Gradient

Video lesson · from 17:274 worked examples

One lesson video covers all of The Straight Line, so it opens at 17:27 for this topic — not from the beginning.

The gradient of a line measures its steepness. It is the ratio of vertical change to horizontal change.

m = \frac{y_2 - y_1}{x_2 - x_1}

The gradient is also related to the angle $\theta$ the line makes with the positive x-axis:

m = \tan\theta

Positive

Negative

Zero

Undefined

Keep the order consistent: whichever point you call the first, use it first in both the numerator and the denominator. Swapping halfway through flips the sign.
Rearrange before reading off $m$ : from $3x+2y=12$ you must make $y$ the subject. The coefficient of $x$ in the original is not the gradient.
Leave gradients as fractions: decimals make later comparisons unreliable, particularly when testing whether two gradients are equal.
Angle questions use $m = \tan\theta$ : and if the angle is measured from the $y$ -axis, subtract from $90^\circ$ first.

Example 1

Calculate the gradient of the straight line shown in the diagram:

\begin{aligned} m &= \tan\theta \\ &= \tan 32^\circ \\ &\approx 0.62 \end{aligned}

Example 2

Find the angle that the line joining $P(-2, -2)$ and $Q(1,7)$ makes with the positive direction of the x-axis.

\begin{aligned} m_{PQ} &= \frac{7 - (-2)}{1 - (-2)} \\ &= \frac{9}{3} \\ &= 3 \end{aligned}

\tan\theta = 3

\theta = \tan^{-1}(3) \approx 71.6^\circ

Example 3

The line AB makes an angle of $60^\circ$ with the y-axis. Find the exact value of the gradient of AB.

The angle with the x-axis would be $90^\circ - 60^\circ = 30^\circ$ .

\begin{aligned} m &= \tan 30^\circ \\ &= \frac{1}{\sqrt{3}} \end{aligned}

Example 4

Find the size of the angle $\theta$ shown in the diagram:

The gradient $m = 5$ tells us the angle with the x-axis.

\tan\alpha = 5

\alpha = \tan^{-1}(5) \approx 78.7^\circ

The angle $\theta$ with the y-axis is $90^\circ - 78.7^\circ = 11.3^\circ$ .