Slope of a Line

Gradient (or slope) of a Line, and Inclination

Application: Road sign, indicating a steep gradient.
A 15% road gradient is equivalent to m=0.15.

The gradient (also known as slope) of a line is defined as

gradient=​horizontal run​​vertical rise​​

In the following triangle, the gradient of the line is given by: ​b​​a​​

In general, for the line joining the points (x₁, y₁) and (x₂,y₂), we have:
We can now write the fomula for the slope of a line.

Gradient of a Line Formula

We see from the diagram above, that the gradient (usually written m) is given by:

m=​x​2​​−x​1​​​​y​2​​−y​1​​​​

Interactive graph - slope of a line

You can explore the concept of slope of a line in the following JSXGraph (it's not a fixed image).
Drag either point A or point B to investigate how the gradient formula works. The numbers will update as you interact with the graph.
Notice what happens to the sign (plus or minus) of the slope when point B is above or below A.

0,0

– o + ← ↓ ↑ →

A

B

C

slope m
= rise / run
= 7 / 10
= 0.7

(-9, -4)

(1, 3)

(1, -4)

y₂ − y₁
= 3 − -4
= 7

x₂ − x₁
= 1 − -9
= 10

5

10

-5

-10

5

-5

x

y

You can move the graph up-down, left-right if you hold down the "Shift" key and then drag the graph.
Sometimes the explanation boxes overlap. It can't be helped!
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Example

Find the slope of the line joining the points (-4, -1) and (2, -5).

Answer

Positive and Negative Slopes

In general, a positive slope indicates the value of the dependent variable increases as we go left to right:

[The dependent variable (usually x) in the above graph is the y-value.]

A negative slope means that the value of the dependent variable (usually y) is decreasing as we go left to right:

Inclination

We have a line with slope m and the angle that the line makes with thex-axis is α.
From trigonometry, we recall that the tan of angle α is given by:

tan α=​adjacent​​opposite​​

Now, since slope is also defined as opposite/adjacent, we have:

This gives us the result:

tan α = m

Then we can find angle α using

α = arctan m

(That is, α = tan^-1 m)

This angle α is called the inclination of the line.