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
x→a, f→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.
Figure 3.1: The leading coefficient test can be used to see how a polynomial
function behaves as x increases and decreases without bound.
Here is a chart that outlines the steps and possibilities of the leading
coefficient test.
Figure 3.2: The leading coefficient test, in chart form.
If the leading coefficient test gets confusing, just think of the graphs of
y = x2 and
y = - x2, as well as
y = x3 and
y = - x3. 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.
