2.4 Using Linear Models

1. Modeling Real-World Data

2. Predicting with Linear Models

1) Modeling Real-World Data

Big idea…

Use linear equations to create graphs of real-world situations. Then use these graphs to make predictions about past and future trends.

Example 1:

There were 174 words typed in 3 minutes. There were 348 words typed in 6 minutes. How many words were typed in 5 minutes?

1) Modeling Real-World Data

1) Modeling Real-World Data

x = independenty = dependent

(x, y) = (time, words typed )

(x1, y1) = (3, 174)

(x2, y2) = (6, 348)

(x3, y3) = (5, ?)

Solution:

Time (minutes)1 2 3 4 5 6

100

200

300

400

Example 2:

Suppose an airplane descends at a rate of 300 ft/min from an elevation of 8000ft. Draw a graph and write an equation to model the plane’s elevation as a function of the time it has been descending. Interpret the vertical intercept.

1) Modeling Real-World Data

1) Modeling Real-World Data

Time (minutes)

(x, y) = (time, height)

(x1, y1) = (0, 8000)

(x2, y2) = (10, ?)

(x3, y3) = (20, ?)

10 20 30

6000

2000

4000

8000

1) Modeling Real-World Data

Time (minutes)

Equation:

Remember… y = mx + b

10 20 30

6000

2000

4000

8000

2) Predicting with Linear Models

• You can extrapolate with linear models to make predictions based on trends.

Example 1:

After 5 months the number of subscribers to a newspaper was 5730. After 7 months the number of subscribers was 6022. Write an equation for the function. How many subscribers will there be after 10 months?

2) Predicting with Linear Models

2) Predicting with Linear Models

(x, y) = (months, subscribers)

(x1, y1) = (5, 5730)

(x2, y2) = (7, 6022)

(x3, y3) = (10, ?)

Equation: y = mx + b

Time (months)

2 4 6 8 10

2000

4000

6000

8000

2) Predicting with Linear Models

(x, y) = (months, subscribers)

(x1, y1) = (5, 5730)

(x2, y2) = (7, 6022)

(x3, y3) = (10, ?)

Equation: y = mx + b

Time (months)

2 4 6 8 10

2000

4000

6000

8000

2) Predicting with Linear Models

(x, y) = (months, subscribers)

(x1, y1) = (5, 5730)

(x2, y2) = (7, 6022)

(x3, y3) = (10, ?)

Equation: y = mx + b

Time (months)

2 4 6 8 10

2000

4000

6000

8000

2) Predicting with Linear Models

(x, y) = (months, subscribers)

(x1, y1) = (5, 5730)

(x2, y2) = (7, 6022)

(x3, y3) = (10, 7000)

Equation: y = mx + b

Time (months)

2 4 6 8 10

2000

4000

6000

8000

y-intercept

run = 4

rise = 1000

Scatter Plots• Connect the dots with a trend line to see

if there is a trend in the data

Types of Scatter Plots

Strong, positive correlation Weak, positive correlation

Types of Scatter Plots

Strong, negative correlation Weak, negative correlation

Types of Scatter Plots

No correlation

Scatter Plots

Example 1:

The data table below shows the relationship between hours spent studying and student grade.

a) Draw a scatter plot. Decide whether a linear model is reasonable.

b) Draw a trend line. Write the equation for the line.

Hours studying

3 1 5 4 1 6

Grade (%)

65 35 90 74 45 87

Scatter Plots

Hours studying 1 2 3 4 5 6

40

50

70

60

90

80

100

(x, y) = (hours studying, grade)

(3, 65)

(1, 35)

(5, 90)

(4, 74)

(1, 45)

(6, 87)

Equation: y = mx + b30

Scatter Plots

Hours studying 1 2 3 4 5 6

40

50

70

60

90

80

100

(x, y) = (hours studying, grade)

(3, 65)

(1, 35)

(5, 90)

(4, 74)

(1, 45)

(6, 87)

a) Based on the graph, is a linear model reasonable?

30

Scatter Plots

Hours studying 1 2 3 4 5 6

40

50

70

60

90

80

100

(x, y) = (hours studying, grade)

(3, 65)

(1, 35)

(5, 90)

(4, 74)

(1, 45)

(6, 87)

b) Equation: y = mx + b30

Rise = 20

Run = 2

Assignment

p.81 #1-3, 8, 11, 12, 13, 19,