Abstract: Newton’s transformation family f w（z）=z-1wz w-1 containing only one complex parameter w（w≠0 or 1） is constructed from the transcendental mapping z→e z w+c . The number of the critical point of f w（z） is countably infinite, and a method （ vcps ） is presented to generate the valid critical point set about the parameter w , vcps = z k|- π ＜ arg （z k）≤ π , f ′ w（z k）=0, k∈ Z , z k∈ C. The Mandelbrot （M） set of f w（z） is constructed according to the w in the parameter plane, about which the orbits of all of the critical points in the vcps are bounded.The other generalized M sets of f w（z） , such as M F, M S, M O and so on, are constructed from the parameter w , in whose vcps the orbit of the n th critical point is bounded. The dynamical characters of f w（z） of the different parameter are analyzed. A great number of novel images of filled-in Julia sets are generated based on the attracting time in which the orbit of a point in the dynamical plane is attracted to the orbit of a critical point of the parameter w .