graph by sketching the axis of symmetry
and then plotting the vertex and the
intercept point, (0, 6). Use the axis of
symmetry to find another point on the
parabola. Notice that (0, 6) is 2 units left
of the axis of symmetry. The point on the
parabola symmetrical to (0, 6) is 2 units
right of the axis at (4, 6).
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B Consider the function f (x) = -4x 2 - 12x - 3.
a. Determine whether the graph opens upward or downward.
Because a is negative, the parabola opens downward.
When a is positive,
the parabola is
happy ( ). When
a is negative, the
parabola is sad ( ).
b. Find the axis of symmetry.
b.
The axis of symmetry is given by x = -_
2a
(-12)
3 Substitute -12 for b and -4 for a.
x = -_ = -_
2
2(-4)
3 , or x = -1.5.
The axis of symmetry is the line x = -_
2
c. Find the vertex.
The vertex lies on the axis of symmetry, so the x-coordinate is -1.5.
The y-coordinate is the value of the function at this x-value,
or f (-1.5).
f (-1.5) = -4(-1.5)2 - 12(-1.5) - 3 = 6
The vertex is (-1.5, 6).
d. Find the y-intercept.
Because c = -3, the y-intercept is -3.
e. Graph the function.
Graph by sketching the axis of symmetry
and then plotting the vertex and the
intercept point, (0, -3). Use the axis of
symmetry to find another point on the
parabola. Notice that (0, -3) is 1.5 units
right of the axis of symmetry. The point
on the parabola symmetrical to (0, -3) is
1.5 units left of the axis at (-3, -3).
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For each function, (a) determine whether the graph opens
upward or downward, (b) find the axis of symmetry, (c) find the
vertex, (d) find the y-intercept, and (e) graph the function.
2a. f (x) = -2x 2 - 4x
2b. g(x) = x 2 + 3x - 1
5- 2 Properties of Quadratic Functions in Standard Form
325
Substituting any real value of x into a quadratic equation results in a real
number. Therefore, the domain of any quadratic function is all real numbers, .
The range of a quadratic function depends on its vertex and the direction that
the parabola opens.
Minimum and Maximum Values
OPENS UPWARD
OPENS DOWNWARD
When a parabola opens downward, the
y-value of the vertex is the maximum
value.
When a parabola opens upward, the
y-value of the vertex is the minimum
value.






R: y|y ≥ k







R: y|y ≤ k


D: x|x 
D: x|x 
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The domain is all real numbers, . The
The domain is all real numbers, . The
range is all values greater than or equal range is all values less than or equal to
the maximum.
to the minimum.
EXAMPLE
3
Finding Minimum or Maximum Values
Find the minimum or maximum value of f (x) = 2x 2 - 2x + 5. Then state
the domain and range of the function.
Step 1 Determine whether the function has a minimum or maximum
value. Because a is positive, the graph opens upward and has a
minimum value.
Step 2 Find the x-value of the vertex.
(-2) _
b = -_
1
x = -_
= 2 =_
4 2
2a
2(2)
The minimum (or
maximum) value
is the y-value of
the vertex. It is not
the ordered pair
representing the
vertex.
Substitute -2 for b and 2 for a.
( )
b .
Step 3 Then find the y-value of the vertex, f -_
2a
2
1
f 1 = 2 1 - 2 1 + 5 = 4_
2
2
2
2
(_) (_) (_)
The minimum value is 4__12 , or 4.5. The domain is all real numbers, .


The range is all real numbers greater than or equal to 4.5, or y | y ≥ 4.5.


