There are three types of the cubic functions - the classification criteria diagram
         The graphs of the source cubic functions
The graphs of the source cubic functions
There are three types (shapes) of cubic functions whose graphs of the source functions are shown in the figure below:
type 1 y = a3x3 + a2x2 + a1x + a0    or    y - y0 = a3(x - x0)3,    - a22 + 3a3a1 = 0 or a1 = 0.
therefore, its source function  y = a3x3,  and the tangent line through the point of inflection is horizontal.
type 2/1 y = a3x3 + a2x2 + a1x + a0     or      y - y0 = a3(x - x0)3 + a1(x - x0), where  a3a1> 0
whose slope of the tangent line through the point of inflection is positive and equals a1.
type 2/2 y = a3x3 + a2x2 + a1x + a0     or      y - y0 = a3(x - x0)3 + a1(x - x0), where  a3a1< 0
whose slope of the tangent line through the point of inflection is negative and is equal a1
The graph of its source function has three zeros or roots at  
and two turning points at
The graphs of the source cubic functions - the classification criteria diagram
Pre-calculus contents E
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