Slope of a
Straight Line
Using two ordered pairs and the slope
formula to find slope of the line
Stapel, Elizabeth. "Slope of a Straight Line." Purplemath. Available from
http://www.purplemath.com/modules/slope.htm. Accessed 19 January 2009
Description

One of the most important
properties of a straight line
is its angle from the horizon
or the slope
Sometimes called it’s
steepness

Question?
Where is the
horizon on a
coordinate
graph?
Let’s look at the equation and
the graph of a straight line
y = ( 2/3 )x – 4
Or y = 2/3x – 4
Finding SLOPE using
the slope formula
Why the
m
for slope?
Answer: There are no definite answers by mathematicians. Some believe it
comes from the French word ‘monter’, to climb. Most believe that this is an
urban legend. JUST DON’T FORGET THAT m STANDS FOR SLOPE IN OUR
FORMULA!
If you are asked to find
slope… you could use
the slope formula
Question?
What could you
do to find
two ordered
pairs? (x,y)
Analyze the formula.
It uses x and y.
 What do we know
about x and y?
 If we pick an x (input) can we
get the matching y (output)?
We need two ordered pairs (or
points) on the line. HOW?

Complete the table
NOTE: When the coefficient is a fraction use 0, the denominator,
and the opposite of the denominator as the inputs (domain).
x
y = 2/3x - 4
y
(x,y)
0
( ,
)
3
( ,
)
Your ordered pairs should
be (0,-4) and (3,-2)
Let’s check the graph
Let’s use those
points in the formula
x
y
f(x)
0 -4
3 -2
Δ y = (-4) – (-2) = -2 = 2
3
Δ x = (0) – (3) -3
Our slope is 2/3
Slope as a ratio

The ratio that describes the tilt of
a line is the slope of that line
Remembering ratios: a comparison
of two quantities by division
 Ratios can be written three ways

2 to 3

2:3
2/3
We will use the fraction form 2/3
You try it…
Find the slope of each line.
1. y = -2x +3
x
y
f(x)
2. y = 3/5x - 2
x
y
f(x)
m= _______
m= _______
The slopes are -2 and 3/5.
What do you notice about the slope and the equation?
Slope-Intercept Form
Look at those equations again.
1. y = -2x +3
2. y = 3/5x - 2
Each slope that we found (using the slope formula) appears just
before the x in the equation.
y = mx + b
This will help you understand the slope-intercept form.
Let’s examine this equation. X means inputs. Y means outputs.
That’s the T-chart numbers.
M means the slope (tilt of the line) and b means y-intercept.
We’ll discuss the y-intercept another day.
Let’s look at the
graphs of these lines
1. y = -2x +3
1.
2. y = 3/5x - 2
2.
This relationship is always true: Increasing lines have positive
slopes, and decreasing lines have negative slopes. Always!
This fact can help you check
your calculations: if you
calculate a slope as being
negative, but you can see
from the graph that the line
is increasing (so the slope
must be positive), you know
you need to re-do your
calculations. Being aware of
this connection can save you
points on a test because it
will enable you to check your
work before you hand it in.
Special equations,
graphs, and slopes
Increasing lines have
positive slopes;
decreasing lines have
negative slopes. With
this in mind, consider
the following horizontal
line:
y=4
Its graph is shown to
the right.
What’s the slope
(or, tilt of the line)?
Is the horizontal line going up; that is, is it an
increasing line? No, so its slope won't be positive.
Is the horizontal line going down; that is, is it a
decreasing line? No, so its slope won't be
negative. What number is neither positive nor
negative? Zero! So the slope of this horizontal
line is zero. Let's do the calculations to confirm
this value. Using the points (–3, 4) and (5, 4), the
slope is:
This relationship is true for every horizontal line:
a slope of zero means the line is horizontal, and a
horizontal line means you'll get a slope of zero.
(By the way, all horizontal lines are of the form "y
= some number", and the equation "y = some
number" always graphs as a horizontal line.)
Describe this airplane’s slope (or, tilt with the horizon)?
What’s the slope
(or, tilt of the line)?
Now consider the vertical line x = 4:
Is the vertical line going up on one end? Well,
kind of. Is the vertical line going down on the
other end? Well, kind of. Is there any number
that is both positive and negative? Nope.
Verdict: vertical lines have NO SLOPE. In
particular, the concept of slope simply does not
work for vertical lines. The slope doesn't exist!
Let's do the calculations. I'll use the points
(4, 5) and (4, –3); the slope is:
(We can't divide by zero, which is of course
why this slope value is "undefined".)
Describe the slope
of these airplanes
(or, tilt with the
horizon)?
This relationship is always true: a vertical line will
have no slope, and "the slope is undefined"
means that the line is vertical. (By the way, all
vertical lines are of the form "x = some number",
and "x = some number" means the line is
vertical. Any time your line involves an undefined
slope, the line is vertical, and any time the line is
vertical, you'll end up dividing by zero if you try
to compute the slope.)
y=
Slope = ______
x=
Slope = ______
WARNING
It is very common to confuse these
two lines and their slopes, but they
are very different. Just as
"horizontal" is not at all the same
as "vertical", so also "zero slope" is
not at all the same as "no slope".
The number "zero" exists, so
horizontal lines do indeed have a
slope. But vertical lines don't have
any slope; "slope" just doesn't
have any meaning for vertical lines.
It is very common for tests to
contain questions regarding
horizontals and verticals.
Don't mix them up!
The 4 Types of Slopes
Negative Slope
Positive Slope
Undefined Slope/
No slope
Zero Slope
Review
Find slope from a graph
1. Locate two or three
good points on the line.
2. Write each ordered pair.
3. Use the slope formula
to calculate the slope of
the line.
4. Check the tilt of the line
and the slope that you
calculated for any
mistakes.
Find slope from a graph
Find slope from a graph
y-intercept and x-intercept
The y-intercept means where
a line will cross the y-axis.
What is the y-intercept of this
line? The variable b is used to
represent this intercept in the
slope-intercept form of an
equation.
y = mx + b
The x-intercept is where a line will
cross the x-axis. It is not indicated
in the slope-intercept form.
Identify the slope (m) and
y-intercept (b) of each line
1.
2.
3.
4.
5.
6.
y= 5x +3
y = -1/2x – 8
y = 3/2x
y = -x + 2
y=3
x = -3
1.
2.
3.
4.
5.
6.
m= 5
b=3
m= -1/2 b=-8
m=3/2 b=0
m=-1
b=2
m=0
b=3
m=no slope