Quadratic and Exponential Functions
Then
Now
Why?
In Chapter 8, you solved quadratic equations by factoring and by using the Square Root Property.
In Chapter 9, you will:
FINANCE The value of a certain company’s stock can be modeled by the function f (x ) = x 2  12x + 75. By graphing this quadratic function, we can make an educated guess as to how the stock will perform in the near future.
Solve quadratic equations by graphing, completing the square, and using the Quadratic Formula. Graph exponential functions. Identify geometric sequences.
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Vocabulary
eGlossary
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Virtual Manipulatives
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Foldables
SelfCheck Practice
Worksheets
Tennessee Curriculum Standards CLE 3102.3.8
Get Ready for the Chapter Diagnose Readiness
1

You have two options for checking prerequisite skills.
Textbook Option Take the Quick Check below. Refer to the Quick Review for help.
QuickCheck
QuickReview Example 1
Use a table of values to graph each equation. (Lesson 31) 1. y = x + 3
2. y = 2x + 2
3. y = 2x  3
4. y = 0.5x  1
x
y = 3x + 1
y
5. 4x  3y = 12
6. 3y = 6 + 9x
1
3(1) + 1
2
0
3(0) + 1
1
1
3(1) + 1
4
2
3(2) + 1
7
Use a table of values to graph y = 3x + 1.
7. SAVINGS Jack has $100 to buy a game system. He plans to save $10 each week. Graph an equation to show the total amount T Jack will have in w weeks.
y
y = 3x + 1 x
Example 2
Determine whether each trinomial is a perfect square trinomial. Write yes or no. If so, factor it. (Lesson 86) 2
8. a + 12a + 36
9. x + 5x + 25
10. x 2  12x + 32
11. x 2 + 20x + 100
2
Determine whether x 2  10x + 25 is a perfect square trinomial. Write yes or no. If so, factor it.
2
1. Is the first term a perfect square? yes
2
12. 4x + 28x + 49
13. k  16k + 64
2. Is the last term a perfect square? yes
14. a 2  22a + 121
15. 5t 2  12t + 25
3. Is the middle term equal to 2(1x)(5)? yes x 2  10x + 25 = (x  5) 2
Example 3
Find the next three terms of each arithmetic sequence. (Lessons 35)
16. 16, 4, 8, 20, …
17. 2, 10, 18, 26, …
18. 5, 2, 1, 4, …
19. 3, 5, 7, 9, …
Find the next three terms of the arithmetic sequence 5, 9, 13, 17, … . Find the common difference by subtracting a term from the next term. 95=4
20. GEOMETRY Write a formula that can be used to find the perimeter of a figure containing n squares.
Add to find the next three terms. 17 + 4 = 21, 21 + 4 = 25, 25 + 4 = 29 P=4
2
P=6
P=8
The next three terms are 21, 25, 29.
Online Option Take an online selfcheck Chapter Readiness Quiz at connectED.mcgrawhill.com. 523
Get Started on the Chapter You will learn several new concepts, skills, and vocabulary terms as you study Chapter 9. To get ready, identify important terms and organize your resources. You may wish to refer to Chapter 0 to review prerequisite skills.
StudyOrganizer Quadratic and Exponential Functions Make this Foldable to help you organize your Chapter 9 notes about quadratic functions. Begin with a sheet of notebook paper.
1
2
3
Fold the sheet of paper along the length so that the edge of the paper aligns with the margin rule on the paper.
Fold the sheet twice widthwise to form four sections.
Unfold the sheet, and cut along the folds on the front flap only.
NewVocabulary English
Español
axis of symmetry
p. 525
eje de simetría
maximum
p. 525
máximo
minimum
p. 525
mínimo
parabola
p. 525
parábola
quadratic function
p. 525
función cuadrática
vertex
p. 525
vértice
double root
p. 538
doble raíz
transformation
p. 544
transformación
completing the square
p. 552
completar el cuadrado
Quadratic Formula
p. 558
Formula cuadrática
discriminant
p. 561
discriminante
exponential function
p. 567
función exponencial
compound interest
p. 574
interés compuesto
common ratio
p. 578
proporción común
geometric sequence
p. 578
secuencia geométrica
ReviewVocabulary domain p. 38 dominio all the possible values of the independent variable, x
4
Label each section as shown.
leading coefficient p. 425 coeficiente delantero the coefficient of the first term of a polynomial written in standard form range p. 38 rango all the possible values of the dependent variable, y In the function represented by the table, the domain is {0, 2, 4, 6}, and the range is {3, 5, 7, 9}.
524  Chapter 9  Quadratic and Exponential Functions
x
y
0
3
2
5
4
7
6
9
Graphing Quadratic Functions Then
Now
Why?
You graphed linear functions.
1
Analyze the characteristics of graphs of quadratic functions.
2
Graph quadratic functions.
The Innovention Fountain in Epcot’s Futureworld in Orlando, Florida, is an elaborate display of water, light, and music. The sprayers shoot water in shapes that can be modeled by quadratic equations. You can use graphs of these equations to show the path of the water.
(Lesson 32)
NewVocabulary quadratic function standard form parabola axis of symmetry vertex minimum maximum symmetry
1 Characteristics of Quadratic Functions
Quadratic functions are nonlinear and can be written in the form f(x) = ax 2 + bx + c, where a ≠ 0. This form is called the standard form of a quadratic function. The shape of the graph of a quadratic function is called a parabola. Parabolas are symmetric about a central line called the axis of symmetry. The axis of symmetry intersects a parabola at only one point, called the vertex.
KeyConcept Quadratic Functions
Tennessee Curriculum Standards CLE 3102.3.6 Understand and use relations and functions in various representations to solve contextual problems. ✔ 3102.3.19 Explore the characteristics of graphs of various nonlinear relations and functions including inverse variation, quadratic, and square root function. Use technology where appropriate. SPI 3102.3.11 Analyze nonlinear graphs including quadratic and exponential functions that model a contextual situation.
Parent Function:
f(x ) = x 2
Standard Form:
f(x ) = ax 2 + bx + c
Type of Graph:
parabola
Axis of Symmetry:
b x = _
yintercept:
c
f(x) axis of symmetry 0
x
2a
vertex
When a > 0, the graph of y = ax 2 + bx + c opens upward. The lowest point on the graph is the minimum. When a < 0, the graph opens downward. The highest point is the maximum. The maximum or minimum is the vertex.
Example 1 Graph a Parabola Use a table of values to graph y = 3x 2 + 6x  4. State the domain and range. y Graph the ordered pairs, and x y 8 connect them to create a smooth 6 1 5 (−3, 5) (1, 5) curve. The parabola extends to 4 0 4 2 infinity. The domain is all real numbers. The range is {yy ≥ 7}, 1 7 −8−6−4 0 2 4 6 8 x because 7 is the minimum. (−2, −4) (0, −4) 2 4 3
5
(−1, −7)
GuidedPractice 1. Use a table of values to graph y = x 2 + 3. State the domain and range.
connectED.mcgrawhill.com
525
y 8 6 4 2
Figures that possess symmetry are those in which each half of the figure matches exactly. A parabola is symmetric about the axis of symmetry. Every point on the parabola to the left of the axis of symmetry has a corresponding point on the other half.
−8−6−4
0
x = −1 axis of symmetry
y = x 2 + 2x  5
2 4 6 8x
−8
(−1, −6) vertex
When identifying characteristics from a graph, it is often easiest to locate the vertex first. It is either the maximum or minimum point of the graph.
Example 2 Identify Characteristics from Graphs Find the vertex, the equation of the axis of symmetry, and the yintercept of each graph. y
a.
Step 1 Find the vertex. Because the parabola opens upward, the vertex is located at the minimum point of the parabola. It is located at (1, 0). Step 2 Find the axis of symmetry. The axis of symmetry is the line that goes through the vertex and divides the parabola into congruent halves. It is located at x = 1.
x
0
Step 3 Find the yintercept. The yintercept is the point where the graph intersects the yaxis. It is located at (0, 1), so the yintercept is 1.
y
b.
Step 1 Find the vertex. The parabola opens downward, so the vertex is located at its maximum point, (2, 3). x
0
Step 2 Find the axis of symmetry. The axis of symmetry is located at x = 2. Step 3 Find the yintercept. The yintercept is where the parabola crosses the yaxis. It is located at (0, 1), so the yintercept is 1.
GuidedPractice y
2A.
0
y
2B.
x
0
526  Lesson 91  Graphing Quadratic Functions
x
StudyTip Function Characteristics When identifying characteristics of a function, it is often easiest to locate the axis of symmetry first.
Example 3 Identify Characteristics from Functions Find the vertex, the equation of the axis of symmetry, and the yintercept of each function. a. y = 2x 2 + 4x  3 b x = _
2a _ x= 4 2·2
x = 1
Formula for the equation of the axis of symmetry a = 2 and b = 4 Simplify.
The equation for the axis of symmetry is x = 1. To find the vertex, use the value you found for the axis of symmetry as the xcoordinate of the vertex. To find the ycoordinate, substitute that value for x in the original equation. Original equation y = 2x 2 + 4x 3 2 = 2(1) + 4(1)  3 x = 1 = 5 Simplify. The vertex is at (1, 5). The yintercept always occurs at (0, c). So, the yintercept is 3. b. y = x 2 + 6x + 4 b x = _
Formula for the equation of the axis of symmetry
2a _ x= 6 2(1)
a = 1 and b = 6
x=3
StudyTip yintercept The ycoordinate of the yintercept is also the constant term (c) of the quadratic function in standard form.
Simplify.
The equation of the axis of symmetry is x = 3. y = x 2 + 6x + 4 = (3) 2 + 6(3) + 4 = 13
Original equation x=3 Simplify.
The vertex is at (3, 13). The yintercept is 4.
GuidedPractice 3A. y = 3x 2 + 6x  5
3B. y = 2x 2 + 2x + 2
There are general differences between linear functions and quadratic functions. Linear Functions
Quadratic Functions 2
Standard Form
y = ax + b
y = ax + bx + c; a ≠ 0
Degree
1; Notice that all of the variables are to the ﬁrst power.
2; Notice that the independent variable, x, is squared in the ﬁrst term. The coefﬁcient a can not equal 0, or the equation would be linear.
Example
y = 2x + 6
y = 3x 2 + 5x  4
Graph
line
parabola
Next you will learn how to identify whether the parabola opens up or down and whether the vertex is a maximum or a minimum point. connectED.mcgrawhill.com
527
KeyConcept Maximum and Minimum Values Words
The graph of f(x) = ax 2 + bx + c, where a ≠ 0: • opens upward and has a minimum value when a > 0, and • opens downward and has a maximum value when a < 0. • The range of a quadratic function is all real numbers greater than or equal to the minimum, or all real numbers less than or equal to the maximum.
Examples
a is positive.
a is negative. f(x)
f(x)
maximum minimum
0
x
x
0
Example 4 Maximum and Minimum Values Consider f(x) = 2x 2  4x + 6. a. Determine whether the function has a maximum or minimum value.
WatchOut! Minimum and Maximum Values Don’t forget to find both coordinates of the vertex (x, y). The minimum or maximum value is the ycoordinate.
For f(x) = 2x 2  4x + 6, a = 2, b = 4, and c = 6. Because a is negative the graph opens down, so the function has a maximum value. b. State the maximum or minimum value of the function. The maximum value is the ycoordinate of the vertex. b 4 or _ or 1. The xcoordinate of the vertex is _ 2a
f(x) = 2x 2  4x + 6
ReviewVocabulary Domain and Range The domain is the set of all of the possible values of the independent variable x. The range is the set of all of the possible values of the dependent variable y.
2(2)
Original function
f(1) = 2(1) 2  4(1) + 6
x = 1
f(1) = 8
Simplify.
The maximum value is 8. c. State the domain and range of the function. The domain is all real numbers. The range is all real numbers less than or equal to the maximum value, or {yy ≤ 8}.
GuidedPractice Consider g(x) = 2x 2  4x  1. 4A. Determine whether the function has a maximum or minimum value. 4B. State the maximum or minimum value. 4C. State the domain and range of the function.
528  Lesson 91  Graphing Quadratic Functions
2 Graph Quadratic Functions
You have learned how to find several important characteristics of quadratic functions.
KeyConcept Graph Quadratic Functions Step 1 Find the equation of the axis of symmetry. Step 2 Find the vertex, and determine whether it is a maximum or minimum. Step 3 Find the yintercept. Step 4 Use symmetry to find additional points on the graph, if necessary. Step 5 Connect the points with a smooth curve.
StudyTip Symmetry and Points When locating points that are on opposite sides of the axis of symmetry, they are not only the same distance to the left and right of the axis of symmetry. They are also the same number of spaces up or down from the vertex.
Example 5 Graph Quadratic Functions Graph f(x) = x 2 + 4x + 3. Step 1 Find the equation of the axis of symmetry. b x=_
2a _ x = 4 2·1
x = 2
Formula for the equation of the axis of symmetry a = 1 and b = 4 Simplify.
Step 2 Find the vertex, and determine whether it is a maximum or minimum. f(x) = x 2 + 4x + 3
Original equation
= (2) 2 + 4(2) + 3
x = 2
= 1
Simplify.
The vertex lies at (2, 1). Because a is positive the graph opens up, and the vertex is a minimum. Step 3 Find the yintercept. f(x) = x 2 + 4x + 3
Original equation
= (0) 2 + 4(0) + 3
x=0
=3
Simplify.
The yintercept is 3. Step 4 The axis of symmetry divides the parabola into two equal parts. So if there is a point on one side, there is a corresponding point on the other side that is the same distance from the axis of symmetry and has the same yvalue. Step 5 Connect the points with a smooth curve.
f(x) 2 units
2 units 0
1 unit
x 1 unit
GuidedPractice Graph each function. 5A. f(x) = 2x 2 + 2x  1
5B. f(x) = 3x 2  6x + 2 connectED.mcgrawhill.com
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You have used what you know about quadratic functions, parabolas, and symmetry to create graphs. You can analyze these graphs to solve realworld problems.
RealWorld Example 6 Use a Graph of a Quadratic Function SCHOOL SPIRIT The cheerleaders at Lake High School launch Tshirts into the crowd every time the Lakers score a touchdown. The height of the Tshirt can be modeled by the function h(x) = 16x 2 + 48x + 6, where h(x) represents the height in feet of the Tshirt after x seconds. a. Graph the function. b x = _
Equation of the axis of symmetry
2a 3 _ x =  48 or _ 2 2(16)
a = 16 and b = 48
3 The equation of the axis of symmetry is x = _ . Thus, the xcoordinate for 2
3 the vertex is _ . 2
2
y = 16x + 48x + 6
RealWorldLink About 1 in 17 high school seniors playing football will go on to play football at an NCAA school. Source: National Collegiate Athletic Association
Original equation
(_32 ) + 48(_32 ) + 6 9 3 = 16(_ + 48(_ +6 2) 4)
(_32 ) = _94
= 36 + 72 + 6 or 42
Simplify.
= 16
2
_
x= 3 2
2
40 36 32 28 24 20 16 12 8 4
3 The vertex is at _ , 42 . 2
(
)
Let’s find another point. Choose an xvalue of 0 and substitute. Our new point is at (0, 6). The point paired with it on the other side of the axis of symmetry is (3, 6). Repeat this and choose an xvalue of 1 to get (1, 38) and its corresponding point (2, 38). Connect these points and create a smooth curve.
−2
y
0
4
x
b. At what height was the Tshirt launched? The Tshirt is launched when time equals 0, or at the yintercept. So, the Tshirt was launched 6 feet from the ground. c. What is the maximum height of the Tshirt? When was the maximum height reached? The maximum height of the Tshirt occurs at the vertex. 3 So the Tshirt reaches a maximum height of 42 feet. The time was _ 2 or 1.5 seconds after launch.
GuidedPractice 6. TRACK Emilio is competing in the javelin throw. The height of the javelin can be modeled by the equation y = 16x 2 + 64x + 6, where y represents the height in feet of the javelin after x seconds. A. Graph the path of the javelin. B. At what height is the javelin thrown? C. What is the maximum height of the javelin?
530  Lesson 91  Graphing Quadratic Functions
Check Your Understanding Example 1
Example 2
= StepbyStep Solutions begin on page R12.
Use a table of values to graph each equation. State the domain and range. 1. y = 2x 2 + 4x  6
2. y = x 2 + 2x  1
3. y = x 2  6x  3
4. y = 3x 2  6x  5
Find the vertex, the equation of the axis of symmetry, and the yintercept of each graph. y
5.
y
6.
0
7.
4 −8
0
x
x
0
y
y
8. 4
8x
−4 −8
0
x
−12
Example 3
Find the vertex, the equation of the axis of symmetry, and the yintercept of the graph of each function. 9. y = 3x 2 + 6x  1 11. y = x 2  4x + 5
Example 4
10. y = x 2 + 2x + 1 12. y = 4x 2  8x + 9
Consider each function. a. Determine whether the function has maximum or minimum value. b. State the maximum or minimum value. c. What are the domain and range of the function?
Example 5
Example 6
13 y = x 2 + 4x  3
14. y = x 2  2x + 2
15. y = 3x 2 + 6x + 3
16. y = 2x 2 + 8x  6
Graph each function. 17. f(x) = 3x 2 + 6x + 3
18. f(x) = 2x 2 + 4x + 1
19. f(x) = 2x 2  8x  4
20. f(x) = 3x 2  6x  1
21. JUGGLING A juggler is tossing a ball into the air. The height of the ball in feet can be modeled by the equation y = 16x 2 + 16x + 5, where y represents the height of the ball at x seconds. a. Graph this equation. b. At what height is the ball thrown? c. What is the maximum height of the ball? connectED.mcgrawhill.com
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Practice and Problem Solving Example 1
Example 2
Extra Practice begins on page 815.
Use a table of values to graph each equation. State the domain and range. 22. y = x 2 + 4x + 6
23. y = 2x 2 + 4x + 7
24. y = 2x 2  8x  5
25. y = 3x 2 + 12x + 5
26. y = 3x 2  6x  2
27. y = x 2  2x  1
Find the vertex, the equation of the axis of symmetry, and the yintercept of each graph. y
28. 0
x
0
x
y
30.
y
29.
y
31.
x
0
x
0
y
32.
0
Example 3
Example 4
y
33.
x
0
x
Find the vertex, the equation of the axis of symmetry, and the yintercept of each function. 34. y = x 2 + 8x + 10
35 y = 2x 2 + 12x + 10
36. y = 3x 2  6x + 7
37. y = x 2  6x  5
38. y = 5x 2 + 20x + 10
39. y = 7x 2  28x + 14
40. y = 2x 2  12x + 6
41. y = 3x 2 + 6x  18
42. y = x 2 + 10x  13
Consider each function. a. Determine whether the function has a maximum or minimum value. b. State the maximum or minimum value. c. What are the domain and range of the function?
Example 5
43. y = 2x 2  8x + 1
44. y = x 2 + 4x  5
45. y = 3x 2 + 18x  21
46. y = 2x 2  16x + 18
47. y = x 2  14x  16
48. y = 4x 2 + 40x + 44
49. y = x 2  6x  5
50. y = 2x 2 + 4x + 6
51. y = 3x 2  12x  9
52. y = 3x 2 + 6x  4
53. y = 2x 2  4x  3
54. y = 2x 2  8x + 2
55. y = x 2 + 6x  6
56. y = x 2  2x + 2
57. y = 3x 2  12x + 5
Graph each function.
532  Lesson 91  Graphing Quadratic Functions
Example 6
58. BOATING Miranda has her boat docked on the west side of Casper Point. She is boating over to the Casper Marina. The distance traveled by Miranda over time can be modeled by the equation d = 16t 2 + 66t, where d is the number of feet she travels in t minutes.
E
a. Graph this equation. b. What is the maximum number of feet north that she traveled?
U
c. How long did it take her to reach Casper Marina?
B
GRAPHING CALCULATOR Graph each equation. Use the TRACE feature to find the vertex on the graph. Round to the nearest thousandth if necessary. 59. y = 4x 2 + 10x + 6
60. y = 8x 2  8x + 8
61. y = 5x 2  3x  8
62. y = 7x 2 + 12x  10
63. GOLF The average amateur golfer can hit a ball with an initial velocity of 31.3 meters per second. If the ball is hit straight up, the height can be modeled by the equation h = 4.9t 2 + 31.3t, where h is the height of the ball, in meters, after t seconds. a. Graph this equation. b. At what height is the ball hit? c. What is the maximum height of the ball? d. How long did it take for the ball to hit the ground? e. State a reasonable range and domain for this situation. 64. FUNDRAISING The marching band is selling poinsettias to buy new uniforms. Last year the band charged $5 each, and they sold 150. They want to increase the price this year, and they expect to lose 10 sales for each $1 increase. The sales revenue R, in dollars, generated by selling the poinsettias is predicted by the function R = (5 + p)(150  10p), where p is the number of $1 price increases. a. Write the function in standard form. b. Find the maximum value of the function. c. At what price should the poinsettias be sold to generate the most sales revenue? Explain your reasoning. 65 FOOTBALL A football is kicked up from ground level at an initial upward velocity of 90 feet per second. The equation h = 16t 2 + 90t gives the height h of the football after t seconds. a. What is the height of the ball after one second? b. When is the ball 126 feet high? c. When is the height of the ball 0 feet? What do these points represent in the context of the situation?
