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2.6: Oblique Asymptotes

Difficulty Level: At Grade Created by: CK-12
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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?

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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 g(x)=x3x2+1\begin{align*}g(x)=\frac{x^{3}}{x^{2}+1}\end{align*}, we can see that there is no horizontal asymptote of this 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 r(x)=f(x)D(x)\begin{align*}r(x)=\frac{f(x)}{D(x)}\end{align*}:

• If the degree of the denominator is greater than the degree of the numerator, then the line y=0\begin{align*}y=0\end{align*} is a horizontal asymptote.
• If the degree of the numerator and the denominator are equal, then the line y=ab\begin{align*}y=\frac{a}{b}\end{align*} is a horizontal asymptote, where a\begin{align*}a\end{align*} is the leading coefficient of f(x)\begin{align*}f(x)\end{align*}, the numerator, and b\begin{align*}b\end{align*} is the leading coefficient of D(x)\begin{align*}D(x)\end{align*}, the denominator.
• If the degree of the numerator is larger than the degree of the denominator, then the quotient function, Q(x)\begin{align*}Q(x)\end{align*}, found by dividing the numerator and denominator of the rational function, is an oblique asymptote. Recall that for any rational function f(x)D(X)\begin{align*}\frac{f(x)}{D(X)}\end{align*}, you can use polynomial division to re-write that function in the form f(x)D(x)=Q(x)+R(x)D(x)\begin{align*}\frac{f(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}\end{align*} where Q(x)\begin{align*}Q(x)\end{align*} is the quotient and R(x)\begin{align*}R(x)\end{align*} is the remainder.

Example A

Graph

g(x)=x21x2\begin{align*}g(x)=\frac{x^{2}-1}{x-2}\end{align*}

Solution

First observe that the vertical asymptote is at x=2\begin{align*}x=2\end{align*}. Notice that the degree of the numerator is greater than the degree of the denominator. We can change the form of the rational expression by long division. You may recall from algebra that polynomials can be divided (just like real numbers), and any rational functions can be written as

f(x)D(x)=Q(x)+R(x)D(x)\begin{align*}\frac{f(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}\end{align*}

Doing the long division here,

x+2x2 )x2+0x1¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯x22x  2x12x4 3\begin{align*}& \qquad \qquad x + 2 \\ & x-2 \ \big ) \overline{x^{2} + 0x -1 }\\ & \qquad \quad \underline{x^{2} -2x \ \ \downarrow}\\ & \qquad \qquad \quad 2x -1\\ & \qquad \qquad \quad \underline{2x -4}\\ & \qquad \qquad \qquad \quad \ 3 \end{align*}

So in this case, the function g(x)\begin{align*}g(x)\end{align*} can be rewritten as

g(x)=x21x2=x+2+3x2\begin{align*}g(x)=\frac{x^{2}-1}{x-2}=x+2+\frac{3}{x-2}\end{align*}

The above equation tells us that as x±\begin{align*}x\to\pm\infty\end{align*}, the graph of g(x)=x+2+3x2\begin{align*}g(x)=x+2+\frac{3}{x-2}\end{align*} gets closer and closer to the line y=x+2\begin{align*}y=x+2\end{align*}. Why? Suppose we let x\begin{align*}x\end{align*} be a big number, i.e. x=1,000,000\begin{align*}x=1,000,000\end{align*}. Then the remainder of this rational function becomes 3999,9980\begin{align*}\frac{3}{999,998}\approx 0\end{align*} and we are left with x+2\begin{align*}x+2\end{align*}. We call this line an oblique asymptote and it is indicated by the dashed line in the image below.

Example B

Identify the oblique asymptote of g(x)=x3x2+1\begin{align*}g(x)=\frac{x^{3}}{x^{2}+1}\end{align*}

Solution

By polynomial division we have,

x Quotientx2+1 )x3+0x2+0x+0¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ Dividend x3+x x Remainder\begin{align*}& \qquad \quad \ \ x \qquad \qquad \qquad \qquad \ \leftarrow\text{Quotient}\\ & x^2+1 \ \big ) \overline{x^3 + 0x^2 + 0x + 0} \quad \ \leftarrow \text{Dividend}\\ & \qquad \quad \ \underline{x^3 \qquad \quad + x}\\ & \qquad \qquad \qquad \quad \ -x \qquad \quad \ \leftarrow \text{Remainder}\end{align*}

So \begin{align*}g(x)=x-\frac{x}{x^{2}+1}\end{align*}. This tells us that the line \begin{align*}y=x\end{align*} is an oblique asymptote of \begin{align*}g(x)\end{align*}.

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 \begin{align*}Q(x)\end{align*} in \begin{align*}\frac{f(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}\end{align*}.

Example C

Find the oblique asymptote of \begin{align*}f(x)=\frac{3x^{3}-5x^{2}+2}{x^{2}-3x}\end{align*}. Sketch a graph of \begin{align*}f(x)\end{align*}.

Solution

Using polynomial long division, \begin{align*}f(x)=3x+4+\frac{4x^{2}}{x^{2}-3x}\end{align*}. Thus, the line \begin{align*}y=3x+4\end{align*} is an oblique asymptote of \begin{align*}f(x)\end{align*}.

