Characteristcs of rational functions

Rational functions are in the form f(x)= P(x)/Q(x) where P(x) and Q(x) are polynomial functions and Q(x) is degree 1 or higher. The simplest rational function is f(x)= 1/x.

As you can see the graph does not cross the x or y axis and so we have vertical and horizontal asymptotes. An ASYMPTOTE is a line that the graph approaches more and more closely but never touches.

When we compare graphs of the base function y=x to y=1/x there are many patterns. Whatever the x intercept of the linear function is will equal the vertical asymptote of the rational function. The section where there are positive intervals are always the same for the linear and rational function. Also if there are all increasing intervals (so the linear function is going up) than the reciprocal function will have the opposite, all decreasing. However, x will never be able to equal where the asymptote is. For example in f(x)= 1/x x cannot equal 0.