Find the average gradient between two points on a curve and estimate the gradient at a point by drawing a tangent. It is Higher only, aimed at grade 8. Free questions with answers are on this page. Worked examples show each line before you try the questions.
Objectives
Work out an average gradient from two points.
Estimate the gradient at a point from a tangent.
Say what a gradient means in a real situation.
Key points
Explain that a curve has a different gradient at different points.
Work out the average gradient between two points as change in y divided by change in x.
Draw a tangent at a point and use two points far apart on it to estimate the gradient there.
Say what a gradient means in context: speed on a distance-time graph, acceleration on a speed-time graph.
Worked examples
Example 1
The curve y = x^2 passes through (1, 1) and (3, 9). Work out the average gradient between the two points.
2 marks
The rise is 9 - 1 = 8
The run is 3 - 1 = 2
The gradient is 8 / 2 = 4
Answer: 4
Common slips
Slip: The rise is 9 + 1 = 10 Subtract the y-values.
Slip: The run is 3 + 1 = 4 Subtract the x-values in the same order.
Slip: The gradient is 2 / 8 It is the rise over the run.
Example 2
A cyclist is 120 m from the start after 10 s and 200 m from the start after 18 s. Work out the average speed in m/s.
2 marks
The change in distance is 200 - 120 = 80
The change in time is 18 - 10 = 8
80 / 8 = 10
Answer: 10
Common slips
Slip: The change in distance is 200 + 120 Subtract the distances.
Slip: The change in time is 18 Subtract the times as well.
Slip: 8 / 80 Distance divided by time is speed.
Example 3
A tangent to y = x^2 at (2, 4) passes through (0, -4) and (3, 8). Estimate the gradient of the curve at (2, 4).
3 marks
Use the two points on the tangent. The rise is 8 - (-4) = 12
The run is 3 - 0 = 3
12 / 3 = 4
Answer: 4
Common slips
Slip: Use (2, 4) and (3, 8) Use two points on the tangent line, far apart, not a point of the curve.
Slip: The run is 3 + 0 Subtract the x-values.
Slip: 3 / 12 Rise over run again.
Questions and answers
GCSE Maths gradient of a curve questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
A curve passes through (0, 1) and (2, 9). Work out the average gradient between the points.
Show answer
Answer: 4
The rise is 8.
The run is 2.
M1 — The rise is 8.
A1 — Answer 4
Question 2 Easy2 marks
A curve passes through (1, 3) and (4, 12). Work out the average gradient.
Show answer
Answer: 3
The rise is 9.
The run is 3.
M1 — The rise is 9.
A1 — Answer 3
Question 3 Medium2 marks
A tangent to a curve passes through (1, 2) and (5, 10). Estimate the gradient of the curve where the tangent touches it.
Show answer
Answer: 2
Rise 8, run 4.
The tangent has the same gradient as the curve at the point.
M1 — Rise 8, run 4.
A1 — Answer 2
Question 4 Medium2 marks
A tangent passes through (0, 7) and (4, 3). Estimate the gradient of the curve at the point of contact.
Show answer
Answer: -1
The rise is 3 - 7 = -4.
The run is 4.
M1 — The rise is 3 - 7 = -4.
A1 — Answer -1
Question 5 Hard2 marks
For y = x^2, work out the gradient of the chord from (2, 4) to (2.5, 6.25).
Show answer
Answer: 4.5
The rise is 2.25.
The run is 0.5.
M1 — The rise is 2.25.
A1 — Answer 4.5
Question 6 Hard2 marks
For y = x^2, work out the gradient of the chord from (2, 4) to (2.1, 4.41).
Show answer
Answer: 4.1
The rise is 0.41.
The run is 0.1.
M1 — The rise is 0.41.
A1 — Answer 4.1
Question 7 Exam-style4 marks
The height of a ball is 0 m when t = 0 and 15 m when t = 1, where t is in seconds. A tangent to the height-time graph at t = 1 passes through (0, 5) and (2, 25). Work out the average speed between t = 0 and t = 1, then use the tangent to estimate the speed at t = 1.
Show answer
Answer: 15 m/s and 10 m/s
The change in height is 15 and the time is 1, so the average speed is 15 m/s.
The tangent has a rise of 20 and a run of 2.
The speed at t = 1 is 10 m/s.
M1 — Divides the change in height by the change in time
A1 — Average speed 15 m/s
M1 — Uses two points on the tangent for the rise over the run
A1 — Estimate 10 m/s
Common mistakes
Most lost marks on this topic are one slip.
Using a long chord when the question asks for the gradient at a point
Draw a tangent at the point and use two points on it
Subtracting the coordinates in a different order on the top and the bottom
Subtract in the same order, rise over run
Leaving off the unit of the gradient
Write the vertical unit per the horizontal unit, such as m/s
Spot the slip
A student finds the gradient of a curve at x = 2 by joining the points at x = 0 and x = 4. What is the problem?
A long chord gives an average, not the gradient at the point
The points should have been at x = 2 and x = 4
Curves have no gradient
Show the slip
A long chord gives an average, not the gradient at the point At a point you draw a tangent. A long chord is an average over a whole stretch of the curve.
Exam tips
This topic can appear on a calculator or a non-calculator paper. Show enough working for a method mark.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.