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Math 112
Section 3.1 Exponential Functions and Graphs
Exponential Function
An exponential function with base a is defined as
f ( x)  a x , where a > 0 and a  0.
x
Example 1:
f ( x)  2
1
x
Example 2: g ( x)     2
 2
x
x
f x  = 2x
8
f(x)
g(x)
f x  =
1
2x
8
0
6
1
6
4
2
4
-1
2
2
-2
-5
-5
a > 1 (increasing)
5
0 < a < 1 (decreasing)
Compound Amount
If P dollars is invested at a yearly rate of interest r per year, compounded m times per year for t years,
the compound amount is
r 

A  P 1 

m

tm
dollars.
Example 3: If $82,000 is invested at 4.5% compounded quarterly, find the sum in 5 years.
Example 4: Interest is compounded semiannually. Find the amount in the account and the interest
earned if the principal is $3,000, the rate of interest is 7%, and the time is 3 years.
r 
Definition of e: As m becomes larger and larger, 
1 


number e, whose approximate value is 2.7182818.
Example 5: Calculate e21, e-5
m
m
becomes closer and closer to the
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Shifting exponential functions:
Example 6:
f ( x)  2 x
Example 7:
f ( x)  2 x  2
hx  = 2x+2
8
f x  = 2x
8
6
6
4
4
2
2
-5
-5
Example 8: f ( x)  2  2
x
Example 9: f ( x)  2
x
rx  = 2x+2
s x  = -2x
8
5
6
-2
4
-4
2
-6
-8
-5
Example 10: f ( x)  e
 x 5
Example 11: f ( x)  2
x4
1