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| Polynomial and/or Polynomial
Functions and Equations |
Quartic
function |
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Transformation of the quartic
polynomial from the general to source form and vice versa |
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The coordinates of translations formulas |
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The values of the
coefficients, a2
and a1
of the source quartic function y
= a4x4 + a2x2
+ a1x |
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The
basic classification criteria diagram for quartic function |
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The graphs of quartic functions
and their characteristic points |
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The quartic
type 1, y
-
y0 = a4(x -
x0)4,
a2
= 0 and a1
= 0 |
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The quartic type 2,
y
-
y0 = a4(x -
x0)4 + a1(x
-
x0),
a2
= 0 |
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The quartic type 3/1,
y
-
y0 = a4(x -
x0)4 + a2(x
-
x0)2,
a1
= 0 and
a4a2
> 0 |
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The quartic type 3/2,
y
-
y0 = a4(x -
x0)4 + a2(x
-
x0)2,
a1
= 0 and
a4a2
< 0 |
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| Quartic
function
y = a4x4
+ a3x3 + a2x2
+ a1x + a0 |
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Transformation of the quartic
polynomial from the general to source form and vice versa |
| 1)
Calculate the
coordinates of translations by plugging
n
= 4
into |
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| 2)
To
get the source quartic function we should plug the coordinates
of translations (with changed signs)
into the general form
of the quartic,
i.e., |
| y
+ y0
= a4(x
+ x0)4
+
a3(x
+ x0)3
+
a2(x
+ x0)2
+
a1(x
+ x0) +
a0, |
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| after
expanding and reducing obtained is the source
quartic function |
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| 3)
Inversely, by plugging the coordinates of translations into the source
quartic |
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y
-
y0
=
a4(x
-
x0)4
+
a2(x
-
x0)2
+
a1(x
-
x0), |
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after
expanding and reducing we obtain |
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y
= a4x4
+ a3x3
+
a2x2
+
a1x + a0
the
quartic function
in the general form. |
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| The
coordinates of translations formulas and the values of the
coefficients of the source quartic function |
| Thus,
y
= a4x4
+ a3x3
+
a2x2
+
a1x + a0
or y
-
y0
=
a4(x
-
x0)4
+
a2(x
-
x0)2
+
a1(x
-
x0), |
 |
 |
| by
setting x0
=
0 and y0
= 0 we get
the source quartic y
=
a4x4
+
a2x2
+
a1x. |
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| The
basic classification criteria diagram of quartic function |
| By
setting the coefficients a2
and a1
of the source quartic to zero, interchangeably, obtained is the
basic classification for quartic function shown in the diagram. |
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| The
graphs of quartic functions and their characteristic points |
| There are
ten types (different shapes of the graphs) of quartic functions |
| type
1 |
y
= a4x4
+ a3x3
+
a2x2
+
a1x + a0
or y
-
y0
=
a4(x
-
x0)4,
a2
= 0 and a1
=
0. |
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| The
zeroes or roots: |
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| type
2 |
y
= a4x4
+ a3x3
+
a2x2
+
a1x + a0
or y
-
y0
=
a4(x
-
x0)4
+ a1(x
-
x0),
a2
= 0. |
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| The
zeroes of the source function: |
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| The
zeroes of the translated function we get
by adding x0
to the solution of the
equation a4x4
+ a1x +
y0
= 0. |
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| type
3 |
y
= a4x4
+ a3x3
+
a2x2
+
a1x + a0
or y
-
y0
=
a4(x
-
x0)4
+ a2(x
-
x0)2,
a1
= 0. |
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T
(x0,
y0). |
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| Functions
contents C |
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