I know this is not preferable but since I don't know how to create graphs, I've just uploaded images of my work.
I first checked for symmetries (that's what the initial table is). Then I graphed the function on the $r\theta-$ plane
Then I graphed it on the $xy-$ plane. This is where I'm running into trouble. 
From 0 to $\pi$ the graph goes from radius 0 to 1. I put that on the left side of the $xy$ plane since that is where an angle of $\pi$ radians would be.
Then from $\pi$ to $2\pi$ it goes from radius 1 to radius 0. I think I got this part correct.
Where I'm struggling is from $2\pi$ to $3\pi$ because here the graph goes from radius 0 to radius -1. I marked (in green) everything I graphed from $2\pi$ to $4\pi$ and it should just be a repeat of what is happening from $0$ to $2\pi$.
I marked where $3\pi$ is on the graph, is this correct? Because the solutions show that this is the final image:
What am I doing wrong?