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3.13.13.13.1
Exponential and Logistic Exponential and Logistic FunctionsFunctions
Quick Review
3
3
4 / 3
2-3
5
Evaluate the expression without using a calculator.
1. -125
272.
643. 27
Rewrite the expression using a single positive exponent.
4.
Use a calculator to evaluate the expression.
5. 3.71293
a
Quick Review Solutions
6
3
3
4 / 3
2-3
Evaluate the expression without using a calculator.
1. -125
272.
643. 27
Rewrite the expression using a single positive e
-5
3
481
1
xponent.
4.
Use a calculator to evaa
a
5
luate the expression.
5. 3.71293 1.3
What you’ll learn about
• Exponential Functions and Their Graphs• The Natural Base e• Logistic Functions and Their Graphs• Population Models
… and whyExponential and logistic functions model
many growth patterns, including the growth of human and animal populations.
Exponential Functions Let and be real number constants. An in is a
function that can be written in the form ( ) , where is nonzero,
is positive, and 1. The constant is the
x
a b x
f x a b a
b b a initial v
exponential function
of (the value
at 0), and is the .
alue f
x b base
Example Finding an Exponential Function from its
Table of ValuesDetermine formulas for the exponential function and whose values are
given in the table below.
g h
Example Finding an Exponential Function from its
Table of ValuesDetermine formulas for the exponential function and whose values are
given in the table below.
g h
1
Because is exponential, ( ) . Because (0) 4, 4.
Because (1) 4 12, the base 3. So, ( ) 4 3 .
x
x
g g x a b g a
g b b g x
1
Because is exponential, ( ) . Because (0) 8, 8.
1Because (1) 8 2, the base 1/ 4. So, ( ) 8 .
4
x
x
h h x a b h a
h b b h x
Exponential Growth and Decay
For any exponential function ( ) and any real number ,
( 1) ( ).
If 0 and 1, the function is increasing and is an
. The base is its .
If 0 an
xf x a b x
f x b f x
a b f
b
a
exponential
growth function growth factor
d 1, the function is decreasing and is an
. The base is its .
b f
b
exponential
decay function decay factor
Example Transforming Exponential Functions
-2Describe how to transform the graph of ( ) 2 into the graph of ( ) 2 .x xf x g x
Example Transforming Exponential Functions
-2Describe how to transform the graph of ( ) 2 into the graph of ( ) 2 .x xf x g x
-2The graph of ( ) 2 is obtained by translating the graph of ( ) 2 by
2 units to the right.
x xg x f x
Example Transforming Exponential Functions
-2Describe how to transform the graph of ( ) 2 into the graph of ( ) 2 .x xf x g x
Example Transforming Exponential Functions
-Describe how to transform the graph of ( ) 2 into the graph of ( ) 2 .x xf x g x
The graph of ( ) 2 is obtained by reflecting the graph of ( ) 2 across
the -axis.
x xg x f x
y
The Natural Base e 1
lim 1x
xe
x
Exponential Functions and the Base e
Any exponential function ( ) can be rewritten as ( ) ,
for any appropriately chosen real number constant .
If 0 and 0, ( ) is an exponential growth function.
If 0 and 0, (
x kx
kx
f x a b f x a e
k
a k f x a e
a k f
) is an exponential decay function.kxx a e
Exponential Functions and the Base e
Example Transforming Exponential Functions
3Describe how to transform the graph of ( ) into the graph of ( ) .x xf x e g x e
Example Transforming Exponential Functions
3Describe how to transform the graph of ( ) into the graph of ( ) .x xf x e g x e
3The graph of ( ) is obtained by horizontally shrinking the graph of
( ) by a factor of 3.
x
x
g x e
f x e
Logistic Growth Functions
Let , , , and be positive constants, with 1. A
in is a function that can be written in the form ( ) or 1
( ) where the constant is the 1
x
kx
a b c k b
cx f x
a bc
f x ca e
logistic growth function
limit to gr
owth.