Investigation of the Function with High Degree of Freedom for the S-shape Curve

The S-shape curve function with high degree of freedom is proposed based on the analysis of the basic characteristics of the Logistic function, and the S-shape curve function is y=A_2+\fracA_1-A_21+\left(\fracx-x_0\Delta x_1\right)^p_1+\left(\fracx-x_0\Delta x_2\right)^p_2. The characteristic parameters of the function are extreme values of A₁ and A₂, and the location of extreme value A₁ is x₀. The shape modulatory parameters of function are Δx₁, p₁, Δx₂ and p₂. The conditions assuring the S shape of the function’s curve are A₁≠A₂, Δx₁>0, Δx₂>0, p₁>1 and p₂>1. The functions of the first derivative (the slope of the tangent), the slope of the normal and the second derivative of the modulatory Logistic function are derived.