Quadratic graphs · Year 12

In Year 12 you learn: solving equations with graphs.

The graph of a quadratic y = ax² + bx + c is a smooth curve called a parabola: ∪-shaped when a is positive and ∩-shaped when a is negative. Its key features — roots, y-intercept and turning point — can all be found from the equation.

Key facts

  • y-intercept: put x = 0, so it is the point (0, c).
  • Roots (x-intercepts): solve ax² + bx + c = 0.
  • The line of symmetry is x = −b/(2a), halfway between the roots.
  • The turning point lies on the line of symmetry; complete the square to find it.
  • a > 0 gives a ∪ shape with a minimum; a < 0 gives a ∩ shape with a maximum.
  • The discriminant b² − 4ac tells you whether the graph meets the x-axis twice, once or not at all.

Worked examples

Example 1

Sketch y = x² − 2x − 3, showing the roots, the y-intercept and the turning point.

Answer: Roots −1 and 3; y-intercept −3; turning point (1, −4)

Example 2

Find the turning point of y = −x² + 4x + 1.

Answer: (2, 5), a maximum

Example 3

A parabola has roots x = −1 and x = 4 and passes through (0, −8). Find its equation.

Answer: y = 2x² − 6x − 8

Other years