Section 2.6 Rational Functions and their Graphs

Definition A rational function is in the form where P(x) and Q(x) are polynomials and Q(x) is not equal to 0.

Graphing a Rational Function Graphing a rational function involves several steps. Before we look more closely at those steps, here is a very important definition: An asymptote is an imaginary line that a rational function usually does not cross. Asymptotes may be horizontal, vertical, or oblique (slanted). They are represented on the graph by dotted lines.

Four Examples We will analyze all four functions, but only graph the first one.

Step 1 Find the vertical asymptote(s) by setting the denominator equal to 0. If the denominator is linear (first degree) isolate the variable. If the denominator is quadratic (second degree), solve by factoring. If your solutions are imaginary, the function has no vertical asymptote (this is rare).

Step 2 To determine whether or not the function has a horizontal asymptote, compare the degree of the numerator (deg num) to the degree of the denominator (deg denom). If deg num = deg denom, your horizontal asymptote is y = the ratio of the lead coefficients. If deg num < deg denom, your horizontal asymptote is y = 0. If deg num > deg denom, you have an oblique (slanted) asymptote, which must be found by long or synthetic division.

Steps 3 and 4 Find the x-intercept(s) by setting the numerator equal to 0. Find the y-intercept by finding f(0).

Step 5 Find additional points by creating a table of values. Choose values of x that are to the left of and to the right of (and where appropriate, in between) vertical asymptote(s). Ideally, you will have at least three points on either side of your vertical asymptote(s).

The Difference Between My 5 Steps and the Book’s 7 Steps My steps are in a different order. I don’t care about symmetry as it relates to graphing rational functions. Their 7 th step is connecting the dots.

Limit Notation Limit notation is to rational functions what end behavior is to polynomials functions. We consider four different types of limits:

An Example

Vertical Asymptotes and Holes Recall that, because a rational function has a denominator, we must consider what values need to be omitted from the domain of the function (we do this by setting the denominator equal to zero). Typically, these values indicate the location of vertical asymptotes. However, sometimes an omitted value indicates the location of a “hole” in the function. This happens when the original function can be reduced (by factoring and cancelling).

Examples

Oblique Asymptotes Re-visited We said earlier that a rational function does not have a horizontal asymptote if the degree of the numerator is greater than the degree of the denominator. More specifically, a rational function has an oblique asymptote if the degree of the numerator is 1 more than the degree of the denominator. Use synthetic (or long) division to find the equation of the oblique asymptote.

Examples