exponential functions
DESCRIPTION
Exponential Functions. 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. (1, 6). (1, 3). (-1, 1/6). (-1, 1/3). (0, 1). Summary of the characteristics of the graph of. a >1 . - PowerPoint PPT PresentationTRANSCRIPT
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Exponential Functions
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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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n
1 22 2.254 2.44140625
12 2.61303529022365 2.714567482028760 2.71812669063
525600 2.7182792154
n
n)11(
e
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The number e is defined as the number that the expression
In calculus this expression is expressed using limit notation as
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3 2 1 0 1 2 3
2
4
6
y x2
y x3 y ex
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Sketching the graph of an exponential function: Advanced Graph the function: .
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