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8.7
Exponential Decay Functions
What will your car be worth after 8 years?
Goal
Write and graph
exponential decay
functions.
In Lesson 8.6 you used exponential
functions to model values that
were increasing. Exponential
functions can also be used to
model values that are decreasing.
In Examples 1–3 you will analyze
a car’s value that is decreasing
exponentially over time.
Key Words
• exponential decay
• decay rate
• decay factor
A quantity is decreasing exponentially if it decreases by the same percent r in
each unit of time t. This is called exponential decay. Exponential decay can be
modeled by the equation
y C(1 r)t
where C is the initial amount (the amount before any decay occurs), r is the
decay rate (as a decimal), t represents time, and where 0 < r < 1. The expression
(1 r) is called the decay factor.
EXAMPLE
1
Write an Exponential Decay Model
You bought a car for $16,000. You expect the car to lose value, or
depreciate, at a rate of 12% per year. Write an exponential decay model to
represent this situation.
CARS
Solution
Let y be the value of the car and let t be the number of years of ownership. The
initial value of the car C is $16,000. The decay rate r is 12%, or 0.12.
y = C(1 r)t
Write exponential decay model.
= 16,000(1 0.12)t
t
= 16,000(0.88)
ANSWER Substitute 16,000 for C and 0.12 for r.
Subtract.
The exponential decay model is y 16,000(0.88)t.
Write an Exponential Decay Model
1. Your friend bought a car for $24,000. The car depreciates at the rate of 10%
per year. Write an exponential decay model to represent the car’s value.
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Chapter 8
Exponents and Exponential Functions
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Use an Exponential Decay Model
2
EXAMPLE
Use the model in Example 1 to find the value of your car after 8 years.
Solution To find the value after 8 years, substitute 8 for t.
y 16,000(0.88)t
Write exponential decay model.
16,000(0.88)8
Substitute 8 for t.
≈ 5754
Use a calculator.
ANSWER Your car will be worth about $5754 after 8 years.
Graph an Exponential Decay Model
3
EXAMPLE
a. Graph the exponential decay model in Example 1.
b. Use the graph to estimate the value of your car after 5 years.
Solution
a. Make a table of values, plot the points in a coordinate plane, and draw a
smooth curve through the points.
t
0
2
4
6
8
y
16,000
12,390
9595
7430
5754
Value (dollars)
y (0, 16,000)
16,000
(2, 12,390)
(4, 9595)
(6, 7430)
8000
y 16,000(0.88)t
12,000
(8, 5754)
4000
0
0
2
4
6
8 t
Years of ownership
b. According to the graph, the value of your car after 5 years will be about
$8400. You can check this answer by using the model in Example 1.
Graph and Use an Exponential Decay Model
Use the model in Checkpoint 1.
2. Find the value of your friend’s car after 6 years.
3. Graph the exponential decay model.
4. Use the graph to estimate the value of your friend’s car after 5 years.
8.7
Exponential Decay Functions
483
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In Lesson 8.3 you learned that for b > 0 a function of the form y ab x is an
exponential function. In the model for exponential growth, b is replaced by 1 r
where r > 0. In the model for exponential decay, b is replaced by 1 r where
0 < r < 1. Therefore you can conclude that an exponential model y Cbt
represents exponential growth if b > 1 and exponential decay if 0 < b < 1.
EXAMPLE
4
Compare Growth and Decay Models
Classify the model as exponential growth or exponential decay. Then identify
the growth or decay factor and graph the model.
3 t
b. y 30 , where t ≥ 0
5
a. y 30(1.2)t, where t ≥ 0
Solution
3
b. Because 0 < < 1, the model
5
3 t
y 30 is an exponential
5
a. Because 1.2 > 1, the
model y 30(1.2)t is an
exponential growth model.
The growth factor (1 r)
is 1.2. The graph is shown
below.
decay model. The decay factor
3
(1 r) is . The graph is shown
5
below.
y
y
70
35
50
25
y 30(1.2)t
30
15
10
5
1
3
5
7 t
y 30
1
35t
3
5
7 t
Compare Growth and Decay Models
Classify the model as exponential growth or exponential decay. Then
identify the growth or decay factor and graph the model.