C
66. REASONING Let f(x) = x 2  9. a. What is the domain of f(x)? b. What is the range of f(x)? c. For what values of x is f(x) negative? d. When x is a real number, what are the domain and range of f(x) = √ x2  9? connectED.mcgrawhill.com
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67
MULTIPLE REPRESENTATIONS In this problem, you will investigate solving quadratic equations using tables. a. Algebraic Determine the related function for each equation. Copy and complete the table below. Equation
Related Function
Zeros
yValues
2
x  x = 12 x 2 + 8x = 9 x 2 = 14x  24 x 2 + 16x = 28
b. Graphical Graph each related function with a graphing calculator. c. Analytical The number of zeros is equal to the degree of the related function. Use the table feature on your calculator to determine the zeros of each related function. Record the zeros in the table above. Also record the values of the function one unit less than and one unit more than each zero. d. Verbal Compare the signs of the function values for xvalues just before and just after a zero. What happens to the sign of the function value before and after a zero?
H.O.T. Problems
Use HigherOrder Thinking Skills
68. OPEN ENDED Write a quadratic function for which the graph has an axis of 3 symmetry of x = _ . Summarize your steps. 8
69. ERROR ANALYSIS Chase and Jade are finding the axis of symmetry of a parabola. Is either of them correct? Explain your reasoning.
Chase
Jade
2
y = x – 4x + 6 b x = _ 2a
y = x 2 – 4x + 6 b x=_ 2a
x= 4
x =  4
x=2
x = 2
_ 2(1)
_ 2(1)
70. CHALLENGE Using the axis of symmetry and one xintercept, write an equation for the graph shown. 71. REASONING The graph of a quadratic function has a vertex at (2, 0). One point on the graph is (5, 9). Find another point on the graph. Explain how you found it. 72. OPEN ENDED Describe a realworld situation that involves a quadratic equation. Explain what the vertex represents.
y
(3, 25)
20 (0, 16) 4 (−2, 0) −8 −4 0
(8, 0) 4
−8
73. REASONING Provide a counterexample to the following statement. The vertex of a parabola is always the minimum of the graph. 74. WRITING IN MATH Explain how to find the axis of symmetry from an equation for a quadratic function. What other characteristics of the graph can you derive from the equation? Explain.
534  Lesson 91  Graphing Quadratic Functions
x
SPI 3102.3.8, SPI 3102.1.5, SPI 3102.3.7
Standardized Test Practice 75. Which of the following is an equation for the line that passes through (2, 5) and is perpendicular to 2x + 4y = 8? A y = 2x + 10
C y = 2x  9
1 x4 B y = _ 2
D y = 2x  1
F 1296π units G 144π units 2
1 ? f(x) = 4x 2  _ 2
⎧ 1⎫ A ⎨all integers less than or equal to _ ⎬ 2 ⎭ ⎩ B {all nonnegative integers} C {all real numbers}
⎧
⎫
⎩
2⎭
1 D ⎨all real numbers less than or equal to _ ⎬
76. GEOMETRY The area of the circle is 36π square units. If the radius is doubled, what is the area of the new circle? 2
77. What is the range of the function
r
A = 36π
H 72π units
2
J 9π units 2
78. SHORT RESPONSE Dylan delivers newspapers for extra money. He starts delivering the newspapers at 3:15 p.m. and finishes at 5:05 p.m. How long does it take Dylan to complete his route?
Spiral Review Determine whether each trinomial is a perfect square trinomial. Write yes or no. If so, factor it. (Lesson 86) 79. 4x 2 + 4x + 1
80. 4x 2  20x + 25
81. 9x 2 + 8x + 16
Factor each polynomial if possible. If the polynomial cannot be factored, write prime. (Lesson 85) 82. n 2  16
83. x 2 + 25
84. 9  4a 2
86. (c  6)(c  5)
87. (2x  1)(x + 9)
Find each product. (Lesson 77) 85. (b  7)(b + 3)
88. MULTIPLE BIRTHS The number of quadruplet births Q in the United States in recent years can be modeled by Q = 0.5t 3 + 11.7t 2 21.5t + 218.6, where t represents the number of years since 1992. For what values of t does this model no longer allow for realistic predictions? Explain your reasoning. (Lesson 74) Use elimination to solve each system of equations. (Lesson 64) 89. 2x + y = 5 3x  2y = 4
90. 4x  3y = 12 x + 2y = 14
91. 2x  3y = 2 5x + 4y = 28
92. HEALTH About 20% of the time you sleep is spent in rapid eye movement (REM), which is associated with dreaming. If an adult sleeps 7 to 8 hours, how much time is spent in REM sleep? (Lesson 54)
Skills Review Find the xintercept of the graph of each equation. (Lesson 31) 93. x + 2y = 10
94. 2x  3y = 12
95. 3x  y = 18 connectED.mcgrawhill.com
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Algebra Lab
Rate of Change of a Quadratic Function OBJECTIVE Investigate the rate of change for a quadratic function. A model rocket is launched from the ground with an upward velocity of 144 feet per second. The function y = 16x 2 + 144x models the height y of the rocket in feet after x seconds. Using this function, we can investigate the rate of change of a quadratic function.
Activity Step 1 Copy the table below. x
0
y
0
Rate of Change
–
0.5
1.0
1.5
…
9.0
Step 2 Find the value of y for each value of x from 0 through 9. Step 3 Graph the ordered pairs (x, y) on grid paper. Connect the points with a smooth curve. Notice that the function increases when 0 < x < 4.5 and decreases when 4.5 < x < 9. Step 4 Recall that the rate of change is the change in y divided by the change in x. Find the rate of change for each half second interval of x and y.
300 275 250 225 200 175 150 125 100 75 50 25 0
y
1 2 3 4 5 6 7 8 9 x
Exercises Use the quadratic function y = x 2. 1. Make a table, similar to the one in the Activity, for the function using x = 4, 3, 2, 1, 0, 1, 2, 3, and 4. Find the values of y for each xvalue. 2. Graph the ordered pairs on grid paper. Connect the points with a smooth curve. Describe where the function is increasing and where it is decreasing. 3. Find the rate of change for each column starting with x = 3. Compare the rates of change when the function is increasing and when it is decreasing. 4. CHALLENGE If an object is dropped from 100 feet in the air and air resistance is ignored, the object will fall at a rate that can be modeled by the equation f(x) = 16x 2 + 100, where f(x) represents the object’s height in feet after x seconds. Make a table like that in Exercise 1, selecting appropriate values for x. Fill in the xvalues, the yvalues, and rates of change. Compare the rates of change. Describe any patterns that you see.
536  Extend 91  Algebra Lab: Rate of Change of a Quadratic Function
Solving Quadratic Equations by Graphing Then
Now
Why?
You solved quadratic equations by factoring.
1
Solve quadratic equations by graphing.
2
Estimate solutions of quadratic equations by graphing.
Dorton Arena at the state fairgrounds in Raleigh, North Carolina, has a shape created by two intersecting parabolas. The shape of one of the parabolas can be modeled by the equation y = x 2 + 127x, where x represents the width of the parabola in feet, and y represents the length of the parabola in feet. The xintercepts of the graph of this function can be used to determine the distance between the points where the parabola meets the ground.
(Lesson 83)
NewVocabulary double root
Tennessee Curriculum Standards CLE 3102.3.8 Solve and understand solutions of quadratic equations with real roots. ✔ 3102.3.32 Recognize the connection among factors, solutions (roots), zeros of related functions, and xintercepts in equations that arise from quadratic functions. SPI 3102.3.10 Find the solution of a quadratic equation and/or zeros of a quadratic function. Also addresses ✓3102.3.30.
1 Solve by Graphing
A quadratic equation can be written in the standard form ax 2 + bc + c = 0, where a ≠ 0. To write a quadratic function as an equation, replace y or f(x) with 0. Recall that the solutions or roots of an equation can be identified by finding the xintercepts of the related graph. Quadratic equations may have two, one, or no solutions.
KeyConcept Solutions of Quadratic Equations y
y 0
y
x
x
0
two unique real solutions
x
0
one unique real solution
no real solutions
Example 1 Two Roots Solve x 2  2x  8 = 0 by graphing.
8
Graph the related function f(x) = x 2  2x  8. The xintercepts of the graph appear to be at 2 and 4, so the solutions are 2 and 4. CHECK Check each solution in the original equation. x 2  2x  8 = 0 Original equation 2 (2)  2(2)  8 0 x = 2 or x = 4 Simplify. 0=0
y
4 0
2
x
x 2  2x  8 = 0 (4)  2(4)  8 0 0=0 2
GuidedPractice Solve each equation by graphing. 1A. x 2  3x + 18 = 0
1B. x 2  4x + 3 = 0 connectED.mcgrawhill.com
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The solutions in Example 1 were two distinct numbers. Sometimes the two roots are the same number, called a double root.
Example 2 Double Root Solve x 2  6x = 9 by graphing. Step 1 Rewrite the equation in standard form. x 2  6x = 9 x  6x + 9 = 0 2
Original equation Add 9 to each side. y
Step 2 Graph the related function f(x) = x 2  6x + 9. Step 3 Locate the xintercepts of the graph. Notice that the vertex of the parabola is the only xintercept. Therefore, there is only one solution, 3.
WatchOut! Exact Solutions Solutions found from the graph of an equation may appear to be exact. Check them in the original equation to be sure.
CHECK Solve by factoring. x 2  6x + 9 = 0 (x  3)(x  3) = 0 x  3 = 0 or x  3 = 0 x=3 x=3 The only solution is 3.
Original equation
x
0
Factor. Zero Product Property Add 3 to each side.
GuidedPractice Solve each equation by graphing. 2A. x 2 + 25 = 10x
2B. x 2 = 8x  16
Sometimes the roots are not real numbers.
Example 3 No Real Roots Solve 2x 2  3x + 5 = 0 by graphing. y
Step 1 Rewrite the equation in standard form. This equation is written in standard form. Step 2 Graph the related function f(x) = 2x 2  3x + 5. Step 3 Locate the xintercepts of the graph. This graph has no xintercepts. Therefore, this equation has no real number solutions. The solution set is ∅.
0
CHECK Solve by factoring. There are no factors of 10 that have a sum of 3, so the expression is not factorable. Thus, the equation has no real number solutions.
GuidedPractice Solve each equation by graphing. 3A. x 2  3x = 5
538  Lesson 92  Solving Quadratic Equations by Graphing
3B. 2x 2  8 = 6x
x
2 Estimate Solutions
The real roots found thus far have been integers. However, the roots of quadratic equations are usually not integers. In these cases, use estimation to approximate the roots of the equation.
Example 4 Approximate Roots with a Table
StudyTip Location of Zeros Since quadratic functions are continuous, there must be a zero between two xvalues for which the corresponding yvalues have opposite signs.
Solve x 2 + 6x + 6 = 0 by graphing. If integral roots cannot be found, estimate the roots to the nearest tenth.
y
Graph the related function f(x) = x 2 + 6x + 6.
x
0
The xintercepts are located between 5 and 4 and between 2 and 1. Make a table using an increment of 0.1 for the xvalues located between 5 and 4 and between 2 and 1. Look for a change in the signs of the function values. The function value that is closest to zero is the best approximation for a zero of the function. x
4.9
4.8
4.7
4.6
4.5
4.4
4.3
4.2
4.1
y
0.61
0.24
0.11
0.44
0.75
1.04
1.31
1.56
1.79
x
1.9
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
y
1.79
1.56
1.31
1.04
0.75
0.44
0.11
0.24
0.61
For each table, the function value that is closest to zero when the sign changes is 0.11. Thus, the roots are approximately 4.7 and 1.3.
GuidedPractice 4. Solve 2x 2 + 6x  3 = 0 by graphing. If integral roots cannot be found, estimate the roots to the nearest tenth.
Approximating the xintercepts of graphs is helpful for realworld applications.
RealWorld Example 5 Approximate Roots with a Calculator SOCCER A goalie kicks a soccer ball with an upward velocity of 65 feet per second, and her foot meets the ball 1 foot off the ground. The quadratic function h = 16t 2 + 65t + 1 represents the height of the ball h in feet after t seconds. Approximately how long is the ball in the air?
RealWorldCareer The game of soccer, called “football” outside of North America, began in 1863 in Britain when the Football Association was founded. Soccer is played on every continent of the world. Source: Sports Know How
You need to find the roots of the equation 16t 2 + 65t + 1 = 0. Use a graphing calculator to graph the related function f(x) = 16t 2 + 65t + 1.
[4, 7] scl: 1 by [10, 70] scl: 10
The positive xintercept of the graph is approximately 4. Therefore, the ball is in the air for approximately 4 seconds.
GuidedPractice 5. If the goalie kicks the soccer ball with an upward velocity of 55 feet per second and his foot meets the ball 2 feet off the ground, approximately how long is the ball in the air?
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Check Your Understanding
= StepbyStep Solutions begin on page R12.
Examples 1–3 Solve each equation by graphing.
Example 4
Example 5
1. x 2 + 3x  10 = 0
2. 2x 2  8x = 0
3. x 2 + 4x = 4
4. x 2 + 12 = 8x
Solve each equation by graphing. If integral roots cannot be found, estimate the roots to the nearest tenth. 5. x 2  5x + 1 = 0
6. 9 = x 2
7. x 2 = 25
8. x 2  8x = 9
9. SCIENCE FAIR Ricky built a model rocket. Its flight can be modeled by the equation shown, where h is the height of the rocket in feet after t seconds. About how long was Ricky’s rocket in the air?
BVODI WFMPDJUZ GUT I=U+U
Practice and Problem Solving
Extra Practice begins on page 815.
Examples 1–3 Solve each equation by graphing.
Example 4
Example 5
B
10. x 2 + 7x + 14 = 0
11 x 2 + 2x  24 = 0
12. x 2  16x + 64 = 0
13. x 2  5x + 12 = 0
14. x 2 + 14x = 49
15. x 2 = 2x  1
16. x 2  10x = 16
17. 2x 2  8x = 13
18. 2x  16x = 30
19. 2x 2 = 24x  72
20. 3x 2 + 2x = 15
21. x 2 = 2x + 80
2
Solve each equation by graphing. If integral roots cannot be found, estimate the roots to the nearest tenth. 22. x 2 + 2x  9 = 0
23. x 2  4x = 20
24. x 2 + 3x = 18
25. 2x 2  9x = 8
26. 3x 2 = 2x + 7
27. 5x = 25  x 2
28. SOFTBALL Sofia hits a softball straight up. The equation h = 16t 2 + 90t models the height h, in feet, of the ball after t seconds. How long is the ball in the air? 29. RIDES A skyrocket roller coaster takes riders straight up and then returns straight down. The equation h = 16t 2 + 185t models the height h, in feet, of the coaster after t seconds. How long is it until the coaster returns to the bottom? Use factoring to determine how many times the graph of each function intersects the xaxis. Identify each zero. 30. y = x 2  8x + 16
31. y = x 2 + 3x + 4
32. y = x 2 + 2x  24
33. y = x 2 + 12x + 32
34. NUMBER THEORY Use a quadratic equation to find two numbers that have a sum of 9 and a product of 20. 35. NUMBER THEORY Use a quadratic equation to find two numbers that have a sum of 1 and a product of 12. 36. GOLF The height of a golf ball in the air can be modeled by the equation h = 16t 2 + 60t + 3, where h is the height in feet of the ball after t seconds. a. How long was the ball in the air? b. What is the ball’s maximum height? c. When will the ball reach its maximum height?
540  Lesson 92  Solving Quadratic Equations by Graphing
C
37 SNOWBOARDING Stefanie is in a snowboarding competition. The equation h = 16t 2 + 30t + 10 models Stefanie’s height h, in feet, in the air after t seconds. a. How long is Stefanie in the air? b. When will Stefanie reach a height of 15 feet? c. To earn bonus points in the competition, you must reach a height of 20 feet. Will Stefanie earn bonus points? 38.
MULTIPLE REPRESENTATIONS In this problem, you will explore how to further interpret the relationship between quadratic functions and graphs. a. Graphical Graph y = x 2. b. Analytical Name the vertex and two other points on the graph. c. Graphical Graph y = x 2 + 2, y = x 2 + 4, and y = x 2 + 6 on the same coordinate plane as the previous graph. d. Analytical Name the vertex and two points from each of these graphs that have the same xcoordinates as the first graph. e. Analytical What conclusion can you draw from this?
GRAPHING CALCULATOR Find the factors of each polynomial. Solve by graphing. 39. f(x) = x 3  3x 2  6x + 8
H.O.T. Problems
40. g(x) = x 3  8x 2 + 15x
Use HigherOrder Thinking Skills y
41. ERROR ANALYSIS Iku and Zachary are finding the number of real zeros of the function graphed at the right. Iku says that the function has no real zeros because there are no xintercepts. Zachary says that the function has one real zero because the graph has a yintercept. Is either of them correct? Explain your reasoning. 42. OPEN ENDED Describe a realworld situation in which a thrown object travels in the air. Write an equation that models the height of the object with respect to time, and determine how long the object travels in the air.
0
43. REASONING The graph shown is that of a quadratic inequality. Analyze the graph, and determine whether the yvalue of a solution of the inequality is sometimes, always, or never greater than 2. Explain.
y
0
x
x
44. CHALLENGE Write a quadratic equation that has the roots described. a. one double root b. one rational (nonintegral) root and one integral root c. two distinct integral roots that are additive opposites. 45. CHALLENGE Find the roots of x 2 = 2.25 without using a calculator. Explain your strategy. 46. WRITING IN MATH Explain how to approximate the roots of a quadratic equation when the roots are not integers. connectED.mcgrawhill.com
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SPI 3102.1.3, SPI 3102.2.3, SPI 3102.4.3, SPI 3102.1.4
Standardized Test Practice 49. EXTENDED RESPONSE Two boats leave a dock. One boat travels 4 miles east and then 5 miles north. The second boat travels 12 miles south and 9 miles west. Draw a diagram that represents the paths traveled by the boats. How far apart are the boats in miles?
47. Adrahan earned 50 out of 80 points on a test. What percentage did Adrahan score on the test? A 62.5% B 16%
C 6.25% D 1.6%
48. Ernesto needs to loosen a bolt. He needs a
1 2 50. The formula s = _ at represents the distance s 2 in meters that a freefalling object will fall near a planet or the Moon in a given time t in seconds. Solve the formula for a, the acceleration due to gravity.
7 inch wrench, wrench that is smaller than a _ 8 3 _ but larger than a inch wrench. Which of the 4
following sizes should Ernesto use? 3 F _ inch
8 5 inch G _ 8
13 H _ inch
1 2 A a=_ t s
16 15 J _ inch 16
2
B a = 2s  t
2
1 2 C a=s_ t 2
2s D a=_ 2 t
Spiral Review Write the equation of the axis of symmetry, and find the coordinates of the vertex of the graph of each function. Identify the vertex as a maximum or minimum. Then graph the function. (Lesson 91) 51. y = 3x 2
52. y = 4x 2  5
53. y = x 2 + 4x  7
54. y = x 2  6x  8
55. y = 3x 2 + 2x + 1
56. y = 4x 2  8x + 5
Solve each equation. Check the solutions. (Lesson 86) 57. 2x 2 = 32
58. (x  4)2 = 25
59. 4x 2  4x + 1 = 16
60. 2x 2 + 16x = 32
61. (x + 3)2 = 5
62. 4x 2  12x = 9
Find each sum or difference. (Lesson 75) 63. (3n 2  3) + (4 + 4n 2)
64. (2d 2  7d  3)  (4d 2 + 7)
65. (2b 3  4b 2 + 4)  (3b 4 + 5b 2  9)
66. (8  4h 2 + 6h 4) + (5h 2  3 + 2h 3)
67. GEOMETRY Supplementary angles are two angles with measures that have a sum of 180°. For the supplementary angles in the figure, the measure of the larger angle is 24° greater than the measure of the smaller angle. Write and solve a system of equations to find these measures. (Lesson 65)
Z Y
Write an equation in pointslope form for the line that passes through each point with the given slope. (Lesson 43) 69. (3, 6), m = 7
1 70. (1, 2), m = _
71. y = x 2 + 5
72. y = x 2  8
73. y = 2x 2  7
74. y = x 2 + 2
75. y = 0.5x 2  3
76. y = (x)2 + 1
68. (2, 5), m = 3
2
Skills Review Graph each function. (Lesson 91)
542  Lesson 92  Solving Quadratic Equations by Graphing
Graphing Technology Lab
Quadratic Inequalities Recall that the graph of a linear inequality consists of the boundary and the shaded half plane. The solution set of the inequality lies in the shaded region of the graph. Graphing quadratic inequalities is similar to graphing linear inequalities.
Activity 1 Shade Inside a Parabola Graph y ≥ x 2  5x + 4 in the standard viewing window. First, clear all functions from the Y= list. To graph y ≥ x 2  5x + 4, enter the equation in the Y= list. until shading Then use the left arrow to select =. Press above the line is selected. 5
KEYSTROKES:
4
[10, 10] scl: 1 by [10, 10] scl: 1
6
All ordered pairs for which y is greater than or equal to x 2  5x + 4 lie above or on the line and are solutions.
