Algebra

We can plug in the points into the standard form of a parabola to get $a$, $b$, and $c$.

 

First, notice that $c$ must be equal to $0$ because when you plug in $(0,0)$ in, you get $c=0$. That way, we only have to solve for $2$ variables.

 

Now we can plug in the rest of the points, and we get the equations $8=4a-2b$ and $8=4a+2b$.

 

We can set the equations equal to each other and then we get $b=0$ as well.

 

Plugging $b=0$ into the equations, we get $a=2$. So $a+b+c$ is $2+0+0$ which is $2$