  We study GIT quotients parametrizing n-pointed conics that generalize the GIT
quotients $(\mathbb{P}^1)^n//SL2$. Our main result is that $\overline{M}_{0,n}$
admits a morphism to each such GIT quotient, analogous to the well-known result
of Kapranov for the simpler $(\mathbb{P}^1)^n$ quotients. Moreover, these
morphisms factor through Hassett's moduli spaces of weighted pointed rational
curves, where the weight data comes from the GIT linearization data.
