The graphs of the cubic function
Translated cubic functions
Drawing graphs of translated cubic functions
Cubic function    y = a3x3 + a2x2 + a1x + a0
There are three types (shapes) of cubic functions whose graphs 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
Translated cubic functions
 type 1 y = a3x3 + a2x2 + a1x + a0     or      y - y0 = a3(x - x0)3- x0)3- x0)3- x0)3   where,
 The root The point of inflection  I(x0, y0).
 type 2/1 y = a3x3 + a2x2 + a1x + a0     or    y - y0 = a3(x - x0)3 + a1(x - x0),    a3 · a1 > 0,
 I(x0, y0).

 type 2/2 y = a3x3 + a2x2 + a1x + a0     or    y - y0 = a3(x - x0)3 + a1(x - x0),    a3 · a1 < 0,
 If  | y0 | > | yT | | | | if  | y0 | < | yT |
 The turning points The point of inflection  I(x0, y0).
Calculus contents A