End Behavior of Polynomial Functions

A close look at polynomials shows a wide variety of interesting behavior. This Demonstration shows the opposite—the predicable, eventual behavior of a polynomial. Indeed, when the range is maximized, there seem to be only four different graphs:

Up, up: highest nonzero coefficient is even and positive.

Down, down: highest nonzero coefficient is even and negative.

Up, down: highest nonzero coefficient is odd and negative.

Down, up: highest nonzero coefficient is odd and positive.