Quartic function
The coordinates of translations formulas of quartic function
The basic classification criteria diagram for quartic function
The graphs and classification criteria diagram for the quartics types 4/1 to 4/6
Graphing the quartic function example
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 and classification criteria diagram for the quartic function types 4/1 to 4/6
Remaining six types of quartic polynomial satisfy the criteria shown in the diagram below.
Graphing the quartic function example
Example:  Given is quartic function  y = x4 - 4x3 + 4x2 + x - 4, find its source or original function and calculate the coordinates of translations, the zero points, the turning points and the points of inflection.
Draw graphs of the source and the given quartic.
Solution:  1)  Calculate the coordinates of translations
y0 f (x0)   =>     y0 = f (1) = 14 -  4 · 1 + 4 · 1 + 1 - = - 2,         y0 = - 2
2)  To get the source quartic, plug the coordinates of translations into the general form of the quartic,
y + y0 =  a4(x + x0)4 + a3(x + x0)3 + a2(x + x0)2 + a1(x + x0) + a0   thus,
y - 2 = (x + 1)4 - 4(x + 1)3 + 4(x + 1)2 + (x + 1) - 4  so,   y = x4 - 2x2 + x  is the source function.
Since the coefficients of the source quartic  y = a4x4 + a2x2 + a1x  satisfy the conditions,
the given quartic function is the type 4/6 whose source graph has four zeroes, three turning points and two points of inflection.
The zeroes of the source quartic   y = x4 - 2x2 + x are,
.
The abscissas of the turning points of the given quartic  y = x4 - 4x3 + 4x2 + x - 4,
xT1 = 1.837,   xT2 = - 0.107 and  xT3 = 1.269.
 The points of inflection of the given quartic,
Functions contents C