Name ___________________________________________________
Date __________________
Exponential Functions
Graphing Exponential Functions
An exponential function is a nonlinear function of the form y = abx, where a ≠ 0, b ≠ 1, and b > 0.
• When a > 0 and b > 1, the function is an exponential growth function.
• When a > 0 and 0 < b < 1, the function is an exponential decay function.
y
y
The graphs of the parent exponential
functions y = b x are shown.
f(x) = b x
(b > 1)
f(x) = b x
(0 < b < 1)
(1, b)
(0, 1)
(1, b)
(0, 1)
x
x
Example 1 Tell whether each function represents exponential growth or exponential decay. Then graph the function.
( )x
a. f(x) = 2 —13
b. f(x) = 4(3)x
Because a = 2 is positive and b = —13 is greater than 0
and less than 1, the function is an exponential decay
function. Use a table to graph the function.
−1
x
y
6
0
1
2
2
2
—3
2
—9
Because a = 4 is positive and b = 3 is greater than 1,
the function is an exponential growth function. Use a
table to graph the function.
y
(−1, 6)
( 13 )
6
x
f(x) = 2
x
−2
−1
0
1
y
4
—9
4
—3
4
12
y
( )
( )
(
2
(0, 2) 1, 3
2
2
2, 9
−2
−1,
(−2, 49 )
4x
2
(1, 12)
12
4
3
8
)4
f(x) = 4(3)x
(0, 4)
−2
Practice
2
x
Check your answers at BigIdeasMath.com.
Tell whether the function represents exponential growth or exponential decay.
Then graph the function.
( )x
1
1. f(x) = —4
exponential decay
16
(−2, 16)
( )x exponential growth
4
2. f(x) = —3
y
y
()
8
4
(−1, 4)
(0, 1)
−4
x
8
3
12
1 x
f(x) = 4
(4 )
f(x) = 3
3. f(x) = 0.5(4)x exponential growth
(1, 14 )
(2, 161 )
−2
2
4 x
(
(
3
−1, 4
9
−2, 16
−4
)
)
2
y
(2, 8)
6
4
(2, 169)
(0, 1)
(1, 43 )
−2
2
4 x
(−1, 18 )
−4
f(x) = 0.5(4)x
(1, 2)
2
(0, 12 )
−2
2
4 x
4. f(x) = 3(0.75)x exponential decay 5. f(x) = 2(0.8)x exponential decay 6. f(x) = 5(2)x exponential growth
8
(−2, 163 )
(−1, 4)
−2
f(x) = 2(0.8)x8
4
y
16
6
6
2
−4
y
(−2, 258 )
f(x) = 3(0.75)x
(0, 3)
(1, 94 )
2
(
4 x
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−4
5
−1, 2
−2
)
(0, 2)
2
( )
8
1, 5
4 x
(−1, 52 )
(−2, 54 )
(−3, 58 )
−4
−2
12
8
y
f(x) = 5(2)x
(1, 10)
(0, 5)
2
x
Topic 9.5
Name ___________________________________________________
Date __________________
Exponential Functions
Rewriting Exponential Functions
Exponential growth occurs when a quantity increases by the same factor over equal intervals of time, whereas
exponential decay occurs when a quantity decreases by the same factor over equal intervals of time.
Exponential Growth Model
rate of growth (in decimal form)
initial amount
initial amount
time
y = a(1 + r)t
final amount
Exponential Decay Model
rate of decay (in decimal form)
y = a(1 − r)t
final amount
growth factor
time
decay factor
Example 1 Rewrite the function y = 120(1.25)t/12 to determine whether it represents exponential growth or
exponential decay. Then find the percent rate of change.
y = 120(1.25)t/12
Write the function.
1/12 t
= 120[(1.25)
]
≈ 120(1.02)t
Power of a Power Property
Evaluate the power.
t
= 120(1 + 0.02)
Rewrite in the form y = a(1 + r)t.
So, the function represents exponential growth and the growth rate is about 0.02, or 2%.
Practice
Check your answers at BigIdeasMath.com.
Rewrite the function to determine whether it represents exponential growth or
exponential decay. Then find the percent rate of change.
2. y = 67(1.13)t/4
1. y = 80(0.85)2t
y = 80(1 − 0.28)t, exponential decay; 28%
( 25 )
( 32 )
3. y = 5 —
−8t
4. y = 17 —
y = 5(1 − 0.96)t, exponential decay; 96%
5. y = 4(0.5)t/88
0.65t
y = 17(1 − 0.45)t, exponential decay; 45%
6. y = 31(1.02)4t
t
y = 4(1 − 0.01) , exponential decay; 1%
y = 31(1 + 0.08)t, exponential growth; 8%
8. y = 750(0.88)t/3
7. y = 9(1.12)0.3t
y = 9(1 + 0.03)t, exponential growth; 3%
y = 750(1 − 0.04)t, exponential decay; 4%
10. y = 6(0.82)−0.25t
9. y = (0.64)5t
t
y = (1 − 0.89) , exponential decay; 89%
2
y = 67(1 + 0.03)t, exponential growth; 3%
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y = 6(1 + 0.05)t, exponential growth; 5%