Curve Sketching Part II

So far we learned to sketch certain types of polynomials by examining the first and second derivatives. In this section we are going to practice sketching functions that have vertical and horizontal asymptotes

What is an asymptote? Informally, an asymptote is the line a function approaches without the function ever touching the line. The function gets closer and closer to the line, but it never reaches it. We can see an example of a function that has asymptotes below:


The preceding graph plotted the fuction y=1/x. As you can see, f(x) gets closer to the x and y axis but will never reach it no matter how far out we go. Here the horizontal and vertical asymptotes are the line x = 0 and y = 0, respectively.

How do we find horizontal and vertical asymptotes? A lot of the time you can find them by finding the value of x that makes the denominator of f(x) equal 0. For example, in the above graph of f(x), the value 0 makes the denominator 0, and dividing by zero is a no-no. You can try to substitute values if you are not convinced. Using values close to 0 from the positive and negative side cause the function to take on very small and very large values very fast. However, setting the denominator equal to 0 will not always let you find all the asymptotes. Often we need to see how the function behaves x approaches positive and negative infinity. Continuing with the existing example, as x goes to negative infinity the value of f(x) approaches 0 incredibly close without it ever reaching 0. It goes vice versa for positive infinity. As x approaches positive infinity, the value of f(x) approaches 0 very close but will never reach it. That is what makes x = 0 an asymptote for the function f(x)=1/x.

Let's practice sketching the graph of a different function for practice. Suppose we had the function, . Here we have an x in the numerator and denominator which will cause us a little bit of trouble finding the second asymptote, but let's get started setting the denominator equal to 0 and solving for x. x = 1 causes the graph of f(x) to behave very wildly. Putting in values closer and closer to 1 from the positive side causes the denominator to be very close to 0 which will blow up the value f(x). Putting in values closer and closer to 1 from the negative side causes f(x) to take on very small values. Kind of as an aside, we can use limit notation to express what we are trying to say: and .

OK we found a vertical asymptote at x = 1. Are there any others? What happens to f(x) as x approaches plus and minus infinity? With the way the function is written it won't tell us a whole lot about that. However, there is a trick which is a little bit sly because it's difficult to think of, but if we multiply the numerator by 1/x, we will have f(x) in a more illuminating form. Doing that we now have:

.

So now we know there is an horizontal asymptote at x = 1, and we can sketch the graph of f(x) which looks like this(the asymptotes are shown in green):