4.5 Finding
The Slope of a
Line
What is the meaning of this sign?
1.
2.
3.
4.
Icy Road Ahead
Steep Road Ahead
Curvy Road Ahead
Trucks Entering
Highway Ahead
Slopes are commonly
associated with mountains.
The slope we are studying is
associated with the graph of a
line.
Definitions of Slope
• The tilt or inclination of a line
• The ratio of vertical change to horizontal
change.
• The change in y over the change in x
Δy
Δx
Tilt or Inclination
FROM LEFT TO RIGHT!!!!
SLOPE IS ZERO
y=#
POSITIVE
SLOPE
SLOPE IS
UNDEFINED
NEGATIVE
SLOPE
x=#
Tell me slope of each line
POSITIVE
ZERO
NEGATIVE
POSITIVE
Slope can be expressed different ways:
( y2  y1 ) rise
vertical change
m


( x2  x1 ) run horizontal change
UP or DOWN
RIGHT
A line has a positive slope if it is
going uphill from left to right.
A line has a negative slope if it is
going downhill from left to right.
Graph (3,2) and (-1,-1)
Draw a line through the points.
Rise
Run
4
THE SLOPE IS
3
4
3
Find the slope of the following
line.
The slope is…
1
2
Find the slope of the line.
The slope is…
-3
1) Determine the slope of the line.
When given the graph, it is easier to apply
“rise over run”. 3 1
6

2
Determine the slope of the line shown.
1.
2.
3.
4.
-2
-½
½
2
Determine the slope of the line.
-2
1
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.
Find the slope of these lines.
• Black line
3
• Red Line
1
• Blue Line
-1/2
Find the slope of these lines
• Orange line = 0
• Green Line = Undefined
Determining Slope
Rise=1
Run =2
Rise=12
Run =4
Rise=6
Run =2
Slope=3
1
Slope=
2
Let’s find the slope of a line
with two points, WITHOUT
LOOKING AT A GRAPH.
REMEMBER THE SLOPE
FORMULA???!!!!!
y 2  y1
m
x 2  x1
Determining Slope without
graphing
•Pick 2 points on the line •Find the change in the Ycoordinates by
(2, 5)
subtracting(rise)
•Find the change in the Xcoordinates by
subtracting(run)
(-4, -6)
5-(-6) = 11
2-(-4)
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 )
y2 is the y coordinate of the 2nd ordered pair (y2 = 1)
y1 is the y coordinate of the 1st ordered pair (y1 = -2)
(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).
x1
y1
x2
y2
y2  y1
m
x2  x1
1 3
m
25
1 3
m
2  (5)
2
m
7
Sample Problems:
• Find the slope of the line thru the points given:
(-3,-1) and (-2,4)
x1 y1
x2 y2
4  1 5

m
 5
2  3 1
(-3,4) and (2,-2)
x1 y1
x2 y2

m

2  4  6
2  3 5
Independent Practice: Calculate the slope for
each pair of points. Classify each slope as
positive, negative, zero, or undefined.
1. (2, 2) and (3, 5)
Slope: 3
Classification: Positive
3. (-2, -1) and (-1, -4)
Slope: -3
2. (0, 0) and (3, 0)
Slope: 0
Classification: Horizontal
4. (2, 3) and (2, 7)
Classification: Negative
Slope: Undefined
Classification: Vertical
5. (-1, -1) and (5, 5)
6. (8, 4) and (6, 4)
Slope: 1
Classification: Positive
Slope: 0
Classification: Zero
EVERYONE GET A
COMMUNICATOR!!!
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.
If you know the slope and one point on the
line you can draw the line.
Example: slope is 1/3 and the point (-2,-1)
Now you draw a line.
• Given: slope is 2/3 and
the point is (1,0)
• Given: slope is –3 and
the point is (–2,4)
• Given: slope is 1 and the
point is (2,3)
Graph the line y = 3x +1,
then tell me the slope of that line after
you graphed it. The slope is 3
x
y
-2
-5
-1
-2
0
1
1
4
2
7
Graph the line 2x + y = 4,
then tell me the slope of that line after
you
graphed
it.
y = -2x + 4
The slope is -2
x
y
-2
8
-1
6
0
4
1
2
2
0
Graph the line y = -5,
then tell me the slope of that line after
you graphed it. The slope is 0
x
y
-2
-5
-1
-5
0
-5
1
-5
2
-5
Graph the line y = 2x,
then tell me the slope of that line after
you graphed it. The slope is 2
x
y
-2
-4
-1
-2
0
0
1
2
2
4
Graph the line y = 3x - 2,
then tell me the slope of that line after
you graphed it. The slope is 3
x
y
-2
-8
-1
-5
0
-2
1
1
2
4
Graph the line 2x + 3y = 6,
then tell me the slope of that line after
2
The slope is you
graphed
it.
y
x 2
2/3
3
x
y
-6
6
-3
4
0
2
3
0
6
-2
Graph the line y = -x - 2,
then tell me the slope of that line after
you graphed it. The slope is -1
x
y
-2
0
-1
-1
0
-2
1
-3
2
-4
Graph the line 3x – 4y = 12,
then tell me the slope of that line after
3
you
graphed
it.
y  x 3
The slope is 3/4
4
x
y
-8
-9
-4
-6
0
-3
4
0
8
3

Graph the line x – y = 5,
then tell me the slope of that line after
you
graphed
it.
y=x-5
The slope is 1
x
y
-2
-7
-1
-6
0
-5
1
-4
2
-3
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)