Gradient definition
Gradient is a measure of how steep a line or curve is. On a straight-line graph, the gradient tells you how much the y-value changes when the x-value increases by 1. A steep line has a large gradient, while a flatter line has a smaller gradient. Gradient also shows direction: a line rising from left to right has a positive gradient, and a line falling from left to right has a negative gradient.
The common formula for gradient is change in y divided by change in x. This is sometimes described as rise over run. If a line goes from the point 2, 3 to the point 6, 11, the change in y is 8 and the change in x is 4. The gradient is 8 divided by 4, which equals 2. This means y increases by 2 for every increase of 1 in x.
Gradient is used in algebra, coordinate geometry, graphs, functions, trigonometry, rates of change, calculus, mechanics and real-life modelling. At GCSE level, students find gradients from graphs, calculate gradients from two points, compare slopes and use the equation of a straight line. At A-Level, gradient becomes linked to differentiation, tangent lines, velocity, acceleration and curve sketching.
In the straight-line equation y = mx + c, the letter m represents the gradient. For example, in y = 3x + 5, the gradient is 3. This means the line rises by 3 units for every 1 unit moved to the right. The value c is the y-intercept, which tells you where the line crosses the y-axis. Together, m and c describe the line completely.
Positive, negative and zero gradients have different meanings. A positive gradient means the line rises as you move from left to right. A negative gradient means the line falls as you move from left to right. A horizontal line has gradient 0 because the y-value does not change. A vertical line does not have a normal numerical gradient because the change in x is 0, and division by 0 is undefined.
Gradient can also describe real-life rates. On a distance-time graph, the gradient represents speed because it shows change in distance divided by change in time. On a cost graph, the gradient may represent cost per item. In science and business, gradients help show how one quantity changes compared with another. This makes gradient a powerful idea beyond pure graph work.
For curves, the gradient changes from point to point. At A-Level, differentiation is used to find the gradient of a curve at an exact point. This is the gradient of the tangent to the curve. The idea is still connected to change in y divided by change in x, but calculus allows the change to be measured at a single point rather than across a wider interval.
Related ideas include slope, coordinate geometry, straight-line graphs, y = mx + c, rate of change, tangent, differentiation, parallel lines and perpendicular lines. A clear definition of gradient helps students understand that a graph is not just a picture. The steepness of the graph has mathematical meaning and can describe relationships, movement and change.