Straight-Line Graphs

• The gradient m of a line measures its slope: m = (y₂−y₁)/(x₂−x₁)
• Three ways to write a line's equation: y = mx + c (gradient-intercept), y − y₁ = m(x − x₁) (point-gradient), and ax + by + d = 0 (general form)
• The midpoint of two points is just the average of their x's and the average of their y's
• The length of a line segment comes from Pythagoras: d = √[(x₁−x₂)² + (y₁−y₂)²]
• Parallel lines have the exact same gradient (m₁ = m₂)
• Perpendicular lines have gradients that multiply to −1 (m₁ × m₂ = −1) — i.e. they're negative reciprocals of each other

What "gradient" actually means

Think of the gradient as the answer to the question: "For every step I take to the right, how much does the line go up (or down)?" It's a single number that completely describes how steep and in which direction a line tilts.

A gradient of 1 means: move 1 unit right → the line rises 1 unit. A gradient of −2 means: move 1 unit right → the line falls 2 units (the negative sign tells you it's going downhill as you move right). The bigger the number (ignoring the sign), the steeper the line. A gradient of 0 means the line is perfectly flat (horizontal).

  y = mx + c — Gradient-Intercept Form

This is the most useful form because you can read the gradient and y-intercept straight off it — no rearranging needed. Here, m is the gradient, and c is the y-intercept, meaning the line crosses the y-axis at the point (0, c).

  y − y₁ = m(x − x₁) — Point-Gradient Form

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