Polynomial and/or Polynomial Functions and Equations
      Definition of a polynomial or polynomial function
         Source or original polynomial function
         Translating (parallel shifting) of the polynomial function
         Coordinates of translations and their role in the polynomial expression
         Transformations of the polynomial function applied to the quadratic and cubic functions
Definition of a polynomial or polynomial function
A polynomial and/or polynomial function in the variable x is an expression of the general (or standard) form 
f (x) = anxn + an-1xn-1 + . . . + a1x + a0
consisting of n + 1 terms each of which is a product of a real coefficient ai and the variable x raised to a non-negative integral power.
If the leading coefficient of a polynomial an is not 0 then, the degree of the polynomial is n. 
The constant term a0 is the y-intercept of the polynomial.
Source or original polynomial function
Any polynomial  f (x) of degree n > 1 in the general form, consisting of n + 1 terms, shown graphically,
represents translation of its source (original) function in the direction of the coordinate axes.
The source polynomial function
fs(x) = anxn + an-2xn-2 + . . . + a2x2 + a1x
has n - 1 terms lacking second and the constant term, since its coefficients, an-1 = 0 and a0 = 0 while the leading coefficient an, remains unchanged.
Therefore, the source polynomial function passes through the origin.
A coefficient ai of the source function is expressed by the coefficients of the general form.
Translating (parallel shifting) of 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. 
Inversely, by plugging the coordinates of translations into a given polynomial function f(x), that is 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.
Therefore, each polynomial missing only second term (an-1 = 0), represents the source polynomial whose graph is translated in the direction of the y-axis by  y0 = a0.
Transformations of the polynomial function applied to the quadratic and cubic functions
The application of the above theory to the quadratic and cubic polynomial functions.
Quadratic function  f (x) = 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.
Cubic function  f (x) = a3x3 + a2x2 + a1x + a0
Applying the same method we can examine the third degree polynomial called cubic function.
1)  Calculate the coordinates of translations
substitute n = 3 in  
then
2)  To get the source cubic function we should plug the coordinates of translations (with changed signs)
     into the general form of the cubic, i.e.,
after expanding and reducing obtained is
  the source cubic function.
3)  Inversely, by plugging the coordinates of translations into the source cubic
                                     y - y0 = a3(x - x0)3 + a1(x - x0),
   
after expanding and reducing we obtain
                                     y = a3x3 + a2x2 + a1x + a0   the cubic function in the general form.
According to mathematical induction we can examine any n-degree polynomial function using shown method.
Intermediate algebra contents
Copyright 2004 - 2020, Nabla Ltd.  All rights reserved.