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
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= tan at .
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 E
Copyright 2004 - 2020, Nabla Ltd.  All rights reserved.