12.3: Horizontal and Vertical Asymptotes
Suppose you had a function like
Try This
To explore more graphs of rational functions, try the applets available at http://www.analyzemath.com/rational/rational1.html.
Watch This
CK12 Foundation: 1203S Horizontal and Vertical Asymptotes
Guidance
We said that a horizontal asymptote is the value of
If we plug in a large value of
We can see that the beginning terms in the numerator and denominator are much bigger than the other terms in each expression. One way to find the horizontal asymptote of a rational function is to ignore all terms in the numerator and denominator except for the highest powers.
In this example the horizontal asymptote is
In the function above, the highest power of
As before, we ignore all the terms except the highest power of
For large values of
To summarize:
 Find vertical asymptotes by setting the denominator equal to zero and solving for
x .  For horizontal asymptotes, we must consider several cases:
 If the highest power of
x in the numerator is less than the highest power ofx in the denominator, then the horizontal asymptote is aty=0 .  If the highest power of
x in the numerator is the same as the highest power ofx in the denominator, then the horizontal asymptote is aty=coefficient of highest power of xcoefficient of highest power of x .  If the highest power of
x in the numerator is greater than the highest power ofx in the denominator, then we don’t have a horizontal asymptote; we could have what is called an oblique (slant) asymptote, or no asymptote at all.
 If the highest power of
Example A
Find the vertical and horizontal asymptotes for
Solution
Vertical asymptotes:
Set the denominator equal to zero.
Horizontal asymptote:
Keep only the highest powers of
Example B
Find the vertical and horizontal asymptotes for
Solution
Vertical asymptotes:
Set the denominator equal to zero.
Horizontal asymptote:
Keep only the highest powers of
Example C
Find the vertical and horizontal asymptotes for
Solution
Vertical asymptotes:
Set the denominator equal to zero:
Factor:
Solve:
Horizontal asymptote. There is no horizontal asymptote because the power of the numerator is larger than the power of the denominator.
Notice the function in part d had more than one vertical asymptote. Here’s another function with two vertical asymptotes.
Example D
Graph the function
Solution
We find that the function is undefined for
We can also find the horizontal asymptote by the method we outlined above. It’s at
So, we start plotting the function by drawing the vertical and horizontal asymptotes on the graph.
Now, let’s make a table of values. Because our function has a lot of detail we must make sure that we pick enough values for our table to determine the behavior of the function accurately. We must make sure especially that we pick values close to the vertical asymptotes.
\begin{align*}y = \frac {x^2}{x^24}\end{align*} 


\begin{align*}5\end{align*} 
\begin{align*}y = \frac {(5)^2}{(5)^24} = \frac {25}{21} = 1.19\end{align*} 
4  \begin{align*}y = \frac {(4)^2}{(4)^24} = \frac {16}{12} = 1.33\end{align*} 
3  \begin{align*}y = \frac {(3)^2}{(3)^24} = \frac {9}{5} = 1.8\end{align*} 
2.5  \begin{align*}y = \frac {(2.5)^2}{(2.5)^24} = \frac {6.25}{2.25} = 2.8\end{align*} 
1.5  \begin{align*}y = \frac {(1.5)^2}{(1.5)^24} = \frac {2.25}{1.75} = 1.3\end{align*} 
1  \begin{align*}y = \frac {(1)^2}{(1)^24} = \frac {1}{3} = 0.33\end{align*} 
0  \begin{align*}y = \frac {0^2}{(0)^24} = \frac {0}{4} = 0\end{align*} 
1  \begin{align*}y= \frac {1^2}{(1)^24} = \frac {1}{3} = 0.33\end{align*} 
1.5  \begin{align*}y = \frac {1.5^2}{(1.5)^24} = \frac {2.25}{1.75} = 1.3\end{align*} 
2.5  \begin{align*}y = \frac {2.5^2}{(2.5)^24} = \frac {6.25}{2.25} = 2.8\end{align*} 
3  \begin{align*}y = \frac {3^2}{(3)^24} = \frac {9}{5} = 1.8\end{align*} 
4  \begin{align*}y = \frac {4^2}{(4)^24} = \frac {16}{12} = 1.33\end{align*} 
5  \begin{align*}y = \frac {5^2}{(5)^24} = \frac {25}{21} = 1.19\end{align*} 
Here is the resulting graph.
Watch this video for help with the Examples above.
CK12 Foundation: Horizontal and Vertical Asymptotes
Guided Practice
Find the vertical and horizontal asymptotes for \begin{align*}y=\frac {x^22}{2x^2+3}\end{align*}.
Solution
Vertical asymptotes:
Set the denominator equal to zero: \begin{align*} 2x^2+3 = 0 \Rightarrow 2x^2 = 3 \Rightarrow x^2 = \frac{3}{2}\end{align*}. Since there are no solutions to this equation, there is no vertical asymptote.
Horizontal asymptote:
Keep only the highest powers of \begin{align*}x\end{align*}. \begin{align*}y=\frac {x^2}{2x^2} \Rightarrow y= \frac {1}{2}\end{align*} is the horizontal asymptote.
Explore More
Find all the vertical and horizontal asymptotes of the following rational functions.
 \begin{align*}y=\frac {4}{x+2}\end{align*}
 \begin{align*}y=\frac {5x1}{2x6}\end{align*}
 \begin{align*}y=\frac {10}{x}\end{align*}
 \begin{align*}y=\frac {2}{x}5\end{align*}
 \begin{align*}y=\frac {x + 1}{x^2}\end{align*}
 \begin{align*}y=\frac {4x^2}{4x^2+1}\end{align*}
 \begin{align*}y=\frac {2x}{x^29}\end{align*}
 \begin{align*}y=\frac {3x^2}{x^24}\end{align*}
 \begin{align*}y=\frac {1}{x^2+4x+3}\end{align*}
 \begin{align*}y=\frac {2x+5}{x^22x8}\end{align*}
Answers for Explore More Problems
To view the Explore More answers, open this PDF file and look for section 12.3.
Asymptotes
An asymptote is a line on the graph of a function representing a value toward which the function may approach, but does not reach (with certain exceptions).Hole
A hole exists on the graph of a rational function at any input value that causes both the numerator and denominator of the function to be equal to zero.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 and axes. An intercept is a point at which the curve intersects the axis. A intercept is a point at which the curve intersects the axis.Intercepts
The intercepts of a curve are the locations where the curve intersects the and axes. An intercept is a point at which the curve intersects the axis. A intercept is a point at which the curve intersects the 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.points of discontinuity
The points of discontinuity for a function are the input values of the function where the function is discontinuous.Image Attributions
Description
Learning Objectives
Here you'l learn how to find the horizontal and vertical asymptotes of radical equations.
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Date Created:
Oct 01, 2012Last Modified:
Jan 31, 2016Vocabulary
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