In this paper we are interested in a general class of hyperbolic balance laws. Within the frame work of existence of a convex entropy function that symmetrizes the system in certain sense and the Kawashima-Shizuta condition, we study the large time behavior of the solution to the Cauchy problem in one space dimension. For a solution around a constant equilibrium state, we predetermine the asymptotic solution as a superposition of the constant state and diffusion waves along the equilibrium characteristic directions. We estimate the remainder in the pointwise sense both in space and in time. The decay rates in various directions are optimal, which further give the optimal Lp rates with 1≤p≤∞. Applications to several examples including the Kerr-Debye model are given.