Polynomial and/or Polynomial Functions and Equations
      Cubic functions
         Transformation of the cubic polynomial from the general to source form and vice versa
         Coordinates of the point of inflection coincide with the coordinates of translations
         The source cubic functions are odd functions
      There are three types of the cubic functions - the classification criteria diagram
         The graphs of the source cubic functions
Cubic function    y = a3x3 + a2x2 + a1x + a0
Transformation of the cubic polynomial from the general to source form and vice versa
Applying the same method we can examine the third degree polynomial called cubic function.
1)  Calculate the coordinates of translations
substitute n = 3 in    
and
2)  To get the source cubic function we should plug the coordinates of translations (with changed signs) into the general form of the cubic, i.e.,
y + y0 = a3(x + x0)3 + a2(x + x0)2 + a1(x + x0) + a0,
after expanding and reducing obtained is
  the source cubic function.
3)  Inversely, by plugging the coordinates of translations into the source cubic
                                     y - y0 = a3(x - x0)3 + a1(x - x0),
   
after expanding and reducing we obtain
                                     y = a3x3 + a2x2 + a1x + a0   the cubic function in the general form.
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 .
Coordinates of the point of inflection coincide with the coordinates of translations, i.e.,  I (x0, y0). 
 
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.
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
Functions contents C
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