We study the worst-case performance of the maximal matching heuristic applied to the Minimum Vertex Cover and Minimum Maximal Matching problems, through a careful analysis of tight examples. We show that the tight worst-case approximation ratio is asymptotic to ${\rm min}\, \{2, 1/(1-\sqrt{1-\epsilon})\}$ for graphs with an average degree at least en and to min {2, 1/e} for graphs with a minimum degree at least en.