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Lesson 2-4

Using Linear Models

Lesson Objectives 1 Modeling real-world data 2 Predicting with linear models

Vocabulary.

y

y

y

x

Weak

x

Strong

correlation

y

correlation

y

x

Weak correlation

A scatter plot is

Strong

x

x

correlation

correlation

Examples. 1 Transportation Suppose an airplane descends at a rate of 300 ft/min from an elevation of 8000 ft. Write and graph an equation to model the plane’s elevation as a function of the time it has been descending. Interpret the intercept at which the graph intersects the vertical axis.

d 8000 6000 4000 2000 t O

Relate

plane’s elevation



rate



time



10

20

30

starting elevation

Define Let t  time (in minutes) since the plane began its descent. Let d  the plane’s elevation. Write

d



300



An equation that models the plane’s elevation is

(

The d-intercept is 0, was 30



t





.

). This tells you that the elevation of the plane

ft at the moment it began its descent.

Algebra 2 Lesson 2-4

Daily Notetaking Guide

© Pearson Education, Inc., publishing as Pearson Prentice Hall.

A trend line is

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2 Using a Linear Model A spring has a length of 8 cm when a 20-g mass is hanging at the bottom end. Each additional gram stretches the spring another 0.15 cm. Write an equation for the length y of the spring as a function of the mass x of the attached weight. Step 1 Identify two points as (x1, y1) and (x2, y2). Adding another 20 g of mass at the end of the spring will give a total mass of 40 g and a length of 8  0.15(20)  11 cm. Use the points (x1, y1)  20, and (x2, y2)  40, to find the linear equation.

(

)

(

)

Step 2 Find the slope of the line. y 2y

m 5 x22 2 x11 

Use the slope formula.

11 

Substitute.

40  20

m

,

or

0.15

Simplify.

Step 3 Use one of the points and the point-slope form to write an equation for the line. y  y1  m(x  x1) Use point-slope form. y8

(x  

Substitute. Solve for y.

An equation of the line that models the length of the spring is y 

.

3 Determining Whether a Linear Model Is Reasonable An art expert guessed the selling prices of five paintings. Then, she checked the actual prices. The data points (guess, actual) show the results, where each number is in thousands of dollars. Graph the data points. Decide whether a linear model is reasonable. If so, draw a trend line and write its equation. {(12, 11), (7, 8.5), (10, 12), (5, 3.8), (9, 10)} Actual (thousands \$)

© Pearson Education, Inc., publishing as Pearson Prentice Hall.

y

)

y

A linear model seems since the points fall close to a line.

12 8

A possible trend line is the line through (6, 6) and (10.5, 11). Using these two points, the equation in

4

slope-intercept form is y 

O



.

x 4 8 12 Guess (thousands \$)

Daily Notetaking Guide

Algebra 2 Lesson 2-4

31

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Quick Check. 1. Suppose a balloon begins descending at a rate of 20 ft/min from an elevation of 1350 ft. a. Write an equation to model the balloon’s elevation as a function of time. What is true about the slope of this line?

b. Graph the equation. Interpret the h-intercept.

3. Graph the data points. Decide whether a linear model is reasonable. If so, draw a trend line and write its equation. {(7.5, 19.75), (2, 9), (0, 6.5), (1.5, 3), (4, 1.5)}

32

Algebra 2 Lesson 2-4

Daily Notetaking Guide

© Pearson Education, Inc., publishing as Pearson Prentice Hall.

2. A candle is 7 in. tall after burning for 1 h, and 5 in. tall after burning for 2 h. Write a linear equation to model the height of the candle.

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