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Cubic
function
y
=
a3x3
+
a2x2
+
a1x + a0 |
Applying
the same method we can examine the third degree polynomial
called cubic function. |
1)
Calculate the
coordinates of translations |
substitute
n
= 3
in |
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and |
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2)
To
get the source cubic function we should plug the coordinates
of translations (with changed signs) |
into the general form
of the cubic,
i.e., |
y
+ y0
= a3(x
+ x0)3
+
a2(x
+ x0)2
+
a1(x
+ x0)
+
a0, |
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after
expanding and reducing obtained is |
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the source
cubic function. |
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3)
Inversely, by plugging the coordinates of translations into the source
cubic |
y
-
y0
= a3(x
-
x0)3
+
a1(x
-
x0), |
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after
expanding and reducing we obtain |
y
=
a3x3
+ a2x2
+ a1x
+ a0
the cubic function
in the general form. |
Thus,
y
=
a3x3
+ a2x2
+ a1x
+ a0
or
y
-
y0
= a3(x
-
x0)3
+
a1(x
-
x0), |
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by
setting x0
=
0 and y0
= 0 we get
the source cubic function
y
= a3x3
+
a1x
where a1=
tanat
. |
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Coordinates
of the point of inflection coincide with the coordinates of
translations, i.e., I
(x0,
y0).
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The source cubic functions are
odd functions. |
Graphs of odd functions are
symmetric about the origin that is, such functions change
the sign but not absolute value when the sign of the independent variable is
changed, so that f
(x)
=
-
f (-x). |
Therefore,
since f (x)
= a3x3
+
a1x
then -
f
(-x)
= -
[a3(-x)3
+
a1(-
x)]
= a3x3
+
a1x
=
f (x). |
That
is, change of the sign of the independent variable of a function
reflects the graph of the function about the y-axis,
while change of the sign of a function reflects the graph of the
function about the x-axis. |
The
graphs of
the translated cubic functions are symmetric about its
point of inflection. |
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There are three types
(shapes) of cubic
functions whose graphs of the source functions are shown in the figure below: |
type
1 |
y
=
a3x3
+ a2x2
+ a1x
+ a0
or y
-
y0
= a3(x
-
x0)3,
-
(a2)2
+ 3a3a1
= 0 or a1
= 0. |
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therefore,
its source function y
=
a3x3,
and the tangent line through the point of
inflection is horizontal. |
type
2/1 |
y
=
a3x3
+ a2x2
+ a1x
+ a0
or
y
-
y0
= a3(x
-
x0)3
+
a1(x
-
x0),
where a3a1>
0 |
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whose
slope of the tangent line through the point of inflection is
positive and equals a1. |
type
2/2 |
y
=
a3x3
+ a2x2
+ a1x
+ a0
or
y
-
y0
= a3(x
-
x0)3
+
a1(x
-
x0),
where a3a1<
0 |
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whose
slope of the tangent line through the point of inflection is
negative and is equal a1. |
The
graph of its source function has three zeros or roots at |
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and two turning
points at |
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Graphs
of cubic functions |
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Translated
cubic functions |
type
1 |
y
=
a3x3
+ a2x2
+ a1x
+ a0
or
y
-
y0
= a3(x
-
x0)3-
x0)3
where, |
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The
root |
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The point of inflection I
(x0,
y0). |
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type
2/1 |
y
=
a3x3
+ a2x2
+ a1x
+ a0 or
y
-
y0
= a3(x
-
x0)3
+ a1(x
-
x0),
a3
· a1
>
0, |
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I
(x0,
y0). |
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type
2/2 |
y
=
a3x3
+ a2x2
+ a1x
+ a0 or
y
-
y0
= a3(x
-
x0)3
+ a1(x
-
x0),
a3
· a1
<
0, |
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If
| y0
| > |
yT
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if
| y0
| < |
yT
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The turning points |
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The
point of
inflection
I
(x0,
y0). |
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