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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.

Figure %: 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 %: 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.

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