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Explain 1 Modeling Exponential Growth
Recall that a function of the form y = a b x represents exponential growth when a > 0 and b > 1. If b is replaced by
1 + r and x is replaced by t, then the function is the exponential growth model y = a
(
1 + r
)
t , where a is the initial
amount, the base
(
1 + r
)
is the growth factor, r is the growth rate, and t is the time interval. The value of the model
increases with time.
Example 1 Write an exponential growth function for each situation. Graph each
function and state its domain, range and an asymptote. What does the
y-intercept represent in the context of the problem?
A A painting is sold for $1800, and its value increases by 11%
each year after it is sold. Find the value of the painting in 30 years.
Write the exponential growth function for this situation.
y = a
(
1 + r
)
t
= 1800
(
1 + 0.11
)
t
= 1800
(
1.11
)
t
Find the value in 30 years.
y = 1800
(
1.11
)
t
= 1800
(
1.11
)
30
≈ 41,206.13
After 30 years, the painting will be worth approximately $41,206.
Create a table of values to graph the function.
t y
(
t, y
)
0 1800 (0, 1800)
8 4148 (8, 4148)
16 9560 (16, 9560)
24 22,030 (24, 22,030)
32 50,770 (32, 50,770)
Determine the domain, range and an asymptote of the function.
The domain is the set of real numbers t such that t ≥ 0.
The range is the set of real numbers y such that y ≥ 1800.
An asymptote for the function is y = 0.
The y-intercept is the value of y when t = 0, which is the value of the painting when it was sold.
(0, 1800)
(8, 4148)
(16, 9560)
(24, 22,030)
(32, 50,770)
11,000
22,000
33,000
44,000
y
16
Time (years)
Value (dollars)
248 32
t
0
Module 15 723 Lesson 2
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