A rational function is a function of one variable that can be represented as f(x)= p(x)/q(x). Rational functions are also called “rational polynomials”. In other words, a rational function has a numerator and a denominator with degrees of at least 1.

Graphs of Rational Functions

Graphs of rational functions often have strange features, like asymptotes and holes. These are discontinuities, points that must be excluded from the domain of the function.

Horizontal Asymptote

The horizontal asymptote is a line that the graph of the function approaches but never touches. This can happen in a parent function with an x -value y that is -3 or any other x -value that makes the value of y undefined.

Slant Asymptote – Irrational Function

A slant asymptote occurs when the degree of the numerator is equal to the degree of the denominator. This is because the graph of the parent function f(x) will get closer and closer to but never touch the slant asymptote, a line that is at a y=3x.

Vertical Asymptote – Irrational Exponents

A vertical asymptote is a straight line that the graph of the function approaches but does not touch. This can happen in a parent function that has an x -value y that increases or decreases without bound.

In this case, the parent function f(x) increases without limit as x and its graph approaches a y=3x line. The graph of this function has no horizontal asymptote, but its slant asymptote will have a line that is at a x=3x.

Graphs of rational functions may have many vertical asymptotes, but they have at most one horizontal asymptote. This is because a graph of a rational function will behave similar to a graph of g(x) as its inputs grow large and will not level off.

It is possible to find all of the asymptotes in a parent function by dividing the function and ignoring its remainder. This method will only work if the function has degrees of the numerator and denominator that are sufficiently high. This method will give you a set of asymptotes that you can use to figure out where to draw the graphs of the parent function. In addition, it will help you determine what kind of slant asymptote a graph will have.

Amy Shelton