Linear Gradients and Constant Rates of Change

From a ratio to a gradient

Suppose a tank gains 6 litres of water in 2 minutes. The ratio of volume gained to time taken is $6:2$, or $3:1$. This means it gains 3 litres for every minute. On a graph with time along the horizontal axis and volume up the vertical axis, that same ratio is the line’s gradient:

$\text{gradient}=\frac{\text{change in vertical value}}{\text{change in horizontal value}}=\frac{6\text{ litres}}{2\text{ minutes}}=3\text{ litres per minute}.$

The vertical change is sometimes called the rise and the horizontal change the run. You can find them between any two points on a straight line. A larger positive gradient means the line rises more for the same move to the right; a negative gradient means it falls.

Data for A straight line showing a constant filling rate

A straight line means a constant rate

Each equal step to the right along a straight line produces the same change upwards or downwards. Its gradient is therefore constant. If the tank continues filling at the rate shown, its volume increases by 3 litres in each minute, whether you compare the first two minutes or a later two-minute interval.

For points $(x_1,y_1)$ and $(x_2,y_2)$ on a straight line, write this as

$\text{gradient}=\frac{y_2-y_1}{x_2-x_1}.$

Subtract in the same order on the top and bottom. The units matter: if the vertical axis shows pounds and the horizontal axis shows hours, the gradient is in pounds per hour. A horizontal line has gradient zero because its vertical value does not change. A vertical line has no defined gradient because its horizontal change is zero.

Reading the rate in context

A gradient describes what changes per unit of the quantity on the horizontal axis. On a distance–time graph, it is distance travelled per unit time: a speed, or velocity when direction is included. The numerical gradient alone is not a complete interpretation; its meaning and units come from the axis labels.