Question 975339
{{{abs(x^2+2x)-x^2}}}{{{""=""}}}{{{abs(y^2+2y)-y^2}}} 
<pre>
There are four cases:

Case 1:
{{{matrix(1,3,x^2+2x>=0,and, y^2+2y>=0)}}}

{{{abs(x^2+2x)-x^2}}}{{{""=""}}}{{{abs(y^2+2y)-y^2}}}
{{{(x^2+2x)-x^2}}}{{{""=""}}}{{{(y^2+2y)-y^2}}}
{{{x^2+2x-x^2}}}{{{""=""}}}{{{y^2+2y-y^2}}}
{{{2x}}}{{{""=""}}}{{{2y}}}
{{{x}}}{{{""=""}}}{{{y}}}

That's this graph:  
{{{graph(400,400,-5,5,-5,5,x)}}}


Case 2:
{{{matrix(1,3,x^2+2x>=0,and, y^2+2y<=0)}}}

{{{abs(x^2+2x)-x^2}}}{{{""=""}}}{{{abs(y^2+2y)-y^2}}}
{{{(x^2+2x)-x^2}}}{{{""=""}}}{{{-(y^2+2y)-y^2}}}
{{{x^2+2x-x^2}}}{{{""=""}}}{{{-y^2-2y-y^2}}}
{{{2x}}}{{{""=""}}}{{{-2y^2-2y}}}
{{{x}}}{{{""=""}}}{{{-y^2-y}}}

That's a parabola throu the origin opening left with vertex
{{{(matrix(1,3,1/4,",",-1/2))}}}  
{{{drawing(400,400,-5,5,-5,5,graph(400,400,-5,5,-5,5,x),
red(arc(-1,-1/2,2.5,-2.5,350,370)),
graph(400,400,-5,5,-5,5,(sqrt(1-4x)-1)/2),
graph(400,400,-5,5,-5,5,(-sqrt(1-4x)-1)/2)  )}}}

Case 3:
{{{matrix(1,3,x^2+2x<=0,and, y^2+2y>=0)}}}

{{{abs(x^2+2x)-x^2}}}{{{""=""}}}{{{abs(y^2+2y)-y^2}}}
{{{-(x^2+2x)-x^2}}}{{{""=""}}}{{{(y^2+2y)-y^2}}}
{{{-x^2-2x-x^2}}}{{{""=""}}}{{{y^2+2y-y^2}}}
{{{2x^2-2x}}}{{{""=""}}}{{{2y}}}
{{{-x^2-x}}}{{{""=""}}}{{{y}}}

That's a parabola thru the origin opening down with vertex
{{{(matrix(1,3,-1/2,",",1/4))}}}  
{{{drawing(400,400,-5,5,-5,5,graph(400,400,-5,5,-5,5,x),
red(arc(-1,-1/2,2.5,-2.5,350,370)),
graph(400,400,-5,5,-5,5,(sqrt(1-4x)-1)/2),
graph(400,400,-5,5,-5,5,(-sqrt(1-4x)-1)/2),
graph(400,400,-5,5,-5,5,-x^2-x)

  )}}}

Case 4:
{{{matrix(1,3,x^2+2x<=0,and, y^2+2y<=0)}}}

{{{abs(x^2+2x)-x^2}}}{{{""=""}}}{{{abs(y^2+2y)-y^2}}}
{{{-(x^2+2x)-x^2}}}{{{""=""}}}{{{-(y^2+2y)-y^2}}}
{{{-x^2-2x-x^2}}}{{{""=""}}}{{{-y^2-2y-y^2}}}
{{{-2x^2-2x}}}{{{""=""}}}{{{-2y^2-2y}}}
{{{x^2+x}}}{{{""=""}}}{{{y^2-y}}}

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

{{{x^2+x+1/4}}}{{{""=""}}}{{{y^2+x+1/4)}}}
{{{(x+1/2)^2}}}{{{""=""}}}{{{(y+1/2)^2}}}
{{{x+1/2}}}{{{""=""}}}{{{"" +- (y+1/2)}}}

Using the +

{{{x+1/2}}}{{{""=""}}}{{{y+1/2}}}
{{{x=y}}} which was also part of case 1, so we have already graphed it.

Using the -

{{{x+1/2}}}{{{""=""}}}{{{-(y+1/2)}}}
{{{x+1/2}}}{{{""=""}}}{{{-y-1/2}}}
{{{y}}}{{{""=""}}}{{{-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:
  
{{{drawing(400,400,-5,5,-5,5,graph(400,400,-5,5,-5,5,x),
red(arc(-1,-1/2,2.5,-2.5,350,370)),
graph(400,400,-5,5,-5,5,(sqrt(1-4x)-1)/2),
graph(400,400,-5,5,-5,5,(-sqrt(1-4x)-1)/2),
graph(400,400,-5,5,-5,5,-x^2-x),
graph(400,400,-5,5,-5,5,-1-x)

  )}}}

You must admit! -- that graph is COOL!

Edwin</pre>