1. 3.1 Exponential Functions and Their Graphs
CW: Exponential Growth and Decay

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. 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. The Graph of f(x) = ax, a > 1y
Range: (0, )
(0, 1) x
Horizontal Asymptote y = 0
Domain: (–, )

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

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

7. 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, )

8. Example: Sketch the graph of g(x) = 2x+1 + 2
Example: Sketch the graph of g(x) = 2x 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 left one unit and up two units. 4 2
y = 2 x
Domain: (–, )
Range: (2, )

9. 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, )

10. The irrational number e, where
is used in applications involving growth and decay.
Using techniques of calculus, it can be shown that

11. The Graph of f(x) = ex x f(x) -2 0.14 -1 0.38 1 2.72 2 7.39 y x 6 4 21
2.72 2
7.39 6 4 2 x –2 2

12. CW: Exponential Growth and Decay
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