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*âÜ’*a*, *f*âÜ’*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 %: The leading coefficient test can be used to see how a polynomial
function behaves as
*x*
increases and decreases without bound.

Figure %: The leading coefficient test, in chart form.

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.

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