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

Holt Algebra 2

7-8 Curve Fitting with Exponentialand Logarithmic Models7-8 Curve Fitting with Exponential and Logarithmic Models

Holt Algebra 2

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

Holt Algebra 2

7-8 Curve Fitting with Exponentialand Logarithmic Models

Warm UpPerform a quadratic regression on the following data:

x 1 2 6 11 13

f(x) 3 6 39 120 170

f(x) ≈ 0.98x2 + 0.1x + 2.1

Holt Algebra 2


Model data by using exponential and logarithmic functions.

Use exponential and logarithmic models to analyze and predict.

Objectives

Holt Algebra 2


exponential regressionlogarithmic regression

Vocabulary

Holt Algebra 2


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

Look at this table of values for the exponential function f(x) = 2(3x).

Holt Algebra 2


For linear functions (first degree), first differences are constant. For quadratic functions, second differences are constant, and so on.

Remember!

Notice that the ratio of each y-value and the previous one is constant. Each value is three times the one before it, so the ratio of function values is constant for equally spaced x-values. This data can be fit by an exponential function of the form f(x) = abx.

Holt Algebra 2


Determine whether f is an exponential function of x of the form f(x) = abx. If so, find the constant ratio.

Example 1: Identifying Exponential Data

A.

+1 +2 +3 +4

x –1 0 1 2 3

f(x) 2 3 5 8 12

B. x –1 0 1 2 3

f(x) 16 24 36 54 81

+8 +12 +18 +27FirstDifferences

SecondDifferences

+1 +1 +1 +4 +6 +9

Ratio 81 54

=54 36

=24 16

=36 24

= 3 2

Second differences are constant; f is a quadratic function of x.

This data set is exponential, with a constant ratio of 1.5.

Holt Algebra 2


Determine whether y is an exponential function of x of the form f(x) = abx. If so, find the constant ratio.a.

x –1 0 1 2 3

f(x) 2.6 4 6 9 13.5

b.

x –1 0 1 2 3

f(x) –3 2 7 12 17

+5 +5 +5 +5FirstDifferences

SecondDifferences

+0.66 +1 +1.5

Ratio13.5 9 =

9 6 =

6 4 =

2 1.3

First differences are constant; y is a linear function of x.

This data set is exponential, with a constant ratio of 1.5.

Check It Out! Example 1

+1.34 +2 +3 +4.5

4 2.6

=

Holt Algebra 2


In Chapters 2 and 5, you used a graphing calculator to perform linear progressions and quadratic regressions to make predictions. You can also use an exponential model, which is an exponential function that represents a real data set.

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

150

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.


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

00 15


Holt Algebra 2


Many natural phenomena can be modeled by natural log functions. You can use a logarithmic regression to find a function

Most calculators that perform logarithmic regression use ln rather than log.

Helpful Hint

Holt Algebra 2


Find a natural log model for the data. According to the model, when will the global population exceed 9,000,000,000?

Enter the into the two lists in a graphing calculator. Then use the logarithmic regression feature. Press CALC 9:LnReg. A logarithmic model is f(x) ≈ 1824 + 106ln x, where f is the year and x is the population in billions.

Global Population Growth

Population (billions)

Year

1 1800

2 1927

3 1960

4 1974

5 1987

6 1999

Example 3: Application

Holt Algebra 2


The calculated value of r2 shows that an equation fits the data.

Example 3 Continued

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

Use the value feature to find y when x is 9. The population will exceed 9,000,000,000 in the year 2058.

0

2500

0 15

Holt Algebra 2


Time (min) 1 2 3 4 5 6 7

Speed (m/s) 0.5 2.5 3.5 4.3 4.9 5.3 5.6

Use logarithmic regression to find a function that models this data. When will the speed reach 8.0 m/s?


Enter the into the two lists in a graphing calculator. Then use the logarithmic regression feature. Press CALC 9: LnReg. A logarithmic model is f(x) ≈ 0.59 + 2.64 ln x, where f is the time and x is the speed.

Holt Algebra 2



The calculated value of r2 shows that an equation fits the data.

Graph the data and function model to verify that it fits the data. an equation fits the data.

Use the intersect feature to find y when x is 8. The time it will reach 8.0 m/s is 16.6 min.

0

10

200

Holt Algebra 2


Lesson Quiz: Part I

Determine whether f is an exponential function of x. If so, find the constant ratio.

x –1 0 1 2

f(x) 10 9 8.1 7.291.

x –1 0 1 2 3

f(x) 3 6 12 21 33

yes; constant ratio = 0.9

no; second difference are constant; f is quadratic.

2.

Holt Algebra 2


Lesson Quiz: Part II3. Find an exponential model for the data. Use the model to estimate when the insurance value will drop below $2000.

Insurance Value

Year(1990 = year 0) Value

0 10,000

2 9,032

5 7,753

9 6,290

11 5,685

f(x) ≈ 10,009(0.95)t; value will dip below 2000 in year 32 or 2022.

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

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