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Rational Functions

Graphing Rational Functions

A rational function is a function that contains a rational expression. That is, any function that can be written in the form where and are polynomial functions. For any values of where the rational function is undefined. One example of a simple rational function is

Concept

Rational Functions and Asymptotes

An asymptote of a function is a line that the function's graph approaches as the distance to the origin approaches infinity. It is very common for rational functions to have at least one asymptote. These can arise when extends to the right or left infinitely, and around -values making the denominator equal to zero. The rational function has two asymptotes, the -axis and the -axis. The shape of this function's graph is called a hyperbola, which means that it consists of two mirror images with bow-like appearances.

As and as the function value approaches Thus, the -axis, is a horizontal asymptote of Similarly, as the function value approaches either positive or negative infinity. This means that the -axis, is a vertical asymptote of

Concept

Simple Rational Function

Rational functions of the form can be called simple rational functions, as their parent function is

When is equal to the denominator becomes Thus, these functions have a vertical asymptote at As and the fraction tends toward This gives the horizontal asymptote

Method

Graphing Simple Rational Functions

The identifiable asymptotes of simple rational functions can be used to simplify the process of graphing them. As an example, consider the function

1

Draw the asymptotes

Comparing the function rule to the general form of a simple rational function makes it possible to identify the asymptotes and For these are and These can be drawn in a coordinate plane.


2

Plot points around the vertical asymptote

Plot some points both to the left and the right of the vertical asymptote. For the example, it would be appropriate to use, for instance, the -values and

The points can be added to the coordinate plane to begin to see the behavior of


3

Draw the graph


The graph can now be drawn by connecting the points with a smooth curve. It must approach both the horizontal and vertical asymptotes.


Rule

Another Form of Rational Functions

Another common form of rational functions is where and are linear functions with different -intercepts. Just like simple rational functions, these are hyperbolas and have horizontal and vertical asymptotes. The horizontal asymptote is the line That is, the quotient of the leading coefficients of and The vertical asymptote is the line

which is the -value that makes the denominator
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Exercise

Identify the asymptotes of the given rational function. Then, rewrite it in the form, and draw its graph.

Show Solution
Solution
The function is currently written in the form The horizontal asymptote can be identified as the quotient of the leading coefficients. The vertical asymptote is located at the -value for which the denominator equals There is more than one viable strategy to rewrite the expression. Two of which are long division and synthetic division. However, here, we'll use inspection. Notice that, if the numerator was the quotient would be Let's manipulate the expression to make that happen.

The fraction can now be split into two, where one numerator is and the other is

As a last step in writing the function in the form we need to rewrite the denominator. We have now written the function in the desired form. The asymptotes and can be confirmed here as well. To graph this function, we start by drawing the asymptotes in a coordinate plane.

Next, we can make a table to find points around the vertical asymptote.

Plotting the points gives the graph.

We can connect the points with two curves that tend toward the asymptote as and extend in both directions.

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Exercise

Below, two rational functions are graphed.

Match the graphs with their corresponding function rule.

Show Solution
Solution

We match the graphs with their corresponding rule by identifying the asymptotes. Observing the first graph, it looks as though the asymptotes are For the second graph, Let's now identify the asymptotes of the given rules. For rational functions in the form the vertical asymptote is and the horizontal asymptote is For the ones written as their vertical asymptote is and the horizontal asymptote is

Function Vertical asymptote Horizontal asymptote

The function has the same asymptotes as Graph I, and has the same asymptotes as Graph II. Thus, the graphs' corresponding function rules are

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