In this lesson, Graphing Asymptotes in a Nutshell, our instructor Vincent Selhorst-Jones will give you the basics of how to easily graph asymptotes. First, Vincent will explain the process for graphing and then dive right into the 7 steps. Youll also learn how to test your intervals and then have a chance to practice with four examples.
This lesson is all about learning how to graph rational functions. It's strongly recommended that you watch the previous lessons beforehand, because we'll be pulling from that work. Also, while we won't be going over it in the lesson, using a graphing
utility (calculator or program) can be a great way to understand how rational functions work. Playing with function graphs can quickly build your intuition.
Here is a step-by-step process to create an accurate graph for any rational function:
1. Begin by factoring the numerator polynomial and the denominator polynomial of the rational function you're working with.
2. Find the domain of the function by looking for where the denominator equals 0. Each of these "forbidden" locations will become one of two things: a vertical asymptote (if the zero does not occur in the numerator) or a hole in the graph (if the
zero occurs in the numerator).
3. Now that we've found the function's domain, simplify the function by canceling out any factors that are in both the numerator and the denominator. [It's important that we don't do this in step #1 (factoring), otherwise we won't be able to find
all the "forbidden" x-values in step #2 (domain).]
4. Once the function is simplified, we can find the vertical asymptotes. The vertical asymptotes occur at all the x-values that still cause the denominator (after simplifying) to become 0.
5. Find the horizontal/slant asymptotes by looking at the degree of the numerator, n, and the degree of the denominator, m. There are a total of four possible cases:
n < m ⇒ horizontal asymptote at y=0.
n = m ⇒ horizontal asymptote at a height given by ratio of leading coefficients in numerator and denominator.
n=m+1 ⇒ slant asymptote, which can be found by using polynomial division.
n > m+1 ⇒ no horizontal or slant asymptote.
6. Find the x- and y-intercepts so we have some useful points that we can graph from the start. [It's possible for a rational function to be missing one type or both.]
7. Finally, using all this information, draw the graph. Place the asymptotes (drawn as dashed lines) and intercepts. You will probably need some more points, so plot more points as necessary until you see how to draw in the appropriate curves.
A useful concept for graphing is the idea of test intervals. Because a rational function is continuous between vertical asymptotes, we know that the function can only change signs at x-intercepts or vertical asymptotes. This means we can put
our x-intercepts and vertical asymptote locations in order and break the x-axis into intervals. In each of these intervals, we can test just one point to find if the function is + or − in the interval.
Graphing Asymptotes in a Nutshell
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.