jmc

The graph of the original parabola ($A$) and its final image ($A^{\prime }$) after rotation and translation is shown below:



Shifting the original parabola 1 unit to the right changes its equation to $y=-x^2+1$. Shifting this last parabola 5 units down changes its equation to $y=-x^2-4$. Rotating it by 180 degrees changes its equation to $y=x^2-4$. So the equation of $A^{\prime }$ is $y=x^2-4$. To find the zeros of this parabola, we set $y=0$ to get $0=x^2-4$. Factoring the right hand side, we get $0=(x-2)(x+2)$, so either $x-2=0\Rightarrow x=2$ or $x+2=0\Rightarrow x=-2$. Thus, $a=-2$ and $b=2$, so $b-a=\boxed{4}$.