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SectionShort-Run Behavior of Rational Functions

SubsectionVertical Asymptotes and Holes

As we saw in Section 12.2, a polynomial function is defined for all values of \(x\text{,}\) and its graph is a smooth curve without any breaks or holes. The graph of a rational function, on the other hand, will have breaks or holes at those \(x\)-values where it is undefined.


Investigate the graph of \(f(x) = \displaystyle{\frac{2}{(x - 3)^2}}\) near \(x = 3\text{.}\)


This function is undefined for \(x = 3\text{,}\) so there is no point on the graph with \(x\)-coordinate \(3\text{.}\) However, we can make a table of values for other values of \(x\text{.}\) Plotting the ordered pairs in the table results in the points shown in Figure335.

\(x\) \(y\)
\(0\) \(\dfrac{2}{9}\)
\(1\) \(\dfrac{1}{2}\)
\(2\) \(2\)
\(3\) undefined
\(4\) \(2\)
\(4\) \(\dfrac{1}{2}\)
\(6\) \(\dfrac{2}{9}\)
graph showing 6 points about a vertical asymptote

Next, we make a table showing \(x\)-values close to \(3\text{,}\) as in Figure336a. As we choose \(x\)-values closer and closer to \(3\text{,}\) \((x - 3)^2\) gets closer to \(0\text{,}\) so the fraction \(\dfrac{2}{(x-3)^2}\) gets very large. This means that the graph approaches, but never touches, the vertical line \(x=3\text{.}\) In other words, the graph has a vertical asymptote at \(x=3\text{.}\) We indicate the vertical asymptote by a dashed line, as shown in Figure336b.

table of values and graph of rational function

The vertical line \(x=3\) in the example above is an example of a vertical asymptote. In general, we have the following result.

Vertical Asymptotes

If \(Q(a) = 0\) but \(P(a) \ne 0\text{,}\) then the graph of the rational function \(f(x) = \displaystyle{\frac{P(x)}{Q(x)}}\) has a vertical asymptote at \(x=a\text{.}\)

However, if both the numerator and denominator of the rational function share a root, we no longer call this a vertical asymptote.

If \(P(a)\) and \(Q(a)\) are both zero in the rational function \(f(x)=\frac{P(x)}{Q(x)}\text{,}\) but the reduced expression for \(f(x)\) is defined at \(x=a\text{,}\) then the graph of \(f(x)\) has a hole at \(x=a\) rather than an asymptote.

Find the vertical asymptotes of \(G(x) = \displaystyle{\frac{4x^2}{x^2 - 4}}\text{.}\)

Near a vertical asymptote, the graph of a rational function has one of the four characteristic shapes, illustrated in Figure338. Locating the vertical asymptotes can help us make a quick sketch of a rational function.

4 cases of behavior near vertical asymptote

Locate the vertical asymptotes and sketch the graph of \(g(x) = \displaystyle{\frac{x}{x + 1}}\text{.}\)


The denominator, \(x+1\text{,}\) equals zero when \(x = -1\text{.}\) Because the numerator does not equal zero when \(x = -1\text{,}\) there is a vertical asymptote at \(x = -1\text{.}\) The asymptote separates the graph into two pieces.

We can use the Table feature of a calculator to evaluate \(g(x)\) for several values of \(x\) on either side of the asymptote, as shown in Figure340a. We plot the points found in this way; then connect the points on either side of the asymptote to obtain the graph shown in Figure340b.

table and graph of rational funciton
  1. Find the vertical asymptotes of \(f(x) = \displaystyle{\frac{1}{x^2 - 4}}\) . Locate any \(x\)-intercepts.

  2. Evaluate the function at \(x = -3\text{,}\) \(-1\text{,}\) \(1\text{,}\) and \(3\text{.}\) Sketch a graph of the function.