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8. Classic Cars The data give the estimated value in dollars of a model of classic car over several years. 15,300 16,100 17,300 18,400 19,600 20,700 INTEGRATE MATHEMATICAL PROCESSES Focus on Modeling 22,000 a. Find an approximate exponential model for the car’s value by averaging the successive ratios of the value. Then make a scatter plot of the data, graph your model with the scatter plot, and assess its fit to the data. Let x = 0 represent the year corresponding to the 24 From the point (0, 15.3), a = 15.3. Estimate b by averaging the value ratios from year to year: 1.052 + 1.074 + 1.064 + 1.065 + 1.056 + 1.063 6 ≈ 1.062 _____ An approximate model is f(x) = 15.3(1.062) . x Value (thousand $) value $15,300. Let f(x) be in thousands of dollars. When modeling a problem in which an exponential regression equation applies, have students start with a scatter plot of the data. This will help them remember that the data points should be increasing (or decreasing) at a rapid rate, depending on the original real-world problem. f(x) 22 20 18 16 x 0 1 2 3 4 5 6 7 8 9 10 The model fits the data very well. Year b. In the last year of the data, a car enthusiast spends $15,100 on a car of the given model that is in need of some work. The owner then spends $8300 restoring it. Use your model to create a table of values with a graphing calculator. How long does the function model predict the owner should keep the car before it can be sold for a profit of at least $5000? © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©B.A.E. Inc./Alamy The owner spent a total of $23,400 for the car. To make a profit of at least $5000, the selling price must be at least $28,400. The table shows the value exceeding $28,400 at year 11. Because the owner bought the car at year 6 of the model, this means that the model predicts that the owner needs to wait for about 5 years. Module 14 A2_MTXESE353947_U6M14L1.indd 788 788 Lesson 1 1/13/15 12:44 PM Fitting Exponential Functions to Data 788 9. INTEGRATE MATHEMATICAL PROCESSES Focus on Reasoning When analyzing a regression equation, ask students to make sure the equation makes sense with respect to the original data set. Have them check several data points in the model to see if the model fits the points fairly closely. Caution them that the actual values of a and b in an exponential model of the form f(x) = ab x may not appear anywhere in the actual data set. Movies The table shows the average price of a movie ticket in the United States from 2001 to 2010. Year 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Price ($) 5.66 5.81 6.03 6.21 6.41 6.55 6.88 7.18 7.50 7.89 a. Make a scatter plot of the data. Then use the first point and another point on the plot to find an approximate exponential model for the average ticket price. Then graph the model with your scatter plot and assess its fit to the data. Possible answer: Let x represent years after 2001. From the point (0, 5.66), a = 5.66. Choose the point for 2009, (8, 7.50) and substitute the coordinates into the model. f(x) = a ⋅ b x (_) An approximate model is f(x) = 5.66(1.036) . x © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Ocean/ Corbis Overall, the model fits the trend fairly well, 7 6 5 4 middle of the data, and passes below the last 0 x 1 2 3 4 5 6 7 8 9 10 11 Years after 2001 b. Use a graphing calculator to find a regression model for the data, and graph the model with the scatter plot. How does this model compare to your previous model? Regression model: f(x) = 5.59(1.037) x The regression model is very close to the previous model. Its graph starts just a bit below the first model’s graph, and it rises a tiny bit more steeply, but the models are very close together at the end of the period of the data. A2_MTXESE353947_U6M14L1.indd 789 Lesson 14.1 f(x) 8 though it overestimates the price a little in the data point. Module 14 789 Average price ($) 9 7.50 = 5.66 ⋅ b 8 1 7.50 _8 =b 5.66 1.036 ≈ b 789 Lesson 1 1/13/15 12:59 PM What does the regression model predict for the average cost in 2014? How does this compare with the actual 2014 cost of about $8.35? A theater owner uses the model in 2010 to project income for 2014 assuming average sales of 490 tickets per day at the predicted price. If the actual price is instead $8.35, did the owner make a good business decision? Explain. c. CONNECT VOCABULARY For students familiar with the term line of best fit as used in linear regression models, it is useful to connect this idea to a notion of “curve of best fit” for exponential (or quadratic) regression. Modeling these types of regression models may help them understand the idea of a “curve of best fit.” Prediction: f(13) = 5.59(1.037) ≈ $8.96; this is about $.616 more per ticket than the actual. No, for the year, the revenue shortfall would be about 490(365)(0.66) ≈ $109,000, which is a large amount of money. $0.61 accounts for about 7% of the individual ticket price which is a large amount. 13 10. Pharmaceuticals A new medication is being studied to see how quickly it is metabolized in the body. The table shows how much of an initial dose of 15 milligrams remains in the bloodstream after different intervals of time. Hours Since Administration Amount Remaining (mg) 0 15 1 14.3 2 13.1 3 12.4 4 11.4 5 10.7 6 10.2 7 9.8 a. Use a graphing calculator to find a regression model. Use the calculator to graph the model with the scatter plot. How much of the drug is eliminated each hour? x © Houghton Mifflin Harcourt Publishing Company a. Model: f(x) = 15.0(0.937) The decay factor is 0.937, so the hourly decay rate is 1 - 0.937 = 0.063, or 6.3%. (Answers might vary slightly due to rounding.) b. The half-life of a drug is how long it takes for half of the drug to be broken down or eliminated from the bloodstream. Using the Table function, what is the half-life of the drug to the nearest hour? b. The half-life is the time to reduce the amount of medication to 7.5 mg. The half-life is about 11 hours. Module 14 A2_MTXESE353947_U6M14L1.indd 790 790 Lesson 1 1/13/15 12:47 PM Fitting Exponential Functions to Data 790