SOLUTION: 2. Check for symmetries, find intercepts and asymptotes, analyze the behavior near the graph of: 4y^2 + x – xy^2 – 1 = 0 (first solve for y^2) Is this thee graph of a func

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: 2. Check for symmetries, find intercepts and asymptotes, analyze the behavior near the graph of: 4y^2 + x – xy^2 – 1 = 0 (first solve for y^2) Is this thee graph of a func      Log On


   

Question 135709: 2. Check for symmetries, find intercepts and asymptotes, analyze the behavior near
the graph of: 4y^2 + x – xy^2 – 1 = 0 (first solve for y^2) Is this thee graph of a function?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!




4y%5E2+%2B+x-xy%5E2-1+=+0 Start with the given equation


4y%5E2-xy%5E2=+1-x Subtract x from both sides


y%5E2%284-x%29=+1-x Factor out the GCF y%5E2


y%5E2+=++%281-x%29%2F%284-x%29+ Divide both sides by 4-x




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y%5E2+=++%281-0%29%2F%284-0%29+ To find the y-intercept, plug in x=0



y%5E2+=++1%2F4+ Simplify

y+=++0%2B-sqrt%281%2F4%29+ Take the square root of both sides


y=1%2F2 or y=-1%2F2 Simplify

So the y-intercepts are (0,1%2F2) and (0,-1%2F2)

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y%5E2+=++%281-x%29%2F%284-x%29+ Go back to the original equation


0%5E2+=++%281-x%29%2F%284-x%29+ To find the x-intercept, plug in y=0


0+=++%281-x%29%2F%284-x%29+ Square 0 to get 0


0%284-x%29+=++1-x+ Multiply both sides by 4-x


0+=++1-x+ Multiply



x=++1+ Add x to both sides


So the x-intercept is (1,0)


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4-x=0 To find the vertical asymptote(s), simply set the denominator equal to zero


x=4 Solve for x

So the vertical asymptote is at x=4


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y%5E2+=++%281-x%29%2F%284-x%29+ Go back to the original equation


y%5E2+=++%28-x%2B1%29%2F%28-x%2B4%29+ Rearrange the terms


Notice how the coefficient of "x" for the numerator and denominator is -1. So the horizontal asymptote is the ratio -1%2F-1=1. Since we are dealing with a square in y%5E2, this means that the final equation looks like y+=++0%2B-sqrt%28%281-x%29%2F%284-x%29%29. So there are two final parts y+=++sqrt%28%281-x%29%2F%284-x%29%29 and y+=++-sqrt%28%281-x%29%2F%284-x%29%29


This means that there is symmetry with respect with the x-axis and that there are two horizontal asymptotes y=1 and y=-1




Notice if we graph both y+=++sqrt%28%281-x%29%2F%284-x%29%29 and y+=++-sqrt%28%281-x%29%2F%284-x%29%29, we can visually verify our answer







Notice how that if you pass a vertical line through the graph, the line will intersect with the graph more than once. So this tells us that this graph is not a function since it fails the vertical line test.