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.

- 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)?*f(x) = (x - 3)/(x + 2)**h(x) = (x*^{2}- 1)/(x - 2)*g(x) = (2x*^{2}+ 8x)/(x^{2}+ 2x - 24)

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

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?

- 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. - Evaluate
*f(1000)*using a calculator or the Evaluate option in the Tools menu of Graphmatica. - How could you
have estimated the value of
*f(1000)*by looking at the graph? - The graph of the rational function
*g(x) = (2x*has vertical asymptotes at^{2}+ 8x)/(x^{2}+ 2x - 24)*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)*. - Use a
calculator or Graphmatica to calculate g(1000); was your estimate correct?
- Clear your screen and graph the rational function
*h(x) = (x*. The line^{2}- 1)/(x - 2)*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? - Read page 327
of your text. What do you think the textbook means by "
*x*approaches*0*from the left"?