The graphs of algebraic functions
         The polynomial of 4th degree or quartic function    f (x) = a4x4 + a3x3 + a2x2 + a1x + a0
         The graphs of the quartic function
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.
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
 
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.
College algebra contents B
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