

Polynomial and/or Polynomial
Functions and Equations 
Transition of the polynomial
expression from the general to source form and vice versa 
Deriving
the coordinates of translations formulas and the coefficients of
the source function 
Quadratic
function 
Cubic
function 
Quartic 





Transition of the polynomial
expression from the general to source form and vice versa 
Deriving
the coordinates of
translations formulas and the coefficients of the source function 
Quadratic
function f
(x)
=
a_{2}x^{2}
+ a_{1}x
+ a_{0} 
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
=
a_{2}x^{2}
the source quadratic function 
3)
Inversely, by plugging the coordinates of translations into the source quadratic function 
y

y_{0}
= a_{2
}(x

x_{0})^{2}, 

and
after
expanding and reducing we obtain 
y
=
a_{2}x^{2}
+ a_{1}x
+ a_{0} the quadratic function
in the general form. 

Cubic
function f
(x)
= a_{3}x^{3}
+
a_{2}x^{2}
+ a_{1}x
+ a_{0} 
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

y_{0}
= a_{3}(x

x_{0})^{3}
+
a_{1}(x

x_{0}), 

after
expanding and reducing we obtain 
y
=
a_{3}x^{3}
+ a_{2}x^{2}
+ a_{1}x
+ a_{0} the cubic function
in the general form. 

Quartic
function y
=
a_{4}x^{4}
+ a_{3}x^{3}
+
a_{2}x^{2}
+
a_{1}x + a_{0} 
1)
Calculate the
coordinates of translations by plugging
n
= 4
into 

2)
To
get the source quartic function we should plug the coordinates
of translations (with changed signs) into the general form
of the quartic,
i.e., 
y
+ y_{0}
= a_{4}(x
+ x_{0})^{4}
+
a_{3}(x
+ x_{0})^{3}
+
a_{2}(x
+ x_{0})^{2}
+
a_{1}(x
+ x_{0}) +
a_{0}, 

after
expanding and reducing obtained is the source
quartic function 

3)
Inversely, by plugging the coordinates of translations into the source
quartic 
y

y_{0}
=
a_{4}(x

x_{0})^{4}
+
a_{2}(x

x_{0})^{2}
+
a_{1}(x

x_{0}), 

after
expanding and reducing we obtain 
y
= a_{4}x^{4}
+ a_{3}x^{3}
+
a_{2}x^{2}
+
a_{1}x + a_{0}
the
quartic function
in the general form. 
Thus,
y
= a_{4}x^{4}
+ a_{3}x^{3}
+
a_{2}x^{2}
+
a_{1}x + a_{0}_{
}or_{ }y

y_{0}
=
a_{4}(x

x_{0})^{4}
+
a_{2}(x

x_{0})^{2}
+
a_{1}(x

x_{0}), 


by
setting x_{0}
=
0 and y_{0}
= 0 we get
the source quartic _{ }y
=
a_{4}x^{4}
+
a_{2}x^{2}
+
a_{1}x. 


According
to mathematical induction we can treat any ndegree
polynomial function using shown method. 








Functions
contents C 



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