1. Exponential Functions and Their Graphs

2. Definition of Exponential Function
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.
Definition of Exponential Function

3. Graph of Exponential Function (a > 1)
The graph of f(x) = ax, a > 1 y 4
Range: ???
(0, 1) x 4
Horizontal Asymptote y = 0
Domain: ???
Graph of Exponential Function (a > 1)

4. Graph of Exponential Function (0 < a < 1)
The graph of f(x) = ax, 0 < a < 1 y 4
Range: ???
Horizontal Asymptote y = 0
(0, 1) x 4
Domain: ???
Graph of Exponential Function (0 < a < 1)

6. 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: ???
Example: Translation of Graph

12. 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