This paper studies the large-t behavior of the boundary generated by the method of images for the first-passage-time problem. We show that this behavior is characterized by certain properties of the Laplace transform of the input measure. Such properties also determine the asymptotic behavior of the first-passage-time density. Most of the paper assumes a positive input measure, which generates a concave boundary. The last section, however, discusses a non-positive measure. We obtain a sufficient condition for the boundary to be convex. © 2012 Springer Science+Business Media New York.