Reciprocal Function
 As $x$ approaches positive or negative infinity, $y$ approaches zero. The graph gets closer to the $x$axis, but does not touch it. The $x$axis is a horizontal asymptote of the function.
 As $x$ approaches zero from the left side, $y$ approaches negative infinity. As $x$ approaches zero from the right side, $y$ approaches positive infinity. The graph gets closer to the $y$axis but does not touch it. The $y$axis is a vertical asymptote of the function.
 The graph is symmetric with respect to the origin. It passes through the points $(1,1)$ and $(1,1)$.
Indirect Variation
Variation describes a relationship between two or more variables and a nonzero constant $k$ known as the constant of proportionality. There are several types of variation:
 Direct variation relationships can be written as $y = kx$ for variables $x$ and $y$ and constant $k$. Since one variable changes in proportion to the other variable, the value of the variables $x$ and $y$ either increase or decrease together.
 Inverse variation is a relationship for variables $x$ and $y$ and constant $k$ that can be written as $y = \frac{k}{x}$ . Since the variables vary inversely with each other, the value of one variable increases while the value of the other variable decreases.
 An example of a joint variation relationship is $z = kxy$ for variables $x$, $y$, and $z$ and constant $k$. Joint variation is similar to direct variation, but involves more than two variables.
 An example of a combined variation relationship is $z = \frac{kx}{y}$ for variables $x$, $y$, and $z$ and constant $k$. Combined variation is a combination of direct or joint variation and inverse variation.
The graph of an indirect variation function is a stretch or compression of the parent rational function. The $x$axis is a horizontal asymptote, the $y$axis is a vertical asymptote, and the graph passes through the point $(1,k)$. Variation equations often represent realworld relationships, in which $x$values are usually positive.
Write the variation equation.
Let $x$ represent the time it takes to mow a lawn and $y$ represent Brittany's hourly rate. They are inversely proportional, which means:Interpret the meaning of the asymptotes.
The vertical asymptote shows that Brittany's hourly rate increases as her time approaches zero. It is impossible to mow the lawn in zero hours, so the hourly rate is not defined for $x=0$.
The horizontal asymptote shows that her hourly rate decreases as her time approaches infinity. The longer she takes to mow the lawn, the less she makes per hour. Her hourly rate cannot be zero. So, the graph never touches the $x$axis.
Identify two points on the graph.
The point $(1,30)$ represents the hourly rate of $30 if she mows the lawn in one hour.
If it takes her two hours to mow the lawn, then her hourly rate is $15 per hour:If the volume of a pyramid increases, but the area of its base stays the same, the height of the pyramid increases. If the area of the base of a pyramid increases, but the volume of the pyramid stays the same, the height of the pyramid decreases. In other words, the height of a pyramid varies directly as the volume and inversely as the area of the base.
A pyramid with a volume of 100 cubic centimeters (cm^{3}) and a base area of 50 square centimeters (cm^{2}) has a height of 6 centimeters (cm). What is the height of the pyramid if the volume and base area are both doubled?
Write the variation equation. Let $h$ represent the height of the pyramid, $V$ represent the volume, and $B$ represent the area of the base. Let $k$ be the constant of variation.
$V$ is in the numerator because the height varies directly as the volume.
$B$ is in the denominator because the height varies inversely as the area of the base.
So, the variation equation is:Use the given values to calculate $k$.
A pyramid with a volume of 100 cm^{3} and a base area of 50 cm^{2} has a height of 6 cm.Transformations of the Reciprocal Function
Transformations can be used to make changes to the graph of a parent function. These include translations (shifts), stretches or compressions, and reflections.
For any function $f(x)$, the function can be translated vertically $k$ units and translated horizontally $h$ units. Note that horizontal translations are opposite in direction from the sign: Subtracting $h$ from the input translates the graph in the positive direction, and adding $h$ to the input translates it in the negative direction.
A function $f(x)$ can also be stretched or compressed by a factor of $a$ and reflected across the $x$axis.
Translations of the Reciprocal Function
Vertical Translations  Horizontal Translations 

For the graph of $f(x)=\frac{1}{x}$ and $k>0$:

For the graph of $f(x)=\frac{1}{x}$ and $h>0$:

Stretches, Compressions, and Reflections of the Reciprocal Function
Stretches and Compressions  Reflections 

For the graph of $f(x)=\frac{1}{x}$ and $a\gt 0$:

For the graph of $f(x)$:

