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

Perform a quadratic regression on the following data: Warm Up Perform a quadratic regression on the following data: x 1 2 6 11 13 f(x) 3 39 120 170 f(x) ≈ 0.98x2 + 0.1x + 2.1

Objectives Model data by using exponential and logarithmic functions. Use exponential and logarithmic models to analyze and predict.

Vocabulary exponential regression logarithmic regression

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).

Notice that the ratio of each y-value and the previous one is constant 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. For linear functions (first degree), first differences are constant. For quadratic functions, second differences are constant, and so on. Remember!

Example 1: Identifying Exponential Data Determine whether f is an exponential function of x of the form f(x) = abx. If so, find the constant ratio. x –1 1 2 3 f(x) 5 8 12 B. x –1 1 2 3 f(x) 16 24 36 54 81 A. +8 +12 +18 +27 +1 +2 +3 +4 First Differences +1 +1 +1 Second Differences +4 +6 +9 81 54 = 54 36 = 36 24 = 24 16 = 3 2 Ratio Second differences are constant; f is a quadratic function of x. This data set is exponential, with a constant ratio of 1.5.

First differences are constant; y is a linear function of x. Check It Out! Example 1 Determine whether y is an exponential function of x of the form f(x) = abx. If so, find the constant ratio. a. b. x –1 1 2 3 f(x) 2.6 4 6 9 13.5 x –1 1 2 3 f(x) –3 7 12 17 +5 +5 +5 +5 +1.34 +2 +3 +4.5 First Differences +0.66 +1 +1.5 Second Differences 13.5 9 = 9 6 = 6 4 = 4 2.6 = 2 1.3 Ratio First differences are constant; y is a linear function of x. This data set is exponential, with a constant ratio of 1.5.

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.

If you do not see r2 and r when you calculate regression, and turn these on by selecting DiagnosticOn. Remember!

Tuition of the University of Texas Example 2: College Application Find an exponential model for the data. Use the model to predict when the tuition at U.T. Austin will be $6000. Tuition of the University of Texas Year Tuition 1999–00 $3128 2000–01 $3585 2001–02 $3776 2002–03 $3950 2003–04 $4188 Step 1 Enter data into two lists in a graphing calculator. Use the exponential regression feature. 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.

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.

The tuition will be about $6000 when t = 9 or 2008–09. Example 2 Continued 7500 15 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.

Check It Out! Example 2 Use exponential regression to find a function that models this data. When will the number of bacteria reach 2000? Time (min) 1 2 3 4 5 Bacteria 200 248 312 390 489 610 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 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.

Check It Out! Example 2 Continued 2500 15 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.

Many natural phenomena can be modeled by natural log functions 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

Global Population Growth Population (billions) Example 3: Application Find a natural log model for the data. According to the model, when will the global population exceed 9,000,000,000? Global Population Growth Population (billions) Year 1 1800 2 1927 3 1960 4 1974 5 1987 6 1999 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.

The calculated value of r2 shows that an equation fits the data. Example 3 Continued The calculated value of r2 shows that an equation fits the data. 2500 15 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.

Check It Out! Example 4 Use logarithmic regression to find a function that models this data. When will the speed reach 8.0 m/s? 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 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.

Check It Out! Example 4 Continued The calculated value of r2 shows that an equation fits the data. 10 20 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.

no; second difference are constant; f is quadratic. Lesson Quiz: Part I Determine whether f is an exponential function of x. If so, find the constant ratio. 1. x –1 1 2 f(x) 10 9 8.1 7.29 yes; constant ratio = 0.9 2. x –1 1 2 3 f(x) 6 12 21 33 no; second difference are constant; f is quadratic.

f(x) ≈ 10,009(0.95)t; value will dip below 2000 in year 32 or 2022. Lesson Quiz: Part II 3. 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 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.