12.2: Graphs of Rational Functions
In the previous lesson, you learned the basics of graphing an inverse variation function. The hyperbola forms two branches in opposite quadrants. The axes are asymptotes to the graph. This lesson will compare graphs of inverse variation functions. You will also learn how to graph other rational equations.
Example: Graph the function \begin{align*}f(x)=\frac{k}{x}\end{align*}
\begin{align*}k=2, 1, \frac{1}{2}, 1, 2, 4\end{align*}
Each graph is shown separately then on one coordinate plane.
As mentioned in the previous lesson, if \begin{align*}k\end{align*}
Rational Functions
A rational function is a ratio of two polynomials (a polynomial divided by another polynomial). The formal definition is:
\begin{align*}f(x)=\frac{g(x)}{h(x)}, \text{where} \ h(x) \neq 0\end{align*}
An asymptote is a value for which the equation or function is undefined. Asymptotes can be vertical, horizontal, or oblique. This text will focus on vertical asymptotes; other math courses will also show you how to find horizontal and oblique asymptotes. A function is undefined when the denominator of a fraction is zero. To find the asymptotes, find where the denominator of the rational function is zero. These are called points of discontinuity of the function.
The formal definition for asymptote is as follows.
An asymptote is a straight line to which, as the distance from the origin gets larger, a curve gets closer and closer but never intersects.
Example: Find the points of discontinuity and the asymptote for the function \begin{align*}y=\frac{6}{x5}\end{align*}
Solution: Find the value of \begin{align*}x\end{align*}
\begin{align*}0=x5 \rightarrow x=5\end{align*}
The point at which \begin{align*}x=5\end{align*}
Look at the graph of the function. There is a clear separation of the branches at the vertical line five units to the right of the origin.
The domain is “all real numbers except five” or symbolically written \begin{align*}x \neq 5\end{align*}
Example 1: Determine the asymptotes of \begin{align*}t(x)=\frac{2}{(x2)(x+3)}\end{align*}
Solution: Using the Zero Product Property, there are two cases for asymptotes, where each set of parentheses equals zero.
\begin{align*}x2 &= 0 \rightarrow x=2\\ x+3 &= 0 \rightarrow x=3\end{align*}
The two asymptotes for this function are \begin{align*}x=2\end{align*}
Check your solution by graphing the function.
The domain of the rational function above has two points of discontinuity. Therefore, its domain cannot include the numbers 2 or –3. The domain: \begin{align*}x \neq 2, x \neq 3\end{align*}
Horizontal Asymptotes
Rational functions can also have horizontal asymptotes. The equation of a horizontal asymptote is \begin{align*}y=c\end{align*}
Example: Identify the vertical and horizontal asymptotes of \begin{align*}f(x)=\frac{3}{(x4)(x+8)}5\end{align*}
Solution: The vertical asymptotes occur where the denominator is equal to zero.
\begin{align*}x4 &= 0 \rightarrow x=4\\ x+8 &= 0 \rightarrow x=8\end{align*}
The vertical asymptotes are \begin{align*}x=4\end{align*}
The rational function has been shifted down five units \begin{align*}f(x)=\frac{3}{(x4)(x+8)}5\end{align*}
Therefore, the horizontal asymptote is \begin{align*}y=5\end{align*}
Multimedia Link: For further explanation about asymptotes, read through this PowerPoint presentation presented by North Virginia Community College or watch this CK12 Basic Algebra: Finding Vertical Asymptotes of Rational Functions
 YouTube video.
RealWorld Rational Functions
Electrical circuits are commonplace is everyday life. For instance, they are present in all electrical appliances in your home. The figure below shows an example of a simple electrical circuit. It consists of a battery that provides a voltage (\begin{align*}V\end{align*}
For resistors placed in a series, the total resistance is just the sum of the resistances of the individual resistors.
\begin{align*}R_{tot} =R_1+R_2\end{align*}
For resistors placed in parallel, the reciprocal of the total resistance is the sum of the reciprocals of the resistances of the individual resistors.
\begin{align*}\frac{1}{R_c}=\frac{1}{R_1}+\frac{1}{R_2}\end{align*}
Ohm’s Law gives a relationship between current, voltage, and resistance. It states that:
\begin{align*}I=\frac{V}{R}\end{align*}
Example: Find the value of \begin{align*}x\end{align*}
Solution: Using Ohm’s Law, \begin{align*}I=\frac{V}{R}\end{align*}
\begin{align*}2= \frac{12}{R}\end{align*}
Using the cross multiplication of a proportion yields:
\begin{align*}2R=12 \rightarrow R=6 \ \Omega\end{align*}
Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following videos. Note that there is not always a match between the number of the practice exercise in the videos and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both.
CK12 Basic Algebra: Asymptotes (21:06)
CK12 Basic Algebra: Another Rational Function Graph Example (8:20)
CK12 Basic Algebra: A Third Example of Graphing a Rational Function (11:31)
 What is a rational function?
 Define asymptote. How does an asymptote relate algebraically to an rational equation?
 Which asymptotes are described in this lesson? What is the general equation for these asymptotes?
Identify the vertical and horizontal asymptotes of each rational function.

\begin{align*}y=\frac{4}{x+2}\end{align*}
y=4x+2 
\begin{align*}f(x)=\frac{5}{2x6}+3\end{align*}
f(x)=52x−6+3  \begin{align*}y=\frac{10}{x}\end{align*}
 \begin{align*}g(x)=\frac{4}{4x^2+1}2\end{align*}
 \begin{align*}h(x)=\frac{2}{x^29}\end{align*}
 \begin{align*}y=\frac{1}{x^2+4x+3}+\frac{1}{2}\end{align*}
 \begin{align*}y=\frac{3}{x^24}8\end{align*}
 \begin{align*}f(x)=\frac{3}{x^22x8}\end{align*}
Graph each rational function. Show the vertical asymptote and horizontal asymptote as a dotted line.
 \begin{align*}y=\frac{6}{x}\end{align*}
 \begin{align*}y=\frac{x}{2x^2}3\end{align*}
 \begin{align*}f(x)=\frac{3}{x^2}\end{align*}
 \begin{align*}g(x)=\frac{1}{x1}+5\end{align*}
 \begin{align*}y=\frac{2}{x+2}6\end{align*}
 \begin{align*}f(x)=\frac{1}{x^2+2}\end{align*}
 \begin{align*}h(x)=\frac{4}{x^2+9}\end{align*}
 \begin{align*}y=\frac{2}{x^2+1}\end{align*}
 \begin{align*}j(x)=\frac{1}{x^21}+1\end{align*}
 \begin{align*}y=\frac{2}{x^29}\end{align*}
 \begin{align*}f(x)=\frac{8}{x^216}\end{align*}
 \begin{align*}g(x)=\frac{3}{x^24x+4}\end{align*}
 \begin{align*}h(x)=\frac{1}{x^2x6}2\end{align*}
Find the quantity labeled \begin{align*}x\end{align*} in the following circuit.
Mixed Review
 A building 350 feet tall casts a shadow \begin{align*}\frac{1}{2}\end{align*}mile long. How long is the shadow of a person five feet tall?
 State the Cross Product Property.
 Find the slope between (1, 1) and (–4, 5).
 The amount of refund from soda cans in Michigan is directly proportional to the number of returned cans. If you earn $12.00 refund for 120 cans, how much do you get per can?
 You put the letters from VACATION into a hat. If you reach in randomly, what is the probability you will pick the letter \begin{align*}A\end{align*}?
 Give an example of a sixthdegree binomial.