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| 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
n-degree
polynomial function using shown method. |
| Therefore,
the polynomial f
(x) =
y = anxn
+ an-1xn-1
+ an-2xn-2
+
.
. .
+
a2x2
+
a1x + a0 |
| we can
write as
|
 |
 |
|
while, for k = 0, an
=
an, |
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and from which, for k =
n,
a0
=
f
(x0)
= y0. |
|
| Thus,
expanded form of the above
sum is |
| y
- y0
= an(x
- x0)n
+ an-2(x
- x0)n-2
+
.
. . +
a2(x
- x0)2
+ a1(x
- x0) |
| where
x0
and y0
are coordinates of translations
of the graph of the source polynomial |
| fs(x)
= anxn
+ an-2xn-2
+
.
. . +
a2x2
+ a1x
|
| in
the direction of the x-axis
and the y-axis
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 x0
and y0
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
- y0
= an(x
- x0)n,
x0
= -
an-1/(
n · an)
and y0
=
f(x0). |
| 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 y-axis,
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). |
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