Gradients and intercepts of straight lines
Linear functions are tested every paper, every tier. This topic is the foundation for graph reading, real-life modelling, and equation finding.
Equation of a straight line
The standard form is y = mx + c, where:
- m = gradient (slope).
- c = y-intercept (the y value where the line crosses the y-axis).
Computing the gradient
Given two points (x₁, y₁) and (x₂, y₂):
m = (y₂ − y₁) / (x₂ − x₁) = "rise / run".
A positive gradient slopes up from left to right. A negative gradient slopes down. Zero gradient is horizontal. Undefined (division by zero) is vertical.
Reading from a graph
- y-intercept: where the line crosses the y-axis. Read off directly.
- x-intercept (root): where the line crosses the x-axis. Set y = 0 in y = mx + c.
Real-life interpretation
In context, the gradient is a RATE (e.g. £/hour, m/s) and the intercept is a STARTING VALUE (e.g. fixed callout fee).
Example: cost C = 30 + 25t (t in hours).
- y-intercept c = 30 → fixed fee of £30 (e.g. callout).
- gradient m = 25 → £25 per hour.
Parallel and perpendicular lines
- Parallel lines have EQUAL gradients.
- Perpendicular lines have gradients that multiply to −1: m₁ × m₂ = −1. Equivalently, one gradient is the negative reciprocal of the other.
Finding the equation from data
Given gradient m and a point (x₁, y₁), use point–slope form: y − y₁ = m(x − x₁), then rearrange to y = mx + c if needed.
WJEC exam tip
When the question asks "interpret the gradient" or "interpret the y-intercept" in context, write a SENTENCE with units. "The cost increases by £25 for each additional hour" — not just "25". The units carry the A1 communication mark.
AI-generated · claude-opus-4-7 · v3-wjec-maths-leaves