GCSE · Graphs of functions

Rates of change from graphs (tangents, chords and the area under a graph)Exam-style questions

Read a rate off a curved graph: the gradient of a chord is the average rate of change over an interval, the gradient of a tangent is the rate at one instant, and the units on the axes say what that rate is (a speed, an acceleration, litres per minute). Then the other half: the area under a speed–time graph is the distance travelled, found exactly from straight pieces or estimated with strips, with a reason why the estimate is too big or too small.

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20 questions · GCSE (Higher) level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
A motorbike is stopped at a junction. It pulls away along a straight road.

The graph shows how its velocity, $v$ m/s, changes over the first $8$ seconds.(a) Estimate the gradient of the curve at the point where $t = 4$

You must show how you get your answer.

(3 marks)
(b) What does your answer to part (a) tell you about the motorbike?

(1 mark)
(c) Use $3$ strips of equal width to estimate how far the motorbike travels between $t = 2$ and $t = 8$

(3 marks)
Level 4 · 7 marksTry it
2
A delivery van is travelling along a straight road at $18$ m/s.

At time $t = 0$ the driver starts to brake, and the van slows down until it stops.

The graph shows the van's velocity, $v$ m/s, from $t = 0$ until it stops.(a) Estimate the acceleration of the van $2$ seconds after the driver starts to brake.

You must show all your working.

(3 marks)
(b) Use $3$ strips of equal width to estimate the distance the van travels while it is braking.

(3 marks)
(c) Is your answer to part (b) an overestimate or an underestimate?

Give a reason for your answer.

(1 mark)
Level 4 · 7 marksTry it
3
A motorboat switches off its engine and drifts in a straight line.

The graph shows the distance, $d$ metres, the boat has travelled $t$ seconds after the engine is switched off.(a) Estimate the speed of the boat when $t = 6$

You must show working to support your estimate.

(3 marks)
(b) Work out the boat's average speed between $t = 6$ and $t = 12$

(2 marks)
(c) Is the speed of the boat at $t = 12$ more than or less than your answer to part (b)?

Explain how you know.

(1 mark)
Level 4 · 6 marksTry it
4
The grid shows a curve for values of $x$ from $0$ to $4.5$ (a) Find the average rate of change of $y$ between $x = 0$ and $x = 2$

(2 marks)
(b) Estimate the gradient of the curve at the point where $x = 2$

You must show working to support your estimate.

(4 marks)
(c) Write down the two values of $x$ at which the rate of change of $y$ is zero.

(1 mark)
Level 4 · 7 marksTry it
5
At time $t = 0$ a tap is opened and water starts to drain out of a tank.

$R$ litres per minute is the rate at which the water drains out $t$ minutes after the tap is opened. The graph shows $R$ for the first $9$ minutes.(a) Work out an estimate of the volume of water that drains out in the first $9$ minutes.

Split the area under the graph into $3$ strips of equal width.

(3 marks)
(b) Is your estimate more than or less than the actual volume?

Tick a box.

☐ More than $\qquad$ ☐ Less than

Give a reason for your answer.

(1 mark)
(c) Work out an estimate of the mean rate of flow during the first $9$ minutes.

(2 marks)
Level 4 · 6 marksTry it

1 more question on rates of change from graphs (tangents, chords and the area under a graph)

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