Algebra and Functions
This domain examines how well you move between the symbolic, graphical and tabular forms of real functions - above all the second-degree (quadratic) function - and how you solve the equations, inequalities and systems that model a situation.
Forms of the quadratic function
- General form: f(x) = ax² + bx + c.
- Canonical (vertex) form: f(x) = a(x − h)² + k, with vertex (h, k) and axis of symmetry x = h.
- Factored (root) form: f(x) = a(x − x₁)(x − x₂). The parameter a fixes the direction of opening and the stretch; the discriminant Δ = b² − 4ac gives the number of real zeros (2, 1 or 0). Move between forms by factoring, by completing the square, or by using x = −b/(2a) for the vertex's x-coordinate.
Equations, inequalities and systems
Factor before dividing: x² = 3x gives x = 0 or x = 3, never x = 3 alone. For inequalities, build a sign table from the zeros and the sign of a, or isolate the variable - if you multiply or divide by a negative number, reverse the symbol. Systems (linear–quadratic, quadratic–quadratic) are solved by substitution, and the number of solutions equals the number of intersection points. Always state restrictions: denominator ≠ 0, radicand ≥ 0 for even roots.
Worked example
A ball is thrown upward; its height in metres after t seconds is h(t) = −5t² + 20t + 1. When is it at least 16 m high? Vertex: t = −b/(2a) = −20/(−10) = 2 s, and h(2) = −5(4) + 40 + 1 = 21 m (the maximum). Inequality: −5t² + 20t + 1 ≥ 16 → −5t² + 20t − 15 ≥ 0. Divide by −5 and reverse: t² − 4t + 3 ≤ 0 → (t − 1)(t − 3) ≤ 0. The parabola opens upward, so it is negative between its zeros: 1 ≤ t ≤ 3. Check: h(1) = −5 + 20 + 1 = 16 ✓. The ball is at least 16 m high from 1 s to 3 s.
Traps that cost marks
- Dropping the solution x = 0 when dividing both sides by x.
- Forgetting to reverse the inequality sign, or giving two isolated values instead of an interval.
- Confusing the zeros with the vertex - reporting t = 2 as the maximum height instead of 21 m.
- Ignoring restrictions, or solving a system without checking the pair in both original equations.