Using multiplicity of factors to characterize graphs of rational functions

Rational functions can be scary because there are so many details to manage. Check other posts on this website for information on how to graph rational functions. In this post, I look at one small clue that can help you figure out the behavior of a rational function as it approaches the vertical asymptotes. All you need to do is check the multiplicity of the factor in the denominator.

If the multiplicity of the factor is even, then the graph approaches +∞ from both sides of the asymptote, or it approaches -∞ from both sides of the asymptote.

If the multiplicity of the factor is odd, then the graph approaches +∞ on one side of the asymptote and approaches -∞ on the other side.

Here is an example that demonstrates this property:

$\text{Graph } \dfrac {(x-2)(x+1)}{(x-1)(x+2)^2}$

There are two vertical asymptotes for this function, at $x=-2$ and at $x=1.$ The $(x+2)$ factor is multiplicity 2 (even), so the graph approaches the same limit from both sides of the asymptote. The $(x-1)$ factor is multiplicity 1 (odd), so the graph approaches opposite limits on either side of the asymptote. Here is the graph of the function, demonstrating this property:

Graphing rational functions—4. Drawing the graph

In previous posts, I described the steps you follow to analyze a rational function. If you follow these steps, you can be a rational function superstar too. In this post, I put all the pieces together to show how you use the information you’ve obtained to plot the graph. [For details on how to execute the various steps, please see other posts on this website.]

Example: Graph the following rational function.

$f(x)=\dfrac{2x^3-16x^2+38x-24}{x^3-4x^2+x+6}$

Solution: We break this into many steps.

1. Find the $y$-intercept:

$f(0)=\dfrac{-24}{6}=-4$

2. Fully factor the function:

$f(x)=\dfrac{2x^3-16x^2+38x-24}{x^3-4x^2+x+6}=\dfrac{2(x-1)(x-3)(x-4)}{(x+1)(x-2)(x-3)}$

3. Find the vertical asymptotes, if any:

The vertical asymptotes are at $x=-1$ and $x=2$.

4.Find the $x$-intercept(s), if any:

$f(x)=0$ @ $x=1$ and $x=4$

5. Find the hole(s), if any:

There is a hole at the point $(3,-1)$

6. Find the horizontal or oblique asymptote, if any, or characterize the end behavior:

The horizontal asymptote is at $y=2$

7. Place all of this information on the graph:

8. Use all the information you have plotted to complete the graph of the function:

Graphing rational functions—3. Finding horizontal or oblique asymptotes

In a previous post, I listed the steps you need to follow to analyze and graph a rational function. In this post, I show you how to find the horizontal or oblique asymptotes, if they exist. If there are no horizontal or oblique asymptotes, then you can determine the end behavior of the function. [Check other posts on this website for other steps in analyzing and graphing a rational function.]

The horizontal and oblique asymptotes, if they exist, tell you the end behavior of the function. That is, they describe what the function is doing as $x$ goes to ±∞. In order to determine the end behavior, examine the degrees of the polynomials in the numerator and denominator. There are four possible cases:

Case 1: The degree of the denominator is greater than the degree of the numerator.

When the denominator has the higher degree, the denominator grows much faster than the numerator as $x$ gets very large. Like all fractions, as the denominator gets very large, the fraction gets very small and approaches zero. So the asymptote is $y=0$.

Example 1: Find the end behavior for the following rational function:

$f(x)=\dfrac{x^2+3x+2}{x^3+4x^2+x-6}$

Solution: Because the degree of the numerator is $2$ and the degree of the denominator is $3$, this function has a horizontal asymptote $y=0$.

Case 2: The degree of the denominator is equal to the degree of the numerator.

When the degrees of the numerator and denominator are the same, the function approaches a finite non-zero number. This number is the ratio of the leading coefficients of the two polynomials and this is the value of the horizontal asymptote.

Example 2: Find the end behavior for the following rational function:

$f(x)=\dfrac{6x^2+7x-4}{5x^2-4x-6}$

Solution: Because the degree of the numerator and denominator are the same, this function has a horizontal asymptote. Divide the leading coefficients ($6$ and $5$) and the horizontal asymptote is $y=6/5$.

Case 3: The degree of the numerator is exactly one higher than the degree of the denominator.

In this case, there is an oblique asymptote (called a slant asymptote in some textbooks). To find the asymptote, divide the denominator into the numerator using long division. The quotient is the oblique asymptote. (Ignore the remainder; it has no effect on the asymptote.)

Example 3: Find the end behavior for the following rational function:

$f(x)=\dfrac{x^3+4x^2+x-6}{x^2+3x+2}$

Here, the numerator (degree 3) is exactly one higher than the denominator (degree 2). To find the asymptote, perform the following long division:

The quotient is the oblique asymptote:

$y=x+1$

Case 4: The degree of the numerator is more than one higher than the degree of the denominator.

