In this work, a class of generalized quadratic Bernstein-like functions having controlling functions is constructed. It contains many particular cases from earlier papers. Regarding the controlling functions, sufficient conditions are given. Corner cutting algorithms and the accompanying quadratic Bézier curves are discussed. A class of generalized quadratic B-splines possessing controlling functions is proposed. Some important properties for curve and surface design are proved. Sufficient conditions for $C^2$ continuity, $C^3$ continuity and $C^n$ continuity are also given. Some applications of the constructed B-splines in ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$ are presented, which show the ability to adjust the shape of the curves flexibly and locally. These applications show that generalized quadratic B-splines can be easily implemented and serve as an alternative strategy for modeling curves.