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 power is even with a positive coefficient.

Down, down: highest nonzero power is even with a negative coefficient.

Up, down: highest nonzero power is odd with a negative coefficient.

Down, up: highest nonzero power is odd with a positive coefficient.