I was experimenting with binary operators, and devised a commutative operator with function $f$ such that the operator mapped $f \colon \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{R}$ in order to make up some problem (e.g. if $f(3,10)=30$ and $f(4,3)=10$, find $f(4,6)$, etc.).
My question is, are there any functions $f$ other than $a \otimes b = f(a,b) = qab+r(a+b)$ where $q,r \in \mathbb{R}$?
Also, is it always true that the operator is also distributive if $a \otimes 0 = 0$?