1. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 2.2, Slide 1 Chapter 2 Modeling with Linear Functions

2. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 2.2, Slide 2 2.2 Finding Equations of Linear Models

3. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 2.2, Slide 3 Example: Finding an Equation of a Linear Model by Using Data Described in Words The number of times airplanes have struck birds has increased approximately linearly from 2.6 thousand strikes in 1992 to 10.0 thousand strikes in 2011 (Source: Federal Aviation Administration). Let n be the number (in thousands) of bird strikes in the year that is t years since 1990. Find an equation of a linear model.

4. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 2.2, Slide 4 Solution Values of t and n are shown in the table. Use the data points (2, 2.6) and (21, 10.0) to find the slope of the model.

5. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 2.2, Slide 5 Solution Substitute 0.39 for m in the equation n = mt + b: n = 0.39t + b

6. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 2.2, Slide 6 Solution Find the constant b by substituting the coordinates of the point (2, 2.6) into the equation n = 0.39t + b:

7. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 2.2, Slide 7 Solution Substitute 1.82 for b in the equation n = 0.39t + b: n = 0.39t + 1.82 Verify the equation using TRACE on a graphing calculator to check that the line approximately contains the points (2, 2.6) and (21, 10.0)

8. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 2.2, Slide 8 Example: Find an Equation of a Linear Model by Using Data Displayed in a Table The numbers of Apple stores are shown in the table for various years. Let n be the number of Apple Stores at t years since 2000. Find an equation of a line that comes close to the points in a scattergram of the data.

9. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 2.2, Slide 9 Solution View the scattergram using a graphing calculator to save time and improve accuracy in plotting the points..

10. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 2.2, Slide 10 Solution We want to find an equation of a line that comes close to the data points. Recall that it is not necessary to use two data points to find an equation, although it is often convenient and satisfactory to do so. Note on the next slide that the red line does not come close to the other data points. However, the green line that passes through points (9, 249) and (12, 358) does. We will find the equation of the green line.

11. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 2.2, Slide 11 Solution

12. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 2.2, Slide 12 Solution Use the points (9, 249) and (12, 358) to find m: Substitute 36.33 for m in the equation n = mt + b: n = 36.33t + b

13. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 2.2, Slide 13 Solution To find b, substitute (9, 249) into the equation: Substitute –77.97 for b in the equation: n = 36.33t – 77.97

14. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 2.2, Slide 14 Solution Check the correctness of our equation using a graphing calculator to verify that our line approximately contains the points (9, 249) and (12, 358).

15. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 2.2, Slide 15 Constructing a Scattergram Warning It is a common error to skip creating a scattergram. However there are many benefits: 1. We can determine whether the data are approximately linearly related. 2. If so, it can help us choose two good points with which to find a model. 3. We can assess whether the data fits the model well.

16. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 2.2, Slide 16 Finding an Equation of a Linear Model To find an equation of a linear model, given some data, 1. Create a scattergram of the data. 2. Determine whether there is a line that comes close to the data points. If so, choose two points (not necessarily data points) you can use to find the equation of a linear model.

17. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 2.2, Slide 17 Finding an Equation of a Linear Model 3. Find an equation of the line you identified in step 2. 4. Use a graphing calculator to verify the graph of your equation comes close to the points of the scattergram.

18. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 2.2, Slide 18 Linear Regression Definition Most graphing calculators have a built-in linear regression feature for finding an equation of a linear model. A linear equation found by linear regression is called a linear regression equation, and the function described by the equation is called a linear regression function. The graph is called a regression line.