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
−= = =
−
Slide 5Copyright © 2017, 2013, 2009 Pearson Education, Inc.
Find the slope of the line through (4, –3 ) and
(2, 2). Graph the line.
Example
Slide 6Copyright © 2017, 2013, 2009 Pearson Education, Inc.
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.
Slide 8Copyright © 2017, 2013, 2009 Pearson Education, Inc.
Example
Find the slope of the line through (–2, 1) and
(3, 5). Graph the line.
Slide 9Copyright © 2017, 2013, 2009 Pearson Education, Inc.
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
Slide 11Copyright © 2017, 2013, 2009 Pearson Education, Inc.
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).
Slide 12Copyright © 2017, 2013, 2009 Pearson Education, Inc.
Example
Find the slope and y-intercept of the line whose
equation is 53.
9y x= +
Slide 13Copyright © 2017, 2013, 2009 Pearson Education, Inc.
Find the slope of the line –3x + 2y = 11.
Example
Slide 14Copyright © 2017, 2013, 2009 Pearson Education, Inc.
Find the slope of the line –y = 6x – 7.
Example
Slide 15Copyright © 2017, 2013, 2009 Pearson Education, Inc.
Finding Slopes of Horizontal
and Vertical Lines
Objective 3
Slide 16Copyright © 2017, 2013, 2009 Pearson Education, Inc.
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
Slide 17Copyright © 2017, 2013, 2009 Pearson Education, Inc.
Find the slope of the line y = 3.
Example
Slide 18Copyright © 2017, 2013, 2009 Pearson Education, Inc.
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.
Slide 19Copyright © 2017, 2013, 2009 Pearson Education, Inc.
Find the slope of the line x = –2.
Example
Slide 20Copyright © 2017, 2013, 2009 Pearson Education, Inc.
Slopes of Parallel and
Perpendicular Lines
Objective 4
Slide 21Copyright © 2017, 2013, 2009 Pearson Education, Inc.
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
Slide 22Copyright © 2017, 2013, 2009 Pearson Education, Inc.
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 .
Slide 25Copyright © 2017, 2013, 2009 Pearson Education, Inc.
Determine whether the following lines are parallel, perpendicular, or neither.
–5x + y = –6
x + 5y = 5
Example
Slide 26Copyright © 2017, 2013, 2009 Pearson Education, Inc.
Slope as a Rate of Change
Objective 5
Slide 27Copyright © 2017, 2013, 2009 Pearson Education, Inc.
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%.