Proportion graphs and gradient as a rate

Graphs show proportion at a glance, and the gradient of a straight-line graph tells you a rate.

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Direct proportion graphs

If y is directly proportional to x, the graph is a straight line through the origin: y = kx.
The gradient k is the multiplier. A conversion graph from miles to kilometres is an example.

Inverse proportion graphs

If y is inversely proportional to x, y = kx. As x doubles, y halves.
The graph is a curve that gets closer to the axes but never touches them.

Gradient as a rate

The gradient of a straight-line graph is a rate of change: change in y ÷ change in x, with units.
A tank goes from 200 litres to 80 litres in 15 minutes.
Gradient = (80 − 200)15 = −8 litres per minute: it empties at 8 litres per minute.
3 6 9 12 15 40 80 120 160 200 Time (minutes) Volume (litres) −120 litres 15 minutes gradient = −120 ÷ 15= −8 litres per minute
Gradient as a rate: (80 − 200) ÷ 15 = −8 litres per minute.
Worked example

y is directly proportional to x. The graph passes through (4, 10). Find y when x = 10.

  1. y = kx, so 10 = 4k and k = 2.5.
  2. y = 2.5 × 10 = 25

Answer: y = 25

Key idea

Direct proportion: a straight line through the origin. Inverse proportion: a curve y = k/x that never touches the axes. The gradient is a rate: units of y per unit of x.

Key words
Gradient
The steepness of a line: change in y ÷ change in x. It is m in y = mx + c.
Direct proportion
Two quantities are in direct proportion when one is a constant multiple of the other: y = kx.
Inverse proportion
As one quantity increases, the other decreases so that their product is constant: y = k/x.

All GCSE and International GCSE Mathematics key terms and definitions →

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  1. Which equation has a graph that shows y is directly proportional to x?

    1. y = 2.5x + 1
    2. y = x2
    3. y = 2.5x
    4. y = 2.5x
    Show the answer

    Answer: y = 2.5x

    y = kx gives a straight line through (0, 0).

  2. Which equation has a graph that is a curve getting closer to both axes, for x > 0?

    1. y = 20 − x
    2. y = x2 + 20
    3. y = 20x
    4. y = 20x
    Show the answer

    Answer: y = 20x

    Inverse proportion y = kx approaches both axes but never touches them.

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Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

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