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Exponential Functions
MM3A2e Investigate characteristics: domain and range, asymptotes, zeros, intercepts, intervals of increase and decrease, rate of change.
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An exponential function is a function of the form
where a is a positive real number (a > 0) and . The domain of f is the set of all real numbers.
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3 2 1 0 1 2 3
2
4
6
(0, 1)
(1, 3)
(1, 6)
(-1, 1/3) (-1, 1/6)
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Summary of the characteristics of the graph of
a >1
• The domain is all real numbers. Range is set of positive numbers.
• No x-intercepts; y-intercept is 1.
• The x-axis (y=0) is a horizontal asymptote as
• a>1, is an increasing function and is one-to-one.
• The graph contains the points (0,1); (1,a), and (-1, 1/a).
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3 2 1 0 1 2 3
2
4
6
(-1, 3)
(-1, 6)
(0, 1) (1, 1/3) (1, 1/6)
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Summary of the characteristics of the graph of
0 <a <1
• The domain is all real numbers. Range is set of positive numbers.
• No x-intercepts; y-intercept is 1.• The x-axis (y=0) is a horizontal asymptote as
• 0<a<1, is a decreasing function and is one-to-one.
• The graph contains the points (0,1); (1,a), and (-1, 1/a).•The graph is smooth continuous with no corners or gaps.
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0
5
10
(0, 1)
(1, 3)
y x3
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0
5
10
(0, 1)(-1, 3)
y x 3
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0
5
10
(0, 3)
(-1, 5)
y = 2
y x 3 2
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Horizontal Asymptote: y = 2
Range: { y | y >2 } or
Domain: All real numbers
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3 2 1 0 1 2 3
2
4
6
y x2
y x3 y ex
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Characteristics of Exponential Functions f(x) = ax
• Domain: (-∞,∞)
• y-intercept is the point (0,1)
• Continuous
• Increasing if a > 1
• Range: (0,∞)
• No x-intercept
• Horizontal asymptote y=0
• Decreasing if a < 1
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