A similar procedure will be used to graph an inequality in which the shading is outside of the parabola.
Activity 2 Shade Outside a Parabola Graph y  4 ≤ x 2  5x in the standard viewing window. First, clear the graph that is displayed. KEYSTROKES:
Then rewrite y  4 ≤ x 2  5x as y ≤ x 2  5x + 4, and graph it. [10, 10] scl: 1 by [10, 10] scl: 1
KEYSTROKES:
5
4
All ordered pairs for which y is less than or equal to x 2  5x + 4 lie below or on the line and are solutions.
Exercises 1. Compare and contrast the two graphs shown above. 2. Graph y  2x + 6 ≥ 5x 2 in the standard viewing window. Name three solutions of the inequality. 3. Graph y  6x ≤ x 2  3 in the standard viewing window. Name three solutions of the inequality. connectED.mcgrawhill.com
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Transformations of Quadratic Functions
y
Then
Now
Why?
You graphed quadratic functions by using the vertex and axis of symmetry.
1 2
The graphs of the parabolas shown at the right are the same size and shape, but notice that the vertex of the red parabola is higher on the yaxis than the vertex of the blue parabola. Shifting a parabola up and down is an example of a transformation.
(Lesson 91)
NewVocabulary transformation translation dilation reflection
Apply translations to quadratic functions. Apply dilations and reflections to quadratic functions.
0
1 Translations
A transformation changes the position or size of a figure. One transformation, a translation, moves a figure up, down, left, or right. When a constant c is added to or subtracted from the parent function, the graph of the resulting function f(x) ± c is the graph of the parent function translated up or down.
The parent function of the family of quadratics is f(x) = x 2. All other quadratic functions have graphs that are transformations of the graph of f(x) = x 2. Tennessee Curriculum Standards ✔ 3102.1.14 Apply graphical transformations that occur when changes are made to coefficients and constants in functions. SPI 3102.1.5 Recognize and express the effect of changing constants and/or coefficients in problem solving. SPI 3102.3.11 Analyze nonlinear graphs including quadratic and exponential functions that model a contextual situation. Also addresses ✓3102.3.17.
KeyConcept Vertical Translations The graph of f(x) = x 2 + c is the graph of f(x) = x 2 translated vertically. If c > 0, the graph of f(x) = x is translated ⎪c⎥ units up.
y c>0 c=0
2
x
O
If c < 0, the graph of f(x) = x 2 is translated ⎪c⎥ units down.
c<0
Example 1 Describe and Graph Translations Describe how the graph of each function is related to the graph of f(x) = x 2. a. h(x) = x 2 + 3 b. g(x) = x 2  4 The value of c is 3, and 3 > 0. The value of c is 4, and 4 < 0. Therefore, the graph of Therefore, the graph of y = x 2  4 2 is a translation of the graph of y = x + 3 is a translation of the graph of y = x 2 up 3 units. y = x 2 down 4 units. y
h(x)
y
O
f(x)
f(x)
x
g(x) O
x
GuidedPractice 1A. f(x) = x 2  7
544  Lesson 93
1B. g(x) = 5 + x 2
1C. h(x) = 5 + x 2 1D. f(x) = x 2 + 1
x
2 Dilations and Reflections
Another type of transformation is a dilation. A dilation makes the graph narrower than the parent graph or wider than the parent graph. When the parent function f(x) = x 2 is multiplied by a constant a, the graph of the resulting function f(x) = ax 2 is either stretched or compressed vertically.
KeyConcept Dilations The graph of g(x) = ax 2 is the graph of f(x) = x 2 stretched or compressed vertically.
f(x)
If ⎪a⎥ > 1, the graph of f(x) = x 2 is stretched vertically. If 0 < ⎪a⎥ < 1, the graph of f(x) = x 2 is compressed vertically.
x
O
StudyTip
Example 2 Describe and Graph Dilations
Compress or Stretch When the graph of a quadratic function is stretched vertically, the shape of the graph is narrower than that of the parent function. When it is compressed vertically, the graph is wider than the parent function.
Describe how the graph of each function is related to the graph of f(x) = x 2.
_
a. h(x) = 1 x 2
b. g(x) = 3x 2 + 2
2
The function can be written 1 . Since h(x) = ax 2, where a = _ 2
1 1 2 < 1, the graph of y = _ x is a 0<_ 2
2
dilation of the graph of y = x 2 that is compressed vertically.
The function g(x) = ax 2 + c, where a = 3 and c = 2. Since 2 > 0 and 3 > 1, the graph of y = 3x 2 + 2 translates the graph y = x 2 up 2 units and stretches it vertically. y
h(x)
h(x)
f(x) = x O
g(x)
2
2
f(x) = x
x
x
O
GuidedPractice 2A. j(x) = 2x 2
StudyTip Reflection A reflection of f(x) = x 2 across the yaxis results in the same function, because f(x) = (x)2 = x 2.
2B. h(x) = 5x 2  2
1 2 2C. g(x) = _ x +2 3
A reflection flips a figure across a line. When f(x) = x 2 or the variable x is multiplied by 1, the graph is reflected across the x or yaxis.
KeyConcept Reflections f(x)
The graph of f(x) is the reflection of the graph of f(x) = x 2 across the xaxis. The graph of f(x) is the reflection of the graph of f(x) = x 2 across the yaxis.
f(x)
f(x) 0
x f(x)
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WatchOut! Transformations The graph of f(x) = ax 2 can result in two transformations of the graph of f(x) = x 2: a reflection across the xaxis if a > 0 and either a compression or expansion depending on the absolute value of a.
Example 3 Describe and Graph Reflections Describe how the graph of g(x) = 2x 2  3 is related to the graph of f(x) = x 2. y
Three separate transformations are occurring. The negative sign of the coefficient of x 2 causes a reflection across the xaxis. Then a dilation occurs and finally a translation down 3 units.
2
y=x
x
0
2
2
So the graph of y = 2x  3 is reflected across the xaxis, compressed, and translated down 3 units.
y = 2x  3
GuidedPractice Describe how the graph of each function is related to the graph of f(x) = x 2. 1 2 3B. g(x) = _ x +3
3A. h(x) = 2(x) 2  9
5
You can use what you know about the characteristics of graphs of quadratic equations to match an equation with a graph. SPI 3102.3.11
Test Example 3 y
Which is an equation for the function shown in the graph? 1 2 A y=_ x 5 2
B y = 2x 2  5
x
O
1 2 C y = _ x +5 2
D y = 2x 2 + 5 Read the Test Item You are given the graph of a parabola. You need to find an equation of the graph. Solve the Test Item Notice that the graph opens downward. Therefore, the graph of y = x 2 has been reflected across the xaxis. The leading coefficient should be negative, so eliminate choices A and D. The parabola is translated up 5 units, so c = 5. Look at the equations. Only choices C and D have c = 5. The answer is C.
GuidedPractice 4. Which is the graph of y = 3x 2 + 1? y
F
y
G 0
0
x
546  Lesson 93  Transformations of Quadratic Functions
y
H x
0
y
J x
0
x
Check Your Understanding
= StepbyStep Solutions begin on page R12.
Examples 1–3 Describe how the graph of each function is related to the graph of f(x) = x 2.
Example 4
1. g(x) = x 2  11
1 2 2. h(x) = _ x
3. h(x) = x 2 + 8
4. g(x) = x 2 + 6
5. g(x) = 4x 2
6. h(x) = x 2  2
2
g(x)
7. MULTIPLE CHOICE Which is an equation for the function shown in the graph? 1 2 A g(x) = _ x +2 5
B g(x) = 5x 2  2
x
0
1 2 C g(x) = _ x 2 5
1 2 D g(x) = _ x 2 5
Practice and Problem Solving
Extra Practice begins on page 815.
Examples 1–3 Describe how the graph of each function is related to the graph of f(x) = x 2. 8. g(x) = 10 + x 2 10. g(x) = 2x 2 + 8
2 2 11. h(x) = 6 + _ x
4 2 x 12. g(x) = –5  _ 3
5 2 13. h(x) = 3 + _ x 2
14. g(x) = 0.25x 2  1.1
15. h(x) = 1.35x 2 + 2.6
3 2 _ 16. g(x) = _ x +5
17. h(x) = 1.01x 2  6.5
3
6
4
Example 4
9 h(x) = 7  x 2
Match each equation to its graph. y
A
x
0
y
B
y
C
x
0
x
0 y
D
y
E
y
F 0
0
x
0
1 2 18. y = _ x 4 3
2
21. y = 3x  2
x
1 2 19. y = _ x 4 3
2
22. y = x + 2
x
1 2 20. y = _ x +4 3
23. y = 3x 2 + 2
24. SQUIRRELS A squirrel 12 feet above the ground drops an acorn from a tree. The function h = 16t 2 + 12 models the height of the acorn above the ground in feet after t seconds. Graph the function and compare this graph to the graph of its parent function. connectED.mcgrawhill.com
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B
List the functions in order from the most stretched vertically to the least stretched vertically graph. 1 2 25. g(x) = 2x 2, h(x) = _ x
2 2 26. g(x) = 3x 2, h(x) = _ x
27. g(x) = 4x 2, h(x) = 6x 2, f(x) = 0.3x 2
5 2 28. g(x) = x 2, h(x) = _ x , f(x) = 4.5x 2
3
2
3
29 ROCKS A rock drops from a cliff 20,000 inches above the ground. Another rock drops from a cliff 30,000 inches above the ground. a. Write two functions that model the heights h of the rocks after t seconds. b. Which rock will reach the ground first? 30. SPRINKLERS The path of water from a sprinkler can be modeled by quadratic functions. The following functions model paths for three different sprinklers. Sprinkler A: y = 0.35x 2 + 3.5 Sprinkler C: y = 0.08x 2 + 2.4
Sprinkler B: y = 0.21x 2 + 1.7
a. Which sprinkler will send water the farthest? Explain. b. Which sprinkler will send water the highest? Explain. c. Which sprinkler will produce the narrowest path? Explain. Describe the transformations to obtain the graph of g(x) from the graph of f(x). 31. f(x) = x 2 + 3 g(x) = x 2  2
C
34.
32. f(x) = x 2  4 g(x) = x 2 + 7
33. f(x) = 6x 2 g(x) = 3x 2
MULTIPLE REPRESENTATIONS In this problem, you will investigate another type of transformation using your graphing calculator. a. Graphical Graph the following family of equations: y = x 2, y = (x  2)2, y = (x – 4)2, y = (x + 3) 2, and y = (x + 5)2 on the same screen. Describe how the graphs of the functions are related to the graph of f(x) = x 2. b. Algebraic Write a concept for quadratic functions, similar to the concept for vertical translations, to describe the effect of a value being added to or subtracted from x inside the parentheses. c. Analytical Predict where the graphs of y = (x  7)2 and y = (x + 4)2 will be located. Verify your answer by graphing each equation.
H.O.T. Problems
Use HigherOrder Thinking Skills
35. REASONING Are the following statements sometimes, always, or never true? Explain. a. The graph of y = x 2 + c has its vertex at the origin. b. The graphs of y = ax 2 and of y = ax 2 are the same width. c. The graph of y = x 2 + c opens downward. 36. CHALLENGE Write a function of the form y = ax 2 + c with a graph that passes through the points (2, 3) and (4, 15). 37. REASONING Determine whether all quadratic functions that are reflected across the yaxis produce the same graph. Explain your answer. 38. OPEN ENDED Write a quadratic function that opens downward and is wider than the parent graph. 39.
E WRITING IN MATH Describe how the values of a and c affect the graphical and tabular representations for the functions y = ax 2, y = x 2 + c, and y = ax 2 + c.
548  Lesson 93  Transformations of Quadratic Functions
SPI 3102.1.2, SPI 3102.3.11, SPI 3102.4.4
Standardized Test Practice 40. SHORT RESPONSE A tutor charges a flat fee of $55 and $30 for each hour of work. Write a function that represents the total charge C, in terms of the number of hours h worked.
42. Candace is 5 feet tall. If 1 inch is about 2.54 centimeters, how tall is Candace to the nearest centimeter? F 13 cm G 26 cm
41. Which best describes the graph of y = 2x 2? A a line with a yintercept of (0, 2) and an xintercept at the origin B a parabola with a minimum point at (0, 0) and that is twice as wide as the graph of y = x 2 when y = 2 C a parabola with a maximum point at (0, 0) and that is half as wide as the graph of y = x 2 when y = 2 D a parabola with a minimum point at (0, 0) and that is half as wide as the graph of y = x 2 when y = 2
H 123 cm J 152 cm
43. While in England, Imani spent 49.60 British pounds on a pair of jeans. If this is equivalent to $100 in U.S. currency, how many British pounds would Imani have spent on a sweater that cost $60? A B C D
2976 pounds 29.76 pounds 19.84 pounds 8.26 pounds
Spiral Review Solve each equation by graphing. (Lesson 92) 44. x 2 + 6 = 0
45. x 2  10x = 24
46. x 2 + 5x + 4 = 0
47. 2x 2  x = 3
48. 2x 2  x = 15
49. 12x 2 = 11x + 15
Find the vertex, the equation of the axis of symmetry, and the yintercept of each graph. (Lesson 91) 50.
12 10 8 6 4 2
y
−8−6−4−20
y
51.
2 4 6 8x
0
y
52.
x
0
x
53. CLASS TRIP Mr. Wong’s American History class will take taxis from their hotel in Washington, D.C., to the Lincoln Memorial. The fare is $2.75 for the first mile and $1.25 for each additional mile. If the distance is m miles and t taxis are needed, write an expression for the cost to transport the group. (Lesson 76) Solve each inequality. Check your solution. (Lesson 53) 54. 3t + 6 ≤ 3
55. 59 > 5  8f
d 56. 2  _ < 23 5
Skills Review Determine whether each trinomial is a perfect square trinomial. If so, factor it. (Lesson 86) 57. 16x 2  24x + 9
58. 9x 2 + 6x + 1
59. 25x 2  60x + 36
60. x 2  8x + 81
61. 36x 2  84x + 49
62. 4x 2  3x + 9 connectED.mcgrawhill.com
549
Graphing Technology Lab
Systems of Linear and Quadratic Equations You can use a graphing calculator to solve systems involving linear and quadratic equations. Tennessee Curriculum Standards ✔ 3102.3.12 Recognize and articulate when an equation has no solution, a single solution, or all real numbers as solutions.
Activity 1 Use a graphing calculator to solve the system of equations. y = x2  x  6 y=x3 Step 1 Enter each equation in the Y= list. Enter the quadratic equation as Y1 and the linear equation as Y2. KEYSTROKES:
Step 2 Graph the system.
KEYSTROKES:
The solutions of the system are the intersection points. The graphs intersect at two points. So, there are two solutions.
3
6
[10, 10] scl: 1 by [10, 10] scl: 1
Step 3 Find the first intersection of the graphs by using the CALC menu. KEYSTROKES:
[CALC] 5
On the screen, notice the question “First Curve?” The cursor should be on . the parabola. Press
Step 4 Move the cursor to the second intersection. Find the second intersection by repeating Step 3. The intersection is at (3, 0). Therefore, the solutions of the system of equations are (–1, 4) and (3, 0).
Notice that the question changes to “Second curve?” and the cursor jumps . to the line. Press Use the arrow keys to move the cursor as close as possible to the intersection point again. in Quadrant III. Press The intersection is the point at (1, 4).
[10, 10] scl: 1 by [10, 10] scl: 1
[10, 10] scl: 1 by [10, 10] scl: 1
550  Extend 93  Graphing Technology Lab: Systems of Linear and Quadratic Equations
Activity 2 Use a graphing calculator to solve the system of equations. y = x 2  8x + 19 y = 2x  6 Step 1 Enter each equation in the Y= list. Enter the quadratic equation as Y1 and the linear equation as Y2. Step 2 Graph the system. In this case, the graphs of the equations intersect at only one point. Therefore, there is only one solution of this system of equations. Step 3 Find the intersection of the graphs of the equations.
[10, 10] scl: 1 by [10, 10] scl: 1
The intersection is the point at about (5, 4). Thus, the solution of the system of equations is about (5, 4).
Activity 3 Use a graphing calculator to solve the system of equations. y = x 2  4x  6
_
y = 1x + 4 3
Step 1 Enter each equation in the Y= list. Enter the quadratic equation as Y1 and the linear equation as Y2. Step 2 Graph the system. The graphs of the equations do not intersect. Thus, this system of equations has no solution.
[10, 10] scl: 1 by [10, 10] scl: 1
Exercises Use factoring to solve each system of equations. Then use a graphing calculator to check your solutions. 1. y = x 2 + 7x + 12 y = 2x + 8
2. y = x 2  x  20 y = 3x + 12
3. y = 3x 2  x  2 y = 2x + 2
Use a graphing calculator to solve each system of equations. 4. y = x 2 y = 2x 7. y = x 2 + 5x + 4 y = x  8
5. y = x 2  6x  3 y=6 1 2 8. y = _ x 4 2
y = 3x + 4
6. y = x 2 + 4 1 x+5 y=_ 2
9. y = x 2 y = 2x  1 connectED.mcgrawhill.com
551
Solving Quadratic Equations by Completing the Square Then
Now
Why?
You solved quadratic equations by using the square root property.
1
Complete the square to write perfect square trinomials.
2
Solve quadratic equations by completing the square.
In competitions, skateboarders may launch themselves from a half pipe into the air to perform tricks. The equation h = 16t 2 + 20t + 12 can be used to model their height, in feet, after t seconds.
(Lesson 86)
NewVocabulary completing the square
To find how long a skateboarder is in the air if he is 25 feet above the half pipe, you can solve 25 = 16t 2 + 20t + 12 by using a method called completing the square.
1 Complete the Square
In Lesson 86, you solved equations by taking the square root of each side. This method worked only because the expression on the lefthand side was a perfect square. In perfect square trinomials in which the leading coefficient is 1, there is a relationship between the coefficient of the xterm and the constant term. (x + 5) 2 = x 2 + 2(5)(x) + 5 2
Tennessee Curriculum Standards CLE 3102.3.8 Solve and understand solutions of quadratic equations with real roots. ✔ 3102.3.30 Solve quadratic equations using multiple methods: factoring, graphing, quadratic formula, or square root principle. SPI 3102.3.10 Find the solution of a quadratic equation and/or zeros of a quadratic function.
= x 2 + 10x + 25
( )2
10 Notice that _ = 25. To get the constant term, divide the coefficient of the xterm 2 by 2 and square the result. Any quadratic expression in the form x 2 + bx can be made into a perfect square by using a method called completing the square.
KeyConcept Completing the Square Words
To complete the square for any quadratic expression of the form x 2 + bx, follow the steps below.
Step 1 Find one half of b, the coefficient of x. Step 2 Square the result in Step 1. Step 3 Add the result of Step 2 to x 2 + bx. Symbols
2
( 2 ) = (x + _2b )
b x 2 + bx + _
2
Example 1 Complete the Square Find the value of c that makes x 2 + 4x + c a perfect square trinomial. Method 1 Use algebra tiles. Arrange the tiles for x 2 + 4x so that the two sides of the figure are congruent.
552  Lesson 94
Y Y Y
2
Y Y
Y Y Y
2
Y Y 1 1
1 1
To make the figure a square, add 4 positive 1tiles.
StudyTip
Method 2 Use complete the square algorithm.
Algorithms An algorithm is a series of steps for carrying out a procedure or solving a problem.
1 Step 1 Find _ of 4.
_4 = 2
Step 2 Square the result in Step 1.
22 = 4
Step 3 Add the result of Step 2 to x 2 + 4x.
x 2 + 4x + 4
2
2
Thus, c = 4. Notice that x 2 + 4x + 4 = (x + 2) 2.
GuidedPractice 1. Find the value of c that makes r 2  8r + c a perfect square trinomial.
2 Solve Equations by Completing the Square
You can complete the square to solve quadratic equations. First, you must isolate the x2 and bxterms.
Example 2 Solve an Equation by Completing the Square Solve x2  6x + 12 = 19 by completing the square. x 2  6x + 12 = 19 x 2  6x = 7 2
x  6x + 9 = 7 + 9 (x  3)2 = 16 x  3 = ±4 x=3±4 x = 3 + 4 or x = 3  4 =7 = 1
Original equation Subtract 12 from each side.
(_2 )
2 Since 6 = 9, add 9 to each side.
Factor x 2  6x + 9. Take the square root of each side. Add 3 to each side. Separate the solutions.
The solutions are 7 and 1.
GuidedPractice 2. Solve x 2  12x + 3 = 8 by completing the square.
To solve a quadratic equation in which the leading coefficient is not 1, divide each term by the coefficient. Then isolate the x 2 and xterms and complete the square.
Example 3 Equation with a ≠ 1
WatchOut!
Solve 2x 2 + 8x  18 = 0 by completing the square.
Leading Coefficient Remember that the leading coefficient has to be 1 before you can complete the square.
