  We study critical orbits and bifurcations within the moduli space of
quadratic rational maps on $\mathbb{P}^1$. We focus on the family of curves,
$Per_1(\lambda)$ for $\lambda$ in $\mathbb{C}$, defined by the condition that
each $f\in Per_1(\lambda)$ has a fixed point of multiplier $\lambda$. We prove
that the curve $Per_1(\lambda)$ contains infinitely many postcritically-finite
maps if and only if $\lambda = 0$; addressing a special case of [BD2,
Conjecture 1.4]. We also show that the two critical points of a map $f$ define
distinct bifurcation measures along $Per_1(\lambda)$.
