Behavior Near Points Not in the Domain
We are now getting closer to understanding the properties of rational functions. We have discussed how to find the domain, end behavior, and -intercepts. Finding the -intercept just involves plugging into the function. But there is one major feature of rational functions we still need to discuss before we can understand them fully. What happens to the graph of a rational function near the points which are not in the domain?
Recall that:
Rational functions can have two different kinds of behavior near a point where the denominator is zero.
We know from investigating that one possibility is a vertical asymptote. Recall that:
Here, as means for points near but less than and as means for points near but greater than .
When you put points into a rational function that are close to the point where it is undefined, it will make the bottom of the fraction very small. Usually, that will make the resulting fraction very big, resulting in a vertical asymptote on the graph. The exception is when the top of the fraction is also getting very small. Then, it is not quite clear what is happening to the value of the fraction overall. Let’s investigate.
First, we need to determine the points which are not in the domain of . That is, which values of make undefined? Note that we can factor the denominator of this rational function. Setting the denominator equal to zero will let us find the points where is undefined. This means or , giving us that is undefined for and . Now lets investigate what happens near each of these points.
Notice that we may write that .
This means that for all values of other than , gives the same -value as . This means that the graph of and the graph of should look identical except for at .
At , is not defined, but is not a special point at all for . The graph will continue as normal on either side of and there will be no asymptote there. Therefore, the conclusion is that will have a hole at . We can even find the height (-value) at which to draw the hole by finding out what the -value of is at . Since we keep referencing it, let’s give a name. It is not the same function as because it has a different domain. Therefore, it needs a different name. Let’s say . Then we have that Our conclusion is that the graph of the function will have a hole at the point .
Let’s let’s compare that with the behavior of near . Recall that for all values of other than , gives the same -value as . Since we are not investigating near one, we can just use the simpler function, . Both and will have the same behavior everyone other than at !. But comparing the expression with what we have previously seen for the function , we see that as and as , so that the line is a vertical asymptote for .
Finding Vertical Asymptotes, Holes, and Zeros
You may notice that the function in the previous example played an important role in our ability to determine whether our original function had a hole or an asymptote at each point which was not in the domain. The function was special because it had no factors in common between the numerator and denominator of the rational function. We give rational functions like this a special name.
This allows us to state the following theorem.
For rational functions which are not in lowest terms, we will want to factor the numerator and denominator completely to determine what cancels. We can the write the rational function in lowest terms, being careful to keep track of any points which are not in the domain.
Let’s try applying this theorem to an example.
Determine where has vertical asymptotes, holes, and zeros.
- Step 1:
- The first step is to make sure that both the numerator and denominator
are completely factored into linear and irreducible quadratic terms.
Recall that a quadratic is irreducible when the discriminant, . Let’s look
at the two quadratic factor above.
The factor is irreducible. It has a disciminant of . This means it cannot be factored and never equals zero. Essentially, for the purposes of this problem, this factor is not going to contribute to any zeros, holes, or vertical asymptotes. We can ignore it.
The factor is a difference of squares. . Thus, we want to replace with in our formula for .
- Step 2:
- The next step is to determine the points which are not in the domain of . To do this, we set the denominator equal to zero. whenever one of the factors equals zero. From the Polynomial Remainder Theorem, we can just read off the zeros for each term. The zeros will be coming from the factor of , which comes from the factor , which comes from the factor , which comes from the factor and which comes from the factor .
- Step 3:
- Now that we know the domain, we can rewrite in lowest terms, provided we keep track of which points are not in the domain.
- Step 4:
- Now that is written in lowest terms, anywhere the denominator is zero is a vertical asymptote. Therefore, we want to consider . This gives us vertical asymptotes at and .
- Step 5:
- Now, let’s determine the holes. There is a hole at any -values which are not in the domain but which are not vertical asymptotes. Since the -values and 5 are not in the domain and only and are vertical asymptotes, we have that and will be -values where holes are in the graph.
- Step 6:
- Finally, let’s determine the zeros of . A rational function equals zero whenever the numerator equals zero, provided those points are in the domain of the function. We can consider the numerator of version of the formula written in lowest terms. We have . We already determined that . Thus, we just have coming from the term . We check to see whether is in the domain of . It is not. Therefore, it is not a zero. has no zeros.
In conclusion, we have interesting phenomena happening whenever the numerator or the denominator of the function equal zero. For this function, we have vertical asymptotes at and , holes at and , and no zeros.
Graphing a Rational Function
Now let’s return the previous example and see if we can draw the graph of . We have already determined:
- As and (or a borizontal asymptote at )
- As and (or a borizontal asymptote at )
- The graph will have a hole at
- The graph will have a vertical asymptote at .
Now let’s find a few additional points to help us graph the function.
To find the -intercept of , . To find any -intercepts, we set , meaning we are looking at A fraction equals zero when the numerator equals zero, so we need
But, is not in the domain of our function so it is not actually an -intercept! There are no -intercepts!It is usually a good idea to find a point between each place where the function is zero or undefined. This will help us see where the function is. There is a result in calculus that the graph of a rational function can only change from being postive (above the -axis) to negative (below the -axis) or vice versa when the function is zero or undefined. We will explore this more later, but this idea motivated the points we are going to choose to plot.
Putting all of this together, we are able to draw the following graph:
It is important to note that many graphing calculators such as Desmos will not show the hole in the graph, but it is important to know that it is there.