features
Rational functions are fractions of polynomials
where the and are real numbers and and .
Again, we prefer polynomials in factored form.
Therefore, usually our first step is to transform the rational function to look like
And, again, we will be able to obtain a product of linear factors with the addition of complex numbers. With real numbers we can get these products to consist only of linear and irreducible quadratics. However, this discussion is about the roots of the polynomial. Therefore, we will ignore the quadratic factors and leave those for a later discussion.
Finally, we would like to clean up the factors and group them together
Reduced Form
Shared Roots
Suppose the numerator and denominater share a root, . Then we can reduce the expression but we must remember that is not in the domain.
With that reduction in mind, let’s assume that we have reduced the rational
expression and there are no shared roots between the numerator and denominator.
No Shared Roots
In this case, all of the roots are distinct
- for all possible and
- for all possible and
- for all possible and
With this we can analyze our rational function from the perspective of the roots.
Numerator
The numerator is a polynomial. Each factor gives a root or zero of the function,
which corresponds to an intercept on the graph.
If the multiplicity is odd, then the function changes sign and the graph crosses the
axis at this intercept.
If the multiplicity is even, then the graph does not cross the axis at this intercept and
the function maintains its sign.
Denominator
The denominator is a polynomial. Each factor gives a root or zero of the
denominator, which corresponds to a singularity of the rational function and a
vertical asymptote on the graph.
The factors in the denominator still affect the sign of the function values.
If the multiplicity is odd, then the function changes sign and the graph jumps to the
other end of the vertical asymptote.
If the multiplicity is even, then the function does not change sign and the graph does
not jump to the other end of the vertical asymptote.
The graph of
The numerator has the factor , which has multiplicity , odd. Therefore has as a root and the graph has as its only intercept, the only place where the graph crosses the horizontal -axis.
The denominator has two factors and . Both have odd even multiplicity, therefore, the function changes sign over these singularities and the graph jumps to the other end of the vertical asymptotes.
The degree of the denominator is larger than the degree of the numerator, therefore the -axis is a horizontal asymptote.
The end-behavior is
Vertical Asymptotes
The factor corresponds to the vertical asymptote described by .
Near , on either side of , the factors and do not change sign.
- The value of is near (negative), when is near
- The value of is near (negative), when is near .
Both factors take on negative values when is near .
The factor does change sign, since its multiplicity is odd.
- On the left side, where , we have , negative.
- On the right side, where , we have , positive.
Therefore, on the left side of , . Plus, the denominator is approaching making the whole fraction get bigger and bigger negatively.
We can see this in the graph. The graph moves down the left side of the vertical asymptote.
Since this singularity has an odd multiplicity, we also know that changes sign across this vertical asymptote.
We can now walk through the sign changes of .
- is a zero of with odd multipicity. The sign of changes across this zero. The graph crosses the -axis at the intercept.
- has an odd singularity. The sign of changes across this singularity. The graph jumps to the other infinity across the vertical asymptote.
We can now categorize the rate of change for .
- is decreasing on .
- is decreasing on .
- is decreasing on .
has no global or local maximums or minimums.
The range of is all real numbers.
Sign Changes
For rational functions, the sign can change only at a zero of odd multiplicity or a singularity of odd multiplicity. This is helpful when graphing. These requirements force the graph to go in particular directions.
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more examples can be found by following this link
More Examples of Analysis