Rational functions are a mixed bag. Sometimes easy to deal with, and sometimes quite tricky. Asymptotes are a vital part of this process, and understanding how they contribute to solving and graphing rational functions can make a world of difference.
Asymptotes are ghost lines drawn on the graph of a rational function to help show where the function either cannot exist or where the graph changes direction. The three types of asymptotes are vertical, horizontal, and oblique asymptotes. They all collectively influence the shape of the graph of the function.
Really learning and nailing down this concept can help you become more proficient in algebra and other higher math classes.
So, let’s try to break it all down into lovely bite-sized pieces that we can consume.
What are rational functions and asymptotes?
Before we jump into the definition of asymptotes, let’s refresh our memories on what rational functions are and how the two concepts are related.
A rational function is a polynomial divided by a polynomial. A ratio of polynomials. Just so that we aren’t confused, here are some other things that qualify a function as a rational function.
- The exponents or degrees of a rational function are whole numbers, not fractions.
- Both the numerator and denominator are functions of the same variable.
- A rational function can consist of a single number over a polynomial, but not a polynomial over a single number.
- The denominator cannot be equal to zero.
Asymptotes are ghost lines on a graph that either guide or shape the function or show where the function is undefined. They are handy in showing how different parts of the function influence the graph.
The three types of asymptotes are vertical asymptote, horizontal asymptote, and oblique asymptote.
The best place to start is with vertical asymptotes.
What are vertical asymptotes?
Rational functions work like fractions. The denominator should not have a zero value in it or should not be equal to zero at any time.
That sounds easy, but there is one step that many people miss: to reduce the rational function before actually seeking the values that create a zero in the denominator.
When we reduce the function first, we find values that could mistakenly be called vertical asymptotes. These values are actually referred to as removable discontinuities. They are different from the vertical asymptotes in that they can be factored out. If we mistakenly leave them in, our graph will take on a whole new shape.
Removable discontinuities are often graphed as holes in the graph. If they are part of the function themselves, they will appear as holes along the graph’s curve. I wrote a post on the difference between removable discontinuities and vertical asymptotes if you need more help.
Once the function has been reduced, we can find the vertical asymptotes. We know that any fraction with a zero in the denominator is undefined. We cannot have zeros lurking around in the denominator. Any value of x that sets the denominator equal to zero is not allowed.
But to find the vertical asymptote for our rational function, we have to find what values of x create this zero in our denominator. This is quickly done by setting the factored polynomial to zero and solving for x.
For instance, if we have a polynomial with x-2 in the denominator, we know that our x cannot equal 2 because the equation x-2=0 will give us a zero in the denominator. The x=2 shows us where our function is undefined. It also shows us where our vertical asymptote exists.
Here is another example. Let’s look at the following function:
The first step that we have to take is to reduce this function. Remember, we must reduce the function to differentiate the removable discontinuities from our vertical asymptotes.
When we break down both the numerator and the denominator, we find that we have common factors. The x-1 shows us where the removable discontinuity is for our function.
Before we cancel it out, we find that the discontinuity is at the point (1, -3/2).
Once we cancel out the common terms, our reduced function will be:
To find the vertical asymptote, we must look at the denominator. Our vertical asymptote is our denominator set to zero. In this case, it would be x+1=0. Our vertical asymptote is at x = -1.
What are horizontal asymptotes?
Horizontal asymptotes are a bit trickier. It feels like the difficulty level increases with each asymptote. It’s still doable but not as easy as finding the vertical asymptote.
We find the horizontal asymptote by looking at the highest degree in both the numerator and the denominator. We have three different cases that we look at to find the horizontal asymptote. We’ll take a look at all three instances.
The first one occurs if both degrees in the numerator are equal. If both degrees are equal, then we take the coefficients of both.
In the example below, the numerator and denominator share the same degree. We then take the numerator’s coefficient and the denominator’s coefficient, and we create the horizontal asymptote.
If the numerator degree is higher than the degree in the denominator, we have no horizontal asymptote.
In the example below, we find that the degree in the numerator is 3, and the degree in the denominator is 2. Because the numerator degree is higher, this function has no horizontal asymptote.
If the degree in the numerator is smaller than the degree in the denominator, then the horizontal asymptote equals zero.
In the final example, we have the numerator degree equal to 1, while the denominator’s degree equals 2. Since the numerator’s degree is smaller, the horizontal asymptote is y=0.
Here are a few more examples. Let’s take a look at the following equation. The degree in the numerator is a zero (x0), and the degree in the denominator is a 1. Because the numerator’s degree is less than the denominator’s degree, the horizontal asymptote is a line at y=0.
In the following example, we see that the degree in the numerator is the same as the degree in the denominator. Since the numerator and denominator share the same degree, the horizontal asymptote is the ratio of the numerator and denominator coefficients. In this function, the coefficients of both are 5 and -4. The horizontal asymptote is at y=-5/4.
In this last example, the degree in the numerator is more than the degree in the denominator. In this case, we know that the horizontal asymptote does not exist for this function.
Now we can move on to the final asymptote, the oblique asymptote.
What are oblique asymptotes?
The last asymptote that we will look at is the oblique asymptote.
The equation for an oblique asymptote is y=ax+b, which is also the equation of a line. The biggest confusion is extracting or digging out the oblique asymptote from our rational function.
The method we use to get to the oblique asymptote is long division. Long division is a method of dividing a polynomial into another polynomial. In our case, we are dividing the denominator into our numerator, but there is one catch.
The highest degree in the numerator has to be one degree higher than the highest degree in the denominator. If it isn’t, then there is no oblique asymptote.
When we use long division on our numerator and denominator, the result we get should be the equation y=ax+b. This is why the degree in the numerator needs to be one degree higher than the one in the denominator.
Let’s go through a few examples to see how this works and what this process looks like.
For the first example, we have this equation:
The first step in finding the oblique asymptote is to make sure that the degree in the numerator is one degree higher than the one in the denominator. The degree in the numerator is 2, and the degree in the denominator is 1. This requirement checks out.
The next step is to divide the denominator into the numerator via long division.
As we can see from this example, we divide x-1 into x2+6x+9.
Using long division, we see that the resulting equation is y=x+7. This is the line that represents the oblique asymptote of our function.
Here is a graph of the function (in blue) graphed along with the oblique asymptote (in orange). As we can see, the oblique asymptote shapes the graph. The function approaches the asymptote but never crosses it.
In our last example, we’ll take a look at the following function:
Once again, the degree in the numerator is one degree higher than the degree in the denominator. We can then perform long division, dividing the denominator into the numerator.
For this problem, we’ll divide x into x2-1.
The result of performing long division is that y=x. This line represents the oblique asymptote for our rational function. It is the line that will shape our function’s graph.
In the image above, the blue line represents the oblique asymptote. The red lines represent the graph of our rational function. Once again, the asymptote shapes the graph of our rational function. The graph approaches the asymptote but never crosses it.
Most of these problems can seem complicated at first, but keep trying and keep practicing.
And never be afraid or ashamed to ask for help. You can do this! 😁