How is the end behavior of a polynomial graph determined?

The end behavior of a polynomial graph is determined by the degree and leading coefficient of the polynomial. The degree of the polynomial indicates how the graph behaves as ( x ) approaches positive or negative infinity. Specifically, if the polynomial has an even degree, the ends of the graph will either both rise or both fall; if it has an odd degree, one end will rise while the other falls.

Additionally, the leading coefficient affects the direction of the ends. If the leading coefficient is positive, the ends will behave in an upward direction, while a negative leading coefficient indicates that the ends will behave downward. This combination of the degree and leading coefficient helps to predict the overall shape and direction of the polynomial graph as far out on the x-axis as one can analyze.

The other factors listed, such as the last term of the polynomial, the number of x-intercepts, and the y-intercept value, contribute information about the polynomial but do not directly influence how the graph behaves at the extremes. Therefore, understanding the degree and leading coefficient is crucial for assessing the end behavior of any polynomial function.