The graphs of the polynomial functions
      Quartic function
         The basic classification diagram for the quartic function
         The graphs of the quartic function types, 1, 2, 3/1 and 3/2
Quartic function    y = a4x4 + a3x3 + a2x2 + a1x + a0
1)  Calculate the coordinates of translations by plugging n = 4 into
2)  To get the source quartic function we should plug the coordinates of translations (with changed signs) into the general form of the quartic, i.e.,
y + y0 = a4(x + x0)4 + a3(x + x0)3 + a2(x + x0)2 + a1(x + x0) + a0,
after expanding and reducing obtained is the source quartic function
3)  Inversely, by plugging the coordinates of translations into the source quartic
                                     y - y0 = a4(x - x0)4 + a2(x - x0)2 + a1(x - x0),
after expanding and reducing we obtain
                                     y = a4x4 + a3x3 + a2x2 + a1x + a0   the quartic function in the general form.
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 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)4a2 = 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)2a1 = 0.
type 3/1 a4a2 > 0
type 3/2 a4a2 < 0
type 3/1
a4a2 > 0
T (x0, y0).
type 3/2
a4a2 < 0
 
Functions contents C
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