Zeroes on a graph

The big idea: the zeroes of p(x) are the x-coordinates of the points where the graph of y = p(x) meets the x-axis. On the x-axis, y = 0, so p(x) = 0 there.

Linear: y = 2x + 3 is a straight line. It crosses the x-axis once, at (–3/2, 0). One line, one zero.

Quadratic: the graph of y = ax² + bx + c is a curve called a parabola. It opens upwards (like a cup) when a > 0, and downwards (like a cap) when a < 0. It can meet the x-axis in three ways:

So a quadratic has at most two zeroes, and sometimes none. Not 'always two'.

Cubic: y = x³ – 4x meets the x-axis at –2, 0 and 2: three zeroes. But y = x³ meets it only at 0, and y = x³ – x² only at 0 and 1. A cubic has at most three zeroes.

The rule: a polynomial of degree n has at most n zeroes. And turning it round: if a graph meets the x-axis at 3 points, its degree is at least 3.

How to read a graph: count the points where the curve cuts or touches the x-axis. Don't count where it crosses the y-axis.

You will not be asked to draw these curves. The book says plotting quadratic and cubic graphs is not for you to do, only to read.