In this lesson, our instructor Vincent Selhorst-Jones will teach you about Rational Functions and Vertical Asymptotes. Starting with the definition of a rational function, Vincent will then explain the domain of a rational function and how you can go about investigating a fundamental function. Youll explore the idea of a vertical asymptote, as well as, learn how to find them and graph them. The lesson ends with four worked-out examples.
A rational function is the quotient of two polynomials:
where N(x) and D(x) are polynomials and D(x) ≠ 0.
Since a rational function f(x) = [N(x)/D(x)] is inherently built out of the operation of division, we must watch out for the possibility of dividing by zero. The domain of a rational function is all real numbers except the zeros of D(x).
As the denominator of a rational function goes to 0 (and assuming the numerator is not also 0), the fraction becomes very large. While it can't actually divide by 0, as it gets extremely close to 0, the function "blows out" to very large values. We call
this location a vertical asymptote. A vertical asymptote is a vertical line x = a where as x gets close to a, |f(x)| becomes arbitrarily large. Symbolically, we show this as
x → a ⇒ f(x) → ∞ or f(x) → − ∞.
On a graph, we show the location of a vertical asymptote with a dashed line. This aids us in drawing the graph and in understanding the graph later.
To find the vertical asymptotes of a rational function, we need to find the x-values where the denominator becomes 0 (the roots of the denominator function). However, not all of these zeros will give asymptotes. It's possible for the numerator to go
to 0 at the same time, which will cause the function to just have a hole at that x-value, but not "blow out" to infinity.
We can find the vertical asymptotes of a rational function by following these steps:
1. Begin by figuring out what x-values are not in the domain of f: these are all the zeros of D(x).
2. Determine if N(x) and D(x) share any common factors. If so, cancel out those factors. [Alternately, this step can be done by checking that the zeros to D(x) are not also zeros to N(x).]
3. After canceling out common factors, determine any zeros (roots) left in the denominator. These are the vertical asymptotes.
(4.) If graphing the function, figure out which way the graph goes (+∞ or −∞) on either side of each asymptote. Do this by using test values very close to the asymptote. For example, if the asymptote is at x=2, look at f(1.99) and f(2.01).
[You can also do this in your head by thinking in terms of positive vs. negative, which we discuss in the Examples.]
Rational Functions and Vertical Asymptotes
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.