smc

If we take the parabola $a{x}^2+bx+c$ and reflect it over the x - axis, we have the parabola $-a{x}^2-bx-c$. Without loss of generality, let us say that the parabola is translated 5 units to the left, and the reflection to the right. Then: $$\begin{array}{r}f(x)=a(x+5{)}^2+b(x+5)+c=a{x}^2+(10a+b)x+25a+5b+c\\ g(x)=-a(x-5{)}^2-b(x-5)-c=-a{x}^2+10ax-bx-25a+5b-c\end{array}$$ Adding them up produces: $$\begin{array}{r}(f+g)(x)=a{x}^2+(10a+b)x+25a+5b+c-a{x}^2+10ax-bx-25a+5b-c=20ax+10b\end{array}$$ This is a line with slope $20a$. Since $a$ cannot be $0$ (because $a{x}^2+bx+c$ would be a line) we end up with $\boxed{\text{a non-horizontal line}}$