Curve Sketching

By the end of this lecture, you should be able to sketch the shape of a curve, including all relative and absolute minimums/maximums,proper curvature, asymptotes, and intercepts, by using nothing but algebra techniques on the equations for f(x), f '(x) and f ''(x).

What information do we need to draw the graph of a function?

If we think about the general shape of some functions, we can begin to think about what information we would need to describe all the places where something "interesting" happens on the curve. Here are some things we may want to know:

Knowing the answers to these questions is not sufficient for us to know everything that it is possible to know about the shape of a function, but they give us a pretty good foundation for identifying all the major features of a function so that we could make a good general sketch with most of the important details. But how do we answer each of these questions?

Now that we know how to find the domain and any holes, intercepts, asymptotes, local maximums and minimums, and inflection points, we have all the information we need to sketch the basic shape of the graph of most functions that we might encounter. We can put all of these calculations together to graph functions, and we call this curve sketching. We use the word sketching here because we don't have enough information to be sure that every single point on our graphs are exactly right, but we do have enough information for the general shape of the graph to be accurate. To see a number of examples of curve sketching, take a look here: