Quick questions on Quadratic functions and the parabola for HSC Maths Advanced: sketching by factoring to find the x-intercepts, completing the square for the vertex form and the turning point, the quadratic formula and the discriminant for the number and nature of the roots, the axis of symmetry, and the maximum or minimum value, with worked examples and practice questions

Every quadratic graph is a parabola, and three features fix it. The concavity is set by the sign of $a$: if $a > 0$ the parabola is concave up (opens upward, like a valley, with a lowest point), and if $a < 0$ it is concave down (opens downward, like a hill, with a highest point). The size of $a$ controls how narrow or wide the curve is, but not which way it opens. The $y$-intercept is always $c$, found by putting $x = 0$.

The fastest sketch comes from the factored form. The $x$-intercepts (also called the zeros or roots) are where the curve meets the $x$-axis, so they are the values of $x$ that make $y = 0$. If the quadratic factors, set each factor to zero. For $y = x^2 - 2x - 8$, factor to $y = (x - 4)(x + 2)$, so $y = 0$ when $x = 4$ or $x = -2$; the $x$-intercepts are $(-2, 0)$ and $(4, 0)$.

Factoring is fast but limited; completing the square works on every quadratic and hands you the vertex directly. The aim is to rewrite $y = ax^2 + bx + c$ in vertex form

Quadratics are the natural model for anything that rises and then falls (or falls and then rises), and the vertex is almost always the point a worded question is hunting for. Projectile motion is the classic case: a stone thrown from a $15$ m cliff with height $h = -5t^2 + 10t + 15$ metres after $t$ seconds is a concave-down parabola, so its greatest height is the maximum value at the vertex. The axis is $t = -\dfrac{10}{2 \times (-5)} = 1$ second, and substituting gives $h = 20$ m, so the stone peaks at $20$ m after one second; setting $h = 0$ and solving $t^2 - 2t - 3 = 0$ gives $t = 3$, the time it hits the sea. The coefficient $-5$ comes from gravity, the $+10t$ from the launch speed, and the $+15$ from the starting height, and each maps onto a feature of the curve.

Each parabola opens upward, but they sit at different heights relative to the $x$-axis. With $\Delta > 0$ the curve dips below the axis and crosses it twice; with $\Delta = 0$ the vertex sits exactly on the axis, giving one repeated root; with $\Delta < 0$ the whole curve stays above the axis, so there are no real roots.