Slide 1Copyright © 2017, 2013, 2009 Pearson Education, Inc.

Chapter 2

Graphing

Chapter 3

Slide 2Copyright © 2017, 2013, 2009 Pearson Education, Inc.

Slope and Rate of

Change

Section 3.4

Slide 3Copyright © 2017, 2013, 2009 Pearson Education, Inc.

Finding the Slope of a Line

Given Two Points of the

Line

Objective 1

Slide 4Copyright © 2017, 2013, 2009 Pearson Education, Inc.

Slope

Slope of a Line

The slope m of the line containing the points

(x1, y1) and (x2, y2) is given by

2 1

2 1

2 1

change in

change in

rise y y ym

run x x x

x x

−= = =

−

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Find the slope of the line through (4, –3 ) and

(2, 2). Graph the line.

Example

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Example (cont)

(4, –3 ) and (2, 2)

Rise –5

Run 2

3 2 5

4 2 2m

− − −= =

−

Slide 7Copyright © 2017, 2013, 2009 Pearson Education, Inc.

Helpful Hint

When finding slope, it makes no difference which

point is identified as (x1, y1) and which is

identified as (x2, y2). Just remember that whatever

y-value is first in the numerator, its corresponding

x-value is first in the denominator.

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Example

Find the slope of the line through (–2, 1) and

(3, 5). Graph the line.

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Slope of Lines

Positive Slope

Line goes up to the right

x

yLines with positive

slopes go upward as

x increases.

Negative Slope

Line goes downward to

the right

x

y Lines with negative

slopes go downward

as x increases.

m > 0

m < 0

Slide 10Copyright © 2017, 2013, 2009 Pearson Education, Inc.

Finding the Slope of a Line

Given Its Equation

Objective 2

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Slope-Intercept Form of a Line

Slope-Intercept Form

When a linear equation in two variables is

written in the slope-intercept form,

y = mx + b

m is the slope and (0, b) is the y-intercept of the

line.

y = 3x – 4

The slope is 3. The y-intercept

is (0, -4).

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Example

Find the slope and y-intercept of the line whose

equation is 53.

9y x= +

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Find the slope of the line –3x + 2y = 11.

Example

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Find the slope of the line –y = 6x – 7.

Example

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Finding Slopes of Horizontal

and Vertical Lines

Objective 3

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For any two points, the y values will be equal to the same

real number.

The numerator in the slope formula = 0 (the difference of

the y-coordinates), but the denominator ≠ 0 (two different

points would have two different x-coordinates).

Slope of a Horizontal Line

Zero Slope

Horizontal Line

x

y Horizontal lines

have a slope of 0.

m = 0

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Find the slope of the line y = 3.

Example

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For any two points, the x values will be equal to the same

real number.

The denominator in the slope formula = 0 (the difference of

the x-coordinates), but the numerator ≠ 0 (two different

points would have two different y-coordinates).

So the slope is undefined (since you can’t divide by 0).

Slope of a Vertical Line

Undefined Slope

Vertical Linex

yA vertical line has an

undefined slope.

m is undefined.

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Find the slope of the line x = –2.

Example

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Slopes of Parallel and

Perpendicular Lines

Objective 4

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Two lines that never intersect are called

parallel lines.

Parallel lines have the same slope. (Unless

they are vertical lines, which have no slope.)

Vertical lines are also parallel.

Parallel Lines

x

y

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Two lines that intersect at right angles are

called perpendicular lines.

Two nonvertical perpendicular lines have

slopes that are negative reciprocals of each

other.

The product of their slopes will be –1.

Perpendicular Lines

Horizontal and vertical

lines are perpendicular to

each other.

slope a

x

y

1slope

a−

Slide 23Copyright © 2017, 2013, 2009 Pearson Education, Inc.

Example

Determine whether the line 6x + 2y = 9 is parallel

to –3x – y = 3.

Slide 24Copyright © 2017, 2013, 2009 Pearson Education, Inc.

Example

Determine whether the line x + 3y = –15 is

perpendicular to –3x + y = – 1 .

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Determine whether the following lines are parallel, perpendicular, or neither.

–5x + y = –6

x + 5y = 5

Example

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Slope as a Rate of Change

Objective 5

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Example

Becky decided to take a bike ride up a mountain trail.

The trail has a vertical rise of 90 feet for every 250 feet of

horizontal change. In percent, what is the grade of the

trail?

The grade of the trail is given by rise

.run

The grade of the trail is 90 feet250 feet

0.36 36%= =

The slope of a line can also be interpreted as the average

rate of change. It tells us how fast y is changing with

respect to x.

Slide 28Copyright © 2017, 2013, 2009 Pearson Education, Inc.

Example

Find the grade of the road:

risegrade

run=

30.15

20= =

The grade of the road is 15%.