A revealing insight into the polynomial function - continue
Third method
Sigma notation of the polynomial, where the coefficients a of the source polynomial are represented by a recursive formula.
Finally, the polynomial 
 f (x) =  y = anxn + an - 1 xn - 1 + an - 2 xn - 2 + . . . + a2x2 + a1x + a0   we can write
 
                        while,  for   k = 0,             an = an,
     and from an- k,    for   k = n,            a0 = f (x0) = y0.
Thus, the expanded form of the above sum is
y - y0 = an(x - x0)n + an - 2(x - x0)n - 2 + . . . + a2(x - x0)2 + a1(x - x0)
where x0 and y0 are coordinates of translations of the graph of the source polynomial
fs(x) = anxn + an - 2 xn - 2 + . . . + a2x2 + a1x
in the direction of the coordinate axes.
Use of the properties of the polynomial
Therefore, we can write linear function or the polynomial of the first degree
 f (x) = y = a1x + a0   or    y = a1(x - x0)    or    y - y0 = a1x,
where
By setting  x0 = 0  or  y0 = 0  we get  y = a1x  the source linear function.
Polynomial of the second degree or quadratic function
 f (x) = a2x2 + a1x + a0   or     y - y0 = a2(x - x0)2,
where  The turning point T (x0,  y0).
If  x0 = 0 and  y0 = 0  then,  y = a2x2  is the source quadratic. 
Polynomial of the third degree or cubic function
 f (x) = a3x3 + a2x2 + a1x + a0   or     y - y0 = a3(x - x0)3 + a1(x - x0)
where  
By setting  x0 = 0  and  y0 = 0 we get the source cubic function
 y = a3x3 + a1x  where  a1= tan at
therefore, the coefficient  a1 shows the slope of the tangent line at the point of inflection I (x0, y0).
Polynomial of the fourth degree or quartic
 f (x) = a4x4 + a3x3 + a2x2 + a1x + a0   or    y - y0 = a4(x - x0)4 + a2(x - x0)2 + a1(x - x0)
where,  
By setting  x0 = 0  and  y0 = 0 we get
 y = a4x4 + a2x2 + a1x,  the source quartic function.
Thus, we can proceed with, quintic, sextic or an nth degree polynomial.
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