MATH100: Graphs of Rational Functions

This worksheet will not be graded. The purpose of this exercise is to explore how the equations of rational functions are related to the shapes of their graphs; in particular, this exercise includes several examples of asymptotes.
Zeros and Vertical Asymptotes

In this exercise you will graph several functions and observe their zeros and domains. Find a partner to work with. Either you or your partner should have a laptop with Graphmatica installed.

  1. For each of the rational functions listed below answer the following questions:
    a) Factor the numerator (if necessary); when is it equal to 0?
    b) Factor the denominator (if necessary); when is it equal to 0?
    c) What are the zeros (x-intercepts) of the function?
    d) What is the domain of the function (what inputs avoid division by zero)?
  2. The rational function f(x) = (x - 3)/(x + 2) is undefined at x = -2. Use Graphmatica to graph f(x) and the line x = -2 and observe the relationship between these two graphs (you may wish to "zoom out" once or twice). We call the line x = -2 a vertical asymptote of the graph of f(x). In your own words, describe what the graph of f(x)looks like near this vertical asymptote.





  3. Why do you think the output of f(x) gets larger when the value of x is close to the value -2?




  4. Use your answers to questions (b) and (d) above to predict what the vertical asymptotes of the functions h(x) and g(x) will be. Use Graphmatica to check your work.





Vertical, Horizontal and Slant Asymptotes

In the last exercise you investigated the "local" phenomenon of the location of the zeros and vertical asymptotes of a rational function. In this exercise you study "global" behavior -- how does the function behave for very large inputs?

  1. Graph the rational function f(x) = x/(x - 1). You know that the line x = 1 will be a vertical asymptote of this graph. The line y = 1 is a horizontal asymptote of the graph. Graph f(x) together with the line y = 1 (zoom out if necessary). In your own words, describe how the two graphs are related.





  2. Evaluate f(1000) using a calculator or the Evaluate option in the Tools menu of Graphmatica.



  3. How could you have estimated the value of f(1000) by looking at the graph?





  4. The graph of the rational function g(x) = (2x2 + 8x)/(x2 + 2x - 24) has vertical asymptotes at x = -6 and x = 4 and a horizontal asymptote at y = 2. Use this information and, if you like, the graph of g(x) to estimate g(1000).


  5. Use a calculator or Graphmatica to calculate g(1000); was your estimate correct?


  6. Clear your screen and graph the rational function h(x) = (x2 - 1)/(x - 2). The line x = 2 is a vertical asymptote of this graph. The line y = x + 2 is a slant asymptote. Graph the function together with its asymptotes. How is a slant asymptote similar to other asymptotes? How is it different?







  7. Read page 327 of your text. What do you think the textbook means by "x approaches 0 from the left"?