The graphs of algebraic functions
       The graphs of the polynomial functions
          The source or original polynomial function
          Translating (parallel shifting) of the polynomial function
          Coordinates of translations and their role in the polynomial expression
       The graphs of the quartic function
Polynomial function
  The polynomial function   f (x) =  yan xn + an-1 xn-1 + an-2 xn-2 + . . . + a2 x2 + a1 x + a0
                                                    y a1x + a0                                                   - Linear function 
                                                    y = a2x2 + a1x + a0                                                      - Quadratic function 
                                                    y = a3x3 + a2x2 + a1x + a0                                       - Cubic function
                                                    y = a4x4 + a3x3 + a2x2 + a1x + a0                        - Quartic function
                                                    y = a5x5 + a4x4 + a3x3 + a2x2 + a1x + a0         - Quintic function
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The graphs of the polynomial functions
The graph of a function ƒ is drawing on the Cartesian plane, plotted with respect to coordinate axes, that shows functional relationship between variables. The points (x, ƒ (x)) lying on the curve satisfy this relation.
The source or original polynomial function
Any polynomial f (x) of degree n > 1 in the general form, consisting of n + 1 terms, shown graphically, represents translation of its source (original) function in the direction of the coordinate axes.
The source polynomial function
fs (x) = an xn + an-2xn-2 + . . . + a2x2 + a1 x
has n - 1 terms lacking second and the constant term, since its coefficients, an-1 = 0 and a0 = 0 while the leading coefficient an, remains unchanged.
Therefore, the source polynomial function passes through the origin.
A coefficient ai of the source function is expressed by the coefficients of the general form.
Translating (parallel shifting) of the polynomial function
Thus, to obtain the graph of a given polynomial function f (x) we translate (parallel shift) the graph of its source function in the direction of the x-axis by x0 and in the direction of the y-axis by y0.
Inversely, to put a given graph of the polynomial function beck to the origin, we translate it in the opposite direction, by taking the values of the coordinates of translations with opposite sign.
Coordinates of translations and their role in the polynomial expression
The coordinates of translations we calculate using the formulas,
Hence, by plugging the coordinates of translations into the source polynomial function fs(x), i.e.,
y - y0 = an (x - x0)n + an-2 (x - x0)n-2 + . . .  + a2 (x - x0)2 + a1 (x - x0)
and by expanding above expression we get the polynomial function in the general form 
f (x) =  yan xn + an-1 xn-1 + an-2 xn-2 + . . . + a2 x2 + a1 x + a0.
Inversely, by plugging the coordinates of translations into the given polynomial f (x) expressed in the general form, i.e.,
y + y0 = an (x + x0)n + an-1 (x + x0)n-1 + . . .  + a1 (x + x0) + a0
and after expanding and reducing above expression we get its source polynomial function.
Note that in the above expression the signs of the coordinates of translations are already changed.
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, we get the basic classification of the quartic function.
Calculus contents A
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