Polynomial and/or Polynomial Functions and Equations
      Quartic function
         Transformation of the quartic polynomial from the general to source form and vice versa
         The coordinates of translations formulas
         The values of the coefficients, a2 and a1 of the source quartic function y = a4x4 + a2x2 + a1x
      The basic classification criteria diagram for quartic function
         The graphs of quartic functions and their characteristic points
         The quartic  type 1,   y - y0 = a4(x - x0)4a2 = 0 and a1 = 0
         The quartic  type 2,   y - y0 = a4(x - x0)4 + a1(x - x0)a2 = 0
         The quartic  type 3/1,   y - y0 = a4(x - x0)4 + a2(x - x0)2a1 = 0 and  a4a2 > 0
         The quartic  type 3/2,   y - y0 = a4(x - x0)4 + a2(x - x0)2a1 = 0 and  a4a2 < 0
Quartic function   y = a4x4 + a3x3 + a2x2 + a1x + a0
Transformation of the quartic polynomial from the general to source form and vice versa
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.
The coordinates of translations formulas and the values of the coefficients of the source quartic function
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.
The basic classification criteria diagram of quartic function
By setting the coefficients a2 and a1 of the source quartic to zero, interchangeably, obtained is the basic classification for quartic function shown in the diagram.
The graphs of quartic functions and their characteristic points
There are ten types (different shapes of the graphs) of quartic functions
type 1 y = a4x4 + a3x3 + a2x2 + a1x + a0    or    y - y0 = a4(x - x0)4a2 = 0 and a1 = 0.
The zeroes or roots:
type 2 y = a4x4 + a3x3 + a2x2 + a1x + a0    or    y - y0 = a4(x - x0)4 + a1(x - x0)a2 = 0.
The zeroes of the source function:  
 The zeroes of the translated function we get by adding x0 to the solution of the equation   a4x4 + a1x + y0 = 0.  
type 3 y = a4x4 + a3x3 + a2x2 + a1x + a0    or    y - y0 = a4(x - x0)4 + a2(x - x0)2a1 = 0.
type 3/1 a4a2 > 0
type 3/2 a4a2 < 0
type 3/1
a4a2 > 0
T (x0, y0).
type 3/2
a4a2 < 0
 
Functions contents C
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