Check Graph f (x) = 2x 2 - 2x + 5 on a graphing calculator. The graph and
table support the answer.
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Find the minimum or maximum value of each function. Then
state the domain and range of the function.
3a. f (x) = x 2 - 6x + 3
3b. g(x) = -2x 2 - 4
326
Chapter 5 Quadratic Functions
EXAMPLE
4
Transportation Application
Steering wheel
The power p in horsepower (hp) generated
by a high-performance speedboat engine
operating at r revolutions per minute
(rpm) can be modeled by the function
p(r) = -0.0000147r 2 + 0.18r - 251. What
is the maximum power of this engine to
the nearest horsepower? At how many
revolutions per minute must the engine
be operating to achieve this power?
Hull
Engine
Propeller
The maximum value will be at the vertex (r, p(r)).
Step 1 Find the r-value of the vertex using a = -0.0000147 and b = 0.18.
b = -__
0.18
r = -_
≈ 6122
2a
2(-0.0000147)
Step 2 Substitute this r-value into p to find the corresponding
maximum, p(r).
p(r) = -0.0000147r 2 + 0.18r - 251
p(6122) = -0.0000147(6122) 2 + 0.18(6122) - 251 Substitute 6122 for r.
p(6122) ≈ 300
Use a calculator.
The maximum power is about 300 hp at
6122 rpm.
Check Graph the function on a graphing
calculator. Use the maximum feature
under the CALCULATE menu to
approximate the maximum. The
graph supports your answer.
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4. The highway mileage m in miles per gallon for a compact car
is approximated by m(s) = -0.025s 2 + 2.45s - 30, where s is
the speed in miles per hour. What is the maximum mileage for
this compact car to the nearest tenth of a mile per gallon? What
speed results in this mileage?
THINK AND DISCUSS
1. Explain whether a quadratic function can have both a maximum value
and a minimum value.
2. Explain why the value of f (x) = x 2 + 2x - 1 increases as the value of x
decreases from -1 to -10.
3. GET ORGANIZED Copy
and complete the graphic
organizer. In each box,
write the criteria or
equation to find each
property of the parabola
for f (x) = ax 2 + bx + c.
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5- 2 Properties of Quadratic Functions in Standard Form
327
5-2
Exercises
KEYWORD: MB7 5-2
KEYWORD: MB7 Parent
GUIDED PRACTICE
1. Vocabulary If the graph of a quadratic function opens upward, the y-value of the
vertex is a ? value. (maximum or minimum)
−−−
SEE EXAMPLE
1
2. f (x) = -2(x - 2) 2 - 4
p. 323
SEE EXAMPLE
Identify the axis of symmetry for the graph of each function.
2
p. 324
For each function, (a) determine whether the graph opens upward or downward,
(b) find the axis of symmetry, (c) find the vertex, (d) find the y-intercept, and
(e) graph the function.
5. f (x) = -x 2 - 2x - 8
SEE EXAMPLE
3
p. 326
p. 327
6. g(x) = x 2 - 3x + 2
7. h(x) = 4x - x 2 - 1
Find the minimum or maximum value of each function. Then state the domain and
range of the function.
8. f (x) = x 2 - 1
SEE EXAMPLE 4
4. h(x) = (x + 5) 2
3. g(x) = 3x 2 + 4
9. g(x) = -x2 + 3x - 2
10. h(x) = -16x 2 + 32x + 4
11. Sports The path of a soccer ball is modeled by the function h(x) = -0.005x 2 +
0.25x, where h is the height in meters and x is the horizontal distance that the ball
travels in meters. What is the maximum height that the ball reaches?
PRACTICE AND PROBLEM SOLVING
Independent Practice
For
See
Exercises Example
12–14
15–23
24–29
30
1
2
3
4
Extra Practice
Skills Practice p. S12
Application Practice p. S36
Identify the axis of symmetry for the graph of each function.
12. f (x) = -x 2 + 4
14. h(x) = 2(x + 1)2 - 3
13. g(x) = (x - 1) 2
For each function, (a) determine whether the graph opens upward or downward,
(b) find the axis of symmetry, (c) find the vertex, (d) find the y-intercept, and
(e) graph the function.
15. f (x) = x 2 + x - 2
16. g(x) = -3x 2 + 6x
17. h(x) = 0.5x 2 - 2x - 4
18. f (x) = -2x 2 + 8x + 5
19. g(x) = 3x 2 + 2x - 8
21. f (x) = -(2 + x 2)
22. g(x) = 0.5x 2 + 3x - 5
20. h(x) = 2x - 1 + x 2
1 x2 + x + 2
23. h(x) = _
4
Find the minimum or maximum value of each function. Then state the domain and
range of the function.
25. g(x) = 6x - x 2
26. h(x) = x 2 - 4x + 3
24. f (x) = -2x 2 + 7x - 3
1 x2 - 4
27. f (x) = -_
28. g(x) = -x 2 - 6x + 1
29. h(x) = x 2 + 8x + 16
2
30. Weather The daily high temperature in Death Valley, California, in 2003 can be
modeled by T (d) = -0.0018d 2 + 0.657d + 50.95, where T is temperature in degrees
Fahrenheit and d is the day of the year. What was the maximum temperature in 2003
to the nearest degree?
31. Sports The height of a golf ball over time
can be represented by a quadratic function.
Graph the data in the table. What is the
maximum height that the ball will reach?
Explain your answer in terms of the axis of
symmetry and vertex of the graph.
328
Chapter 5 Quadratic Functions
Golf Ball Height
Time (s)
0
0.5
1
2
3
Height (ft)
0
28
48
64
48
Biology
Spittlebugs are insects
that feed on the sap of
plants. Young spittlebug
larvae use sap to produce
a layer of bubbles around
themselves. The bubbles
help to keep the larvae
from drying out and
may protect them from
predators.
32. Manufacturing A roll of aluminum with a width of 32 cm is to be bent into rain
gutters by folding up two sides at 90° angles. A rain gutter’s greatest capacity, or
volume, is determined by the gutter’s greatest cross-sectional area, as shown.
a. Write a function C to describe
(32 – 2x ) cm
x cm
the cross-sectional area in terms
of the width of the bend x.
b. Make a table, and graph the
x cm
function.
Cross-sectional
c. Identify the meaningful domain
area
and range of the function.
32 cm
d. Find the value of x that
maximizes the cross-sectional area.
33. Biology The spittlebug is the world’s highest jumping animal relative to its
body length of about 6 mm. The height h of a spittlebug’s jump in millimeters
can be modeled by the function h(t) = -4000t 2 + 3000t, where t is the time in
seconds.
a. What is the maximum height that the spittlebug will reach?
b. What is the ratio of a spittlebug’s maximum jumping height to its body length?
In the best human jumpers, this ratio is about 1.38. Compare the ratio for
spittlebugs with the ratio for the best human jumpers.
c. What if...? Suppose humans had the same ratio of maximum jumping height
to body length as spittlebugs. How high would a person with a height of 1.8 m
be able to jump?
34. Gardening The function A(x) = x(10 - x) describes the area A of a rectangular
flower garden, where x is its width in yards. What is the maximum area of the
garden?
Graphing Calculator Once you have graphed a function, the graphing calculator can
automatically find the minimum or maximum value. From the CALC menu, choose the
minimum or maximum feature.
Use a graphing calculator to find the approximate minimum or maximum value of
each function.
35. f (x) = 5.23x 2 - 4.84x - 1.91
36. g(x) = -12.8x 2 + 8.73x + 11.69
5 x2 + _
9 x+_
1 x2 - _
4x + _
2
21
37. h(x) = _
38. j(x) = -_
5
4
3
12
3
10
39. Critical Thinking Suppose you are given a parabola with two points that have the
same y-value, such as (-7, 11) and (3, 11). Explain how to find the equation for the
axis of symmetry of this parabola, and then determine this equation.
40. Write About It Can a maximum value for a quadratic function be negative? Can a
minimum value for a quadratic function be positive? Explain by using examples.
41. This problem will prepare you for the Multi-Step Test Prep on page 364.
A baseball is thrown with a vertical velocity of 50 ft/s from an initial height of 6 ft.
The height h in feet of the baseball can be modeled by h(t) = -16t 2 + 50t + 6, where
t is the time in seconds since the ball was thrown.
a. Approximately how many seconds does it take the ball to reach its maximum
height?
b. What is the maximum height that the ball reaches?
5- 2 Properties of Quadratic Functions in Standard Form
329
Use the graph for exercises 42 and 43.
42. What is the range of the function graphed?
All real numbers
y ≥ -2
{
y≤2
-2 ≤ y ≤ 2
Þ
Ó
Ý
43. The graph shown represents which quadratic function?
f(x) = x 2 + 2x - 2
{
ä
Ó
Ó
Ó
f(x) = -x 2 + 4x - 2
f(x) = x 2 - 4x - 2
f(x) = -x 2 - 2x + 2
44. Which of the following is NOT true of the graph of the function
f(x) = -x 2 - 6x + 5?
Its vertex is at (-3, 14).
Its maximum value is 14.
Its axis of symmetry is x = 14.
Its y-intercept is 5.
45. Which equation represents the axis of symmetry for f(x) = 2x 2 - 4x + 5?
x = -4
x=5
x=1
x=2
46. Short Response Explain how to find the maximum value or minimum value of a
quadratic function such as f(x) = -x 2 - 8x + 4.
CHALLENGE AND EXTEND
47. Write the equations in standard form for two quadratic functions that have the same
vertex but open in different directions.
48. The graph of a quadratic function passes through the point (-5, 8), and its axis of
symmetry is x = 3.
a. What are the coordinates of another point on the graph of the function? Explain
how you determined your answer.
b. Can you determine whether the graph of the function opens upward or
downward? Explain.
49. Critical Thinking What conclusions can you make about the axis of symmetry
and the vertex of a quadratic function of the form f (x) = ax 2 + c?
50. Critical Thinking Given the quadratic function f and the fact that f (-1) = f (2),
how can you find the axis of symmetry of this function?
SPIRAL REVIEW
Simplify each expression. (Lesson 1-3)
· √180
51. √40
52. 2 √
8 · 4 √
3
(_)
53. √
54 ÷ √
30
54. √
304
For each function, evaluate f (0), f 1 , and f (-2). (Lesson 1-7)
2
1 2
55. f (x) = (x - 3) 2 + 1
56. g(x) = 2 x - _
2
3
57. f (x) = -4(x + 5)
58. g(x) = x - 4x + 8
(
)
Write the equation of each line with the given properties. (Lesson 2-4)
59. a slope of 3 passing through (1, -4)
61. a slope of -2 passing through (3, 5)
330
Chapter 5 Quadratic Functions
60. passing through (-3, 5) and (-1, -7)
62. passing through (4, 6) and (-2, 1)
Factoring Quadratic
Expressions
Previous Courses
Review the methods of factoring quadratic expressions in the examples
below. Recall that the standard form of a quadratic expression is
ax 2 + bx + c.
Examples
Factor each expression.
1 x 2 - 3x - 10
Because a = 1, use a table to find the factors of -10
that have a sum of -3. These factors are 2 and -5.
Factors of -10
Sum
Rewrite the expression as a product of binomial
factors with 2 and -5 as constants.
-2 and 5
-3 ✘
-1 and 10
-9 ✘
x 2 - 3x - 10 = (x + 2)(x - 5)
Check your answer by multiplying.
1 and -10
-9 ✘
2 and -5
-3 ✔
(x + 2)(x - 5) = x 2 - 5x + 2x - 10
= x 2 - 3x - 10 ✔
3 -x 2 + 3x + 4
2 6x 2 - 15x
Find the greatest common factor (GCF)
of the terms.
6x = 2 · 3 · x · x
2
15x = 3 · 5 · x
The GCF is 3x.
Because a is negative, factor out -1.
-x 2 + 3x + 4 = -1(x 2 - 3x - 4)
Use the method from Example 1 to factor
the expression in parentheses.
Factor 3x from both terms.
-(x 2 - 3x - 4) = -(x + 1)(x - 4)
6x 2 - 15x = 3x (2x - 5)
Check your answer by multiplying.
Check your answer by multiplying.
3x(2x - 5) = 3x (2x) - 3x (5)
-(x + 1)(x - 4) = -(x 2 - 3x - 4)
= -x 2 + 3x + 4 ✔
= 6x 2 - 15x ✔
Try This
Factor each expression.
1. 4x 2 + 10x
2. 16x - 2x 2
3. x 2 - 6x + 8
4. x 2 + 4x + 3
5. x 2 - 8x + 15
6. x 2 + 10x - 24
7. x 2 - x - 56
8. x 2 - 6x + 9
9. x 2 + 48x - 100
10. -x 2 + 12x - 32
11. -x 2 + x + 20
12. -x 2 - 14x - 13
13. 4x 2 + 6x
14. x 2 + 14x + 24
15. x 2 - 16
16. 2x 2 - x - 3
17. 3x 2 + 16x + 5
18. 2x 2 - 9x + 7
Connecting Algebra to Previous Courses
331
5-3
Explore Graphs and Factors
You can use graphs and linear factors to find the x-intercepts of a parabola.
Use with Lesson 5-3
KEYWORD: MB7 Lab5
Activity
Graph the lines y = x + 4 and y = x - 2.
1. Press
, and enter X + 4 for Y1 and X - 2 for Y2. Graph the
and choosing
functions in the square window by pressing
5 : ZSquare.
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2. Identify the x-intercept of each line. The x-intercepts are -4 and 2.
Ó]Êä®
3. Find the x-value halfway between the two x-intercepts. This
-4 + 2
x-value is the average of the x-intercepts: _____
= -1.
2
Graph the quadratic function y = (x + 4)(x - 2), which is the product
of the two linear factors graphed above.
4. Press
and enter
(X + 4)(X - 2) for Y3.
Press
.
5. Identify the x-intercepts
of the parabola. The
x-intercepts are -4 and 2.
Notice that they are the
same as those of the two
linear factors.
6. Examine the parabola at
x = -1 (the x-value that
is halfway between the
x-intercepts). The axis of
symmetry and the vertex of
the parabola occur at this
x-value.
Try This
Graph each quadratic function and each of its linear factors. Then
identify the x-intercepts and the axis of symmetry of each parabola.
1. y = (x - 2)(x - 6)
2. y = (x + 3)(x - 1)
3. y = (x - 5)(x + 2)
4. y = (x + 4)(x - 4)
5. y = (x - 5)(x - 5)
6. y = (2x - 1)(2x + 3)
7. Critical Thinking Use a graph to determine whether the quadratic function y = 2x 2 + 5x - 12
is the product of the linear factors 2x - 3 and x + 4.
8. Make a Conjecture Make a conjecture about the linear factors, x-intercepts, and axis of
symmetry of a quadratic function.
332
Chapter 5 Quadratic Functions
5-3
*/
Solving Quadratic Equations
by Graphing and Factoring
IA-3.3
Carry
out
a procedure
to
solve
quadratic
equationscompleting
algebraically
A2.1.1
zeros,
domain
and
range
of a function.
2.3a Find
Solvethe
quadratic
equations
by
graphing,
factoring,
the square
(including
factoring,
completing the square, and applying the quadratic formula).
and quadratic
formula.
Why learn this?
You can use quadratic functions to model
the height of a football, baseball, or soccer
ball. (See Example 3.)
Objectives
Solve quadratic equations
by graphing or factoring.
Determine a quadratic
function from its roots.
Vocabulary
zero of a function
root of an equation
binomial
trinomial
When a soccer ball is kicked into the air, how
long will the ball take to hit the ground? The
height h in feet of the ball after t seconds
can be modeled by the quadratic function
h(t) = -16t 2 + 32t. In this situation, the value
of the function represents the height of the
soccer ball. When the ball hits the ground, the
value of the function is zero.
A zero of a function is a value of the input
x that makes the output f (x) equal zero.
The zeros of a function are the x-intercepts.
vÝ®ÊÊ>ÝÓÊ ÊLÝÊ ÊV
Unlike linear functions, which have no
more than one zero, quadratic functions
can have two zeros, as shown at right.
These zeros are always symmetric about
the axis of symmetry.
EXAMPLE
1
/ iÊÝVÀ`>ÌiÃ
>ÀiÊÌ iÊâiÀÃ°
Finding Zeros by Using a Graph or Table
Find the zeros of f (x) = x 2 + 2x - 3 by using a graph and table.
Method 1 Graph the function f (x) = x 2 + 2x - 3.
The graph opens upward because a > 0. The y-intercept is -3 because
c = -3.
b = -_
2 = -1
The x-coordinate of the
Find the vertex: x = -_
2a
b
2 (1)
vertex is -__
.
2a
Find f (-1): f (x) = x 2 + 2x - 3
Recall that for the
graph of a quadratic
function, any pair
of points with the
same y-value are
symmetric about the
axis of symmetry.
f (-1) = (-1) 2 + 2(-1) - 3
Substitute -1 for x.
f (-1) = -4
The vertex is (-1, -4).
Plot the vertex and the y-intercept. Use
symmetry and a table of values to find
additional points.
x
f(x)
-3
-2
0
-3
-1
0
-4
-3
{
Þ
1
0
Ý
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ä
£]Êä®
{
The table and the graph indicate that the
zeros are -3 and 1.
5- 3 Solving Quadratic Equations by Graphing and Factoring
333
Find the zeros of f (x) = x 2 + 2x - 3 by using a graph and table.
Method 2 Use a calculator.
Enter y = x 2 + 2x - 3 into a graphing calculator.
Both the table and the graph show that y = 0 at x = -3 and x = 1.
These are the zeros of the function.
1. Find the zeros of g(x) = -x 2 - 2x + 3 by using a graph and
a table.
• Functions have zeros
or x-intercepts.
• Equations have
solutions or roots.
You can also find zeros by using algebra. For example, to find the zeros of
f (x) = x 2 + 2x - 3, you can set the function equal to zero. The solutions to
the related equation x 2 + 2x - 3 = 0 represent the zeros of the function.
The solutions to a quadratic equation of the form ax 2 + bx + c = 0 are roots. The
roots of an equation are the values of the variable that make the equation true.
You can find the roots of some quadratic equations by factoring and applying the
Zero Product Property.
Zero Product Property
For all real numbers a and b,
WORDS
If the product of two quantities
equals zero, at least one of the
quantities equals zero.
EXAMPLE
2
NUMBERS
ALGEBRA
3(0 ) = 0
If ab = 0, then a = 0
or b = 0.
0 (4) = 0
Finding Zeros by Factoring
Find the zeros of each function by factoring.
A f (x) = x 2 - 8x + 12
x 2 - 8x + 12 = 0
(x - 2 )(x - 6 ) = 0
x - 2 = 0 or x - 6 = 0
x = 2 or x = 6
Set the function equal to 0.
Factor: Find factors of 12 that add to -8.
Apply the Zero Product Property.
Solve each equation.
Check
x 2 - 8x + 12 = 0
−−−−−−−−−−−−−
(2) 2 - 8(2) + 12 0
4 - 16 + 12 0
0 0✔
334
Chapter 5 Quadratic Functions
2
x
- 8x + 12 = 0
−−−−−−−−−−−
−−−−−−−−−−−
2
(6) - 8(6) + 12 0
36 - 48 + 12 0
0 0✔
Substitute each
value into
the original
equation.
Find the zeros of each function by factoring.
B g(x) = 3x 2 + 12x
3x 2 + 12x = 0
Set the function equal to 0.
3x(x + 4) = 0
Factor: The GCF is 3x.
3x = 0 or x + 4 = 0
x = 0 or x = -4
Apply the Zero Product Property.
Solve each equation.
n
Check Check algebraically and by graphing.
3x 2 + 12x = 0
3x 2 + 12x = 0
−−−−−−−−−−−−
−−−−−−−−−−−−−−
2
2
3(0) + 12(0) 0
3(-4) + 12(-4) 0
0+0 0✔
48 - 48 0 ✔
£x°Ó
£x°Ó
£Ó
Find the zeros of each function by factoring.
2a. f (x) = x 2 - 5x - 6
2b. g(x) = x 2 - 8x
Any object that is thrown or launched into the air, such as a baseball, basketball,
or soccer ball, is a projectile. The general function that approximates the height h
in feet of a projectile on Earth after t seconds is given below.
Constant due to Earth's
gravity in ft/s2
Initial height
in ft (at t 0)
Initial vertical velocity
in ft/s (at t 0)
Note that this model has limitations because it does not account for air
resistance, wind, and other real-world factors.
EXAMPLE
3
Sports Application
A soccer ball is kicked from ground level with an initial vertical velocity of
32 ft/s. After how many seconds will the ball hit the ground?
h(t) = -16t 2 + v 0t + h 0
Write the general projectile function.
h(t) = -16t + 32t + 0
Substitute 32 for v 0 and 0 for h 0.
2
The ball will hit the ground when its height is zero.
-16t 2 + 32t = 0
Set h(t) equal to 0.
-16t(t - 2) = 0
Factor: The GCF is -16t.
-16t = 0 or (t - 2) = 0
Apply the Zero Product Property.
t = 0 or t = 2
Solve each equation.
The ball will hit the ground in 2 seconds.
Notice that the height is also zero when
t = 0, the instant that the ball is kicked.
Check The graph of the function
h(t) = -16t 2 + 32t shows its
zeros at 0 and 2.
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£
{
x
5- 3 Solving Quadratic Equations by Graphing and Factoring
335
3. A football is kicked from ground level with an initial vertical
velocity of 48 ft/s. How long is the ball in the air?
Quadratic expressions can have one, two, or three terms, such as -16t 2,
-16t 2 + 25t, or -16t 2 + 25t + 6. Quadratic expressions with two terms are
binomials . Quadratic expressions with three terms are trinomials . Some
quadratic expressions with perfect squares have special factoring rules.
Special Products and Factors
Perfect-Square Trinomial
Difference of Two Squares
2
a 2 - 2ab + b 2 = (a - b)
a 2 - b 2 = (a + b)(a - b)
EXAMPLE
4
a 2 + 2ab + b 2 = (a + b)
2
Finding Roots by Using Special Factors
Find the roots of each equation by factoring.
A 9x 2 = 1
9x 2 - 1 = 0
Rewrite in standard form.
(3x ) - ( 1 ) = 0
2
2
Write the left side as a 2 - b 2.
(3x + 1)(3x - 1) = 0
Factor the difference of squares.
3x + 1 = 0 or 3x - 1 = 0
Apply the Zero Product Property.
1 or x = _
1
x = -_
3
3
Solve each equation.
Check Graph each side of the equation
on a graphing calculator. Let Y1
equal 9x 2, and let Y2 equal 1.
The graphs appear to intersect
at x = -__13 and at x = __13 .
3
-4.5
B 40x = 8x 2 + 50
-3
8x 2 - 40x + 50 = 0
2(4x 2 - 20x + 25) = 0
A quadratic equation
can have two roots
that are equal, such
as x = __52 and x = __52 .
Two equal roots are
sometimes called a
double root.
4x - 20x + 25 = 0
2
(2x) - 2(2x)(5) + (5) = 0
2
2
(2x - 5) 2 = 0
Rewrite in standard form.
Factor. The GCF is 2.
Divide both sides by 2.
Write the left side as a 2 - 2ab + b 2.
Factor the perfect-square trinomial: (a - b) .
2x - 5 = 0 or 2x - 5 = 0
Apply the Zero Product Property.
5
5
_
_
x = or x =
Solve each equation.
2
2
5 into the original equation.
Check Substitute the root _
2
40x = 8x 2 + 50
−−−−−−−−−−−
() ()
5 8 _
5 2 + 50
40 _
2
2
100 100 ✔
Find the roots of each equation by factoring.
4a. x 2 - 4x = -4
4b. 25x 2 = 9
336
4.5
Chapter 5 Quadratic Functions
2
If you know the zeros of a function, you can work backward to write a rule
for the function.
EXAMPLE
5
Using Zeros to Write Function Rules
Write a quadratic function in standard form with zeros 2 and -1.
x = 2 or x = -1
Write the zeros as solutions for two equations.
x - 2 = 0 or x + 1 = 0
Rewrite each equation so that it equals 0.
(x - 2)(x + 1) = 0
Apply the converse of the Zero Product
Property to write a product that equals 0.
x2 - x - 2 = 0
Multiply the binomials.
f (x) = x - x - 2
2
Replace 0 with f(x).
x
Check Graph the function
f (x) = x 2 - x - 2 on a
calculator. The graph shows
the original zeros of 2 and -1.
Ç°È
Ç°È
x
5. Write a quadratic function in standard form with
zeros 5 and -5.
Note that there are many quadratic functions
with the same zeros. For example, the functions
f (x) = x 2 - x - 2, g(x) = -x 2 + x + 2, and
h(x) = 2x 2 - 2x - 4 all have zeros at 2 and -1.
x
Ç°È
Ç°È
x
THINK AND DISCUSS
1. Describe the zeros of a function whose terms form a perfect square
trinomial.
2. Compare the x- and y-intercepts of a quadratic function with those of a
linear function.
3. A quadratic equation has no real solutions. Describe the graph of the
related quadratic function.
4. GET ORGANIZED Copy and complete the graphic organizer. In each
box, give information about special products and factors.
>i
,Õi
Ý>«i
À>«
vviÀiViÊvÊ
/ÜÊ-µÕ>ÀiÃ
*iÀviVÌ-µÕ>ÀiÊ
/À>
5- 3 Solving Quadratic Equations by Graphing and Factoring
337
5-3
Exercises
KEYWORD: MB7 5-3
KEYWORD: MB7 Parent
GUIDED PRACTICE
1. Vocabulary The solutions of the equation 3x 2 + 2x + 5 = 0 are its ? .
−−−
(roots or zeros)
SEE EXAMPLE
1
2. f (x) = x 2 + 4x - 5
p. 333
SEE EXAMPLE
2
p. 334
SEE EXAMPLE
3
p. 335
SEE EXAMPLE 4
5
p. 337
3. g(x) = -x 2 + 6x - 8
4. f (x) = x 2 - 1
Find the zeros of each function by factoring.
5. f (x) = x 2 - 7x + 6
6. g(x) = 2x 2 - 5x + 2
8. f (x) = x 2 + 9x + 20
9. g(x) = x 2 - 6x - 16
7. h(x) = x 2 + 4x
10. h(x) = 3x 2 + 13x + 4
11. Archery The height h of an arrow in feet is modeled by h(t) = -16t 2 + 63t + 4,
where t is the time in seconds since the arrow was shot. How long is the arrow
in the air?
Find the roots of each equation by factoring.
12. x 2 - 6x = -9
p. 336
SEE EXAMPLE
Find the zeros of each function by using a graph and table.
13. 5x 2 + 20 = 20x
14. x 2 = 49
Write a quadratic function in standard form for each given set of zeros.
15. 3 and 4
16. -4 and -4
17. 3 and 0
PRACTICE AND PROBLEM SOLVING
Independent Practice
For
See
Exercises Example
18–20
21–26
27
28–33
34–36
1
2
3
4
5
Extra Practice
Skills Practice p. S12
Application Practice p. S36
Find the zeros of each function by using a graph and table.
18. f (x) = -x 2 + 4x - 3
19. g(x) = x 2 + x - 6
20. f (x) = x 2 - 9
Find the zeros of each function by factoring.
21. f (x) = x 2 + 11x + 24
22. g(x) = 2x 2 + x - 10
23. h(x) = -x 2 + 9x
24. f (x) = x 2 - 15x + 54
25. g(x) = x 2 + 7x - 8
26. h(x) = 2x 2 - 12x + 18
27. Biology A bald eagle snatches a fish from a lake and flies to an altitude of 256 ft.
The fish manages to squirm free and falls back down into the lake. Its height h in
feet can be modeled by h(t) = 256 - 16t 2, where t is the time in seconds. How many
seconds will the fish fall before hitting the water?
Find the roots of each equation by factoring.
28. x 2 + 8x = -16
29. 4x 2 = 81
30. 9x 2 + 12x + 4 = 0
31. 36x 2 - 9 = 0
32. x 2 - 10x + 25 = 0
33. 49x 2 = 28x - 4
Write a quadratic function in standard form for each given set of zeros.
34. 5 and -1
35. 6 and 2
36. 3 and 3
Find the zeros of each function.
338
37. f (x) = 6x - x 2
38. g(x) = x 2 - 25
39. h(x) = x 2 - 12x + 36
40. f (x) = 3x 2 - 12
41. g(x) = x 2 - 22x + 121
42. h(x) = 30 + x - x 2
43. f (x) = x 2 - 11x + 30
44. g(x) = x 2 - 8x - 20
45. h(x) = 2x 2 + 18x + 28
Chapter 5 Quadratic Functions
Entertainment
The Guinness world
record for the greatest
number of people
juggling at one time
was set in 1998 by
1508 people, each of
whom juggled at
least 3 objects for
10 seconds.
46. Movies A stuntwoman jumps from a building 73 ft high and lands on an air bag
that is 9 ft tall. Her height above ground h in feet can be modeled by h (t) = 73 - 16t 2,
where t is the time in seconds.
a. Multi-Step How many seconds will the stuntwoman fall before touching the air
bag? (Hint: Find the time t when the stuntwoman’s height above ground is 9 ft.)
b. What if...? Suppose the stuntwoman jumps from a building that is half as tall.
Will she be in the air for half as long? Explain.
47. Entertainment A juggler throws a ball into the air from a height of 5 ft with an
initial vertical velocity of 16 ft/s.
a. Write a function that can be used to model the height h of the ball in feet
t seconds after the ball is thrown.
b. How long does the juggler have to catch the ball before it hits the ground?
Find the roots of each equation.
48. x 2 - 2x + 1 = 0
49. x 2 + 6x = -5
50. 25x 2 + 40x = -16
51. 9x 2 + 6x = -1
52. 5x 2 = 45
53. x 2 - 6 = x
For each function, (a) find its vertex, (b) find its y-intercept, (c) find its zeros, and
(d) graph it.
54. f (x) = x 2 + 2x - 8
55. g(x) = x 2 - 16
56. h(x) = x 2 - x - 12
57. f (x) = -2x 2 + 4x
58. g(x) = x 2 - 5x - 6
59. h(x) = 3x 2 + x - 4
60. Geometry The hypotenuse of a right triangle is 2 cm longer than one leg and 4 cm
longer than the other leg.
a. Let x represent the length of the hypotenuse. Use the Pythagorean Theorem to
write an equation that can be solved for x.
b. Find the solutions of the equation from part a.
c. Are both solutions reasonable in the context of the problem situation? Explain.
Geometry Find the dimensions of each rectangle.
61.
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ÝÊ Ê£È
Ý
62.
63.
ÊÊÓ£äÊVÓ Ý
ÝÊ Ê£
ÊÊxäÊÓ
ÝÊÊÎ
ÝÊ ÊÓ
64. Critical Thinking Will a function whose rule can be factored as a binomial
squared ever have two different zeros? Explain.
65. Write About It Explain how the Zero Product Property can be used to help
determine the zeros of quadratic functions.
66. This problem will prepare you for the Multi-Step Test Prep on page 364.
A baseball player hits a ball toward the outfield. The height h of the ball in feet is
modeled by h(t) = -16t 2 + 22t + 3, where t is the time in seconds. In addition, the
function d(t) = 85t models the horizontal distance d traveled by the ball.
a. If no one catches the ball, how long will it stay in the air?
b. What is the horizontal distance that the ball travels before it hits the ground?
5- 3 Solving Quadratic Equations by Graphing and Factoring
339
68. Which function has -7 as its only zero?
f(x) = x(x - 7)
h(x) = (x - 7) 2
>Êi} Ì
i} ÌÊvÌ®
67. Use the graph provided to choose the best
description of what the graph represents.
A ball is dropped from a height of 42 feet and
lands on the ground after 3 seconds.
A ball is dropped from a height of 42 feet and
lands on the ground after 1.5 seconds.
A ball is shot up in the air and reaches a height
of 42 feet after 1 second.
A ball is shot up in the air, reaches a height
of 42 feet, and lands on the ground after
1.5 seconds.
{ä
Îä
Óä
£ä
ä°x
£
£°x
/iÊÃ®
g(x) = (x + 1)(x + 7)
j(x) = (x + 7) 2
69. Which expression is a perfect square trinomial?
25y 2 - 40y + 16
25y 2 - 16
25y 2 - 20y + 16
25y 2 - 10y + 16
70. Gridded Response Find the positive root of x 2 + 4x - 21 = 0.
CHALLENGE AND EXTEND
Find the roots of each equation by factoring.
71. 3(x 2 - x) = x 2
1x
72. x 2 = _
3
3x + _
1 =0
73. x 2 - _
4
8
74. x 2 + x + 0.21 = 0
75. Another special factoring case involves perfect cubes. The sum of two cubes
can be factored by using the formula a 3 + b 3 = (a + b)(a 2 - ab + b 2).
a. Verify the formula by multiplying the right side of the equation.
b. Factor the expression 8x 3 + 27.
c. Use multiplication and guess and check to find the factors of a 3 - b 3.
d. Factor the expression x 3 - 1.
SPIRAL REVIEW
Evaluate each expression. Write the answer in scientific notation. (Lesson 1-5)
76.
78.
(1.4 × 10 8)(6.1 × 10 -3)
(3.5 × 10 6)
__
(1.4 × 10 -4)
77.
79.
Solve each proportion. (Lesson 2-2)
n
w
1.2 = _
12 = _
80. _
81. _
7.5
5
4.8 8.8
(2.7 × 10 10)(3.2 × 10 2)
(3.12 × 10 -6)
__
(4.8 × 10 3)
6.8 = _
r
82. _
4.5 90
Using the graph of f (x) = x 2 as a guide, describe the transformations, and then
graph each function. (Lesson 5-1)
83. h(x) = 0.5x 2
340
Chapter 5 Quadratic Functions
84. d(x) = x 2 + 2
85. g(x) = (x + 1) 2
Ó
5-4
*/
Completing the
Square
IA-3.3
Carry out
a procedure
to be
solve
quadratic
equations
algebraically
A2.3.5
problems
that
can
modeled
using
quadratic
equations
and
functions,
2.3a Solve
quadratic
equations
by
graphing,
factoring,
completing
the
square
(including
factoring,
completing
the square,
andthe
applying
theare
quadratic
formula).
interpret
the
solutions,
and determine
whether
solutions
reasonable.
and quadratic
formula.
Objectives
Solve quadratic equations
by completing the square.
Why learn this?
You can solve quadratic equations to find how
long water takes to fall from the top to the
bottom of a waterfall. (See Exercise 39.)
Write quadratic equations
in vertex form.
Vocabulary
completing the square
Many quadratic equations contain expressions
that cannot be easily factored. For equations
containing these types of expressions, you can
use square roots to find roots.
Square-Root Property
WORDS
NUMBERS
x = 15
2
To solve a quadratic equation,
you can take the square root of
both sides. Be sure to consider the
positive and negative square roots.
EXAMPLE
1
⎪x⎥ = √
15
15
x = ± √
ALGEBRA
If x = a and a is a
nonnegative real
number, then
x = ± √
a.
2
Solving Equations by Using the Square Root Property
Solve each equation.
A 3x 2 - 4 = 68
3x 2 = 72
Add 4 to both sides.
x 2 = 24
Divide both sides by 3 to isolate the squared term.
x = ± √
24
Take the
square root of both sides.
x = ±2 √
6
Read ± √
a as “plus
or minus square root
of a.”
Simplify.
Check Use a graphing calculator.
B x 2 - 10x + 25 = 27
(x - 5) 2 = 27
x - 5 = ± √
27
Factor the perfect square trinomial.
Take the square root of both sides.
x = 5 ± √
27
Add 5 to
x = 5 ± 3 √
3
Simplify.
both sides.
Check Use a graphing calculator.
Solve each equation.
1a. 4x 2 - 20 = 5
1b. x 2 + 8x + 16 = 49
5- 4 Completing the Square
341
The methods in the previous examples can be used only for expressions that
are perfect squares. However, you can use algebra to rewrite any quadratic
expression as a perfect square.
You can use algebra tiles to model a perfect square
trinomial as a perfect square. The area of the square
at right is x 2 + 2x + 1. Because each side of the square
measures x + 1 units, the area is also (x + 1)(x + 1),
or (x + 1) 2. This shows that (x + 1) 2 = x 2 + 2x + 1.
If a quadratic expression of the form x 2 + bx cannot model a square, you can add
a term to form a perfect square trinomial. This is called completing the square.
Completing the Square
WORDS
NUMBERS
ALGEBRA
x + 6x +
x + bx +
2
To complete the square of
2
b .
x 2 + bx, add _
2
()
x 2 + 6x +
2
(_26)
2
x 2 + bx +
(x + _b2 )
x 2 + 6x + 9
(x + 3) 2
Ý ÓÊ ÊÈÝ
The model shows completing
the square for x 2 + 6x by
adding 9 unit tiles. The
resulting perfect square
trinomial is x 2 + 6x + 9.
Note that completing the
square does not produce an
equivalent expression.
EXAMPLE
2
2
(_b2 )
2
Ý ÓÊ ÊÈÝÊ Ê
LÊÊÈ
Ú
Ó
Ú
Ó
L
Ê Ê ÊÊÊÊ Ê ÊÊÊÊ ÊÈÊÊÊÊ Ê ÊÊ
Ó
Ó
Completing the Square
Complete the square for each expression. Write the resulting expression as
a binomial squared.
A x 2 - 2x +
( )
-2
_
2
2
(2)
B
2
b .
= (-1) 2 = 1 Find _
x 2 - 2x + 1
Add.
(x - 1) 2
Factor.
Check Find the square of the
binomial.
x 2 + 5x +
25
5 2=_
_
4
2
()
()
25
x 2 + 5x + _
4
2
5
x+_
2
(
)
Add.
Factor.
Check Find the square of the
binomial.
(x + _52 ) = (x + _52 )(x + _52 )
2
(x - 1) 2 = (x - 1)(x - 1)
25
= x 2 + 5x + _
4
= x 2 - 2x + 1
Complete the square for each expression. Write the resulting
expression as a binomial squared.
2a. x 2 + 4x +
342
Chapter 5 Quadratic Functions
2
b .
Find _
2
2b. x 2 - 4x +
2c. x 2 + 3x +
You can complete the square to solve quadratic equations.
Solving Quadratic Equations ax 2 + bx + c = 0 by Completing the Square
1. Collect variable terms on one side of the equation and constants on the other.
2. As needed, divide both sides by a to make the coefficient of the x 2-term 1.
()
b
3. Complete the square by adding __
2
2
to both sides of the equation.
4. Factor the variable expression as a perfect square.
5. Take the square root of both sides of the equation.
6. Solve for the values of the variable.
EXAMPLE
3
Solving a Quadratic Equation by Completing the Square
Solve each equation by completing the square.
A x 2 = 27 - 6x
x 2 + 6x = 27
x 2 + 6x +
To keep the equation
balanced, you
b 2
to
must add __
2
both sides of the
equation.
()
x 2 + 6x +
Collect variable terms on one side.
= 27 +
Set up to complete the square.
(_26 ) = 27 + (_26 )
2
2
x 2 + 6x + 9 = 27 + 9
()
b
Add _
2
2
to both sides.
Simplify.
2
(x + 3) = 36
Factor.
x + 3 = ± √
36
Take the square root of both sides.
x + 3 = ±6
Simplify.
x + 3 = 6 or x + 3 = -6
Solve for x.
x = 3 or x = -9
B 2x 2 + 8x = 12
x 2 + 4x = 6
2
x + 4x +
=6+
(_)
x 2 + 4x + 4
2
Divide both sides by 2.
2
Set up to complete the square.
(_)
=6+ 4
2
2
x 2 + 4x + 4 = 6 + 4
2
(x + 2) = 10
x + 2 = ± √
10
x = -2 ± √
10
()
b
Add _
2
2
to both sides.
Simplify.
Factor.
Take the square root of both sides.
Solve for x.
Solve each equation by completing the square.
3a. x 2 - 2 = 9x
3b. 3x 2 - 24x = 27
Recall the vertex form of a quadratic function from Lesson 5-1:
2
f (x) = a(x - h) + k, where the vertex is (h, k).
You can complete the square to rewrite any quadratic function
in vertex form.
5- 4 Completing the Square
343
EXAMPLE
4
Writing a Quadratic Function in Vertex Form
Write each function in vertex form, and identify its vertex.
A f (x) = x 2 + 10x - 13
f (x) = (x 2 + 10x +
In Example 3, the
equation was
balanced by adding
_b_ 2 to both sides.
2
(2)
and subtracting _b_
on one side.
2
Set up to complete the square.
10
(_)  - 13 - (_
2 )

