Quadratic function    y = a2x2 + a1x + a0
         Graphing source and translated quadratic function
Quadratic function    y = a2x2 + a1x + a0
1)  Let calculate the coordinates of translations of quadratic function using the formulas,
substitute n = 2 in    
then    
2)  To get the source quadratic function we should plug the coordinates of translations (with changed signs) into the general form of the quadratic, i.e.,
after expanding and reducing obtained is
                                     y = a2x2   the source quadratic function.
3)  Inversely, by plugging the coordinates of translations into the source quadratic function
                                     y - y0 = a2(x - x0)2,
   
and after expanding and reducing we obtain
                                     y = a2x2 + a1x + a0   the quadratic function in the general form.
Graphing source and translated quadratic function
f (x) = y = a2x2 + a1x + a0   or   y - y0a2(x + x0)2,
where
By setting  x0 = 0  and  y0 = 0  obtained is 
the source quadratic  y a2x2 .
If  a2 y0 < 0  then the roots are
The turning point  V(x0 , y0 ).
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.
Pre-calculus contents D
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