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Question 1191207: The function f is of the form f(x) = (ax + b) / (2x + c), for x ≠ -(c/2), where a, b, and c ∈ ℤ. Given that the graph of y = f(x) has asymptotes x = -5 and y = 2, and that the point P(1, -1/12) lies on the graph, what are the values of a, b, and c?
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
The vertical asymptote is x = -5, which occurs because of a division by zero error.
Plug x = -5 into the denominator to get
2x+c = 2(-5)+c = -10+c
Set that result equal to 0 and isolate c
-10+c = 0
c = 0+10
c = 10
Having 2x+10 in the denominator will lead to a denominator of zero when x = -5.
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The horizontal asymptote y = 2 is the result of dividing the leading terms. This applies because the degree of the numerator and denominator are the same (both are of degree 1).
This works because the leading terms are the largest when x gets really big (meaning that the smaller terms b and c effectively get so small we can ignore them).
Anyways, dividing the leading terms gets us this
(ax)/(2x) = a/2
where x is nonzero
Setting that equal to the horizontal asymptote value and solving for 'a' yields the following
a/2 = 2
a = 2*2
a = 4
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We found that
a = 4
c = 10
Let's update the function from
f(x) = (ax+b)/(2x+c)
to
f(x) = (4x+b)/(2x+10)
We're told that P(1,-1/12) is on the f(x) curve.
This means that the input x = 1 pairs with the output y = f(x) = -1/12
We'll use these two new values to find b
f(x) = (4x+b)/(2x+10)
-1/12 = (4(1)+b)/(2(1)+10)
-1/12 = (4+b)/(12)
-1 = 4+b .... multiplied both sides by 12
4+b = -1
b = -1-4
b = -5
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Answers:
a = 4
b = -5
c = 10
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