Mathematics (9709)
Topic 2 of 4Cambridge A Levels

Quadratics & Functions

Completing the square, discriminant, domain and range, inverse functions

Quadratic equations (ax² + bx + c = 0):

  • Factorising: Find two numbers that multiply to ac and add to b
  • Completing the square: x² + bx = (x + b/2)² − (b/2)²
  • Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a

    • The discriminant (b² − 4ac) determines the number of roots:

  • b² − 4ac > 0 → two distinct real roots
  • b² − 4ac = 0 → one repeated root (touches x-axis)
  • b² − 4ac < 0 → no real roots

    • Functions: f(x) maps each input to exactly one output. **Domain** = set of valid inputs. **Range** = set of possible outputs.


    Composite functions: fg(x) means apply g first, then f. **Inverse function:** f⁻¹(x) reverses f. To find: swap x and y, solve for y. The graph of f⁻¹ is a reflection of f in the line y = x.


    Modulus function: |f(x)| reflects all negative parts above the x-axis.

    Key Points to Remember

    • 1Discriminant b²−4ac determines number of roots
    • 2Completing the square for vertex form
    • 3Composite: fg(x) = f(g(x)), apply g first
    • 4Inverse: swap x and y, reflect in y = x

    Pakistan Example

    Projectile Height — Quadratics in Pakistani Cricket

    The height of a cricket ball hit for a six follows h = −5t² + 30t + 1.5. Using the discriminant to find when h = 0 (ball lands): −5t² + 30t + 1.5 = 0. Completing the square or using the formula gives t ≈ 6.05 seconds. The vertex (maximum height) is at t = 3 seconds, h = 46.5m.

    Test Your Knowledge!

    3 questions to check if you understood this topic.

    Start Quiz
    Quadratics & FunctionsDifferentiation