Introduction
To Slope
Slope is a
measure of
Steepness.
Types of Slope
Zero
Negative
Positive
Undefined
or
No Slope
• Slope is sometimes referred
to as the “rate of change”
between 2 points.
• The letter “m” is always
used to represent slope.
A. FORMULA
If given 2 points on
a line, you may find
the slope using the
Formula… m = y2 – y1
x2 – x1
NOTE:
The formula may
sometimes be written
as m =∆y .
∆x
Slope can be expressed in some other
different ways:
( y2  y1 ) rise
vertical change
m


( x2  x1 ) run horizontal change
1.) Find the slope of the
line through the
points (3,7) and (5, 19).
x1 y1
x2 y2
m = y2 – y1 m = 19 – 7 m = 12
x2 – x1
2
5–3
m=6
2) Find the slope of the line that passes
through the points (-2, -2) and (4, 1).
When given points, it is easier to use the formula!
( y2  y1 )
m
( x2  x1 )
(1  (2)) (1  2) 3 1
m

 
(4  (2)) (4  2) 6 2
3) Find the slope of the line that goes
through the points (-5, 3) and (2, 1).
y2  y1
m
x2  x1
1 3
m
2  (5)
2
m
7
4) Find the slope of the line that
passes through (3, 5) and (-1, 4).
1.
2.
3.
4.
4
-4
¼
-¼
B. Determine the slope
of a line with a graph
When given the graph, it is easier to apply
“rise over run”.
Determine the slope of the line.
Start with the lower point and count how
much you rise and then how much you run
to get to the other point!
6
3
rise
3
1
=
=
run
6
2
• This is called the
Triangle Method
• The slope is positive since
the line is increasing
Determine the slope of the line shown.
Determine the slope of the line.
-1
2
Find points on the graph.
Use two of them and
apply rise over run.
rise 2

 2
run 1
The line is decreasing (slope is negative).
What is the slope of a horizontal line?
The line doesn’t rise!
0
m
0
number
All horizontal lines have a slope of 0.
What is the slope of a vertical line?
The line doesn’t run!
number
m
 undefined
0
All vertical lines have an undefined slope.
C. Creating a Graph
Draw a line through the point (2,0)
that has a slope of 3.
1
3
1. Graph the ordered pair (2, 0).
2. From (2, 0), apply rise over
run (write 3 as a fraction).
3. Plot a point at this location.
4. Draw a straight line through
the points.
Ratio Tables
x
3
6
9
12
y
2
4
6
8
• Sometimes, instead of giving coordinate
points, they will give a ratio table.
• You can still find the slope from this by using
them as coordinate points.
D. Finding the Slope
with Variables
The slope of a line that goes through
the points (r, 6) and (4, 2) is 4. Find r.
To solve this, plug the given
information into the formula
( y2  y1 )
m
.
(x2  x1 )
26
4
4r
To solve for r, simplify and write as
a proportion.
26
4
4r
4
4

1 4r
Cross multiply.
4
4

1 4r
1(-4) = 4(4 – r)
Simplify and solve the equation.
1(-4) = 4(4 – r)
-4 = 16 – 4r
-16 -16
-20 = -4r
-4 -4
5=r
The ordered pairs are (5, 6) and (4, 2)
The
End