The graphs of the elementary functions
       Sigma notation of the polynomial
          Coefficients of the source polynomial in the form of a recursive formula
         Translated power function
Sigma notation of the polynomial
Coefficients of the source polynomial in the form of a recursive formula
According to mathematical induction we can examine any n-degree polynomial function using shown method.
Therefore, the polynomial   f (x) =  yanxn + an-1xn-1 + an-2xn-2 + . . .  + a2x2 + a1x + a0
we can write as
                        while,  for   k = 0,            an = an,
            and from which, for  k = n,            a0 = f (x0) = y0.
Thus, 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-2xn-2 + . . . + a2x2 + a1x
in the direction of the x-axis and the y-axis of a Cartesian coordinate system.
Therefore, every given polynomial written in the general form can be transformed into translatable form by calculating the coordinates of translations x0 and y and the coefficients a of its source function.
Translated power function
If we set all coefficients a in the above expanded form of the polynomial to zero, we get
y - y0 = an(x - x0)n,   x0- an-1/( n an) and  y0 = f(x0).
translated power (or monomial) function, the exponent of which is an odd or an even positive integer.
When the exponent is even, i.e., of the form n = 2m,  m N, the graph of the source power function is symmetric about the y-axis, that is  f (- x) =  f (x).
When the exponent is odd, i.e., of the form n = 2m + 1,  m N, the graph of the source power function is symmetric about the origin, that is  f (- x) = - f (x).
   
College algebra contents B
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