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We examine quintessence models for dark energy in which the scalar field ɸ evolves near the vicinity of a local maximum or minimum in the potential V(ɸ), so that V(ɸ) can be approximated by a quadratic function of ɸ with no linear term. We generalize previous studies of this type by allowing the initial value of dɸ/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(ɸ). [ABSTRACT FROM AUTHOR]