1. Exponential Functions and Their Graphs

2. Definition of Exponential Function
The exponential function f with base b is defined by
f(x) = abx
where b > 0, b  1, and x is any real number.
** when b> 1; b is considered a growth factor.
For instance,
f(x) = 3x and g(x) = 0.5x
are exponential functions.
Definition of Exponential Function

3. The value of f(x) = 3x when x = 2 is9
The value of f(x) = 3x when x = –2 is
f(–2) = 3–2 =
The value of g(x) = 0.5x when x = 4 is
g(4) = 0.54 =
0.0625

4. Graph of Exponential Function (a > 1)
The graph of f(x) = abx, b > 1 y 4
Range: (0, )
(0, 1) x 4
Horizontal Asymptote y = 0
Domain: (–, )
Graph of Exponential Function (a > 1)

5. Graph of Exponential Function (0 < a < 1)
The graph of f(x) = abx, 0 < b < 1
Since a< 1; a is decay factor. y 4
Range: (0, )
Horizontal Asymptote y = 0
(0, 1) x 4
Domain: (–, )
Graph of Exponential Function (0 < a < 1)

6. Example: Sketch the graph of f(x) = 2x.y
x f(x) -2 -1 1 2
2-2 = 1/4 4
2-1 = 1/2 2
20 = 1
21 = 2 x
22 = 4 –2 2
Example: Graph f(x) = 2x

7. Example: Translation of Graph
Example: Sketch the graph of g(x) = 2x – 1. State the domain and range. y
f(x) = 2x
The graph of this function is a vertical translation of the graph of f(x) = 2x down one unit . 4 2
Domain: (–, ) x
y = –1
Range: (–1, )
Example: Translation of Graph

8. Example: Reflection of Graph
Example: Sketch the graph of g(x) = 2-x. State the domain and range. y
f(x) = 2x
The graph of this function is a reflection the graph of f(x) = 2x in the y-axis. 4
Domain: (–, ) x –2 2
Range: (0, )
Example: Reflection of Graph

9. Example: Reflection of Graph
Example: Sketch the graph of g(x) = 4x
State the domain and range. y
Make a table.
x y 4 x –2 2
Domain: (–, )
Range: (3, ) or y > 3
Example: Reflection of Graph

10. Write an exponential function y = abx for the graph that includes the given points.
(2, 2) and (3, 4)
2 = ab2
y = abx
Substitute in (2,2) for x and y.
Solve for a
2 = a b2
Now substitute (3,4) in for x and y and 2/b2 in for a
4 = (2)b3 b2
4 = (2)b
4 = b
y = ½ (4)x

11. Write an exponential function y = abx for the graph that includes the given points.
(2, 4) and (3, 16)
y = abx

12. The Natural Base e The irrational number e, where e  2.718281828…
is used in applications involving growth and decay.
Using techniques of calculus, it can be shown that
The number e

13. Graph of Natural Exponential Function f(x) = ex
The graph of f(x) = ex y x
f(x) -2
0.14 -1
0.38 1
2.72 2
7.39 6 4 2 x –2 2
Graph of Natural Exponential Function f(x) = ex

14. Example: Sketch the graph of g(x) = ex-5 + 2.
State the domain and range. y
Make a table.
x y 4 x –2 2
Domain: (–, )
Range: (2, ) or y > 2

15. One to One Property
If the bases are the same, then set the exponents equal.
Example—
3x = 32
x = ___
Example—
3x+ 1 = 9
x = ___
Example—
2x - 1 = 1/32
x = ___
Example—
e3x + 2 = e3
x = ___
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

16. Interest Applications
Formulas for Compound Interest—
1.) compound per year -- A = P 1 + r nt n
r is the rate
n is the number times you
compound your money per
year
t is time.
Balance in account
Principal ($ you invest)
2. Compounded continuously– A = Pert

17. A total of $12000 is invested at an annual interest rate of 9%
A total of $12000 is invested at an annual interest rate of 9%. Find the balance after 5 years if it is compounded a. quarterly b. monthly c. continuously