Graphs of Polynomials

Starting point: what you already know

• The roots of a factorised polynomial: $(x-2)(x+5)=0$ gives $x=2$ or $x=-5$.
• The y-intercept is the value at $x=0$.
• $y=x^2$ is a parabola that touches the axis at 0. $y=x^3$ passes through 0 and flattens there, then keeps going.
• Factor theorem (previous chapter): if $P(a)=0$, then $(x-a)$ is a factor.

Step 1: the ends of the graph

For very large $x$, only the leading term matters. So look at the degree and the sign of the leading coefficient.

Step 2: what happens at each root

The power on a factor is called the multiplicity of that root.

• Single root $(x-a)$: the graph crosses the axis.
• Double root $(x-a{)}^2$: the graph touches the axis and turns back, like $y=x^2$.
• Triple root $(x-a{)}^3$: the graph crosses but flattens as it passes, like $y=x^3$.

Step 3: put it together

To sketch $P(x)$ from its factors:

1. Mark the roots and note each multiplicity.
2. Find the y-intercept, $P(0)$.
3. Use the leading term to start from the correct end.
4. Draw from left to right, crossing, touching or flattening at each root.

Quick summary

• The ends depend on the degree (odd or even) and the sign of the leading coefficient.
• Single roots cross, double roots touch, triple roots flatten as they cross.
• To sketch: roots with multiplicity, y-intercept, start from the correct end.
• To find the equation from a graph, write the factors with an unknown $a$ and use a known point.

Quiz