Question 185936
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: 
{{{graph(400,300,-10,10,-20,30,(3x^2+4x+4)/(x^2-5x-6))}}}
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Cheers,
Stan H.