(after Test 2 on Thursday)
Topics: Rational Functions and their graphs
• Domain: the domain of a rational function will be all real numbers except the x-values which make the denominator 0.
• Vertical asymptotes: these occur where the denominator is 0 and the numerator is not 0 (and see also the second box on the handout).
• Holes in the graph (also called removable singularities): these occur where both the numerator and denominator are 0 to the same multiplicity (and see also the second box on the handout).
• Horizontal asymptotes: these represent the “end behavior” of the graph, so they depend only on the leading terms of the numerator and denominator. See the discussion of Example 11.2(a-d) for details.
• y-intercept: if 0 is in the domain, the y-intercept is f(0)
• x-intercept(s): these are where f(x) = 0, which means that the numerator of the rational function is 0 (and the denominator is not 0).
Before looking at the graph on your calculator, determine all of the above algebraically, and then consider what viewing window will be appropriate. Be careful in interpreting what you see on your calculator display! The graphing calculator will sometimes connect parts of the graph which are actually separated by vertical asymptotes. This is another reason that you need to know what you expect the graph to look like before you ever put it into your graphing calculator!
Note: there is a corrected and improved version of the handout on Graphing Rational Functions posted over on Piazza.
Homework:
• In Tuesday’s class I told you if you still need to put an email address in WeBWorK. Please take care of this today if you have not done it yet!
• Review the discussion of Example 11.2(a-d) as it pertains to the vertical and horizontal asymptotes. You will probably also want to study the Examples 11.5.
• Do the assigned problems from Session 11
• Do the WeBWorK: due by Monday evening 11 PM. Start early!
• Do the Warm-Up: also due by Monday 11 PM.
