jmc

When $x=-1$, we have $y=a-b+c$. The graph appears to pass through $(-1,-2)$. Since $a$, $b$, and $c$ are integers, we know that $y$ is an integer when $x=-1$, so the graph does indeed pass through $(-1,-2)$. Therefore, $y=-2$ when $x=-1$, so $a-b+c=\boxed{-2}$.