4.3B Exponential Functions
January 15, 2009
4.3B Exponential Functions
Objectives: Graph exponential functions
Solve exponential equations
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4.3B Exponential Functions
January 15, 2009
The Basic Exponential Function
(1, a)
x
f(x) = a
(­1, 1/a)
(0,1)
a > 0 and a = 1 a is a positive real number and is not equal to 1.
D : (All Reals) The domain of f(x) is the set of all real numbers.
R : y > 0 The range of f(x) is the set of y greater than zero.
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4.3B Exponential Functions
January 15, 2009
The Basic Exponential Function
Examples
x
1
f(x) = 2 g(x) = ( /3)
The base is 2.
x
The base is (1/3).
The base is a constant number only the exponent is a variable.
x ­2
­1
0
1
2
3
y
x
y
2­2 = 1/22 = 1/4
1
/2
1
2
4
8
­2
­1
0
1
2
3
(1/3)­2 = 1/3­2 = 32 = 9
3
1
1
/3
1
/9
1
/27
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4.3B Exponential Functions
January 15, 2009
Properties of the Exponential Function
f(x) = ax a > 1
1) The Domain : { x | x = (All Reals)}
Range : { y | y > 0}
2) There are no x intercepts and the y intercept is 1.
3) The x axis (y=0) is a horizontal asymptote as x ­ .
4) f(x) = ax , a >1, is an increasing funtion and is one­to­one.
5) The graph of f contains the points (0,1), (1,a), (­1,1/a).
(1, a)
(­1, 1/a)
(0,1)
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4.3B Exponential Functions
January 15, 2009
Properties of the Exponential Function
f(x) = ax 0 < a < 1
1) The Domain : { x | x = (All Reals)}
Range : { y | y > 0}
2) There are no x intercepts and the y intercept is 1.
3) The x axis (y=0) is a horizontal asymptote as x .
4) f(x) = ax , a >1, is an decreasing funtion and is one­to­one.
5) The graph of f contains the points (0,1), (1,a), (­1,1/a).
(­1, 1/a)
(0,1)
(1, a)
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4.3B Exponential Functions
January 15, 2009
Lets try together!
f(x) = 3x
The base is 3. So a = 3.
So we know that the points (1, 3), (0,1), (­1,1/3).
(1,3)
(­1,1/3)
(0,1)
(1,3)
(­1,1/3)
(0,1)
Now we can draw the line.
Jan 11­10:39 AM
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4.3B Exponential Functions
January 15, 2009
Lets try together!
f(x) = (1/4)x
The base is 1/4. So a = 1/4.
So we know that the points (1, 1/4), (0,1), (­1,4).
(­1,4)
(0,1)
(1,1/4)
(0,1)
Now we can draw the line.
Jan 11­10:39 AM
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4.3B Exponential Functions
January 15, 2009
Lets try together!
f(x) = 2­x ­ 3
The base is 2. So a = 2.
So we know that the points (1, 2), (0, 1), (­1, 1/2).
(1, 2)
(­1, 1/2)
(0, 1)
(0, 1)
Now we can draw the line.
Now we can graph the transformations
on the next page.
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4.3B Exponential Functions
January 15, 2009
Lets try together!
­x
f(x) = 2 ­ 3
f(x) = 2­x ­ 3 flips the graph
about the y­axis.
So we know that the points (­1, 2), (0, 1), (1, 1/2).
(­1, 2)
(0, 1)
(1, 1/2)
f(x) = 2­x ­ 3 shifts the graph down 3 units.
(­1, ­1)
(0, ­2)
(1, ­21/2)
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4.3B Exponential Functions
January 15, 2009
Homework:
page 282 (6, 7, 10, 29 - 36, 72, 73, 75, 78, 79, 98)
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