Polynomial and/or Polynomial Functions and Equations
      Transition of the polynomial expression from the general to source form and vice versa
         Deriving the coordinates of translations formulas and the coefficients of the source function
         Quadratic function
         Cubic function
         Quartic function
Transition of the polynomial expression from the general to source form and vice versa
Deriving the coordinates of translations formulas and the coefficients of the source function
Quadratic function  f (x) = a2x2 + a1x + a0
1)  Let calculate the coordinates of translations of quadratic function using the formulas,
substitute n = 2 in  
then    
2)  To get the source quadratic function we should plug the coordinates of translations (with changed signs)
     into the general form of the quadratic, i.e.,
after expanding and reducing obtained is
                                     y = a2x2   the source quadratic function
3)  Inversely, by plugging the coordinates of translations into the source quadratic function
                                     y - y0 = a2(x - x0)2,
   
and after expanding and reducing we obtain
                                     y = a2x2 + a1x + a0   the quadratic function in the general form.
Cubic function  f (x) = a3x3 + a2x2 + a1x + a0
Applying the same method we can examine the third degree polynomial called cubic function.
1)  Calculate the coordinates of translations
substitute n = 3 in  
then
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.,
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.
Quartic    y = a4x4 + a3x3 + a2x2 + a1x + a0
1)  Calculate the coordinates of translations by plugging n = 4 into
2)  To get the source quartic function we should plug the coordinates of translations (with changed signs)
     into the general form of the quartic, i.e.,
y + y0 = a4(x + x0)4 + a3(x + x0)3 + a2(x + x0)2 + a1(x + x0) + a0,
after expanding and reducing obtained is the source quartic function
3)  Inversely, by plugging the coordinates of translations into the source quartic
                                     y - y0 = a4(x - x0)4 + a2(x - x0)2 + a1(x - x0),
after expanding and reducing we obtain
                                     y = a4x4 + a3x3 + a2x2 + a1x + a0   the quartic function in the general form.
Thus,      y = a4x4 + a3x3 + a2x2 + a1x + a0    or    y - y0 = a4(x - x0)4 + a2(x - x0)2 + a1(x - x0),
by setting  x0 = 0  and  y0 = 0 we get the source quartic   y = a4x4 + a2x2 + a1x.
According to mathematical induction we can treat any n-degree polynomial function using shown method.
Pre-calculus contents F
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