1. Holt McDougal Algebra 2 4-8 Curve Fitting with Exponential and Logarithmic Models 4-8 Curve Fitting with Exponential and Logarithmic Models Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz Holt McDougal Algebra 2

2. 4-8 Curve Fitting with Exponential and Logarithmic Models Warm Up Perform a quadratic regression on the following data: x 1261113 f(x)f(x) 3639120170 f(x) ≈ 0.98x 2 + 0.1x + 2.1

3. Holt McDougal Algebra 2 4-8 Curve Fitting with Exponential and Logarithmic Models Model data by using exponential and logarithmic functions. Use exponential and logarithmic models to analyze and predict. Objectives

4. Holt McDougal Algebra 2 4-8 Curve Fitting with Exponential and Logarithmic Models exponential regression logarithmic regression Vocabulary

5. Holt McDougal Algebra 2 4-8 Curve Fitting with Exponential and Logarithmic Models 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(3 x ).

6. Holt McDougal Algebra 2 4-8 Curve Fitting with Exponential and Logarithmic Models 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 suggests the data can be fit by an exponential function of the form f(x) = ab x.

7. Holt McDougal Algebra 2 4-8 Curve Fitting with Exponential and Logarithmic Models Determine whether f is an exponential function of x of the form f(x) = ab x. If so, find the constant ratio. Example 1: Identifying Exponential Data A. +1 +2 +3 +4 x –10123 f(x)f(x) 235812 B. x –10123 f(x)f(x) 1624365481 +8 +12 +18 +27 First Differences Second Differences +1 +1 +1 +4 +6 +9 Ratio 81 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.

8. Holt McDougal Algebra 2 4-8 Curve Fitting with Exponential and Logarithmic Models Determine whether y is an exponential function of x of the form f(x) = ab x. If so, find the constant ratio. a. x –10123 f(x)f(x) 2.646913.5 b. x –10123 f(x)f(x) –3271217 Check It Out! Example 1

9. Holt McDougal Algebra 2 4-8 Curve Fitting with Exponential and Logarithmic Models In Chapters 1 and 2, you used a graphing calculator to perform linear regressions and quadratic regressions to make predictions. You can also use an exponential regression, which gives 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 ab x, so translations cannot be modeled.

10. Holt McDougal Algebra 2 4-8 Curve Fitting with Exponential and Logarithmic Models If you do not see r 2 and r when you calculate regression, and turn these on by selecting DiagnosticOn. Remember!

11. Holt McDougal Algebra 2 4-8 Curve Fitting with Exponential and Logarithmic Models 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 YearTuition 1999–00$3128 2000–01$3585 2001–02$3776 2002–03$3950 2003–04$4188 Have your calculator store the regression equation as Y1.

12. Holt McDougal Algebra 2 4-8 Curve Fitting with Exponential and Logarithmic Models 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 The exponential model is f(t) ≈ 3236(1.07 t ), where f(t) represents the tuition and t is the number of years after the 1999–2000 year.

13. Holt McDougal Algebra 2 4-8 Curve Fitting with Exponential and Logarithmic Models We can use the model to find the time when the tuition will equal $6000. First, set up the equation 3236(1.07 t ) = 6000. Now solve graphically. 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 = 9.073 or in 2008–09. Example 2 Continued 7500 15 0 0

14. Holt McDougal Algebra 2 4-8 Curve Fitting with Exponential and Logarithmic Models Use exponential regression to find a function that models this data. When will the number of bacteria reach 2000? Check It Out! Example 2 Time (min)012345 Bacteria200248312390489610 2500 0 015

15. Holt McDougal Algebra 2 4-8 Curve Fitting with Exponential and Logarithmic Models 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

16. Holt McDougal Algebra 2 4-8 Curve Fitting with Exponential and Logarithmic Models 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 11800 21927 31960 41974 51987 61999 Example 3: Application

17. Holt McDougal Algebra 2 4-8 Curve Fitting with Exponential and Logarithmic Models The calculated value of r 2 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 trace feature to find y when x is 9. The population will exceed 9,000,000,000 in the year 2058. 0 2500 0 15

18. Holt McDougal Algebra 2 4-8 Curve Fitting with Exponential and Logarithmic Models Time (min)1234567 Speed (m/s)0.52.53.54.34.95.35.6 Use logarithmic regression to find a function that models this data. When will the speed reach 8.0 m/s? Check It Out! Example 4 0 10 20 0

19. Homework pp. 292-294 8-16, 22