2x 2 + 8x  18 = 0
2x + 8x  18 0 __ =_
Original equation
2
2
2
2
Divide each side by 2.
x  4x + 9 = 0 Simplify. 2 x  4x = 9 Subtract 9 from each side. 4 2 2 x  4x + 4 = 9 + 4 Since _ = 4, add 4 to each side. 2 2 2 (x  2) = 5 Factor x  4x + 4. No real number has a negative square. So, this equation has no real solutions.
( )
GuidedPractice 3. Solve 3x 2  9x  3 = 21 by completing the square. connectED.mcgrawhill.com
553
RealWorld Example 4 Use a Graph of a Quadratic Function JERSEYS The senior class at Bay High School buys jerseys to wear to the football games. The cost of the jerseys can be modeled by the equation C = 0.1x 2 + 2.4x + 25, where C is the amount it costs to buy x jerseys. How many jerseys can they purchase for $430? The seniors have $430, so set the equation equal to 430 and complete the square. 0.1x 2 + 2.4x + 25 = 430
Original equation
0.1x + 2.4x + 25 430 __ =_ 2
0.1
RealWorldLink The oldest public high school rivalry takes place between Wellesley High School and Needham Heights High School in Massachusetts. The first football game between them took place on Thanksgiving morning in 1882 in Needham.
Divide each side by 0.1.
0.1
x 2 + 24x + 250 = 4300
Simplify.
x 2 + 24x + 250  250 = 4300  250
Subtract 250 from each side.
x 2 + 24x = 4050
Simplify.
x + 24x + 144 = 4050 + 144
24 Since _
x 2 + 24x + 144 = 4194
Simplify.
(2)
2
(x + 12) 2 = 4194
= 144, add 144 to each side.
Factor x 2 + 24x + 144.
x + 12 = ± √ 4194
Take the square root of each side.
x = 12 ± √ 4194
Source: USA Football
2
Subtract 12 from each side.
Use a calculator to approximate each value of x. x = 12 + √4194 ≈ 52.8
or
x = 12  √ 4194 ≈ 76.8
Separate the solutions. Evaluate.
Since you cannot buy a negative number of jerseys, the negative solution is not reasonable. The seniors can afford to buy 52 jerseys.
GuidedPractice 4. If the senior class were able to raise $620, how many jerseys could they buy?
Check Your Understanding Example 1
= StepbyStep Solutions begin on page R12.
Find the value of c that makes each trinomial a perfect square. 1 x 2  18x + c
2. x 2 + 22x + c
3. x 2 + 9x + c
4. x 2  7x + c
Examples 2–3 Solve each equation by completing the square. Round to the nearest tenth if necessary.
Example 4
5. x 2 + 4x = 6
6. x 2  8x = 9
7. 4x 2 + 9x  1 = 0
8. 2x 2 + 10x + 22 = 4
9. CONSTRUCTION Collin is building a deck on the back of his family’s house. He has enough lumber for the deck to be 144 square feet. The length should be 10 feet more than its width. What should the dimensions of the deck be?
554  Lesson 94  Solving Quadratic Equations by Completing the Square
Practice and Problem Solving Example 1
Extra Practice begins on page 815.
Find the value of c that makes each trinomial a perfect square. 10. x 2 + 26x + c
11. x 2  24x + c
12. x 2  19x + c
13. x 2 + 17x + c
14. x 2 + 5x + c
15. x 2  13x + c
16. x 2  22x + c
17. x 2  15x + c
18. x 2 + 24x + c
Examples 2–3 Solve each equation by completing the square. Round to the nearest tenth if necessary.
Example 4
19 x 2 + 6x  16 = 0
20. x 2  2x  14 = 0
21. x 2  8x  1 = 8
22. x 2 + 3x + 21 = 22
23. x 2  11x + 3 = 5
24. 5x 2  10x = 23
25. 2x 2  2x + 7 = 5
26. 3x 2 + 12x + 81 = 15
27. 4x 2 + 6x = 12
28. 4x 2 + 5 = 10x
29. 2x 2 + 10x = 14
30. 3x 2  12 = 14x
31. FINANCIAL LITERACY The price p in dollars for a particular stock can be modeled by the quadratic equation p = 3.5t  0.05t 2, where t represents the number of days after the stock is purchased. When is the stock worth $60? GEOMETRY Find the value of x for each figure. Round to the nearest tenth if necessary. 32. A = 45 in 2
33. A = 110 ft 2 (x + 5) ft
x in. 2x ft (x + 8) in.
34. NUMBER THEORY The product of two consecutive even integers is 224. Find the integers. 35. NUMBER THEORY The product of two consecutive negative odd integers is 483. Find the integers. 36. GEOMETRY Find the area of the triangle below. x+6 x
30
B
Solve each equation by completing the square. Round to the nearest tenth if necessary. 37. 0.2x 2  0.2x  0.4 = 0
38. 0.5x 2 = 2x  0.3
3 11 39. 2x 2  _ x = _
5 2 2 _ 40. _ x  4x = _
3 1 2 41. _ x + 2x = _
1 2 2 42. _ x + 2x = _
5
4
8
10
3
5
3
6
5
connectED.mcgrawhill.com
555
43 ASTRONOMY The height of an object t seconds after it is dropped is given by the 1 2 gt + h 0, where h 0 is the initial height and g is the acceleration due equation h = _ 2
to gravity. The acceleration due to gravity near the surface of Mars is 3.73 m/s 2, while on Earth it is 9.8 m/s 2. Suppose an object is dropped from an initial height of 120 meters above the surface of each planet. a. On which planet would the object reach the ground first? b. How long would it take the object to reach the ground on each planet? Round each answer to the nearest tenth. c. Do the times that it takes the object to reach the ground seem reasonable? Explain your reasoning. 44. Find all values of c that make x 2 + cx + 100 a perfect square trinomial. 45. Find all values of c that make x 2 + cx + 225 a perfect square trinomial. 46. PAINTING Before she begins painting a picture, Donna stretches her canvas over a wood frame. The frame has a length of 60 inches and a width of 4 inches. She has enough canvas to cover 480 square inches. Donna decides to increase the dimensions of the frame. If the increase in the length is 10 times the increase in the width, what will the dimensions of the frame be?
C
47.
MULTIPLE REPRESENTATIONS In this Number Trinomial b 2  4ac problem, you will investigate a property of Roots of quadratic equations. 0 1 x 2  8x + 16 a. Tabular Copy the table shown and 2x 2  11x + 3 complete the second column. 3x 2 + 6x + 9 b. Algebraic Set each trinomial equal x 2  2x + 7 to zero, and solve the equation by x 2 + 10x + 25 completing the square. Complete the last column of the table with the x 2 + 3x  12 number of roots of each equation. c. Verbal Compare the number of roots of each equation to the result in the b 2  4ac column. Is there a relationship between these values? If so, describe it. d. Analytical Predict how many solutions 2x 2  9x + 15 = 0 will have. Verify your prediction by solving the equation.
H.O.T. Problems
Use HigherOrder Thinking Skills
48. CHALLENGE Given y = ax 2 + bx + c with a ≠ 0, derive the equation for the axis of symmetry by completing the square and rewriting the equation in the form y = a(x  h) 2 + k. b 2 49. REASONING Determine the number of solutions x 2 + bx = c has if c <  _ . 2 Explain.
()
50. WHICH ONE DOESN’T BELONG? Identify the expression that does not belong with the other three. Explain your reasoning. n2  n + _ 1 4
n2 + n + _ 1 4
n 2  _n + _ 2 3
1 9
n 2 + _n + _ 1 3
1 9
51. OPEN ENDED Write a quadratic equation for which the only solution is 4. 52.
E
WRITING IN MATH Compare and contrast the following strategies for solving x  5x  7 = 0: completing the square, graphing, and factoring. 2
556  Lesson 94  Solving Quadratic Equations by Completing the Square
SPI 3102.3.5, SPI 3102.5.5
Standardized Test Practice 53. The length of a rectangle is 3 times its width. The area of the rectangle is 75 square feet. Find the length of the rectangle in feet. A 25
B 15
C 10
55. GRIDDED RESPONSE The population of a town can be modeled by P = 22,000 + 125t, where P represents the population and t represents the number of years from 2000. How many years after 2000 will the population be 26,000?
D 5
54. PROBABILITY At a festival, winners of a game draw a token for a prize. There is one token for each prize. The prizes include 9 movie passes, 8 stuffed animals, 5 hats, 10 jump ropes, and 4 glow necklaces. What is the probability that the first person to draw a token will win a movie pass? 1 G _
9 F _
9
61
1 H _ 4
56. Percy delivers pizzas for Pizza King. He is paid $6 an hour plus $2.50 for each pizza he delivers. Percy earned $280 last week. If he worked a total of 30 hours, how many pizzas did he deliver? A B C D
1 J _ 36
250 pizzas 184 pizzas 40 pizzas 34 pizzas
Spiral Review Describe how the graph of each function is related to the graph of f(x) = x 2. (Lesson 93)
57. g(x) = 12 + x 2
58. h(x) = 2  x 2
59. g(x) = 2x 2 + 5
2 2 60. h(x) = 6 + _ x
4 2 61. g(x) = 6 + _ x
3 62. h(x) = 1  _ x
3
3
2
2
63. RIDES A popular amusement park ride whisks riders to the top of a 250foot tower and drops them. A function for the height of a rider is h = 16t 2 + 250, where h is the height and t is the time in seconds. The ride stops the descent of the rider 40 feet above the ground. Write an equation that models the drop of the rider. How long does it take to fall from 250 feet to 40 feet? (Lesson 92) Simplify. Assume that no denominator is equal to zero. (Lesson 72) a6 64. _ 3
c 3d 4 66. _ 7
47 65. _ 5
a
cd
4
( ) 2
4h g 67. _ 5
0
2 6
5q t 68. _ 2 4
69. b 3(m 3)(b 6)
70. y  2 > 7
71. z + 5 < 3
72. 2b + 7 ≤ 6
73. 3  2y ≥ 8
74. 9  4m < 1
75. 5c  2 ≤ 13
2g
10q t
Solve each open sentence. (Lesson 55)
Skills Review 2  4ac for each set of values. Round to the nearest tenth if necessary. (Lesson 12) Evaluate √b
76. a = 2, b = 5, c = 2
77. a = 1, b = 12, c = 11
78. a = 9, b = 10 , c = 1
79. a = 1, b = 7, c = 3
80. a = 2, b = 4, c = 6
81. a = 3, b = 1, c = 2 connectED.mcgrawhill.com
557
Solving Quadratic Equations by Using the Quadratic Formula Then
Now
Why?
You solved quadratic equations by completing the square. (Lesson 94)
1
Solve quadratic equations by using the Quadratic Formula.
2
Use the discriminant to determine the number of solutions of a quadratic equation.
For adult women, the normal systolic blood pressure P in millimeters of mercury (mm Hg) can be modeled by P = 0.01a 2 + 0.05a + 107, where a is age in years. This equation can be used to approximate the age of a woman with a certain systolic blood pressure. However, it would be difficult to solve by factoring, graphing, or completing the square.
NewVocabulary Quadratic Formula discriminant
1 Quadratic Formula
Completing the square of the quadratic equation ax 2 + bx + c = 0 produces a formula that allows you to find the solutions of any quadratic equation. This formula is called the Quadratic Formula.
KeyConcept The Quadratic Formula Tennessee Curriculum Standards CLE 3102.3.8 Solve and understand solutions of quadratic equations with real roots. ✔ 3102.3.31 Determine the number of real solutions for a quadratic equation including using the discriminant and its graph. SPI 3102.3.10 Find the solution of a quadratic equation and/or zeros of a quadratic function. Also addresses ✓3102.3.30.
The solutions of a quadratic equation ax 2 + bx + c = 0, where a ≠ 0, are given by the Quadratic Formula.
b ± √ b 2  4ac 2a
x = __
You will derive this formula in Lesson 102.
Example 1 Use the Quadratic Formula Solve x 2  12x = 20 by using the Quadratic Formula. Step 1 Rewrite the equation in standard form. x 2  12x = 20 x 2  12x + 20 = 0
Original equation Add 20 to each side.
Step 2 Apply the Quadratic Formula. 2  4ac b ± √b 2a (12) ± √ (12) 2  4(1)(20)
x = __
Quadratic Formula
= ___
a = 1, b = 12, and c = 20
12 ± √ 144  80 = __
Multiply.
2(1)
2 12 ± √64 12 ± 8 = _ or _ 2 2 12 +8 12 8 x = _ or x = _ 2 2
=2
= 10
Subtract and take the square root. Separate the solutions.
The solutions are 2 and 10.
GuidedPractice 1. Solve 2x 2 + 9x = 18 by using the Quadratic Formula.
558  Lesson 95
The solutions of quadratic equations are not always integers.
Example 2 Use the Quadratic Formula Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary. a. 3x 2 + 5x  12 = 0 For this equation, a = 3, b = 5, and c = 12. b ± √ b 2  4ac x = __
Quadratic Formula
2a
(5) ±
√ (5) 2  4(3)(12)
= ___
a = 3, b = 5, and c = 12
2(3)
5 ± √25 + 144 6
Multiply.
5 ± √ 169 6
Add and simplify.
= __ = _ or _ 5 ± 13 6
5  13 x=_ or x = _
5 + 13 6 4 = 3 =_ 3 4 The solutions are 3 and _ . 3
Separate the solutions.
6
Simplify.
b. 10x 2  5x = 25 Step 1 Rewrite the equation in standard form. 10x 2  5x = 25
Original equation
2
10x  5x  25 = 0
Subtract 25 from each side.
Step 2 Apply the Quadratic Formula. b ± √ b 2  4ac 2a
x = __ (5) ±
Quadratic Formula
√ (5) 2  4(10)(25)
= ___ 2(10)
5 ± √25 + 1000 20
Multiply.
5 ± √ 1025 20
Add.
= __ =_
StudyTip Exact Answers In Example 2, the number √ 1025 is irrational, so the calculator can only give you an approximation of its value. So, the exact answer in 5± √ 1025 Example 2 is _. 20
The numbers 1.4 and 1.9 are approximations.
a = 10, b = 5, and c = 25
= _ or _
5 + √ 1025 20
Separate the solutions.
≈ 1.4
≈ 1.9
Simplify.
5  √ 1025 20
The solutions are about 1.4 and 1.9.
GuidedPractice 2A. 4x 2  24x + 35 = 0
2B. 3x 2  2x  9 = 0
You can solve quadratic equations by using many different methods. No one way is always best. connectED.mcgrawhill.com
559
WatchOut! Solutions No matter what method is used to solve a quadratic equation, all of the methods should produce the same solution(s).
Example 3 Solve Quadratic Equations Using Different Methods Solve x 2  4x = 12. Method 1 Graphing Rewrite the equation in standard form. x 2  4x = 12 x 2  4x  12 = 0
2
Original equation
−8−6−4−20
Subtract 12 from each side. 2
Graph the related function f(x) = x  4x  12. Locate the xintercepts of the graph. The solutions are 2 and 6.
y 2 4 6 8x
4 −6 8 10 12 14
Method 2 Factoring x 2  4x = 12 x 2  4x  12 = 0 (x  6)(x + 2) = 0 x  6 = 0 or x + 2 = 0 x=6 x = 2
Original equation Subtract 12 from each side. Factor. Zero Product Property Solve for x.
Method 3 Completing the Square The equation is in the correct form to complete the square, since the leading coefficient is 1 and the x 2 and x terms are isolated. x 2  4x = 12 x 2  4x + 4 = 12 + 4 2) 2
(x = 16 x  2 = ±4 x=2±4 x = 2 + 4 or x = 2  4 =6 = 2
Original equation 4 Since _
(2)
2
= 4, add 4 to each side.
2
Factor x  4x + 4. Take the square root of each side. Add 2 to each side. Separate the solutions. Simplify.
Method 4 Quadratic Formula From Method 1, the standard form of the equation is x 2  4x  12 = 0. b ± √ b 2  4ac 2a (4) ± (4)2  4(1)(12) √ ___
x = __ =
2(1) 4 ± √16 + 48 2
= __ 4 ± √ 64 4±8 2 2 4+8 48 _ _ x= or x = 2 2
= _ or _
= 2
=6
Quadratic Formula a = 1, b = 4, and c = 12 Multiply. Add and simplify. Separate the solutions. Simplify.
GuidedPractice Solve each equation. 3A. 2x 2  17x + 8 = 0
3B. 4x 2  4x  11 = 0
560  Lesson 95  Solving Quadratic Equations by Using the Quadratic Formula
ConceptSummary Solving Quadratic Equations Method M th d
When Wh tto Use U
Factoring
Use when the constant term is 0 or if the factors are easily determined. Not all equations are factorable.
Graphing
Use when an approximate solution is sufficient.
Using Square Roots
Use when an equation can be written in the form x 2 = n. Can only be used if the equation has no xterm.
Completing the Square
Can be used for any equation ax 2 + bx + c = 0, but is simplest to apply when b is even and a = 1.
Quadratic Formula
Can be used for any equation ax 2 + bx + c = 0.
2 The Discriminant
In the Quadratic Formula, the expression under the radical sign, b 2  4ac, is called the discriminant. The discriminant can be used to d determine the number of real solutions of a quadratic equation.
StudyTip Discriminant Recall that when the left side of the standard form of an equation is a perfect square trinomial, there is only one solution. Therefore, the discriminant of a perfect square trinomial will always be zero.
KeyConcept Using the Discriminant Equation
x 2 + 2x + 5 = 0
x 2 + 10x + 25 = 0
2x 2  7x + 2 = 0
Discriminant
b 2  4ac = 16 negative
b 2  4ac = 0 zero
b 2  4ac = 33 positive
y
y
y
x
0
Graph of Related Function
0 x 0
x
0 xintercepts
1 xintercept
2 xintercepts
0
1
2
Real Solutions
Example 4 Use the Discriminant State the value of the discriminant of 4x 2 + 5x = 3. Then determine the number of real solutions of the equation. Step 1 Rewrite in standard form.
4x 2  5x = 3
4x 2  5x + 3 = 0
Step 2 Find the discriminant. b 2  4ac = (5) 2  4(4)(3) = 23
a = 4, b = 5, and c = 3 Simplify.
Since the discriminant is negative, the equation has no real solutions.
GuidedPractice 4A. 2x 2 + 11x + 15 = 0
4B. 9x 2  30x + 25 = 0 connectED.mcgrawhill.com
561
Check Your Understanding
= StepbyStep Solutions begin on page R12.
Examples 1–2 Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary.
Example 3
1. x 2  2x  15 = 0
2. x 2  10x + 16 = 0
3. x 2  8x = 10
4. x 2 + 3x = 12
5. 10x 2  31x + 15 = 0
6. 5x 2 + 5 = 13x
Solve each equation. State which method you used. 7. 2x 2 + 11x  6 = 0 9. 9x 2 = 25
Example 4
8. 2x 2  3x  6 = 0 10. x 2  9x = 19
State the value of the discriminant for each equation. Then determine the number of real solutions of the equation. 11. x 2  9x + 21 = 0
12. 2x 2  11x + 10 = 0
13. 9x 2 + 24x = 16
14. 3x 2  x = 8
15. TRAMPOLINE Eva is jumping on a trampoline. Her height h in feet can be modeled by the equation h = 16t 2 + 2.4t + 6, where t is time in seconds. Use the discriminant to determine if Eva will ever reach a height of 20 feet. Explain.
Practice and Problem Solving
Extra Practice begins on page 815.
Examples 1–2 Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary. 16. 4x 2 + 5x  6 = 0
17 x 2 + 16 = 0
18. 6x 2  12x + 1 = 0
19. 5x 2  8x = 6
20. 2x 2  5x = 7
21. 5x 2 + 21x = 18
22. 81x 2 = 9
23. 8x 2 + 12x = 8
24. 4x 2 = 16x  16
25. 10x 2 = 7x + 6
26. 3x 2 = 8x  12
27. 2x 2 = 12x  18
28. AMUSEMENT PARKS The Demon Drop at Cedar Point in Ohio takes riders to the top of a tower and drops them 60 feet. A function that approximates this ride is h = 16t 2 + 64t  60, where h is the height in feet and t is the time in seconds. About how many seconds does it take for riders to drop from 60 feet to 0 feet? Example 3
Example 4
Solve each equation. State which method you used. 29. 2x 2  8x = 12
30. 3x 2  24x = 36
31. x 2  3x = 10
32. 4x 2 + 100 = 0
33. x 2 = 7x  5
34. 12  12x = 3x 2
State the value of the discriminant for each equation. Then determine the number of real solutions of the equation. 35. 0.2x 2  1.5x + 2.9 = 0
36. 2x 2  5x + 20 = 0
4 37. x 2  _ x=3
38. 0.5x 2  2x = 2
39. 2.25x 2  3x = 1
5 3 40. 2x 2 = _ x+_
5
2
2
41. INTERNET The percent of U.S. households with highspeed Internet h can be estimated by h = 0.2n 2 + 7.2n + 1.5, where n is the number of years since 1990. a. Use the Quadratic Formula to determine when 20% of the population will have highspeed Internet. b. Is a quadratic equation a good model for this information? Explain.
