Gradients of curves and areas under graphs

A curve has a different gradient at every point. You can estimate it with a tangent, and estimate the area under a curve with trapezia.

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Average and instantaneous rates

The average rate of change between two points is the gradient of the chord joining them.
y = x2 from x = 1 to x = 3: (9 − 1)(3 − 1) = 4.
The instantaneous rate of change at a point is the gradient of the tangent there. Draw a tangent with a ruler, pick two points far apart on it, and work out rise ÷ run.

What gradients and areas mean

Distance-time graph: gradient = speed.
Velocity-time graph: gradient = acceleration, area under the graph = distance travelled.
A graph of money against time: gradient = rate of earning or spending.

Area under a curve

Split the area into vertical strips and treat each as a trapezium: area = 12(a + b)h.
Velocities 0, 8, 12, 14 m/s at t = 0, 5, 10, 15 s:
2.5 × (0 + 8) + 2.5 × (8 + 12) + 2.5 × (12 + 14) = 135 m.
If the curve bends upwards between the points, trapezia give an overestimate; if it bends downwards (like this one), an underestimate.
Worked example

A velocity-time graph goes in straight lines from (0, 0) to (10, 20) then to (30, 20). Find the distance travelled.

  1. Triangle: 12 × 10 × 20 = 100 m
  2. Rectangle: 20 × 20 = 400 m
  3. Total = 500 m

Answer: 500 m

Key idea

Gradient of a chord = average rate. Gradient of a tangent = rate at that instant. Area under a velocity-time graph = distance; estimate it with trapezia.

Key words
Gradient
The steepness of a line: change in y ÷ change in x. It is m in y = mx + c.
Estimate
An approximate answer, usually found by rounding each number to 1 significant figure.
Chord
A straight line joining two points on a circle.
Speed
speed = distance ÷ time

All GCSE and International GCSE Mathematics key terms and definitions →

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  1. Find the average rate of change of y = x2 between x = 1 and x = 5.

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    Answer: 6

    (52 − 12)(5 − 1) = 244 = 6.

  2. A tangent to a curve passes through (2, 7) and (4, 3). Estimate the gradient of the curve at the point of contact.

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    Answer: -2

    (3 − 7)(4 − 2) = −2.

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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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