SOLUTION: Find the asymptotes for {{{y=(3x^2-2)/(x+3))}}}

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Question 138683: Find the asymptotes for y=%283x%5E2-2%29%2F%28x%2B3%29%29
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
y=%283x%5E2-2%29%2F%28x%2B3%29%29 Start with the given function



Looking at the numerator 3x%5E2-2, we can see that the degree is 2 since the highest exponent of the numerator is 2. For the denominator x%2B3, we can see that the degree is 1 since the highest exponent of the denominator is 1.


Oblique Asymptote:

Since the degree of the numerator (which is 2) is greater than the degree of the denominator (which is 1), there is no horizontal asymptote. In this case, there's an oblique asymptote

To find the oblique asymptote, simply use polynomial division to find it 


    __3x_____-9____
x+3 | 3x^2+0x-2
      3x^2+9x
     ----------
          -9x-2  
          -9x-27
        ----------
              25


So the oblique asymptote is y=3x-9



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

x%2B3=0 Set the denominator equal to zero


x=0-3Subtract 3 from both sides


x=-3 Combine like terms on the right side


So the vertical asymptote is x=-3


Notice if we graph y=%283x%5E2-2%29%2F%28x%2B3%29, we can visually verify our answers:

Graph of y=%283x%5E2-2%29%2F%28x%2B3%29%29 with the oblique asymptote y=3x-9 (blue line) and the vertical asymptote x=-3 (green line)