Chapter 3.2 Rational Functions
In this section, we focus on rational functions. We will learn how rational functions form a rigid collection and how we use this information to sketch rational functions.
Sketching Reciprocals of Polynomials
Rational functions are quotients of polynomial functions and inherit their rigidity from the rigidity of polynomial functions.
Reduced Form
A rational function is written in simplest form (or reduced form) if it is written as a quotient of polynomials that do not have any common zeros.
It is important that we work with rational functions in reduced form. Make sure you understand what it means for a rational function to be in simplest form with this next example.
Example 1
Put a check by each rational function below that is given in simplest form:
; ; ; ; .
Not in simplest form;
is a common zero.Not in simplest form: the numerator can be rewritten as
, so both the numerator and denominator have as a common zero.It is in simplest form because there are no common zeros.
It is in simplest form because there are no common zeros.
Not in simplest form;
is a common zero.
Just like with polynomials, understanding the zeros is crucial. However, because we are dividing by a polynomial, it is possible a rational function is undefined at specific real numbers. At this places, the rational function exhibits interesting behavior.
To articulate this, we need the following words.
Zero, Pole, Local Behavior
Take
A zero of
A pole of
The zeros and poles of
Understand the difference between zeros and poles with this example.
Example 2
List all zeros and poles of
The zero
The pole
Inversion of the
:
- Sketch the polynomial in the denominator;
- Invert the
-axis.
The transformation principle used to simplify the problem is: Use the
Here is another word we will use to describe poles.
Vertical Asymptote
If
The function
In this example, we will use the pole to sketch this simple rational function.
Example 3
Take
Pole of
So by using
In this more complicated example, we will first graph what the rational function’s denominator looks like and then use
Example 4
Take
Poles of
The asymptotic behavior was
So by using
Asymptotic Behavior
In the previous examples, we specialized to rational functions that were of the form
where
To be able to make sense of how this may affect the graph, we need to understand how this affects its asymptotic behavior.
Here are two words we will use to describe asymptotic behavior:
Horizontal Asymptote and Slant Asymptote
A rational function
For large values of
The function
If
Let’s understand this definition and also revisit vertical asymptotes with this example.
Example 5
List, classify, and sketch all asymptotes of the rational function
The vertical asymptotes are
The horizontal asymptote can be determine by writing
So the vertical asymptotes of
Again, understand how to find asymptotes of a rational function with this example.
Example 6
List, classify, and sketch all asymptotes of the rational function
The vertical asymptote is
To find the horizontal asymptote we need to write
First, expand the numerator:
Hence we need to solve
Hence we get
or
Thus the slant asymptote is
The asymptotes look like this
Take
It can be difficult to precisely determine
Asymptotically Equal To
The function
All we need to do is identify the leading term of the numerator and the denominator and then divide them.
Example 7
Find a
; ; .
- The leading term of the numerator is
and the leading term of the denominator is . Thus is asymptotically equal to
- The leading term of the numerator is
and the leading term of the denominator is . Thus is asymptotically equal to
- The leading term of the numerator is
and the leading term of the denominator is . Thus is asymptotically equal to
Sketching Rational Functions
Rational functions with linear denominators readily break into manageable pieces and so can be sketched more accurately than more general rational functions.
Decompose such functions into a sum of a polynomial and the reciprocal of a polynomial of degree 1.
Find the zeros by finding the zeros of the numerator of the original function.
Overlay sketches of the polynomial part of the quotient and the part that is a proper fraction separately to approximate how they add.
Sketch the quotient of linear functions by simply scaling and shifting the reciprocal of the identity function.
Here is an example.
Example 8
Sketch
The function
The function
The function
The function therefore looks like this:
Let’s see how this rational function looks.
Example 9
Sketch
The function
The function
The function
The function therefore looks like this:
More general rational functions may still be written as a sum of a polynomial and a proper rational function.
Finding the quotient requires a more involved calculation and there is not a great payoff in making such a calculation since knowing the asymptotic behavior is usually sufficient for sketching the function.
Remember that determining the asymptotic behavior only involves finding the degree of the quotient.
Example 10
Write the rational function
The function can be rewritten like
The function
In general, it is straightforward to roughly sketch the rational functions of the form
This is done by finding the zeros and their orders (local behavior), the poles and their orders (local behavior), and noticing that (global behavior) the function asymptotically behaves like
Warning: We may miss certain ``wiggles’’ in the shape of
Example 11
Sketch
The zero is
The pole is
The function
The function looks like this:
In this next example, pay attention to how the information comes together.
Example 12
Sketch
The zero is
The pole is
The function
The function looks like this:
In this final example, use the graphical information to extract as much as possible what features the rational function may have.
Example 13
Below is a sketch of a rational function,
Zeros
Pole
No horizontal asymptote or slant asymptote.