Question 0.2.3: Sketching the Graph of a Rational Function Sketch a graph of...

Your initial graph should look something like Figure 0.32a or 0.32b. From either graph, it should be clear that something unusual is happening near x = 2. Zooming in closer to x = 2 should yield a graph like that in Figure 0.32c.

In Figure 0.32c, it appears that as x increases up to 2, the function values get more and more negative, while as x decreases down to 2, the function values get more and more positive. This is also observed in the following table of function values.

Note that at x = 2, f(x) is undefined. However, as x approaches 2 from the left, the graph veers down sharply. In this case, we say that f(x) tends to −∞. Likewise, as x approaches 2 from the right, the graph rises sharply. Here, we say that f(x) tends to ∞ and there is a vertical asymptote at x = 2. (We’ll define this more carefully in Chapter 1.) It is common to draw a vertical dashed line at x = 2 to indicate this. (See Figure 0.33.) Since f(2) is undefined, there is no point plotted at x = 2.