Gradient (or slope) of a Line, and Inclination

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

`"gradient"= text(vertical rise)/text(horizontal run`

In the following diagram, the gradient of the line AB is given by: `a/b`

In general, for the line joining the points (x₁, y₁) and (x₂, y₂), we have:

We can now write the formula 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=(y_2-y_1)/(x_2-x_1`

Interactive graph - slope of a line

You can explore the concept of slope of a line in the following interactive graph (it's not a fixed image).

Drag either point A (x₁, y₁) or point B (x₂, y₂) 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.

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 (usually y) increases as we go left to right:

The dependent variable in the above graph is the y-value, while the independent variable is x.

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 the x-axis is α.

From trigonometry, we recall that the tan of angle α is given by:

`tan\ alpha=text(opposite)/text(adjacent)`

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.

Exercise 1

Find the inclination of the line with slope `2`.

Answer

NOTE: The size of angle α is (by definition) only between `0°` and `180°`.

Exercise 2

Find the slope of the line with inclination α = 137°.

Answer