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
Quartic function   y = a4x4 + a3x3 + a2x2 + a1x + a0
Transformation of the quartic polynomial function 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 for the quartic function
By setting the coefficients a2 and a1 of the source quartic to zero, interchangeably, obtained is the basic classification shown in the diagram.
Thus, there are ten types (different shapes of the graphs) of quartic polynomial functions.
Pre-calculus contents E