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