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Shape of a Polynomial Graph

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If you have an arbitrary polynomial P(x) of the nth degree in x, and you graph the equation y = P(x), describe the overall shape of this graph. What happens when x gets large in the positive and negative directions, and how many peaks and valleys does the graph have? How many times does the graph cross the x-axis? (To simplify, let the sign of the highest power term in P(x) be positive.)

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Solution Summary

This solution describes the shape of a graph of a polynomial of any power. The explanation of the shape includes:

1) Limits at small and large values for x.
2) Number of peaks
3) Number of valleys
4) Number of times the graph crosses the x-axis.

Solution Preview

Any polynomial has a curve. The shape of a polynomial is dependent on whether the power is even or odd.

If the polynomial has an even power:
When x gets large in each direction, the y value ...

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