Strange graph problem

Question

I'm trying to figure out this question: in the Cartesian coordinate system, how many grid points (x,y) satisfy $(|x|-1)^2 + (|y|-1)^2 < 2$? When I plugged it in to a graphing calculator, it didn't let me use the < sign, so I changed it to the = sign, and got a graph that looked like four connected semicircles, with centers at (1,1), (1,-1), (-1,1), and (-1,-1). The resulting image looked like a four-leafed clover.

Answer 1

Hint: Notice that if $(x,y)$ is on the boundary of the region, then so are the points $(\pm x,\pm y)$. This means that the region is symmetric with respect to the $x$ and $y$ axes. Hence, you may graph it in the first quadrant (where $x=|x|$ and $y=|y|$), and reflect to the other three quadrants to obtain the whole shape.

From a quick visualization, I believe that the answer is $16$. You should check this on your own to make sure it is right.

Answer 2

Hint: Notice the powers of 2. What does squaring a number do? Try listing out all possible values of $y$ for $x = -2, -1, \ldots, 2$.