To sketch the graph we can find the vertical asymptotes by setting the denominator equal to zero,

\begin{align*}x^{2}-3x & = 0\\ x(x-3) & = 0\end{align*}

So the two vertical asymptotes are \begin{align*}x=0\end{align*} and \begin{align*}x=3\end{align*}.

\begin{align*}f(0)\end{align*} is undefined, so there is no \begin{align*}y-\end{align*}intercept. Also, there is no simple way to solve for the roots (setting the numerator equal to zero), but we can see by inspection that \begin{align*}f(1)=0\end{align*}. To get an idea of the shape of the graph we will make a table of a few test points. We used a calculator to evaluate decimal values of \begin{align*}x\end{align*} in \begin{align*}f(x)\end{align*}.

\begin{align*}& x && -2 && -1 && -0.1 && 1 && \ \ \ 2 && \ 4 && 3.5 && \ 5 && \ 6\\ & f(x) && -4.2 && -1.5 && 66.48 && 0 && -3 && 28.5 && 39.6 && 25.2 && 26.1\end{align*}

Finally we use all of this information to make a sketch of the graph of \begin{align*}f(x)\end{align*}:

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.

Vocabulary

Oblique (or 'Slant') Asymptote: A diagonal line marking a specific range of values toward which the graph of a function may approach, but will never reach. A slant asymptote exists when the numerator is exactly one degree greater than the denominator. A slant asymptote may be found through long division.

Guided Practice

Questions

1) Graph \begin{align*}f(x)=\frac{x^{2}-x-2}{x-1}\end{align*}

2) Find the asymptote(s) of \begin{align*}\frac{x^2 + 4x -21}{x + 3}\end{align*}

3) Identify the asymptote(s) of \begin{align*}\frac{1+x^2}{x}+\frac{x^2-1}{x}+\frac{x}{x^2-1}\end{align*}

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 x-intercepts, Factor the numerator:
\begin{align*}f(x)=\frac{x^{2}-x-2}{x-1}=\frac{(x-2)(x+1)}{x-1}\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{(x-2)(x+1)}{x-1} = 0\end{align*} true.
To identify the oblique asymptote, we divide \begin{align*}\frac{x^{2}-x-2}{x-1}\end{align*} using polynomial long division, yielding:
\begin{align*}f(x)=x-\frac{2}{x-1}\end{align*}
Recall that the whole (non-fractional) 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 x-intercepts, Factor the numerator:
\begin{align*}f(x) = \frac{(x-3)(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{(x-3)(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 (non-fractional) 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^2-1}{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^2-1}{x}\end{align*} we can see a hole in the graph at (0, 0)
The sketch of the graph would look like this:

Practice

1. 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:

1. \begin{align*} y = \frac{3x^3}{x^2 - 1}\end{align*}
2. \begin{align*} y = \frac{2x^2}{x + 1}\end{align*}
3. \begin{align*} y = \frac{2x^3 - 7x^2 - 4}{(x+3)(x - 1)}\end{align*}
4. \begin{align*} y = \frac{(2x)(x + 11)}{x - 4}\end{align*}
5. \begin{align*} y = \frac{x^3 - x + 3}{x^2 + x - 2}\end{align*}
6. \begin{align*}f(x) = \frac{x^2 - 4}{x}\end{align*}
7. \begin{align*}f(x) = \frac{x^3 - 3}{x^2}\end{align*}
8. \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.

1. \begin{align*} f(x)= \frac{2x^2}{1 - x}\end{align*}
2. \begin{align*} f(x)= \frac{x^3 - 3x^2}{x^2 - 1}\end{align*}
3. \begin{align*} f(x)= \frac{x^3 - 1}{x^2 - x - 2}\end{align*}
4. \begin{align*} f(x)= \frac{x^3 - 1}{2(x^2 - 1)}\end{align*}
5. \begin{align*} y = \frac{2x^2}{x - 3}\end{align*}
6. \begin{align*} y = \frac{3x^2}{x+2}\end{align*}

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Vocabulary Language: English

Degree

The degree of a polynomial is the largest exponent of the polynomial.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that indicates where a function flattens out as the independent variable gets very large or very small. A function may touch or pass through a horizontal asymptote.

Intercept

The intercepts of a curve are the locations where the curve intersects the $x$ and $y$ axes. An $x$ intercept is a point at which the curve intersects the $x$-axis. A $y$ intercept is a point at which the curve intersects the $y$-axis.

Oblique Asymptote

An oblique asymptote is a diagonal line marking a specific range of values toward which the graph of a function may approach, but generally never reach. An oblique asymptote exists when the numerator of the function is exactly one degree greater than the denominator. An oblique asymptote may be found through long division.

Oblique Asymptotes

An oblique asymptote is a diagonal line marking a specific range of values toward which the graph of a function may approach, but generally never reach. An oblique asymptote exists when the numerator of the function is exactly one degree greater than the denominator. An oblique asymptote may be found through long division.

Rational Equation

A rational equation is an equation that contains a rational expression.

Rational Function

A rational function is any function that can be written as the ratio of two polynomial functions.

Slant Asymptote

A slant asymptote is a diagonal line marking a specific range of values toward which the graph of a function may approach, but will never reach. A slant asymptote exists when the numerator of the function is exactly one degree greater than the denominator. A slant asymptote may be found through long division.

Transformations

Transformations are used to change the graph of a parent function into the graph of a more complex function.

Vertical Asymptote

A vertical asymptote is a vertical line marking a specific value toward which the graph of a function may approach, but will never reach.

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