Video Examples

The video examples below will take you through how to approach sketching several kinds of rational functions. You'll find a lot of variety in rational functions, and sometimes even small changes in a function can dramatically change its graph.

Steps for sketching the graphs of rational functions

  1. Find the roots (if any) by finding the zeros of the numerator
  2. Find any vertical asymptote(s) (VA) by finding the zeros of the denominator
  3. Factor both numerator and denominator to check for holes. These don't show up on your calculator graph!
  4. Find the y-intercept, if it exists, by calculating f(0)
  5. Find the horizontal asymptote(s) (HA) by exploring the behavior of the function as x → ∞.
    1. degree of numerator > degree of denominator → slanted or curved asymptote
    2. degree of numerator = degree of denominator → HA at ratio of leading coefficients
    3. degree of numerator < degree of denominator → HA at y = 0
  6. Find f(x) at selected points to help make decisions about tricky parts of the graph.

The functions and graphs below get progressively more complicated as you work from top to bottom. Good luck!

Being able to "sketch" a graph means having some rough picture, either in your head or on a scrap of paper, of what the graph ought to look like when drawn more accurately by a computer. Of course we want to use computers to draw graphs of functions, but we also want to be able to recognize when a computer-drawn graph is inaccurate. We do that by developing a sense of what kinds of features the graph of a function can have.

Example 1

This function has a horizontal asymptote, but no vertical asymptote(s), and it has no real roots. You might think that would make it difficult to graph, but remember that knowing that something doesn't exist (like roots) is information, too.


Example 2

This function graph has two vertical asymptotes and a horizontal asymptote at y = 0. Notice that the graph crosses the horizontal asymptote as x = 0. That's OK. Remember that the question we ask when finding horizontal asymptotic behavior is "what does the function do when x gets very large, either in the negative or positive directions." We don't ask about what happens in between, and in fact a graph can cross that/those lines in between x = ±∞.


Example 3

This function also has two vertical asymptotes, but its behavior in between is a little different than the last example. The graph is totally consistent with the information we calculate and infer from the function.


Example 4

This graph has a hole at x = 3 because the binomial (x - 3) occurs both in the numerator and the denominator. That means when x → 3, the both of these binomials go to zero at the same rate. They divide to one (and thus have no effect on the graph) for all values of x except x = 3, and there the function is infinite. We say that there's a "hole" at x = 3; the function just has no value there. Later we'll refer to that kind of discontinuity in the graph as a "removable discontinuity."


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