

The
graphs of the
polynomial functions 
The
source
or original polynomial function 
Translating
(parallel shifting) of the polynomial function 
Coordinates of translations
and their role in the polynomial expression 
Coefficients
of the source polynomial function are related to its derivative
at x_{0} 





Elementary
functions are, Algebraic functions
and Transcendental functions 
Algebraic
functions 
· The
polynomial function
f (x) =
y = a_{n}x^{n}
+ a_{n}_{1}x^{n}^{}^{1}
+ a_{n}_{2}x^{n}^{}^{2}
+
.
. . +
a_{2}x^{2}
+
a_{1}x + a_{0} 
y
=
a_{1}x
+ a_{0}
 Linear
function 
y
=
a_{2}x^{2}
+
a_{1}x + a_{0}_{
}_{
}
Quadratic
function 
y
=
a_{3}x^{3}
+
a_{2}x^{2}
+
a_{1}x + a_{0}_{
}
Cubic
function 
y
=
a_{4}x^{4}
+
a_{3}x^{3}
+
a_{2}x^{2}
+
a_{1}x + a_{0}_{
}
Quartic
function 
y
=
a_{5}x^{5}
+ a_{4}x^{4}
+
a_{3}x^{3}
+
a_{2}x^{2}
+
a_{1}x + a_{0}_{
}
Quintic
function 
   
        
       
        
·
Rational
functions  a ratio of two polynomials 



Reciprocal function 



Translation of the reciprocal function,
called linear rational function. 



The
graphs of the
polynomial functions 
The
graph of a function f
is drawing on the Cartesian plane, plotted with respect to
coordinate axes, that shows functional relationship between
variables. The points (x,
f (x)) lying on the curve
satisfy this relation. 

The
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 

f_{s
}(x)
= a_{n}x^{n}
+ a_{n}_{}_{2}x^{n}^{}^{2}
+
.
. . +
a_{2}x^{2}
+ a_{1}x



has
n
 1 terms
lacking second and the constant term, since its coefficients, a_{n}_{}_{1
}=
0
and a_{0
}=
0
while
the leading coefficient a_{n},
remains unchanged. 
Therefore,
the source polynomial function passes through the
origin. 
A
coefficient a_{i
}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 xaxis
by x_{0}
and in the direction of the yaxis
by y_{0}. 
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 f_{s}(x),
i.e., 

y
 y_{0}
= a_{n}(x
 x_{0})^{n}
+ a_{n}_{}_{2}(x
 x_{0})^{n}^{}^{2}
+
.
. .
+
a_{2}(x
 x_{0})^{2}
+ a_{1}(x
 x_{0}) 


and
by expanding above expression we get the polynomial function in
the general form 
f
(x) =
y = a_{n}x^{n}
+ a_{n}_{1}x^{n}^{}^{1}
+ a_{n}_{}_{2}x^{n}^{}^{2}
+
.
. . +
a_{2}x^{2}
+
a_{1}x + a_{0}. 
Inversely, by plugging the coordinates of translations into
the given polynomial f(x)
expressed in the general form,
i.e., 

y
+ y_{0}
= a_{n}(x
+ x_{0})^{n}
+ a_{n}_{}_{1}(x
+ x_{0})^{n}^{}^{1}
+
.
. .
+ a_{1}(x
+ x_{0})
+ a_{0} 


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. 

Coefficients
of the source polynomial function are related to its derivative
at x_{0} 
The
coefficients of the source polynomial are related to
corresponding value of its derivative at x_{0}
like the coefficients
of the Taylor polynomial in Taylor's or Maclaurin's formula,
thus 



Such
for example, the coefficient a_{1}
of the source cubic of
f (x) =
a_{3}x^{3}
+
a_{2}x^{2}
+
a_{1}x + a_{0} 
since
f ' (x)
=
3a_{3}x^{2}
+ 2a_{2}x
+
a_{1}
and x_{0}
=

a_{2}/(3a_{3})
then 









Functions
contents C 



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