10
f (x) =  x 2 + 10x +
2

()
Here, the equation is
balanced by adding
) - 13 2
2
f (x) = (x + 5) 2 - 38
()
b 2
Add and subtract __
.
2
Simplify and factor.
Because h = -5 and k = -38, the vertex is (-5, -38).
Check Use the axis of symmetry formula to confirm the vertex.
b = -_
10 = -5
x = -_
y = f (-5) = (-5) 2 + 10(-5) - 13 = -38 ✔
2a
2(1)
B g(x) = 2x 2 - 8x + 3
g(x) = 2(x 2 - 4x) + 3
g(x) = 2(x 2 - 4x +
(
Factor so the coefficient of x 2 is 1.
)+3-
Set up to complete the square.
-4
(_) ) + 3 - 2(_
2 )
g(x) = 2 x 2 - 4x + -4
2
2
2
()
()
b 2
b
Add __
. Because __
2
2
2
is
multiplied by 2, you must
()
b 2
subtract 2 __
.
2
g(x) = 2(x 2 - 4x + 4) - 5
Simplify.
g(x) = 2(x - 2) 2 - 5
Factor.
Because h = 2 and k = -5, the vertex is (2, -5).
Check A graph of the function on
a graphing calculator
supports your answer.
Write each function in vertex form, and identify its vertex.
4a. f (x) = x 2 + 24x + 145
4b. g(x) = 5x 2 - 50x + 128
THINK AND DISCUSS
1. Explain two ways to solve x 2 = 25.
2. Describe how to change a quadratic function from standard form to
vertex form by completing the square.
3. GET ORGANIZED Copy and
complete the graphic organizer.
Compare and contrast two
methods of solving quadratic
equations.
344
Chapter 5 Quadratic Functions
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5-4
Exercises
KEYWORD: MB7 5-4
KEYWORD: MB7 Parent
GUIDED PRACTICE
1. Vocabulary What must you add to the expression x 2 + bx to complete the square?
SEE EXAMPLE
1
2. (x - 2) 2 = 16
p. 341
SEE EXAMPLE
Solve each equation.
2
p. 342
3. x 2 - 10x + 25 = 16
Complete the square for each expression. Write the resulting expression as a
binomial squared.
5. x 2 + 14x +
SEE EXAMPLE
3
p. 343
6. x 2 - 12x +
p. 344
7. x 2 - 9x +
Solve each equation by completing the square.
8. x 2 - 6x = -4
9. x 2 + 8 = 6x
11. x 2 = 24 - 4x
SEE EXAMPLE 4
4. x 2 - 2x + 1 = 3
10. 2x 2 - 20x = 8
12. 10x + x 2 = 42
13. 2x 2 + 8x - 15 = 0
Write each function in vertex form, and identify its vertex.
14. f (x) = x 2 + 6x - 3
15. g(x) = x 2 - 10x + 11
16. h(x) = 3x 2 - 24x + 53
17. f (x) = x 2 + 8x - 10
18. g(x) = x 2 - 3x + 16
19. h(x) = 3x 2 - 12x - 4
PRACTICE AND PROBLEM SOLVING
Independent Practice
For
See
Exercises Example
20–22
23–25
26–31
32–37
1
2
3
4
Extra Practice
Skills Practice p. S12
Application Practice p. S36
Solve each equation.
20. (x + 2) 2 = 36
22. (x - 3) 2 = 5
21. x 2 - 6x + 9 = 100
Complete the square for each expression. Write the resulting expression as a
binomial squared.
1x +
23. x 2 - 18x +
24. x 2 + 10x +
25. x 2 - _
2
Solve each equation by completing the square.
26. x 2 + 2x = 7
27. x 2 - 4x = -1
28. 2x 2 - 8x = 22
29. 8x = x 2 + 12
30. x 2 + 3x - 5 = 0
31. 3x 2 + 6x = 1
Write each function in vertex form, and identify its vertex.
32. f (x) = x 2 - 4x + 13
33. g(x) = x 2 + 14x + 71
34. h(x) = 9x 2 + 18x - 3
35. f (x) = x 2 + 4x - 7
36. g(x) = x 2 - 16x + 2
37. h(x) = 2x 2 + 6x + 25
38. Engineering The height h above the roadway of the main cable of the Golden Gate
7
1
x 2 - __
x + 500, where x is the
Bridge can be modeled by the function h(x) = ____
15
9000
distance in feet from the left tower.
x
h
a. Complete the square, and write the function in vertex form.
b. What is the vertex, and what does it represent?
c. Multi-Step The left and right towers have the same height. What is the
distance in feet between them?
5- 4 Completing the Square
345
Caracas
Venezuela
Angel
Falls
39. Waterfalls Angel Falls in Venezuela is the tallest waterfall in the world. Water falls
uninterrupted for 2421 feet before entering the river below. The height h above
the river in feet of water going over the edge of the waterfall is modeled by
h(t) = -16t 2 + 2421, where t is the time in seconds after the initial fall.
a. Estimate the time it takes for the water to reach the river.
b. Multi-Step Ribbon Falls in California has a height of 1612 ft. Approximately
how much longer does it take water to reach the bottom when going over Angel
Falls than when going over Ribbon Falls?
40. Sports A basketball is shot with an initial vertical velocity of 24 ft/s from 6 ft above
the ground. The ball’s height h in feet is modeled by h(t) = -16t 2 + 24t + 6, where t
is the time in seconds after the ball is shot. What is the maximum height of the ball,
and when does the ball reach this height?
Solve each equation using square roots.
41. x 2 - 1 = 2
42. 25x 2 = 0
43. 8x 2 - 200 = 0
44. -3x 2 + 6 = -1
45. (x + 13) 2 = 7
46.
48. x 2 + 14x + 49 = 64
49. 9x 2 + 18x + 9 = 5
25
(x + _32 ) = _
2
2
47.
50.
9 =0
(x + _14 ) - _
16
2
Two attempts to write f (x) = 2x 2 - 8x in vertex form are
shown. Which is incorrect? Explain the error.
/////ERROR ANALYSIS/////
J \ \ ¦\
J \ \ ¦\
J \ \ ¦\ ¦
J \ \¦ ¦
J \ \ \
J \ \ \
J \ \ \ J \ \ Solve each equation by completing the square.
Sports
Acapulco, Mexico, is
famous for its cliff-diving
shows. Divers perform
complicated acrobatic
dives from heights of up
to 80 feet.
346
51. x 2 + 8x = -15
52. x 2 + 22x = -21
53. 3x 2 + 4x = 1
54. 2x 2 = 5x + 12
55. x 2 - 7x - 2 = 0
56. x 2 = 4x + 11
57. x 2 + 6x + 4 = 0
58. 5x 2 + 10x - 7 = 0
59. x 2 - 8x = 24
60. Sports A diver’s height h in meters above the water is approximated by
h(t) = h 0 - 5t 2, where h 0 is the initial height in meters, -5 is a constant based on
the acceleration due to gravity in m/s 2, and t is the time in seconds that the diver
falls through the air.
a. Find the total time that the diver falls through
Dive Heights
the air for each type of dive in the table.
Type
Height (m)
b. How high is a dive that keeps the diver in the
Platform
5
air twice as long as a 5-meter dive?
Platform
10
c. The speed of a diver entering the water can be
approximated by s = 18t, where s is the speed
Cliff
20
in kilometers per hour and t is the time in
Cliff
30
seconds. Using your results from part a, find
the speed of the diver entering the water for
each dive height.
d. How many times as high is a dive that results in a speed that is twice as fast?
Chapter 5 Quadratic Functions
61. This problem will prepare you for the Multi-Step Test Prep on page 364.
The height h in feet of a baseball hit from home plate can be modeled by the
function h(t) = -16t 2 + 32t + 5.5, where t is the time in seconds since the ball was
hit. The ball is descending when it passes 7.5 ft over the head of a 6 ft player
standing on the ground.
a. To the nearest tenth of a second, how long after the ball is hit does it pass over
the player’s head?
b. The horizontal distance between the player and home plate is 120 ft. Use your
answer from part a to determine the horizontal speed of the ball to the nearest
foot per second.
62. Estimation A bag of grass seed will cover 525 square feet. Twenty bags of seed are
used to cover an area shaped like a square. Estimate the side length of the square.
Check your answer with a calculator.
63. Critical Thinking The functions f and g are defined by f (x) = x 2 + 2x - 2 and
g (x) = (x + 1) 2 - 3. Use algebra to prove that f and g represent the same function.
64. Sports A player bumps a volleyball with an initial
vertical velocity of 20 ft/s.
a. Write a function h in standard form for the ball’s
height in feet in terms of the time t in seconds
after the ball is hit.
b. Complete the square to rewrite h in vertex form.
c. What is the maximum height of the ball?
d. What if...? Suppose the volleyball were hit under
the same conditions, but with an initial velocity of
32 ft/s. How much higher would the ball go?
? ft
4 ft
Graphing Calculator Use a graphing calculator to approximate the roots of each
equation to the nearest thousandth.
65. x 2 - 15 = 40
66. x 2 = 2.85
67. 1.4x 2 = 24.6
68. (x + 0.6) 2 = 7.4
x2 = _
1
69. _
7
3
70.
(x + _14 ) = _65
2
71. Critical Thinking Why do equations of the form x 2 = k have no real solution
when k < 0?
72. Write About It Compare the methods of factoring and completing the square for
solving quadratic equations.
73. Which gives the solution to 3x 2 = 33?
± √3
± √11
11
74. Which equation represents the graph
at right?
Þ
È
y = (x - 2) 2 + 1
y = (x - 2) 2 - 1
y = (x + 2) 2 + 1
y = (x + 2) 2 - 1
121
{
Ó
Ý
Ó
ä
Ó
{
5- 4 Completing the Square
347
75. Which gives the vertex of the graph of y = 3(x - 1) 2 - 22?
(1, -22)
(3, -22)
(-1, -22)
(-3, -22)
76. Which number should be added to x 2 + 14x to make a perfect square trinomial?
196
7
14
49
77. Gridded Response What is the positive root of the equation 2x 2 - x = 10?
78. Extended Response Solve the quadratic equation x 2 - 6x = 16 by completing
the square. Explain each step of the solution process, and check your answer.
CHALLENGE AND EXTEND
Find the value of b in each perfect square trinomial.
79. x 2 - bx + 144
80. 4x 2 - bx + 16
81. 3x 2 + bx + 27
82. ax 2 + bx + c
Find the zeros of each function.
84. f (x) = x 2 + 6x √
3 + 23
83. f (x) = x 2 - 4x √
5 + 19
85. Farming To create a temporary grazing area, a farmer is using
1800 feet of electric fencing to enclose a rectangular field and then to
subdivide the field into two plots. The fence that divides the field into
two plots is parallel to the field’s shorter sides.
a. What is the largest area of the field that the farmer can enclose?
b. What are the dimensions of the field with the largest area?
c. What if...? What would be the largest area of a square field that
the farmer could enclose and divide into two plots?
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*ÌÊÓ
SPIRAL REVIEW
Express each set of numbers using set-builder notation. (Lesson 1-1)
86.
(72, ∞)
87. numbers within 10 units of 4
88. positive multiples of 4
89.
Î Ó £
ä
£
Ó
Î
{
x
È
Use the table for Exercises 90–93. (Lesson 4-1)
Monthly Budget
Food
Housing
Auto
Aboline family
$352
$895
$426
Hernandez family
$675
$1368
$642
Walker family
$185
$615
$295
90. Display the data in the form of a matrix B.
91. What are the dimensions of the matrix?
92. What is the address of the entry that has the value 185?
93. What is the value of the matrix entry with the address b 22? What does it represent?
Identify the axis of symmetry and the vertex of the graph of each function.
(Lesson 5-2)
2 x2 - 1
94. f (x) = 3(x - 2) 2
95. g (x) = _
96. h(x) = 6x 2 + 2.5
5
348
Chapter 5 Quadratic Functions
Areas of Composite Figures
Geometry
Quadratic equations can be used to solve problems involving the areas of
composite figures. Write an equation that represents the information given
in the problem. Then solve the equation.
Example
The diagram shows a rectangular garden surrounded by a walkway. The
garden measures 10 m by 34 m. The total area of the garden and walkway
is 640 m 2. What is the width x of the walkway?
The total area is equal to the total length multiplied by the total width.
The total length is 2x + 34 m, and the total width is 2x + 10 m.
A=×w
Ý
£äÊ
Ý
Ý
Î{Ê
Write the formula for total area.
640 = (2x + 34)(2x + 10)
Substitute.
640 = 4x 2 + 88x + 340
Multiply the binomials.
Ý
0 = 4x 2 + 88x - 300
Subtract 640 from both sides.
0 = x 2 + 22x - 75
Divide both sides by 4.
0 = (x - 3)(x + 25)
Factor.
x - 3 = 0 or x + 25 = 0
/Ì>Ê>Ài>ÊÊÈ{äÊÓ
Use the Zero Product Property.
x = 3 or x = -25
Solve for x.
The width cannot be negative. Therefore, the width of the walkway is 3 m.
Try This
Write an equation that represents each problem. Then solve.
1. Use figure 1 below. A ring of grass with an area of 314 yd 2 surrounds a circular flower bed.
Find the width x of the ring of grass.
2. Use figure 2 below. Sid cuts four congruent squares from the corners of a 30-in.-by-50-in.
rectangular piece of cardboard so that it can be folded to make a box. Find the side length
s of the squares, given that the area of the bottom of the box is 200 in 2.
3. Use figure 3 below. Harriet has 80 m of fencing materials to enclose three sides of a rectangular
garden. She will use the side of her garage as a border for the fourth side. Find the width x of the
garden if its area is to be 700 m 2.
Figure 1
Ý
£äÊÞ`
Figure 2
Figure 3
xäÊ°
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Ý
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Connecting Algebra to Geometry
349
5-5
IA-3.3
Carry
out
a that
procedure
solve
quadratic
algebraically
A2.3.2
quadratic
equations
in the
complexequations
number
system.
1.3a Solve
Recognize
to
solvetocertain
problems
and
equations,
number (including
systems
factoring,
the square,
and applying
the quadratic
need to becompleting
extended from
real numbers
to complex
numbers.formula).
Why learn this?
Complex numbers can be used to
describe the zeros of quadratic
functions that have no real
zeros. (See Example 4.)
Objectives
Define and use imaginary
and complex numbers.
Solve quadratic equations
with complex roots.
Vocabulary
imaginary unit
imaginary number
complex number
real part
imaginary part
complex conjugate
Andrew Toos/CartoonResource.com
*/
Complex Numbers
and Roots
You can see in the graph of
f (x) = x 2 + 1 below that f has no real
zeros. If you solve the corresponding
equation 0 = x 2 + 1, you find that
x = ± √
-1 , which has no real solutions.
However, you can find solutions if you define
the square root of negative numbers, which is
why imaginary numbers were invented. The
imaginary unit i is defined as √
-1 . You can use
the imaginary unit to write the square root of any
negative number.
Þ
vÝ®ÊÊÝÓÊ Ê£
Ý
ÊÝÌiÀVi«ÌÃ
Imaginary Numbers
WORDS
An imaginary number is the square
root of a negative number.
Imaginary numbers can be written
in the form bi, where b is a real
number and i is the imaginary unit.
NUMBERS
√
-1 = i
If b is a positive
real number,
= i √
√-2
= √
-1 √
2 = i √
2 then √-b
b
√
-4 = √
-1 √
4 = 2i
The square of an imaginary number
is the original negative number.
EXAMPLE
1
ALGEBRA
and √
-b 2 = bi.
2
( √
-1 ) = i 2 = -1
2
( √
-b ) = -b
Simplifying Square Roots of Negative Numbers
Express each number in terms of i.
A 3 √
-16
B - √
-75
(16)(-1) Factor out -1.
3 √
(75)(-1)
- √
Factor out -1.
3 √
16 √
-1 Product Property
- √
75 √
-1
Product Property
3 · 4 √
-1
Simplify.
- √
25 √
3 √
-1
Product Property
12 √
-1
Multiply.
-5 √
3 √
-1
Simplify.
12i
Express in terms
of i.
-5 √
3 i = -5i √
3 Express in terms
Express each number in terms of i.
1a. √
-12
1b. 2 √
-36
350
Chapter 5 Quadratic Functions
of i.
1 √
1c. -_
-63
3
EXAMPLE
2
Solving a Quadratic Equation with Imaginary Solutions
Solve each equation.
A x 2 = -81
B 3x 2 + 75 = 0
-81 Take square
x = ± √
x = ±9i
3x 2 = -75
roots.
Express in
terms of i.
x 2 = -25 Divide both sides by 3.
x = ± √
-25 Take square roots.
Check
x = ±5i Express in terms of i.
Check −−−−−−−−−−−−
3x 2 + 75 = 0
2
3(±5i) + 75 0
3(25)i 2 + 75 0
75(-1) + 75 0 ✔
x 2 = -81
x 2 = -81
−−−−−−−−−
−−−−−−−−−
2
(9i) -81
(-9i) 2 -81
81i 2 -81
81i 2 -81
81(-1) -81 ✔ 81(-1) -81 ✔
Solve each equation.
2a. x 2 = -36
2b. x 2 + 48 = 0
A complex number is a number
that can be written in the form
a + bi, where a and b are real
numbers and i = √
-1 . The set of
real numbers is a subset of the set
of complex numbers .
2c. 9x 2 + 25 = 0
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£
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е
°ÊÈÊÊÊÊÊÊÊȖÓÊ
Ê
Every complex number has a real
part a and an imaginary part b.
Real part
Add -75 to both sides.
>}>ÀÞ
ÕLiÀÃ
ÊÊÊÊÎÊÊÊÊx
е
ÊȖÇÊ
еÊ
Imaginary part
Real numbers are complex numbers where b = 0. Imaginary numbers are
complex numbers where a = 0 and b ≠ 0. These are sometimes called pure
imaginary numbers.
Two complex numbers are equal if and only if their real parts are equal and their
imaginary parts are equal.
EXAMPLE
3
Equating Two Complex Numbers
Find the values of x and y that make the equation 3x - 5i = 6 - (10y)i true.
Real parts
3x - 5i = 6 - (10y)i
Imaginary parts
3x = 6
x=2
Equate the real parts.
Solve for x.
-5 = -(10y) Equate the imaginary parts.
1 =y
_
Solve for y.
2
Find the values of x and y that make each equation true.
3a. 2x - 6i = -8 + (20y)i
3b. -8 + (6y)i = 5x - i √
6
5- 5 Complex Numbers and Roots
351
EXAMPLE
4
Finding Complex Zeros of Quadratic Functions
Find the zeros of each function.
A f (x) = x 2 - 2x + 5
B
x 2 - 2x + 5 = 0
x 2 - 2x +
x 2 + 10x + 35 = 0
Set equal to 0.
= -5 +
x 2 + 10x +
Rewrite.
()
g (x) = x 2 + 10x + 35
2
= -35 +
x - 2x + 1 = -5 + 1
b
Add __
.
x + 10x + 25 = -35 + 25
(x - 1) = -4
Factor.
(x + 5) 2 = -10
2
2
2
x - 1 = ± √
-4
2
x + 5 = ± √
-10
Take square roots.
x = 1 ± 2i
x = -5 ± i √
10
Simplify.
Find the zeros of each function.
4a. f (x) = x 2 + 4x + 13
4b. g(x) = x 2 - 8x + 18
When given one
complex root, you
can always find the
other by finding its
conjugate.
EXAMPLE
The solutions -5 + i √
10 and -5 - i √
10 in Example 4B are related. These
solutions are a complex conjugate pair. Their real parts are equal and their
imaginary parts are opposites. The complex conjugate of any complex number
a + bi is the complex number a - bi.
If a quadratic equation with real coefficients has nonreal roots, those roots are
complex conjugates.
5
Finding Complex Conjugates
Find each complex conjugate.
A 2i - 15
-15 + 2i
-15 - 2i
B
Write as a + bi.
Find a - bi.
Find each complex conjugate.
5a. 9 - i
5b. i + √
3
-4i
0 + (-4)i
0 - (-4)i
4i
Write as a + bi.
Find a - bi.
Simplify.
5c. -8i
THINK AND DISCUSS
1. Given that one solution of a quadratic equation is 3 + i, explain how to
determine the other solution.
2. Describe a number of the form a + bi in which a ≠ 0 and b = 0.
Then describe a number in which a = 0 and b ≠ 0. Are both numbers
complex? Explain.
3. GET ORGANIZED Copy and complete
the graphic organizer. In each box or
oval, give a definition and examples of
each type of number.
352
Chapter 5 Quadratic Functions
«iÝÊ ÕLiÀÃ
,i>Ê
ÕLiÀÃ
>}>ÀÞÊ
ÕLiÀÃ
5-5
Exercises
KEYWORD: MB7 5-5
KEYWORD: MB7 Parent
GUIDED PRACTICE
1. Vocabulary The number 7 is the ? part of the complex number √
5 + 7i. (real or
̶̶̶
imaginary)
SEE EXAMPLE
1
p. 350
SEE EXAMPLE
2
p. 351
SEE EXAMPLE
6. x 2 = -9
3
p. 351
7. 2x 2 + 72 = 0
p. 352
p. 352
5. √
-144
8. 4x 2 = -16
9. x 2 + 121 = 0
11. -4 + (y)i = -12x - i + 8
Find the zeros of each function.
12. f (x) = x 2 - 12x + 45
5
4. - √
-32
Find the values of x and y that make each equation true.
10. -2x + 6i = (-24y)i - 14
SEE EXAMPLE 4
SEE EXAMPLE
Express each number in terms of i.
1 √
2. 5 √
-100
3. _
-16
2
Solve each equation.
13. g(x) = x 2 + 6x + 34
Find each complex conjugate.
14. -9i
15. √
5 + 5i
16. 8i - 3
17. 6 + i √
2
PRACTICE AND PROBLEM SOLVING
Independent Practice
For
See
Exercises Example
18–21
22–25
26–27
28–31
32–35
1
2
3
4
5
Extra Practice
Skills Practice p. S13
Application Practice p. S36
Express each number in terms of i.
1 √-90

