In this article, we have studied the behavior of the potential function along some geodesic in a Ricci soliton under some curvature restriction. In particular, we haveshowed that under some curvature restriction, the potential function is reduced to a parabola along some geodesic. Furthermore, we have investigated the change of intersecting angles between the potential vector field and a geodesic in a Ricci soliton. Further, we have deduced the condition when the potential function becomes convex in a shrinking Ricci soliton. Finally, we have concluded the paper by showing the non-existence of convex potential in an expanding Ricci soliton having non-negative Ricci curvature.