Vertical and Horizontal Asymptotes


  1. Factor both the numerator N(x) and the denominator D(x).
  2. Cancel any common factors and simplify the function.
  3. Equate the canceled factor to zero. This will give you the x-value of the hole.
  4. Plug this x-value in the simplified function to find the y-value of the hole.
  5. Plot that hole (or holes) on the graph.


  1. Plot the remaining zeros of D(x) on the x-axis. Draw vertical dotted lines through them. These are the locations for the vertical asymptotes.
  2. Find the sign of f(x), just before and after the dotted line. This you can do by finding the signs of the factors in the simplified function and resolving them. This will tell you if the graph is going asymptotic upwards or downwards near the dotted vertical line.
  3. Find remaining zeros of N(x) on the x-axis. These are points where the graph crosses the x-axis.


  1. Horizontal asymptotes occurs at either end of the graph as x goes to plus or minus infinity.
  2. For n < m, the horizontal asymptote is y = 0 (the x-axis).
  3. For n = m, the horizontal asymptote is y = an / bm
  4. For n = m+1, the asymptote is a slanted line, y = kx, found by dividing N(x) by D(x).
  5. For n > m+1, there are no asymptotes;
    when n – m is even, both ends of the graph rise up
    when n – m is odd, the left end goes down while the right end rises up.

From the above data you can sketch a rough approximation of the shape of the graph.


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