Full Curve Sketching

Given a function, especially a rational function, our goal is to make a sketch of the function which exhibits the rough positions of all of the following features which describe the qualitative behavior of a function and are exactly those needed for many applications:

Asymptotes (vertical and horizontal)
Symmetry (even and odd functions, other symmetry)
Points of discontinuity.
Local Extrema and inflection points.
Intercepts (points where the curve crosses either axis.

The method is to go down the list and get information by using the tools from Stages 3 and 4 as well as those just developed in this Stage. In doing so, there are generally multiple approaches to each piece of the puzzle - you could find out if a vertical asymptote goes up or down by looking at values of the function, by looking at the algebraic representation of the function, by looking at the concavity, and so on. Here is an example in which we stress methods involving the second derivative.

Example: Let's sketch the curve y = f(x) where f(x) = x²/(x + 3). We start by listing features and questions we must answer about these features.

Vertical Asymptotes. There is one factor of (x + 3) in the denominator and none in the numerator, so there is a vertical asymptote at x = 3.

To sketch the curve we will need to know the nature of this asymptote. Specifically, does the function go to

or -

as x approaches 3 from the left and from the right.

Horizontal Asymptotes. Since the degree of the numerator is larger than the degree of the denominator, there are no horizontal asymptotes.

However, if we divide the numerator and denominator by x we get that

	x
f(x) =
	1 + 3/x

so that for large (and large negative) x, f(x) is very close to x. This means that the curve y = f(x) is asymptotic to the line y = x. Such an asymptote is called an oblique asymptote.

To sketch the curve we will need to know if it approaches the line y = x from below or from above.

Symmetry. The equation does not exhibit any clear symmetry.

Continuity. The function and its derivatives are rational functions with various powers of (x + 3) in the denominator. Such functions are continuous everywhere except at x = -3.

Critical Points. f(x) is differentiable for x 3. This domain contains no endpoints. Hence critical points other than x = -3 can only occur at places where the derivative is zero. We computed that f'(x) = (x² + 6 x)/(x + 3). The critical points are then x = -3, x = -6 and x = 0.

To sketch the curve we will need to know whether the given critical points are maxima, minima, or inflection points. The values of f at these points are: f(-6) = - 12 and f(0) = 0.

Most of the questions raised above can be answered by looking at the sign of the second derivative f''(x) = 18/(x + 3)³.

f''(x) is negative for x < -3 and positive for x > -3. This means that the graph of f(x) is concave down for x > 3 and concave up for x < 3. The consequences of this are:

The vertical asymptotes must go to - to the left and to the right of u = 3.
The critical point at x = 6 is a local maximum and the critical point at x = 0 is a local minimum.

In addition, we note that x/(x + 3) < 1 for x > 0 and for x < -3. Since f(x) can be written as x*[x/(x + 3)] this implies that f(x) < x for large positive and for large negative x. In other words, the graph of f(x) is below its oblique asymptote y = x for large positive and large negative x. In sketching, we have to keep in mind that the curve is concave up for large x even though it is approaching the oblique asymptote y = x from below.

Computer-generated graph of y = x²/(x + 3)
One of the interesting attributes of curve sketches is that the sketches we make by hand are rarely to scale and can grossly exaggerate features of interest. We are NOT trying to reproduce the work of a graphing calculator or computer. Rather, we are looking for qualitative statements which may or may not be apparent from a more sophisticated graph and we are thinking our way into the function's behavior.

Equipment Check: Curve sketching is time-consuming, but the only way to learn it is by doing it. Here are two more examples which you should try on your own with pencil and paper before you look at the solutions. They are chosen to be workable by hand. It is best to come up with a set routine such as going down the checklist given towards the top of this page.

Sketch the curve y = x/(x² - 4).