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| The
cubic
function
y
=
a3x3
+
a2x2
+
a1x + a0 |
| Thus,
y
=
a3x3
+ a2x2
+ a1x
+ a0
or
y
-
y0
= a3(x
-
x0)3
+
a1(x
-
x0), |
 |
| by
setting x0
=
0 and y0
= 0 we get
the source cubic function
y
= a3x3
+
a1x
where a1=
tanat
. |
|
|
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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. |
|
| 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 |
|
| 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 |
|
| 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 |
 |
|
| and two turning
points at |
 |
|
| The
classification diagram of the cubic function |
| The
graphs
of the source cubic functions |
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