Modeling Real-World Data 9-6 Modeling Real-World Data Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2

Warm Up quadratic: y ≈ 2.13x2 – 2x + 6.12 1. Use a calculator to perform quadratic and exponential regressions on the following data. x 3 5 8 13 y 19 50 126 340 quadratic: y ≈ 2.13x2 – 2x + 6.12 exponential: y ≈ 10.57(1.32)x

Objectives Apply functions to problem situations. Use mathematical models to make predictions.

Much of the data that you encounter in the real world may form a pattern. Many times the pattern of the data can be modeled by one of the functions you have studied. You can then use the functions to analyze trends and make predictions. Recall some of the parent functions that you have studied so far.

Because the square-root function is the inverse of the quadratic function, the constant differences for x- and y-values are switched. Helpful Hint

Example 1A: Identifying Models by Using Constant Differences or Ratios Use constant differences or ratios to determine which parent function would best model the given data set. Time (yr) 5 10 15 20 25 Height (in.) 58 93 128 163 198 Notice that the time data are evenly spaced. Check the first differences between the heights to see if the data set is linear.

Example 1A Continued Height (in.) 58 93 128 163 198 First differences 35 35 35 35 . Because the first differences are a constant 35, a linear model will best model the data.

Example 1B: Identifying Models by Using Constant Differences or Ratios Time (yr) 4 8 12 16 20 Population 10,000 9,600 9,216 8,847 8,493 Notice that the time data are evenly spaced. Check the first differences between the populations. Population 10,000 9,600 9,216 8,847 8,493 First differences –400 –384 –369 –354 Second differences 16 15 15

Example 1B Continued Neither the first nor second differences are constant. Check ratios between the volumes. 9216 9600 8847 = 0.96, 10,000 ≈ 0.96, and ≈ 0.96. 8493 8847

Example 1B Continued Because the ratios between the values of the dependent variable are constant, an exponential function would best model the data. Check A scatter plot reveals a shape similar to an exponential decay function.

Example 1C: Identifying Models by Using Constant Differences or Ratios Time (s) 1 2 3 4 5 Height (m) 132 165 154 99 Notice that the time data are evenly spaced. Check the first differences between the heights. Height (m) 132 165 154 99 First differences 33 –11 –55 –99 Second differences –44 –44 –44

Example 1C Continued Because the second differences of the dependent variables are constant when the independent variables are evenly spaced, a quadratic function will best model the data. Check A scatter reveals a shape similar to a quadratic parent function f(x) = x2.

Check It Out! Example 1a Use constant differences or ratios to determine which parent function would best model the given data set. x 12 48 108 192 300 y 10 20 30 40 50 Notice that the data for the y-values are evenly spaced. Check the first differences between the x-values.

Check It Out! Example 1a Continued 12 48 108 192 300 First differences 36 60 84 108 Second differences 24 24 24 Because the second differences of the independent variable are constant when the dependent variables are evenly spaced, a square-root function will best model the data.

Check It Out! Example 1b x 21 22 23 24 y 243 324 432 576 Notice that x-values are evenly spaced. Check the differences between the y-values. y 243 324 432 576 First differences 81 108 144 Second differences 27 36

Check It Out! Example 1b Continued Neither the first nor second differences are constant. Check ratios between the values. 324 243 ≈ 1.33, and ≈ 1.33, ≈ 1.33 432 576

Check It Out! Example 1b Continued Because the ratios between the values of the dependent variable are constant, an exponential function would best model the data. Check A scatter reveals a shape similar to an exponential decay function.

To display the correlation coefficient r on some calculators, you must turn on the diagnostic mode. Press , and choose DiagnosticOn. Helpful Hint

Example 2: Economic Application A printing company prints advertising flyers and tracks its profits. Write a function that models the given data. Flyers Printed 100 200 300 400 500 600 Profit ($) 10 70 175 312 720 Step 1 Make a scatter plot of the data. The data appear to form a quadratic or exponential pattern.

Example 2 Continued Step 1 Analyze differences. Profit ($) 10 70 175 312 500 720 First differences 60 105 137 188 220 Second differences 45 32 51 32 Because the second differences of the dependent variable are close to constant when the independent variables are evenly spaced, a quadratic function will best model the data.

Example 2 Continued Step 3 Use your graphing calculator to perform a quadratic regression. A quadratic function that models the data is f(x) = 0.002x2 + 0.007x – 11.2. The correlation r is very close to 1, which indicates a good fit.

Check It Out! Example 2 Write a function that models the given data. x 12 14 16 18 20 22 24 y 110 141 176 215 258 305 356 Step 1 Make a scatter plot of the data. The data appear to form a quadratic or exponential pattern.

Check It Out! Example 2 Continued Step 1 Analyze differences. y 110 141 176 215 258 305 356 First differences 31 35 39 43 47 51 Second differences 4 4 4 4 4 Because the second differences of the dependent variable are constant when the independent variables are evenly spaced, a quadratic function will best model the data.

Check It Out! Example 2 Continued Step 3 Use your graphing calculator to perform a quadratic regression. A quadratic function that models the data is f(x) = 0.5x2 + 2.5x + 8. The correlation r is 1, which indicates a good fit.

When data are not ordered or evenly spaced, you may to have to try several models to determine which best approximates the data. Graphing calculators often indicate the value of the coefficient of determination, indicated by r2 or R2. The closer the coefficient is to 1, the better the model approximates the data.

Example 3: Application The data shows the population of a small town since 1990. Using 1990 as a reference year, write a function that models the data. Year 1990 1993 1997 2000 2002 2005 2006 Population 400 490 642 787 901 1104 1181 The data are not evenly spaced, so you cannot analyze differences or ratios.

• • • • • • • Example 3 Continued Population year 1990 2006 1997 2002 2005 1993 400 800 1200 2000 • Create a scatter plot of the data. Use 1990 as year 0. The data appear to be quadratic, cubic or exponential. • • • • • •

Example 3 Continued Use the calculator to perform each type of regression. Compare the value of r2. The exponential model seems to be the best fit. The function f(x) ≈ 399.94(1.07)x models the data.

Write a function that models the data. Check It Out! Example 3 Write a function that models the data. Fertilizer/Acre (lb) 11 14 25 31 40 50 Yield/Acre (bushels) 245 302 480 557 645 705 The data are not evenly spaced, so you cannot analyze differences or ratios.

Check It Out! Example 3 Continued Create a scatter plot of the data. The data appear to be quadratic, cubic or exponential.

Check It Out! Example 3 Continued Use the calculator to perform each type of regression. Compare the value of r2. The cubic model seems to be the best fit. The function f(x) ≈ –9.3x3 –0.2x2 + 23.97x + 5.46.

2. Write a function that models the given data. Lesson Quiz 1. Use finite differences or ratios to determine which parent function would best model the given data set, and write a function that models the data. Length (ft) 5 10 15 20 25 Cost ($) 102.50 290.00 602.50 1040.00 1602.50 quadratic; C(l) = 2.5x2 + 40 2. Write a function that models the given data. Time (min) 2 4 6 8 10 Volume (cm2) 1.2 3.9 15.7 64.2 256.5 1023.8 f(x) = 1.085(1.977)x