The graphs of the quartic function
The basic classification diagram for the quartic polynomial function
The transition of quartic function expression
 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 diagram for the quartic polynomial function
By setting the coefficients a2 and a1 of the source quartic to zero, interchangeably, obtained is the basic classification shown in the diagram.
There are ten types (shapes of the graphs) of quartic functions.
 type 1 y = a4x4 + a3x3 + a2x2 + a1x + a0    or    y - y0 = a4(x - x0)4,  a2 = 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)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 D