  Let $f \in Q(z)$ be a polynomial or rational function of degree 2. A special
case of Morton and Silverman's Dynamical Uniform Boundedness Conjecture states
that the number of rational preperiodic points of $f$ is bounded above by an
absolute constant. A related conjecture of Silverman states that the canonical
height $\hat{h}_f(x)$ of a non-preperiodic rational point $x$ is bounded below
by a uniform multiple of the height of $f$ itself. We provide support for these
conjectures by computing the set of preperiodic and small height rational
points for a set of degree 2 maps far beyond the range of previous searches.
