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.
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
Practice questions for Gradients of Curves
How do you find the gradient of a curve at a specific point?
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.
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.