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What is the graph of $x^2+y^2=|x|+|y|$. I tried solving this but I have don't understand how we know that the graph is symmetric to the axes. I read the solutions to this Area enclosed by the curve $x^2+y^2=|x|+|y|$. btw

user501887
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    Write it as $(|x| - 1/2)^2 + (|y| - 1/2)^2 = 1/2$ (completing the square twice; this is also what you would do if the absolute value signs weren't there and then you'd just have a circle). – Qiaochu Yuan Nov 18 '18 at 03:57

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Because of the squares and the absolute value signs, changing the sign on $x$ or $y$ will not change either side of the equation. That means that if you find one solution $(x,y)$ you know that $(-x,y), (x,-y)$ and $(-x,-y)$ are also solutions. You can find the solution in the first quadrant (for example) and reflect it in both axes to get the solution in the other quadrants.

Ross Millikan
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