A straight line can be written as y = mx + c. Those two letters, m and c, are like the DNA of the line — they tell you everything about it, so once you spot them you can picture the line in your head or draw it on a grid without plotting dozens of points.
Why this matters: straight-line graphs turn up whenever something changes at a steady rate — a phone tariff with a fixed fee plus a cost per minute, distance covered at constant speed, the fuel left in a tank as you drive. Reading m and c off the equation lets you compare deals, predict values and understand the story a graph is telling at a glance.
m = the gradient (how steep the line is). c = the y-intercept (where the line crosses the y-axis, i.e. the value of y when x = 0).
In y = 4x − 3: m = 4 so it rises steeply, and c = −3 so it crosses the y-axis at (0, −3). In a tariff, c would be the fixed starting charge and m the cost added per unit.
Gradient — "rise over run"
The gradient measures steepness: how far the line goes up for every step across — just like the slope of a hill or a ramp. A big gradient is a steep climb; a gradient of 0 is flat.
gradient = (change in y) ÷ (change in x)
Through (1, 2) and (4, 11): y changes by 11 − 2 = 9, x changes by 4 − 1 = 3, so gradient = 9 ÷ 3 = 3.
| Equation | Gradient m | y-intercept c | Crosses y-axis at |
|---|---|---|---|
| y = 2x + 5 | 2 | 5 | (0, 5) |
| y = 4x − 3 | 4 | −3 | (0, −3) |
| y = −x + 4 | −1 | 4 | (0, 4) |
A positive gradient climbs uphill left-to-right; a negative gradient slopes downhill. Spotting the sign alone tells you which way the line tilts.
Parallel lines have the same gradient — they're equally steep, so they run alongside each other and never meet. y = 2x + 1 and y = 2x − 4 are parallel: same m, different c.
Common mistake: read the gradient as the number attached to x, not just any number in the equation. In y = 5 − 3x the gradient is −3 (the coefficient of x), and c = 5 — don't be caught out when the terms are written in a different order.
In a nutshell: y = mx + c — m is steepness (rise ÷ run, with a sign for its direction), c is where it crosses the y-axis. Same gradient means parallel lines.