6.4Gradient: Rise Over Run

The gradient of a line measures how steeply it climbs or falls: the ratio of the vertical change (rise) to the horizontal change (run) between any two points on it, and it is the same wherever you choose those points.

Gradient formula

\displaystyle \begin{aligned} m &= \dfrac{\text{rise}}{\text{run}} \\[6pt] &= \dfrac{y_2 - y_1}{x_2 - x_1} \end{aligned}

Rise and run between

P(2, 1)

and

Q(6, 13)

: the run is the horizontal change of 4 and the rise is the vertical change of 12, so

m = \frac{12}{4} = 3

What the sign of

m

does to the line, read left to right.

Now you try

Find the gradient of the line passing through $(0,\, 5)$ and $(3,\, 10)$ .

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\displaystyle \text{Gradient}

\displaystyle = \dfrac{y_2 - y_1}{x_2 - x_1}

\displaystyle = \frac{10 - 5}{3 - 0}

\displaystyle = \frac{5}{3}

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Gradient from two points

Find the gradient of the line through $\displaystyle A(-1, 7)$ and $\displaystyle B(5, -5)$ .

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\displaystyle m

\displaystyle = \dfrac{-5 - 7}{5 - (-1)}

\displaystyle = \dfrac{-12}{6}

\displaystyle = -2

The negative gradient confirms the line falls from left to right.