Quadratic graphs

The graph of a quadratic y = ax² + bx + c is a parabola: ∪-shaped if a is positive, ∩-shaped if it’s negative. Its key features are where it crosses the axes and its turning point.

Key facts

  • y-intercept: put x = 0, so it’s c.
  • Roots (x-intercepts): solve ax² + bx + c = 0.
  • The line of symmetry is halfway between the roots: x = −b/(2a).
  • The turning point lies on the line of symmetry.

Worked examples

Example 1

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

  1. (x + 1)(x − 3) = 0 gives x = −1 or x = 3.
  2. y-intercept: y = −3.
  3. Line of symmetry halfway between −1 and 3: x = 1.
  4. Turning point: y = 1 − 2 − 3 = −4. (1, −4)

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

Example 2

How many roots does y = x² + 2x + 5 have?

  1. b² − 4ac = 4 − 20 = −16
  2. Negative, so the parabola doesn’t reach the x-axis: no real roots.

Answer: None