The graphs of the polynomial functions
          Translating (parallel shifting) of the polynomial function
          Coordinates of translations and their role in the polynomial expression
       Coefficients of the source polynomial function are related to its derivative at x0
The graphs of the polynomial functions
Translating or parallel shifting 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) =  yanxn + an-1xn-1 + an-2xn-2 + . . . + a2x2 + a1x + 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.
Coefficients of the source polynomial function are related to its derivative at x0
The coefficients of the source polynomial are related to corresponding value of its derivative at x0 like the coefficients of the Taylor polynomial in Taylor's or Maclaurin's formula, thus
where, an = anan - 1 = 0a0 = f (x0), and where  f (n - k) (x0)  denotes (n - k)th derivative at x0.
Such for example, the coefficient a1 of the source cubic of   f (x) = a3x3 + a2x2 + a1x + a0 
since   f ' (x) = 3a3x2 + 2a2x + a1   and  x0 = - a2/(3a3)  then 
Thus, if    y = a4x4 + a3x3 + a2x2 + a1x + a0    or    y - y0 = a4(x - x0)4 + a2(x - x0)2 + a1(x - x0),
where or    f ' (x0) = 1! a1  and   f '' (x0) = 2! a2
then, a1 and a2  define vertical translations of the successive derivatives, as shows below figure.
Calculus contents A
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