Why should we learn this?
The Slope of a Line
Objectives: To find slope of a line
given two points, and to graph a line
using the slope and the y-intercept.
VOCABULARY
Rate – a comparison of two quantities
measured in different units
Rate of change – allows you to see the
relationship between two changing
quantities
RATE OF CHANGE FORMULA:
change in dependent variable
change in independent variable
Rate of Change and Slope
Below is a graph of the distance traveled by a motorcycle
from its starting point. Find the rate of change. Explain what this rate of
change means.
One real-world connection is
to find the rate of change in
an airplane’s altitude.
Rate of Change and Slope
For the data in the table, is the rate of change the same for each pair of
consecutive mileage amounts?
Fee for Miles Driven
Miles
100
Fee
$30
150
$42
200
$54
250
rate of change =
42 – 30
12
=
150–100 50
54 – 42
12
=
200–150 50
Cost depends on the
number of miles.
66 – 54
12
=
200–250
50
The rate of change for each pair of consecutive mileage
amounts is $12 per 50 miles. The rate of change is the same
for all the data.
Rate of Change and Slope
(continued)
Using the points (0,0) and (20, 400), find the rate of change.
rate of change =
Choose two points
On the graph (0,0) and
(20, 400)
$66
change in cost
change in number of miles
=
=
=
vertical change
horizontal change
400 – 0
change in distance
change in time
Use two points.
20 – 0
400
20
20
Divide the vertical change by
the horizontal change.
Simplify.
The rate of change is 20 m/s.
The motorcycle is traveling 20 meters each
second.
1
Rate of Change and Slope
More Vocabulary
SLOPE -- vertical change
horizontal change
A.K.A.
Find the slope of each line.
a.
slope =
“rise over run”
RISE
RUN
rise
run
=
4–1
0–2
=
3
–2
=–3
2
The slope of the line is – 3 .
2
6-1
Rate of Change and Slope
(continued)
b. Find the slope of the line.
slope =
=
rise
run
–1 – 1
–2 – (–2)
= –2
–1
= 2
“rise over run”
For positive slope, rise up and run
right…or move down and to the left.
For negative slope, move down and to
the right or up and to the left.
The slope of the line is 2.
Finding slope when
given a graph of a line.
REMEMBER
Locate two points on the line.
2. Count straight up or down from one
point until you’re “even” with the other
point. (This is your rise.)
3. Count across left or right until you get
to the second point. (This is your run.)
4. Simplify this fraction, if needed.
If a line SLANTS UPWARD from left to
right, it has a POSITIVE slope. /
1.
If a line SLANTS DOWNWARD from left
to right, it has a NEGATIVE slope. \
2
Rate of Change and Slope
FORMULA
Find the slope of the line through E(3, –2) and F(–2, –1).
SLOPE FORMULA -- y2 – y1
x2 – x1
(where x2 – x1 does not equal 0.)
y2 – y1
slope = x2 – x1
=
–1 – (–2)
Substitute (–2, –1) for (x2, y2) and
(3, –2) for (x1, y1).
–2 – 3
1
= –5
Simplify.
= – 15
Symbol for slope = m.
The slope of EF is – 1 .
5
Find the slope of the line
through each pair of points.
SOLUTION: Use the Slope formula.
Find the slope of the line
through each pair of points.
a)
(2,5) and (4,7)
c) (a,b) & (c,d)
a)
(2,5) and (4,7)
m=
b)
(-1,4) & (3,2)
b)
Find the slope of each line.
a.
slope =
=
y2 – y1
x2 – x1
2–2
1 – (–4)
= 0
5
d −b
c−a
2 − 4 −2 −1
=
=
3 − −1 4
2
Rate of Change and Slope
(continued)
b. Find the slope of the line.
Substitute (1, 2) for
(x2, y2) and (–4, 2)
for (x1, y1).
Simplify.
m=
(-1,4) & (3,2)
m=
Rate of Change and Slope
7−5 2
= =1
4−2 2
c) (a,b) & (c,d)
y –y
slope = x2 – x1
2
1
=
1 – (–4)
2–2
= 5
0
Substitute (2, 1) for
(x2, y2) and (2, –4)
for (x1, y1).
Simplify.
= 0
The slope of the horizontal line is 0.
Division by zero is undefined.
The slope of the vertical line in undefined.
3
Special Lines
Find the slope.
The slope of a horizontal line is zero.
Equation is y = b. (No “x” in equation.)
The slope of a vertical line is undefined.
Equation is x = a. (No “y” in equation.)
y=5
x = -2
10
y
10
x
-10
-10
10
Find the slope.
Solution
y = -5
Since there is no x
in the equation, this
is a horizontal line.
The slope of a
horizontal is zero.
So m = 0.
x=3
y
10
x
-10
-10
y = -5
Graphing a Line using the
y-intercept and Slope
x= 3
Since there is no y in
the equation, this is
a vertical line. The
slope of a vertical
line is undefined.
So m = undefined.
Reminder: The y-intercept is the point
where a line crosses the y-axis.
STEPS
1. Plot the y-intercept. (referred to as “b”)
2. Use the slope and move rise/run to plot
at least two more points.
3. Graph the line.
Example -- Graph the line given
the slope and y-intercept.
Example -- Graph the line given
the slope and y-intercept.
y-intercept = -3, m = -1/2
y-intercept = -3, m = -1/2
y
y
5
5
x
-5
1. Plot the y-intercept, (0,-3).
2. From this point, use
m = -1/2 and rise/run to
get additional points.
3. Graph the line.
x
-5
-5
5
-5
5
4
Example -- Graph the line given
the slope and y-intercept.
Example -- Graph the line given
the slope and y-intercept.
y-intercept = 2, m = 2/3
y-intercept = 2, m = 2/3
y
y
5
5
x
-5
1. Plot the y-intercept, (0,2).
2. From this point, use
m = 2/3 and rise/run to
get additional points.
3. Graph the line.
x
-5
-5
5
-5
Example -- Graph the line given
the slope and y-intercept.
5
Example -- Graph the line given
the slope and y-intercept.
y-intercept = 0, m = -1
y
5
y-intercept = 0, m = -1
1. Plot the y-intercept, (0,0).
2. From this point, use
m = -1/1 and rise/run to
get additional points.
3. Graph the line.
y
5
x
-5
x
-5
-5
5
-5
5
SUMMARY
The slope of a line represents the rate
of change in the x and y values.
The slope formula is y2 – y1
x2 – x1
5