Determine end behavior

As we have already learned, the behavior of a graph of a polynomial function of the form

[latex]f(x)={a}_{n}{x}^{n}+{a}_{n - 1}{x}^{n - 1}+…+{a}_{1}x+{a}_{0}[/latex]

will either ultimately rise or fall as x increases without bound and will either rise or fall as decreases without bound. This is because for very large inputs, say 100 or 1,000, the leading term dominates the size of the output. The same is true for very small inputs, say –100 or –1,000.

Recall that we call this behavior the end behavior of a function. As we pointed out when discussing quadratic equations, when the leading term of a polynomial function, [latex]{a}_{n}{x}^{n}[/latex], is an even power function, as x increases or decreases without bound, [latex]f(x)[/latex] increases without bound. When the leading term is an odd power function, as x decreases without bound, [latex]f(x)[/latex] also decreases without bound; as x increases without bound, [latex]f(x)[/latex] also increases without bound. If the leading term is negative, it will change the direction of the end behavior. The table below summarizes all four cases.

Even Degree Odd Degree
11 12
13 14