Asymptotes are a vital part of graphing a rational function. They help show the direction the graph moves in and where it is defined and undefined.
Removable discontinuities and vertical asymptotes are undefined areas of a rational function. Both bring shape and direction to the graph of a rational function. Vertical asymptotes are invisible or ghost lines that show where a rational function is not allowed. A removable discontinuity is a hole along the curve of a function in a rational function graph. It is an undefined point instead of a line. The discontinuity is not as stark as the vertical asymptote, but it is still undefined at that particular point.
As we continue, we’ll break down how to find each one so that the concepts are a bit clearer. We will dive deeper into finding vertical asymptotes and how to find removable discontinuities. We’ll start with removable discontinuities.
What is a removable discontinuity?
Both removable discontinuities and vertical asymptotes are undefined parts of a rational function. They are both areas on the graph where the function cannot exist since they create a zero in the denominator. Because of the nature of a removable discontinuity, they are less severe than a vertical asymptote.
A removable discontinuity is a puddle that the graph merely jumps over. It appears as a hole in the graph. On the other hand, a vertical asymptote is the same as an invisible wall. The graph is not allowed to pass.
The first step in our process is to see if the rational function has any removable discontinuities. We do this by finding out if there are any common factors in the numerator and denominator.
For example, if we have the following rational function.
The first step that we have to take is to reduce this function. We have to see if there are any common factors in the numerator and denominator.
When we factor the rational function, we find a common factor in the numerator and denominator. The common factor shows us where the removable discontinuity is.
Our removable discontinuity exists at a point (x, y). To find this point, we take the common factor out from our function and set it equal to zero. Our common factor is x+6. Once we set it equal to zero, the result is x=-6.
Now to find the y-coordinate of our removable discontinuity.
To find the y-coordinate, we plug the x back into the reduced function. The result is y = -1/12.
The discontinuity exists at the point (-6, -1/12). When we graph this rational function, there will be a hole at that one point.
Remember that removable discontinuity won’t always exist in a rational function. Removable discontinuities or holes only exist when there is a common factor in the numerator and denominator.
What are vertical asymptotes?
After reducing the rational function, we can find the vertical asymptotes.
Keep in mind that there cannot be a zero in the denominator of any rational functions. The denominator is where we will look to find the vertical asymptotes.
Let’s take the example from above. Remember that the vertical asymptote shows where our function is undefined, so we must find the values that create a zero in the denominator. Once we have reduced the function, we can find the vertical asymptotes.
Our denominator contains x-6. We set x-6 equal to zero and solve.
We find that our vertical asymptote is at x=6. The function, shown in red, is being influenced by the vertical asymptote. You can’t see it here, but the function approaches the asymptote but never crosses it.
Here are a few more examples.
The first example is for a function that doesn’t have any common factors in the numerator and denominator. Since the function cannot be reduced, there are no removable discontinuities.
Our next step is to find the vertical asymptotes. Looking at the denominator, we find the function:
Vertical asymptotes exist where the denominator is equal to zero. If we look at this function, we see that this function can never be zero. If we put in any negative number because we squared it, it will always be positive. That means that no value can create a zero in the denominator.
Therefore, there are no vertical asymptotes.
This function has no removable discontinuities and no vertical asymptotes.
For the following example, we see the function:
With this function, there may be a possibility to find common factors. But once we factor the polynomials, we find that the numerator and denominator do not share a common factor.
Now we can move on to find the vertical asymptotes. Concentrating on the denominator, we find that we have 2x-3 and x+2.
Once we set each of these equations to zero, we find x=3/2 and x=-2. These two x-values are the vertical lines that will guide our rational function. The vertical asymptotes are at these x-values.
For the last example, we see this function:
We immediately see that this function can be reduced. The common factor in the numerator and denominator is the x. Since it’s just x, we can set the x equal to zero. That’s the x-coordinate of our removable discontinuity.
To find the y-coordinate, we put 0 back into our reduced function and solve.
Our removable discontinuity is at (0,4).
Now to find the vertical asymptote. Looking at the denominator, we see that we have x-2 and x+2.
Setting both of these equal to zero, we find that we have vertical asymptotes at x=2 and x=-2.
Hopefully, this has been helpful. Keep practicing, and don’t give up!