Gradient of a Line

Gradient is a number that measures how steep a line is. You find it by picking two points on the line, then measuring how far it goes up (the rise) and how far it goes across (the run), and dividing: gradient equals rise over run. In the picture the line rises 2 while running 2, so the gradient is 1 — meaning for every 1 across, it climbs 1. A bigger gradient means a steeper line: a gradient of 3 climbs three times as fast as a gradient of 1. The order is fixed, with rise on top. Writing run over rise gives the reciprocal, which is the most common slip in this topic.

Lines come in four kinds. A positive gradient goes uphill from left to right. A negative gradient goes downhill. A horizontal line has zero rise, so its gradient is zero divided by the run, which is zero. And a vertical line has zero run — and since dividing by zero is impossible, its gradient is undefined. That last distinction matters and is often muddled: a horizontal line has a gradient, and that gradient is the number zero. A vertical line has no gradient at all. 'Zero gradient' and 'no gradient' describe different lines.

You do not need a picture to find a gradient — two points are enough. The rise is the difference in the y values and the run is the difference in the x values. From (2, 3) to (6, 11): the rise is 11 minus 3, which is 8, and the run is 6 minus 2, which is 4. So the gradient is 8 over 4, which is 2. The rule that keeps you safe is to subtract in the same order top and bottom. Doing 11 minus 3 over 6 minus 2 gives 2, and doing 3 minus 11 over 2 minus 6 also gives 2. But mixing the orders gives negative 2 — the right size with the wrong sign.

A downhill line gives a negative gradient, and the arithmetic needs care. From (−1, 4) to (3, −4): the rise is negative 4 minus 4, which is negative 8. The run is 3 minus negative 1 — and subtracting a negative adds, so that is 3 plus 1, which is 4, not 2. The gradient is negative 8 over 4, which is negative 2, meaning the line falls 2 for every 1 across. Two habits protect you here. Write the subtraction out rather than doing it mentally, and sense-check the sign at the end: the line clearly goes downhill, so a negative answer is right.

In a real situation the gradient is a rate, and that is where it becomes genuinely useful. On a distance-time graph, a line rising 160 km over 2 hours has a gradient of 80 — and that 80 is the speed in kilometres per hour. A steeper line means faster, and a flat line means not moving at all. The units always come from the axes: the y-unit per the x-unit. So cost against number of items gives dollars per item, water against time gives litres per minute, and pay against hours gives dollars per hour. Whenever a question asks what the gradient represents, name the rate and give its units.