Straight-Line Graphs (y = mx + c)

The equation of a straight line

y = mx + c

• m = gradient (steepness / rate of change)
• c = y-intercept (where the line crosses the y-axis)

Finding the gradient

            change in y      y₂ − y₁
gradient =  ───────────  =  ─────────
            change in x      x₂ − x₁

Through (1, 3) and (3, 11): m = (11 − 3)/(3 − 1) = 8/2 = 4.

• Positive gradient slopes up; negative slopes down.

Finding the equation from two points

1. Work out the gradient m.

2. Substitute m and one point into y = mx + c to find c.

Example: gradient 4 through (1, 3): 3 = 4(1) + c → c = −1 → y = 4x − 1.

Parallel and perpendicular

• Parallel lines have the same gradient.
• Perpendicular gradients multiply to −1 (negative reciprocal): if m = 2, perpendicular gradient = −½.

Midpoint and length

• Midpoint = average of coordinates: ((x₁+x₂)/2, (y₁+y₂)/2)
• Length = √[(x₂−x₁)² + (y₂−y₁)²] (Pythagoras)

Exam tip

To sketch quickly: plot c on the y-axis, then step using the gradient ("up m, right 1").