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

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 2.2, Slide 1

Chapter 2Modeling with Linear Functions


2.2 Finding Equations of Linear Models


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.


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.

2.621

7.40.39

02

0.9

11

m


Solution

Substitute 0.39 for m in the equation n = mt + b:

n = 0.39t + b


Solution

Find the constant b by substituting the coordinates of the point (2, 2.6) into the equation n = 0.39t + b:

0.39(2.6 )2 b

2.6 0.78 b 2.6 0.780.78 0.78b

1.82 b


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)


Example: Find an Equation of a Linear Model by Using Data Displayed in a

TableThe 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.


Solution

View the scattergram using a graphing calculator to save time and improve accuracy in plotting the points..


SolutionWe 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.


Solution


Solution

Use the points (9, 249) and (12, 358) to find m:

9123 109

36.358 249

33

m

Substitute 36.33 for m in the equation n = mt + b:

n = 36.33t + b


Solution

To find b, substitute (9, 249) into the equation:

36.33(249 )9 b

249 326.97 b

77.97 b 249 326.97326.97 326.97b

Substitute –77.97 for b in the equation:

n = 36.33t – 77.97


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).


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.


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.


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.


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.

Documents

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