Recently, the Voronoi diagram for moving objects has been investigated in connection with motion planning in robotics and geometric optimization problems in computational geometry. In this paper, we consider the Voronoi diagram for moving points parametrized by t in the plane, whose coordinates are polynomials or rational functions of t. We show that the dynamic Voronoi diagram has the combinatorial complexity of O(n^2λ_<s+2>(n)) and can be computed in O(n^2λ_<s+1>(n)log n) time and O(n) space, where s is some fixed number and λ_s(n) is the maximum length of (n, s) Davenport-Schinzel sequence.