Slope

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What is slope?

The slope of a straight line is a number that indicates how steep the line is.

You can calculate the slope of a line by dividing the vertical change, called the

rise,

by the horizontal change, called the

run,

between two distinct points on the line.

slope

change in

rise

run

Let's try it! Find the slope of this line. Pick two points on the line, and calculate the rise over

run.

To get from (4, 4) to (8, 5), you go up 1 unit. The rise is
1.
You also go to the right 4 units. The run is
4.
Write that as a fraction.
slope
=
rise
run
=
1
4

So, the slope is

Any two points you choose on a line will result in the same slope.

Let's try it! Pick two different points on that same line, and calculate the slope.

To get from
(
0
,
3
)
to
(
8
,
5
)
,
you go up 2 units. The rise is
2.
You also go to the right 8 units. The run is
8.
Write that as a fraction. Simplify.
slope
=
rise
run
=
2
8
=
1
4

So, the slope is

You can see that the slope of a line is the same no matter which points you pick to calculate the rise and run.

Try some practice problems!

Find the slope from a graph

What can slope tell you?

The slope of a line can tell you how steep the line is and whether it tilts up or down. Read more to see how!

Steepness

The greater the slope, the steeper the line goes upward. Compare the three lines below.

slope

rise

run

slope

rise

run

slope

rise

run

As you can see, steeper lines have larger slopes.

Positive and negative slopes

Lines that travel up from left to right have positive slopes. Lines that travel down from left to right have negative slopes.

Let's calculate another slope. Since this line travels down from left to right, it has a negative slope.

To get from (3, 6) to (5, 4), you go down 2 units. The rise is
-2.
You also go to the right 2 units. The run is
2.
Write that as a fraction. Simplify.
slope
=
rise
run
=
–
2
2
=
–
1

So, the slope of this line is

–

Tip:

When graphing, you can use the sign of the slope to help check your work!

Horizontal lines

A horizontal line does not go up or down, so what is its slope?

To get from
(
3
,
5
)
to
(
7
,
5
)
,
you don't move up or down at all. The rise is
0.
You go to the right 4 units. The run is
4.
Write that as a fraction. Simplify.
slope
=
rise
run
=
0
4
=
0

Any fraction with a numerator of 0 simplifies to 0. So, the slope of a horizontal line will always be 0.

Vertical lines

A vertical line goes straight up and down, so what is its slope?

To get from
(
2
,
3
)
to
(
2
,
8
)
,
you move up 5. The rise is
5.
You don't move left or right at all. The run is
0.
Write that as a fraction.
slope
=
rise
run
=
5
0
=
undefined

Since it is impossible to divide by 0, the slope of a vertical line will always be undefined.

Slope formula

You don't need a graph to find the slope! You can find the slope of a line that goes through points

(

)

and

(

)

using this formula.

slope

(

–

)

(

–

)

This formula is the same thing as rise over

run,

just written using the coordinates of points on the line.

Let's try it! Find the slope of the line that passes through

(

)

and

(

)

Let's choose

(

)

to be the point

(

)

and

(

)

to be the point

(

)

Use the formula for slope.

slope

–

slope

–

So, the slope of the line is

It doesn't matter which point you call

(

)

and which you call

(

)

as long as you stay consistent throughout the calculation. Let's try it with the points reversed!

This time choose

(

)

to be the point

(

)

and

(

)

to be the point

(

)

slope

–

slope

–

The slope is still

Try some practice problems!

Find the slope from two points

Skills

Find the slope from a graph Find the slope from two points Find a missing coordinate using slope Graph a line using slope

Videos(4)

Find the slope from a graph

Find the slope from two points

Find a missing coordinate using slope

Graph a line using slope

Lessons

Slope-intercept form

What is slope?

Try some practice problems!

What can slope tell you?

Steepness

Positive and negative slopes

Horizontal lines

Vertical lines

Slope formula

Try some practice problems!

Related

Skills

Videos(4)

Lessons