GCSE Maths revision · Higher: number and algebra

Gradients and areas under graphs

Higher only

The idea

On a speed-time graph, the gradient is the acceleration and the area underneath is the distance travelled. Estimate a curved area by splitting it into trapezia.

Gradient $= \frac{\text{change in } y}{\text{change in } x}$ with units.
For area, split into triangles, rectangles or trapezia and add.
For a curve, say whether the estimate is too big or too small.

Watch out The area under a velocity-time graph is the distance, not the speed.

See it worked

Worked exampleNon-calculator4 marks

The velocity-time graph shows a cyclist's journey over 40 seconds.

(a) Work out the acceleration during the first 10 seconds, in $\text{m/s}^2$ .

(b) Work out the total distance travelled.

(a) Acceleration is the gradient: $\frac{20 - 0}{10 - 0} = 2\ \text{m/s}^2$ .
(b) Distance is the area under the graph, a trapezium with parallel sides 40 and 20 and height 20.
$\frac{1}{2}(40 + 20) \times 20 = 600$

Answer (a) $2\ \text{m/s}^2$ ; (b) $600$ m

Your turn

Question 1Non-calculator3 marks

The table shows the speed of a vehicle at four times.

Use three trapezia, each 2 seconds wide, to estimate the distance travelled from 0 to 6 seconds.

Table of speeds. Time 0 s: 1 m/s. 2 s: 5 m/s. 4 s: 8 m/s. 6 s: 6 m/s.
Time (s)	0	2	4	6
Speed (m/s)	1	5	8	6

Question 2Non-calculator4 marks

An object moves along a straight line through a fixed point O. The graph shows its distance from O during 7 seconds. A tangent has been drawn to the curve at $t = 4$ .

(a) Use the tangent to estimate the instantaneous speed at $t = 4$ .

(b) Use the graph to work out the average speed over the first 4 seconds.

Question 3Non-calculator4 marks

The diagram shows a sketch of the speed-time graph for a car journey.

The total distance travelled is 750 m.

Work out the value of $T$ .

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