Isolate "a" in a quadratic function

Question

You have a quadratic function: $ax^2 + bx + c = y$. If you know $b$ and $c$, are able to plug any domain value $x$ into this blackbox equation and receive a range value $y$, and do not know the vertex of the equation, how could you find the unknown variable $a$.

Answer 1

The solution for $a$ is $$a=\begin{cases}\dfrac{y-bx-c}{x^2} & x\neq0\\ \text{unknown value} & x=0\end{cases}$$ How that is derived can be seen by rearranging and dividing.