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Use the regression feature to find an exponential function that best fits the data in the table.

f ( x ) = 731.92 ( 0.738 ) x

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Write the exponential function as an exponential equation with base e .

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Graph the exponential equation on the scatter diagram.

Graph of a scattered plot with an estimation line.
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Use the intersect feature to find the value of x for which f ( x ) = 250.

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For the following exercises, refer to [link] .

x f(x)
1 5.1
2 6.3
3 7.3
4 7.7
5 8.1
6 8.6

Use a graphing calculator to create a scatter diagram of the data.

Graph of the table’s values.
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Use the LOGarithm option of the REGression feature to find a logarithmic function of the form y = a + b ln ( x ) that best fits the data in the table.

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Use the logarithmic function to find the value of the function when x = 10.

f ( 10 ) 9.5

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Graph the logarithmic equation on the scatter diagram.

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Use the intersect feature to find the value of x for which f ( x ) = 7.

When f ( x ) = 7 , x 2.7.

Graph of the intersection of a scattered plot with an estimation line and y=7.
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For the following exercises, refer to [link] .

x f(x)
1 7.5
2 6
3 5.2
4 4.3
5 3.9
6 3.4
7 3.1
8 2.9

Use a graphing calculator to create a scatter diagram of the data.

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Use the LOGarithm option of the REGression feature to find a logarithmic function of the form y = a + b ln ( x ) that best fits the data in the table.

f ( x ) = 7.544 2.268 ln ( x )

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Use the logarithmic function to find the value of the function when x = 10.

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Graph the logarithmic equation on the scatter diagram.

Graph of a scattered plot with an estimation line.
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Use the intersect feature to find the value of x for which f ( x ) = 8.

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For the following exercises, refer to [link] .

x f(x)
1 8.7
2 12.3
3 15.4
4 18.5
5 20.7
6 22.5
7 23.3
8 24
9 24.6
10 24.8

Use a graphing calculator to create a scatter diagram of the data.

Graph of the table’s values.
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Use the LOGISTIC regression option to find a logistic growth model of the form y = c 1 + a e b x that best fits the data in the table.

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Graph the logistic equation on the scatter diagram.

Graph of a scattered plot with an estimation line.
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To the nearest whole number, what is the predicted carrying capacity of the model?

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Use the intersect feature to find the value of x for which the model reaches half its carrying capacity.

When f ( x ) = 12.5 , x 2.1.

Graph of the intersection of a scattered plot with an estimation line and y=12.
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For the following exercises, refer to [link] .

x f ( x )
0 12
2 28.6
4 52.8
5 70.3
7 99.9
8 112.5
10 125.8
11 127.9
15 135.1
17 135.9

Use a graphing calculator to create a scatter diagram of the data.

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Use the LOGISTIC regression option to find a logistic growth model of the form y = c 1 + a e b x that best fits the data in the table.

f ( x ) = 136.068 1 + 10.324 e 0.480 x

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Graph the logistic equation on the scatter diagram.

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To the nearest whole number, what is the predicted carrying capacity of the model?

about 136

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Use the intersect feature to find the value of x for which the model reaches half its carrying capacity.

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Extensions

Recall that the general form of a logistic equation for a population is given by P ( t ) = c 1 + a e b t , such that the initial population at time t = 0 is P ( 0 ) = P 0 . Show algebraically that c P ( t ) P ( t ) = c P 0 P 0 e b t .

Working with the left side of the equation, we see that it can be rewritten as a e b t :

c P ( t ) P ( t ) = c c 1 + a e b t c 1 + a e b t = c ( 1 + a e b t ) c 1 + a e b t c 1 + a e b t = c ( 1 + a e b t 1 ) 1 + a e b t c 1 + a e b t = 1 + a e b t 1 = a e b t

Working with the right side of the equation we show that it can also be rewritten as a e b t . But first note that when t = 0 , P 0 = c 1 + a e b ( 0 ) = c 1 + a . Therefore,

c P 0 P 0 e b t = c c 1 + a c 1 + a e b t = c ( 1 + a ) c 1 + a c 1 + a e b t = c ( 1 + a 1 ) 1 + a c 1 + a e b t = ( 1 + a 1 ) e b t = a e b t

Thus, c P ( t ) P ( t ) = c P 0 P 0 e b t .

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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