5. y (2)t
6. y (0.5)t
7. y 5(0.2)t
8. y 0.7(1.1)t
SUMMARY
EXPONENTIAL
y
GROWTH MODEL
y C(1 r)t,
where 1 r > 1
and t ≥ 0
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Chapter 8
(0, C )
t
Exponents and Exponential Functions
EXPONENTIAL
y
DECAY MODEL
(0, C )
y C (1 r)t,
where 0 < 1 r < 1
and t ≥ 0
t
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8.7 Exercises
Guided Practice
Vocabulary Check
Skill Check
1. In the exponential decay model, y C(1 r)t, what is the decay factor?
2. BUSINESS A business earned $85,000 in 1990. Then its earnings decreased
by 2% each year for 10 years. Write an exponential decay model to represent
the decreasing annual earnings of the business.
CARS You buy a used car for $7000. The car depreciates at the rate of
6% per year. Find the value of the car after the given number of years.
3. 2 years
4. 5 years
5. 8 years
7. CHOOSE A MODEL Which model best
6. 10 years
y
represents the decay curve shown in the
graph at the right?
50
A. y 60(0.08)t
B. y 60(1.20)t
30
C. y 60(0.40)t
D. y 60(1.05)t
10
1
3
5
t
Classify the model as exponential growth or exponential decay.
8. y 0.55(3)t
9. y 3(0.55)t
10. y 55(3)t
11. y 55(0.3)t
Practice and Applications
EXPONENTIAL DECAY MODEL Identify the initial amount and the decay
factor in the exponential function.
12. y 10(0.2)t
13. y 18(0.11)t
1
14. y 2 4
t
5
15. y 0.5 8
t
WRITING EXPONENTIAL MODELS Write an exponential model to
represent the situation. Tell what each variable represents.
16. A $25,000 car depreciates at a rate of 9% each year.
17. A population of 100,000 decreases by 2% each year.
18. A new sound system, valued at $800, decreases in value by 10% each year.
Student Help
HOMEWORK HELP
FINANCE Write an exponential decay model for the investment.
Example 1: Exs. 12–21
Example 2: Exs. 22–30
Example 3: Exs. 31–41
Example 4: Exs. 42–53
19. A stock is valued at $100. Then the value decreases by 9% per year.
20. $550 is placed in a mutual fund. Then the value decreases by 4% per year.
21. A bond is purchased for $70. Then the value decreases by 1% per year.
8.7
Exponential Decay Functions
485
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TRUCKS You buy a used truck for $20,000. The truck depreciates 7% per
year. Find the value of the truck after the given number of years.
Careers
22. 3 years
23. 8 years
24. 10 years
25. 12 years
PHARMACEUTICALS In Exercises 26–28, use the following information.
The amount of aspirin y (in milligrams) in a person’s blood can be modeled by
y A(0.8)t where A represents the dose of aspirin taken (in milligrams) and t
represents the number of hours since the aspirin was taken. Find the amount of
aspirin remaining in a person’s blood for the given dosage and time.
26. Dosage: 250 mg
27. Dosage: 500 mg
Time: after 2 hours
28. Dosage: 750 mg
Time: after 3.5 hours
Time: after 5 hours
BASKETBALL In Exercises 29 and 30, use the following information.
PHARMACISTS must
INT
understand the use,
composition, and effects
of pharmaceuticals.
NE
ER T
More about
pharmacists at
www.mcdougallittell.com
At the start of a basketball tournament consisting of six rounds, there are
64 teams. After each round, one half of the remaining teams are eliminated.
29. Write an exponential decay model showing the number of teams left in the
tournament after each round.
30. How many teams remain after 3 rounds? after 4 rounds?
GRAPHING Graph the exponential decay model.
31. y 15(0.9)t
32. y 72(0.85)t
1
33. y 10 2
t
3
34. y 55 4
t
GRAPHING AND ESTIMATING Write an exponential decay model for the
situation. Then graph the model and use the graph to estimate the value
at the end of the given time period.
35. A $22,000 investment decreases in value by 9% per year for 8 years.
36. A population of 2,000,000 decreases by 2% per year for 15 years.
37. You buy a new motorcycle for $10,500. It’s value depreciates by 10% each
year for the 10 years you own it.
CABLE CARS In Exercises 38–41, use the following information.
From 1894 to 1903 the number of miles of cable car track in the United States
decreased by about 11% per year. There were 302 miles of track in 1894.
38. Write an exponential decay model showing the number of miles of cable car
track left each year.
39. Copy and complete the table. You may want to use a calculator.
Year
Miles of track
1894
1896
1898
1900
1902
?
?
?
?
?
40. Graph the results.
41. Use your graph to estimate the number of miles of cable car track in 1903.
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MATCHING Match the equation with its graph.
