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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:

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In this problem, we were asked to study the function y with the additional term xy:

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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:

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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 numeratorís degree is larger than the denominatorís degree. Finally, we must check the slant asymptote.Since the numeratorís degree is one more than the denominatorís 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:

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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.


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