562  Lesson 95  Solving Quadratic Equations by Using the Quadratic Formula
B
42. TRAFFIC The equation d = 0.05v 2 + 1.1v models the distance d in feet it takes a car traveling at a speed of v miles per hour to come to a complete stop. The speed limit on some highways is 65 miles per hour. If Hannah’s car stopped after 250 feet, was she speeding? Explain your reasoning. Without graphing, determine the number of xintercepts of the graph of the related function for each equation. 3 2 44. x 2 + _ =_ x 45. 0.25x 2 + x = 1 43. 4.25x + 3 = 3x 2 25
5
Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary. 47. 2.3x 2  1.4x = 6.8 48. x 2  2x = 5 46. 2x 2  7x = 1.5 49 POSTER Bartolo is making a poster for the dance. He wants to cover three fourths of the area with text.
20 in. 4x in.
a. Write an equation for the area of the section with text. b. Solve the equation by using the Quadratic Formula.
25 in.
4KMDIB'GDIB x in.
'MD?
x in.
KH
c. What should be the margins of the poster?
3x in.
C
50.
MULTIPLE REPRESENTATIONS In this problem, you will investigate exponential functions.
Time (hours)
Number of Bacteria
a. Tabular Copy and complete the table.
0
1 = 20
b. Graphical Construct a graph from the information given in the table using the points (time, number of bacteria). Is the graph linear or quadratic?
1
2 = 21
2
4 = 22
c. Analytical What happens to the number of bacteria after every hour? Write a function that models the pattern in the table.
H.O.T. Problems
3 4 5 6
Use HigherOrder Thinking Skills
51. CHALLENGE Find all values of k such that 2x 2  3x + 5k = 0 has two solutions. 52. REASONING Use factoring techniques to determine the number of real zeros of f(x) = x 2  8x + 16. Compare this method to using the discriminant. REASONING Determine whether there are two, one, or no real solutions. 53. The graph of a quadratic function does not have an xintercept. 54. The graph of a quadratic function is tangent at the xaxis. 55. The graph of a quadratic function intersects the xaxis twice. 56. Both a and b are greater than 0 and c is less than 0 in a quadratic equation. 57. OPEN ENDED Write a quadratic function that has a positive discriminant, one with a negative discriminant, and one with a zero discriminant. 58. WRITING IN MATH Describe the advantages and disadvantages of each method of solving quadratic equations. Which method do you prefer, and why? connectED.mcgrawhill.com
563
SPI 3102.2.3, SPI 3108.1.4, SPI 3102.3.10
Standardized Test Practice 59. If n is an even integer, which expression represents the product of three consecutive even integers? A B C D
n(n + 1)(n + 2) (n + 1)(n + 2)(n + 3) 3n + 2 n(n + 2)(n + 4)
61. Which statement best describes the graph of x = 5? F G H J
It is parallel to the xaxis. It is parallel to the yaxis. It passes through the point (2, 5). It has a yintercept of 5.
62. What are the solutions of the quadratic equation 6h 2 + 6h = 72?
60. SHORT RESPONSE The triangle shown is an isosceles triangle. What is the value of x?
64°
A 3 or 4 B 3 or 4
x°
C no solution D 12 or 48
Spiral Review Solve each equation by completing the square. Round to the nearest tenth if necessary. (Lesson 94) 63. 6x 2  17x + 12 = 0
64. x 2  9x = 12
65. 4x 2 = 20x  25
Describe the transformations needed to obtain the graph of g(x) from the graph of f(x). (Lesson 93) 66. f(x) = 4x 2
67. f(x) = x 2 + 5
68. f(x) = x 2  6
g(x) = 2x 2
g(x) = x 2  1
g(x) = x 2 + 3
Determine whether each graph shows a positive, a negative, or no correlation. If there is a positive or negative correlation, describe its meaning in the situation. (Lesson 45)
Electronic Tax Returns 60 50 40 30 20 0
Atlantic Hurricanes
70.
Number
Number (millions)
69.
’01 ’02 ’03 ’04
10 8 6 4 2 0
’92 ’94 ’96 ’98 ’00 ’02 ’04 ’06
Year
Year
71. ENTERTAINMENT Coach Washington wants to take her softball team out for pizza and soft drinks after the last game of the season. A large pizza costs $12 and a pitcher of a soft drink costs $3. She does not want to spend more than $60. Write an inequality that represents this situation and graph the solution set. (Lesson 56)
Skills Review Evaluate a(b x) for each of the given values. (Lesson 12) 72. a = 1, b = 2, x = 4
73. a = 4, b = 1, x = 7
74. a = 5, b = 3, x = 0
75. a = 0, b = 6, x = 8
76. a = 2, b = 3, x = 1
77. a = 3, b = 5, x = 2
564  Lesson 95  Solving Quadratic Equations by Using the Quadratic Formula
Graphing Technology Lab
Cubic Functions You have studied linear functions and monomials. Some functions can be defined by the sums of monomials. One function that can be defined this way is a cubic function. A cubic equation has the form ax 3 + bx 2 + cx + d = 0, where a ≠ 0. All cubic equations have at least one but no more than three real roots.
Activity Solve x 3 − 6x 2 + 3x + 10 = 0 by graphing. Step 1 Enter the related function in the Y= list. KEYSTROKES: 3 6
3
10
Step 2 Graph the function in the standard viewing window. KEYSTROKES:
6
Step 3 Find the zeros of the function by determining where the graph crosses the xaxis. Notice that this graph crosses the xaxis three times. Therefore, there are 3 real solutions for the equation. [CALC] 2 KEYSTROKES: [10, 10] scl: 1 by [10, 10] scl: 1 Press the left arrow to move to the left of the intercept closest to the origin. . Press Press the right arrow to move to the right of the intercept. . Do not go past another intercept. Press Notice the arrows above the intercept. The intercept you are finding should be between these two arrows. . Press the left arrow to move as close as possible to the intercept. Press
[10, 10] scl:1 by [10, 10] scl:1
[10, 10] scl:1 by [10, 10] scl:1
[10, 10] scl:1 by [10, 10] scl:1
[10, 10] scl:1 by [10, 10] scl:1
One root is x = 1. Step 4 Repeat Step 3 for each additional root. The solutions for x 3  6x 2 + 3x + 10 = 0 are x = 1, 2, and 5.
Exercises Solve each equation by graphing. 1. x 3  4x 2  9x + 36 = 0
2. x 3  6x 2  6x  7 = 0
3. x 3 + x 2 + x  3 = 0
4. x 3 − 5x 2  2x + 24 = 0 connectED.mcgrawhill.com
565
MidChapter Quiz
Tennessee Curriculum Standards
Lessons 91 through 95
SPI 3102.3.11
Use a table of values to graph each equation. State the domain and range. (Lesson 91)
Describe how the graph of each function is related to the graph of f (x ) = x 2. (Lesson 93)
1. y = x 2 + 3x + 1
15. g(x ) = x 2 + 3
2. y = 2x 2  4x + 3
16. h (x ) = 2x 2
3. y = x 2  3x  3
17. g(x ) = x 2  6
4. y = 3x 2  x + 1
18. MULTIPLE CHOICE Which is an equation for the function shown in the graph? (Lesson 93)
Consider y = x 2  5x + 4. (Lesson 91)
y
5. Write the equation of the axis of symmetry.
O
x
6. Find the coordinates of the vertex. Is it a maximum or minimum point? 7. Graph the function. 8. SOCCER A soccer ball is kicked from ground level with an initial upward velocity of 90 feet per second. The equation h = 16t 2 + 90t gives the height h of the ball after t seconds. (Lesson 91) a. What is the height of the ball after one second? b. How many seconds will it take for the ball to reach its maximum height? c. When is the height of the ball 0 feet? What do these points represent in this situation? Solve each equation by graphing. If integral roots cannot be found, estimate the roots to the nearest tenth. (Lesson 92) 2
A y = 2x 2 B y = 2x 2 + 1 C y = x2  1 D y = 2x 2 + 1 Solve each equation by completing the square. Round to the nearest tenth. (Lesson 94) 19. x 2 + 4x + 2 = 0 20. x 2  2x  10 = 0 21. 2x 2 + 4x  5 = 7
9. x + 5x + 6 = 0 10. x 2 + 8 = 6x
Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary. (Lesson 95)
11. x 2 + 3x  1 = 0
22. x 2  3x  18 = 0
12. x 2 = 12
23. x 2  10x = 24
13. BASEBALL Juan hits a baseball. The equation h = 16t 2 + 120t models the height h, in feet, of the ball after t seconds. How long is the ball in the air? (Lesson 92) 14. CONSTRUCTION Christopher is repairing the roof on a shed. He accidentally dropped a box of nails from a height of 14 feet. This is represented by the equation h = 16t 2 + 14, where h is the height in feet and t is the time in seconds. Describe how the graph is related to h = t 2. (Lesson 93)
566  Chapter 9  MidChapter Quiz
24. 2x 2 + 5x  3 = 0 25. PARTIES Della’s parents are throwing a Sweet 16 party for her. At 10:00, a ball will slide 25 feet down a pole and light up. A function that models the drop is h = t 2 + 5t + 25, where h is height in feet of the ball after t seconds. How many seconds will it take for the ball to reach the bottom of the pole? (Lesson 95)
25 ft
Then
Now
Why?
You simplified numerical expressions involving exponents.
1 2
Tarantulas can appear scary with their large hairy bodies and legs, but they are harmless to humans. The graph shows a tarantula spider population that increases over time. Notice that the graph is neither linear nor quadratic.
(Lesson 12)
Graph exponential functions. Identify data that display exponential behavior.
The graph represents the function y = 3(2) x. This is an example of an exponential function.
Tarantulas (hundreds)
Exponential Functions 35 30 25 20 15 10 5 −2−1O
y
1 2 3 4 5 6x
Years Since 2010
NewVocabulary exponential function
1
Graph Exponential Functions An exponential function is a function of the
form y = ab x, where a ≠ 0, b > 0, and b ≠ 1. Notice that the base is a constant and the exponent is a variable. Exponential functions are nonlinear and nonquadratic.
KeyConcept Exponential Function Tennessee Curriculum Standards CLE 3102.3.6 Understand and use relations and functions in various representations to solve contextual problems. CLE 3102.3.9 Understand and use exponential functions to solve contextual problems. SPI 3102.3.11 Analyze nonlinear graphs including quadratic and exponential functions that model a contextual situation. Also addresses ✓3102.3.33, ✓3102.3.34, and ✓3103.3.35.
Words
An exponential function is a function that can be described by an equation of the form y = ab x, where a ≠ 0, b > 0, and b ≠ 1.
Examples
y = 2(3) x
1 y= _
(2)
y = 4x
x
Example 1 Graph with a > 0 and b > 1 a. Graph y = 3 x. Find the yintercept, and state the domain and range. The graph crosses the yaxis at 1, so the yintercept is 1. The domain is all real numbers, and the range is all positive real numbers.
x
3x
y
2
3 2
_1
1
1
_1
3
9 3
0
30
1
1
31
3
2
2
9
3
y
x
y=3
0
x
b. Use the graph to approximate the value of 3 0.7. The graph represents all real values of x and their corresponding values of y for y = 3 x. So, when x = 0.7, y is about 2. Use a calculator to confirm this value: 3 0.7 ≈ 2.157669.
GuidedPractice 1A. Graph y = 7 x. Find the yintercept, and state the domain and range. 1B. Use the graph to approximate the value of y = 7 0.5 to the nearest tenth. Use a calculator to confirm the value. connectED.mcgrawhill.com
567
The graphs of functions of the form y = ab x, where a > 0 and b > 1, all have the same shape as the graph in Example 1. The greater the base or bvalue, the faster the graph rises as you move from left to right on the graph. The graphs of functions of the form y = ab x, where a > 0 and 0 < b < 1, also have the same general shape.
StudyTip a < 0 If the value of a is less than 0, the graph will be reflected across the xaxis.
Example 2 Graph with a > 0 and 0 < b < 1 1 x a. Graph y = _ . Find the yintercept, and state the domain and range.
(3)
x (_1 )
x
3
2
(_13 )
2
_1 0
0
(3)
2
(_13 )
2
y
y 9 y = _1
x
(3)
1
_1 9
x
0
The yintercept is 1. The domain is all real numbers, and the range is all positive real numbers. Notice that as x increases, the yvalues decrease less rapidly. 1 b. Use the graph to approximate the value of _
(3)
1.5
.
When x = 1.5, the value of y is about 5. Use a calculator to confirm this value: 1
KEYSTROKES:
3
1.5
5.196152
GuidedPractice x
1 2A. Graph y = _  1. Find the yintercept, and state the domain and range.
(2)
1 2B. Use the graph to approximate the value of _ 2 Use a calculator to confirm the value.
()
2.5
 1 to the nearest tenth.
Exponential functions occur in many real world situations.
RealWorld Example 3 Use Exponential Functions to Solve Problems SODA The consumption of soda has increased each year since 2000. The function C = 179(1.029) t models the amount of soda consumed in the world, where C is the amount consumed in billions of liters and t is the number of years since 2000.
RealWorldLink The United States is the largest soda consumer in the world. In a recent year, the United States accounted for one third of the world’s total soda consumption.
a. Graph the function. What values of C and t are meaningful in the context of the problem? Since t represents time, t > 0. At t = 0, the consumption is 179 billion liters. Therefore, in the context of this problem, C > 179 is meaningful.
Source: Worldwatch Institute
568  Lesson 96  Exponential Functions
[50, 50] scl: 10 by [0, 350] scl: 25
b. How much soda was consumed in 2005? Original equation C = 179(1.029) t = 179(1.029) 5 t=5 ≈ 206.5 Use a calculator. The world soda consumption in 2005 was approximately 206.5 billion liters.
GuidedPractice 3. A certain bacteria population doubles every 20 minutes. Beginning with 10 cells in a culture, the population can be represented by the function B = 10(2) t, where B is the number of bacteria cells and t is the time in 20 minute increments. How many will there be after 2 hours?
2 Identify Exponential Behavior
Recall from Lesson 33 that linear functions have a constant rate of change. Exponential functions do not have constant rates of change, but they do have constant ratios.
ProblemSolvingTip Make an Organized List Making an organized list of xvalues and corresponding yvalues is helpful in graphing the function. It can also help you identify patterns in the data.
Example 4 Identify Exponential Behavior Determine whether the set of data shown below displays exponential behavior. Write yes or no. Explain why or why not. x
0
5
10
15
20
25
y
64
32
16
8
4
2
Method 1 Look for a pattern. The domain values are at regular intervals of 5. Look for a common factor among the range values. 64 32 16 8 4 2
_ _ _ _ _
×1 ×1 ×1 ×1 ×1 2
2
2
2
2
1 The range values differ by the common factor of _ . 2
Since the domain values are at regular intervals and the range values differ by a positive common factor, the data are probably exponential. Its equation 1 x . may involve _
(2)
Method 2 Graph the data.
64 56 48 40 32 24 16 8
Plot the points and connect them with a smooth curve. The graph shows a rapidly decreasing value of y as x increases. This is a characteristic of exponential behavior in which the base is between 0 and 1.
StudyTip Checking Answers The graph of an exponential function may resemble part of the graph of a quadratic function. Be sure to check for a pattern as well as to look at a graph.
−50
GuidedPractice
y
5 10 15 20 25 30 35x
4. Determine whether the set of data shown below displays exponential behavior. Write yes or no. Explain why or why not. x
0
3
6
9
12
15
y
12
16
20
24
28
32
connectED.mcgrawhill.com
569
Check Your Understanding
= StepbyStep Solutions begin on page R12.
Examples 1–2 Graph each function. Find the yintercept and state the domain and range. Then use the graph to determine the approximate value of the given expression to the nearest tenth. Use a calculator to confirm the value. 1. y = 2 x; 2 1.5
2. y = 5 x; 5 0.5
x
1 1 3. y =  _ ; _
(5) (5)
x
0.5
1 1 4. y = 3 _ ;3 _
(4) (4)
0.5
Graph each function. Find the yintercept, and state the domain and range. 5. f(x) = 6 x + 3 Example 3
6. f(x) = 2  2 x
7. BIOLOGY The function f(t) = 100(1.05) t models the growth of a fruit fly population, where f(t) is the number of flies and t is time in days. a. What values for the domain and range are reasonable in the context of this situation? Explain. b. After two weeks, approximately how many flies are in this population?
Example 4
Determine whether the set of data shown below displays exponential behavior. Write yes or no. Explain why or why not. 8.
x
1
2
3
4
5
6
y
4
2
0
2
4
6
9.
x
2
4
6
8
10
12
y
1
4
16
64
256
1024
Practice and Problem Solving
Extra Practice begins on page 815.
Examples 1–2 Graph each function. Find the yintercept and state the domain and range. Then use the graph to determine the approximate value of the given expression to the nearest tenth. Use a calculator to confirm the value. 10. y = 2 · 8 x, 2(8) 0.5
1 1 x 11. y = 2 · _ ;2 _
13. y = 3 · 9 x, 3(9) 0.5
14. y = 4 · 10 x, 4(10) 0.5 15. y = 3 · 11 x, 3(11) 0.2
(6) (6)
1 x _ 12. y = _ ; 1
1.5
( 12 ) ( 12 )
0.5
Graph each function. Find the yintercept and state the domain and range. 1( x 17. y = _ 2  8)
16. y = 4 x + 3 Example 3
2
18. y = 5(3 x) + 1
19. y = 2(3 x) + 5
20. BIOLOGY A population of bacteria in a culture increases according to the model p = 300(2.7) 0.02t, where t is the number of hours and t = 0 corresponds to 9:00 a.m. a. Use this model to estimate the number of bacteria at 11 a.m. b. Graph the function and name the pintercept. Describe what the pintercept represents, and describe a reasonable domain and range for this situation.
Example 4
Determine whether the set of data shown below displays exponential behavior. Write yes or no. Explain why or why not. 21
23.
x
4
0
4
8
12
y
2
4
8
16
32
x
8
6
4
2
y
0.25
0.5
1
2
570  Lesson 96  Exponential Functions
22.
24.
x
6
3
0
3
y
5
10
15
20
x
20
30
40
50
60
y
1
0.4
0.16
0.064
0.0256
B
25 PHOTOGRAPHY Jameka is enlarging a photograph to make a poster for school. She will enlarge the picture repeatedly at 150%. The function P = 1.5 x models the new size of the picture being enlarged, where x is the number of enlargements. How many times as big is the picture after 4 enlargements? 26. FINANCIAL LITERACY Daniel invested $500 into a savings account. The equation A = 500(1.005) 12t models the value of Daniel’s investment A after t years. How much will Daniel’s investment be worth in 8 years? Identify each function as linear, quadratic, or exponential. y
27.
y
28.
y
29.
0 0
0
30. y = 4 x
x
x
x
31. y = 2x(x  1)
32. 5x + y = 8
33. GRADUATION The number of graduates at a high school has increased by a factor of 1.055 every year since 2001. In 2001, 110 students graduated. The function N = 110(1.055) t models the number of students N expected to graduate t years after 2001. How many students will graduate in 2012?
C
Describe the graph of each equation as a transformation of the graph of y = 2 x. 34. y = 2 x + 6 37. y = 3 + 2 x
35. y = 3(2) x 1 38. y = _
(2)
x
1 ( )x 36. y = _ 2 4
39. y = 5(2) x
40. DEER The deer population at a national park doubles every year. In 2000, there were 25 deer in the park. The function N = 25(2) t models the number of deer N in the park t years after 2000. What will the deer population be in 2015?
H.O.T. Problems
Use HigherOrder Thinking Skills
41. CHALLENGE Write an exponential function for which the graph passes through the points at (0, 3) and (1, 6). 42. REASONING Determine whether the graph of y = ab x, where a ≠ 0, b > 0, and b ≠ 1, sometimes, always, or never has an xintercept. Explain your reasoning. 43. OPEN ENDED Find an exponential function that represents a realworld situation, and graph the function. Analyze the graph. 44. REASONING Compare and contrast a function of the form y = ab x + c, where a ≠ 0, b > 0, and b ≠ 1 and a quadratic function of the form y = ax 2 + c. 45. WRITING IN MATH Explain how to determine whether a set of data displays exponential behavior. connectED.mcgrawhill.com
571
SPI 3102.3.10, SPI 3108.4.7, SPI 3102.3.3
Standardized Test Practice 46. SHORT RESPONSE What are the zeros of the function graphed below? y
48. GEOMETRY Ayana placed a circular piece of paper on a square picture as shown below. If the picture extends 4 inches beyond the circle on each side, what is the perimeter of the square picture? 20 in.
x
0
47. Hinto invested $300 into a savings account. The equation A = 300(1.005)12t models the amount in Hinto’s account A after t years. How much will be in Hinto’s account after 7 years? A $25,326 B $456.11
C $385.01 D $301.52
F 64 in. G 80 in.
H 94 in. J 112 in.
