curve fitting with exponential and logarithmic … algebra 2 7-8 curve fitting with exponential and...

Holt Algebra 2

7-8 Curve Fitting with Exponential

and Logarithmic Models

Analyzing data values can identify a pattern, or repeated relationship, between two quantities.

Below is the table of values for the exponential function f(x) = 2(3x).

Holt Algebra 2



A table of values with a constant ratio between its f(x) values can be modeled by an exponential function of the form f(x) = abx.

x –1 0 1 2 3

f(x) 16 24 36 54 81

b value: the ratio 81 54

= 54 36

= 24 16

= 36 24

= 3 2 b =

3

2

a value: where x = 0. a = 24

f(x) = 𝑎𝑏𝑥

f(x) = 243

2

𝑥

Holt Algebra 2



x 0 1 2 3 4

f(x) 3 12 48 192 768


= 192 48

= 12 3

= 48 12

= 4 b = 4

a value: where x = 0. a = 3

f(x) = 𝑎𝑏𝑥

f(x) = 3 4 𝑥

Holt Algebra 2



x 4 5 6 7 8

f(x) 48 24 12 6 3


= 6 12

= 24

48 =

12 24

= b = 1

2

a value: f(x) = abx

48 = a1

2

4

48 = a1

16

16 x 48 = a

768 = a

f(x) = 𝑎𝑏𝑥

f(x) = 7681

2

𝑥

1 2

Holt Algebra 2



x -1 0 1 2 3

f(x) 1 6 11 16 21


16 11

≠ 6 1

≠ 11 6

≠

The function is not exponential

Holt Algebra 2



Once you know that data are exponential, you can use ExpReg (exponential regression) on your calculator to find a function that fits. This method of using data to find an exponential model is called an exponential regression. The calculator fits exponential functions to abx, so translations cannot be modeled.

Holt Algebra 2



If you do not see r2 and r when you calculate

regression, and turn these on by

selecting DiagnosticOn.

Remember!

Holt Algebra 2



Find an exponential model for the data. Use the model to predict when the tuition at U.T. Austin will be $6000.

Example 2: College Application

Step 1 Enter data into two lists in a graphing calculator. Use the exponential regression feature.

Tuition of the

University of Texas

Year Tuition

1999–00 $3128

2000–01 $3585

2001–02 $3776

2002–03 $3950

2003–04 $4188

An exponential model is f(x) ≈ 3236(1.07t), where f(x) represents the tuition and t is the number of years after the 1999–2000 year.

Holt Algebra 2



Example 2 Continued

Step 2 Graph the data and the function model to verify that it fits the data.

To enter the regression equation as Y1

from the screen, press , choose

5:Statistics, press , scroll to select the

EQ menu, and select 1:RegEQ.

Holt Algebra 2



Enter 6000 as Y2. Use the intersection feature. You may need to adjust the dimensions to find the intersection.

The tuition will be about $6000 when t = 9 or 2008–09.

Example 2 Continued

7500

15

0

0

Holt Algebra 2



Use exponential regression to find a function that models this data. When will the number of bacteria reach 2000?

Step 1 Enter data into two lists in a graphing calculator. Use the exponential regression feature. An exponential model is f(x) ≈ 199(1.25t), where f(x) represents the tuition and t is the number of minutes.

Check It Out! Example 2

Time (min) 0 1 2 3 4 5

Bacteria 200 248 312 390 489 610

Holt Algebra 2



Step 2 Graph the data and the function model to verify that it fits the data.

Check It Out! Example 2 Continued

To enter the regression equation as Y1

from the screen, press , choose

5:Statistics, press , scroll to select the

EQ menu, and select 1:RegEQ.

Holt Algebra 2



Enter 2000 as Y2. Use the intersection feature. You may need to adjust the dimensions to find the intersection.

The bacteria count at 2000 will happen at approximately 10.3 minutes.

2500

0

0 15

Check It Out! Example 2 Continued

Holt Algebra 2



HW pg. 548

#’s 6, 12, 13, 18, 19

curve fitting with exponential and logarithmic … algebra 2 7-8 curve fitting with exponential and...

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