SOLUTION: Discuss completely (as in the textbook) and draw the graphs of the following rational functions. 1. {{{y=(2x^2-2)/(x^2-9))}}}

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Question 132441: Discuss completely (as in the textbook) and draw the graphs of the following rational functions.

1.

Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
I'm not sure as to what you mean by "Discuss completely (as in the textbook)", but I'm assuming that you want to find the asymptotes right?


Start with the given function



Looking at the numerator , we can see that the degree is since the highest exponent of the numerator is . For the denominator , we can see that the degree is since the highest exponent of the denominator is .


Horizontal Asymptote:
Since the degree of the numerator and the denominator are the same, we can find the horizontal asymptote using this procedure:

To find the horizontal asymptote, first we need to find the leading coefficients of the numerator and the denominator.

Looking at the numerator , the leading coefficient is

Looking at the denominator , the leading coefficient is

So the horizontal asymptote is the ratio of the leading coefficients. In other words, simply divide by to get


So the horizontal asymptote is





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Vertical Asymptote:
To find the vertical asymptote, just set the denominator equal to zero and solve for x

Set the denominator equal to zero


Add 9 to both sides


Combine like terms on the right side


Take the square root of both sides

Notice if we graph , we can visually verify our answers:

Graph of with the horizontal asymptote (blue line) and the vertical asymptotes and (green lines)

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