Rational Functions

A function    is a rational function if  
where    and    are polynomials.
The domain of    is the set of all real numbers   
such that   .
Assume that all rational functions are reduced to lowest terms,
that is, any factors common to the numerator and to the denominator
have been canceled.
The function is discontinuous at a point where the denominator is zero.
Division by zero is a meaningless operation.
At the point where the denominator is zero, the graph of the function will
have a hole or a break.

See Example 1, page 323.


Vertical and Horizontal Asymptotes
Generally, the asymptote of a curve is a straight line that the curve approaches
closer and closer such that the separation between the curve and the straight line
tends to zero.
The asymptote may be vertical line, a horizontal, or an oblique line.
It is always a straight straight, so it's equation will be a linear equation
     .
Remember that any point on the graph is given by two coordinates   ,
where the first coordinate, or abscissa,    gives the horizontal distance
of the point from the y-axis; the second coordinate, or ordinate,    gives
the vertical distance of the point from the x-axis.

Typical behavior is exhibited by the simple rational function   .
          

As a point move up along the graph in the first quadrant, the point
gets arbitrarily closer and closer to the y-axis.
This situation is algebraically symbolized by
     ,
then the corresponding values of   , which is given by  ,
[ Remember that   and    represent the same values. ]
increase beyond all bounds, becoming arbitrarily bigger and bigger.
This behavior of is algebraically symbolized by writing
        or     .
The y-axis, then, is a vertical asymptote of the function
because the point gets arbitrarily close to it and the function
itself blows up, that is, tends to infinity.

See Table 1, page 323, and Table 2, page 324.


Vertical Asymptotes
In a rational function, if the denominator is zero at a point   
and the numerator is not zero at that point, then
     
is the equation of a vertical asymptote to the graph.

In the above example,    is a vertical asymptote.
This is the equation of the y-axis.

In general, to find the vertical asymptotes, if any exist, of a function,
determine where the denominator becomes zero.

Example
The vertical asymptotes of    
occur where
       .
The equations of the vertical asymptotes are

     
     
     .


Horizontal Asymptotes
Let    .
To determine the existence of any horizontal asymptotes,
examine the behavior of the function as a point on the graph moves
arbitrarily farther and farther away from the y-axis.
Algebraically, this means that    if the point moves away
to the right of the y-axis (the positive direction for values of   ),
or     if the point moves away to the left of the y-axis
(the negative direction for values of   ).
If the values of the function get closer and closer to some number,
say,    then the graph of the function    has a horizontal
asymptote whose equation is   .
Algebraically,
                    .


In the example above,   .
As  ,   , so a horizontal asymptote
is given by the equation   .

Notice that if the function blows up, that is, tends to positive or negative
infinity, then the function does not have a horizontal asymptote.

Example
      .
Square a big number and a bigger number results.
As    ,    .

The graph of      can not be bounded above or below
by any horizontal line because it shoots up past any finite horizontal value;
that is what    going to positive infinity means.
The graph has no bounding horizontal asymptote.


               
                            curve with no horizontal asymptote



Example
The following problem illustrates a method for calculating the limit
of a rational function as  .

Let     .
Divide the numerator and the denominator by the highest power of   , namely,   .

     

As    or           ( )

both   and   ,
so the numerator tends to   and the denominator tends to   .
.
The fraction itself tends to   .

         is a finite number,
so the function has a horizontal asymptote whose equation is    .


Summary for Horizontal Asymptotes
Let
         .

1)
If the degree of the numerator is less than the degree of the denominator
(  ),
the line    is a horizontal asymptote.

2)
If the degree of the numerator is equal to the degree of the denominator
(  ),
the line     is a horizontal asymptote.

3)
If the degree of the numerator is greater than the degree of the denominator
(  ),
the graph will increase or decrease without bound.
There are no horizontal asymptotes.

See Example 2, page 327.


Graphing Rational Functions
     

Step 1)
Plot the x- and y-intercepts.

Step 2)
Plot any vertical asymptotes as dashed lines.

Step 3)
Plot any horizontal asymptotes as dashed lines.

Step 4)
Make a sign chart, showing where the function is positive
(the graph is above the x-axis)
or negative
(the graph is below the x-axis).

Step 5)
Complete the graph of the function by plotting more points
and joining these points with a continuous curve over each
interval of the domain of the function.
Do not cross any vertical asymptotes.

NOTE:
The graph of a function can cross a horizontal asymptote.
Below, the graph of
crosses its asymptote   an infinite number of times,
yet    .

          

See Examples 3 -5, pages 328 - 333.


Oblique Asymptotes

Example
               .
Does the function have an asymptote that is neither vertical nor horizontal?
Do the long division,

     

Now let    , the remainder term    .

This means that as a point on the graph of the function moves out
towards infinity, the graph itself behaves more and more like   ,
which is a linear function. The graph of a linear function is a straight line.
So, we conclude that the line
     
is an oblique asymptote of the function.

          
The slanting line is the oblique asymptote of the graph of the function.
The vertical line is the vertical asymptote,  .
There is no horizontal asymptote because         as    .


A rational function has an oblique asymptote if the degree
of the numerator is exactly one more than the degree of
the denominator.

See Example 5, pages 332 - 333.


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