What if you had a function like \begin{align*}y=\frac{3x^2 - 2x + 1}{x + 2}\end{align*}? How could you rewrite it to find its asymptotes? After completing this Concept, you'll be able to rewrite rational functions like this one using division.

### Watch This

CK-12 Foundation: 1206S Rewriting Rational Functions Using Division

### Guidance

In the last section we saw how to find vertical and horizontal asymptotes. Remember, the horizontal asymptote shows the value of \begin{align*}y\end{align*} that the function approaches for large values of \begin{align*}x\end{align*}. Let’s review the method for finding horizontal asymptotes and see how it’s related to polynomial division.

When it comes to finding asymptotes, there are basically four different types of rational functions.

**Case 1:** The polynomial in the numerator has a lower degree than the polynomial in the denominator.

#### Example A

*Find the horizontal asymptote of \begin{align*}y=\frac{2}{x-1}\end{align*}.*

**Solution:**

We can’t reduce this fraction, and as \begin{align*}x\end{align*} gets larger the denominator of the fraction gets much bigger than the numerator, so the whole fraction approaches zero.

**The horizontal asymptote is \begin{align*}y = 0\end{align*}.**

**Case 2:** The polynomial in the numerator has the same degree as the polynomial in the denominator.

#### Example B

*Find the horizontal asymptote of \begin{align*}y=\frac{3x+2}{x-1}\end{align*}.*

**Solution:**

In this case we can divide the two polynomials:

\begin{align*}& \overset{\qquad \qquad \ 3}{x-1 \overline{ ) 3x+2 \;}}\\ & \qquad \underline{-3x+3}\\ & \qquad \qquad \quad 5\end{align*}

So the expression can be written as \begin{align*}y=3+\frac{5}{x-1}\end{align*}.

Because the denominator of the remainder is bigger than the numerator of the remainder, the remainder will approach zero for large values of \begin{align*}x\end{align*}. Adding the 3 to that 0 means the whole expression will approach 3.

**The horizontal asymptote is \begin{align*}y = 3\end{align*}.**

**Case 3:** The polynomial in the numerator has a degree that is one more than the polynomial in the denominator.

#### Example C

*Find any asymptotes of \begin{align*}y=\frac{4x^2+3x+2}{x-1}\end{align*}.*

**Solution:**

We can do long division once again and rewrite the expression as \begin{align*}y=4x+7+\frac{9}{x-1}\end{align*}. The fraction here approaches zero for large values of \begin{align*}x\end{align*}, so the whole expression approaches \begin{align*}4x + 7\end{align*}.

When the rational function approaches a straight line for large values of \begin{align*}x\end{align*}, we say that the rational function has an **oblique asymptote.** In this case, then, **the oblique asymptote is \begin{align*}y = 4x + 7\end{align*}.**

**Case 4:** The polynomial in the numerator has a degree that is two or more than the degree in the denominator.

#### Example D

*Find any asymptotes of \begin{align*}y=\frac{x^3}{x-1}\end{align*}.*

This is actually the simplest case of all: **the polynomial has no horizontal or oblique asymptotes.**

Notice that a rational function will either have a horizontal asymptote, an oblique asymptote or neither kind. In other words, a function can’t have both; in fact, it can’t have more than one of either kind. On the other hand, a rational function can have any number of *vertical* asymptotes at the same time that it has horizontal or oblique asymptotes.

Watch this video for help with the Examples above.

CK-12 Foundation: Rewriting Rational Functions Using Division

### Vocabulary

- When the rational function approaches a straight line for large values of \begin{align*}x\end{align*}, we say that the rational function has an
**oblique asymptote.**

### Guided Practice

*Find the horizontal or oblique asymptotes of the following rational functions.*

a) \begin{align*}y=\frac{3x^2}{x^2+4}\end{align*}

b) \begin{align*}y=\frac{x-1}{3x^2-6}\end{align*}

c) \begin{align*}y=\frac{x^4+1}{x-5}\end{align*}

d) \begin{align*}y=\frac{x^3-3x^2+4x-1}{x^2-2}\end{align*}

**Solution**

a) When we simplify the function, we get \begin{align*}y=3-\frac{12}{x^2+4}\end{align*}. **There is a horizontal asymptote at \begin{align*}y = 3\end{align*}.**

b) We cannot divide the two polynomials. **There is a horizontal asymptote at \begin{align*}y = 0\end{align*}.**

c) The power of the numerator is 3 more than the power of the denominator. **There are no horizontal or oblique asymptotes.**

d) When we simplify the function, we get \begin{align*}y=x-3+\frac{6x-7}{x^2-2}\end{align*}. **There is an oblique asymptote at \begin{align*}y = x - 3\end{align*}.**

### Practice

Find all asymptotes of the following rational functions:

- \begin{align*}\frac{x^2}{x-2}\end{align*}
- \begin{align*}\frac{1}{x+4}\end{align*}
- \begin{align*}\frac{x^2-1}{x^2+1}\end{align*}
- \begin{align*}\frac{x-4}{x^2-9}\end{align*}
- \begin{align*}\frac{x^2+2x+1}{4x-1}\end{align*}
- \begin{align*}\frac{x^3+1}{4x-1}\end{align*}
- \begin{align*}\frac{x-x^3}{x^2-6x-7}\end{align*}
- \begin{align*}\frac{x^4-2x}{8x+24}\end{align*}

Graph the following rational functions. Indicate all asymptotes on the graph:

- \begin{align*}\frac{x^2}{x+2}\end{align*}
- \begin{align*}\frac{x^3-1}{x^2-4}\end{align*}
- \begin{align*}\frac{x^2+1}{2x-4}\end{align*}
- \begin{align*}\frac{x-x^2}{3x+2}\end{align*}