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Section5.2
Exponential Functions and Graphs
Section5.2 Exponential Functions and Graphs Graphing Definition - - PowerPoint PPT Presentation
Section5.2 Exponential Functions and Graphs Graphing Definition - - PowerPoint PPT Presentation
Section5.2 Exponential Functions and Graphs Graphing Definition The exponential function with base a is given by f ( x ) = a x Definition The exponential function with base a is given by f ( x ) = a x Here a must be positive, and also a = 1
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Definition
The exponential function with base a is given by f (x) = ax
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Definition
The exponential function with base a is given by f (x) = ax Here a must be positive, and also a = 1
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Graphing Exponential Functions
Let’s graph y = 2x
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Graphing Exponential Functions
Let’s graph y = 2x x y −4 2−4 = 1
24 = 1 16
−3 2−3 = 1
23 = 1 8
−2 2−2 = 1
22 = 1 4
−1 2−1 = 1
21 = 1 2
20 = 1 1 21 = 2 2 22 = 4 3 23 = 8 4 24 = 16
SLIDE 7
Graphing Exponential Functions
Let’s graph y = 2x x y −4 2−4 = 1
24 = 1 16
−3 2−3 = 1
23 = 1 8
−2 2−2 = 1
22 = 1 4
−1 2−1 = 1
21 = 1 2
20 = 1 1 21 = 2 2 22 = 4 3 23 = 8 4 24 = 16
−4 −3 −2 −1 1 2 3 4 5 10 15
SLIDE 8
Graphing Exponential Functions
Let’s graph y = 2x x y −4 2−4 = 1
24 = 1 16
−3 2−3 = 1
23 = 1 8
−2 2−2 = 1
22 = 1 4
−1 2−1 = 1
21 = 1 2
20 = 1 1 21 = 2 2 22 = 4 3 23 = 8 4 24 = 16
−4 −3 −2 −1 1 2 3 4 5 10 15
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Graphing Exponential Functions
Let’s graph y = 2x x y −4 2−4 = 1
24 = 1 16
−3 2−3 = 1
23 = 1 8
−2 2−2 = 1
22 = 1 4
−1 2−1 = 1
21 = 1 2
20 = 1 1 21 = 2 2 22 = 4 3 23 = 8 4 24 = 16
−4 −3 −2 −1 1 2 3 4 5 10 15
Notice that the graph has a horizontal asymptote along the x-axis.
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Graphing Exponential Functions (continued)
Let’s graph y = 1
3
x
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Graphing Exponential Functions (continued)
Let’s graph y = 1
3
x x y −3 1
3
−3 = 33 = 27 −2 1
3
−2 = 32 = 9 −1 1
3
−1 = 31 = 3 1
3
0 = 1 1 1
3
1 = 1
3
2 1
3
2 = 1
9
3 1
3
3 = 1
27
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Graphing Exponential Functions (continued)
Let’s graph y = 1
3
x x y −3 1
3
−3 = 33 = 27 −2 1
3
−2 = 32 = 9 −1 1
3
−1 = 31 = 3 1
3
0 = 1 1 1
3
1 = 1
3
2 1
3
2 = 1
9
3 1
3
3 = 1
27
−4 −3 −2 −1 1 2 3 4 5 10 15 20 25
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Graphing Exponential Functions (continued)
Let’s graph y = 1
3
x x y −3 1
3
−3 = 33 = 27 −2 1
3
−2 = 32 = 9 −1 1
3
−1 = 31 = 3 1
3
0 = 1 1 1
3
1 = 1
3
2 1
3
2 = 1
9
3 1
3
3 = 1
27
−4 −3 −2 −1 1 2 3 4 5 10 15 20 25
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Graphing Exponential Functions (continued)
Let’s graph y = 1
3
x x y −3 1
3
−3 = 33 = 27 −2 1
3
−2 = 32 = 9 −1 1
3
−1 = 31 = 3 1
3
0 = 1 1 1
3
1 = 1
3
2 1
3
2 = 1
9
3 1
3
3 = 1
27
−4 −3 −2 −1 1 2 3 4 5 10 15 20 25
Again, notice that the graph has a horizontal asymptote along the x-axis.
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Graphing Exponential Functions (continued)
In general, there are just two basic shapes for f (x) = ax:
(0,1) (1,a)
a > 1
(0,1) (1,a)
a < 1
Again, they both have horizontal asymptote along the x-axis. Domain: (−∞, ∞) Range: (0, ∞)
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The Natural Exponential Function
The natural exponential function is given by f (x) = ex
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The Natural Exponential Function
The natural exponential function is given by f (x) = ex e is an irrational number e ≈ 2.7182818
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The Natural Exponential Function (continued)
Let’s graph y = ex
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The Natural Exponential Function (continued)
Let’s graph y = ex x y −3 e−3 ≈ 0.0498 −2 e−2 ≈ 0.1353 −1 e−1 ≈ 0.3679 e0 = 1 1 e1 ≈ 2.7183 2 e2 ≈ 7.3891 3 e3 ≈ 20.0855
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The Natural Exponential Function (continued)
Let’s graph y = ex x y −3 e−3 ≈ 0.0498 −2 e−2 ≈ 0.1353 −1 e−1 ≈ 0.3679 e0 = 1 1 e1 ≈ 2.7183 2 e2 ≈ 7.3891 3 e3 ≈ 20.0855
−4 −3 −2 −1 1 2 3 4 5 10 15 20
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The Natural Exponential Function (continued)
Let’s graph y = ex x y −3 e−3 ≈ 0.0498 −2 e−2 ≈ 0.1353 −1 e−1 ≈ 0.3679 e0 = 1 1 e1 ≈ 2.7183 2 e2 ≈ 7.3891 3 e3 ≈ 20.0855
−4 −3 −2 −1 1 2 3 4 5 10 15 20
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Examples
- 1. y = 4 − 3x
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Examples
- 1. y = 4 − 3x
−4 −3 −2 −1 1 2 3 4 −20 −15 −10 −5 5 10
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Examples
- 1. y = 4 − 3x
−4 −3 −2 −1 1 2 3 4 −20 −15 −10 −5 5 10
- 2. y = ex − 2
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Examples
- 1. y = 4 − 3x
−4 −3 −2 −1 1 2 3 4 −20 −15 −10 −5 5 10
- 2. y = ex − 2
−4 −3 −2 −1 1 2 3 4 −5 5 10 15 20
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Modeling
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Example
A company begins an Internet advertising campaign to market a new
- telephone. The percentage of the target market that buys a product
is generally a function of the length of the advertising campaign. The estimated percentage is given by the exponential function f (t) = 100(1 − e−0.04t), where t is the number of days of the campaign. Find f (25), the percentage of the target market that has bought the product after a 25-day advertising campaign.
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Example
A company begins an Internet advertising campaign to market a new
- telephone. The percentage of the target market that buys a product
is generally a function of the length of the advertising campaign. The estimated percentage is given by the exponential function f (t) = 100(1 − e−0.04t), where t is the number of days of the campaign. Find f (25), the percentage of the target market that has bought the product after a 25-day advertising campaign. 63.2%