Gradient as a rate of change; proportion graphs
A gradient on a graph is more than a number — it carries physical meaning. In real-world contexts, gradient = "how much y changes per unit increase in x".
Gradient = rate of change
For a straight-line graph, gradient = (rise)/(run) = Δy/Δx. The units of gradient are (units of y)/(units of x).
| Graph | Gradient meaning |
|---|---|
| Distance–time | speed (m/s, km/h) |
| Velocity–time | acceleration (m/s²) |
| Cost–quantity | unit price (£ per item) |
| Conversion graph | exchange rate |
| Mass–volume | density |
Worked example: a distance–time graph passes through (0, 0) and (4, 80). Gradient = 80/4 = 20 km/h — the speed.
Proportion graphs
A direct proportion (y ∝ x) graphs as a straight line through the origin. The gradient = the constant of proportionality k.
Worked example: a graph of cost (£) vs litres of fuel passes through (0, 0) and (10, 14). Gradient = 14/10 = £1.40 per litre. Equation: C = 1.4ℓ.
Negative gradient
A negative gradient means y decreases as x increases.
Worked example: a tank empties uniformly. Volume–time graph passes through (0, 50) and (10, 30). Gradient = (30 − 50)/(10 − 0) = −2 (litres per minute).
Reading gradient between two points
Gradient = (y₂ − y₁) / (x₂ − x₁).
⚠Common mistakes
- Forgetting units — gradient has units; state them in context.
- Confusing rise and run — rise is vertical change (y), run is horizontal (x).
- Reading axes incorrectly — check the scale of each axis, not just the squares.
- Negative gradient sign error — y decreasing → negative gradient.
- Treating proportion graphs that don't pass through origin as direct — check it does pass through (0, 0).
➜Try this— Quick check
A line passes through (3, 7) and (8, 22). Find the gradient and interpret as a rate.
- (22 − 7)/(8 − 3) = 15/5 = 3 — y increases by 3 per unit increase in x.
AI-generated · claude-opus-4-7 · v3-deep-ratio