Perform the stretch and the reflection.
 Since $a=3$ and $a>1$, the graph is vertically stretched by a factor of 3 compared to the graph of the parent function.
 Since $a<0$, the graph is reflected across the $x$axis.
Perform the translation.
 Since $h=1$ and this value is added, the graph is translated 1 unit to the left.
 Since $k=2$ and this value is added, the graph is translated 2 units up.
Holes and Asymptotes
The end behavior of the graph determines whether there is a horizontal asymptote. If the function approaches a certain value as $x$ approaches positive or negative infinity, then there is a horizontal asymptote at that value. Horizontal asymptotes are determined by comparing the degree of the numerator and denominator. It is possible for a graph to cross a horizontal asymptote, but never a vertical one.
Determining Holes and Asymptotes
Holes and Vertical Asymptotes  Horizontal Asymptotes 

Let $q(x)$ have a zero at $x=a$ of multiplicity $k$ in the rational function:
$f(x)=\frac{p(x)}{q(x)}$

Let $n$ be the degree of $p(x)$ and $m$ be the degree of $q(x)$.

To determine whether the values are holes or vertical asymptotes, simplify the fraction. Divide the numerator and denominator by the common factor $(x+2)$ to get:
 The value $x=2$ is in the domain of the simplified function, but not in the domain of the original function. Therefore, there is a hole in the graph at $x=2$. Since this hole is not in the simplified form of the function, it is called a removable discontinuity.
 The value $x=4$ is not in the domain of either the original function or the simplified rule. As $x$ approaches 4, the function approaches positive or negative infinity. Therefore, $f(x)$ has a vertical asymptote at $x=4$.
 The end behavior of each polynomial function is determined by the term with the highest degree. So, look at the rational function $g$ made up of only the first term of the numerator and denominator:
 This can be simplified to $g(x)=4$, or $y=4$.
 The graph of $y=4$ is a horizontal line. Therefore, $f(x)$ has a horizontal asymptote at $y=4$.
Holes and Asymptotes
Graphing Rational Functions
To graph a rational function by hand:
1. Analyze the function to determine the location of any holes, vertical asymptotes, horizontal asymptotes, and intercepts.
2. Sketch the asymptotes, and plot any holes or intercepts.
3. Draw curves to complete the graph.
Identify any $x$intercepts.
The function has an $x$intercept for each real value of $x$ for which the numerator is equal to zero. The numerator is zero at $x=5$ and $x=5$. These are the $x$intercepts.
Identify any vertical asymptotes.
The denominator of the function is zero at $x=3$ and $x=4$. The numerator is not zero at these values. So, there are two vertical asymptotes, one at $x=3$ and one at $x=4$.
Identify any holes.
The graph of the function has no holes because the numerator and denominator have no zeros in common. The zeros of the numerator are –5 and 5, and the zeros of the denominator are –3 and 4.
Identify the horizontal asymptote, if any.
Look at the degrees of the numerator and denominator. They have the same degree. So, divide the leading coefficient of the numerator by the leading coefficient of the denominator. Both have a leading coefficient of 1. So, the horizontal asymptote is at:Identify any $x$intercepts.
The numerator is zero at $x=0$, $x=3$, and $x=5$. The denominator is also zero at $x=3$. So that value of $x$ cannot be an $x$intercept. The $x$intercepts are at zero and 5.
Identify any vertical asymptotes.
The denominator is zero at $x=3$ and $x=4$. The numerator is zero at $x=3$, but not at $x=4$. So, the only vertical asymptote is at $x=4$.
Identify any holes.
At $x=3$, the numerator and denominator both have a zero of multiplicity 1. So, there is a hole at this value of $x$. To see where to plot the hole, simplify the rule for $f$ by dividing out $(x+3)$ in the numerator and denominator to make a new function:The graph has a hole at approximately $(3,3.4)$.
Identify the horizontal asymptote, if any.
Look at the degrees of the numerator and denominator. The numerator has a greater degree. So, there is no horizontal asymptote.
Identify any vertical asymptotes.
The denominator is zero at $x=0$, $x=3$, and $x=2$. The numerator is zero at $x=2$, but not at $x=0$ or $x=3$. So, there are vertical asymptotes at $x=0$ and $x=3$.
Identify any holes.
At $x=2$, look at the multiplicity of the zeros (the number of times the related factor appears in the polynomial) of the numerator and denominator. Both the numerator and denominator have a zero of multiplicity 1. The multiplicities are equal, so there is a hole at this value of $x$.