Quadratics and Parabolas

Starting point: Years 9 and 10

A quadratic has the form $y=a{x}^2+bx+c$ with $a\ne 0$. Its graph is a parabola.

• If $a>0$, the parabola opens upward (a smile) and has a minimum.
• If $a<0$, it opens downward (a frown) and has a maximum.

Solving quadratic equations

To solve $a{x}^2+bx+c=0$, get zero on one side first. Then use one of three methods.

1. Factorise. If two things multiply to zero, one of them must be zero.

$$x^2-2x-3=0\Rightarrow (x-3)(x+1)=0\Rightarrow x=3\text{ or }x=-1$$

2. Quadratic formula. It always works.

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

3. Completing the square. Halve the middle number, square it, add and subtract it.

$$x^2+6x+2=(x+3{)}^2-9+2=(x+3{)}^2-7$$

Sketching a parabola

Find these four things:

1. Shape: up or down from the sign of $a$.
2. y-intercept: put $x=0$, so it is $c$.
3. x-intercepts: solve $y=0$.
4. Vertex: the turning point. Its x-value is $x=-\frac{b}{2a}$, which is also the axis of symmetry.

Quick summary

• A quadratic $y=a{x}^2+bx+c$ graphs as a parabola. The sign of $a$ decides up or down.
• Solve by factorising, the formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$, or completing the square.
• The vertex is at $x=-\frac{b}{2a}$, halfway between the roots.
• The discriminant $\Delta =b^2-4ac$ counts the real roots.

Quiz