Curve Sketching

When sketching curves of a polynomial function

1. try to factorise it first, if it is not already factorised.Writing a polynomial as a product of factors --for example - makes it easier to identify the roots – the pointswhereTheroots will be points on the x – axis, since each root is asolution of the equation

2. Find the–intercept by substitutinginto

3. Decide whether the curve tends toorasx tends toorIfthe coefficient of the highest power ofispositive when the expression is expanded, then as tendstosodoesandif the coefficient is negative, then astendsto tendsto

4. Each distinct root --with no power - will give rise to a point on the–axis where the curve CROSSES the–axis, and each repeated root - given by a factor– will give rise to a point on the curve which touches the–axis but does not cross it ifiseven, or which forms a tangent to the–axis and crosses it ifisodd. Examples are shown below.

For example, to sketch

The roots are the solutions to

These are

Substitutingintothe expression gives

The highest power ofisandthe coefficient ofis2 (considersoas

Each root is distinct, so the graph crosses the–axis at each root. The graph is sketch below.

To sketch

The roots are given by

Substitutingintothe expression gives

The highest power ofisandthe coefficient ofis-1 (considersoas

The root atisa double root, so the curve touches theaxisatbutdoes not cross it, and the root atisa single root so the graph crosses the–axis there.

The curve is sketched below.