Summary

We now have a lot of tools for understanding the graph of a function: we know how to explore the domain, the range, the asymptotes, the extrema, and most recently the points of inflection. We now want to put it all together, to produce a graph of a function. In the old days this was how it was done, before the advent of the computer. This skill is still useful, because technology routinely lies to us (e.g. ``papering'' over a vertical asymptote, for example, making it appear to be simply a steep spot on a graph, maybe with corners).

Here are the steps as given in our text:

1. Find the domain.
2. Find the intercepts, both x and y.
3. Find asymptotes, vertical and horizontal.
4. Calculate the derivative. Locate any critical points (places where f'(x) is zero or doesn't exist).
5. Calculate the second derivative. Find relative extrema, and determine where f is increasing and decreasing (in conjunction with first derivative). Determine concavity and points of inflection (where or does not exist).
6. Plot all points determined so far, along with a few additional points where appropriate.
7. Connect the points with a smooth curve in accordance with concavity, watching for the correct handling of vertical asymptotes and other points where the function isn't defined.
8. Check your plot against your calculator, noting any differences (in case your calculator has found your mistakes!).

We'll do a couple of examples, including one that has an oblique or slant asymptote: that is an asymptote that is neither vertical nor horizontal, but rather travels off towards some other line.

Examples:

4. #38, p. 319