Section 6.4: Sketching Rational Functions

In the previous section, we discussed how to divide and factor polynomials.  While this is an interesting algebraic technique in its own right, we have seen that knowing how a polynomial factors allows us to quickly sketch a rough graph.  In this section we will see that we may generalize this technique to sketch rational functions.

Remember that a  rational function is simply a quotient of polynomials, such as p(x) = (x3 - 3x - 2)/(-x2 - x + 2).  Right away this tells us that near ±¥ the graph is approximately x3/(-x2) = -x and that the vertical intercept is -2/2 = -1.  If we rewrite such a function in two different ways:

we can discover a lot more.  The divided form of the function tells us more precisely that the graph is approximately -x + 1 + 4/(-x2) » -x + 1 on the "ends".  On the other hand, the factored form tells us that:

Putting all this information together, we can make the following analysis, as we move left-to-right through the graph:

Points Approximation Description
x » -¥ p(x) » -x+ 1 The graph approaches the "slant" asymptote -x + 1 as we go to the left,
and comes down from the left (i.e., from x = -¥).
x » -2 p » a negative, odd degree power graph Since the graph is above the axis to the left of this point,
 on the left it must go up towards an asymptote at -2, 
and go down towards the asymptote on the right.
x » -1 p » a positive, even degree power graph Since the graph is below the axis to the left of this point, 
it must go up, touch the axis at -1 and go back down.
x = 0 p(0) = -1 It crosses the vertical axis at -1.
x » 1 p(x) » a negative, odd degree power graph Since the graph is now below the axis to the left of this point, 
 on the left it must go down towards an asymptote at 1, 
and go up towards the asymptote on the right.
x » 2 p(x) » a linear graph Since the graph is now above the axis to the left of this point, 
it goes down through the axis at 2.
x » ¥ p(x) » -x+ 1 The graph approaches the "slant" asymptote -x + 1 and continuing down as we move right (i.e., towards x = ¥).

Plotting each piece of information, i.e., sketching the graph near each intercept and asymptote gives:

Connecting-the-dots, gives a rough sketch of the graph:

You can verify our analysis by selecting the different Examples in XFunctions:

We can summarize this technique, as follows:

  1. Plug in 0 to find the vertical intercept and plot it.
  2. Perform long division to determine the approximate graph at the "ends" (i.e., at x = ±¥), which is given by the quotient; sketch  the asymptote given by the quotient and sketch the "ends" of the graph so that they approach the asymptote as x approaches ±¥.
  3. Factor the numerator and denominator:
    1. Factors of the numerator correspond to horizontal intercepts; sketch the 
    2. Factors of the denominator correspond to vertical asymptotes.
  4. Work from left to right through the intercepts and asymptotes and sketch pieces of the graph near each:
    1. Near a horizontal intercept, the graph will look like an even or odd power graph near 0, depending on the power of the corresponding factor; you will need to decide whether or not to flip the graph vertically, so that it can connect with the part of the graph to the left that you have already sketched.
    2. Near a vertical asymptotes, the graph will look like the reciprocal of an even or odd power graph near its asymptote, depending on the power of the corresponding factor; you will again need to decide whether or not to flip the graph vertically, so that it can connect with the part of the graph to the left that you have already sketched.
  5. "Connect-the-dots" to achieve a sketch of the entire graph.

Note: You may find it helpful to make a chart of the key points, in order, as we have done, to summarize your analysis before trying to sketch the graph.

Make sure that you can use this technique to sketch rational functions, by completing the following exercises. 

Go to Introduction.

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