IGCSE Maths / Direct and inverse proportion
Graphs of direct and inverse proportion
Edexcel International GCSE Maths (4MA1), chapter 21: Direct and inverse proportion, specification point 2.5a-f. Free lesson.
Read the revision notes · IGCSE Maths past papers
Direct proportion has a straight-line graph through the origin. Inverse proportion has a falling curve that approaches both axes but never touches them.
How to recognise each graph
The graph’s shape shows how the two values change together. In direct proportion, doubling x doubles y, so the graph rises steadily. In inverse proportion, doubling x halves y, so the graph falls and then flattens. Direct proportion is a climbing straight line that passes through the origin, (0, 0). Inverse proportion is a falling curve that approaches both axes without touching them. A straight line that misses the origin is not direct proportion. It may be a linear relationship, but it is not proportional.
How a direct proportion graph works
If y is directly proportional to x, write y = kx. The constant k is the gradient, which tells you how steep the line is. The graph passes through the origin because when x = 0, y = 0. A larger k gives a steeper line. Use a known pair of values to find k, then use the equation to find another value.
How an inverse proportion graph works
If y is inversely proportional to x, write y = k/x. As x increases, y decreases. The curve drops steeply at first, then flattens as it approaches the axes. The curve never reaches either axis: y = k/x does not equal zero for a fixed non-zero k. Every pair of values on the curve gives the same product: xy = k. This is a quick way to check a table of values.
Check the origin before deciding
A straight line is not enough to prove direct proportion. Direct proportion must have y = 0 when x = 0. If the line crosses the vertical axis at another value, it has a constant added and misses the origin. For example, y = 4x passes through the origin, but y = 4x + 5 crosses the vertical axis at 5. An inverse curve approaches the x-axis but never touches or crosses it.
Use the graph’s shape to answer questions
A curve that falls from the top left, then flattens as it approaches both axes, shows inverse proportion. If x doubles on this curve, y halves. Direct: straight line through the origin; gradient is k. Inverse: falling curve; every x and y pair multiplies to k.