As the degree of a polynomial increases, it becomes increasingly hard to
sketch it accurately and analyze it completely. There are a few things we can
do, though.

Using the Leading Coefficient Test, it is possible to predict the end behavior
of a polynomial function of any degree. Every polynomial function either
approaches infinity or negative infinity as x increases and decreases without
bound. Which way the function goes as x increases and decreases without bound
is called its end behavior. End behavior is symbolized this way: as xâÜ’a, fâÜ’b; "As x approaches a, f of x approaches
b."

If the degree of the polynomial function is even, the function behaves the same
way at both ends (as x increases, and as x decreases). If the leading
coefficient is positive, the function increases as x increases and
decreases. If the leading coefficient is negative, the function decreases as
x increases and decreases.

If the degree of the polynomial function is odd, the function behaves
differently at each end (as x increases, and as x decreases). If the
leading coefficient is positive, the function increases as x increases, and
decreases as x decreases. If the leading coefficient is negative, the
function decreases as x increases and increases as x decreases. The figure
below should make this all clearer.

Here is a chart that outlines the steps and possibilities of the leading
coefficient test.
If the leading coefficient test gets confusing, just think of the graphs of y = x^{2} and y = - x^{2}, as well as y = x^{3} and y = - x^{3}. The behavior of these
graphs, which hopefully by now you can picture in your head, can be used as a
guide for the behavior of all higher polynomial functions.

Besides predicting the end behavior of a
function, it is possible to sketch a function,
provided that you know its roots. By evaluating the function at a test point
between roots, you can find out whether the function is positive or negative for
that interval. Doing this for every interval between roots will result in a
rough, but in many ways accurate, sketch of a function.