18. 8 √
-4
19. -_
3
20. 6 √
-12
21. √
-50
24. 3x 2 + 27 = 0
1 x 2 = -32
25. _
2
Solve each equation.
22. x 2 + 49 = 0
23. 5x 2 = -80
Find the values of x and y that make each equation true.
26. 9x + (y)i - 5 = -12i + 4
27. 5(x - 1) + (3y)i = -15i - 20
Find the zeros of each function.
28. f (x) = x 2 + 2x + 3
29. g(x) = 4x 2 - 3x + 1
30. f (x) = x 2 + 4x + 8
31. g(x) = 3x 2 - 6x + 10
Find each complex conjugate.
32. i
√
3
33. -_ - 2i
2
34. -2.5i + 1
36. What if...? A carnival game asks participants to strike a
spring with a hammer. The spring shoots a puck upward
toward a bell. If the puck strikes the bell, the participant
wins a prize. Suppose that a participant strikes the spring
and shoots the puck according to the model
d(t) = 16t 2 - 32t + 18, where d is the distance
in feet between the puck and the bell and t is the
time in seconds since the puck was struck. Is it
possible for the participant to win a prize?
Explain your answer.
i -1
35. _
10
18 ft
5- 5 Complex Numbers and Roots
353
Given each solution to a quadratic equation, find the other solution.
5i
37. 1 + 14i
38. _
5
39. 4i - 2 √
7
17i
_
40. -12 - i
41. 9 - i √
2
42. 3
Find the values of c and d that make each equation true.
43. 2ci + 1 = -d + 6 - ci
44. c + 3ci = 4 + di
45. c 2 + 4i = d + di
48. 2x 2 + 12.5 = 0
Solve each equation.
The Granger Collection, New York
Math History
46. 8x 2 = -8
1 x 2 + 72 = 0
49. _
2
52. x 2 - 4x + 8 = 0
1 x 2 = -27
47. _
3
50. x 2 = -30
51. 2x 2 + 16 = 0
53. x 2 + 10x + 29 = 0
54. x 2 - 12x + 44 = 0
55. x 2 + 2x = -5
56. x 2 + 18 = -6x
57. -149 = x 2 - 24x
Tell whether each statement is always, sometimes, or never true. If sometimes true,
give examples to support your answer.
58. A real number is an imaginary number.
59. An imaginary number is a complex number.
The Swiss mathematician
Leonhard Euler (1707–
1783) was the first to
use the notation i to
-1 . He also
represent √
introduced the notation
f(x) to represent the
value of a function f at x.
60. A rational number is a complex number.
61. A complex number is an imaginary number.
62. An integer is a complex number.
63. Quadratic equations have no real solutions.
64. Quadratic equations have roots that are real and complex.
65. Roots of quadratic equations are conjugate pairs.
Find the zeros of each function.
66. f (x) = x 2 - 10x + 26
67. g(x) = x 2 + 2x + 17
68. h(x) = x 2 - 10x + 50
69. f (x) = x 2 + 16x + 73
70. g(x) = x 2 - 10x + 37
71. h(x) = x 2 - 16x + 68
72. Critical Thinking Can you determine the zeros of f (x) = x 2 + 64 by using a graph?
Explain why or why not.
73. Critical Thinking What is the complex conjugate of a real number?
74. Write About It Explain the procedures you can use to solve for nonreal
complex roots.
75. This problem will prepare you for the Multi-Step Test Prep on page 364.
A player throws a ball straight up toward the roof of an indoor baseball stadium.
The height h in feet of the ball after t seconds can be modeled by the function
h(t) = -16t 2 + 112t.
a. The height of the roof is 208 ft. Solve the equation 208 = -16t 2 + 112t.
b. Based on your answer to part a, does the ball hit the roof? Explain
your answer.
c. Based on the function model, what is the maximum height that the ball will
reach?
354
Chapter 5 Quadratic Functions
76. What is the complex conjugate of -2 + i ?
2-i
2+i
i-2
-2 - i
in terms of i.
77. Express √-225
15i
-15i
i √15
-i √15
78. Find the zeros of f(x) = x 2 - 2x + 17.
4±i
1 ± 4i
-1 ± 4i
-4 ± i
79. What value of c makes the equation 3 - 4i - 5 = (9 + ci) - 11 true?
4
-2
-4
2
80. Which of the following equations has roots of -6i and 6i?
1 x2 = 6
-_
6
1 x2 = 9
_
4
x 2 - 30 = 6
20 - x 2 = -16
81. Short Response Explain the types of solutions that equations of the form x 2 = a
have when a < 0 and when a > 0.
CHALLENGE AND EXTEND
82. Find the complex number a + bi such that 5a + 3b = 1 and -5b = 7 + 4a.
83. Can a quadratic equation have only one real number root? only one imaginary root?
only one complex root? Explain.
84. Given the general form of a quadratic equation x 2 + bx + c = 0, determine the effect
of each condition on the solutions.
a. b = 0
b. c ≤ 0
c. c > 0
d. What is needed for the solutions to have imaginary parts?
SPIRAL REVIEW
Use the following matrices for Exercises 85–88. Evaluate, if possible. (Lesson 4-2)
⎡ 1 -5 ⎤
S =⎢
⎣ -2 0 ⎦
85. T 2
86. TV
T=
⎡ 10 1 ⎤
⎡ -4 1 -2 ⎤
V
=
0 -3 1
0 -1
⎣ -5 5 ⎦
⎣ 2 -2 2 ⎦
⎢
⎢
88. S 2
87. ST
For each function, (a) determine whether the graph opens upward or downward,
(b) find the axis of symmetry, (c) find the vertex, (d) find the y-intercept, and
(e) graph the function. (Lesson 5-2)
1 x 2 + x - 10
90. f (x) = -x 2 + 3
89. f (x) = _
5
1 x 2 + 3x + 1
91. f (x) = 2x 2 + 4x - 3
92. f (x) = -_
2
Find the roots of each equation by factoring. (Lesson 5-3)
93. x 2 + 5x = 14
94. 6x 2 = -x + 2
95. 4x 2 + 9 = 15x
96. 4x 2 = 1
97. x 2 + 11x = -24
98. x 2 = -7x
5- 5 Complex Numbers and Roots
355
5-6
*/
The Quadratic Formula
IA-3.3
Carry
a procedure
to solve
quadratic
equations
algebraically
A2.3.2
quadratic
equations
in
complex
number completing
system.
2.3a Solve
Solve out
quadratic
equations
bythe
graphing,
factoring,
the (including
square
factoring,
completing
and quadratic
formula.the square, and applying the quadratic formula).
Objectives
Solve quadratic equations
using the Quadratic
Formula.
Classify roots using the
discriminant.
Vocabulary
discriminant
Who uses this?
Firefighting pilots can use the Quadratic Formula to
estimate when to release water on a fire. (See
Example 4.)
You have learned several methods for solving quadratic
equations: graphing, making tables, factoring, using
square roots, and completing the square. Another
method is to use the Quadratic Formula, which allows
you to solve a quadratic equation in standard form.
By completing the square on the standard form of a
quadratic equation, you can determine the Quadratic Formula.
Numbers
Algebra
ax 2 + bx + c = 0 (a ≠ 0)
3x 2 + 5x + 1 = 0
5x + _
1=0
x2 + _
3
3
5 x = -_
1
x2 + _
3
3
_
( )
5x + 5
x2 + _
3
2(3)
2
Divide by a.
_
( )
1+ 5
= -_
3
2(3)
2
(_)
b
b
Complete x 2 + _
a x + 2a
the square.
5=±
x+_
6
( )
x=-
b - 4ac
_
4a 2
2
Factor.
13
√_
36
(_b )
c
= -_
a+
2
2a
b
b =_
c
-_
(x + _
a
2a )
4a
b =± _
b - 4ac
x+_
√
2a
4a
2
2
2
Take square roots.
√
13
_5 ± _
6
2
2
2
_
c
b
x2 + _
ax = - a
c
Subtract _
a.
25 _
-1
(x + _56 ) = _
3
36
To subtract fractions,
you need a common
denominator.
c
b2 - _
_
a
4a 2
c _
4a
b2 - _
_
a 4a
4a 2
b
c
_
x2 + _
ax + a = 0
6
-5 ± √
13
x= _
6
b.
Subtract _
2a
Simplify.
2
√
b - 4ac
b
_
±_
x=-
2
2a
2a
-b ± √
b 2 - 4ac
x = __
2a
Ó
The symmetry of a quadratic function is evident in the next
√
b 2 - 4ac
b
to last step, x = -___
± ________
. These two zeros are the
2a
2a
√
b 2 - 4ac
________
,
2a
same distance,
away from the axis of symmetry,
b
x = -___
,
with
one
zero
on
either side of the vertex.
2a
£
Ý
Ó
The Quadratic Formula
If ax 2 + bx + c = 0 (a ≠ 0), then the solutions, or roots, are
b 2 - 4ac
-b ± √
x = __.
2a
356
Chapter 5 Quadratic Functions
£
£
You can use the Quadratic Formula to solve any quadratic equation that is written
in standard form, including equations with real solutions or complex solutions.
EXAMPLE
1
Quadratic Functions with Real Zeros
Find the zeros of f (x) = x 2 + 10x + 2 by using the Quadratic Formula.
x 2 + 10x + 2 = 0
Set f(x) = 0.
-b ± √b 2 - 4ac
x = __
2a
Write the Quadratic Formula.
-10 ± √(10 ) 2 - 4(1)(2)
x = ___
2(1)
Substitute 1 for a, 10 for b, and 2 for c.
-10 ± √100 - 8
-10 ± √92
x = __ = __
2
2
Simplify.
-10 ± 2 √23
x = __ = -5 ± √23
Write in simplest form.
2
Check Solve by completing the square.
x 2 + 10x + 2 = 0
x 2 + 10x = -2
x 2 + 10x + 25 = -2 + 25
(x + 5) 2 = 23
x = -5 ± √23 ✔
Find the zeros of each function by using the Quadratic Formula.
1a. f (x) = x 2 + 3x - 7
1b. g (x) = x 2 - 8x + 10
EXAMPLE
2
Quadratic Functions with Complex Zeros
Find the zeros of f (x) = 2x 2 - x + 2 by using the Quadratic Formula.
2x 2 - x + 2 = 0
Set f(x) = 0.
-b ± √b 2 - 4ac
x = __
2a
Write the Quadratic Formula.
-(-1) ± √(-1) 2 - 4(2)(2)
x = ___
2(2)
Substitute 2 for a, -1 for b, and 2 for c.
1 ± √1 - 16
1 ± √-15
x = __ = _
4
4
Simplify.
√15
1 ± i √15
1 ±_
x=_=_
i
4
4
4
Write in terms of i.
2.
Find the zeros of g (x) = 3x 2 - x + 8 by using the Quadratic
Formula.
The discriminant is part of the
Quadratic Formula that you can
use to determine the number of
real roots of a quadratic equation.
-b ± √b - 4ac
x = __
2a
2
Discriminant
5- 6 The Quadratic Formula
357
Discriminant
The discriminant of the quadratic equation ax 2 + bx + c = 0 (a ≠ 0) is b 2 - 4ac.
b 2 - 4ac > 0
b 2 - 4ac = 0
b 2 - 4ac < 0
two distinct real solutions one distinct real solution
EXAMPLE
3
two distinct nonreal
complex solutions
Analyzing Quadratic Equations by Using the Discriminant
Find the type and number of solutions for each equation.
A x 2 - 6x = -7
Make sure the
equation is in
standard form before
you evaluate the
discriminant,
b 2 - 4ac.
B
x 2 - 6x = -9
C
x 2 - 6x = -11
x 2 - 6x + 7 = 0
x 2 - 6x + 9 = 0
x 2 - 6x + 11 = 0
b 2 - 4ac
b 2 - 4ac
b 2 - 4ac
(-6) 2 - 4(1)(7)
(-6) 2 - 4(1)(9)
(-6) 2 - 4(1)(11)
36 - 28 = 8
36 - 36 = 0
36 - 44 = -8
b - 4ac > 0;
the equation
has two distinct
real solutions.
b - 4ac = 0;
the equation
has one distinct
real solution.
b 2 - 4ac < 0; the
equation has two
distinct nonreal
complex solutions.
2
2
Find the type and number of solutions for each equation.
3a. x 2 - 4x = -4
3b. x 2 - 4x = -8
3c. x 2 - 4x = 2
The graph shows the related functions
for Example 3. Notice that the number
of real solutions for the equation can be
changed by changing the value of the
constant c.
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Double-Checking Roots
If I get integer roots when I use the Quadratic Formula, I know that I can quickly
factor to check the roots. Look at my work for the equation x 2 - 7x + 10 = 0.
Quadratic Formula:
Factoring:
2
x=
Terry Cannon,
Carver High School
358
-(-7) ± √(-7) - 4(1)(10)
___
2(1)
7 ± √9
10
=
or 4 = 5 or 2
=
2
2
2
Chapter 5 Quadratic Functions
_ _ _
x 2 - 7x + 10 = 0
(x - 5)(x - 2) = 0
x = 5 or x = 2
EXAMPLE
4
Aviation Application
The pilot of a helicopter plans to
release a bucket of water on a forest
fire. The height y in feet of the water
t seconds after its release is modeled
by y = -16t 2 - 2t + 500. The horizontal
distance x in feet between the water
and its point of release is modeled by
x = 91t. At what horizontal distance
from the fire should the pilot start
releasing the water in order to hit
the target?
Path of
water
Release
point
Target
x ft
Step 1 Use the first equation to determine how long it will take the water
to hit the ground. Set the height of the water equal to 0 feet, and use the
quadratic formula to solve for t.
y = -16t 2 - 2t + 500
0 = -16t 2 - 2t + 500
Set y equal to 0.
-b ± √b 2 - 4ac
t = __
2a
Use the Quadratic Formula.
-(-2) ± √(-2) 2 - 4(-16)(500)
t = ___
2(-16)
Substitute for a, b, and c.
2 ± √32,004
t = __
-32
Simplify.
t ≈ -5.65 or t ≈ 5.53
The time cannot be negative, so the water lands on the target about
5.5 seconds after it is released.
Once you have found
the value of t, you
have solved only part
of the problem. You
will use this value to
find the answer you
are looking for.
Step 2 Find the horizontal distance that the water will have traveled in
this time.
x = 91t
x = 91(5.5)
Substitute 5.5 for t.
x = 500.5
Simplify.
The water will have traveled a horizontal distance of about 500 feet.
Therefore, the pilot should start releasing the water when the horizontal
distance between the helicopter and the fire is 500 feet.
Check Use substitution to check that the water hits the ground after about
5.53 seconds.
y = -16t 2 - 2t + 500
y = -16(5.53) 2 - 2(5.53) + 500
y ≈ -0.3544 ✔ The height is approximately equal to 0 when
t = 5.53.
Use the information given above to answer the following.
4. The pilot’s altitude decreases, which changes the function
describing the water’s height to y = -16t 2 - 2t + 400. To the
nearest foot, at what horizontal distance from the target should
the pilot begin releasing the water?
5- 6 The Quadratic Formula
359
Summary of Solving Quadratic Equations
Method
Graphing
When to Use
Examples
2x 2 + 5x - 14 = 0
Only approximate
solutions or the
number of real
solutions is needed.
x ≈ -4.2 or x ≈ 1.7
Factoring
No matter which
method you use to
solve a quadratic
equation, you should
get the same answer.
Square roots
c = 0 or the
expression is easily
factorable.
x 2 + 4x + 3 = 0
(x + 3)(x + 1) = 0
x = -3 or x = -1
(x - 5)2 = 24
The variable side of
the equation is a
perfect square.
(x - 5) 2 = ± √
24
√
x - 5 = ±2 √
6
x = 5 ± 2 √
6
Completing the
square
a = 1 and b is an
even number.
x 2 + 6x = 10
x 2 + 6x +
= 10 +
()
6
x 2 + 6x + _
2
2
()
6
= 10 + _
2
2
(x + 3) 2 = 19

