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.