

The
graphs of the elementary functions 
Sigma
notation of the polynomial 
Coefficients of the source
polynomial in the form of a recursive formula 
Translated
power function 





Sigma
notation of the polynomial 
Coefficients of the source
polynomial in the form of a recursive formula 
According
to mathematical induction we can examine any
ndegree
polynomial function using shown method. 
Therefore,
the polynomial 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} 
we can
write as



while, for k = 0, a_{n}
=
a_{n}, 

and from which, for k =
n,
a_{0}
=
f
(x_{0})
= y_{0}. 

Thus,
expanded form of the above
sum is 
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}) 
where
x_{0}
and y_{0}
are coordinates of translations
of the graph of the source polynomial 
f_{s}(x)
= a_{n}x^{n}
+ a_{n}_{}_{2}x^{n}^{}^{2}
+
.
. . +
a_{2}x^{2}
+ a_{1}x

in
the direction of the xaxis
and the yaxis
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 x_{0}
and y_{0 }
and the coefficients a
of its source function. 

Translated
power function 
If
we set all coefficients a in the above expanded form of the
polynomial to zero, we get 
y
 y_{0}
= a_{n}(x
 x_{0})^{n},
x_{0}
= 
a_{n}_{1}/(
n · a_{n})
and y_{0}
=
f(x_{0}). 
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 is symmetric about the yaxis,
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 is symmetric about the origin, that
is f (
x)
=

f (x). 









College
algebra contents B




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