x = -3 ± √19
Quadratic Formula
Numbers are large or
complicated, and the
expression does not
factor easily.
5x - 7x - 8 = 0
2
(-7) 2 - 4(5)(-8)
-(-7) ± √
x = ___
2(5)
7 ± √
209
x=_
10
THINK AND DISCUSS
1. Describe how the graphs of quadratic functions illustrate the type and
number of zeros.
2. Describe the values of c for which the equation x 2 + 8x + c = 0 will have
zero, one, or two distinct solutions.
3. GET ORGANIZED
Copy and complete
the graphic organizer.
Describe the possible
solution methods for each
value of the discriminant.
360
Chapter 5 Quadratic Functions
��
��
��
��
��
��
��
��
��
5-6
Exercises
KEYWORD: MB7 5-6
KEYWORD: MB7 Parent
GUIDED PRACTICE
1. Vocabulary What information does the value of the discriminant give about a
quadratic equation?
SEE EXAMPLE
1
p. 357
SEE EXAMPLE
2
p. 357
SEE EXAMPLE
3
p. 358
SEE EXAMPLE 4
p. 359
Find the zeros of each function by using the Quadratic Formula.
2. f (x) = x 2 + 7x + 10
3. g(x) = 3x 2 - 4x - 1
4. h(x) = 3x 2 - 5x
5. g(x) = -x 2 - 5x + 6
6. h(x) = 4x 2 - 5x - 6
7. f (x) = 2x 2 - 19
8. f (x) = 2x 2 - 2x + 3
9. r(x) = x 2 + 6x + 12
10. h(x) = 3x 2 + 4x + 3
11. p(x) = x 2 + 4x + 10
12. g(x) = -5x 2 + 7x - 3
13. f (x) = 10x 2 + 7x + 4
Find the type and number of solutions for each equation.
14. 4x 2 + 1 = 4x
15. x 2 + 2x = 10
16. 2x - x 2 = 4
17. Geometry One leg of a right triangle is 6 in. longer than the other leg. The
hypotenuse of the triangle is 25 in. What is the length of each leg to the
nearest inch?
PRACTICE AND PROBLEM SOLVING
Independent Practice
For
See
Exercises Example
18–23
24–29
30–35
36
1
2
3
4
Extra Practice
Skills Practice p. S13
Application Practice p. S36
Find the zeros of each function by using the Quadratic Formula.
18. f (x) = 3x 2 - 10x + 3
19. g(x) = x 2 + 6x
20. h(x) = x(x - 3) - 4
21. g(x) = -x 2 - 2x + 9
22. p(x) = 2x 2 - 7x - 8
23. f (x) = 7x 2 - 3
24. r (x) = x 2 + x + 1
25. h(x) = -x 2 - x - 1
26. f (x) = 2x 2 + 8
27. f (x) = 2x 2 + 7x - 13
28. g(x) = x 2 - x - 5
29. h(x) = -3x 2 + 4x - 4
Find the type and number of solutions for each equation.
30. 2x 2 + 5 = 2x
31. 2x 2 - 3x = 8
32. 2x 2 - 16x = -32
33. 4x 2 - 28x = -49
34. 3x 2 - 8x + 8 = 0
35. 3.2x 2 - 8.5x + 1.3 = 0
36. Safety If a tightrope walker falls, he will
land on a safety net. His height h in feet after
a fall can be modeled by h(t) = 60 - 16t 2,
where t is the time in seconds. How many
seconds will the tightrope walker fall before
landing on the safety net?
60 ft
37. Physics A bicyclist is riding at a speed of
20 mi/h when she starts down a long hill.
The distance d she travels in feet can be
11 ft
modeled by the function d(t) = 5t 2 + 20t,
where t is the time in seconds.
a. The hill is 585 ft long. To the nearest second,
how long will it take her to reach the bottom?
b. What if...? Suppose the hill were only half as long. To the nearest second,
how long would it take the bicyclist to reach the bottom?
5- 6 The Quadratic Formula
361
Find the zeros of each function. Then graph the function.
Aerospace
SpaceShipOne was the
winner of the Ansari X
Prize competition. The X
Prize was awarded to the
first nongovernmental
spacecraft to reach
an altitude of at least
100 km twice within
a 2 week period.
38. f (x) = 3x 2 - 4x - 2
39. g(x) = 2x 2 - 2x - 1
40. h(x) = 2x 2 + 6x + 5
41. p(x) = 2x 2 + 3x - 1
42. h(x) = 3x 2 - 5x - 4
43. r (x) = x 2 - x + 22
44. Aerospace In 2004, the highest spaceplane flight was made by Brian Binnie
in SpaceShipOne. A flight with this altitude can be modeled by the function
h(t) = -0.17t 2 + 187t + 61,000, where h is the altitude in meters and t is flight
time in seconds.
a. Approximately how long did the
Earth’s Atmosphere
flight last?
Layer
Altitude (in km)
b. What was the highest altitude to the
Troposphere
0 to 10
nearest thousand meters?
c. The table shows the altitudes of layers
Stratosphere
10 to 50
of Earth’s atmosphere. According to
Mesosphere
50 to 85
the model, which of these layers did
Thermosphere
85 to 600
SpaceShipOne enter, and at what time(s)
did the spaceplane enter them?
Solve each equation by any method.
45. x 2 - 3x = 10
46. x 2 - 16 = 0
47. 4x 2 + 4x = 15
48. x 2 + 2x - 2 = 0
49. x 2 - 4x - 21 = 0
50. 4x 2 - 4x - 1 = 0
51. 6x 2 = 150
52. x 2 = 7
53. x 2 - 16x + 64 = 0
54. Critical Thinking If you are solving a real-world
problem involving a quadratic equation, and the
discriminant is negative, what can you conclude?
55. Multi-Step The outer dimensions of a picture frame
are 25 inches by 20 inches. If the area inside the
picture frame is 266 square inches, what is the width
w of the frame?
w
Critical Thinking Find the values of c that make each
equation have one real solution.
56. x 2 + 8x + c = 0
57. x 2 + 12x = c
58. x 2 + 2cx + 49 = 0
59. Write About It What method would you use to solve the equation
-14x 2 + 6x = 2.7? Why would this method be easier to use than the
other methods?
60. This problem will prepare you for the Multi-Step Test Prep on page 364.
An outfielder throws a baseball to the player on third base. The height h of the ball
in feet is modeled by the function h(t) = -16t 2 + 19t + 5, where t is time in seconds.
The third baseman catches the ball when it is 4 ft above the ground.
a. To the nearest tenth of a second, how long was the ball in the air before it
was caught?
b. A player on the opposing team starts running from second base to third base
1.2 s before the outfielder throws the ball. The distance between the bases
is 90 ft, and the runner’s average speed is 27 ft/s. Will the runner reach third
base before the ball does? Explain.
362
Chapter 5 Quadratic Functions
61. Which best describes the graph of a quadratic function with a discriminant of -3?
Parabola with two x-intercepts
Parabola with no x-intercepts
Parabola that opens upward
Parabola that opens downward
62. What is the discriminant of the equation 2x 2 - 8x = 14?
48
176
-176
-48
63. Which function has zeros of 3 ± i?
f(x) = x 2 + 6x + 10
f(x) = x 2 + 6x - 10
g(x) = x 2 - 6x + 10
h(x) = x 2 - 6x - 10
64. Which best describes the discriminant of
the function whose graph is shown?
Positive
Zero
Þ
Negative
Undefined
Ý
CHALLENGE AND EXTEND
65. Geometry The perimeter of a right triangle is 40 cm, and its hypotenuse measures
17 cm. Find the length of each leg.
66. Geometry The perimeter of a rectangle is 88 cm.
a. Find the least possible value of the length of the diagonal. Round to the
nearest tenth of a centimeter.
b. What are the dimensions of the rectangle with this diagonal?
Write a quadratic equation whose solutions belong to the indicated sets.
67. integers
68. irrational real numbers
69. complex numbers
70. A quadratic equation has the form ax 2 + bx + c = 0 (a ≠ 0).
a. What is the sum of the roots of the equation? the product of the roots?
b. Determine the standard form of a quadratic equation whose roots have a sum
of 2 and a product of -15.
71. Describe the solutions to a quadratic equation for which a = b = c.
SPIRAL REVIEW
72. Biology The length of a human hair is a linear function of time. Juan’s hair grows
2.1 cm in 60 days. Express the growth in centimeters of Juan’s hair as a function of
the number of days since his last haircut. (Lesson 2-4)
Write the augmented matrix, and use row reduction to solve. (Lesson 4-6)
 3y = 2x + 7
 2x = -3y + 12
 4x + 5y = -1
73. 
74. 
75. 
 x - 6y = 1
 x + y = 14
 9 + 7y = 2x
Solve each equation by completing the square. (Lesson 5-4)
76. x 2 - 5x = 1
77. 2x 2 = 16x - 4
78. 3x = 5x 2 - 12
5- 6 The Quadratic Formula
363
SECTION 5A
Quadratic Functions and Complex Numbers
Ballpark Figures When a baseball is thrown or hit into
the air, its height h in feet after t seconds can be modeled by
h(t) = -16t 2 + v yt + h 0, where v y is the initial vertical velocity
of the ball in feet per second and h 0 is the ball’s initial height.
The horizontal distance d in feet that the ball travels in
t seconds can be modeled by d(t) = v xt, where v x is the ball’s
initial horizontal velocity in feet per second.
1. A short stop makes an error by dropping the ball. As the ball drops, its
height h in feet is modeled by h(t) = -16t 2 + 3. A slow-motion replay
of the error shows the play at half speed. What function describes the
height of the ball in the replay?
2. A player hits a foul ball with an initial
90 ft
vertical velocity of 70 ft/s and an initial
height of 5 ft. To the nearest foot, what is
the maximum height reached by the ball?
3. A pitch will be a strike if its height is
between 2.5 ft and 5 ft when it crosses
home plate. The pitcher throws the ball
from a height of 6 ft with an initial vertical
velocity of 5 ft/s and a
horizontal velocity of
116 ft/s. Could this pitch
be a strike? Explain.
4. The next pitch crosses
home plate 1 ft too high
to be a strike. The pitch is
thrown from a height of
6 ft with an initial vertical
velocity of 8 ft/s. What is the
initial horizontal velocity of
this pitch?
5. A player throws the ball home
from a height of 5.5 ft with
an initial vertical velocity of
28 ft/s. The ball is caught at
home plate at a height of 5 ft.
Three seconds before the ball
is thrown, a runner on third
base starts toward home plate
at an average speed of 25 ft/s.
Does the runner reach home
plate before the ball does?
Explain.
364
Chapter 5 Quadratic Functions
Pitcher’s mound
90 ft
90 ft
60 ft 6 in.
90 ft
Home plate
SECTION 5A
Quiz for Lessons 5-1 Through 5-6
5-1 Using Transformations to Graph Quadratic Functions
Using the graph of f (x) = x 2 as a guide, describe the transformations, and then
graph each function.
1 x2 + 1
1. g (x) = (x + 2)2 - 4
2. g (x) = -4(x - 1)2
3. g (x) = _
2
Use the description to write each quadratic function in vertex form.
4. f (x) = x 2 is vertically stretched by a factor of 9 and translated 2 units left to create g.
5. f (x) = x 2 is reflected across the x-axis and translated 4 units up to create g.
5-2 Properties of Quadratic Functions in Standard Form
For each function, (a) determine whether the graph opens upward or downward,
(b) find the axis of symmetry, (c) find the vertex, (d) find the y-intercept, and
(e) graph the function.
6. f (x)= x 2 - 4x + 3
7. g (x) = -x 2 + 2x - 1
8. h (x) = x 2 - 6x
9. A football kick is modeled by the function h(x) = -0.0075x 2 + 0.5x + 5, where h
is the height of the ball in feet and x is the horizontal distance in feet that the ball
travels. Find the maximum height of the ball to the nearest foot.
5-3 Solving Quadratic Equations by Graphing and Factoring
Find the roots of each equation by factoring.
10. x 2 - 100 = 0
11. x 2 + 5x = 24
12. 4x 2 + 8x = 0
5-4 Completing the Square
Solve each equation by completing the square.
13. x 2 - 6x = 40
14. x 2 + 18x = 15
15. x 2 + 14x = 8
Write each function in vertex form, and identify its vertex.
16. f (x) = x 2 + 24x + 138
17. g(x) = x 2 - 12x + 39
18. h(x) = 5x 2 - 20x + 9
5-5 Complex Numbers and Roots
Solve each equation.
19. 3x 2 = -48
20. x 2 - 20x = -125
21. x 2 - 8x + 30 = 0
5-6 The Quadratic Formula
Find the zeros of each function by using the Quadratic Formula.
22. f (x) = (x + 6)2 + 2
23. g (x) = x 2 + 7x + 15
24. h (x) = 2x 2 - 5x + 3
25. A bicyclist is riding at a speed of 18 mi/h when she starts down a long hill. The
distance d she travels in feet can be modeled by d (t) = 4t 2 + 18t, where t is the time
in seconds. How long will it take her to reach the bottom of a 400-foot-long hill?
Ready to Go On?
365
5-7
Solving Quadratic
Inequalities
Who uses this?
Tour companies and other businesses use
quadratic inequalities to make predictions
of profits. (See Example 4.)
Objectives
Solve quadratic
inequalities by using
tables and graphs.
Solve quadratic
inequalities by using
algebra.
Vocabulary
quadratic inequality in
two variables
Many business profits can be modeled by
quadratic functions. To ensure that the profit
is above a certain level, financial planners
may need to graph and solve quadratic
inequalities.
A quadratic inequality in two variables can be written in one of the following
forms, where a, b, and c are real numbers and a ≠ 0. Its solution set is a set of
ordered pairs (x, y).
y < ax 2 + bx + c
y ≤ ax 2 + bx + c
y > ax 2 + bx + c
y ≥ ax 2 + bx + c
In Lesson 2-5, you solved linear inequalities in two variables by graphing. You
can use a similar procedure to graph quadratic inequalities.
Graphing Quadratic Inequalities
To graph a quadratic inequality
T
Þ
1. Graph the parabola that defines the boundary.
1
2. Use a solid parabola for y ≤ and y ≥ and a dashed
parabola for y < and y >.
Ý
3. Shade above the parabola for y > or ≥ and below the
parabola for y ≤ or <.
EXAMPLE
1
Graphing Quadratic Inequalities in Two Variables
Graph y < -2x 2 - 4x + 6.
Step 1 Graph the boundary of the related
parabola y = -2x 2 - 4x + 6 with a
dashed curve.
Þ
È
Its y-intercept is 6, its vertex is (-1, 8),
and its x-intercepts are -3 and 1.
Step 2 Shade below the parabola because the
solution consists of y-values less than those on
the parabola for corresponding x-values.
Check Use a test point to verify the solution region.
y < -2x 2 - 4x + 6
0 < -2(0) 2 - 4(0) + 6
Try (0, 0).
0<6✔
366
Chapter 5 Quadratic Functions
{
Ó
{
ä]Êä®
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Graph each inequality.
1a. y ≥ 2x 2 - 5x - 2
1b. y < -3x 2 - 6x - 7
Quadratic inequalities in one variable, such as ax 2 + bx + c > 0 (a ≠ 0), have
solutions in one variable that are graphed on a number line.
EXAMPLE
2
Solving Quadratic Inequalities by Using Tables and Graphs
Solve each inequality by using tables or graphs.
A x 2 - 6x + 8 ≤ 3
Use a graphing calculator to graph each side of the inequality.
Set Y1 equal to x 2 - 6x + 8 and Y2 equal to 3. Identify the values of x
for which Y1 ≤ Y2.
The parabola is at or below the line when x is between 1 and
5 inclusive. So, the solution set is 1 ≤ x ≤ 5, or 1, 5. The table
supports your answer.
The number line shows the
solution set.
Ó £
ä
£
Ó
Î
{
x
È
Ç
n
B x 2 - 6x + 8 > 3
Use a graphing calculator to graph each side of the inequality. Set
Y1 equal to x 2 - 6x + 8 and Y2 equal to 3. Identify the values of x for
which Y1 > Y2.
For and statements,
both of the
conditions must
be true. For or
statements, at least
one of the conditions
must be true.
The parabola is above the line y = 3 when x is less than 1 or greater
than 5. So the solution set is x < 1 or x > 5, or (-∞, 1) (5, ∞).
The number line shows the
solution set.
Ó £
ä
£
Ó
Î
{
x
È
Ç
n
Solve each inequality by using tables or graphs.
2a. x 2 - x + 5 < 7
2b. 2x 2 - 5x + 1 ≥ 1
The number lines showing the solution sets in Example 2 are divided into three
distinct regions by the points 1 and 5. These points are called critical values. By
finding the critical values, you can solve quadratic inequalities algebraically.
5- 7 Solving Quadratic Inequalities
367
EXAMPLE
3
Solving Quadratic Inequalities by Using Algebra
Solve the inequality x 2 - 4x + 1 > 6 by using algebra.
Step 1 Write the related equation.
x 2 - 4x + 1 = 6
Step 2 Solve the equation for x to find the critical values.
x 2 - 4x - 5 = 0
(x - 5)(x + 1) = 0
x - 5 = 0 or x + 1 = 0
x = 5 or x = -1
Write in standard form.
Factor.
Zero Product Property
Solve for x.
The critical values are 5 and -1. The critical values divide the number
line into three intervals: x < -1, -1 < x < 5, and x > 5.
Step 3 Test an x-value in each interval.
#RITICAL VALUES
x - 4x + 1 > 6
2
(-2) - 4(-2) + 1 > 6 ✔ Try x = -2.
2
Î Ó £
ä
£
Ó
Î
{
x
È
Ç
{
x
È
Ç
(0) 2 - 4(0) + 1 > 6 ✘ Try x = 0.
4EST POINTS
(6) 2 - 4(6) + 1 > 6 ✔ Try x = 6.
Shade the solution regions on
the number line. Use open
circles for the critical values
because the inequality does not
contain or equal to.
Î Ó £
ä
£
Ó
Î
The solution is x < -1 or x > 5, or (-∞, -1) (5, ∞).
Solve each inequality by using algebra.
3a. x 2 - 6x + 10 ≥ 2
3b. -2x 2 + 3x + 7 < 2
EXAMPLE
4
Problem-Solving Application
1
Chapter 5 Quadratic Functions
Barcelos
ver
zon R i
R
s
e
i
õ
v
Ama
er
Solim
Manaus
Tefé
Understand the Problem
The answer will be the number of
people required for a profit that is
greater than or equal to $5000.
List the important information:
• The profit must be at least $5000.
• The function for the profit is
P(x) = -25x 2 + 1000x -3000.
368
Río N
egro
Belém
Xingu River
A business offers tours to the
Amazon. The profit P that the
company earns for x number
of tourists can be modeled by
P(x) = -25x 2 + 1000x - 3000.
How many people are needed
for a profit of at least $5000?
B R A Z I L
Travel Brazil
2 Make a Plan
Write an inequality showing profit greater than or equal to $5000.
Then solve the inequality by using algebra.
3 Solve
Write the inequality.
-25x 2 + 1000x - 3000 ≥ 5000
Find the critical values by solving the related equation.
-25x 2 + 1000x - 3000 = 5000 Write as an equation.
-25x 2 + 1000x - 8000 = 0
Write in standard form.
-25(x 2 - 40x + 320) = 0
Factor out -25 to simplify.
Use the
-(-40) ± √(
-40) 2 - 4(1)(320)
-b ± √
b 2 - 4ac
__
___
Quadratic
x=
=
2a
2(1)
Formula.
40 ± √
320
= _
2
Simplify.
x ≈ 28.94 or x ≈ 11.06
Test an x-value in each of the three
regions formed by the critical x-values.
x
-25(10) 2 + 1000(10) - 3000 5000
£ä
£x
Óä
Óx
Îä
Îx
Try x = 10.
4500 ≥ 5000 ✘
-25(20) + 1000(20) - 3000 5000
2
Try x = 20.
7000 ≥ 5000 ✓
-25(30) + 1000(30) - 3000 5000
2
Try x = 30.
4500 ≥ 5000 ✘
A compound
inequality such as
12 ≤ x ≤ 28 can
be written as


 x | x ≥ 12 x ≤ 28 ,

or x ≥ 12 and x ≤ 28.
(See Lesson 2-8.)
Write the solution as an inequality. The solution is approximately
11.06 ≤ x ≤ 28.94. Because you cannot have a fraction of a person, round
each critical value to the appropriate whole number.
12 ≤ x ≤ 28
For a profit of at least $5000, from 12 to 28 people are needed.
4 Look Back
Enter y = -25x 2 + 1000x - 3000 into
a graphing calculator, and create a
table of values. The table shows that
integer values of x between 12 and
28 inclusive result in y-values greater
than or equal to 5000.
4.
The business also offers educational tours to Patagonia, a
region of South America that includes parts of Chile and
Argentina. The profit P for x number of persons is
P(x) = -25x 2 + 1250x - 5000. The trip will be rescheduled
if the profit is less than $7500. How many people must have
signed up if the trip is rescheduled?
5- 7 Solving Quadratic Inequalities
369
THINK AND DISCUSS
1. Compare graphing a quadratic inequality with graphing a linear inequality.
2. Explain how to determine if the intersection point(s) is/are included in the solution
set when you solve a quadratic inequality by graphing.
3. GET ORGANIZED Copy and complete the graphic organizer. Compare the solutions
of quadratic equations and inequalities.
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5-7
Exercises
KEYWORD: MB7 5-7
KEYWORD: MB7 Parent
GUIDED PRACTICE
1. Vocabulary Give an example of a quadratic inequality in two variables.
SEE EXAMPLE
1
SEE EXAMPLE
2
3
p. 368
SEE EXAMPLE 4
p. 368
3. y ≤ 2x 2 - 4x - 1
4. y ≤ -3x 2 + x + 3
Solve each inequality by using tables or graphs.
5. x 2 - 5x + 3 ≤ 3
p. 367
SEE EXAMPLE
Graph each inequality.
2. y > -(x + 1) 2 + 5
p. 366
6. 3x 2 - 3x - 1 > -1
7. 2x 2 - 9x + 5 ≤ -4
Solve each inequality by using algebra.
8. x 2 + 10x + 1 ≥ 12
9. x 2 + 13x + 45 < 5
10. -2x 2 + 3x + 12 > 10
11. Business A consultant advises the owners of a beauty salon that their profit p each
month can be modeled by p(x) = -50x 2 + 3500x - 2500, where x is the average
cost that a customer is charged. What range of costs will bring in a profit of at
least $50,000?
PRACTICE AND PROBLEM SOLVING
Graph each inequality.
12. y < x 2 + 2x - 5
15. y ≥ x 2 + 6
1 x2 + 3
13. y > -_
2
16. y < (x + 1)(x + 4)
14. y ≤ 2(x - 1) 2 - 3
17. y ≤ x 2 - 2x + 6
Solve each inequality by using tables or graphs.
370
18. x 2 - x + 5 < 11
19. 2x 2 + 3x + 6 ≥ 5
20. x 2 - 5x + 12 > 6
21. x 2 - 2x - 8 > 0
22. x 2 + 7x + 6 ≤ 6
23. x 2 - 12x + 32 < 12
Chapter 5 Quadratic Functions
Independent Practice
For
See
Exercises Example
12–17
18–23
24–26
27
1
2
3
4
Extra Practice
Skills Practice p. S13
Application Practice p. S36
Solve each inequality by using algebra.
24. x 2 - 11x + 13 ≤ 25
25. -2x 2 + 3x + 4 ≥ -1
26. x 2 - 5x - 4 < -9
27. Sports A football thrown by a quarterback follows a path given by
h(x) = -0.0095x 2 + x + 7, where h is the height of the ball in feet and x is the
horizontal distance the ball has traveled in feet. If any height less than 10 feet
can be caught or knocked down, at what distances from the quarterback can
the ball be knocked down?
Graph each quadratic inequality.
28. y ≤ 2x 2 + 4x - 3
29. y < 3x 2 - 12x - 4
31. y > -2(x + 3) 2 + 1
32. y > -x 2 - 2x - 1
34. Circus The human cannonball is an act
where a performer is launched through
the air. The height of the performer can
be modeled by h(x) = -0.007x 2 + x + 20,
where h is the height in feet and x is the
horizontal distance traveled in feet. The
circus act is considering a flight path
directly over the main tent.
30. y ≥ -3x 2 + 4x
1 x 2 + 2x - 1
33. y ≤ _
3
At least 5 ft
a. If the performer wants at least 5 ft of
vertical height clearance, how tall can
the tent be?
b. How far from the central pole should
the “cannon” be placed?
Solve each inequality by using any method.
35. x 2 - 5x - 24 ≤ 0
36. x 2 - 14 ≥ 2
37. -2x 2 - x + 8 > 6
38. x 2 - 4x - 5 ≤ -9
39. 3x 2 + 6x + 11 < 10
40. 4x 2 - 9 > 0
41. 3x 2 + 5x + 13 ≤ 16
42. -2x 2 + 3x + 17 ≥ 11
43. 5x 2 - 2x - 1 ≥ 0
44. (x - 2)(x + 11) ≥ 2
45. x 2 + 27 > 12x
46. -2x 2 + 3x + 6 > 0
47. Multi-Step A medical office has a rectangular
parking lot that measures 120 ft by 200 ft. The
owner wants to expand the size of the parking
lot by adding an equal distance to two sides as
shown. If zoning restrictions limit the total size
of the parking lot to 35,000 ft 2, what range of
distances can be added?
Ý
£ÓäÊvÌ
Ý
ÓääÊvÌ
Match each graph with one of the following
inequalities.
A. y < x 2 + 2x - 3
48.
B. y > -x 2 - 2x + 3
49.
Þ
Ý
{
Ó
Ó
50.
Þ
È
Þ
Ý
Ó
ä
Ó
C. y < x 2 - 2x + 3
Ó
{
Ó
ä
Ó
Ó
Ó
Ý
Ó
ä
Ó
{
5- 7 Solving Quadratic Inequalities
371
51. This problem will prepare you for the Multi-Step Test Prep on page 390.
A small square tile is placed on top of a larger square tile as shown. This creates four
congruent triangular regions.
a. Write a function for the area A of one of the triangular
regions in terms of x.
Ý
b. For what values of x, to the nearest tenth, is the area of
each triangular region at least 30 cm 2?
ÓäÊV
c. For what values of x, to the nearest tenth, is the area of
each triangular region less than 40 cm 2?
52. Music A manager estimates a band’s profit p
for a concert by using the function
p(t) = -200t 2 + 2500t - c, where t is the price
per ticket and c is the band’s operating cost. The
table shows the band’s operating cost at three
different concert locations. What range of ticket
prices should the band charge at each location
in order to make a profit of at least $1000 at each
concert?
53. Gardening Lindsey has 40 feet of metal
fencing material to fence three sides of a
rectangular garden. A tall wooden fence
serves as her fourth side.
a. Write a function for the area of the
garden A in terms of x, the width in feet.
b. What measures for the width will give an
area of at least 150 square feet?
c. What measures for the width will give an
area of at least 200 square feet?
Band’s Costs
Location
Operating Cost
Freemont Park
$900
Saltillo Plaza
$1500
Riverside Walk
$2500
x
Graphing Calculator Use the intersect feature
of a graphing calculator to solve each inequality
to the nearest tenth.
54. x 2 + 6x - 13 > 4
55. x 2 - 15x + 20 ≤ 7
56. x 2 - 24 < 28
57. 2x 2 + 3x + 5 ≥ 8
58. Business A wholesaler sells snowboards to sporting-good stores. The price per
snowboard varies based on the number purchased in each order. The function
r(x) = -x 2 + 125x models the wholesaler’s revenue r in dollars for an order of
x snowboards.
a. To the nearest dollar, what is the maximum revenue per order?
b. How many snowboards must the wholesaler sell to make at least $1500 in
revenue in one order?
59. Critical Thinking Explain whether the solution to a quadratic inequality in one
variable is always a compound inequality.
60. Critical Thinking Can a quadratic inequality have a solution set that is all real
numbers? Give an example to support your answer.
61. Write About It Explain how the solutions of x 2 - 3x - 4 ≤ 6 differ from the
solutions of x 2 - 3x - 4 = 6.
372
Chapter 5 Quadratic Functions
62. Which is the solution set of x 2 - 9 < 0?
-3 < x < 3
-9 < x < 9
x < -3 or x > 3
x < -9 or x > 9
63. Which is the graph of the solution to x 2 - 7x + 10 ≥ 0?
È x { Î Ó £
ä
£
Ó
Î
{
x
È x { Î Ó £
ä
ä
È
64. Which is the solution set of x 2 - 7x ≤ 0?
0<x<7
0≤x≤7
£
Ó
Î
{
x
ä
È
x < 0 or x > 7
x ≤ 0 or x ≥ 7
65. Short Response Demonstrate the process for solving x 2 + 4x + 4 > 1
algebraically. Justify each step in the solution process.
CHALLENGE AND EXTEND
Graph each system of inequalities.
66.
{
y ≤ x2
y ≥ -x 2 + 5
67.
{
y ≥ x2 - 3
68.
y ≤ -x 2 - 2x + 9
{
y ≥ 2x 2 - 12x + 20
1 x 2 - 2x + 8
y≥_
3
Geometry The area inside a parabola bounded from above or below
by a horizontal line segment is __23 bh, where b is the length of the line
segment and h is the vertical distance from the vertex of the parabola
to the line segment. Find the area bounded by the graphs of each pair
of inequalities.
69. y > x 2 + 5x - 6; y < 8
Þ
Ý
L
70. y < -2x 2 + 3x + 9; y > -5
SPIRAL REVIEW
71. Community Once a month, four teams of teens (lawn team, shopping team,
cleaning team, and laundry team) spend a day assisting elderly residents of their
neighborhood. Lynnette started the assignment chart for June but was interrupted.
Complete the chart. Each home has only one team helping during each shift.
(Previous course)
Shifts
Reed Home
Brown Home
Sondi Home
Clem Home
Lawn
Cleaning
?
?
10:00 A.M.–12:30 P.M.
?
Shopping
?
Lawn
1:00 P.M.–3:30 P.M.
?
?
Laundry
?
4:00 P.M.–6:30 P.M.
Cleaning
?
?
?
7:00 A.M.–9:30 A.M.
Graph each inequality by using intercepts. (Lesson 2-5)
72. 4x - 3y > 15
73. 6x - y ≤ 8
74. 8x + 5y < 40
Find the values of c that make each equation true. (Lesson 5-5)
75. 4 - 2c + 7i = 7i - 14
76. 4c + 2 - 3i + 2(i - 5) = 4(2i - 6) - 9i
5- 7 Solving Quadratic Inequalities
373
5-8
IA-3.5
Analyze
given
information
models)
to for
solve
contextual
A2.3.5
problems
that
be(including
modeled quadratic
using
quadratic
equations
functions,
3.1b Solve
Identify
whether
thecan
model/equation
is a curve
of best
fit
theand
data,
using
problems.
interpret
the solutions
and determine
solutionscalculator.
are reasonable.
various methods
and tools
which maywhether
include the
a graphing
Who uses this?
Film preservationists use quadratic
relationships to estimate film run
times. (See Example 3.)
Objectives
Use quadratic functions
to model data.
Use quadratic models to
analyze and predict.
New York
Recall that you can use differences to
analyze patterns in data. For a set of
ordered pairs with equally spaced
x-values, a quadratic function has
constant nonzero second differences,
as shown below.
r Collection,
Vocabulary
quadratic model
quadratic regression
The Grange
*/
Curve Fitting with
Quadratic Models
Equally spaced x-values
⎧
⎨
⎩
x
-3
-2
-1
0
1
2
3
f (x) = x 2
9
4
1
0
1
4
9
-5
1st differences
2nd differences
-3
2
-1
2
1
2
3
2
Þ
n
È
5
2
{
Constant 2nd differences
vÝ®ÊÊÊÝÊÓÊ
Ó
Ý
Ó
EXAMPLE
1
ä
Ó
Identifying Quadratic Data
Determine whether each data set could represent a quadratic
function. Explain.
A
x
0
2
4
6
8
y
12
10
9
9
10
Find the first and second
differences.
Equally spaced x-values
B
x
y
2
4
8
16
12
10
9
9
10
y
-2 -1
0
1
2
4
8
16
1
1st
2nd
2
1
2
1
4
2
⎧
y
x
8
⎨
6
4
⎩
⎧
⎨
⎩
2
Quadratic function; second
differences are constant
for equally spaced x-values.
2
1
Equally spaced x-values
0
-2 -1 0 1
1 1 1
1
0
Find the first and second
differences.
x
1st
2nd
-2 -1
8
4
Not a quadratic function; second
differences are not constant for
equally spaced x-values.
Determine whether each data set could represent a quadratic
function. Explain.
1a. x 3 4 5 6 7
1b. x 10 9 8 7 6
y
374
Chapter 5 Quadratic Functions
11
21 35 53 75
y
6
8
10 12 14
Just as two points define a linear function, three noncollinear points define
a quadratic function. You can find the three coefficients, a, b, and c, of
f (x) = ax 2 + bx + c by using a system of three equations, one for each point.
The points do not need to have equally spaced x-values.
EXAMPLE
2
Writing a Quadratic Function from Data
Write a quadratic function that fits the points (0, 5), (2, 1), and (3, 2).
Use each point to write a system of equations to find a, b, and c in
f (x) = ax 2 + bx + c.
Collinear points lie
on the same line.
Noncollinear points
do not all lie on the
same line.
(x, y)
f(x) = ax 2 + bx + c
(0, 5)
(2, 1)
(3, 2)
5 = a(0) + b(0) + c
2
1 = a(2) + b(2) + c
2
2 = a(3) + b(3) + c
2
System in a, b, c
c = 5

