College Algebra, 2013a

320 Rational Functions
4.2 Graphs of Rational Functions
In this section, we take a closer look at graphing rational functions. In Section 4.1, we learned that
the graphs of rational functions may have holes in them and could have vertical, horizontal and
slant asymptotes. Theorems 4.1, 4.2 and 4.3 tell us exactly when and where these behaviors will
occur, and if we combine these results with what we already know about graphing functions, we
will quickly be able to generate reasonable graphs of rational functions.
One of the standard tools we will use is the sign diagram which was first introduced in Section 2.4,
and then revisited in Section 3.1. In those sections, we operated under the belief that a function
couldn’t change its sign without its graph crossing through the x-axis. The major theorem we
used to justify this belief was the Intermediate Value Theorem, Theorem 3.1. It turns out the
Intermediate Value Theorem applies to all continuous functions, 1 not just polynomials. Although
rational functions are continuous on their domains, 2 Theorem 4.1 tells us that vertical asymptotes
and holes occur at the values excluded from their domains. In other words, rational functions
aren’t continuous at these excluded values which leaves open the possibility that the function could
change sign without crossing through the x-axis. Consider the graph of y = h(x) from Example
4.1.1, recorded below for convenience. We have added its x-intercept at ( 1
2 , 0) for the discussion
that follows. Suppose we wish to construct a sign diagram for h(x). Recall that the intervals where
h(x) > 0, or (+), correspond to the x-values where the graph of y = h(x) isabove the x-axis; the
intervals on which h(x) < 0, or (−) correspond to where the graph is below the x-axis.
8
7
6
5
4
3
y
1
−4 −3 −2 1 1 2 3 4
−1 2
−2
−3
−4
−5
−6
x
(+) ” (−) 0 (+) ” (+)
−1 1
2
1
As we examine the graph of y = h(x), reading from left to right, we note that from (−∞, −1),
the graph is above the x-axis, so h(x) is (+) there. At x = −1, we have a vertical asymptote, at
which point the graph ‘jumps’ across the x-axis. On the interval ( −1, 1 2)
, the graph is below the
1 Recall that, for our purposes, this means the graphs are devoid of any breaks, jumps or holes
2 Another result from Calculus.