SOLUTION: I seem to be having problems with the graphing portions. I never know what the graph is going to look like based on the equation. For example, y=(3x^2+4x+4)/(x^2-5x-6). With this p

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Question 185936: I seem to be having problems with the graphing portions. I never know what the graph is going to look like based on the equation. For example, y=(3x^2+4x+4)/(x^2-5x-6). With this problem, I need to find the x-intercepts, the vertical asymptote and (if applicable) the horizontal asymptote. Could you please help me solve this problem? Is there a trick to knowing what general shape your graph will have depending on the equation?
Thank you for your time.
Answer by stanbon(75887) (Show Source):
You can put this solution on YOUR website!
y=(3x^2+4x+4)/(x^2-5x-6). With this problem,
I need to find the x-intercepts,
Let y = 0 and solve for "x":
(3x^2+4x+4)/(x^2-5x-6)
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The fraction is zero when the numerator is zero:
Solve 3x^2 + 4x + 4 = 0 using the quadratic formula:
x = [-4 +- sqrt(16 - 4*3*4)]/6
16-48 is negative so there are no Real Number solutions; no x-intercepts
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the vertical asymptote
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You have vertical asymptotes when the denominator is zero and the numerator is not zero.
Solve x^2-5x-6 = 0
(x-6)(x+1) = 0
x = 6 and x=-1 are vertical asymptotes
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the horizontal asymptote of y=(3x^2+4x+4)/(x^2-5x-6)
y = 3x^2/x^2 = 3 is the horizontal asymptote
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Is there a trick to knowing what general shape your graph will have depending on the equation?
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There is no trick. That is why you find the intercepts and the asymptotes.
That way you have a general idea what the graph looks like.
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Graph:

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Cheers,
Stan H.