SOLUTION: I need help graphing |x^2+2x|-x^2= |y^2+2y|-y^2

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Question 975339: I need help graphing |x^2+2x|-x^2= |y^2+2y|-y^2
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
abs%28x%5E2%2B2x%29-x%5E2%22%22=%22%22abs%28y%5E2%2B2y%29-y%5E2
There are four cases:

Case 1:
matrix%281%2C3%2Cx%5E2%2B2x%3E=0%2Cand%2C+y%5E2%2B2y%3E=0%29

abs%28x%5E2%2B2x%29-x%5E2%22%22=%22%22abs%28y%5E2%2B2y%29-y%5E2
%28x%5E2%2B2x%29-x%5E2%22%22=%22%22%28y%5E2%2B2y%29-y%5E2
x%5E2%2B2x-x%5E2%22%22=%22%22y%5E2%2B2y-y%5E2
2x%22%22=%22%222y
x%22%22=%22%22y

That's this graph:  
graph%28400%2C400%2C-5%2C5%2C-5%2C5%2Cx%29


Case 2:
matrix%281%2C3%2Cx%5E2%2B2x%3E=0%2Cand%2C+y%5E2%2B2y%3C=0%29

abs%28x%5E2%2B2x%29-x%5E2%22%22=%22%22abs%28y%5E2%2B2y%29-y%5E2
%28x%5E2%2B2x%29-x%5E2%22%22=%22%22-%28y%5E2%2B2y%29-y%5E2
x%5E2%2B2x-x%5E2%22%22=%22%22-y%5E2-2y-y%5E2
2x%22%22=%22%22-2y%5E2-2y
x%22%22=%22%22-y%5E2-y

That's a parabola throu the origin opening left with vertex
%28matrix%281%2C3%2C1%2F4%2C%22%2C%22%2C-1%2F2%29%29  


Case 3:
matrix%281%2C3%2Cx%5E2%2B2x%3C=0%2Cand%2C+y%5E2%2B2y%3E=0%29

abs%28x%5E2%2B2x%29-x%5E2%22%22=%22%22abs%28y%5E2%2B2y%29-y%5E2
-%28x%5E2%2B2x%29-x%5E2%22%22=%22%22%28y%5E2%2B2y%29-y%5E2
-x%5E2-2x-x%5E2%22%22=%22%22y%5E2%2B2y-y%5E2
2x%5E2-2x%22%22=%22%222y
-x%5E2-x%22%22=%22%22y

That's a parabola thru the origin opening down with vertex
%28matrix%281%2C3%2C-1%2F2%2C%22%2C%22%2C1%2F4%29%29  


Case 4:
matrix%281%2C3%2Cx%5E2%2B2x%3C=0%2Cand%2C+y%5E2%2B2y%3C=0%29

abs%28x%5E2%2B2x%29-x%5E2%22%22=%22%22abs%28y%5E2%2B2y%29-y%5E2
-%28x%5E2%2B2x%29-x%5E2%22%22=%22%22-%28y%5E2%2B2y%29-y%5E2
-x%5E2-2x-x%5E2%22%22=%22%22-y%5E2-2y-y%5E2
-2x%5E2-2x%22%22=%22%22-2y%5E2-2y
x%5E2%2Bx%22%22=%22%22y%5E2-y

Complete the square on both sides by adding 1/4
to both sides:

x%5E2%2Bx%2B1%2F4%22%22=%22%22y%5E2%2Bx%2B1%2F4%29
%28x%2B1%2F2%29%5E2%22%22=%22%22%28y%2B1%2F2%29%5E2
x%2B1%2F2%22%22=%22%22%22%22+%2B-+%28y%2B1%2F2%29

Using the +

x%2B1%2F2%22%22=%22%22y%2B1%2F2
x=y which was also part of case 1, so we have already graphed it.

Using the -

x%2B1%2F2%22%22=%22%22-%28y%2B1%2F2%29
x%2B1%2F2%22%22=%22%22-y-1%2F2
y%22%22=%22%22-x-1 

So we draw that line and the graph is complete:
 
That's two straight lines thru the origin.  The first we have
already graphed in case 1. So we graph y=-x-1:
  


You must admit! -- that graph is COOL!

Edwin