74) Graphing Rational Functions and Their Asymptotes
Professor Dave again, let’s learn about asymptotes.
We just learned how to graph higher degree polynomials, but we also need to graph rational
functions.
These are functions that are expressed as quotients of polynomial functions.
This can make things a bit trickier for a number of reasons.
First, it’s not as immediately apparent what the behavior of the function will be,
and second, the domain will typically not be all real numbers as it was for polynomials.
This is because the denominator of the fraction can’t be zero, and any value for X that
produces zero for one of these terms will therefore not be present in the domain of
the function.
Let’s learn how to graph these kinds of functions, starting with the simplest, one
over X. Immediately we know that X can’t equal zero,
so zero is not part of the domain of this function.
But what happens on either side of X equals zero?
Well, let’s plug in some numbers.
Negative one gives us negative one, so negative one negative one is part of the function.
As we go left, we get closer and closer to zero from the negative direction, with the
limit of reaching zero as X approaches negative infinity, so the function approaches zero
but never quite gets there when we go left.
In this tiny interval from negative one to zero, we can see that plugging in fractions
of negative one will give us very big negative numbers that increase rapidly, heading towards
negative infinity as X approaches zero, so that will look like this.
On the positive side we get something similar.
Plugging in one, we can see that one, one is part of the function.
Then with bigger numbers we get closer and closer to zero, so that gives us this behavior,
approaching zero as X gets infinitely large.
And plugging in fractions of one brings the function closer and closer to infinity as
X approaches zero, so that will look like this.
What we can say about this function is that it has two asymptotes that can be represented
by X equals zero and Y equals zero.
These are lines that the function approaches, getting closer and closer, but never quite
touching.
Asymptotes can be vertical, horizontal, or at an angle, and finding these will be the
key to graphing rational functions, so let’s learn how to find them.
Finding vertical asymptotes is pretty simple, as these are values that are not in the domain
of the function.
That means that these are the zeroes of the denominator.
Take something like X over X squared minus nine.
We know that this denominator can be factored to get X plus three times X minus three, and
that means that three and negative three are the two zeroes of the denominator.
Since those values are not in the domain of the function, we must have vertical asymptotes
at X equals negative three and X equals three.
We can also find out if there is a horizontal asymptote.
We should note that while there can be multiple vertical asymptotes, there can only be one
horizontal asymptote at most, so it’s either one or none.
To find out which, we list the numerator and denominator of the function as polynomials
in standard form, not as products of factors.
Then we look at the leading terms.
If the denominator is a higher degree polynomial than the numerator, Y equals zero will be
an asymptote.
If the numerator is a higher degree polynomial than the denominator, there will be no horizontal
asymptote.
If the two polynomials are of the same degree, then we take the leading coefficient of the
top over the leading coefficient of the bottom, and Y equals that number will be the horizontal
asymptote.
In this case, the denominator is of a higher degree as it is a quadratic, so Y equals zero
is the asymptote.
Then to graph the function, we can test selective values of X to find the behavior of the function
as it approaches these asymptotes from either side, so we pick some near the vertical asymptotes,
and some points as the function approaches positive and negative infinity.
A quick test of five or six points will reveal the following behavior for the function.
Let’s take note that a function can sometimes cross an asymptote, the way this function
crosses the horizontal asymptote in the middle section.
Let’s also quickly point out that another strategy for graphing rational functions is
to use transformations, when applicable.
If we know what one over X looks like, then one over X plus this additional one over here
means it’s just the normal function with a vertical shift of one.
If instead we have a number within the denominator, that’s a horizontal shift.
If this one in the numerator is a two instead, that’s a stretch.
All the transformations we looked at for parabolas will apply to these and any other types of
functions as well.
But as we said, when such an approach does not apply, we have a solid strategy for graphing
rational functions.
We can find vertical asymptotes by finding the zeroes of the denominator.
We can see if there is a horizontal asymptote by looking at the degree of the polynomials
on the top and bottom as well as the coefficients of their leading terms.
We can always find X and Y intercepts to help us sketch, and then selectively picking other
points to plot will help reveal the behavior of the function as it approaches an asymptote.
With all of this information, it should be relatively easy to graph rational functions
with all real roots, so let’s check comprehension.