# 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"= 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 (*x*_{1}, *y*_{1}) and (*x*_{2},
*y*_{2}), we have:

Slope of the line joining the points (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}).

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* (*x*_{1}, *y*_{1}) or point *B* (*x*_{2}, *y*_{2}) 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.

Slope `= (y_2 - y_1)/(x_2 - x_1)`

` =(BC)/(AC)`

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You can move the graph up-down, left-right if you hold down the "Shift" key and then drag the graph.

If you get lost, you can always refresh the page.

### Example

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

Answer

These are the points involved:

So the slope is:

`m=(y_2-y_1)/(x_2-x_1`

`=(-5-(-1))/(2-(-4)`

`=(-4)/6`

`=-2/3`

Note the slope is **negative**. The line is going "down hill" as we move left to right.

### 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= arctanm(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

The situation is as follows:

So the slope is:

m= tan α= tan 137°

= −0.933

Note that the slope is negative.

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