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The
graphs of the
polynomial functions |
The
graph of a function ƒ
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. |
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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 |
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fs(x)
= anxn
+ an-2xn-2
+
.
. . +
a2x2
+ a1x
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has
n
- 1 terms
lacking second and the constant term, since its coefficients, an-1
=
0
and a0
=
0
while
the leading coefficient an,
remains unchanged. |
Therefore,
the source polynomial function passes through the
origin. |
A
coefficient ai
of
the source function is expressed by the coefficients of the general
form. |
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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 x-axis
by x0
and in the direction of the y-axis
by y0. |
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. |
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Coordinates of translations
and their role in the polynomial expression |
The
coordinates of translations we calculate using the formulas, |
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Hence,
by plugging the coordinates of translations into
the source polynomial function fs(x),
i.e., |
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y
- y0
= an(x
- x0)n
+ an-2(x
- x0)n-2
+
.
. .
+
a2(x
- x0)2
+ a1(x
- x0) |
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and
by expanding above expression we get the polynomial function in
the general form |
f(x) =
y = anxn
+ an-1xn-1
+ an-2xn-2
+
.
. . +
a2x2
+
a1x + a0. |
Inversely, by plugging the coordinates of translations into
the given polynomial f
(x)
expressed in the general form,
i.e., |
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y
+ y0
= an(x
+ x0)n
+ an-1(x
+ x0)n-1
+
.
. .
+ a1(x
+ x0)
+ a0 |
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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. |
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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
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while, for k = 0, an
=
an, |
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and from an
- k for
k =
n,
a0
=
f(x0)
= y0. |
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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
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in
the direction of the x-axis
and the y-axis
of a Cartesian coordinate system. |
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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. |
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Coefficients
of the source polynomial function are related to its derivative
at x0 |
The
coefficients of the source polynomial are related to
corresponding value of its derivative at x0
like the coefficients
of the Taylor polynomial in Taylor's or Maclaurin's formula,
thus |
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Such
for example, the coefficient a1
of the source cubic of
f(x) =
a3x3
+
a2x2
+
a1x + a0 |
since
f ' (x)
=
3a3x2
+ 2a2x
+
a1
and x0
=
-
a2/(3a3)
then |
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