Behavior of Graphs

The graph of a polynomial function near its zeros is dependent on the multiplicity of the zero. The same statement can be made for rational functions. Suppose that f(x) is defined by a rational expression in lowest terms. If n is the greatest positive integer such that LaTeX: \left(x-a\right)^n(xa)n is a factor of the numerator of f(x), the graph will behave in the manner illustrated.

How graphs of rational functions behave near zeros, depending upon the multiplicity of the zero.PNG