exponential functions and their graphs

1. Exponential Functions and
Their Graphs
Digital Lesson

2. 2
The exponential function f with base a is
defined by
f(x) = ax
where a > 0, a  1, and x is any real number.
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) = ax, a > 1
y
x
(0, 1)
Domain: (–, )
Range: (0, )
Horizontal Asymptote
y = 0
4
4

5. 5
The graph of f(x) = ax, 0 < a < 1
y
x
(0, 1)
Domain: (–, )
Range: (0, )
Horizontal Asymptote
y = 0
4
4

6. 6
Example: Sketch the graph of f(x) = 2x.
x
x f(x) (x, f(x))
-2 ¼ (-2, ¼)
-1 ½ (-1, ½)
0 1 (0, 1)
1 2 (1, 2)
2 4 (2, 4)
y
2
–2
2
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
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

11. 11
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

12. 12
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

13. 13
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

14. 14
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