49. Which of the following shows 4x 2  8x  12 factored completely? A B C D
4(x  3)(x + 1) 4(x + 3)(x  1) (4x + 12)(x  1) (x  3)(4x + 4)
Spiral Review Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary. (Lesson 95) 50. 6x 2  3x  30 = 0
51. 4x 2 + 18x = 10
52. 2x 2 + 6x = 7
Solve each equation by taking the square root of each side. Round to the nearest tenth if necessary. (Lesson 94) 53. x 2 = 25
54. x 2 + 6x + 9 = 16
55. x 2  14x + 49 = 15
Evaluate each product. Express the results in both scientific notation and standard form. (Lesson 73) 56. (1.9 × 10 2)(4.7 × 10 6)
57. (4.5 × 10 3)(5.6 × 10 4)
58. (3.8 × 10 4)(6.4 × 10 8)
59. DEMOLITION DERBY When a car hits an object, the damage is measured by the collision impact. For a certain car the collision impact I is given by I = 2v 2, where v represents the speed in kilometers per minute. What is the collision impact if the speed of the car is 4 kilometers per minute? (Lesson 71) Use elimination to solve each system of equations. (Lesson 63) 60. x + y = 3 xy=1
61. 3a + b = 5 2a + b = 10
62. 3x  5y = 16 3x + 2y = 10
Skills Review Find the next three terms of each arithmetic sequence. (Lesson 35) 63. 1, 3, 5, 7, …
64. 6, 4, 2, 0, …
65. 6.5, 9, 11.5, 14,
66. 10, 3, 4, 11,
11 1 _ 67. _ , 5 , 2, _ ,… 4 2 4
3 _ 1 68. 1, _ , 1, _ ,…
572  Lesson 96  Exponential Functions
4 2 4
Growth and Decay
Growth of Blogs
Then
Now
Why?
You analyzed exponential functions.
1
Solve problems involving exponential growth.
2
Solve problems involving exponential decay.
The number of Weblogs or blogs increased at a monthly rate of about 13.7% over 21 months. The average number of blogs per month can be modeled by y = 1.1(1 + 0.137) t or y = 1.1(1.137) t, where y represents the total number of blogs in millions and t is the number of months since November 2003.
(Lesson 96)
NewVocabulary exponential growth compound interest exponential decay
1
Number (millions)
14 12 10 8 6 4 2 0
3
v.‘0
No
4 4 5 5 4 r.‘0 ul.‘0 ov.‘0 ar.‘0 ul.‘0 J J N Ma M
Month
Exponential Growth The equation for the number of blogs is in the form y = a(1 + r) t. This is the general equation for exponential growth.
KeyConcept Equation for Exponential Growth a is the initial amount.
t is time.
y = a (1 + r)t Tennessee Curriculum Standards CLE 3102.3.9 Understand and use exponential functions to solve contextual problems. ✔ 3102.3.33 Recognize data that can be modeled by an exponential function. ✔ 3102.3.35 Apply growth/ decay and simple/compound interest formulas to solve contextual problems.
y is the final amount.
r is the rate of change expressed as a decimal, r > 0.
RealWorld Example 1 Exponential Growth CONTEST The prize for a radio station contest begins with a $100 gift card. Once a day, a name is announced. The person has 15 minutes to call or the prize increases by 2.5% for the next day. a. Write an equation to represent the amount of the gift card in dollars after t days with no winners. y = a(1 + r) t y = 100(1 + y=
0.025) t
100(1.025)t
Equation for exponential growth a = 100 and r = 2.5% or 0.025 Simplify.
In the equation y = 100(1.025) t, y is the amount of the gift card and t is the number of days since the contest began. b. How much will the gift card be worth if no one wins after 10 days? Equation for amount of gift card y = 100(1.025) t 10 = 100(1.025) t = 10 ≈ 128.01 Use a calculator. In 10 days, the gift card will be worth $128.01.
GuidedPractice 1. TUITION A college’s tuition has risen 5% each year since 2000. If the tuition in 2000 was $10,850, write an equation for the amount of the tuition t years after 2000. Predict the cost of tuition for this college in 2015. connectED.mcgrawhill.com
573
Compound interest is interest earned or paid on both the initial investment and previously earned interest. It is an application of exponential growth.
KeyConcept Equation for Compound Interest n is the number of times the interest is compounded each year, and t is time in years.
A is the current amount.
A = P (1 + _nr )
nt
P is the principal or initial amount.
r is the annual interest rate expressed as a decimal, r > 0.
RealWorld Example 2 Compound Interest FINANCE Maria’s parents invested $14,000 at 6% per year compounded monthly. How much money will there be in the account after 10 years? r A = P(1 + _ n)
nt
P = 14,000, r = 6% or 0.06, n = 12, and t = 10
= 14,000(1.005) 120
Simplify.
≈ 25,471.55
Use a calculator.
(
RealWorldCareer Financial advisors help people plan their financial futures. A good financial advisor has mathematical, problemsolving, and communication skills. A bachelor’s degree is strongly preferred but not required.
Compound interest equation
0.06 12(10) = 14,000 1 + _ 12
)
There will be about $25,471.55 in 10 years.
GuidedPractice 2. FINANCE Determine the amount of an investment if $300 is invested at an interest rate of 3.5% compounded monthly for 22 years.
Exponential Decay In exponential decay, the original amount decreases by the same percent over a period of time. A variation of the growth equation can be used as the general equation for exponential decay. StudyTip
KeyConcept Equation for Exponential Decay
Growth and Decay Since r is added to 1, the value inside the parentheses will be greater than 1 for exponential growth functions. For exponential decay functions, this value will be less than 1 since r is subtracted from 1.
a is the initial amount.
t is time.
y = a (1  r ) t y is the final amount.
r is the rate of decay expressed as a decimal, 0 < r < 1.
RealWorld Example 3 Exponential Decay SWIMMING A fully inflated child’s raft for a pool is losing 6.6% of its air every day. The raft originally contained 4500 cubic inches of air. a. Write an equation to represent the loss of air. y = a(1  r) t
Equation for exponential decay
= 4500(1  0.066) t
a = 4500 and r = 6.6% or 0.066
= 4500(0.934) t
Simplify.
y = 4500(0.934) t, where y is the air in the raft in cubic inches after t days.
574  Lesson 97  Growth and Decay
b. Estimate the amount of air in the raft after 7 days. Equation for air loss y = 4500(0.934) t = 4500(0.934) 7 t=7 ≈ 2790 Use a calculator. The amount of air in the raft after 7 days will be about 2790 cubic inches.
GuidedPractice 3. POPULATION The population of Campbell County, Kentucky, has been decreasing at an average rate of about 0.3% per year. In 2000, its population was 88,647. Write an equation to represent the population since 2000. If the trend continues, predict the population in 2010.
Check Your Understanding
= StepbyStep Solutions begin on page R12.
Example 1
1. SALARY Ms. Acosta received a job as a teacher with a starting salary of $34,000. According to her contract, she will receive a 1.5% increase in her salary every year. How much will Ms. Acosta earn in 7 years?
Example 2
2. MONEY Paul invested $400 into an account with a 5.5% interest rate compounded monthly. How much will Paul’s investment be worth in 8 years?
Example 3
3. ENROLLMENT In 2000, 2200 students attended Polaris High School. The enrollment has been declining 2% annually. a. Write an equation for the enrollment of Polaris High School t years after 2000. b. If this trend continues, how many students will be enrolled in 2015?
Practice and Problem Solving Example 1
Extra Practice begins on page 815.
4. MEMBERSHIPS The WorkOut Gym sold 550 memberships in 2001. Since then the number of memberships sold has increased 3% annually. a. Write an equation for the number of memberships sold at WorkOut Gym t years after 2001. b. If this trend continues, predict how many memberships the gym will sell in 2020. 5. COMPUTERS The number of people who own computers has increased 23.2% annually since 1990. If half a million people owned a computer in 1990, predict how many people will own a computer in 2015. 6. COINS Camilo purchased a rare coin from a dealer for $300. The value of the coin increases 5% each year. Determine the value of the coin in 5 years.
Example 2
7 INVESTMENTS Theo invested $6600 at an interest rate of 4.5% compounded monthly. Determine the value of his investment in 4 years. 8. COMPOUND INTEREST Paige invested $1200 at an interest rate of 5.75% compounded quarterly. Determine the value of her investment in 7 years. 9. SAVINGS Brooke is saving money for a trip to the Bahamas that costs $295.99. She puts $150 into a savings account that pays 7.25% interest compounded quarterly. Will she have enough money in the account after 4 years? Explain. 10. INVESTMENTS Jin’s investment of $4500 has been losing its value at a rate of 2.5% each year. What will his investment be worth in 5 years? connectED.mcgrawhill.com
575
Example 3
11 POPULATION Hawaii has been experiencing a 1.06% annual increase in population. In 2000, the population was 1,211,537. If this trend continues, what will be the population of Hawaii in 2020? 12. CARS Leonardo purchases a car for $18,995. The car depreciates at a rate of 18% annually. After 6 years, Manuel offers to buy the car for $4500. Should Leonardo sell the car? Explain.
B
13. HOUSING The median house price in the United States increased an average of 1.4% each year between 2005 and 2007. Assume that this pattern continues. a. Write an equation for the median house price for t years after 2004. b. Predict the median house price in 2018.
C
Median House Price
14. ELEMENTS A radioactive element’s halflife is the time it takes for one half of the element’s quantity to decay. The halflife of Plutonium241 is 14.4 years. The number of grams A of Plutonium241 left after t years can be t _
2005
$240,900
2006
$246,500
2007
$247,900
Source: Real Estate Journal
modeled by A = p(0.5) 14.4 , where p is the original amount of the element.
a. How much of a 0.2gram sample remains after 72 years? b. How much of a 5.4gram sample remains after 1095 days? 15. FINANCIAL LITERACY Marta is planning to buy a new car. She will finance $16,000 at an annual interest rate of 7% over a period of 60 months. In the formula, P is the amount of each payment, r is the annual interest rate in decimal form, and t is the time in years of the loan. r 12t ⎤ ⎡ 1 (1 + _ 12 ) __ Amount financed = P ⎢
⎣
r _ 12
⎦
a. Use the formula to find her monthly payment. b. Assuming that she does not pay ahead, what will she have paid on the car?
H.O.T. Problems
Use HigherOrder Thinking Skills
16. REASONING Determine the growth rate (as a percent) of a population that quadruples every year. Explain. 17. CHALLENGE Santos invested $1200 into an account with an interest rate of 8% compounded monthly. Use a calculator to approximate how long it will take for Santos’ investment to reach $2500. 18. REASONING The amount of water in a container doubles every minute. After 8 minutes, the container is full. After how many minutes was the container half full? Explain. 19. OPEN ENDED Create a realworld situation that can be modeled by y = 200(1.05) t. 20. WRITING IN MATH Compare and contrast the exponential growth formula and the exponential decay formula.
576  Lesson 97  Growth and Decay
SPI 3108.4.7, SPI 3102.3.10, SPI 3102.1.4, SPI 3102.1.2
Standardized Test Practice 21. GEOMETRY The parallelogram has an area of 35 square inches. Find the height h of the parallelogram. 2h  3
23. Thi purchased a car for $22,900. The car depreciated at an annual rate of 16%. Which of the following equations models the value of Thi’s car after 5 years? A A = 22,900(1.16) 5
h
B A = 22,900(0.16) 5 C A = 16(22,900) 5
A 3.5 inches B 4 inches
D A = 22,900(0.84) 5
C 5 inches D 7 inches
24. GRIDDED RESPONSE A deck measures 12 feet by 18 feet. If a painter charges $2.65 per square foot, including tax, how much will it cost in dollars to have the deck painted?
22. What are the roots of x 2 + 2x = 48? F 6 and 8 G 6 and 8
H 6 and 8 J 6 and 8
Spiral Review Graph each function. Find the yintercept and state the domain and range. (Lesson 96) 25. y = 3 x
1 26. y = _
(2)
x
27. y = 6 x
Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary. (Lesson 95) 28. 4x 2 + 15x = 25 31. 4x 2 + 16x = 16
29. 3x 2  4x = 5 32. 5x 2 + 5x = 60
30. 2x 2 = 2x + 11 33. 2x 2 = 3x + 15
34. EVENT PLANNING A hall does not charge a rental fee as long as at least $4000 is spent on food. For the prom, the hall charges $28.95 per person for a buffet. How many people must attend the prom to avoid a rental fee for the hall? (Lesson 52) Determine whether the graphs of each pair of equations are parallel, perpendicular, or neither. (Lesson 44) 35. y = 2x + 11 y + 2x = 23
36. 3y = 2x + 14 3x  2y = 2
38. AGES The table shows equivalent ages for horses and humans. Write an equation that relates human age to horse age and find the equivalent horse age for a human who is 16 years old. (Lesson 34)
37. y = 5x y = 5x  18 Horse age (x)
0
1
2
3
4
5
Human age (y)
0
3
6
9
12
15
Find the total price of each item. (Lesson 27) 39. umbrella: $14.00 tax: 5.5%
40. sandals: $29.99 tax: 5.75%
41. backpack: $35.00 tax: 7%
Skills Review Graph each set of ordered pairs. (Lesson 16) 42. (3, 0), (0, 1), (4, 6)
43. (0, 2), (1, 6), (3, 4)
44. (2, 2), (2, 3), (3, 6) connectED.mcgrawhill.com
577
Geometric Sequences as Exponential Functions Then
Now
Why?
You related arithmetic sequences to linear functions.
1 2
You send a chain email to a friend who forwards the email to five more people. Each of these five people forwards the email to five more people. The number of new emails generated forms a geometric sequence.
(Lesson 35)
NewVocabulary geometric sequence common ratio
Tennessee Curriculum Standards SPI 3102.1.1 Interpret patterns found in sequences, tables, and other forms of quantitative information using variables or function notation. CLE 3102.3.1 Use algebraic thinking to analyze and generalize patterns. ✔ 3102.3.1 Recognize and extend arithmetic and geometric sequences. Also addresses ✓3102.3.2.
Identify and generate geometric sequences. Relate geometric sequences to exponential functions.
1
Recognize Geometric Sequences The first person generates 5 emails. If
each of these people sends the email to 5 more people, 25 emails are generated. If each of the 25 people sends 5 emails, 125 emails are generated. The sequence of emails generated, 1, 5, 25, 125, … is an example of a geometric sequence. In a geometric sequence, the first term is nonzero and each term after the first is found by multiplying the previous term by a nonzero constant r called the common ratio. The common ratio can be found by dividing any term by its previous term.
Example 1 Identify Geometric Sequences Determine whether each sequence is arithmetic, geometric, or neither. Explain. a. 256, 128, 64, 32, … Find the ratios of consecutive terms. 256 128 64 128 1 _ =_ 256
2
64 1 _ =_ 128
32 32 1 _ =_
2
64
2
1 Since the ratios are constant, the sequence is geometric. The common ratio is _ . 2
b. 4, 9, 12, 18, … Find the ratios of consecutive terms. 4 9 12
_9 = 2_1 4
4
12 _ 1 = 1_ 9
3
18 18 _ 1 = 1_ 12
2
The ratios are not constant, so the sequence is not geometric. Find the differences of consecutive terms. 4 9 12 94=5
12  9 = 3
18
18  12 = 6
There is no common difference, so the sequence is not arithmetic. Thus, the sequence is neither geometric nor arithmetic.
GuidedPractice 1A. 1, 3, 9, 27, …
578  Lesson 98
1B. 20, 15, 10, 5, … 1C. 2, 8, 14, 22, …
Once the common ratio is known, more terms of a sequence can be generated. The recursive formula can be rewritten as a n = a 1r n  1, where n is a counting number and r is the common ratio.
Example 2 Find Terms of Geometric Sequences Find the next three terms in each geometric sequence.
StudyTip Common Ratio If the terms of a geometric sequence alternate between positive and negative terms or vice versa, the common ratio is negative.
a. 1, 4, 16, 64, … Step 1 Find the common ratio. 4
1 4 _ = 4
16 _ = 4
64 _ = 4
4
1
64
16
16
Step 2 Multiply each term by the common ratio to find the next three terms. 64
1024
256 ×(4)
×(4)
4096 ×(4)
The next three terms are 256, 1024, and 4096. 1 b. 9, 3, 1, _ … 3
Step 1 Find the common ratio. 9
3
_3 = _1 9
3
3
_1
_1 = _1
3
_1
1
_3 = _1
3
3
3
1 The value of r is _ . 3
Step 2 Multiply each term by the common ratio to find the next three terms.
_1
_1
3
_
Math HistoryLink Thomas Robert Malthus (1766–1834) Malthus studied populations and had pessimistic views about the future population of the world. In his work, he stated: “Population increases in a geometric ratio, while the means of subsistence increases in an arithmetic ratio.”
1 _
9
_
×1
81
_
×1
3
1 _
27
×1
3
3
1 _ 1 The next three terms are _ , 1 , and _ . 9 27 81
GuidedPractice 2A. 3, 15, 75, 375, …
2B. 24, 36, 54, 81, …
2 Geometric Sequences and Functions
Finding the nth term of a geometric sequence would be tedious if we used the above method. The table below shows a rule for finding the nth term of a geometric sequence. Position, n Term, a n
1 a1
2 a 1r
3 a 1r
4 2
a 1r
3
…
n
…
a 1r n  1
Notice that the common ratio between the terms is r. The table shows that to get the nth term, you multiply the first term by the common ratio r raised to the power n  1. A geometric sequence can be defined by an exponential function in which n is the independent variable, a n is the dependent variable, and r is the base. The domain is the counting numbers. connectED.mcgrawhill.com
579
KeyConcept nth term of a Geometric Sequence The nth term a n of a geometric sequence with first term a 1 and common ratio r is given by the following formula, where n is any positive integer and a 1, r ≠ 0. a n = a 1r n  1
Example 3 Find the nth Term of a Geometric Sequence a. Write an equation for the nth term of the sequence 6, 12, 24, 48, … . The first term of the sequence is 6. So, a 1 = 6. Now find the common ratio. 6 12 _ = 2
24 _ = 2
6
WatchOut! Negative Common Ratio If the common ratio is negative, as in Example 3, make sure to enclose the common ratio in parentheses. (2) 8 ≠ 2 8
24
12
a n = a 1r
12
n1
a n = 6(2)
48 48 _ = 2
The common ratio is 2.
24
Formula for nth term n1
a 1 = 6 and r = 2
b. Find the ninth term of this sequence. a n = a 1r n  1
Formula for nth term
a 9 = 6(2) 9  1
For the nth term, n = 9.
= 6(2) 8
Simplify.
= 6(256)
(2) 8 = 256
= 1536
GuidedPractice 3. Write an equation for the nth term of the geometric sequence 96, 48, 24, 12, … . Then find the tenth term of the sequence.
RealWorld Example 4 Graph a Geometric Sequence BASKETBALL The NCAA women’s basketball tournament begins with 64 teams. In each round, one half of the teams are left to compete, until only one team remains. Draw a graph to represent how many teams are left in each round. Compared to the previous rounds, one half of the 1 teams remain. So, r = _ . Therefore, the geometric 2
RealWorldLink The first NCAA Division I women’s basketball tournament was held in 1982. The University of Tennessee has won the most national titles with 8 titles as of 2008. Source: NCAA Sports
sequence that models this situation is 64, 32, 16, 8, 4, 2, 1. So in round two, 32 teams compete, in round three 16 teams compete and so forth. Use this information to draw a graph.
64 56 48 40 32 24 16 8 0
1 2 3 4 5 6 7 8
GuidedPractice 4. TENNIS A tennis ball is dropped from a height of 12 feet. Each time the ball bounces back to 80% of the height from which it fell. Draw a graph to represent the height of the ball after each bounce.
580  Lesson 98  Geometric Sequences as Exponential Functions
Check Your Understanding Example 1
Determine whether each sequence is arithmetic, geometric, or neither. Explain. 1. 200, 40, 8, …
Example 2
2. 2, 4, 16, …
3. 6, 3, 0, 3, …
4. 1, 1, 1, 1, …
Find the next three terms in each geometric sequence. 5. 10, 20, 40, 80, …
Example 3
= StepbyStep Solutions begin on page R12.
6. 100, 50, 25, …
1 7. 4, 1, _ ,… 4
8. 7, 21, 63, …
Write an equation for the nth term of each geometric sequence, and find the indicated term. 9. the fifth term of 6, 24, 96, … 10. the seventh term of 1, 5, 25, … 11. the tenth term of 72, 48, 32, … 12. the ninth term of 112, 84, 63, …
Example 4
13. EXPERIMENT In a physics class experiment, Diana drops a ball from a height of 16 feet. Each bounce has 70% the height of the previous bounce. Draw a graph to represent the height of the ball after each bounce.
Practice and Problem Solving Example 1
Example 2
Example 3
Extra Practice begins on page 815.
Determine whether each sequence is arithmetic, geometric, or neither. Explain. 14. 4, 1, 2, …
15. 10, 20, 30, 40, …
16. 4, 20, 100, …
17. 212, 106, 53, …
18. 10, 8, 6, 4, …
19. 5, 10, 20, 40, …
Find the next three terms in each geometric sequence. 20. 2, 10, 50, …
21 36, 12, 4, …
22. 4, 12, 36, …
23. 400, 100, 25, …
24. 6, 42, 294, …
25. 1024, 128, 16, …
26. The first term of a geometric series is 1 and the common ratio is 9. What is the 8th term of the sequence? 27. The first term of a geometric series is 2 and the common ratio is 4. What is the 14th term of the sequence? 28. What is the 15th term of the geometric sequence 9, 27, 81, …? 29. What is the 10th term of the geometric sequence 6, 24, 96, …?
