Approximating continuous functions by polynomials is vital to scientific computing and numerous numerical techniques. On the other hand, polynomials can be characterized in several ways using different bases, where every form of basis has its advantages and power. By a proper choice of basis, several problems will be removed; for instance, stability and efficiency can be improved, and numerous complications can be resolved. In this paper, we provide an explicit formula of the generalized shifted Chebyshev Koornwinder’s type polynomial of the first kind, T r * ( K 0 , K 1 ) ( x ) , using the Bernstein basis of fixed degree. Moreover, a Bézier’s degree elevation was used to rewrite T r * ( K 0 , K 1 ) ( x ) in terms of a higher degree Bernstein basis without altering the shapes. In addition, explicit formulas of conversion matrices between generalized shifted Chebyshev Koornwinder’s type polynomials and Bernstein polynomial bases were given.