Question

The parabola $y = ax^2 + bx + c$ is graphed below. Find $a+b+c$. (The grid lines are one unit apart.)

Answer 1

If $x=0$, then $y=a\cdot0^2+b\cdot0+c=5$, so $c=5$.

If $x=1$, then $y=a\cdot1^2+b\cdot1+c=2$, so $a+b+5=2$ or $a+b=-3$.

If $x=2$, then $y=a\cdot2^2+b\cdot2+c=1$, so $4a+2b+5=1$ or $2a+b=-2$.

We have

$a+b=-3\implies -a-b=3$

$(2a+b)+(-a-b)=-2+3$

$\implies a=1\implies b=-4$

So we end up with

$a+b+c=1+(-4)+5=\boxed{2}$