Curve Sketching
When sketching curves of a polynomial function
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
Find the
–intercept by substituting
into
Decide whether the curve tends to
or
asx tends toor
Ifthe coefficient of the highest power of
ispositive when the expression is expanded, then as tendsto
sodoes
andif the coefficient is negative, then astendsto 
tendsto
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 ifisodd. Examples are shown below.
For example, to sketch
The roots are the solutions to
These are
Substituting
intothe expression gives
The highest power ofis
andthe coefficient ofis2 (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
Substitutingintothe expression gives
The highest power ofisandthe coefficient ofis-1 (consider
so
as
The root at
isa double root, so the curve touches theaxisatbutdoes not cross it, and the root at
isa single root so the graph crosses the–axis there.
The curve is sketched below.