42. y 4 3t
A.
43. y 4(0.6)t
B.
y
y
3
3
1
1
1
1
5 t
3
3 t
1
COMPARING MODELS Classify the model as exponential growth or
exponential decay. Then identify the growth or decay factor and graph
the model.
44. y 24(1.18)t
45. y 14(0.98)t
2
47. y 112 3
Student Help
HOMEWORK HELP
NE
ER T
INT
Extra help with
problem solving in
Exs. 50-52 is available at
www.mcdougallittell.com
46. y 97(1.01)t
t
2
48. y 9 5
t
t
5
49. y 35 4
EXPONENTIAL FUNCTIONS Use a calculator to investigate the effects
of a and b on the graph of y ab x.
50. In the same viewing rectangle, graph y 2(2)x, y 4(2)x, and y 8(2)x.
How does an increase in the value of a affect the graph of y abx?
51. In the same viewing rectangle, graph y 2x, y 4x, and y 8x. How does
an increase in the value of b affect the graph of y abx when b > 1?
1 x
1 x
1 x
52. In the same viewing rectangle, graph y , y , and y .
2
4
8
How does a decrease in the value of b affect the graph of y ab when
0 < b < 1?
x
53. LOGICAL REASONING Choose a positive value for b and graph y bx and
1 x
y . What do you notice about the graphs?
b
54. CHALLENGE A store is having a sale on sweaters. On the first day the price
of the sweaters is reduced by 20%. The price will be reduced another 20%
each day until the sweaters are sold. On the fifth day of the sale will the
sweaters be free? Explain.
Standardized Test
Practice
55. MULTIPLE CHOICE In 1995 you purchase a parcel of land for $8000. The
value of the land depreciates by 4% every year. What will the approximate
value of the land be in 2002?
A
B
$224
$5760
C
D
$6012
56. MULTIPLE CHOICE Which model best
y
represents the decay curve shown in the
graph at the right?
F
H
y 50(0.25)
y 50(1.5)
t
t
G
J
$7999
30
y 50(0.75)
y 50(2)
t
10
t
8.7
1
3
Exponential Decay Functions
5
t
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Mixed Review
VARIABLE EXPRESSIONS Evaluate the expression for the given value of
the variable(s). (Lesson 1.3)
57. x2 12 when x 6
58. 49 4w when w 2
59. 100 rs when r 4, s 7
60. b2 4ac when a 1, b 5, c 3
SOLVING EQUATIONS Solve the equation. Round the result to the
nearest hundredth. (Lesson 3.6)
61. 1.29x 5.22x 3.61
62. 1.33x 7.42 5.48x
63. 10.52x 1.15 1.12x 6.35
64. 8.75x 2.16 18.28x 6.59
WRITING EQUATIONS Write in point-slope form the equation of the line
that passes through the given point and has the given slope. (Lesson 5.2)
Maintaining Skills
65. (2, 5), m 3
66. (0, 3), m 5
67. (1, 4), m 4
68. (6, 3), m 1
69. (1, 7), m 6
70. (4, 5), m 2
DIVIDING DECIMALS Divide. (Skills Review p. 760)
71. 0.5 0.2
72. 4.62 0.4
73. 0.074 0.37
74. 0.084 0.007
75. 0.451 0.082
76. 0.6064 0.758
Quiz 3
COMPOUND INTEREST You deposit $250 in an account that pays
8% interest compounded yearly. Find the balance at the end of the given
time period. (Lesson 8.6)
1. 1 year
2. 3 years
3. 5 years
4. 8 years
5. POPULATION GROWTH An initial population of 50 raccoons doubles each
year for 5 years. What is the raccoon population after 5 years? (Lesson 8.6)
CAR DEPRECIATION You buy a used car for $15,000. The car depreciates
at a rate of 9% per year. Find the value of the car after the given number
of years. (Lesson 8.7)
6. 2 years
7. 4 years
8. 5 years
9. 10 years
10. CAMPERS You buy a camper for $20,000. The camper depreciates at a rate
of 8% per year. Write an exponential decay model to represent this situation.
Then graph the model and use the graph to estimate the value of the camper
after 5 years. (Lesson 8.7)
Classify the model as exponential growth or exponential decay. Then
identify the growth or decay factor and graph the model. (Lesson 8.7)
11. y 6(0.1)t
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Chapter 8
12. y 10(1.2)t
Exponents and Exponential Functions
9
13. y 3 2
t
1
14. y 2 10
t