  We compute curvature-dependent graviton correlation functions and couplings
as well as the full curvature potential $f(R)$ in asymptotically safe quantum
gravity coupled to scalars. The setup is based on a systematic vertex expansion
about metric backgrounds with constant curvatures initiated in arXiv:1711.09259
for positive curvatures. We extend these results to negative curvature and
investigate the influence of minimally coupled scalars. The quantum equation of
motion has two solutions for all accessible numbers of scalar fields. We
observe that the solution at negative curvature is a minimum, while the
solution at positive curvature is a maximum. We find indications that the
solution to the equation of motions for scalar-gravity systems is at large
positive curvature, for which the system might be stable for all scalar
flavours.
