The Study of Rational Functions
By: Amanda Sawyer
In this write up, we will study the effects of adding a term with two different variables to a function. Let us first begin by defining y as:
In this problem, we were asked to study the function y with the additional term xy:
We can view the first equation in pink and the second equation in red. When we study this new equation, we will notice that it is actually a rational function. The best way to see this is by solving the equation for y:
The two equations appear to have the
similar parabolic shape as seen from the two graphs given below. However, we know that since our second
equation is a rational function it can have either a vertical, horizontal, or
slant asymptotes. To determine if it has
a vertical asymptote all we need to do is simplify the expression and set the
denominator equal to zero. Since the
expression is already simplified, to find the vertical asymptote we create the
equation:
This shows that this rational function
has a vertical asymptote at x = 1. Next, we notice that we do not have a
horizontal asymptote because of the numerator’s degree is larger than the
denominator’s degree. Finally, we must check the slant asymptote. Since the numerator’s degree is one more than
the denominator’s degree, we know that this graph has a slant asymptote. We can find this by doing long division as
seen below:
When r represents the remainder, we can see from this division the slant asymptote is:
.
Now let us study what will happen if we change the coefficient in front of xy to a where a is a real number:
We notice that it is still a rational
function with a restriction on its values for x. It also has a vertical and slant
asymptote. It has its vertical asymptote
when the denominator is equal to zero:
Since the vertical asymptote is 
,
we notice that when a>1 the vertical asymptote will stay between 0 and
1. When a<1, the vertical asymptote
will increase to a value larger than 1.
Also notice that when the sign of a changes, the vertical asymptote
changes to the corresponding value on the y axis. Notice how the slant asymptote changes:
Again, we notice a large change in values
when a is greater than or less
than one. When the sign changes the
slant asymptote becomes an increasing function.