The standard means of applying a discrete search to a continuous or hybrid system is the uniform discretization of control actions and action timing. Such discretization is fixed a priori and does not allow search to benefit from information gained at run-time. This paper introduces Information-Based Alpha-Beta Search, a new algorithm that preserves and benefits from the continuous or hybrid nature of the search. In a novel merging of alpha-beta game-tree search and information-based optimization, Information-Based Alpha-Beta Search makes trajectorysampling decisions dynamically based on the maximum-likelihood of search pruning. The result is a search algorithm which, while incurring higher computational overhead for the optimization, manages to so increase the quality of the sampling, that the net effect is a significant increase in performance. We present a new piecewise-parabolic variant of the algorithm and provide empirical evidence of its performance relative to random and uniform discretizations in the context of a variant of the homicidal chauffeur game.