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Question 59578: -2(x+1)(x+4)^2
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(x+2)^2(x-3)
What are the vertical and horizontal asymptotes.
Found 2 solutions by stanbon, Edwin McCravy:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! -2(x+1)(x+4)^2
---------------over
(x+2)^2(x-3)
What are the vertical and horizontal asymptotes.
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Horizontal:
The highest power term in the numerator is -2x^3
The highest power term in the denominator is x^3
The ratio of these terms is -2/1
There is a horizontal asymptote at y=-2
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Vertical:
Vertical asymptotes occur when the denominator is zero at
an x value where the numerator is not also zero.
You have vertical asymptotes at x=-2 and x=3
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Cheers,
Stan H.
Answer by Edwin McCravy(20056)  (Show Source):
You can put this solution on YOUR website!
-2(x+1)(x+4)²
----------------
(x+2)²(x-3)
What are the vertical and horizontal asymptotes.
If the numerator and denominator are relatively
prime polynomials, i.e., have no common factor
other than 1, then:
1. The vertical asymptotes are found by setting the
denominator = 0. For each value obtatined a
vertical asymptote will be obtained whose equation
is x = that value.
denominator = (x+2)²(x-3) = 0
have solutions x = -2, x = 3
So these are the equations of the two horizontal
asymptotes:
x = -2 and x = 3
2. There will be a horizontal asymptote if and only
if the degree of the numerator is not larger than
the degree of the denominator.
A. If the degree of the numerator is less than the
degree of the denominator, the horizontal asymptote
is always the x-axis, which has equation y = 0
B. If the degrees of the numerator and denominator are
equal, then the horizontal asymptote is y = a/b,
where a and b are the leading coefficients of the
numerator and denominator respectively, when the
numerator and denominator are multiplied out.
Now in this case, if we were to multiply the numerator
and denominator out, we would find that both would
have degree 3. So the rule B holds.
We don't need to multiply out the top and bottom,
since all we need is the leading term of each.
We can see that if we were to multiply the numerator
-2(x+1)(x+4)²
out, that the leading term would be -2x³, and the
leading coefficient would be a = -2
We can see that if we were to multiply the denominator
(x+2)²(x-3)
out, that the leading term would be x³, and the
leading coefficient would be b = 1.
Therefore the horizontal asymptote has equation
y = -2/1 or y = -2
The green and blue vertical lines are the vertical
asymptotes and the red horizontal line is the
horizontal asymptote. If we were to extend the graph
farther to the left and to the right, the blue curve
would get closer to the horizontal asymptote, and if
we extended the graph farther upward and downward,
the blue curve would get closer and closer to the
two vertical asymptotes.
Edwin
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