Quadratic Functions

A quadratic is any expression of the form $y = ax^2 + bx + c$ where $a \neq 0$. Its graph is a smooth curve called a parabola. Quadratics appear in almost every P1 paper, so this chapter is worth knowing cold.

The shape of a parabola

If $a > 0$ the parabola is a valley (opens upward) with a minimum point. If $a < 0$ it is a hill (opens downward) with a maximum. Every parabola is symmetric about a vertical line through its turning point — the axis of symmetry.

Three features tell you everything about a sketch:

y-intercept: put $x = 0$, so $y = c$. Here $y = -3$.
roots (x-intercepts): put $y = 0$