# Polynomial and Rational Functions

## Rational Functions

Suppose we know that the cost of making a product is dependent on the number of items, 
  produced. This is given by the equation 
  If we want to know the average cost for producing 
  items, we would divide the cost function by the number of items, 
  
The average cost function, which yields the average cost per item for 
  items produced, is
Many other application problems require finding an average value in a similar way, giving us variables in the denominator. Written without a variable in the denominator, this function will contain a negative integer power.
In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables in the denominator.

### Using Arrow Notation 

We have seen the graphs of the basic reciprocal function and the squared reciprocal function from our study of toolkit functions. Examine these graphs, as shown in , and notice some of their features.
Several things are apparent if we examine the graph of 
  
1. On the left branch of the graph, the curve approaches the x-axis 
  
2. As the graph approaches 
  from the left, the curve drops, but as we approach zero from the right, the curve rises.
3. Finally, on the right branch of the graph, the curves approaches the x-axis 
  
To summarize, we use arrow notation to show that 
  or 
  is approaching a particular value. See .

### Local Behavior of 
  

Let’s begin by looking at the reciprocal function, 
  We cannot divide by zero, which means the function is undefined at 
  so zero is not in the domain. As the input values approach zero from the left side (becoming very small, negative values), the function values decrease without bound (in other words, they approach negative infinity). We can see this behavior in .
We write in arrow notation
As the input values approach zero from the right side (becoming very small, positive values), the function values increase without bound (approaching infinity). We can see this behavior in .
We write in arrow notation
See .
This behavior creates a vertical asymptote, which is a vertical line that the graph approaches but never crosses. In this case, the graph is approaching the vertical line 
  as the input becomes close to zero. See .

### End Behavior of 
  

As the values of 
  approach infinity, the function values approach 0. As the values of 
  approach negative infinity, the function values approach 0. See . Symbolically, using arrow notation
 
  
Based on this overall behavior and the graph, we can see that the function approaches 0 but never actually reaches 0; it seems to level off as the inputs become large. This behavior creates a horizontal asymptote, a horizontal line that the graph approaches as the input increases or decreases without bound. In this case, the graph is approaching the horizontal line 
  See .

### Solving Applied Problems Involving Rational Functions

In , we shifted a toolkit function in a way that resulted in the function 
  This is an example of a rational function. A rational function is a function that can be written as the quotient of two polynomial functions. Many real-world problems require us to find the ratio of two polynomial functions. Problems involving rates and concentrations often involve rational functions.

### Finding the Domains of Rational Functions

A vertical asymptote represents a value at which a rational function is undefined, so that value is not in the domain of the function. A reciprocal function cannot have values in its domain that cause the denominator to equal zero. In general, to find the domain of a rational function, we need to determine which inputs would cause division by zero.

### Identifying Vertical Asymptotes of Rational Functions

By looking at the graph of a rational function, we can investigate its local behavior and easily see whether there are asymptotes. We may even be able to approximate their location. Even without the graph, however, we can still determine whether a given rational function has any asymptotes, and calculate their location.

### Vertical Asymptotes

The vertical asymptotes of a rational function may be found by examining the factors of the denominator that are not common to the factors in the numerator. Vertical asymptotes occur at the zeros of such factors.

### Removable Discontinuities

Occasionally, a graph will contain a hole: a single point where the graph is not defined, indicated by an open circle. We call such a hole a removable discontinuity.
For example, the function 
  may be re-written by factoring the numerator and the denominator.
Notice that 
  is a common factor to the numerator and the denominator. The zero of this factor, 
  is the location of the removable discontinuity. Notice also that 
  is not a factor in both the numerator and denominator. The zero of this factor, 
  is the vertical asymptote. See .

### Identifying Horizontal Asymptotes of Rational Functions

While vertical asymptotes describe the behavior of a graph as the output gets very large or very small, horizontal asymptotes help describe the behavior of a graph as the input gets very large or very small. Recall that a polynomial’s end behavior will mirror that of the leading term. Likewise, a rational function’s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions.
There are three distinct outcomes when checking for horizontal asymptotes:
Case 1: If the degree of the denominator > degree of the numerator, there is a horizontal asymptote at 
  
In this case, the end behavior is 
  This tells us that, as the inputs increase or decrease without bound, this function will behave similarly to the function 
  and the outputs will approach zero, resulting in a horizontal asymptote at 
  See . Note that this graph crosses the horizontal asymptote.
Case 2: If the degree of the denominator < degree of the numerator by one, we get a slant asymptote.
In this case, the end behavior is 
  This tells us that as the inputs increase or decrease without bound, this function will behave similarly to the function 
  As the inputs grow large, the outputs will grow and not level off, so this graph has no horizontal asymptote. However, the graph of 
  looks like a diagonal line, and since 
  will behave similarly to 
  it will approach a line close to 
  This line is a slant asymptote.
To find the equation of the slant asymptote, divide 
  The quotient is 
  and the remainder is 2. The slant asymptote is the graph of the line 
  See .
Case 3: If the degree of the denominator = degree of the numerator, there is a horizontal asymptote at 
  where 
  and 
  are the leading coefficients of 
  and 
  for 
  
