Exponential functions, definituon and examples. Graph and characteristics of exponential functions. Applications of exponential functions.

1. Exponential Functions and
Their Graphs

2. 2
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.

3. 3
The value of f(x) = 3x when x = 2 is
f(2) = 32 =
The value of g(x) = 0.5x when x = 4 is
g(4) = 0.54 =
The value of f(x) = 3x when x = –2 is
9
1
9
f(–2) = 3–2 =
0.0625

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

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

6. 6
Example: Sketch the graph of f(x) = 2x.
x
y
2
–2
2
4
x f(x)
-2
-1
0
1
2
2-2 = 1/4
2-1 = 1/2
20 = 1
21 = 2
22 = 4

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

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

9. 9
Example: Sketch the graph of g(x) = 4x-3 + 3.
State the domain and range.
x
y
Make a table.
Domain: (–, )
Range: (3, ) or y > 3
2
–2
4
x y
3 4
2 3.25
1 3.0625
4 7
5 19

10. 10
Write an exponential function y = abx for the
graph that includes the given points.
(2, 2) and (3, 4)
y = abx Substitute in (2,2) for x and y.
2 = ab2
Solve for a
2 = a
b2
Now substitute (3,4) in for x and y into y = abx
4 = ab3
b3
4 = a
b3
Set them equal to each other:
2 = 4 = a
b3 b2
Now solve for b to get b=2
You must solve for a: 2/2^2 = (1/2) = a
Subst you’re a = (1/2) & b = 2 into either one
of your 2 equations: y = abx
Y = (1/2)(2)^x

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

12. 12
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









 n
e
n
n
as
1
1
The Natural Base e

13. 13
The graph of f(x) = ex
y
x
2
–2
2
4
6
x f(x)
-2 0.14
-1 0.38
0 1
1 2.72
2 7.39

14. 14
Example: Sketch the graph of g(x) = ex-5 + 2.
State the domain and range.
x
y
Make a table.
Domain: (–, )
Range: (2, ) or y > 2
2
–2
4
x y
5 3
6 4.72
7 9.39
4 2.36
3 2.14

15. 15
One to One Property
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
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 = ___

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

17. 17
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