

The
graphs of the
polynomial functions 
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} 






The
graphs of the
polynomial functions 
Translating
or parallel shifting 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 

where, a_{n}
= a_{n},
a_{n

}_{1}
= 0,
a_{0}
= f (x_{0}),
and where f
^{(n }^{}^{
k}^{)}^{
}(x_{0})
denotes (n 
k)th
derivative at x_{0}. 

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 

Thus,
if 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}), 
where 

or
f
' (x_{0})
= 1!
a_{1}_{ }_{
}and f
'' (x_{0})
= 2!
a_{2} 

then,
a_{1}
and
a_{2}
define vertical translations of
the successive derivatives, as shows below figure.












Calculus
contents A 



Copyright
© 2004  2020, Nabla Ltd. All rights reserved. 