In this case, there is no horizontal or oblique asymptote. Instead, the end behavior of the function is the same as the end behavior of a polynomial whose degree is the same as the difference between the degrees of the numerator and the denominator. For example, if the numerator is degree $5$ and the denominator is degree $2$, the rational function will look like a cubic function as $x$ approaches ±∞.

Graphing rational functions—2. Finding holes and vertical asymptotes

In a previous post, I listed the steps you need to follow to analyze and graph a rational function. In this post, I show you how to find the hole(s) and vertical asymptote(s), if they exist. [Check other posts on this website for other steps in analyzing and graphing a rational function.]

In a rational function, holes and vertical asymptotes will only occur where the denominator is equal to zero. So fully factor the function, and look for the roots of the denominator. When a factor in the denominator cancels out with a factor in the numerator, the function will have a hole where the factor equals zero. If the factor in the denominator does not cancel out with a factor in the numerator, there will be a vertical asymptote where the factor equals zero.

Example: Find the holes and vertical asymptotes, if any, for the following rational function:

$f(x)=\dfrac{x^2+3x+2}{x^3+4x^2+x-6}$

First, fully factor the function:

$f(x)=\dfrac{x^2+3x+2}{x^3+4x^2+x-6}=\dfrac{(x+1)(x+2)}{(x-1)(x+2)(x+3)}=\dfrac{(x+1)}{(x-1)(x+3)}$

The roots of the denominator are $-3$, $-2$ and $1$. At $x=-2$, there is a hole. To find the “value” of the hole, evaluate the reduced function:

$f(-2)=\dfrac{(-2+1)}{(-2-1)(-2+3)}=\dfrac{-1}{-3}=\dfrac{1}{3}$

This means that when you graph the function, there will be a hole at the point $(-2, 1/3)$.

The vertical asymptotes will be $x=-3$ and $x=1$.

Graphing rational functions—1. Finding x- and y- intercepts

In a previous post, I listed the steps you need to follow to analyze and graph a rational function. In this and subsequent posts, I will walk you through each of those steps. Usually the first step in plotting a rational function is finding the $x$– and $y$-intercepts. You have done this before for other functions and the process here is no different.

• To find the $y$-intercept, set $x=0$ and solve for $f(x)$.
• To find the $x$-intercept(s), set $f(x)=0$ and solve for $x$. Recall that a rational function is a fraction, and a fraction can only be equal to zero when its numerator is equal to zero (as long as the denominator is not also equal to zero!). So in practice, you find the $x$-intercept(s) by setting the numerator equal to zero and solving for $x$.

Example: Find the $x$– and $y$-intercepts, if any, for the following rational function:

$f(x) = \dfrac{x^3+4x^2+x-6}{x^2+3x+2}$

Finding the $y$-intercept is very easy if the polynomials are written in standard form as above. $f(0)$ is simply the ratio of the constant terms: $\frac{-6}{2} = -3$. If the rational function has already been factored, you need to plug zero in to each term and evaluate.

To find the $x$-intercept (and all the other parts of the rational function), the first thing you need to do is fully factor the numerator and denominator.

$f(x)=\dfrac{x^3+4x^2+x-6}{x^2+3x+2}=\dfrac{(x-1)(x+2)(x+3)}{(x+1)(x+2)}$

From here, it is easy to see that the roots of the numerator are $-3$, $-2$ and $1$. When $x=-2$, the denominator is also zero, so this is not an $x$-intercept; it is a hole. The $x$‑intercepts are at $-3$ and $1$.

Is it possible that a rational function does not have a $y$-intercept? Yes, if the function is not defined for $x=0$. In other words, if a rational function has either a hole or a vertical asymptote at $x=0$, there is no $y$-intercept.

[Check out my other posts in this section that describe how to perform the other steps when analyzing a rational function.]

Graphing rational functions—overview

Graphing rational functions can be scary for a lot of students because there are so many details to manage. But you can make the process simpler by breaking down all the requirements into small steps. In this post I list all of the steps you should follow to analyze a rational function. Note that you can do these steps in any order. In subsequent posts, I will walk you through each of the steps and show you how to execute them.

Let’s start with the definition of a rational function. A rational function is a function that consists of a fraction, where both the numerator and denominator are polynomials. Or in symbols, if g(x) and h(x) are polynomials, then

$f(x)=\dfrac{g(x)}{h(x)}$

is called a rational function.

To analyze and graph a rational function, you need to do all of the following steps:

• Find the y-intercept (if it exists)
• Find the x-intercept(s) (if they exist)
• Determine the location of holes, if any
• Find the vertical asymptotes, if any
• Find the horizontal or oblique asymptotes, if any
• Determine the end behavior (if no horizontal or oblique asymptotes)

[Note that this is a list for algebra 2 and precalc students. When you get to calculus, there will be additional steps to completely analyze the function.]

This probably feels like a long and complicated list, but for the most part, each step by itself is not very difficult. Just perform each step in turn and you can become an expert at rational functions. Check out my other posts in this section that describe how to perform each of these steps.

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