Maple T.A. High School Enrichment Question Repository

Reviewer: Paul Kates Some useful online plotting tools for checking questions:

Path to reviewed questions: 

  High School Enrichment
   > Advanced Functions
     > 4. Rational Functions
       > Group: 4.2 Vertical Asymptotes and Discontinuity

Group 4.1. Q.01 Vertical Asymptote

Comments:

  1. The first question asks to find the vertical asymptote of the given rational functions. E.g. 

      Determine the value of x for the vertical asymptote for the given function:
  (2*x + 2)/(3*x + 3).

    The question is marking incorrect answers as correct because it is checking only the function denominator for zeros.

    The answer marked correct was -1. But, there is no vertical asymptote for this function since both numerator and denominator are 0 at x = -1. And, +/- limits approaching -1 are 2/3.

    See this image:

Similarly with other functions:

  1. (2*x-2)/(x-1) at point x=1.
  2. (4*x-4)/(x-1) at point x=1.

The answer text of the third question State the vertical asympote(s) (see below) explains about the lack of an asymptote.

Example Images:

Group 4.1. Q.02. Points of Discontinuity

This question gives a rational function and its vertical asymptote but unlike question 1 above, it avoids generating rational functions that can evaluate to 0/0 for some x. This check for 0/0 in the question algorithm could be used in question 1 above.

Group 4.1. Q.03. State the vertical asympote(s)

Part of the answer comment for this third question is useful for the question generation algorithm of question 1 above: 

  Comment:
  
  The first step in finding the vertical asymptotes of a rational function
  is to ensure both the numerator and denominator are fully factored. Then,
  if any factors in the denominator and numerator are equal, we can cancel
  them and note that there is a "hole" or discontinuity at that point,
  but it is not a vertical asymptote.
  1. Is a warning about the comma needed in the question?
    A typical question is 
      State the equation(s) of the vertical asympote(s) for the rational function:
  y =  6x + 2
      --------
     (x+1)(x-5)
    The answer format is x=-1,x=5 and half marks are given for half an answer x=-1 or x=5 but the question gives no marks for a typo like x=5, that includes the comma, but is otherwise half correct. And, typos like x=5 x=-1 with no comma but right answers get no marks. Is a warning about the comma needed in the question?

    It is interesting that answers get 50% marks, but answers get no marks.

    And, in the question below, with 2 equal roots in the denominator, and one asymptote, the answer x=2,x=2 is marked wrong whereas above, repeating an answer is sometimes ok.

  2. A rational function is mistyped or mis-generated as 2 x x (answer x=0 is still correctly graded):

  3. A cancelled factor (x+2) in the top and bottom of the rational expression 2x+4/(x-1)(x+2) is mistaken for an asymptote at x=-2 and marked as correct when it isn't an asymptote as part of the answer comment explains.

Contact

Paul Kates
Mathematics Faculty CTE Liaison
pkates@uwaterloo.ca, x37047
Last modification date: Fri Jul 11 14:51:33 2014.