$\begin{align}  & {{x}_{A}}=-2\Rightarrow \frac{-b}{-2a}=-2\Rightarrow -b=4a\Rightarrow b=-4a\,\,(1) \\  & y=-a{{x}^{2}}+bx+1\xrightarrow{A(-2,2)}2=-a{{(-2)}^{2}}+b(-2)+1\Rightarrow 2=-4a-2b+1\Rightarrow 4a+2b=-1\,\,(2) \\  & (1),(2)\Rightarrow 4a+2(-4a)=-1\Rightarrow 4a-8a=-1\Rightarrow -4a=-1\Rightarrow a=\frac{1}{4}\xrightarrow{b=-4a}b=-4(\frac{1}{4})=-1 \\  & a+b=\frac{1}{4}+(-1)=-\frac{3}{4} \\ \end{align}$