The general Mandelbrot sets from the non-analytical complex mapping ()czz+←^-α for 2≥α are studied in this paper. The M-sets’ properties of different parameter α are theoretically analyzed and proved. The parameter equation of the fixed point region’s boundary when α is a positive integer are strictly given. A symmetrical period-checking algorithm, which colors M-sets according to the period of each point in the complex C-plane, is put forward for the first time. The new algorithm takes full advantage of the M-sets’ property during the rendering process and can greatly reduce the number of iterations in calculating the period of all pixel points in the drawing region. The experimental results show that both high quality and high drawing speed of the M-sets’ fractal image can be acquired with the new algorithm. Furthermore, the new algorithm can be generalized to the drawing of other Mandelbrot sets and Julia sets.