3.6 - Graphs of Rational Functions

Let f(x) = p(x) / q(x) where p(x) and q(x) have no common factors.

If p(x) and q(x) have a common factor, then divide out the extra factors so that it is only left in the numerator, or denominator, or not at all. Then look at the section on holes in the lecture notes for section 3.5.

  1. The y-intercept is the value of f(0). That is, substitute 0 in for x in both the numerator and denominator.
  2. The x-intercepts are the zeros of p(x).
  3. The vertical asymptotes are the zeros of q(x).
  4. The horizontal asymptote is that value that f(x) approaches as x increases or decreases without bound. Look at the previous section for specifics.
  5. Determine the behavior between each vertical asymptote or x-intercept based on the Odd Changes, Even stays the Same principle.

When graphing a rational function, do the following (in pretty much this order)

  1. Simplify the function by dividing out any common factors. Be sure that if something is no longer in the implied domain, that you state the restriction.
  2. Identify and graph the x-intercepts. Make a note (possibly mental) of whether the graph crosses (odd exponent) or touches (even exponent) at each intercept.
  3. Identify and draw the vertical asymptotes. Make a note (possibly mental) of whether the graph is asymptotic in the same direction (even exponent) or different directions (odd exponent).
  4. Identify and draw the horizontal or oblique asymptote(s).
  5. Start on the far right of the graph close to the horizontal asymptote.
  6. Move from right to left.
    1. At each x-intercept encountered, touch or cross as noted in step 2.
    2. At each vertical asymptote, you must either go up or go down. The direction you go is determined by which side of the x-axis your on when you get to the vertical asymptote. When picking back up on the left side of the vertical asymptote, use the same or different side of the x-axis based on your observations in step 3.
    3. Watch out for holes.