The cubic function
General or translated form of cubic function
Graphs of the source cubic functions
The cubic function
General or translated form of cubic function
 The general cubic function   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 graphs) 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
Graphs of the source cubic functions
Coordinates of the point of inflection coincide with the coordinates of translations, i.e.,  I (x0, y0).
The source cubic functions are odd functions.
Graphs of odd functions are symmetric about the origin that is, such functions change the sign but not absolute value when the sign of the independent variable is changed, so that  f (x) = - f (-x).
Therefore, since  f (x) = a3x3 + a1x  then  - f (-x) = - [a3(-x)3 + a1(-x)]  = a3x3 + a1x f (x).
That is, change of the sign of the independent variable of a function reflects the graph of the function about the y-axis, while change of the sign of a function reflects the graph of the function about the x-axis.
The graphs of the translated cubic functions are symmetric about its point of inflection.
Pre-calculus contents D