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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 |
|
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 |
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| Quartic
function
y = a4x4
+ a3x3 + a2x2
+ a1x + a0 |
|
Transformation of the quartic
polynomial function from the general to source form and vice versa |
| 1)
Calculate the
coordinates of translations by plugging
n
= 4
into |
|
| 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, |
 |
| 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 |
|
y
-
y0
=
a4(x
-
x0)4
+
a2(x
-
x0)2
+
a1(x
-
x0), |
|
|
after
expanding and reducing we obtain |
|
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 for the quartic function |
| By
setting the coefficients a2
and a1
of the source quartic to zero, interchangeably, obtained is the
basic classification shown in the diagram. |
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| Thus,
there are
ten types (different shapes of the graphs) of quartic polynomial
functions. |
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| Pre-calculus contents
E |
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