Quadratic Functions and Equations
Quadratic Functions and Equations
This domain covers solving quadratic equations by factoring, completing the square, and the quadratic formula; interpreting the discriminant; and analyzing parabolas - vertex, axis of symmetry, intercepts, and maximum or minimum values in applied settings such as projectile height or area optimization.
Core rules
- Standard form y = ax² + bx + c with a ≠ 0; vertex form y = a(x − h)² + k with vertex (h, k).
- Axis of symmetry x = −b/(2a); the vertex's y-value is f(−b/(2a)). The parabola opens up when a > 0 and down when a < 0.
- Quadratic formula: x = (−b ± √(b² − 4ac))/(2a).
- Discriminant b² − 4ac: positive gives two distinct real solutions; zero gives one repeated real solution; negative gives no real solutions.
- Zero product property: if (x − r)(x − s) = 0, then x = r or x = s.
- Completing the square: factor a from the variable terms, then add and subtract (b/2a)² inside the parentheses and rebalance.
Worked example
Rewrite f(x) = 2x² − 8x + 5 in vertex form and state the vertex.
Factor 2 from the variable terms: 2(x² − 4x) + 5. Half of −4 is −2, and squaring gives 4. Add and subtract inside: 2(x² − 4x + 4 − 4) + 5 = 2(x − 2)² − 8 + 5 = 2(x − 2)² − 3. The vertex is (2, −3), the axis of symmetry is x = 2, and −3 is the minimum value.
Traps that cost marks
- Dropping the ± or the denominator 2a in the quadratic formula.
- Sign errors when b is negative, since −b becomes positive.
- Treating standard form as if it already revealed the vertex.
- Reporting the vertex's x-coordinate as the maximum or minimum value.
- Splitting an added radicand: √(a² + b²) is not a + b.