Quadratic functions — A/L Combined Mathematics

What must I remember about Quadratic functions?

• A quadratic function has the standard form f(x) = ax² + bx + c with a ≠ 0; its graph is a parabola opening upwards if a > 0 (minimum vertex) or downwards if a < 0 (maximum vertex).
• The roots of ax² + bx + c = 0 are given by x = [-b ± √(b² - 4ac)] / (2a); nature of roots is determined by discriminant Δ = b² - 4ac (Δ > 0: two distinct real roots; Δ = 0: two equal real roots; Δ < 0: complex conjugate roots / no real roots).
• Relations between roots and coefficients: If α and β are the roots, then α + β = -b/a and αβ = c/a; quadratic equation with roots α, β is x² - (α + β)x + αβ = 0.
• Symmetric functions of roots: (α - β)² = (α + β)² - 4αβ; α² + β² = (α + β)² - 2αβ; α³ + β³ = (α + β)³ - 3αβ(α + β).
• Condition for a quadratic expression to be strictly positive: ax² + bx + c > 0 for all real x if and only if a > 0 and Δ < 0; strictly negative (ax² + bx + c < 0 for all x) if and only if a < 0 and Δ < 0.

Test yourself (1 minute)

1. Vertex coordinates

   Completing the square f(x) = a(x + b/(2a))² + (4ac - b²)/(4a) gives the vertex at (-b/(2a), -(b² - 4ac)/(4a)); line of symmetry is x = -b/(2a).

2. Common root condition

   For two quadratics a1*x² + b1*x + c1 = 0 and a2*x² + b2*x + c2 = 0 to have a common root α, eliminate x² and constant terms using cross-multiplication: (c1*a2 - c2*a1)² = (a1*b2 - a2*b1)(b1*c2 - b2*c1).

3. Sign of roots

   Both roots positive if Δ ≥ 0, α + β > 0, and αβ > 0; both roots negative if Δ ≥ 0, α + β < 0, and αβ > 0; opposite signs if αβ < 0 (no need to check Δ as ac < 0 ensures Δ > 0).

Practise Quadratic functions questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 3

What comes next?

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Where does Quadratic functions fit in the syllabus?

Year 1 · Term 1