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

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) (Show Source): You can put this solution on YOUR website!

Start with the given equation

Subtract x from both sides

Factor out the GCF

Divide both sides by

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To find the y-intercept, plug in

Simplify

Take the square root of both sides

or Simplify

So the y-intercepts are (0,) and (0,)

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Go back to the original equation

To find the x-intercept, plug in

Square 0 to get 0

Multiply both sides by 4-x

Multiply

Add x to both sides

So the x-intercept is (1,0)

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To find the vertical asymptote(s), simply set the denominator equal to zero

Solve for x

So the vertical asymptote is at

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Go back to the original equation

Rearrange the terms

Notice how the coefficient of "x" for the numerator and denominator is -1. So the horizontal asymptote is the ratio . Since we are dealing with a square in , this means that the final equation looks like . So there are two final parts and

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

Notice if we graph both and , 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.
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