Although many phenotypic traits are determined by a large number of genetic variants, how a polygenic trait adapts in response to a change in the environment is not completely understood. In the framework of diffusion theory, we study the steady state and the adaptation dynamics of a large but finite population evolving under stabilizing selection and symmetric mutations when selection and mutation are moderately large. We find that in the stationary state, the allele frequency distribution at a locus is unimodal if its effect size is below a threshold effect and bimodal otherwise; these results are the stochastic analog of the deterministic ones where the stable allele frequency becomes bistable when the effect size exceeds a threshold. It is known that following a sudden shift in the phenotypic optimum, in an infinitely large population, selective sweeps at a large-effect locus are prevented and adaptation proceeds exclusively via subtle changes in the allele frequency; in contrast, we find that the chance of sweep is substantially enhanced in large, finite populations and the allele frequency at a large-effect locus can reach a high frequency at short times even for small shifts in the phenotypic optimum.