Summary: We consider families of optimization problems with quadratic object function and affine linear constraints, which depend smoothly on one real parameter. For a generic subclass of such problems only three different types of (generalized) critical points occur, whereas in the general case (of nonlinear one-parameter families of constrained optimization problems on \(\mathbb{R}^n\)) five types are to be distinguished. We clarify the theoretical background of these phenomena and illustrate the underlying mechanism with simple examples.