1
4a + 2b + c = 1 2
 9a + 3b + c = 2 3
Substitute c = 5 from equation 1 into both equation 2 and equation 3.
2
4a + 2b + c = 1
9a + 3b + c = 2
3
4a + 2b + 5 = 1
9a + 3b + 5 = 2
4a + 2b = -4 4
9a + 3b = -3 5
Solve equation 4 and equation 5 for a and b using elimination.
4
5
3(4a + 2b) = 3(-4) →
12a + 6b = -12
-2(9a + 3b) = -2(-3) → -18a
- 6b = 6
−−−−−−−−−−−−
-6a = -6
a =1
Multiply by 3.
Multiply by -2.
Add the equations.
Substitute 1 for a into equation 4 or equation 5 to find b.
4
4a + 2b = -4
→ 4(1) + 2b = -4
2b = -8
b = -4
Write the function using a = 1, b = -4, and c = 5.
f (x) = ax 2 + bx + c → f (x) = 1x 2 - 4x + 5, or f (x) = x 2 - 4x + 5
Check Substitute or create a table to verify
that (0, 5), (2, 1), and (3, 2) satisfy the
function rule.
2. Write a quadratic function that fits the points (0, -3), (1, 0),
and (2, 1).
You may use any method that you studied in Chapters 3 or 4 to solve
the system of three equations in three variables. For example, you
can use a matrix equation as shown.
c=5
0 0 1a 5 a  1
→
4a + 2b + c = 1
4 2 1
b = 1 → b = -4
 9a + 3b + c = 2  9 3 1   c   2   c   5 


    5- 8 Curve Fitting with Quadratic Models
375
A quadratic model is a quadratic function that represents a real data set.
Models are useful for making estimates.
In Chapter 2, you used a graphing calculator to perform a linear regression and
make predictions. You can apply a similar statistical method to make a quadratic
model for a given data set using quadratic regression.
EXAMPLE
3
Film Application
The table shows approximate
run times for 16 mm films,
given the diameter of the film
on the reel. Find a quadratic
model for the run time given
the diameter. Use the model
to estimate the run time for
a reel of film with a diameter
of 15 in.
The coefficient of
determination R 2
shows how well a
quadratic model fits
the data. The closer
R 2 is to 1, the better
the fit. In this model,
R 2 ≈ 0.996, which
is very close to 1, so
the quadratic model
is a good fit.
Film Run Times (16 mm)
Diameter
(in.)
Reel Length
(ft)
Run Time
(min)
5
200
5.55
7
400
11.12
9.25
600
16.67
10.5
800
22.22
12.25
1200
33.33
13.75
1600
44.45
Step 1 Enter the data into two lists
in a graphing calculator.
Step 2 Use the quadratic regression
feature.
Step 3 Graph the data and function
model to verify that the
model fits the data.
Step 4 Use the table feature to find
the function value at x = 15.
xä
Î
£x
ä
A quadratic model is T (d) ≈ 0.397d 2 - 3.12d + 11.94, where T is the run
time in minutes and d is the film diameter in inches.
For a 15 in. diameter, the model predicts a run time of about 54.5 min, or
54 min 30 s.
Use the information given above to answer the following.
3. Find a quadratic model for the reel length given the diameter
of the film. Use the model to estimate the reel length for an
8-inch-diameter film.
376
Chapter 5 Quadratic Functions
THINK AND DISCUSS
1. Describe how to determine if a data set is quadratic.
2. Explain whether a quadratic function is a good model for the path of an
airplane that ascends, descends, and rises again out of view.
3. GET ORGANIZED
Copy and complete
the graphic organizer.
Compare the
different quadratic
models presented in
the lesson.
5-8
+Õ>`À>ÌVÊì
7 iÊ««À«À>Ìi
*ÀVi`ÕÀi
Ý>VÌÊì
««ÀÝ>Ìi
ì
Exercises
KEYWORD: MB7 5-8
KEYWORD: MB7 Parent
GUIDED PRACTICE
1. Vocabulary How does a quadratic model differ from a linear model?
SEE EXAMPLE
1
p. 374
SEE EXAMPLE
2.
2
p. 375
SEE EXAMPLE
p. 376
Determine whether each data set could represent a quadratic function. Explain.
3
x -2 -1
0
1
2
y
0
-8
-16
16
8
3.
x
1
2
3
4
5
y
1
3
9
27
81
4.
x
2
y
4
4
6
8
10
-5 -8 -5
4
Write a quadratic function that fits each set of points.
5.
(-2, 5), (0, -3), and (3, 0)
6.
(0, 1), (2, -1), and (3, -8)
7.
(-1, 8), (0, 4), and (2, 2)
8.
(-4, 9), (0, -7), and (1, -1)
9.
(2, 3), (6, 3), and (8, -3)
10.
(-1, -12), (1, 0), and (2, 9)
11. Hobbies The cost of mounting
different-sized photos is shown in the
table. Find a quadratic model for the
cost given the average side length.
(For an 8 in. × 10 in. photo, the
8 + 10
average side length is _____
= 9 in.)
2
Estimate the cost of mounting a
24 in. × 36 in. photo.
Costs of Mounting Photos
Size (in.)
Cost ($)
8 × 10
10
14 × 18
16
16 × 20
19
24 × 30
27
32 × 40
39
PRACTICE AND PROBLEM SOLVING
Determine whether each data set could represent a quadratic function. Explain.
12. x
0
f (x) -1
2
4
6
8
2 11 26 47
13.
x
0
1
2
3
4
f (x)
10
9
6
1
-6
14.
x
1
f (x) -3
2
3
4
5
0
3
6
9
5- 8 Curve Fitting with Quadratic Models
377
Independent Practice
For
See
Exercises Example
12–14
15–18
19
1
2
3
Extra Practice
Skills Practice p. S13
Application Practice p. S36
Write a quadratic function that fits each set of points.
15.
(-2, 5), (-1, 0), and (1, -2)
16.
(1, 2), (2, -1), and (5, 2)
17.
(-4, 12), (-2, 0), and (2, -12)
18.
(-1, 2.6), (1, 4.2), and (2, 14)
19. Gardening The table shows the amount spent on
water gardening in the United States between 1999 and
2003. Find a quadratic model for the annual amount
in millions of dollars spent on water gardening based
on number of years since 1999. Estimate the amount
that people in the United States will spend on water
gardening in 2015.
Water Gardening
Amount Spent
(million $)
Year
Write a function rule for each situation, and identify each
relationship as linear, quadratic, or neither.
1999
806
2000
943
2001
1205
2002
1441
2003
1565
20. the circumference C of a bicycle wheel, given its radius r
21. the area of a triangle A with a constant height, given its base length b
22. the population of bacteria P in a petri dish doubling every hour t
23. the area of carpet A needed for square rooms of length s
24. Physics In the past, different
Relative Distance Fallen (units)
mathematical descriptions of
Time Interval Aristotle’s da Vinci’s
Galileo’s
falling objects were proposed.
(s)
Rule
Rule
Rule
a. Which rule shows the
0
0
0
0
greatest increase in the
distance fallen per second and
1
1
1
1
thus the greatest rate
2
2
3
4
of increase in speed?
3
3
6
9
b. Identify each rule as linear,
4
4
10
16
quadratic, or neither.
c. Describe the differences in
da Vinci’s rule, and compare it with the differences in Galileo’s.
d. The most accurate rule is sometimes described as the odd-number
law. Which rule shows an odd-number pattern of first differences
and correctly describes the distance for falling objects?
Find the missing value for each quadratic function.
25.
x
f (x)
-1 0
0
1
1
0
2
3
-8
26.
x
-3 -2 -1 0
f (x) 12
2
0
1
8
27.
x
-2
f (x) -2
0
2
4
6
2
7
14
28. This problem will prepare you for the Multi-Step Test Prep on page 390.
A home-improvement store sells several sizes of
Length (in.)
Area (in 2)
rectangular tiles, as shown in the table.
a. Find a quadratic model for the area of a tile based
4
28
on its length.
6
54
b. The store begins selling a new size of tile with a
8
88
length of 9 in. Based on your model, estimate the
10
130
area of a tile of this size.
378
Chapter 5 Quadratic Functions
29. Food The pizza prices for DeAngelo’s pizza parlor
are shown at right.
a. Find a quadratic model for the price of a pizza
based upon the size (diameter).
b. Use the quadratic model to find the price of a
pizza with an 18 in. diameter.
c. Graph the quadratic function. Does the function
have a minimum or maximum point? What does
this point represent?
d. What if...? According to the model, how much
should a 30 in. pizza cost? How much should an
8 in. pizza cost?
e. Is the quadratic function a good model for
the price of DeAngelo’s pizza? Explain your
reasoning.
Determine whether each data set could represent a quadratic function. If so, find a
quadratic function rule.
30.
x
0
1
2
3
4
31.
y -1 0 -1 -4 -9
33.
x -2 -1
0
1
2
y
0
7
24
16
3
x
1
2
3
4
5
32.
y 10 20 40 60 80
34.
x
0
1
2
3
4
y
9
5
3
1
0
36. Winter Sports The diagram shows the
motion of a skier following a jump. Find a
quadratic model of the skier’s height h in
meters based on time t in seconds. Estimate
the skier’s height after 2 s.
x
2
4
6
8
10
0
1
3
5
x -2 -1
0
1
2
y
9
27
81
y -1
35.
0
3
t = 1.1 s
h = 18.7 m
t=0s
h = 13.2 m
37. Data Collection Use a graphing calculator
and a motion detector to measure the height
of a basketball over time. Drop the ball from
t = 3.0 s
a height of 1 m, and let it bounce several
h=0m
times. Position the motion detector 0.5 m
above the release point of the ball.
a. What is the greatest height the ball reaches during its first bounce?
b. Find an appropriate model for the height of the ball as a function of time during
its first bounce.
38. Safety The light produced
High-Pressure Sodium Vapor Streetlamps
by high-pressure sodium
Energy Use
vapor streetlamps for different
35
50
70
100
150
(watts)
energy usages is shown in
the table.
Light Output
2250 4000 5800 9500 16,000
(lumens)
a. Find a quadratic model
for the light output with
respect to energy use.
b. Find a linear model for the light output with respect to energy use.
c. Apply each model to estimate the light output in lumens of a 200-watt bulb.
d. Which model gives the better estimate? Explain.
5- 8 Curve Fitting with Quadratic Models
379
39. Sports The table lists the average distance that a normal shot
travels for different golf clubs.
2 iron
Average Distance for Normal Shot
Club Iron (no.)
2
3
4
5
6
7
8
9
Loft Angle
16°
20°
24°
28°
32°
36°
40°
44°
Distance (yd)
186
176
166
155
143
132
122
112
16º
9 iron
a. Select three data values (club number, distance), and
use a system of equations to find a quadratic model.
Check your model by using a quadratic regression.
b. Is there a quadratic relationship between club number
and average distance of a normal shot? Explain.
c. Is the relationship between club number and loft
angle quadratic or linear? Find a model of this
relationship.
Math History
Pythagoras made
numerous contributions
to mathematics, including
the Pythagorean Theorem,
which bears his name.
40. Multi-Step Use the table of alloy-steel
chain data.
a. Do each of the last two columns
appear to be quadratic functions
with respect to the nominal chain
size? Explain.
b. Verify your response in part a
by finding each of the quadratic
regression equations. Do the
models fit the data well? Explain.
c. Predict the values for the last two
columns for a chain with a nominal
size of __85 in.
44º
Alloy-Steel Chain Specifications
Nominal
Size (in.)
Maximum
Length 100
Links (in.)
Maximum
Weight 100
Links (lb)
1
_
4
1
_
2
3
_
4
98
84
156
288
208
655
1
277
1170
1
1_
4
371
1765
41. Math History The Greek mathematician
Pythagoras developed a formula for triangular
numbers, the first four of which are shown. Write
a quadratic function that determines a triangular
number t in terms of its place in the sequence n.
(Hint: The fourth triangular number has n = 4.)
42. Critical Thinking Two points define a unique line.
How many points define a unique parabola, and what
restriction applies to the points?
43. Critical Thinking Consider the following data set.
x
10
8
13
9
11
14
6
4
12
7
5
y 9.14 8.14 8.74 8.77 9.29 8.1 6.13 3.1 9.13 7.26 4.74
a.
b.
c.
d.
Create a scatter plot of the data.
Perform a linear regression on the data.
Perform a quadratic regression on the data.
Which model best describes the data set? Explain your answer.
44. Write About It What does it mean when the coefficient a in a
quadratic regression model is zero?
380
Chapter 5 Quadratic Functions
Loft
angle
45. Which of the following would best be modeled by a quadratic function?
Relationship between circumference and diameter
Relationship between area of a square and side length
Relationship between diagonal of a square and side length
Relationship between volume of a cube and side length
46. If (7, 11) and (3, 11) are two points on a parabola, what is the x-value of the vertex
of this parabola?
11
3
5
7
47. If y is a quadratic function of x, which value completes the table?
x -2
0
2
4
y -8
0
12
28
6
48
44
20
12
48. The graph of a quadratic function having the form f (x) = ax 2 + bx + c passes
through the points (0, -8), (3, 10), and (6, 34). What is the value of the function
when x = -3?
-32
-26
-20
10
49. Extended Response Write a quadratic function in standard form that fits the
data points (0, -5), (1, -3), and (2, 3). Use a system of equations, and show all of
your work.
CHALLENGE AND EXTEND
50. Three points defining a quadratic function are (1, 2), (4, 6), and (7, w).
a. If w = 9, what is the quadratic function? Does it have a maximum value or a
minimum value? What is the vertex?
b. If w = 11, what is the quadratic function? Does it have a maximum value or a
minimum value? What is the vertex?
c. If w = 10, what function best fits the points?
51. Explain how you can determine from three points whether the parabola that fits the
points opens upward or downward.
SPIRAL REVIEW
Determine whether each data set could represent a linear function. (Lesson 2-3)
52.
-2
1
4
f (x) -5
7
1
x
53.
x
-8 -6
f (x) -1
0
0
3
Find the inverse of the matrix, if it is defined. (Lesson 4-5)
54.


