Revise Quadratic functions and equations for IB Mathematics: Analysis and Approaches (2.3)

A quadratic function is any function that can be written with $x^2$ as its highest power. It always graphs as a parabola. What makes quadratics so workable is that the very same function can be written three ways, and each form is optimised to reveal a different feature of the graph.

$\text{Standard: } y=ax^2+bx+c \qquad \text{Factored: } y=a(x-p)(x-q) \qquad \text{Vertex: } y=a(x-h)^2+k$

The skill the exam tests is choosing the right form for the question and converting when needed. Multiplying out factored or vertex form returns you to standard form; factorising or completing the square takes you the other way. None of the three is 'more correct' — they are three views of one parabola.

The graph of a quadratic is a parabola, symmetric about a vertical line called the axis of symmetry. The point where the curve turns — its lowest point if $a>0$, its highest if $a<0$ — is the vertex, and it lies on the axis of symmetry. The y-intercept is always $(0,c)$; the x-intercepts, if any, are the roots of $ax^2+bx+c=0$.

$\text{Axis of symmetry: } x=-\frac{b}{2a} \qquad \text{Vertex: } \left(-\frac{b}{2a},\; f\!\left(-\frac{b}{2a}\right)\right)$

Solving $ax^2+bx+c=0$ means finding the x-values that make the function zero — the roots, which are the x-intercepts. Factorising rewrites the quadratic as a product of two brackets; then the zero-product principle says that if a product is zero, at least one factor must be zero. It is the fastest method in Paper 1 whenever the numbers are friendly.

Completing the square rewrites a quadratic in vertex form $a(x-h)^2+k$. It does two jobs at once: it solves the equation (rearrange and take square roots) and it hands you the vertex $(h,k)$. For a monic quadratic, halve the coefficient of $x$, square it, and balance.

$x^2+bx+c=\left(x+\frac{b}{2}\right)^2-\frac{b^2}{4}+c$

When factorising is not obvious or the answer is an awkward surd, the quadratic formula solves any quadratic. It is in your formula booklet, so the marks are in the careful, correct substitution — especially with the signs.

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

The expression under the square root in the formula, $\Delta=b^2-4ac$, is the discriminant. Its sign alone tells you how many real roots the equation has and how the parabola meets the x-axis — all without solving anything. This is why so many exam questions are phrased in terms of the number of roots rather than the roots themselves.

Paper 1 is marked analytically: each mark is tied to a specific line of working. An M mark rewards a correct method — the right approach, even if the arithmetic later slips. An A mark rewards accuracy and is DEPENDENT on the method: you cannot earn the A without the M it hangs off. The engine also applies ISW (ignore subsequent working — once a correct answer appears, later tidying does not lose it), FT/ECF (follow-through — a correct step built on an earlier wrong value still scores), and it accepts any correct exact or equivalent form. Study how each mark below is earned by one specific line.

Quadratics are a template for the rest of the functions unit. Completing the square reappears as the standard way to handle the vertex of any parabola and, later, to derive results by transformation; the discriminant returns whenever a question asks how many times two graphs meet or when a line is tangent to a curve. The three-forms habit — read the question, pick the form that reveals the wanted feature, convert if needed — carries directly into cubics, rational functions and the graph-sketching that dominates Paper 1. Master it here and much of what follows becomes variation on a method you already own.