For extending the representation of the Bézier curve, a class of cubic polynomial basis functions with two shape parameters is presented in this paper firstly, which is an extension of the quadratic Bernstein basis. Then, a modifiable cubic polynomial curve is presented based on the basis functions, and the relation between the curve and the classical Bézier curves is discussed. The curve is an extension of the quadratic Bézier curve, which inherits most properties of the quadratic Bézier curve, and its shape can be local or globally modified by changing the values of the two shape parameters when the control points are not changed. For designing free curves, the continuity condition of the twopiece curves is discussed. Finally, some application examples of the curve in the curve design are presented.