FINDING THE EQUATION OF A TREND LINE

When a linear association is shown by a scatter plot, we can use a line to model the relationship between the variables. A trend line is a straight line that comes closest to the points on a scatter plot. We can use two points on a trend line to write an equation in slope-intercept form for the trend line.

Example 1 :

Lily is getting trained for a 10K race. For some of her training runs, she records the distance she ran and how many minutes she ran.

Distance (miles)

4

2

1

2

3

5

2

4

3

Time (minutes)

38

25

07

16

26

55

20

45

31

Draw a trend line that shows the relationship between distance and time. And also write an equation for the trend line.

Solution :

Step 1 :

Make a scatter plot of Lily’s running data.

Step 2 :

To draw a trend line, use a straight edge to draw a line that has about the same number of points above and below it. Ignore any outliers.

Step 3 :

Find the slope of the trend line.

The line passes through points (2, 20) and (4.5, 45).

Use the slope formula. 

m = (y2 - y1)/(x2 - x1)

Substitute x1, = 2, y1 = 20, x2 = 4,  and y2 = 45.

m = (45 - 20)/(4.5 - 2)

m = 25/2.5

m = 10

Step 4 : 

Find the y-intercept of the trend line.

Slope-intercept form equation of a line : 

y = mx + b

Substitute y = 20, m = 10 and x = 2. 

20 = 10 · 2 + b

20 = 20 + b

Subtract 20 from both sides. 

0 = b

Step 5 : 

Use your slope and y-intercept values to write the equation.

Slope-intercept form equation of a line : 

y = 10x + 0

So, the equation for the trend line is

y = 10x

Example 2 :

David records the number of chapters and the total number of pages for several books as given below. 

No. of chapters

3

4

5

8

10

11

15

17

19

No. of pages

20

44

50

95

90

130

125

170

160

Draw a trend line that shows the relationship between the number of chapters and the number of pages. And also write an equation for the trend line.

Solution :

Step 1 :

Make a scatter plot of the data given above. 

Step 2 :

To draw a trend line, use a straight edge to draw a line that has about the same number of points above and below it. Ignore any outliers.

Step 3 : 

Find the slope of the trend line. The line passes through points (5, 50) and (17, 170).

Use the slope formula. 

m = (y2 - y1)/(x2 - x1)

Substitute x1, = 5, y1 = 50, x2 = 17,  and y2 = 170.

m = (170 - 50)/(17 - 5)

m = 120/12

m = 10

Step 4 : 

Find the y-intercept of the trend line.

Slope-intercept form equation of a line : 

y = mx + b

Substitute y = 50, m = 10 and x = 5. 

50 = 10 · 5 + b

50 = 50 + b

Subtract 50 from both sides. 

0 = b

Step 5 : 

Use your slope and y-intercept values to write the equation.

Slope-intercept form equation of a line : 

y = mx + b

y = 10x + 0

So, the equation for the trend line is

y = 10x

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