1 0
__
3
 -4 1 
 2 -2 
55. 
 1 -1 
56.
-2 0 1 
0 0 1
 4 2 2

57.
 3 -4 
0

-__12 
Find the zeros of each function by using the Quadratic Formula. (Lesson 5-6)
58. f (x) = 2x 2 - 4x + 1
59. f (x) = x 2 + 9
60. f (x) = -3x 2 + 10x + 12
5- 8 Curve Fitting with Quadratic Models
381
5-9
*/
Operations with
Complex Numbers
A2.3.1
add,
subtract,
and
divide
complex
numbers.
Represent complex
IA-3.2
Carry
out
a procedure
todivide,
perform
with complex
numbers
1.3b Define,
Add, subtract,
multiply,multiply
andoperations
simplify
expressions
involving
numbers,
and
the addition,
subtraction
and absolute
value of complex numbers,
(including
addition,
subtraction,
multiplication,
and division).
complex numbers.
in the complex plane.
Why learn this?
Complex numbers can be used in formulas
to create patterns called fractals. (See
Exercise 84.)
Objective
Perform operations with
complex numbers.
Vocabulary
complex plane
absolute value of a
complex number
Just as you can represent real numbers
graphically as points on a number line, you
can represent complex numbers in a special
coordinate plane.
>}>ÀÞÊ>ÝÃ
The complex plane is a set of coordinate axes in which the
horizontal axis represents real numbers and the vertical axis
represents imaginary numbers.
Ó
äÊ Êä
Ó
Ó
,i>
>ÝÃ
Ó
EXAMPLE
1
Graphing Complex Numbers
>}>ÀÞÊ>ÝÃ
Graph each complex number.
ÓÊ {
A -3 + 0i
The real axis
corresponds to
the x-axis, and
the imaginary axis
corresponds to the
y-axis. Think of
a + bi as x + yi.
{
{Ê Î
Ó
B -3i
ÎÊ ä
{
C 4 + 3i
Ó
ä
Ó
{
,i>
>ÝÃ
Ó
Î
D -2 + 4i
{
Graph each complex number.
1a. 3 + 0i
1b. 2i
1c. -2 - i
1d. 3 + 2i
Recall that the absolute value of a real number is its distance from 0 on the real
axis, which is also a number line. Similarly, the absolute value of an imaginary
number is its distance from 0 along the imaginary axis.
Absolute Value of a Complex Number
WORDS
The absolute value
of a complex
number a + bi is
the distance from
the origin to the
point (a, b) in the
complex plane, and
is denoted ⎪a + bi⎥.
ALGEBRA
EXAMPLE
⎪a + bi⎥ = √
a2 + b2
>}>ÀÞÊ>ÝÃ
ÎÊ {
{
{
]ÎÊ {]
Î
ä
Ó
{
,i>
>ÝÃ
⎪3 + 4i⎥ = √
32 + 42
= √
9 + 16
=5
382
Chapter 5 Quadratic Functions
EXAMPLE
2
Determining the Absolute Value of Complex Numbers
Find each absolute value.
A ⎪-9 + i⎥
B
⎪6⎥
C
⎪-4i⎥
⎪-9 + 1i⎥
⎪6 + 0i⎥
⎪0 + (-4)i⎥
(-9) 2 + 1 2
√
√
62 + 02
0 2 + (-4) 2
√
√
81 + 1
√
36
√
16
√
82
6
4
Find each absolute value.
1
2a. ⎪1 - 2i⎥
2b. -_
2
⎪ ⎥
2c. ⎪23i⎥
Adding and subtracting complex numbers is similar to adding and subtracting
variable expressions with like terms. Simply combine the real parts, and combine
the imaginary parts.
The set of complex numbers has all the properties of the set of real numbers.
So you can use the Commutative, Associative, and Distributive Properties to
simplify complex number expressions.
EXAMPLE
3
Adding and Subtracting Complex Numbers
Add or subtract. Write the result in the form a + bi.
A (-2 + 4i) + (3 - 11i)
(-2 + 3) + (4i - 11i)
1 - 7i
B (4 - i) - (5 + 8i)
(4 - i) - 5 - 8i
(4 - 5) + (-i - 8i)
-1 - 9i
Complex numbers
also have additive
inverses. The additive
inverse of a + bi is
-(a + bi), or
-a - bi.
C (6 - 2i) + (-6 + 2i)
(6 - 6) + (-2i + 2i)
0 + 0i
Associative and Commutative Properties
Add real parts and imaginary parts.
Distributive Property
Associative and Commutative Properties
Add real parts and imaginary parts.
Associative and Commutative Properties
Add real parts and imaginary parts.
0
D
(10 + 3i) - (10 - 4i)
(10 + 3i) - 10 - (-4i)
(10 - 10) + (3i + 4i)
0 + 7i
Distributive Property
Associative and Commutative Properties
Add real parts and imaginary parts.
7i
Add or subtract. Write the result in the form a + bi.
3a. (-3 + 5i) + (-6i) 3b. 2i - (3 + 5i) 3c. (4 + 3i) + (4 - 3i)
You can also add complex numbers by using coordinate geometry.
5- 9 Operations with Complex Numbers
383
EXAMPLE
4
Adding Complex Numbers on the Complex Plane
Find (4 + 3i) + (-2 + i) by graphing on the
complex plane.
Step 1 Graph 4 + 3i and -2 + i on the
complex plane. Connect each of these
numbers to the origin with a line
segment.
>}>ÀÞÊ>ÝÃ
ÓÊ {
{
{Ê Î
ÓÊ
{
Step 2 Draw a parallelogram that has
these two line segments as sides.
The vertex that is opposite the origin
represents the sum of the two complex
numbers, 2 + 4i. Therefore, (4 + 3i) +
(-2 + i) = 2 + 4i.
ä
Ó
Ó
{
,i>
>ÝÃ
Ó
{
Check Add by combining the real parts and combining the
imaginary parts.
(4 + 3i) + (-2 + i) = 4 + (-2) + (3i + i) = 2 + 4i
Find each sum by graphing on the complex plane.
4a. (3 + 4i) + (1 - 3i)
4b. (-4 - i) + (2 - 2i)
You can multiply complex numbers by using the Distributive Property and
treating the imaginary parts as like terms. Simplify by using the fact i 2 = -1.
EXAMPLE
5
Multiplying Complex Numbers
Multiply. Write the result in the form a + bi.
A 2i(3 - 5i)
6i - 10i 2
B (5 - 6i)(4 - 3i)
20 - 15i - 24i + 18i 2 Multiply.
Distribute.
6i - 10(-1)
20 - 39i + 18(-1)
Use i 2 = -1.
10 + 6i Write in a + bi form.
C (7 + 2i)(7 - 2i)
Use i 2 = -1.
2 - 39i
D (6i)(6i)
49 - 14i + 14i - 4i 2
Multiply.
36i 2
49 - 4(-1) Use i 2 = -1.
36(-1 ) Use i 2 = -1.
53
-36
Multiply. Write the result in the form a + bi.
5a. 2i(3 - 5i) 5b. (4 - 4i)(6 - i) 5c. (3 + 2i)(3 - 2i)
The imaginary unit i can be raised to higher powers as shown below.
Notice the repeating
pattern in each row
of the table. The
pattern allows you
to express any power
of i as one of four
possible values: i, -1,
-i, or 1.
384
Powers of i
i =i
i5 = i4 · i = 1 · i = i
i9 = i
i 2 = -1
i 6 = i 4 · i 2 = 1 · (-1) = -1
i 10 = -1
i 3 = i 2 · i = -1 · i = -i
i 7 = i 4 · i 3 = 1 · (-i) = -i
i 11 = -i
i 4 = i 2 · i 2 = -1 · (-1) = 1
i8 = i4 · i4 = 1 · 1 = 1
i 12 = 1
1
Chapter 5 Quadratic Functions
EXAMPLE
6
Evaluating Powers of i
A Simplify -3i 12.
-3i 12 = -3(i 2)
6
Rewrite i 12 as a power of i 2.
= -3(-1) 6 = -3(1) = -3
Simplify.
B Simplify i 25.
i 25 = i · i 24
= i · (i 2)
Rewrite as a product of i and an even
power of i.
12
Rewrite i 24 as a power of i 2.
= i · (-1) 12 = i · 1 = i
Simplify.
1 i 7.
6a. Simplify _
2
The complex
conjugate of a
complex number
a + bi is a - bi.
(Lesson 5-5)
EXAMPLE
6b. Simplify i 42.
Recall that expressions in simplest form cannot have square roots in the
denominator (Lesson 1-3). Because the imaginary unit represents a square root,
you must rationalize any denominator that contains an imaginary unit. To do
this, multiply the numerator and denominator by the complex conjugate of the
denominator.
7
Dividing Complex Numbers
A Simplify _.
3 + 7i
8i
3
+
7i
-8i
_
8i
-8i
(_)
-24i - 56i 2
__
-64i 2
-24i + 56
_
64
-3i + 7 _
3i
_
= 7 -_
8 8
8
B Simplify _ .
Multiply by
the conjugate.
Distribute.
Use i 2 = -1.
Simplify.
3 + 8i
7a. Simplify _.
-i
5+i
2 - 4i
5 + i 2 + 4i
_
2 - 4i 2 + 4i
(_)
10
+ 20i + 2i + 4i 2
__
4 + 8i - 8i - 16i 2
10 + 22i - 4
__
4 + 16
6 + 22i _
11 i
_
= 3 +_
10 10
20
3 - i.
7b. Simplify _
2-i
THINK AND DISCUSS
1. Explain when a complex number a + bi and its conjugate are equal.
2. Find the product (a + bi)(c + di), and identify which terms in the
product are real and which are imaginary.
3. GET ORGANIZED Copy and
complete the graphic organizer.
In each box, give an example.
LÃÕÌi
Û>Õi
``}
«iÝ
ÕLiÀÃ
ÕÌ«Þ}
Õ}>ÌiÃ
5- 9 Operations with Complex Numbers
385
5-9
Exercises
KEYWORD: MB7 5-9
KEYWORD: MB7 Parent
GUIDED PRACTICE
1. Vocabulary In the complex number plane, the horizontal axis represents
? numbers, and the vertical axis represents ? numbers. (real, irrational,
−−−
−−−
or imaginary)
SEE EXAMPLE
1
p. 382
SEE EXAMPLE
2
3
p. 383
SEE EXAMPLE 4
5
p. 384
SEE EXAMPLE 6
p. 385
SEE EXAMPLE
7
p. 385
4. 3 + 2i
5. -2 - 3i
Find each absolute value.
6. 4 - 5i
7. -33.3
8. -9i
9. 5 + 12i
10. -1 + i
11. 15i
Add or subtract. Write the result in the form a + bi.
12. (2 + 5i) + (-2 + 5i)
13. (-1 - 8i) + (4 + 3i)
14. (1 - 3i) - (7 + i)
15. (4 - 8i) + (-13 + 23i)
16. (6 + 17i) - (18 - 9i)
17. (-30 + i) - (-2 + 20i)
Find each sum by graphing on the complex plane.
18. (3 + 4i) + (-2 - 4i)
p. 384
SEE EXAMPLE
3. -i
2. 4
p. 383
SEE EXAMPLE
Graph each complex number.
19. (-2 - 5i) + (-1 + 4i)
20. (-4 - 4i) + (4 + 2i)
Multiply. Write the result in the form a + bi.
21. (1 - 2i)(1 + 2i)
22. 3i(5 + 2i)
23. (9 + i)(4 - i)
24. (6 + 8i)(5 - 4i)
2
25. (3 + i)
26. (-4 - 5i)(2 + 10i )
27. -i 9
28. 2i 15
29. i 30
5 - 4i
30. _
i
17
33. _
4+i
11 - 5i
31. _
2 - 4i
45 - 3i
34. _
7 - 8i
8 + 2i
32. _
5+i
-3 - 12i
35. _
6i
Simplify.
PRACTICE AND PROBLEM SOLVING
Independent Practice
For
See
Exercises Example
36–39
40–45
46–51
52–54
55–60
61–63
64–69
1
2
3
4
5
6
7
Extra Practice
Skills Practice p. S13
Graph each complex number.
36. -3
37. -2.5i
38. 1 + i
Find each absolute value.
39. 4 - 3i
 
40. 2 + 3i
41. -18
4i
42. _
5
43. 6 - 8i
44. -0.5i
45. 10 - 4i
Add or subtract. Write the result in the form a + bi.
46. (8 - 9i) - (-2 - i)
47. 4i - (11 - 3i)
48. (4 - 2i) + (-9 - 5i)
49. (13 + 6i) + (15 + 35i)
50. (3 - i) - (-3 + i)
51. -16 + (12 + 9i)
Application Practice p. S36
Find each sum by graphing on the complex plane.
52. (4 + i) + (-3i)
386
Chapter 5 Quadratic Functions
53. (5 + 4i) + (-1 + 2i)
54. (-3 - 3i) + (4 - 3i)
Multiply. Write the result in the form a + bi.
55. -12i (-1 + 4i)
56. (3 - 5i)(2 + 9i)
57. (7 + 2i)(7 - 2i)
2
58. (5 + 6i)
59. (7 - 5i)(-3 + 9i)
60. -4(8 + 12i)
62. -i 11
5 - 2i
65. _
3+i
8 + 4i
_
68.
7+i
63. 5i 10
3
66. _
-1 - 5i
6 + 3i
69. _
2 - 2i
Simplify.
61. i 27
2 - 3i
64. _
i
19 + 9i
_
67.
5+i
Write the complex number represented
by each point on the graph.
>}>ÀÞÊ>ÝÃ
70. A
{
Ó
71. B
{
72. C
ä
Ó
{
,i>
>ÝÃ
Ó
73. D
{
74. E
Find the absolute value of each complex number.
Fractals
Fractals are self-similar,
which means that
smaller parts of a fractal
are similar to the fractal
as a whole. Many objects
in nature, such as the
veins of leaves and snow
crystals, also exhibit selfsimilarity. As a result,
scientists can use fractals
to model these objects.
75. 3 - i
76. 7i
77. -2 - 6i
78. -1 - 8i
3 -_
1i
81. _
2 2
79. 0
80. 5 + 4i
82. 5 - i √
3
83. 2 √
2 - i √
3
84. Fractals Fractals are patterns produced
using complex numbers and the repetition of a
mathematical formula. Substitute the first number
into the formula. Then take the result, put it back
into the formula, and so on. Each complex number
produced by the formula can be used to assign a
color to a pixel on a computer screen. The result is
an image such as the one at right. Many common
fractals are based on the Julia Set, whose formula is
Z n + 1 = (Z n) 2 + c, where c is a constant.
a. Find Z 2 using Z 2 = (Z 1) 2 + 0.25. Let Z 1 = 0.5 + 0.6i.
b. Find Z 3 using Z 3 = (Z 2) 2 + 0.25. Use Z 2 that you obtained in part a.
c. Find Z 4 using Z 4 = (Z 3) 2 + 0.25. Use Z 3 that you obtained in part b.
Simplify. Write the result in the form a + bi.
85. (3.5 + 5.2i) + (6 - 2.3i)
86. 6i - (4 + 5i)
87. (-2.3 + i) - (7.4 - 0.3i)
88. (-8 - 11i) + (-1 + i)
89. i(4 + i)
2
90. (6 - 5i)
2
91. (-2 - 3i)
92. (5 + 7i)(5 - 7i)
93. (2 - i)(2 + i)(2 - i)
94. 3 - i 11
12 + i
97. _
i
1+i
_
100.
-2 + 4i
95. i 52 - i 48
18 - 3i
98. _
i
4
_
101.
2 - 3i
96. i 35 - i 24 + i 18
4 + 2i
99. _
6+i
6
_
102.
√
2-i
5- 9 Operations with Complex Numbers
387
Multi-Step Impedance is a measure of the opposition of
a circuit to an electric current. Electrical engineers find it
convenient to model impedance Z with complex numbers. In a
parallel AC circuit with two impedances Z 1 and Z 2, the equivalent
or total impedance in ohms can be determined by using the
Z 1Z 2
formula Z eq = ______
.
Z +Z
1
��
��
2
��
��
103. Find the equivalent impedance Z eq for Z 1 = 3 + 2i and
Z 2 = 1 - 2i arranged in a parallel AC circuit.
104. Find the equivalent impedance Z eq for Z 1 = 2 + 2i and Z 2 = 4 - i arranged in a
parallel AC circuit.
Tell whether each statement is sometimes, always, or never true. If the statement is
sometimes true, give an example and a counterexample. If the statement is never
true, give a counterexample.
105. The sum of any complex number a + bi and its conjugate is a real number.
106. The difference between any complex number a + bi (b ≠ 0) and its conjugate is a
real number.
107. The product of any complex number a + bi (a ≠ 0) and its conjugate is a positive
real number.
108. The product of any two imaginary numbers bi (b ≠ 0) and di (d ≠ 0) is a positive
real number.
3 are shown. Which is
109. /////ERROR ANALYSIS///// Two attempts to simplify _
2+i
incorrect? Explain the error.
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��
��
� ��
� � ��
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� ��
��
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110. Critical Thinking Why are the absolute value of a complex number and the
absolute value of its conjugate equal? Use a graph to justify your answer.
111. Write About It Discuss how the difference of two squares, a 2 - b 2 =
(a + b)(a - b), relates to the product of a complex number and its conjugate.
112. This problem will prepare you for the Multi-Step Test Prep on page 390.
You have seen how to graph sums of complex numbers
on the complex plane.
a. Find three pairs of complex numbers whose sum
is 4 + 4i.
b. Graph each of the sums on the same complex plane.
c. Describe the results of your graph.
��
��
�
388
Chapter 5 Quadratic Functions
��
��
�
�
��
��
Use the graph for Exercises 113–114.
113. Which point on the graph represents 1 - 2i?
C
A
D
B
>}>ÀÞÊ>ÝÃ
{
Ó
114. What is the value of the complex number
represented in the graph by E?
-2
-2i
ä
{
,i>
>ÝÃ
{
Ó
{
2i
2
Ó
115. Which expression is equivalent to (2 - 5i) - (2 + 5i) ?
4 + 10i
10i
-10i
4 - 10i
116. Which expression is equivalent to (-5 + 3i) ?
16 - 15i
16 - 30i
34 - 15i
34 - 30i
2
CHALLENGE AND EXTEND
117. Consider the powers of i.
a. Complete the table, and look for a pattern.
i1 =
i -1 =
i0 =
i -2 =
i -3 =
i -4 =
i -5 =
b. Explain the pattern that you observed for i raised to negative powers. What
are the only possible values of i raised to a negative integer power?
c. Simplify i -12, i -37, and i -90.
Find the general form of the result for each complex operation.
a + bi
118. (a + bi)(c + di)
119. _
c + di
SPIRAL REVIEW
120. Money The table shows the amount that James spent for lunches each week over
an eight-week period. Make a scatter plot of the data. Sketch a line of best fit, and
find its equation. (Lesson 2-7)
Lunches
Purchased
5
7
3
5
6
2
4
5
Weekly Cost ($)
10
13
8
9
8
5
10
11
Solve each inequality by using algebra. (Lesson 5-7)
121. 0 ≥ 3x 2 - 6x
122. 10 < x 2 - 4x - 11
123. -6 ≥ 2x + 7x - 21
124. 3 - x 2 < 7 - 5x
2
Determine whether each data set could represent a quadratic function. Explain.
(Lesson 5-8)
125.
x
-2
-1
0
1
y
5
-1
-3
-1
126.
x
0
2
4
6
y
18
10
2
-6
5- 9 Operations with Complex Numbers
389
SECTION 5B
Applying Quadratic Functions
Tilted Tiles Mitch and Jacob are making mosaics in an art class. To make
one mosaic, Mitch first divides a wall into a grid made up of squares with a
side length of 20 cm. Then Jacob glues a tile on each square, making sure
that each corner of the tile touches a side of the grid square.
They measure the side length of each tile as well as
the distance x from the upper right corner of the grid
square to a corner of the tile. They find that for each
tile there are two possible values of x, as shown.
1. Complete the table by finding the
area of each tile and the ratio y of
the area of each tile to the area of
the grid square.
2. Make a scatterplot of the ordered
pairs (x, y). Find and graph a
quadratic model for the data. Is the
model a reasonable representation
of the data? Explain.
3. Describe the domain for the
problem situation. Explain why
the domain of the problem situation
is different from the domain of
the model.
4. Use your model to determine the
value of y when x = 3.8. Explain
the meaning of your answer in the
context of the problem.
5. For what values of x does a tile cover
at least 75% of the grid square? Round to
the nearest tenth.
390
Chapter 5
Side Length
of Tile (cm)
x
x
20 cm
Area of
x (cm) Tile (cm 2)
15
6.4
15
13.6
15.5
5.5
15.5
14.5
16
4.7
16
15.3
17
3.3
17
16.7
18
2.1
18
17.9
19
1.1
19
18.9
20
0
y
SECTION 5B
Quiz for Lessons 5-7 Through 5-9
5-7 Solving Quadratic Inequalities
Graph each inequality.
2
1. y > -x 2 + 6x
2. y ≤ -x - x + 2
Solve each inequality by using tables or graphs.
3. x 2 - 4x + 1 > 6
4. 2x 2 + 2x -10 ≤ 2
Solve each inequality by using algebra.
5. x 2 + 4x - 7 ≥ 5
6. x 2 - 8x < 0
7. The function p(r) = -1000r 2 + 6400r - 4400 models the monthly profit p of a small
DVD-rental store, where r is the rental price of a DVD. For what range of rental prices
does the store earn a monthly profit of at least $5000?
5-8 Curve Fitting with Quadratic Models
Determine whether each data set could represent a quadratic function. Explain.
8.
x
5
6
7
8
9
y
13
11
7
1
-7
9.
x
-4
-2
0
2
4
y
10
8
4
8
10
Write a quadratic function that fits each set of points.
10.
(0, 4), (2, 0), and (3, 1)
11.
(1, 3), (2, 5), and (4, 3)
For Exercises 12–14, use the table of maximum load allowances
for various heights of spruce columns.
Maximum Load Allowance
No. 1 Common Spruce
12. Find a quadratic regression equation to model the maximum
load given the height.
13. Use your model to predict the maximum load allowed for a
6.5 ft spruce column.
Height of
Column (ft)
Maximum
Load (lb)
4
7280
5
7100
6
6650
7
5960
14. Use your model to predict the maximum load allowed for an
8 ft spruce column.
5-9 Operations with Complex Numbers
Find each absolute value.
15. -6i
16. 3 + 4i
17. 2 - i
Perform each indicated operation, and write the result in the form a + bi.
18. (3 - 5i) - (6 - i)
19. (-6 + 4i) + (7 - 2i)
20. 3i (4 + i)
21. (3 + i)(5 - i)
22. (1 - 4i)(1 + 4i)
2 - 7i
24. _
-i
23. 3i 15
3-i
25. _
4 - 2i
Ready to Go On?
391
Vocabulary
absolute value of a complex
number . . . . . . . . . . . . . . . . . 382
imaginary number . . . . . . . . . . 350
quadratic model . . . . . . . . . . . . 376
imaginary part . . . . . . . . . . . . . 351
quadratic regression . . . . . . . . 376
axis of symmetry . . . . . . . . . . . 323
imaginary unit . . . . . . . . . . . . . 350
real part . . . . . . . . . . . . . . . . . . . 351
binomial . . . . . . . . . . . . . . . . . . . 336
maximum value . . . . . . . . . . . . 326
root of an equation . . . . . . . . . 334
completing the square . . . . . . 342
minimum value . . . . . . . . . . . . 326
standard form . . . . . . . . . . . . . . 324
complex conjugate . . . . . . . . . . 352
parabola . . . . . . . . . . . . . . . . . . . 315
trinomial. . . . . . . . . . . . . . . . . . . 336
complex number . . . . . . . . . . . 351
quadratic function . . . . . . . . . . 315
vertex form . . . . . . . . . . . . . . . . . 318
complex plane . . . . . . . . . . . . . 382
quadratic inequality in two
variables . . . . . . . . . . . . . . . . 366
vertex of a parabola . . . . . . . . . 318
discriminant . . . . . . . . . . . . . . . 357
zero of a function . . . . . . . . . . . 333
Complete the sentences below with vocabulary words from the list above.
1. The number 5i can be classified as both a(n)
?
and a
?
.
−−−−−−
−−−−−−
2. The value of the input x that makes the output f (x) equal zero is called the
3. The
?
.
−−−−−−
is the point at which the parabola intersects the axis of symmetry.
?
−−−−−−
4. The type and number of solutions to a quadratic equation can be determined by
finding the
?
.
−−−−−−
5. When a parabola opens upward, the y-value of the vertex is the
?
of a
−−−−−−
quadratic function.
5-1 Using Transformations to Graph Quadratic Functions (pp. 315–322)
EXERCISES
EXAMPLES
■
2
Using the graph of f (x) = x as a guide,
describe the transformations, and then graph
1 2
g (x) = __
x + 3.
Þ
2
n
v
}
È
g (x) = __12 x 2 + 3 is f
vertically compressed
by a factor of __12 and
translated 3 units up.
■
{
ä]ÊÎ®
Ý
Ó
ä
Ó
Use the description to write a quadratic
function in vertex form. The function
f (x) = x 2 is translated 1 unit right to create g.
translation 1 unit right:
h=1
2
g (x) = a(x - h) + k → g (x) = (x - 1)2
392
Chapter 5 Quadratic Functions
Graph each function by using a table.
1 x 2 + 3x - 4
6. f (x) = -x 2 - 2x
7. f (x) = _
2
Using the graph of f (x) = x 2 as a guide, describe the
transformations, and then graph each function.
8. g (x) = 4(x - 2)2
9. g (x) = -2(x + 1)2
1 x2 - 3
11. g (x) = -(x + 2)2 + 6
10. g (x) = _
3
Use the description to write each quadratic function
in vertex form.
12. f (x) = x 2 is reflected across the x-axis and
translated 3 units down to create g.
13. f (x) = x 2 is vertically stretched by a factor of 2 and
translated 4 units right to create g.
14. f (x) = x 2 is vertically compressed by a factor of __14
and translated 1 unit left to create g.
5-2 Properties of Quadratic Functions in Standard Form (pp. 323–330)
EXERCISES
EXAMPLE
■
For f (x) = -x 2 + 2x + 3, (a) determine
whether the graph opens upward or
downward, (b) find the axis of symmetry,
(c) find the vertex, (d) find the y-intercept,
and (e) graph the function.
e.
a. Because a < 0, the
parabola opens
downward.
� ��
�
��
b. axis of symmetry:
b = -_
2 =1
x = -_
2a
2(-1)
c. f (1) = -1 2 + 2(1) + 3 = 4
The vertex is (1, 4).
��
�
�
�
��
For each function, (a) determine whether the graph
opens upward or downward, (b) find the axis of
symmetry, (c) find the vertex, (d) find the y-intercept,
and (e) graph the function.
15. f (x) = x 2 - 4x + 3
16. g (x) = x 2 + 2x + 3
1 x 2 - 2x + 4
17. h(x) = x 2 - 3x
18. j(x) = _
2
Find the minimum or maximum value of each
function.
19. f (x) = x 2 + 2x + 6
20. g (x) = 6x - 2x 2
21. f (x) = x 2 - 5x + 1
22. g (x) = -2x 2 - 8x + 10
23. f (x) = -x 2 - 4x + 8
24. g (x) = 3x 2 + 7
d. Because c = 3, the y-intercept is 3.
5-3 Solving Quadratic Equations by Graphing and Factoring (pp. 333–340)
EXERCISES
EXAMPLES
■
■
Find the roots of x 2 + x = 30 by factoring.
x 2 + x - 30 = 0 Rewrite in standard form.
(x - 5)(x + 6) = 0 Factor.
x - 5 = 0 or x + 6 = 0 Zero Product Property.
Solve each equation.
x = 5 or x = -6
Write a quadratic function with zeros
8 and -8.
Write zeros as solutions.
x = 8 or x = -8
x - 8 = 0 or x + 8 = 0 Set equations equal to 0.
(x - 8)(x + 8) = 0 Converse Zero Product
Property
f (x) = x 2 - 64
Replace 0 with f (x).
Find the roots of each equation by factoring.
25. x 2 - 7x - 8 = 0
26. x 2 - 5x + 6 = 0
27. x 2 = 144
28. x 2 - 21x = 0
29. 4x 2 - 16x + 16 = 0
30. 2x 2 + 8x + 6 = 0
31. x 2 + 14x = 32
32. 9x 2 + 6x + 1 = 0
Write a quadratic function in standard form for each
given set of zeros.
33. 2 and -3
34. 1 and -1
35. 4 and 5
36. -2 and -3
37. -5 and -5
38. 9 and 0
5-4 Completing the Square (pp. 342–349)
EXERCISES
EXAMPLE
■
Solve x 2 - 8x = 12 by completing the square.
2
x - 8x + = 12 +
x 2 - 8x + 16 = 12 + 16
(x - 4)2 = 28
x - 4 = ± √
28
x = 4 ± 2 √
7
Set up equation.
Solve each equation by completing the square.
39. x 2 - 16x + 48 = 0
40. x 2 + 20x + 84 = 0
b 2
.
Add __
2
41. x 2 - 6x = 16
()
Factor.
Take square roots.
Solve for x.
42. x 2 - 14x = 13
Write each function in vertex form, and identify
its vertex.
44. g(x) = x 2 + 2x - 7
43. f (x) = x 2 - 4x + 9
Study Guide: Review
393
5-5 Complex Numbers and Roots (pp. 350–355)
EXERCISES
EXAMPLE
■
Solve x 2 - 22x + 133 = 0.
x 2 - 22x +
= -133 +
x 2 - 22x + 121 = -133 + 121
(x - 11 )2 = -12
-12
x - 11 = ± √
x = 11 ± 2i √
3
Rewrite.
b 2
Add __
.
2
()
Factor.
Take square
roots.
Solve.
Solve each equation.
45. x 2 = -81
46. 6x 2 + 150 = 0
47. x 2 + 6x + 10 = 0
48. x 2 + 12x + 45 = 0
49. x 2 - 14x + 75 = 0
50. x 2 - 22x + 133 = 0
Find each complex conjugate.