Example 4
30. PENDULUM The first swing of a pendulum is shown. On each swing after that, the arc length is 60% of the length of the previous swing. Draw a graph that represents the arc length after each swing. GU
31. Find the eighth term of a geometric sequence for which a 3 = 81 and r = 3.
B
32. MAPS At an online mapping site, Mr. Mosley notices that when he clicks a spot on the map, the map zooms in on that spot. The magnification increases by 20% each time. a. Write a formula for the nth term of the geometric sequence that represents the magnification of each zoom level. (Hint: The common ratio is not just 0.2.) b. What is the fourth term of this sequence? What does it represent? connectED.mcgrawhill.com
581
33 ALLOWANCE Danielle’s parents have offered her two different options to earn her allowance for a 9week period over the summer. She can either get paid $30 each week or $1 the first week, $2 for the second week, $4 for the third week, and so on. a. Does the second option form a geometric sequence? Explain. b. Which option should Danielle choose? Explain.
C
34. SIERPINSKI’S TRIANGLE Consider the inscribed equilateral triangles at the right. The perimeter of each triangle is one half of the perimeter of the next larger triangle. What is the perimeter of the smallest triangle? 40 cm
35. If the second term of a geometric sequence is 3 and the third term is 1, find the first and fourth terms of the sequence. 36. If the third term of a geometric sequence is 12 and the fourth term is 24, find the first and fifth terms of the sequence. 37. EARTHQUAKES The Richter scale is used to measure the force of an earthquake. The table shows the increase in magnitude for the values on the Richter scale. a. Copy and complete the table. Remember that the rate of change is the change in y divided by the change in x.
Richter Number (x)
Increase in Magnitude (x)
Rate of Change (slope)
1
1

2
10
9
3
100
4
1000
5
10,000
b. Plot the ordered pairs (Richter number, increase in magnitude). c. Describe the graph that you made of the Richter scale data. Is the rate of change between any two points the same? d. Write an exponential equation that represents the Richter scale.
H.O.T. Problems
Use HigherOrder Thinking Skills
38. CHALLENGE Write a sequence that is both geometric and arithmetic. Explain your answer. 39. ERROR ANALYSIS Haro and Matthew are finding the ninth term of the geometric sequence 5, 10, 20, … . Is either of them correct? Explain your reasoning.
Haro 10 r=_ –5 or –2
a 9 = –5 (–2) 9 – 1 = –5(512) = –2560
Matthew 10 r=_ –5 or –2 a 9 = –5 (–2) 9 – 1 = –5 –256 = 1280
40. REASONING Write a sequence of numbers that form a pattern but are neither arithmetic nor geometric. Explain the pattern. 3 . 41. OPEN ENDED Write a geometric sequence that has a common ratio of _ 4
42. WRITING IN MATH Summarize how to find a specific term of a geometric sequence.
582  Lesson 98  Geometric Sequences as Exponential Functions
SPI 3102.1.1, SPI 3102.3.6, SPI 3102.3.5, SPI 3108.4.7
Standardized Test Practice 43. Find the eleventh term of the sequence 3, 6, 12, 24, … . A 6144 C 33 B 3072 D 6144 44. What is the total amount of the investment shown in the table below if interest is compounded monthly? Principal
$500
Length of Investment
4 years
Annual Interest Rate
5.25%
F $613.56 G $616.00
45. SHORT RESPONSE Gloria has $6.50 in quarters and dimes. If she has 35 coins in total, how many of each coin does she have? 46. A sidewalk is being built along the inside edges of all four sides of a rectangular lawn. The lawn is 32 feet long and 24 feet wide. The remaining lawn will have an area of 425 square feet. How wide will the sidewalk be? A B C D
H $616.56 J $718.75
3.5 feet 17 feet 24.5 feet 25 feet
Spiral Review Find the next three terms in each geometric sequence. (Lesson 97) 47. 2, 6, 18, 54, …
48. 5, 10, 20, 40, …
1 _ 1 49. 1, _ , 1 , _ ,…
50. 3, 1.5, 0.75, 0.375, …
51. 1, 0.6, 0.36, 0.216, …
52. 4, 6, 9, 13.5, …
2 4
8
Graph each function. Find the yintercept and state the domain and range. (Lesson 96) 1 53. y = _
(4)
x
5
1 x 55. y = _ (3 )
54. y = 2(4) x
2
56. LANDSCAPING A blue spruce grows an average of 6 inches per year. A hemlock grows an average of 4 inches per year. If a blue spruce is 4 feet tall and a hemlock is 6 feet tall, write a system of equations to represent their growth. Find and interpret the solution in the context of the situation. (Lesson 62) 57. MONEY City Bank requires a minimum balance of $1500 to maintain free checking services. If Mr. Hayashi is going to write checks for the amounts listed in the table, how much money should he start with in order to have free checking? (Lesson 51)
Check
Amount
750
$1300
751
$947
Write an equation in slopeintercept form of the line with the given slope and yintercept. (Lesson 41) 58. slope: 4, yintercept: 2
2 59. slope: 3, yintercept: _
1 60. slope: _ , yintercept: 5
1 61. slope: _ , yintercept: 9
3 2 , yintercept: _ 62. slope: _ 4 5
63. slope: 6, yintercept: 7
4
3
2
Skills Review Evaluate a(1 + r) t to the nearest hundredth for each of the given values. (Lesson 12) 64. a = 20, r = 0.25, t = 5
65. a = 1000, r = 0.65, t = 4
66. a = 200, r = 0.35, t = 8
67. a = 60, r = 0.2, t = 10
68. a = 8, r = 0.5, t = 2
69. a = 500, r = 0.55, t = 12 connectED.mcgrawhill.com
583
Analyzing Functions with Successive Differences Then
Now
Why?
You graphed linear, quadratic, and exponential functions. (Lessons
1
Identify linear, quadratic, and exponential functions from given data.
2
Write equations that model data.
Every year the golf team sells candy to raise money for charity. By knowing what type of function models the sales of the candy, they can determine the best price of the candy.
32, 91, 96)
Tennessee Curriculum Standards ✔ 3102.1.4 Write a rule with variables that expresses a pattern. SPI 3102.1.1 Interpret patterns found in sequences, tables, and other forms of quantitative information using variables or function notation.
1 Identify Functions
You can use linear functions, quadratic functions, and exponential functions to model data. The general forms of the equations and a graph of each function type are listed below.
ConceptSummary Linear and Nonlinear Functions Linear Function
Quadratic Function
Exponential Function
2
y = ab x, when b > 0
y = ax + bx + c
y = mx + b
y
y
0
y
x
x
0
x
0
Example 1 Choose a Model Using Graphs Graph each set of ordered pairs. Determine whether the ordered pairs represent a linear function, a quadratic function, or an exponential function. a. {(2, 5), (1, 2), (0, 1), (1, 2), (2, 5)} The ordered pairs appear to represent a quadratic function.
⎫ ⎧ b. ⎨ 2, 1 , 1, 1 , (0, 1), (1, 2), (2, 4)⎬
(⎩ _4 ) ( _2 )
y
0
⎭
The ordered pairs appear to represent an exponential function. y
x
0
x
GuidedPractice 1A. (2, 3), (1, 1), (0, 1), (1, 3)
584  Lesson 99
1B. (1, 0.25), (0, 1), (1, 4), (2, 16)
Another way to determine which model best describes data is to use patterns. The differences of successive yvalues are called first differences. The differences of successive first differences are called second differences. • If the differences of successive yvalues are all equal, the data represent a linear function. • If the second differences are all equal, but the first differences are not equal, the data represent a quadratic function. • If the ratios of successive yvalues are all equal and r ≠ 1, the data represent an exponential function.
WatchOut! xValues Before you check for successive differences or ratios, make sure the xvalues are increasing by the same amount.
Example 2 Choose a Model Using Differences or Ratios Look for a pattern in each table of values to determine which kind of model best describes the data. a.
x
2
1
0
1
2
y
8
3
2
7
12
8
First differences:
3 5
2
5
7 5
12 5
Since the first differences are all equal, the table of values represents a linear function. b.
x
1
0
1
2
3
y
8
4
2
1
0.5
8
First differences:
4
2
1
0.5
4 2 1 0.5
The first differences are not all equal. So, the table of values does not represent a linear function. Find the second differences and compare. 4
First differences: Second differences:
2
1
2
1
0.5 0.5
The second differences are not all equal. So, the table of values does not represent a quadratic function. Find the ratios of the yvalues and compare. 8 Ratios:
4
2
_4 = _1 _2 = _1 8
2
4
1
_1
2
0.5
0.5 1 _ =_
2
1
2
The ratios of successive yvalues are equal. Therefore, the table of values can be modeled by an exponential function.
GuidedPractice 2A.
x
3
2
1
0
1
y
3
7
9
9
7
2B.
x
2
1
0
1
2
y
18
13
8
3
2
2 Write Equations
Once you find the model that best describes the data, you can write an equation for the function. For a quadratic function in this lesson, the equation will have the form y = ax 2. connectED.mcgrawhill.com
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Example 3 Write an Equation Determine which kind of model best describes the data. Then write an equation for the function that models the data. Step 1 Determine which model fits the data. 32 18 8 14
First differences: Second differences:
10 4
2 6
4
0 2
4
Since the second differences are equal, a quadratic function models the data.
WatchOut! Finding a In Example 3, the point (0, 0) cannot be used to find the value of a. You will have to divide each side by 0, giving you an undefined value for a.
Step 2 Write an equation for the function that models the data. The equation has the form y = ax 2. Find the value of a by choosing one of the ordered pairs from the table of values. Let’s use (1, 2). y = ax 2
Equation for quadratic function
2 = a(1) 2
x = 1 and y = 2
2=a
An equation that models the data is y = 2x 2.
GuidedPractice 3A.
x
2
1
0
1
2
y
11
7
3
1
5
3B.
x
3
2
1
0
1
y
0.375
0.75
1.5
3
6
RealWorld Example 4 Write an Equation for a RealWorld Situation BOOK CLUB The table shows the number of book club members for four consecutive years. Determine which model best represents the data. Then write a function that models the data. Understand We need to find a model for the data, and then write a function.
Time (years)
0
1
2
3
4
Members
5
10
20
40
80
Plan Find a pattern using successive differences or ratios. Then use the general form of the equation to write a function.
RealWorldLink A poll by the National Education Association found that 87% of all teens polled found reading relaxing, 85% viewed reading as rewarding, and 79% found reading exciting.
Solve The constant ratio is 2. This is the value of the base. An exponential function of the form y = ab x models the data. y = ab x Equation for exponential function 5 = a(2) 0
x = 0, y = 5, and b = 2
5=a
The equation that models the data is y = 5 · 2 x.
Check You used (0, 5) to write the function. Verify that every other ordered pair satisfies the equation.
Source: American Demographics
GuidedPractice 4. ADVERTISING The table shows the cost of placing an ad in a newspaper. Determine a model that best represents the data and write a function that models the data.
586  Lesson 99  Analyzing Functions with Successive Differences
No. of Lines
5
6
7
8
Total Cost ($)
14.50
16.60
18.70
20.80
Check Your Understanding Example 1
Example 2
Graph each set of ordered pairs. Determine whether the ordered pairs represent a linear function, a quadratic function, or an exponential function. 1. (2, 8), (1, 5), (0, 2), (1, 1)
2. (3, 7), (2, 3), (1, 1), (0, 1), (1, 3)
3. (3, 8), (2, 4), (1, 2), (0, 1), (1, 0.5)
4. (0, 2), (1, 2.5), (2, 3), (3, 3.5)
Look for a pattern in each table of values to determine which kind of model best describes the data. 5.
7.
Example 3
6.
x
0
1
2
3
4
y
5
8
17
32
53
x
1
0
1
2
3
y
3
6
12
24
48
8.
x
3
2
1
0
y
6.75
7.5
8.25
9
x
3
4
5
6
7
y
1.5
0
2.5
6
10.5
Determine which kind of model best describes the data. Then write an equation for the function that models the data. 9.
11.
Example 4
= StepbyStep Solutions begin on page R12.
10.
x
1
0
1
2
3
y
1
3
9
27
81
x
3
2
1
0
1
y
1
1.5
2
2.5
3
12.
13. PLANTS The table shows the height of a plant for four consecutive weeks. Determine which kind of function best models the height. Then write a function that models the data.
x
5
4
3
2
1
y
125
80
45
20
5
x
1
0
1
2
y
1.25
1
0.75
0.5
Week
0
1
2
3
4
Height (in.)
3
3.5
4
4.5
5
Practice and Problem Solving Example 1
Extra Practice begins on page 815.
Graph each set of ordered pairs. Determine whether the ordered pairs represent a linear function, a quadratic function, or an exponential function. 14. (1, 1), (0, 2), (1, 3), (2, 2), (3, 1)
15. (1, 2.75), (2, 2.5), (3, 2.25), (4, 2)
16. (3, 0.25), (2, 0.5), (1, 1), (0, 2)
17. (3, 11), (2, 5), (1, 3), (0, 5)
18. (2, 6), (1, 1), (0, 4), (1, 9)
19. (1, 8), (0, 2), (1, 0.5), (2, 0.125)
Examples 2–3 Look for a pattern in each table of values to determine which kind of model best describes the data. Then write an equation for the function that models the data. 20.
22.
24.
21
x
3
2
1
0
y
8.8
8.6
8.4
8.2
x
1
0
1
2
3
y
0.75
3
12
48
192
x
0
1
2
3
4
y
0
4.2
16.8
37.8
67.2
23.
25.
x
2
1
0
1
2
y
10
2.5
0
2.5
10
x
2
1
0
1
2
y
0.008
0.04
0.2
1
5
x
3
2
1
0
1
y
14.75
9.75
4.75
0.25
5.25
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Example 4
B
26. WEBSITES A company launched a new Web site. They tracked the number of visitors to its Web site over a period of 4 days. Determine which kind of model best represents the number of visitors to the Web site with respect to time. Then write a function that models the data. Day
0
1
2
3
4
Visitors (in thousands)
0
0.9
3.6
8.1
14.4
27 LONG DISTANCE The cost of a longdistance telephone call depends on the length of the call. The table shows the cost for up to 6 minutes. Length of call (min) Cost ($)
1
2
3
4
5
6
0.12
0.24
0.36
0.48
0.60
0.72
a. Graph the data and determine which kind of function best models the data. b. Write an equation for the function that models the data. c. Use your equation to determine how much a 10minute call would cost. 28. DEPRECIATION The value of a car depreciates over time. The table shows the value of a car over a period of time. Year Value ($)
0
1
2
3
4
18,500
15,910
13,682.60
11,767.04
10,119.65
a. Determine which kind of function best models the data. b. Write an equation for the function that models the data. c. Use your equation to determine how much the car is worth after 7 years.
C
29. BACTERIA A scientist estimates that a bacteria culture with an initial population of 12 will triple every hour. a. Make a table to show the bacteria population for the first 4 hours. b. Which kind of model best represents the data? c. Write a function that models the data. d. How many bacteria will there be after 8 hours? 30. PRINTING A printing company charges the fees shown to print flyers. Write a function that models the total cost of the flyers, and determine how much 30 flyers would cost.
H.O.T. Problems
Set Up Fee $25 15¢ each flyer
Use HigherOrder Thinking Skills
31. CHALLENGE Write a function that has constant second differences, first differences that are not constant, a yintercept of 5, and passes through the point at (2, 3). 32. REASONING What type of function will have a constant third differences but not constant second differences? Explain. 33. OPEN ENDED Write a linear function that has a constant first difference of 4. 34. REASONING If data can be modeled by a quadratic function, what is the relationship between the coefficient of x 2 and the constant second difference? 35. WRITING IN MATH Summarize how to determine whether a given set of data is modeled by a linear function, a quadratic function, or an exponential function.
588  Lesson 99  Analyzing Functions with Successive Differences
SPI 3102.1.1, SPI 3102.3.8, SPI 3102.3.9, SPI 3108.4.7
Standardized Test Practice 38. The point (r, 4) lies on a line with an equation of 2x + 3y = 8. Find the value of r. F 10 H 2 G 0 J 8
36. SHORT RESPONSE Write an equation that models the data in the table. x
0
1
2
3
4
y
3
6
12
24
48
37. What is the equation of the line below? 2 A y=_ x+2 5
y
2 B y=_ x2 5
5 C y=_ x+2 2
0
x
39. GEOMETRY The rectangle has an area of 220 square feet. Find the length ℓ. A 8 feet B 10 feet + 12 C 22 feet D 34 feet
5 x2 D y=_
2
Spiral Review 40. INVESTMENTS Joey’s investment of $2500 has been decreasing in value at a rate of 1.5% each year. What will his investment be worth in 5 years? (Lesson 98) Write an equation for the nth term of each geometric sequence, and find the seventh term of each sequence. (Lesson 97) 41. 1, 2, 4, 8, …
42. 20, 10, 5, …
43. 4, 12, 36, …
44. 99, 33, 11, …
45. 22, 44, 88, …
1 2 _ 46. _ , 1, _ ,…
47. (x  4) 2
48. (2y + 3) 2
49. (4x  7) 2
50. (a  5)(a + 5)
51. (5x  6y)(5x + 6y)
52. (9c  2d 2)(9c + 2d 2)
3 3 6
Find each product. (Lesson 78)
53. CANOE RENTAL To rent a canoe, you must pay a daily rate plus $10 per hour. Ilia and her friends rented a canoe for 3 hours and paid $45. Write a linear equation for the cost C of renting the canoe for h hours, and determine how much it cost to rent the canoe for 8 hours. (Lesson 42) Determine whether each equation is a linear equation. If so, write the equation in standard form. (Lesson 31) 54. 3x = 5y
55. 6  y = 2x
56. 6xy + 3x = 4
57. y + 5 = 0
58. 7y = 2x + 5x
59. y = 4x 2  1
61. g(x) = ⎪3x + 4⎥
1 62. f(x) = _ x+5
Skills Review Graph each function. (Lesson 47) 60. f(x) = ⎪x  2⎥
⎪2
⎥
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Graphing Technology Lab
Curve Fitting If there is a constant increase or decrease in data values, there is a linear trend. If the values are increasing or decreasing more and more rapidly, there may be a quadratic or exponential trend. Linear Trend
Quadratic Trend
Exponential Trend
[0, 5] scl: 1 by [0, 6] scl: 1
[0, 5] scl: 1 by [0, 6] scl: 1
[0, 5] scl: 1 by [0, 6] scl: 1
Tennessee Curriculum Standards ✔ 3102.5.1 Identify patterns or trends in data. Also addresses ✓3102.5.8 and ✓3102.5.11.
With a graphing calculator, you can find the appropriate regression equation.
Activity CHARTER AIRLINE The table shows the average monthly number of flights made each year by a charter airline that was founded in 2000. Year
2000
2001
2002
2003
2004
2005
2006
2007
Flights
17
20
24
28
33
38
44
50
Step 1 Make a scatter plot. • Enter the number of years since 2000 in L1 and the number of flights in L2. KEYSTROKES:
Review entering a list on page 253.
• Use STAT PLOT to graph the scatter plot. KEYSTROKES:
Review statistical plots on page 254. Use
9 to graph.
[0, 10] scl: 1 by [0, 60] scl: 5
From the scatter plot we can see that the data may have either a quadratic trend or an exponential trend.
590  Extend 99  Graphing Technology Lab: Curve Fitting
Step 2 Find the regression equation. We will check both trends by examining their regression equations. • Select DiagnosticOn from the CATALOG. • Select QuadReg on the KEYSTROKES:
menu.
5
The equation is in the form y = ax 2 + bx + c.
The equation is about y = 0.25x 2 + 3x + 17. R 2 is the coefficient of determination. The closer R 2 is to 1, the better the model. To acquire the exponential equation select ExpReg menu. To choose a quadratic on the or exponential model, fit both and use the one with the R 2 value closer to 1.
Step 3 Graph the quadratic regression equation.
Step 4 Predict using the equation. If this trend continues, we can use the graph of our equation to predict the monthly number of flights the airline will make in a specific year. Let’s check the year 2020. First adjust the window.
• Copy the equation to the Y= list and graph. KEYSTROKES:
1
5 9
1 At x = enter 20
KEYSTROKES:
.
[0, 10] scl: 1 by [0, 60] scl: 5 [0, 25] scl: 1 by [0, 200] scl: 5
There will be approximately 177 flights per month if this trend continues.
Exercises Plot each set of data points. Determine whether to use a linear, quadratic or exponential regression equation. State the coefficient of determination. 1.
x
y
1
2.
x
y
30
0.0
2
40
3
3.
x
y
12.1
0
0.1
9.6
50
0.2
4
55
5 6
4.
x
y
1.1
1
1.67
2
3.3
5
2.59
6.3
4
2.9
9
4.37
0.3
5.5
6
5.6
13
6.12
50
0.4
4.8
8
11.9
17
5.48
40
0.5
1.9
10
19.8
21
3.12
5. BAKING Alyssa baked a cake and is waiting for it to cool so she can ice it. The table shows the temperature of the cake every 5 minutes after Alyssa took it out of the oven. a. Make a scatter plot of the data.
