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
The graphs of quartic functions, type 3/1 and type 3/2
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
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.
The graphs of quartic functions and their characteristic points
There are ten types (different shapes of the graphs) of quartic functions
 type 3 y = a4x4 + a3x3 + a2x2 + a1x + a0    or    y - y0 = a4(x - x0)4 + a2(x - x0)2,  a1 = 0.
 type 3/1 a4·a2 > 0
 type 3/2 a4·a2 < 0
 type 3/1 a4·a2 > 0
T (x0, y0).
 type 3/2 a4·a2 < 0
Pre-calculus contents E