The coordinates of translations formulas
The values of the coefficients, a2 and a1 of the source quartic function y = a4x4 + a2x2 + a1x
Graphing the quartic polynomial 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.
Graphing the quartic polynomial 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 function.
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 function,
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 function   y = a4x4 + a2x2 + a1x  satisfy the conditions,
the given quartic 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,
Pre-calculus contents E