Using Linear ModelsUsing Linear Models

A lot of situations can be A lot of situations can be modeled by a straight line…modeled by a straight line…

Example…Purchasing Example…Purchasing Gasoline.Gasoline.

Gallons Purchased

Total Cost

0 $0

1 $3.41

2 $6.82

10 $34.10

If we were to plot that data, If we were to plot that data, what would it look like?what would it look like?

Gallons

Total Cost

0 $0

1 $3.41

2 $6.82

10 $34.10

•••

•

Perfectly

Linear

How can we find the How can we find the equation of that line?equation of that line?

Gallons

Total Cost

0 $0

1 $3.41

2 $6.82

10 $34.10

Turn any two points into Turn any two points into ordered pairs and then use the ordered pairs and then use the point-slope formula.point-slope formula.

What does the slope indicate What does the slope indicate to you?to you?

What does the y-intercept What does the y-intercept mean to you?mean to you?

Based on the equation, how can Based on the equation, how can you predict how many gallons you you predict how many gallons you can get for $20?can get for $20?

Let’s do the Worksheet on Let’s do the Worksheet on Line of Best Fit together.Line of Best Fit together.

A lot of situations will not be A lot of situations will not be modeled by a straight line…modeled by a straight line…

Example…A photography studio offers Example…A photography studio offers several packages to students posing for several packages to students posing for yearbook photos. Let x represent the yearbook photos. Let x represent the number of pictures, and let y represent number of pictures, and let y represent the price in dollars.the price in dollars.

Pictures, x Price, y in $30 $27

18 $20

15 $17

11 $14

If we were to plot that data, If we were to plot that data, what would it look like?what would it look like?Pictur

esPrice

30 $27

18 $20

15 $17

11 $14

•

•

••

No so Linear

What can we do to find a What can we do to find a linear model in this linear model in this situation?situation?

2

2

When the models are not When the models are not perfectly linear, they are perfectly linear, they are called scatter plots.called scatter plots.In these instances, we In these instances, we determine a line of best fit or determine a line of best fit or a regression line.a regression line.

Pictures

Price

30 $27

18 $20

15 $17

11 $14

•

•

••

Envision a rectangular box Envision a rectangular box drawn around the points.drawn around the points.Draw a line through the center of Draw a line through the center of the box leaning in the general the box leaning in the general direction of the points.direction of the points.

Determine two points that your Determine two points that your line has passed through (they line has passed through (they may or may notmay or may not be ordered be ordered pairs in your data table)pairs in your data table)

•

•

2

2

(10,14)

(28,26)

Now find the slope and the Now find the slope and the equation using either the point-equation using either the point-slope formula or the slope-slope formula or the slope-intercept formula. (Like we did intercept formula. (Like we did yesterday).yesterday).

Explain the real world Explain the real world meaning of the slope of meaning of the slope of

the line.the line.Explain the real world Explain the real world

meaning of the y-meaning of the y-intercept of the line.intercept of the line.

If the studio offers a 50-If the studio offers a 50-print package, what do print package, what do

you think it should you think it should charge?charge?

How many prints do you think How many prints do you think the studio should include in the studio should include in

the package for a $9.99 the package for a $9.99 special?special?

Let’s do one more …Let’s do one more …The data below was collected from 9 The data below was collected from 9 students.students.

Height,x in cm

Forearm, y in cm

185.9 48.5

172.0 44.5

155.0 41.0

191.5 50.5

162.0 43

164.3 42.5

177.5 47.0

180.0 48.0

179.5 47.5

Height,x in cm

Forearm, y in

cm

185.9 48.5

172.0 44.5

155.0 41.0

191.5 50.5

162.0 43

164.3 42.5

177.5 47.0

180.0 48.0

179.5 47.5

What is a good graphing window for your scatter plot?

Let’s plot the data and draw the line of best fit.

Height,x in cm

Forearm, y in

cm

185.9 48.5

172.0 44.5

155.0 41.0

191.5 50.5

162.0 43

164.3 42.5

177.5 47.0

180.0 48.0

179.5 47.5

38394041

150175 200

Write a linear equation Write a linear equation that models the data.that models the data.

What does the calculator What does the calculator say is the line of best fit?say is the line of best fit?

Explain the real world Explain the real world meaning of the slope of meaning of the slope of

the line.the line.

Explain the real world Explain the real world meaning of the y-meaning of the y-

intercept of the line.intercept of the line.

Why doesn’t the real Why doesn’t the real world meaning of the y-world meaning of the y-intercept make sense?intercept make sense?

Use the equation to estimate Use the equation to estimate the height of a student with a the height of a student with a

50 cm forearm.50 cm forearm.

Use the equation to estimate Use the equation to estimate the length of a forearm of a the length of a forearm of a student that is 158 cm tall.student that is 158 cm tall.