Polynomial and/or Polynomial Functions and Equations
Translated monomial (or power) function
Translated monomial function graphs
Translated sextic function example
Translated monomial or power function
If we set all coefficients, an-2  to a1, in the expanded form of the polynomial to zero, we get
y - y0 = an(x - x0)n,   x0- an-1/( n · an)  and  y0 = f (x0).
the 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 Î N, the graph of the source power function             y = anxn  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 Î N, the graph of the source power function        y = anxn  is symmetric about the origin, that is  f (-x) = - f (x).
Example:  Given is sextic  y = (1/4)x6 - 6x5 + 60x4 - 320x3 + 960x2 - 1536x + 1008, find its source or original function and calculate the coordinates of translations, the zero points and the turning point.
Draw graphs of the source and the given sextic.
Solution:  1)  Calculate the coordinates of translations
y0 = f (4) = (1/4) · 46 - 6 · 45 + 60 · 44 - 320 · 43 + 960 · 42 - 1536 · 4 + 1008,     y0 = - 16.
2)  To get the source sextic, plug the coordinates of translations into the general form of the given sextic, to set its graph back to the origin,
y + y0 = a6(x + x0)6 + a5(x + x0)5 + a4(x + x0)4 + a3(x + x0)3 + a2(x + x0)2 + a1(x + x0) + a0  or
y - 16 = (1/4)(x - 4)6 - 6(x - 4)5 + 60(x - 4)4 - 320(x - 4)3 + 960(x - 4)2 - 1536(x - 4) + 1008,
after expanding and reducing above expression obtained is
y = (1/4) x6    - the source sextic function.
Since all the coefficients, a4, a3, a2 and a1, of the source sextic y = a6x6 + a4x4 + a3x3 + a2x2 + a1x, are zero then, the only turning point is T(x0, y0)  or  T( 4, - 16).
3)  Inversely, by plugging the coordinates of translations into the source sextic
y - y0 = a6(x - x0)6    or     y - 16 = (1/4)(x - 4)6
what after expanding yields
y = (1/4) x6 - 6x5 + 60x4 - 320x3 + 960x2 - 1536x + 1008
the given sextic. Therefore, the given sextic is translated monomial or power function.
As the translated monomial or power function has zeros if  a6 ·  y0 < 0 then.
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