GraphsTopic Summary

Knowledge Organiser: Gradients of Curves

Part of Gradients of Curves · GCSE GCSE Mathematics revision

This topic summary covers Knowledge Organiser: Gradients of Curves within Gradients of Curves for GCSE Mathematics. Revise Gradients of Curves in Graphs for GCSE Mathematics with 9 exam-style questions and 10 flashcards. This topic appears less often, but it can still be a useful differentiator on mixed-topic papers. It is section 10 of 10 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.

Topic position

Section 10 of 10

Practice

9 questions

Recall

10 flashcards

Knowledge Organiser: Gradients of Curves

Key Methods

Tangent method: draw tangent at the point, calculate gradient = rise ÷ run
Gradient = (y₂ − y₁) ÷ (x₂ − x₁) using two far-apart points on the tangent
Chord: average rate of change between two points on the curve
Gradient at turning point = 0 (horizontal tangent)

Interpretation by Graph Type

Distance-time → speed (m/s)
Speed-time → acceleration (m/s²)
Positive gradient → increasing quantity
Negative gradient → decreasing quantity
Zero gradient → turning point

Key Vocabulary

Tangent: line touching curve at one point with same gradient
Chord: line joining two points on a curve
Instantaneous rate: rate at a single moment (tangent)
Average rate: rate over an interval (chord)
Rate of change: how quickly one quantity changes with respect to another

Common Errors

Confusing tangent (touches) with chord (joins two points)
Using nearby points on the tangent (magnifies drawing errors)
Omitting units from gradient answers
Using nearby points on the tangent for the gradient calculation — always choose two well-separated points

Key Formulas

Gradient of tangent = gradient of curve at that point
Gradient = rise ÷ run = (y₂ − y₁) ÷ (x₂ − x₁) using two points on the tangent
Velocity from distance-time: gradient of tangent gives instantaneous velocity
Acceleration from velocity-time: gradient of tangent gives instantaneous acceleration

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Keep building this topic

Read this section alongside the surrounding pages in Gradients of Curves. That gives you the full topic sequence instead of a single isolated revision point.

Common Misconceptions How Fast is Fast Right Now?Gradients of Curves — Essential Concepts

Practice Questions for Gradients of Curves

How do you find the gradient of a curve at a specific point?

A. Draw a chord joining two points on the curve and find its gradient
B. Draw a tangent to the curve at that point and find the gradient of the tangent
C. Find the average of the y-values on either side of the point
D. Divide the y-coordinate by the x-coordinate of the point

1 markfoundation

Explain why the gradient of a chord between two points on a curve is only an estimate of the gradient at a point, and how this estimate can be improved.

2 markshigher

Quick Recall Flashcards

Steps to estimate the gradient of a curve at a point

1. Mark the point on the curve 2. Place a ruler so it just TOUCHES the curve at that point (tangent) 3. Make the tangent line extend well across the graph 4. Choose two clear points on the tangent line 5. Calculate: gradient = (y2 - y1)/(x2 - x1) Tip: use points far apart on the tangent for greater accuracy.

What is a tangent to a curve?

A straight line that touches the curve at exactly one point and has the same gradient as the curve at that point. It does NOT cross through the curve at that point — it only touches it. The gradient of the tangent = the gradient of the curve at that point. Used to estimate the rate of change at an instant.

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Common Misconceptions

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Gradients of Curves