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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