Symmetries are a good start: If $f(x)=f(-x)$, then it's even. If $f(y)=f(-y)$, then it has symmetry across the horizontal axis. If $f(x,y)=f(y,x)$ (swap $x$ and $y$), then it has symmetry across the line $y=x$, etc. For example, the first one is
Swap these around, and you get
which you should be familiar with. Since this was done by interchanging $x$ and $y$, the first one should be the graph of $|x|$ transformed that way. i.e, a v-shape opening to the right.
Another way, good for absolute values, is to break it up into cases. For $y>0, |y|=y$, so we have, as one part:
When $y<0,|y|=-y$ (i.e, $|-1|=-(-1)=1$), so the other part is
So sticking these two graphs together in their respective domains (first one on top since it has $y>0$, and the second underneath because it has $y<0$, you get the result. You are correct in saying that none of these are functions, but you can still graph them by looking at pieces that would be functions on their own.