(Course: MHF4U - Grade 12 Advanced Functions)
Asymptotes and intercepts are tricky topics in math! In this article, we will cover how to find the asymptotes and intercepts of rational functions.
First, what is a rational function?
A rational function is defined as a function that is a ratio of two polynomials. In other words, it is a fraction where the numerator and denominator are both polynomials. For example, the function below is an example of a rational function.
If we graph this using Desmos, we get the following.
It's an interesting graph! We can see that the function has a y-intercept at (0, 3) and an x-intercept at (-3/2, 0). It also appears like the graph avoids two imaginary lines, one at x = -1/3 and another at y = 2/3. These two imaginary lines are asymptotes. The x = -1/3 asymptote is a vertical asymptote while y = 2/3 is a horizontal asymptote of the function.
These are not the only types of asymptotes. They can also be oblique, quadratic, cubic, or the shape of any other polynomial! We will be discussing how to find all the asymptotes and intercepts for any type of rational function.
The Box Method
To start off, let's take a look at a simple type of rational: a linear over a linear, like the first example given above. Interestingly, each of the four values described above (x- and y-intercepts, horizontal and vertical asymptotes) can be calculated using two of the coefficients or constants in the function. These pairs of numbers form a box, hence: the Box Method (not to be confused with the box method for factoring trinomials).
These pairs of numbers are the same for all rational functions that are a linear over a linear. For example, the coefficient of x in the numerator divided by the coefficient of x in the denominator will always give you the y-value of the horizontal asymptote.
An Exception and an Important Note
Actually, there's one exception to the Box Method. Take the following function:
Here, since the numerator and denominator cancel out (leaving nothing in the denominator), there is no asymptote. Instead, there's a removable discontinuity—a "hole" in the graph—at x = -1/2 because the function evaluates to 0/0 at that point. This is why it's important to check for common factors first, which is explained in more depth below.
Also, important note: be wary of negative signs! If the rational function has a negative sign before it, the y-intercept and the horizontal asymptote will be flipped. For example, if a rational function has a y-intercept at (0, 2) and a horizontal asymptote at y = 3/4, then adding a negative sign moves the y-intercept to (0, -2) and the horizontal asymptote to y = -3/4.
Keeping those two things in mind, now you know how to factor linear-over-linear rational functions!
More Complex Rational Functions
When dealing with more complex rational functions, your first step should always be to factor both the numerator and denominator. You may need to use the factor theorem to factor higher-degree polynomials. Factoring allows you to see if any factors cancel out and makes it easy to find the zeros of the numerator and denominator. For example:
Note that it is necessary to specify that the function is undefined at x = 1 because although the factor of x - 1 was cancelled out, it still affects the domain of the function. Also note that while the factor of 3x + 2 was cancelled out of the numerator, it still remained in the denominator so there is still a vertical asymptote at x = -2/3.
For the rational function above, after cancelling, you can the Box Method to find its asymptotes and intercepts because it's a linear over a linear!
In fact, three of the four sides of the "box" work for any rational function as long as you've factored and cancelled out any common factors. Let's use the following factored rational function as an example:
Top side (numerator): the x-intercepts are at the x-values where the numerator is equal to 0. Since you've already factored, you can just find when each of the factors of the numerator is equal to 0. In the example, this gives you x-intercepts at: (-2, 0), (7, 0), and (3/4, 0).
Right side: the y-intercept is equal to the constant term of the numerator over the constant term of the denominator. You can quickly find the constant term of each by getting the product of all the constant terms of its factors. In the example, the constant term of the numerator is 2 × (-7) × (-3) = 42 and the constant term of the denominator is (-3) × 5 = -15. Therefore, the y-intercept is at (0, -42/15) which is (0, -14/5).
Bottom side (denominator): the vertical asymptotes are at the x-values where the denominator is equal to 0. Since you've already factored, you can just find when each of the factors of the denominator is equal to 0. In the example, this gives you: x = 3, -5/2.
However, the left side of the box doesn't work for all rational functions. When finding the other asymptote, there are three cases to consider:
1. The degree of the numerator is less than the degree of the denominator
In this case, the only other asymptote is a horizontal asymptote at y = 0.
2. The degree of the numerator is equal to the degree of the denominator
Here, you can use the normal left side of the Box Method to find the other asymptote, which is a horizontal asymptote. The y-value of the horizontal asymptote is equal to the leading coefficient of the numerator over the leading coefficient of the denominator. You could expand the numerator and denominator entirely to do this, but a faster way is to simply get the product of the leading coefficient of all the factors of the numerator and all the factors of the denominator to get the leading coefficient for each.
3. The degree of the numerator is greater than the degree of the denominator.
Here, you need to perform long division. You will need to expand your numerator and denominator and then divide the numerator by the denominator. The result, ignoring the remainder, will give you your asymptote. Note that this asymptote will not be horizontal; it may be oblique or even curvilinear!
Now you know how to find the asymptotes and intercepts of a rational function. You're ready to ace that test!
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