A class of quadratic polynomial basis functions with multiple shape parameters is presented, which is an extension to quadratic classical Bernstein basis functions. Properties of this new basis functions are analyzed and the corresponding quadratic extended B´ezier curve is defined. The shape parameters can adjust the shape of quadratic extended B´ezier curve without changing the control points. The extended curve can degenerate to a point or a line segment or the classical quadratic B´ezier curve. At last, the condition of C2 continuity for smoothly joining the quadratic extended B´ezier curve and B´ezier curve is discussed.