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Exponential Functions and Their Graphs
Digital Lesson
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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.
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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
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The graph of f(x) = ax, a > 1
y
x(0, 1)
Domain: (–, )
Range: (0, )
Horizontal Asymptote y = 0
4
4
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The graph of f(x) = ax, 0 < a < 1
y
x
(0, 1)
Domain: (–, )
Range: (0, )
Horizontal Asymptote y = 0
4
4
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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
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Example: Sketch the graph of g(x) = 2x – 1. State the domain and range.
x
yThe 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
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Example: Sketch the graph of g(x) = 2-x. State the domain and range.
x
yThe 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
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The graph of f(x) = ex
y
x2 –2
2
4
6
x f(x)
-2 0.14
-1 0.38
0 1
1 2.72
2 7.39
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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
ne
n
n
as 1
1
