Working with rational functions is often a matter of finding what the graph isn't as much as finding what it is. By identifying the values that the function cannot have, either as an input or as an output, we restrict the possibilities of what the graph may look like.
You have previously graphed rational functions using transformations and horizontal and vertical asymptotes. You may have noted that horizontal asymptotes appear in rational equations when the degree of the numerator is less than or equal to the degree of the denominator.
What happens when the degree of the numerator is greater than the degree of the denominator? How does this situation appear in the graph of the function?
Watch This
James Sousa: Determining Slant Asymptotes of Rational Functions
Guidance
Oblique Asymptotes
In previous lessons, we have considered both horizontal and vertical asymptotes, but not all asymptotes of rational functions are vertical or horizontal. In this lesson we will consider what happens when the degree of the numerator is one greater than the denominator, resulting in a diagonal line known as an oblique or slant asymptote.
If we look at the graph of the rational function
There is no horizontal asymptote in this function because the degree of the numerator is greater than the degree of the denominator.
As a reminder, the following guidelines can help identify the asymptotes of a rational function
 If the degree of the denominator is greater than the degree of the numerator, then the line
y=0 is a horizontal asymptote.
 If the degree of the numerator and the denominator are equal, then the line
y=ab is a horizontal asymptote, wherea is the leading coefficient off(x) , the numerator, andb is the leading coefficient ofD(x) , the denominator.
 If the degree of the numerator is larger than the degree of the denominator, then the quotient function,
Q(x) , found by dividing the numerator and denominator of the rational function, is an oblique asymptote. Recall that for any rational functionf(x)D(X) , you can use polynomial division to rewrite that function in the formf(x)D(x)=Q(x)+R(x)D(x) whereQ(x) is the quotient andR(x) is the remainder.
Example A
Graph
Solution:
First observe that the vertical asymptote is at
Doing the long division here,
So in this case, the function
The above equation tells us that as
Example B
Identify the oblique asymptote of
Solution:
By polynomial division we have,
So
Notice that the oblique asymptotes of a rational function also describe the end behavior of the function. That is, as you “zoom out” from the graph of a rational function it looks like a line or the function defined by
Example C
Find the oblique asymptote of
Solution:
Using polynomial long division,
To sketch the graph we can find the vertical asymptotes by setting the denominator equal to zero,
So the two vertical asymptotes are
Finally we use all of this information to make a sketch of the graph of
Concept question wrapup: Do you remember the question at the beginning of the lesson? What happens when the degree of the numerator is greater than the degree of the denominator? How does this situation appear in the graph of the function? You should have no difficulty with this question now: When the degree of the numerator is greater, there is no horizontal asymptote, but rather a slant or oblique asymptote. It appears as a diagonal line across the graph of the function. 

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Guided Practice
1) Graph
2) Find the asymptote(s) of
3) Identify the asymptote(s) of \begin{align*}\frac{1+x^2}{x}+\frac{x^21}{x}+\frac{x}{x^21}\end{align*}
Answers
1) The vertical asymptote here is \begin{align*}x=1\end{align*} since a 1 for x in the denominator makes the fraction undefined.
 To find the xintercepts, factor the numerator:
 \begin{align*}f(x)=\frac{x^{2}x2}{x1}=\frac{(x2)(x+1)}{x1}\end{align*}
 Notice that the \begin{align*}x\end{align*}intercepts are at \begin{align*}x=2\end{align*} and \begin{align*}x=1\end{align*} since those are the values which make \begin{align*}\frac{(x2)(x+1)}{x1} = 0\end{align*} true.
 To identify the oblique asymptote, we divide \begin{align*}\frac{x^{2}x2}{x1}\end{align*} using polynomial long division, yielding:
 \begin{align*}f(x)=x\frac{2}{x1}\end{align*}
 Recall that the whole (nonfractional) part of the quotient indicates the oblique asymptote, so we have \begin{align*}y=x\end{align*}.
 Make a table of points, then sketch the graph using those points and the asymptote.

x f(x) 2 0 (1) 0 0 2 3 0
2) The vertical asymptote here is \begin{align*}x=3\end{align*} since a 3 for x in the denominator makes the fraction undefined.
 To find the xintercepts, Factor the numerator:
 \begin{align*}f(x) = \frac{(x3)(x+7)}{x+3}\end{align*}
 Notice that the \begin{align*}x\end{align*}intercepts are at \begin{align*}x = 3\end{align*} and \begin{align*}x = 7\end{align*} since those are the values which make \begin{align*}\frac{(x3)(x+7)}{x+3} = 0\end{align*} true.
 To identify the oblique asymptote, we divide \begin{align*}\frac{x^{2} + 4x 21}{x + 3}\end{align*} using polynomial long division, yielding:
 \begin{align*}f(x)=x + 1 \frac{24}{x + 3}\end{align*}
 Recall that the whole (nonfractional) part of the quotient indicates the oblique asymptote, so we have \begin{align*}y = x + 1\end{align*} as the oblique asymptote.
 The graph would look like this:
3) First, we need to simplify the expressions \begin{align*}\frac{1+x^2}{x}+\frac{x^21}{x}\end{align*}
 Since the denominators are the same, we can just add the numerators, yielding: \begin{align*}\frac{1 + x^2 + x^2 1}{x} ==> 2x\end{align*}
 Now we have: \begin{align*}2x+\frac{x}{x^2  1}\end{align*} convenient! No polynomial division necessary!
 The slant asymptote is \begin{align*}y = 2x\end{align*}
 Looking at \begin{align*}2x+\frac{x}{x^2  1}\end{align*} we can see that the vertical asymptotes are at \begin{align*}x = 1\end{align*} and \begin{align*}x = 1\end{align*}
 Looking at the original form: \begin{align*}\frac{1+x^2}{x}+\frac{x^21}{x}\end{align*} we can see a hole in the graph at (0, 0)
 The sketch of the graph would look like this:
Explore More
 What has to be true of the degree of the numerator and the denominator for a asymptote to be called oblique or slant?
Find the slant asymptotes:
 \begin{align*} y = \frac{3x^3}{x^2  1}\end{align*}
 \begin{align*} y = \frac{2x^2}{x + 1}\end{align*}
 \begin{align*} y = \frac{2x^3  7x^2  4}{(x+3)(x  1)}\end{align*}
 \begin{align*} y = \frac{(2x)(x + 11)}{x  4}\end{align*}
 \begin{align*} y = \frac{x^3  x + 3}{x^2 + x  2}\end{align*}
 \begin{align*}f(x) = \frac{x^2  4}{x}\end{align*}
 \begin{align*}f(x) = \frac{x^3  3}{x^2}\end{align*}
 \begin{align*}y = \frac{3x^3  3}{2x^2}\end{align*}
Find all intercepts and asymptotes for the graphs of the following rational functions and use that information to help you sketch the graphs of the functions.
 \begin{align*} f(x)= \frac{2x^2}{1  x}\end{align*}
 \begin{align*} f(x)= \frac{x^3  3x^2}{x^2  1}\end{align*}
 \begin{align*} f(x)= \frac{x^3  1}{x^2  x  2}\end{align*}
 \begin{align*} f(x)= \frac{x^3  1}{2(x^2  1)}\end{align*}
 \begin{align*} y = \frac{2x^2}{x  3}\end{align*}
 \begin{align*} y = \frac{3x^2}{x+2}\end{align*}
Answers for Explore More Problems
To view the Explore More answers, open this PDF file and look for section 2.6.