The cubic function
The classification diagram of the cubic function
The graphs of the source cubic functions
The cubic function    y = a3x3 + a2x2 + a1x + a0
 Thus,         y = a3x3 + a2x2 + a1x + a0     or      y - y0 = a3(x - x0)3 + a1(x - x0), by setting  x0 = 0  and  y0 = 0 we get the source cubic function  y = a3x3 + a1x  where  a1= tanat .
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,    - (a2)2 + 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 classification diagram of the cubic function
The graphs of the source cubic functions
Calculus contents A