Slopes (or steepness) of lines are seen everywhere.

The steepness of the roof of a house is referred to as the pitch of the roof by home builders.

Give one reason why some homes have roofs which have a greater pitch.

There is less snow build up in the wintertime.

Engineers refer to the slope of a road as the grade.

They often refer to the slope as a percentage.

Slopes and Lines

rise

run

The slope of a line is the steepness of the line.

riseslope =

run

8100

A grade of 8% would mean for every run of 100 units, there is a rise of 8 units.

8slope =

100

= 8%

The steepness of wheelchair ramps is of great importance to handicapped persons.

1

12The slope of wheelchair ramps is usually about

112

If the rise is 1.5 m, what is the run?

Ans: 18 m

Determine the slope of the line.

5 cm

9 cm

5

9m

We use the letter m because in French the word for “to go up” is monter.

Because the slope is a ratio, there are no units such as cm or cm2.

3 m

5 m

Determine the slope (pitch) of the roof.

5

3m

2

3m

3

3m

1m

Determine the slope of the staircase.

23

3

3

3

9

3

9m

1

3m

rise

runm

2

4 6 1080 12 14

4

6

8

10

12

2

Determine the slope.

– 5

75

7m

rise

runm

Determine the slope.

2

4 6 1080 12 14

4

6

8

10

12

2

7

0

7m

rise

runm

0m

Horizontal lines have a slope of zero.2

4 6 1080 12 14

4

6

8

10

12

2

Determine the slope.

6 6

0m

rise

runm

(undefined)

Vertical lines have no slope.2

4 6 1080 12 14

4

6

8

10

12

2

Determine the slope.

rise

runm

positive

negative

zeroundefined

Summary: Types of Slopes

Determine the slope of this line.

Which points will you use to determine rise and run?

5

40 riseslope =

run40

5m

= 8

Determine the slope of the line segment.

60–2

2

60m

1

30

Draw a line which has a slope of 1

2

Draw a line which has a slope of 2

Draw a line which has a slope of 5

6

–5

6

–5

6

Draw a line which has a slope of …

–3

1–3

1

a) – 3 b) 3

3

13

1

3

9

3

9m

1

3m

rise

runm

2

4 6 1080 12 14

4

6

8

10

12

2

Remember this example?

9 6 3

13 4 9m

1

3m

rise

runm

2

4 6 1080 12 14

4

6

8

10

12

2

You can use the coordinates of two points on the line to find the slope.

4,6

13,9

Formula for Slope

When given the coordinates of two points on

the line: and

the slope is the difference of the y-coordinates divided by the difference of the x-coordinates.

To represent the difference, we use the Greek letter for d, delta, in the formula.

Slope = =

1 1,x y 2 2,x y

2 1

2 1

y y

x x

y

x

Summary

Slope = = =

y

x

rise

run2 1

2 1

y y

x x

Examples:

Find the slope of the line passing through the two points. Use your answer to decide whether the line is increasing, decreasing, vertical, or horizontal.

a) and

b) and

c) and 2, 3

1, 8

2, 3

2, 4

2, 3 2, 3

Answers:

a) decreasing

b) vertical

c) horizontal

3 3 00

2 2 4m

3 3 6

2 2 0m undefined

8 4 12

121 2 1

m