In this case, the end behavior is 
  This tells us that as the inputs grow large, this function will behave like the function 
  which is a horizontal line. As 
  resulting in a horizontal asymptote at 
  See . Note that this graph crosses the horizontal asymptote.
Notice that, while the graph of a rational function will never cross a vertical asymptote, the graph may or may not cross a horizontal or slant asymptote. Also, although the graph of a rational function may have many vertical asymptotes, the graph will have at most one horizontal (or slant) asymptote.
It should be noted that, if the degree of the numerator is larger than the degree of the denominator by more than one, the end behavior of the graph will mimic the behavior of the reduced end behavior fraction. For instance, if we had the function
with end behavior
the end behavior of the graph would look similar to that of an even polynomial with a positive leading coefficient.

### Graphing Rational Functions

In , we see that the numerator of a rational function reveals the x-intercepts of the graph, whereas the denominator reveals the vertical asymptotes of the graph. As with polynomials, factors of the numerator may have integer powers greater than one. Fortunately, the effect on the shape of the graph at those intercepts is the same as we saw with polynomials.
The vertical asymptotes associated with the factors of the denominator will mirror one of the two toolkit reciprocal functions. When the degree of the factor in the denominator is odd, the distinguishing characteristic is that on one side of the vertical asymptote the graph heads towards positive infinity, and on the other side the graph heads towards negative infinity. See .
When the degree of the factor in the denominator is even, the distinguishing characteristic is that the graph either heads toward positive infinity on both sides of the vertical asymptote or heads toward negative infinity on both sides. See .
For example, the graph of 
  is shown in .
1. At the x-intercept 
  corresponding to the 
  factor of the numerator, the graph bounces, consistent with the quadratic nature of the factor.
2. At the x-intercept 
  corresponding to the 
  factor of the numerator, the graph passes through the axis as we would expect from a linear factor.
3. At the vertical asymptote 
  corresponding to the 
  factor of the denominator, the graph heads towards positive infinity on both sides of the asymptote, consistent with the behavior of the function 
  
4. At the vertical asymptote 
  corresponding to the 
  factor of the denominator, the graph heads towards positive infinity on the left side of the asymptote and towards negative infinity on the right side.

### Writing Rational Functions

Now that we have analyzed the equations for rational functions and how they relate to a graph of the function, we can use information given by a graph to write the function. A rational function written in factored form will have an x-intercept where each factor of the numerator is equal to zero. (An exception occurs in the case of a removable discontinuity.) As a result, we can form a numerator of a function whose graph will pass through a set of x-intercepts by introducing a corresponding set of factors. Likewise, because the function will have a vertical asymptote where each factor of the denominator is equal to zero, we can form a denominator that will produce the vertical asymptotes by introducing a corresponding set of factors.

### Key Equations


### Key Concepts

1. We can use arrow notation to describe local behavior and end behavior of the toolkit functions 
  and 
  See .
2. A function that levels off at a horizontal value has a horizontal asymptote. A function can have more than one vertical asymptote. See .
3. Application problems involving rates and concentrations often involve rational functions. See .
4. The domain of a rational function includes all real numbers except those that cause the denominator to equal zero. See .
5. The vertical asymptotes of a rational function will occur where the denominator of the function is equal to zero and the numerator is not zero. See .
6. A removable discontinuity might occur in the graph of a rational function if an input causes both numerator and denominator to be zero. See .
7. A rational function’s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions. See , , , and .
8. Graph rational functions by finding the intercepts, behavior at the intercepts and asymptotes, and end behavior. See .
9. If a rational function has x-intercepts at 
  vertical asymptotes at 
  and no 
  then the function can be written in the form
See 


### Section Exercises


### Verbal


### Algebraic

For the following exercises, find the domain of the rational functions.
For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions.
For the following exercises, find the x- and y-intercepts for the functions.
For the following exercises, describe the local and end behavior of the functions.
For the following exercises, find the slant asymptote of the functions.

### Graphical

For the following exercises, use the given transformation to graph the function. Note the vertical and horizontal asymptotes.
For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph.
For the following exercises, write an equation for a rational function with the given characteristics.
For the following exercises, use the graphs to write an equation for the function.

### Numeric

For the following exercises, make tables to show the behavior of the function near the vertical asymptote and reflecting the horizontal asymptote

### Technology

For the following exercises, use a calculator to graph 
  Use the graph to solve 
  

### Extensions

For the following exercises, identify the removable discontinuity.

### Real-World Applications

For the following exercises, express a rational function that describes the situation.
For the following exercises, use the given rational function to answer the question.
For the following exercises, construct a rational function that will help solve the problem. Then, use a calculator to answer the question.