Problem 58 - College Algebra

Question

What is meant by the end behavior of a polynomial function?

Step-by-Step Solution

Answer

The end behavior of a polynomial function refers to the direction the function heads, as the function approaches negative and positive infinity. This behavior is determined by the degree and the leading coefficient of the polynomial.

Step 1: Definition of End Behavior

The end behavior of a polynomial is how the function behaves (values of y) as the function approaches negative infinity \( x \rightarrow -\infty \) and as the function approaches positive infinity \( x \rightarrow +\infty \). It basically tells us what happens to the values of y (the function's output) as the inputs (x-values) get very large in either the positive or negative direction.

Step 2: Determining the End Behavior based on Degree and Leading Coefficient

The degree and the leading coefficient of a polynomial function determine its end behavior. If the degree of the polynomial is even, then the end behavior is the same in both directions (either up or down). If the degree is odd, the function will be down on one end and up on the other. If the leading coefficient is positive, the right end of the graph will point upwards. If the leading coefficient is negative, the right end of the graph will point downwards.