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Graphs of Increasing and Decreasing Functions and Asymptotes Study Guide (page 2)

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Updated on Oct 1, 2011

Example 2

Graph g(x) = .

Solution 2

The domain is x ≠ 2. There is a vertical asymptote at x = 2. The sign diagram or g(x) is shown in Figure 14.11.

Figure 14.11

Thus, and

Because and , there is a horizontal asymptote at y = 1, both to the left and to the right. The derivative g'(x) = has the sign diagram shown in Figure 14.12.

Figure 14.12

The second derivative g"(x) = has sign diagram shown in Figure 14.13.

Figure 14.13

Because we have no points plotted at all, it makes sense to pick one or two to the left and right of the vertical asymptote at x = 2. At x = 1, g(1) = –4, so (1,–4) is a point. At x = 3, g(3) = 6, so (3,6) is another point. At x = –3, g(–3) = 0, so (–3,0) is another nice point to know. Judging by these, it will be useful to have both the x- and y-axes run from – 10 to 10.

To graph g(x), it helps to start with the points and the asymptotes as shown in Figure 14.14.

Figure 14.14

Then we establish the shapes of the lines through these points using the concavity and the intervals of decrease (see Figure 14.15).

Figure 14.15

Example 3

Graph h(x) =

Solution 3

To start, h(x) = Thus, h(x) is undefined with a vertical asymptote at x = 1 and x = –1. The sign diagram for h(x) is shown in Figure 14.16.

Figure 14.16

Note: x2 + 1 can never be zero. The limits at the vertical asymptotes are thus:

Because and there is a horizontal asymptote at y = 1.

The derivative is as follows:

It has the sign diagram shown in Figure 14.17. This indicates that there is a local maximum at x = 0. The corresponding y-value is y = h(0) = – 1 .

Figure 14.17

The second derivative is as follows:

The sign diagram is shown in Figure 14.18. It looks like there ought to be points of inflection at x = –1 and x = 1, but these are asymptotes not in the domain, so there are no points where the concavity changes.

Figure 14.18

Before we graph the function, it will be useful to have a few more points. When x = – 2, then y = h( –2) = and when x = 2, y = h(2) = as well. Thus, it will be useful to have the x- and y-axes run from – 3 to 3. We start with just the points and asymptotes (see Figure 14.19).

Figure 14.19

Then we add in the actual curves, guided by the concavity and the intervals of increase and decrease (see Figure 14.20).

Figure 14.20

Find practice problems and solutions for these concepts at Graphs of Increasing and Decreasing Functions and Asymptotes Practice Questions

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