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

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




-----------------------------------
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,)

----------------------

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)


-----------------------

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


Solve for x

So the vertical asymptote is at


-----------------

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.