  We examine quintessence models for dark energy in which the scalar field,
$\phi$, evolves near the vicinity of a local maximum or minimum in the
potential $V(\phi)$, so that $V(\phi)$ be approximated by a quadratic function
of $\phi$ with no linear term. We generalize previous studies of this type by
allowing the initial value of $d \phi/dt$ to be nonzero. We derive an analytic
approximation for $w(a)$ and show that it is in excellent agreement with
numerical simulations for a variety of scalar field potentials having local
minima or maxima. We derive an upper bound on the present-day value of $w$ as a
function of the other model parameters and present representative limits on
these models from observational data. This work represents a final
generalization of previous studies using linear or quadratic approximations for
$V(\phi)$.
