Gradients and intercepts of linear functions
This is a core Edexcel skill examined on both tiers. Foundation tests gradient as "rise over run" and y-intercept as the value when x = 0. Higher tier extends to interpreting these in real-world contexts and finding equations of parallel/perpendicular lines.
Form y = mx + c
A straight-line equation in the form y = mx + c gives:
- m = gradient (slope).
- c = y-intercept (where the line crosses the y-axis).
Finding the gradient between two points
m = (y₂ − y₁) / (x₂ − x₁) — "change in y over change in x".
Example: through (1, 5) and (4, 14): m = (14 − 5)/(4 − 1) = 9/3 = 3.
Finding the equation from two points
- Find m from the points.
- Substitute one point into y = mx + c to find c.
- Write y = mx + c.
Example: through (2, 7) and (5, 16): m = 3, then 7 = 3(2) + c gives c = 1, so y = 3x + 1.
Parallel and perpendicular lines (Higher only)
- Parallel lines have the same gradient.
- Perpendicular lines have gradients whose product is −1: m₁ × m₂ = −1.
So the line perpendicular to y = 2x + 5 has gradient −1/2.
Real-world interpretation
In a context like cost vs miles for a taxi:
- Gradient = price per mile (rate).
- y-intercept = fixed fee (cost when miles = 0).
Always state the units in the interpretation: "£2 per mile" not just "2".
Edexcel paper alignment
- Paper 1F/1H: equation manipulation (rearrange, find m and c, plot).
- Paper 2H/3H: real-context interpretation, distance-time and cost graphs.
Common Edexcel exam tip
When asked to find an equation, always finish with the equation in the form y = mx + c (or rearranged neatly). Leaving it as "m = 3 and c = 1" without the final equation costs A1.
⚠Common mistakes— Common errors
- Reading the gradient as run/rise (inverted).
- Negative gradient sign errors when the line falls.
- Mixing up perpendicular gradient (−1/m) with reciprocal (1/m).
AI-generated · claude-opus-4-7 · v3-edexcel-maths-leaves