Identifying and interpreting gradients and intercepts in context
Real-world straight-line graphs always model a relationship: distance vs time, cost vs quantity, depth vs minutes. The gradient and y-intercept carry physical meanings — and exam mark schemes expect both number AND units AND interpretation.
What gradient really means
Gradient = change in y per unit change in x. The units of gradient are (units of y) ÷ (units of x).
| Graph type | What gradient means | Units |
|---|---|---|
| Distance vs time | Speed | m/s, mph, km/h |
| Cost vs items | Cost per item (unit price) | £/item |
| Petrol used vs miles | Fuel consumption | gallons per mile |
| Volume vs time | Flow rate | litres/second |
What y-intercept really means
The y-intercept is the value of y when x = 0 — i.e. the starting value before any x has happened.
- A taxi: y-intercept = the call-out fee paid before any miles driven.
- A water tank: y-intercept = initial volume of water in the tank.
- A phone tariff: y-intercept = monthly fixed charge.
✦Worked example
A taxi charges according to C = 1.50d + 3, where C is cost in £ and d is distance in miles.
- Gradient 1.50 means £1.50 per mile.
- Intercept 3 means a £3 fixed/booking charge, regardless of distance.
- For a 12-mile journey: C = 1.50 × 12 + 3 = £21.
✦Worked example— Worked example — reading from a graph
A graph of mass (kg) on the y-axis vs volume (cm³) on the x-axis is a straight line through (0, 0) with gradient 8.4.
- Gradient = 8.4 g/cm³ — wait, check units! kg/cm³ if y is kg. The gradient represents density.
- Through the origin: at zero volume, mass is zero (consistent with physics — no offset).
✦Worked example— Worked example — comparing two tariffs
Phone A: £10 fixed + 5p per minute. Phone B: £4 fixed + 8p per minute.
A: C = 0.05m + 10. B: C = 0.08m + 4.
For low usage B is cheaper; for high usage A is cheaper. Set equal to find break-even:
0.05m + 10 = 0.08m + 4 ⇒ 6 = 0.03m ⇒ m = 200 minutes.
So A is cheaper for usage above 200 min/month.
⚠Common mistakes— Common mistakes (examiner traps)
- Stating the gradient with no units. Always include units; mark schemes require them.
- Misinterpreting the intercept as "the answer when m = 1". It's the value when x = 0.
- Forgetting the meaning in a worded interpretation question. "The gradient is 5" earns less than "5 represents a £5 increase in cost per additional kilometre".
- Negative gradient interpretation. A negative gradient means y is decreasing with x — for distance/time graphs that means returning home; for cost vs subsidy that means spending less per item.
- Mixing up x and y meanings. Always check axis labels — speed graphs sometimes use t on x, sometimes on y.
➜Try this— Quick check
A pool drains according to V = -50t + 800 (V in litres, t in minutes).
Gradient -50 = pool empties at 50 litres per minute. Intercept 800 = initial volume 800 L. Pool empties when V = 0: t = 16 minutes.
AI-generated · claude-opus-4-7 · v3-deep-algebra