Section 4.2
Exponential Functions and
Graphs
Exponential Function
The exponential function is very
important in math because it is used to
model many real life situations.
For example: population growth and
decay, compound interest, economics,
and much more.
Exponent
Remember
x 1
0
Exponential Function
The function f(x) = ax, where x is
a real number, a > 0 and a  1, is
called the exponential function,
base a.
The base needs to be positive in order to avoid
the complex numbers that would occur by taking
even roots of negative numbers.
Examples:
f ( x)  3
x
1
f ( x)   
3
x
f ( x)  (4.23)
x
Graphing Exponential Functions
To graph an exponential function, follow
the steps listed:
1. Compute some function values and list
the results in a table.
2. Plot the points and connect them with a
smooth curve. Be sure to plot enough
points to determine how steeply the
curve rises.
Example
Graph the exponential function y = f(x) = 3x.
x
y = f(x) = 3x
(x, y)
0
1
(0, 1)
1
3
(1, 3)
2
9
(2, 9)
3
27
(3, 27)
1
1/3
(1, 1/3)
2
1/9
(2, 1/9)
3
1/27
(3,
1/27)
Example
Graph y = 3x + 2.
The graph is the graph of y = 3x shifted _____
2 units.
x
y
3
1/3
2
1
1
3
0
9
1
27
2
81
3
243
Example
1
Graph the exponential function y  f ( x)   
3

1
x
x
y  f ( x)   
3
(x, y)
0
1
(0, 1)
1
3
(1, 3)
2
9
(2, 9)
3
27
(3, 27)
1
1/3
(1, 1/3)
2
1/9
(2, 1/9)
3
1/27
(3, 1/27)
x
Example
Graph y = 4  3x
The graph is a reflection of the graph of y = 3x across the
_______, followed by a reflection across the _______ and
then a shift _______ of 4 units.
x
3
2
1
0
1
2
3
y
23
5
1
3
3.67
3.88
3.96
Observing Relationships
Connecting the Concepts
The Number e
e is known as the natural base
(Most important base for exponential
functions.)
e is an irrational number
(can’t write its exact value)
We approximate e
e  2.718281
The Number e
Find each value of ex, to four decimal
places, using the ex key on a calculator.
a) e4
b) e0.25
c) e2
d) e1
Natural Exponential Function
f(x)  e
x
Remember
e is a number
e lies between 2 and 3
Graphs of Exponential Functions,
Base e
Graph f(x) = ex.
x
f(x)
2
0.135
1
0.368
0
1
1
2.718
2
7.389
Example
Graph f(x) = ex+2.
x
f(x)
4
0.135
3
0.368
2
1
1
2.718
0
7.389
1
20.086
Example
Graph f(x) = 2  e3x.
x
f(x)
2
401.43
1
18.09
0
1
1
1.95
2
1.99
Compound Interest Formula
 r
A  P 1  
 n
nt
A  amount accumulated after t years
P  principal dollar s in vested
r  int erest rate (decimal form)
n  number of times money is compounded per year
t  number of years money is compounded
Example
A father sets up a savings account for his daughter.
He puts $1000 in an account that is compounded
quarterly at an annual interest rate of 8%.
How much money will be in the account at the end
of 10 years? (Assume no other deposits were made
after the original one.)