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. Example: Exponential Function
The value of f(x) = 3x when x = 2 is
f(2) = 32 = 9
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
Example: Exponential Function

4. Graph of Exponential Function (a > 1)
The graph of f(x) = ax, a > 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) = ax, 0 < a < 1 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. x f(x) (x, f(x))y x
f(x)
(x, f(x)) -2
(-2, ¼) -1
(-1, ½) 1
(0, 1) 2
(1, 2) 4
(2, 4) 4 2 x –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. 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

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
The number e