## 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 pointswhere Theroots will be points on the x – axis, since each root is asolution of the equation 2. Find the –intercept by substituting into 3. Decide whether the curve tends to or asx tends to or Ifthe coefficient of the highest power of ispositive when the expression is expanded, then as tendsto sodoes andif the coefficient is negative, then as tendsto  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 if iseven, or which forms a tangent to the –axis and crosses it if isodd. Examples are shown below. For example, to sketch The roots are the solutions to These are   Substituting intothe expression gives The highest power of is andthe coefficient of is2 (consider so as 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  Substituting intothe expression gives The highest power of is andthe coefficient of is-1 (consider so as The root at isa double root, so the curve touches the axisat butdoes not cross it, and the root at isa single root so the graph crosses the –axis there.

The curve is sketched below.  