Exponential
Functions
Exponential
Functions and
Their Graphs
Irrational
Exponents
If b is a positive number and x is a
real number, the expression bx
always represents a positive number.
It is also true that the familiar
properties of exponents hold for
irrational exponents.
Example 1:
Use properties of exponents to
simplify
 
a) 3
2
2
Example 1:
Use properties of exponents to simplify
 
a) 3
2
2
3
2 2
3
4
3
2
9
Example 1:
Use properties of exponents to simplify
b) a  a
8
2
Example 1:
Use properties of exponents to simplify
b) a  a
8
2
a
8 2
a
42  2
a
2 2 2
a
3 2
Exponential Functions
An exponential function with base
b is defined by the equation
f  x   b where b  0 , b  1 and
x
x is a real number.
The domain of any exponential
function is the interval
, 


The range is the interval  0,  
Graphing Exponential
Functions
Graph f  x   2x
Graphing Exponential
Functions
1
Graph f  x    
2
x
Example 2:
Graph f  x   4
x
Let’s make a table and
plot points to graph.
Example 2:
Graph f  x   4
x
Example 2:
Graph f  x   4
x
Properties:
Exponential Functions
Example 3:

Given a graph, find the value of b:
Example 3:

Given a graph, find the value of b:
Increasing and
Decreasing Functions
One-to-One
Exponential Functions
Compound Interest
 r
A  P 1  
 k
kt
Example 4:

The parents of a newborn child
invest $8,000 in a plan that earns
9% interest, compounded
quarterly. If the money is left
untouched, how much will the
child have in the account in 55
years?
Example 4 Solution:
Using the compound interest formula:
 r
A  P 1  
 k
kt
 0.09 
A  8000 1 

4 

 8000 1.0225
4 55 
220
 $ 1, 069,103.27
Future value of account
in 55 years
Base e
Exponential Functions
Sometimes called the natural base,
e  2.71828182845.... irrational number 
often appears as the base of an
exponential functions.
It is the base of the continuous
compound interest formula:
A  Pe
rt
Example 5:

If the parents of the newborn
child in Example 4 had invested
$8,000 at an annual rate of 9%,
compounded continuously, how
much would the child have in the
account in 55 years?
Example 5
Solution:
A  Pe
rt
compounded continuously
A  8000 e
 0.09  55
 8000 e
4.95
 $ 1,129,399.71
Future value of account
in 55 years
Graphing

f  x  e
Make a table and plot points:
x
Exponential
Functions





Horizontal asymptote
Function increases
y-intercept (0,1)
Domain all real
numbers
Range: y > 0
Translations
For k>0
 y = f(x) + k
 y = f(x) – k
 y = f(x - k)
 y = f(x + k)
Up k units
Down k units
Right k units
Left k units
Example 6:

On one set of axes, graph
f  x  2
x
and
f  x  2  3
x
Example 6:

On one set of axes, graph
f  x  2
Up 3
x
and
f  x  2  3
x
Example 7:

On one set of axes, graph
f  x   e and f  x   e
Right 3
x
x 3
Non-Rigid
Transformations

Exponential Functions with the
form f(x)=kbx and f(x)=bkx are
vertical and horizontal
stretchings of the graph f(x)=bx.
Use a graphing calculator to
graph these functions.