Time (min)
Temperature (°F)
0
350
5
244
10
178
b. Which regression equation has an R 2 value closest to 1? Is this the equation that best fits the context of the problem? Explain your reasoning.
15
137
20
112
c. Find an appropriate regression equation, and state the coefficient of determination. What is the domain and range?
25
96
30
89
d. Alyssa will ice the cake when it reaches room temperature (70°F). Use the regression equation to predict when she can ice her cake. connectED.mcgrawhill.com
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Study Guide and Review Study Guide KeyConcepts
KeyVocabulary
Graphing Quadratic Functions (Lesson 91) • A quadratic function can be described by an equation of the form y = ax 2 + bx + c, where a ≠ 0. • The axis of symmetry for the graph of y = ax 2 + bx + c, b where a ≠ 0, is x = _ . 2a
Solving Quadratic Equations (Lessons 92, 94, and 95) • Quadratic equations can be solved by graphing. The solutions are the xintercepts or zeros of the related quadratic function. • Quadratic equations can be solved by completing the square. 1 To complete the square for x 2 + bx, find _ of b, square this 2 2 result, and then add the result to x + bx. • Quadratic equations can be solved by using the b ± √ b 2  4ac Quadratic Formula, x = __. 2a
Transformations of Quadratic Functions (Lesson 93) • f (x ) = x 2 + c translates the graph up or down. • f (x ) = ax 2 compresses or expands the graph vertically. Exponential Functions (Lessons 96 and 97) • An exponential function can be described by an equation of the form y = ab x, where a ≠ 0, b > 0 and b ≠ 1. • The general equation for exponential growth is y = a (1 + r ) t, where r > 0, and the general equation for exponential decay is y = a (1  r ) t, where 0 < r < 1. y represents the final amount, a is the initial amount, r represents the rate of change, and t is the time in years.
StudyOrganizer Be sure the Key Concepts are noted in your Foldable.
axis of symmetry (p. 525)
minimum (p. 525)
common ratio (p. 578)
parabola (p. 525)
completing the square (p. 552) Quadratic Formula (p. 558) compound interest (p. 574)
quadratic function (p. 525)
dilation (p. 545)
reflection (p. 545)
discriminant (p. 561)
standard form (p. 525)
double root (p. 538)
symmetry (p. 526)
exponential decay (p. 574)
transformation (p. 544)
exponential function (p. 567)
translation (p. 544)
exponential growth (p. 573)
vertex (p. 525)
geometric sequence (p. 578) maximum (p. 525)
VocabularyCheck State whether each sentence is true or false. If false, replace the underlined term to make a true sentence. 1. The axis of symmetry of a quadratic function can be found b . by using the equation x = _ 2a
2. The vertex is the maximum or minimum point of a parabola. 3. The graph of a quadratic function is a straight line. 4. The graph of a quadratic function has a maximum if the coefﬁcient of the x 2 is positive. 5. A quadratic equation with a graph that has two xintercepts has one real root. 6. The expression b 2  4ac is called the discriminant. 7. An example of an exponential function is y = 3 x. 8. The exponential growth equation is y = C (1  r ) t. 9. The solutions of a quadratic equation are called roots. 10. The graph of the parent function is translated down to form the graph of f (x ) = x 2 + 5.
592  Chapter 9  Study Guide and Review
LessonbyLesson Review
911Graphing Quadratic Functions
CLE 3102.3.6, ✔3102.3.19, SPI 3102.3.11
(pp. 525–535)
Example 1
Consider each equation. a. Determine whether the function has a maximum or minimum value. b. State the maximum or minimum value.
Consider f (x ) = x 2 + 6x + 5. a. Determine whether the function has a maximum or minimum value. For f (x ) = x 2 + 6x + 5, a = 1, b = 6, and c = 5.
c. What are the domain and range of the function?
Because a is positive, the graph opens up, so the function has a minimum value.
11. y = x 2  4x + 4 12. y = x 2 + 3x
b. State the minimum or minimum value of the function.
13. y = x 2  2x  3
The minimum value is the ycoordinate of the vertex.
14. y = x 2 + 2. 15. BASEBALL A baseball is thrown with an upward velocity of 32 feet per second. The equation h = 16t 2 + 32t gives the height of the ball t seconds after it is thrown. a. Determine whether the function has a maximum or minimum value. b. State the maximum or minimum value.
b _ The xcoordinate of the vertex is _ or 6 or 3. 2a
f (x ) = x 2 + 6x + 5 f (3) = (3) 2 + 6(3) + 5 f (3) = 4
2(1)
Original function x = 3 Simplify.
The minimum value is 4. c. State the domain and range of the function.
c. State a reasonable domain and range for this situation.
The domain is all real numbers. The range is all real numbers greater than or equal to the minimum value, or { y y ≥ 4}.
CLE 3102.3.8, ✔3102.3.32, SPI 3102.3.10
922 Solving Quadratic Equations by Graphing
(pp. 537–542)
Example 2
Solve each equation by graphing. If integral roots cannot be found, estimate the roots to the nearest tenth.
Solve x 2  x  6 = 0 by graphing.
16. x 2  3x  4 = 0
Graph the related function f (x ) = x 2  x  6.
17. x 2 + 6x  9 = 0
y
18. x 2  x  12 = 0 19. x 2 + 4x  3 = 0
0
x
20. x 2  10x = 21 21. 6x 2  13x = 15 22. NUMBER THEORY Find two numbers that have a sum of 2 and a product of 15.
The xintercepts of the graph appear to be at 2 and 3, so the solutions are 2 and 3.
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Study Guide and Review Continued ✔3102.1.14, SPI 3102.1.5, SPI 3102.3.11
933 Transformations of Quadratic Functions
(pp. 544–549)
Describe how the graph of each function is related to the graph of f (x ) = x 2. 23. f (x ) = x 2 + 8
24. f (x ) = x 2  3
25. f (x ) = 2x 2
26. f (x ) = 4x 2  18
1 2 x 27. f (x ) = _
1 2 28. f (x ) = _ x
3
4
29. Write an equation for the function shown in the graph. y
Example 3 Describe how the graph of f (x ) = x 2  2 is related to the graph of f (x ) = x 2. The graph of f (x ) = x 2 + c represents a translation up or down of the parent graph. Since c = 2, the translation is down. So, the graph is shifted down from the parent function.
Example 4 Write an equation for the function shown in the graph. y
0
x
30. PHYSICS A ball is dropped off a cliff that is 100 feet high. The function h = 16t 2 + 100 models the height h of the ball after t seconds. Compare the graph of this function to the graph of h = t 2.
0
x
Since the graph opens upward, the leading coefficient must be positive. The parabola has not been translated up or down, so c = 0. Since the graph is stretched vertically, it must be of the form of f (x ) = ax 2 where a > 1. The equation for the function is y = 2x 2.
CLE 3102.3.8, ✔3102.3.30, SPI 3102.3.10
944 Solving Quadratic Equations by Completing the SquareExample 5
(pp. 552–557)
Solve each equation by completing the square. Round to the nearest tenth if necessary. 31. x 2 + 6x + 9 = 16 32. a 2  10a + 25 = 25 33. y 2  8y + 16 = 36 2
34. y  6y + 2 = 0 35. n 2  7n = 5 36. 3x 2 + 4 = 0 37. NUMBER THEORY Find two numbers that have a sum of 2 and a product of 48.
594  Chapter 9  Study Guide and Review
Solve x 2  16x + 32 = 0 by competing the square. Round to the nearest tenth if necessary. Isolate the x 2 and xterms. Then complete the square and solve. x 2  16x + 32 = 0 x 2  16x = 32 2 x  16x + 64 = 32 + 64 (x  8) 2 = 32 32 x  8 = ± √ 32 x = 8 ± √ 2 x = 8 ± 4 √ The solutions are about 2.3 and 13.7.
CLE 3102.3.8, ✔3102.3.31, SPI 3102.3.10
955 Solving Quadratic Equations by Using the Quadratic Formula Example 6
(pp. 558–564)
Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary. 38. x 2  8x = 20
Solve x 2 + 10x + 9 = 0 by using the Quadratic Formula. b ± √ b 2  4ac 2a
Quadratic Formula
10 ± √ 10 2  4(1)(9) 2(1)
a = 1, b = 10, c = 9
10 ± √ 64 2
Simplify.
x = __
39. 21x 2 + 5x  7 = 0 40. d 2  5d + 6 = 0
= __
41. 2f 2 + 7f  15 = 0
=_
42. 2h 2 + 8h + 3 = 3 43. 4x 2 + 4x = 15 44. GEOMETRY The area of a square can be quadrupled by increasing the side length and width by 4 inches. What is the side length?
10 + 8 10  8 = _ or _
Separate the solutions.
= 1 or 9
Simplify.
2
2
CLE 3102.3.6, CLE 3102.3.9, SPI 3102.3.11
966 Exponential Functions
(pp. 567–572)
Graph each function. Find the yintercept, and state the domain and range. 45. y = 2 x
Example 7 Graph y = 3 x + 6. Find the yintercept, and state the domain and range.
46. y = 3 x + 1
x
3x + 6
y
47. y = 4 x + 2
3
3 3 + 6
6.04
48. y = 2 x  3 49. BIOLOGY The population of bacteria in a petri dish increases according to the model p = 550(2.7) 0.008t, where t is the number of hours and t = 0 corresponds to 1:00 P.M. Use this model to estimate the number of bacteria in the dish at 5:00 P.M.
2
3
+6
6.11
1
3 1 + 6
6.33
0
30 + 6
7
14 12 10 8 6 4 2
9
−8−6−4−20
2
1
3 +6
1
y
2 4 6 8x
The yintercept is (0, 7). The domain is all real numbers, and the range is all real numbers greater than 6.
CLE 3102.3.9, ✔3102.3.33, ✔3102.3.35
977 Growth and Decay
(pp. 573–577)
50. Find the final value of $2500 invested at an interest rate of 2% compounded monthly for 10 years. 51. COMPUTERS Zita’s computer is depreciating at a rate of 3% per year. She bought the computer for $1200. a. Write an equation to represent this situation. b. What will the computer’s value be after 5 years?
Example 8 Find the final value of $2000 invested at an interest rate of 3% compounded quarterly for 8 years. r A=P 1+_ n
) nt
(
(
Compound interest equation
)
0.03 = 2000 1 + _ 4
4(8)
P = 2000, r = 0.03, n = 4, and t = 8
≈ $2540.22 Use a calculator. There will be about $2540.22 in 8 years.
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Study Guide and Review Continued SPI 3102.1.1, CLE 3102.3.1, ✔3102.3.1
988 Geometric Sequences as Exponential Functions
(pp. 578–583)
Find the next three terms in each geometric sequence.
Example 9
52. 1, 1, 1, 1, ...
Find the next three terms in the geometric sequence 2, 6, 18, … .
53. 3, 9, 27, ...
Step 1 Find the common ratio. Each number is
54. 256, 128, 64, ...
3 times the previous number, so r = 3.
Write the equation for the nth term of each geometric sequence.
Step 2 Multiply each term by the common ratio to find the
next three terms. 18 × 3 = 54, 54 × 3 = 162, 162 × 3 = 486 The next three terms are 54, 162, and 486
55. 1, 1, 1, 1, ... 56. 3, 9, 27, ...
Example 10
57. 256, 128, 64, ... 58. SPORTS A basketball is dropped from a height of 20 1 its height after each bounce. Draw feet. It bounces to _ 2 a graph to represent the situation.
Write the equation for the nth term of the geometric sequence 3, 12, 48, … . The common ratio is 4. So r = 4. a n = a 1r n  1 a n = 3(4) n  1
Formula for the nth term a 1 = 3 and r = 4
✔3102.1.4, SPI 3102.1.1
999 Analyzing Functions with Successive Differences Look for a pattern in each table of values to determine which kind of model best describes the data. Then write an equation for the function that models the data. 59.
60.
61.
62.
x
0
1
2
3
4
y
0
3
12
27
48
x
0
1
2
3
4
y
1
2
4
8
16
(pp. 584–589)
Example 11 Determine which kind of model best describes the data. Then write an equation for the function that models the data. x
0
1
2
3
4
y
3
4
5
6
7
Step 1 Determine which model fits the data.
First differences: x
0
1
2
3
4
y
0
1
4
9
16
x
0
1
2
3
4
y
3
6
9
12
15
3
4 1
5 1
6 7 1
1
Since the first differences are all equal, a linear function models the data. Step 2 Write an equation for the function that models
63. SCHOOL SPIRIT The table shows the cost to purchase schoolspirit posters. Determine which kind of model best describes the data. Then write the equation. No. of posters
2
4
6
8
Cost
4
7
10
13
596  Chapter 9  Study Guide and Review
the data. The equation has the form y = mx + b. The slope is 1 and the yintercept is 3, so the equation is y = x + 3.
Tennessee Curriculum Standards
Practice Test Use a table of values to graph the following functions. State the domain and range.
SPI 3102.3.11
Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary.
1. y = x 2 + 2x + 5
15. x 2  x  30 = 0
2. y = 2x 2  3x + 1
16. x 2  10x = 15 17. 2x 2 + x  15 = 0
Consider y = x 2  7x + 6. 3. Determine whether the function has a maximum or minimum value. 4. State the maximum or minimum value. 5. What are the domain and range? Solve each equation by graphing. If integral roots cannot be found, estimate the roots to the nearest tenth.
18. BASEBALL Elias hits a baseball into the air. The equation h = 16t 2 + 60t + 3 models the height h in feet of the ball after t seconds. How long is the ball in the air? Graph each function. Find the yintercept, and state the domain and range. 19. y = 2(5) x
6. x 2 + 7x + 10 = 0
20. y = 3(11) x
7. x 2  5 = 3x
21. y = 3x + 2
Describe how the graph of each function is related to the graph of f(x) = x 2. 8. g(x) = x 2  5 9. g(x) = 3x
Find the next three terms in each geometric sequence. 22. 2, 6, 18, …
2
23. 1000, 500, 250, …
1 2 10. h(x) = _ x +4 2
24. 32, 8, 2, …
11. MULTIPLE CHOICE Which is an equation for the function shown in the graph? y
O
x
25. MULTIPLE CHOICE Lynne invested $500 into an account with a 6.5% interest rate compounded monthly. How much will Lynne’s investment be worth in 10 years? F $600.00
H $956.09
G $938.57
J $957.02
26. INVESTMENTS Shelly’s investment of $3000 has been losing value at a rate of 3% each year. What will her investment be worth in 6 years? A y = 3x 2 B y = 3x 2 + 1 C y = x2 + 2 D y = 3x 2 + 2 Solve each equation by completing the square. 12. x 2 + 2x + 5 = 0
27. Graph {(2, 4), (1, 1), (0, 0), (1, 1), (2, 4)}. Determine whether the ordered pairs represent a linear function, a quadratic function, or an exponential function. 28. Look for a pattern in the table to determine which kind of model best describes the data.
13. x 2  x  6 = 0
x
0
1
2
3
4
14. 2x 2  36 = 6x
y
1
3
5
7
9
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Preparing for Standardized Tests Use a Formula A formula is an equation that shows a relationship among certain quantities. Many standardized test problems will require using a formula to solve them.
Strategies for Using a Formula Step 1
Become familiar with common formulas and their uses. You may or may not be given access to a formula sheet to use during the test. • If given a formula sheet, be sure to practice with the formulas on it before taking the test so you know how to apply them. • If not given a formula sheet, study and practice with common formulas such as perimeter, area, and volume formulas, the Distance Formula, the Pythagorean Theorem, the Midpoint Formula, the Quadratic Formula, and others.
Step 2
Choose a formula and solve. • Ask Yourself: What quantities are given in the problem statement? • Ask Yourself: What quantities am I looking for? • Ask Yourself: Is there a formula I know that relates these quantities? • Write: Write the formula out that you have chosen each time. • Solve: Substitute known quantities into the formula and solve for the unknown quantity. • Check: Check your answer if time permits.
SPI 3102.3.10
Test Practice Example Read the problem. Identify what you need to know. Then use the information in the problem to solve. Find the exact roots of the quadratic equation 2x 2 + 6x + 5 = 0. 3 ± √ 17 A_
4 4 ± √ 17 B _ 3
3 ± √ 19 C _ 2 √19 3 ± D _ 4
598  Chapter 9  Preparing for Standardized Tests
Read the problem carefully. You are given a quadratic equation and asked to find the exact roots of the equation. Use the Quadratic Formula to find the roots. 2x 2 + 6x + 5 = 0
Original equation
a = 2, b = 6, c = 5
Identify the coefficients of the equation.
b ± √ b 2  4ac x = __
Quadratic Formula
2a
2
(6) ± √(6)  4(2)(5) = __
a = 2, b = 6, and c = 5
6 ± √ 36  (40) = __
Simplify.
2(2)
4
6 ± √ 76 4
=_ 6 ± 2 √ 19 4
=_
Subtract.
√ 76 = √ 4 · 19 or 2 √ 19 .
2(3 ± √ 19 ) 2(2)
Factor out 2 from the numerator and denominator.
3 ± √ 19 2
Simplify.
= __ =_
3 + √ 19 3  √ 19 The roots of the equation are _ and _. The correct answer is C. 2
2
Exercises Read each problem. Identify what you need to know. Then use the information in the problem to solve.
3. Find the volume of the figure below.
6.5 cm
1. Find the exact roots of the quadratic equation x 2 + 5x  12 = 0. 5 ± √73 A _ 2
3 ± √73 C _ 4
4 ± √ 61 B _ 3
1 ± √ 61 D _ 2
2. The area of a triangle in which the length of the base is 4 centimeters greater than twice the height is 80 square centimeters. What is the length of the base of the triangle?
5 cm
9 cm
A 18.5 cm 3
C 272 cm 3
B 91 cm 3
D 292.5 cm 3
4. Myron is traveling 263.5 miles at an average rate of 62 miles per hour. How long will it take Myron to complete his trip?
F 10
F 4 h 10 min
G 8
G 4 h 15 min
H 16
H 5 h 10 min
J 20
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Standardized Test Practice Cumulative, Chapters 1 through 9 4. Write an expression for the area of the rectangle below.
Multiple Choice Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper.
4 3
2b c  3bc
2
5bc
1. What is the vertex of the parabola graphed below? F 10b 5c 5  3bc
y
G 10b 5c 5  15b 2c 3 H 2b 5c 5  3b 2c 3 0
x
J 10b 4c 6  15bc 2 5. Solve the quadratic equation below by graphing. x 2  2x  15 = 0
A (2, 0) B (0, 2)
A 1, 4
C (2, 2)
B 3, 5
D (2, 2)
C 3, 5 D
2. Write an equation in slopeintercept form with 9 and yintercept of 3. a slope of _ 10
9 F y = 3x + _ 10
9 G y=_ x+3
6. Jason is playing games at a family fun center. So far he has won 38 prize tickets. How many more tickets would he need to win to place him in the gold prize category? Number of Tickets
Prize Category
1–20
bronze
21–40
silver
41–60
gold
61–80
platinum
10 9 H y=_ x3 10 9 J y = 3x  _ 10
F 2 ≤ t ≤ 22 3. Use the Quadratic Formula to find the exact solutions of the equation 2x 2  6x + 3 = 0. 3± √ 3 A _ 2
G 3 ≤ t ≤ 22 H 1 ≤ t ≤ 20 J 3 ≤ t ≤ 20
3± √ 2 B _ 4 2± √ 5 C _ 3
5± √ 2 D _ 2
600  Chapter 9  Standardized Test Practice
TestTakingTip Question 5 If permitted, you can use a graphing calculator to quickly graph an equation and find its roots.
10. The table shows the total cost of renting a canoe for n hours.
Short Response/Gridded Response Record your answers on the answer sheet provided by your teacher or on a sheet of paper.
Number of Hours (n)
Rental Cost (C )
1
$15
2
$20
3
$25
4
$30
7. GRIDDED RESPONSE Misty purchased a car several years ago for $21,459. The value of the car depreciated at a rate of 15% annually. What was the value of the car after 5 years? Round your answer to the nearest whole dollar.
a. Write a function to represent the situation.
8. Use the graph of the quadratic equation shown below to answer each question.
b. How much would it cost to rent the canoe for 7 hours?
y
Extended Response Record your answers on a sheet of paper. Show your work.
x
0
11. Use the equation and its graph to answer each question. a. What is the vertex?
y
b. What is the yintercept? c. What is the axis of symmetry? d. What are the roots of the corresponding quadratic equation?
x
0
9. The cost of 5 notebooks and 3 pens is $9.75. The cost of 4 notebooks and 6 pens is $10.50. Which of the following systems can be used to find the cost of a notebook n and a pen p?
a. Factor x 2  7x + 10. b. What are the solutions of x 2  7x + 10 = 0?
a. Write a system of equations to model the situation.
c. What do you notice about the graph of the quadratic equation and where it crosses the xaxis? How do these values compare to the solutions of x 2  7x + 10 = 0? Explain.
b. Solve the system of equations. How much does each item cost?
Need ExtraHelp? If you missed Question... Go to Lesson... For help with TN SPI...
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3102.3.11
3102.3.8
3102.3.10
3102.3.2
3102.3.10
3102.3.5
3102.3.5
3102.3.11
3102.3.9
3102.3.5
3102.3.10
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