  We study the effect of constant shifts on the zeros of rational harmomic
functions $f(z) = r(z) - \conj{z}$. In particular, we characterize how shifting
through the caustics of $f$ changes the number of zeros and their respective
orientations. This also yields insight into the nature of the singular zeros of
$f$. Our results have applications in gravitational lensing theory, where
certain such functions $f$ represent gravitational point-mass lenses, and a
constant shift can be interpreted as the position of the light source of the
lens.
