 The quartic function
The source or original quartic function
The quartic functions classification diagram The graphs of the quartic functions types 4/1 . . . 4/6
Quartic function   y = a4x4 + a3x3 + a2x2 + a1x + a0
 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.
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 altogether ten types or shapes of graphs of quartic functions.
The graphs of the quartic functions types  4/1 . . . 4/6
Remaining six types of quartic polynomial satisfy the criteria shown in the diagram below. The roots of the source quartic  y = a4x4 + a2x2 + a1x  Types, 4/1, 4/2, 4/3 and 4/4 the roots of the Types, 4/5 and 4/6  The abscissa of the turning point of the Types, 4/1, 4/2 and 4/3 the abscissas of the turning points of the Types, 4/4, 4/5 and 4/6  The abscissas of the points of inflection of the source quartic of  Types 4/2 to 4/6, The roots of the translated quartic Type 4 we get by adding x0 to the solutions of the equation a4x4 + a2x2 + a1x + y0 = 0.   Calculus contents A 