How does the degree of a polynomial affect its end behavior?

To understand the behaviour of a polynomial graphically all one one needs is the degree (order) and leading coefficient. This two components predict what polynomial does graphically as gets larger or smaller indefinitely. This called "end behavior". For example it easy to predict what a polynomial with even degree and +ve leading coefficient will do. All even polynomial have both ends of the graph moving in the same direction with direction dictated by the sign of leading coefficient. On the other hand odd power will get have the polynomial end points moving in opposite direction, with the leading coefficient dictating the direction.

The degree of a polynomial determines the behavior of the polynomial function as the input values approach positive or negative infinity. For polynomials of odd degree, the end behavior is such that as ( x ) approaches positive infinity, the function approaches positive infinity, and as ( x ) approaches negative infinity, the function approaches negative infinity. Conversely, for polynomials of even degree, the end behavior is such that as ( x ) approaches positive infinity, the function approaches positive infinity, and as ( x ) approaches negative infinity, the function approaches positive infinity. Additionally, the leading coefficient of the polynomial also influences the end behavior, determining whether the function increases or decreases without bound as ( x ) approaches positive or negative infinity.