51. 5i - 4
52. 3 + i √5
5-6 The Quadratic Formula (pp. 356–363)
EXERCISES
EXAMPLES
■
Find the zeros of f (x) = 3x 2 - 5x + 3 by using
the Quadratic Formula.
2

- 4ac
-b ± √b
Quadratic
x = __
2a
Find the zeros of each function by using the
Quadratic Formula.
( -5 )2 - 4( 3 )( 3 )
-(-5 ) ± √
x = ___
2(3 )
55. f (x) = 2x 2 - 10x + 18
Formula
√11

5 ± √
-11
5 ±i_
=_=_
6
6
6
■
Substitute.
53. f (x) = x 2 - 3x - 8
54. h(x) = (x - 5)2 + 12
56. g(x) = x 2 + 3x + 3
Simplify.
Find the type and number of solutions for
x 2 + 9x + 20 = 0.
b 2 - 4ac = 9 2 - 4(1)(20)
= 81 - 80 = 1
There are two distinct real roots because the
discriminant is positive.
57. h(x) = x 2 - 5x + 10
Find the type and number of solutions for each
equation.
58. 2x 2 - 16x + 32 = 0
59. x 2 - 6x = -5
60. x2 + 3x + 8 = 0
61. x 2 - 246x = -144
62. x 2 + 5x = -12
63. 3x 2 - 5x + 3 = 0
5-7 Solving Quadratic Inequalities (pp. 366–373)
EXAMPLE
■
Solve x 2 - 4x - 9 ≥ 3 by using algebra.
Graph each inequality.
Write and solve the related equation.
Write in standard form.
x 2 - 4x - 12 = 0
Factor.
(x + 2)(x - 6) = 0
Solve.
x = -2 or x = 6
64. y > x 2 + 3x + 4
The critical values are -2 and 6. These values
divide the number line into three intervals:
x ≤ -2, -2 ≤ x ≤ 6, and x ≥ 6.
Testing an x-value in each interval gives the
solution of x ≤ -2 or x ≥ 6.
394
EXERCISES
Chapter 5 Quadratic Functions
65. y ≤ 2x 2 - x - 5
Solve each inequality by using tables or graphs.
66. x 2 + 2x - 4 ≥ -1
67. -x 2 - 5x > 4
Solve each inequality by using algebra.
68. -x 2 + 6x < 5
69. 3x 2 - 25 ≤ 2
70. x 2 - 3 < 0
71. 3x 2 + 4x - 3 ≤ 1
5-8 Curve Fitting with Quadratic Models (pp. 374–381)
EXERCISES
EXAMPLE
■
Find a quadratic model for the wattage of
fluorescent bulbs F given the comparable
incandescent bulb wattage I. Use the model
to estimate the wattage of a fluorescent bulb
that produces the same amount of light as a
120-watt incandescent bulb.
Write a quadratic function that fits each set of points.
72. (-1, 8), (0, 6), and (1, 2)
73. (0, 0), (1, -1), and (2, -6)
Construction For Exercises 74–77, use the table of
copper wire gauges.
Wattage Comparison
Incandescent (watts)
40
60
75
90
100
Fluorescent (watts)
11
15
20
23
28
Enter the data into two lists in a graphing
calculator. Use the quadratic regression feature.
Common U.S. Copper Wire Gauges
Resistance per
1000 ft (ohms)
Gauge
Diameter (in.)
24
0.0201
25.67
22
0.0254
16.14
20
0.0320
10.15
18
0.0403
6.385
74. Find a quadratic regression equation to model the
diameter given the wire gauge.
75. Use your model to predict the diameter for a
12-gauge copper wire.
The model is F(I ) ≈ 0.0016I 2 + 0.0481I + 6.48.
A 36-watt fluorescent bulb produces about
the same amount of light as a 120-watt
incandescent bulb.
76. Find a quadratic regression equation to model the
resistance given the wire gauge.
77. Use your model to predict the resistance for a
26-gauge copper wire.
5-9 Operations with Complex Numbers (pp. 382–389)
EXERCISES
EXAMPLES
Perform each indicated operation, and write the
result in the form a + bi.
■
⎜-2 + 4i⎟
(-2)2 + 4 2 = √
4 + 16 =
√
■
√
20 = 2 √
5
(3 + 2i)(4 - 5i)
2
12 - 15i + 8i - 10i
12 - 7i - 10(-1 ) = 22 - 7i
■
-5 + 3i
_
1 - 2i
80. ⎜12 - 16i⎟
81. ⎜7i⎟
82. (1 + 5i) + (6 - i)
83. (9 + 4i) - (3 + 2i)
84. (5 - i) - (11 - i)
85. -5i (3 - 4i)
86. (5 - 2i)(6 + 8i)
87. (3 + 2i)(3 - 2i)
88. (4 + i)(1 - 5i)
89. (-7 + 4i)(3 + 9i)
32
-5 + 3i _
-5 - 7i + 6i
_
( 1 + 2i ) = __
2
1 - 2i
Perform each indicated operation, and write the
result in the form a + bi.
78. ⎜-3i⎟
79. ⎜4 - 2i⎟
1 + 2i
1 - 4i 2
7i
-11 - 7i = -_
11 - _
=_
5
5
1+4
90. i
2 + 9i
92. _
-2i
8
_
94. - 4i
1+i
91. -5i 21
5 + 2i
93. _
3 - 4i
-12 + 26i
_
95.
2 + 4i
Study Guide: Review
395
Using the graph of f (x) = x 2 as a guide, describe the transformations, and then
graph each function.
1 x2 + 2
1. g (x) = (x + 1)2 - 2
2. h (x) = -_
2
3. Use the following description to write a quadratic function in vertex form: f(x) = x 2
is vertically compressed by a factor of __12 and translated 6 units right to create g.
For each function, (a) determine whether the graph opens upward or downward,
(b) find the axis of symmetry, (c) find the vertex, (d) find the y-intercept, and
(e) graph the function.
4. f (x) = -x 2 + 4x + 1
5. g (x) = x 2 - 2x + 3
6. The area A of a rectangle with a perimeter of 32 cm is modeled by the function
A(x) = -x 2 + 16x, where x is the width of the rectangle in centimeters. What is the
maximum area of the rectangle?
Find the roots of each equation by using factoring.
7. x 2 - 2x + 1 = 0
8. x 2 + 10x = -21
Solve each equation.
9. x 2 + 4x = 12
10. x 2 - 12x = 25
11. x 2 + 25 = 0
12. x 2 + 12x = -40
Write each function in vertex form, and identify its vertex.
13. f (x) = x 2 - 4x + 9
14. g (x) = x 2 - 18x + 92
Find the zeros of each function by using the Quadratic Formula.
15. f (x) = (x - 1)2 + 7
16. g (x) = 2x 2 - x + 5
17. The height h in feet of a person on a waterslide is modeled by the function h (t) = -0.025t 2 - 0.5t + 50,
where t is the time in seconds. At the bottom of the slide, the person lands in a swimming pool. To the
nearest tenth of a second, how long does the ride last?
18. Graph the inequality y < x 2 - 3x - 4.
Solve each inequality.
19. -x 2 + 3x + 5 ≥ 7
20. x 2 - 4x + 1 > 1
For Exercises 21 and 22, use the table showing the average
cost of LCD televisions at one store.
21. Find a quadratic model for the cost of a television given
its size.
Costs of LCD Televisions
Size (in.)
15
17
23
30
Cost ($)
550
700
1500
2500
22. Use the model to estimate the cost of a 42 in. LCD television.
Perform the indicated operation, and write the result in the form a + bi.
23. (12 - i) - (5 + 2i)
396
24. (6 - 2i)(2 - 2i)
Chapter 5 Quadratic Functions
25. -2i 18
1 - 8i
26. _
4i
FOCUS ON SAT MATHEMATICS SUBJECT TESTS
The SAT Mathematics Subject Tests assess knowledge
from course work rather than ability to learn. The Level 1
test is meant to be taken by students who have completed
two years of algebra and one year of geometry, and it tests
more elementary topics than the Level 2 test.
You will need to use a calculator for some
of the problems on the SAT Mathematics
Subject Tests. Before test day, make sure
that you are familiar with the features of
the calculator that you will be using.
You may want to time yourself as you take this practice test. It should
take you about 8 minutes to complete.
1. For what value of c will 3x 2 - 2x + c = 0 have
exactly one distinct real root?
2
(A) -_
3
4. What is the solution set of y 2 - 2y ≤ 3y + 14?
(A) y ≥ -2
(B) y ≤ 7
(C) y ≤ -2 or y ≥ 7
1
(B) -_
3
(C) 0
(D) -7 ≤ y ≤ 2
(E) -2 ≤ y ≤ 7
1
(D) _
3
2
(E) _
3
5. Which of the following is a factor of
(a - 1)2 - b 2 ?
2. If m and n are real numbers, i 2 = -1, and
(m - n) - 4i = 7 + ni, what is the value of m?
(A) a + b - 1
(B) a - b
(A) -4
(C) a - 1
(B) -3
(D) a - b + 1
(C) 1
(E) 1 - b
(D) 3
(E) 4
6. If z = 5 - 4i and i 2 = -1, what is z ?
(A) 1
3. If x 2 - 5x + 6 = (x - h) + k, what is the value
of k?
25
(A) -_
4
2
5
(B) -_
2
(B) 3
(C) 9
(D) √41
(E) √42
1
(C) -_
4
(D) 0
(E) 6
College Entrance Exam Practice
397
Multiple Choice: Work Backward
When taking a multiple-choice test, you can sometimes work backward to determine
which answer is correct. Because this method can be time consuming, it is best used
only when you cannot solve a problem in any other way.
Which expression is equivalent to 2x 2 - 3x - 14?
(2x + 7)(x + 2)
(2x - 7)(x + 2)
(2x - 7)(x - 2)
(2x + 7)(x - 2)
If you have trouble factoring the quadratic expression given in the question, you
can multiply the binomials in the answer choices to find the product that is the same
as 2x 2 - 3x - 14.
Try Choice A: (2x + 7)(x + 2) = 2x 2 + 11x + 14
Try Choice B: (2x - 7)(x - 2) = 2x 2 - 11x + 14
Try Choice C: (2x - 7)(x + 2) = 2x 2 - 3x - 14
Choice C is the answer.
Note: Trying choice D can help you check your work.
What is the solution set of x 2 - 36 < 0?
x < -6 or x > 6
-36 < x < 36
-6 < x < 6
x < -36 or x > 36
If you have trouble determining the solution set, substitute values of x into the
inequality. Based on whether the values make the inequality true or false, you may be
able to eliminate one or more of the answer choices.
Substitute 0 for x: x 2 - 36 < 0 → (0)2 - 36
0 → -36 < 0 ✔
When x = 0, the inequality is true. Therefore, the solution set must include x = 0.
Because choices F and J do not include x = 0, they can be eliminated.
Substitute 10 for x: x 2 - 36 < 0 → (10)2 - 36
0 → 64
0 ✘
When x = 10, the inequality is false. Therefore, the solution set does not include
x = 10. Because choice H includes x = 10, it can be eliminated.
The only remaining choice is choice G. Therefore, choice G must be correct.
398
Chapter 5 Quadratic Functions
You can also work backward to check whether
the answer you found by another method is
correct or reasonable.
Read each test item, and answer the questions
that follow.
Item A
What are the zeros of the function
g (x) = 6x 2 - 8x - 4, rounded to the nearest
hundredth?
-10.32 and 2.32
1.72 and -0.39
-1.72 and 0.39
10.32 and -2.32
1. Rachel cannot remember how to determine
the zeros of a quadratic function, so she
plans to pick one of the answer choices at
random. What could Rachel do to make a
more educated guess?
2. Describe how to find the correct answer by
working backward.
Item B
A portable television has a screen with a
diagonal of 4 inches. The length of the screen
is 1 inch greater than its width. What are
the dimensions of the screen to the nearest
hundredth?
Item C
Which of the following is a solution of
(x + 4)2 = 25?
x = -9
x=0
x = -1
x=9
5. Explain how to use substitution to determine
the correct answer.
6. Check whether choice A is correct by
working backward. Explain your findings.
What should you do next?
Item D
The height h of a golf ball in feet t seconds
after it is hit into the air is modeled by
h(t) = -16t 2 + 64t. How long is the ball in
the air?
2 seconds
12 seconds
4 seconds
16 seconds
7. The measurements given in the answer
choices represent possible values of which
variable in the function?
8. Describe how you can work backward to
determine that choice F is not correct.
Item E
Ü
{
ÜÊ Ê£
The base of a triangle is 4 in. longer than twice
its height. If the triangle has an area of 24 in 2,
what is its height?
1.28 inches by 2.28 inches
1.28 inches by 3.28 inches
LÊÓ Ê Ê{
2.28 inches by 2.28 inches
2.28 inches by 3.28 inches
3. Can any of the answer choices be eliminated
immediately? If so, which choices and why?
4. Describe how you can determine the correct
answer by using the Pythagorean Theorem
and working backward.
2 in.
6 in.
4 in.
8 in.
9. What equation do you need to solve to find
the value of h?
10. Try choice A by working backward. Explain
your findings. What should you do next?
Test Tackler
399
KEYWORD: MB7 TestPrep
CUMULATIVE ASSESSMENT, CHAPTERS 1– 5
Multiple Choice
5. Which graph represents the function
1 (x - 3) - 4?
f(x) = -_
2
⎡-1 8 2⎤
N=⎢
⎣ 0 1 6⎦
⎡6 -2⎤
1. M = ⎢
⎣3 7⎦
8
What is the matrix product 2MN?
⎡ -24 184
0⎤
⎢
⎣ -12 124 192 ⎦
-8
⎡ -12 92 0⎤
⎢
⎣ -6 62 96⎦
⎡-24 -12⎤
-4
4
8
4
8
y
x
-4
⎡-12 -6 ⎤
92 62
⎣ 0 96⎦
0
(1, -6)
(-3, -4)
-8
2. Which of these functions does NOT have zeros
at -1 and 4?
4
(-5, 0)
f(x) = x 2 - 3x - 4
-8
f(x) = 2x + 6x - 8
2
-4
y
x
0
4
8
-4
f(x) = -x 2 + 3x + 4
f(x) = 2x 2 - 6x - 8
(5, -5)
-8
3. Dawn and Julia are running on a jogging trail.
Dawn starts running 5 minutes after Julia does. If
Julia runs at an average speed of 8 ft/s and Dawn
runs at an average speed of 9 ft/s, how many
minutes after Dawn starts running will she catch
up with Julia?
5 minutes
40 minutes
27 minutes
45 minutes
4. Which equation has intercepts at (20, 0, 0),
(0, 40, 0), and (0, 0, 5)?
(7, 2)
0
4
124
0 192⎦
⎢
4
-4
⎢ 184
⎣
(-1, 6)
y
8
y
(5, 0) x
(-1, 3)
-8
-4
0
4
-4
6. What is the equation of the function graphed
below?
y
20x + 40y + 5z = 0
(-1, 3)
20x + 40y + 5z = 1
2
4x + 8y + z = 5
2x + y + 8z = 40
400
Chapter 5 Quadratic Functions
x
-6
0
-4
(-3, -1) -2
2
y = (x - 3)2 - 1
y = (x - 1)2 - 3
y = (x + 3)2 - 1
y = (x + 1)2 - 3
7. If the relationship between x and y is quadratic,
which value of y completes the table?
x
-3
-1
y
21
7
1
3
5
27
61
Short Response
⎧-4x + 8y - 2z = 8
14. ⎨ 4x - 4y + 2z = -5
⎩ x + 4y - 2z = 15
a. Write the augmented matrix that could be used
3
9
7
17
to solve the system of equations given above.
b. Find the solution of the system, and explain
5(6 - 8i )
8. Which is equivalent to the expression _ ?
-20 + 10i
15 - 40i
15 - 8i
20 - 10i
2-i
how you determined your answer.
15. The graph below shows a feasible region for a set
of constraints.
8
9. What is the inverse of the following matrix?
⎡-2 -4⎤
⎢
⎣ 4 2⎦
y
6
4
⎡ _
1⎤
1 _
3
6
1
_
_
- -1
6⎦
⎣ 3
⎡ _
1⎤
- 1 -_
6 3
1
1 _
_
⎣ 3 6⎦
⎢
⎢
⎡ _
1⎤
- 1 -_
2 4
1
1 _
_
⎣ 4 2⎦
⎢
2
x
0
2
4
6
8
a. Write the constraints for the feasible region.
b. Maximize the objective function P = 3x - 4y
⎡ 2 4⎤
⎢
⎣-4 -2⎦
under these constraints.
In nearly all standardized tests, you cannot enter
a negative value as the answer to a griddedresponse question. If you get a negative value as
an answer to one of these questions, you have
probably made a mistake in your calculations.
Gridded Response
16. Consider the function f(x) = x 2 - 2x - 48.
a. Determine the roots of the function. Show
your work.
b. The function f is translated to produce the
function g. The vertex of g is the point (3, 30).
Write the function rule for g in vertex form,
and explain how you determined your answer.
Extended Response
10. What value of x makes the equation
17. A small alteration store charges $15.00 per hour
x 2 + 64 = 16x true?
11. The table shows the fees that are charged at an
airport parking lot for various lengths of time.
What is the slope of the linear function that
models the parking fee f in dollars for h number
of hours?
plus a $12.50 consulting fee for alterations. A
competing store charges $20.00 per hour but does
not charge a consulting fee.
a. For each store, write a linear function c
that can be used to find the total cost of an
alteration that takes h hours.
b. For which values of h is the small alteration
Time (h)
Parking Fee ($)
1
3
5
7
3.35
5.05
6.75
8.45
12. What is the x-value of the vertex of
f(x) = 2x - 15x + 5?
2
13. What is the value of c given that the following
store less expensive than the competing store?
Explain how you determined your answer.
c. The small store wants to adjust its pricing so
that it is less expensive than the competing
store for any alteration job that takes an hour
or more. By how much should the small store
lower its consulting fee in order to make this
adjustment?
system is dependent?
⎧ 2y - x + 10 = 0
⎨
⎩ 3x - 6y - c = 16
Cumulative Assessment, Chapters 1–5
401

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