Slope of a Line

Chapter 7 Section 3

Learning Objective

• Find the slope of a line

• Recognize positive and negative slopes

• Examine the slopes of horizontal and vertical lines

• Examine the slope of parallel and perpendicular lines

• Key Vocabulary: slope, rise, run, positive slope, negative slope, parallel, perpendicular, negative reciprocals

Find the Slope of a Line

• The slope of a line is a ratio of the vertical change to the horizontal change between any two selected points on the line.

• The slope of a line is a measure of the steepness of the line

• Road has a 8% grade, = 8

100

vertical change (y)slope =

horizontal change (x)

Run 100 ft

Rise 8 ft

Find the Slope of a Line• Slope of the line goes through the points

•

• m is used for the slope and the Greek letter Delta (Δ) means “the change in”

• It makes no difference which two points are selected when finding the slope of the line.

• It makes no difference which points you labeled

• You read the slope of a graph from left to right just as you read a book

1 1 2 2( , ) and ( , )x y x y

2 1

2 1

vertical change (y)slope =

horizontal change (x)

y y y

x x x

1 1 2 2( , ) or ( , )x y x y

ym

x

Change in y

Change in x

Find the Slope of a LineExample: (-2, 1) and (3, 4)

2 1

2 1

vertical change (y)slope =

horizontal change (x)

4 1 3slope

3 ( 2) 5

y yslope

x x

(-2,1)

(3,4)

(0,0)

Increase (up) 3 units on the y-axis, increase (over/right) 5 unit on the x-axis

Positive Slope

Find the Slope of a LineExample: (1, 2) and (3, 6)

2 1

2 1

vertical change (y)slope =

horizontal change (x)

6 2 4 2slope

3 1 2 1

y yslope

x x

(1,2)

(3,6)

(0,0)

Increase (up) 2 units on the y-axis, increase (over/right) 1 unit on the x-axis

Positive Slope

Find the Slope of a LineExample: (1, 0) and (3, 4)

2 1

2 1

vertical change (y)slope =

horizontal change (x)

4 0 4 2slope

3 1 2 1

y yslope

x x

(1,0)

(3,4)

(0,0)

Increase (up) 2 units on the y-axis, increase (over/right) 1 unit on the x-axis. This is a positive slope.

Positive Slope

Find the Slope of a LineExample: Find the slope of the following line

2 1

2 1

vertical change (y)slope =

horizontal change (x)

4 2 2 2

1 4 3 3

y yslope

x x

m

(4,2)

(1,4)

(0,0)

Down (decrease) 2 units on the y-axis, increase (over/right) 3 unit on the x-axis

Negative Slope

Find the Slope of a LineExample: Find the slope of the following line

2 1

2 1

vertical change (y)slope

horizontal change (x)m

y yslope m

x x

(20)

(5, 15)

(0,0)

Down (decrease) 2 units on the y-axis, over (right) 3 unit on the x-axis

(5)

(10)

(30, 5)

(5)

(5, 15) and (30, 10)

(15)

(10) (15) (25)

15 5

5 3010

m 252

m 5

m

Slope of a Horizontal LineEvery horizontal line has a slope of 0

Example: y = 2

2 1

2 1

vertical change (y)slope =

horizontal change (x)

1 1 0 = 0

2 2 4

y yslope

x x

m

(2,2)

(0,0)

(-2,2)

There is no change in y. Slope is 0. Any two points would yield the same slope of 0. Horizontal line is parallel to the x-axis

Slope of a Vertical LineThe slope of a vertical line is undefined

Example: x = 2

2 1

2 1

vertical change (y)slope =

horizontal change (x)

2 2 4 = undefined

2 2 0

y yslope

x x

m

(2,2)

(0,0)

(2,-2)There is no change in x. Slope is undefined. Any two points would yield the same results. Vertical line is parallel to the y-axis

Slope of Parallel LineTwo nonvertical lines with the same slope and different y-intercepts are parallel lines. Any two vertical lines are parallel to each other.

Example: draw two lines with a slope of 1; one through (1,3) and another through (1, -2)

m = 1 (up 1 and right 1)

(1,3)

(0,0)

(1,-2)

These two lines are parallel.

Parallel lines do not intersect.

Two parallel lines will have the same slope, and a different y-intercept

Line 1

Line 2

Slope of Perpendicular Line

• To determine if two lines are perpendicular multiply the slopes of the two lines together. If the product is -1 then the slopes are negative reciprocals, and the lines are perpendicular.

• m1 represents line 1and m2 represents line 2

m1 = 2

m2 = - ½

1 2

1 22 1

2 2m m

Slope of Perpendicular LineTwo lines whose slopes are negative reciprocals of each other are perpendicular lines. Any two vertical lines is perpendicular to any horizontal line.

Example: Draw a line with a slope of 2; through (1,-1) Up 2 right 1

Draw a line with aSlope of – ½ through (-2, 4)Down 1 and right 2

PerpemdicularThe reciprocal of 2 is ½ The reciprocal of -½ is -2

The product of the slopes is -1.

(2)(- ½) = -1

(1,-1)

(0,0)

(-2,4)

Line 1

Line 2

Determining if lines are Parallel, Perpendicular, or neither

1 2 1 2

2 3 2 3 6 and 1 Perpendicular

3 2 3 2 6m m m m

1 2

4 4 and Same slope, the lines are Parallel

7 7m m

1 2 1 2

1 1 3 3 and 3 1

3 3 1 3

Neither, not parallel because slopes are different, and

not perpendicular because when multiplied positive 1

m m m m

Positive SlopesSlopes are read from left to right, just like a book.

Positive slope y increases as x increases

Rises as it moves from left to right

(0,0)

Negative SlopesSlopes are read from left to right, just like a book.

Negative slope y decreases as x decreases

Falls as it moves from left to right

(0,0)

Remember

• The y goes in the numerator.

• The x goes in the denominator.

• Be careful when you subtract negatives. The signs become positive (+). Example: 3 – (-2) = 3 + 2

• The term negative reciprocal may be thought of as opposite reciprocal.

HOMEWORK 7.3

Page 449-450:

#11, 13, 15, 19, 25, 27, 33, 35