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7.7 Choosing the Best Model for Two-Variable Data p. 279

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The following functions have been used to model a set of data. To determine the best model for a set of data pts ( x, y ), make a scatter plot of the data and choose the type of function suggested by the pattern of the data pts.

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Function General Form Graph Linear

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Function General Form Graph Quadratic

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Function General Form Graph Cubic

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Function General Form Graph Exponential

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EXAMPLE 1 Use a linear model Tuition The table shows the average tuition y (in dollars) for a private four-year college in the United States from 1995 to 2002, where x is the number of years since 1995. Use a graphing calculator to find a model for the data.

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EXAMPLE 1 Use a linear model SOLUTION STEP 1 Make: a scatter plot. The points lie approximately on a line. This suggests a linear model. STEP 2 Use: the linear regression feature to find an equation of the model.

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EXAMPLE 1 Use a linear model STEP 3 Graph: the model along with the data to verify that the model fits the data well. A model for the data is y = 933x + 14,600. ANSWER

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EXAMPLE 2 Use an exponential model Cooling Rates You are storing leftover chili in a freezer. The table shows the chili’s temperature y (in degrees Fahrenheit) after x minutes in the freezer. Use a graphing calculator to find a model for the data.

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EXAMPLE 2 Use an exponential model SOLUTION STEP 1 Make: a scatter plot. The points fall rapidly at first and then begin to level off. This suggests an exponential decay model. STEP 2 Use: the exponential regression feature to find an equation of the model.

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EXAMPLE 2 Use an exponential model SOLUTION STEP 3 Graph: the model along with the data to verify that the model fits the data well. A model for the data is y = 98.2(0.969) x.ANSWER

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EXAMPLE 3 Use a quadratic model Fuel Efficiency A study compared the speed x (in miles per hour) and the average fuel efficiency y (in miles per gallon) of cars. The results are shown in the table. Use a graphing calculator to find a model for the data.

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EXAMPLE 3 Use a quadratic model SOLUTION STEP 1 Make a scatter plot. The points form an inverted U-shape. This suggests a quadratic model. STEP 2 Use the quadratic regression feature to find an equation of the model.

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EXAMPLE 3 Use a quadratic model STEP 3Graph the model along with the data to verify that the model fits the data well. ANSWER A model for the data is y = – 0.00793x 2 + 0.727x + 13.8.

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p. 280 #’s 1-3 Plot the points to decide which model works for the data Linear, Quadratic, or Exponential p. 281 #’s 1 – 6

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Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

Example 2 The following table gives the mileage y, in miles per gallon, of a certain car at various speeds x (in miles per hour). a) Use a graphing utility.

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