GCSE Quadratic Graphs Study Notes for Years 9/10/11

Quadratic Graphs Essentials

Quadratic equations follow the pattern y = ax² + bx + c, and they create some really distinctive graph shapes. When 'a' is positive, you get a lovely U-shaped curve with the turning point at the bottom. When 'a' is negative, flip that U upside down - you'll have an upside-down U with the turning point at the top.

The roots (or x-intercepts) show you exactly where your graph crosses the x-axis. To find them, factorise your equation first. For example, with y = x² - 2x - 8, you'd factorise to get y = x + 2$$x - 4, then set each bracket equal to zero: x = -2 and x = 4.

Finding the turning point requires completing the square. Take y = x² - 4x + 20: halve the x coefficient $-4 ÷ 2 = -2$, square it (4), then rearrange to get y = $x - 2$² + 16. Your turning point sits at (2, 16).

Quick Tip: The turning point's x-coordinate always matches the number inside the brackets (but with opposite sign), whilst the y-coordinate is the constant term at the end.