
Polynomial and/or Polynomial
Functions and Equations 
Cubic functions 
Transformation of the cubic
polynomial from the general to source form and vice versa 
Coordinates
of the point of inflection coincide with the coordinates of
translations 
The
source cubic functions are odd functions 
There
are three types of the cubic functions  the classification
criteria diagram 
The graphs
of the source cubic functions 






Cubic
function
y
=
a_{3}x^{3}
+
a_{2}x^{2}
+
a_{1}x + a_{0} 
Transformation of the
cubic polynomial from the general to source form and vice versa 
Applying
the same method we can examine the third degree polynomial
called cubic function. 
1)
Calculate the
coordinates of translations 
substitute
n
= 3
in 



and 


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
+ y_{0}
= a_{3}(x
+ x_{0})^{3}
+
a_{2}(x
+ x_{0})^{2}
+
a_{1}(x
+ x_{0})
+
a_{0}, 

after
expanding and reducing obtained is 


the source
cubic function. 

3)
Inversely, by plugging the coordinates of translations into the source
cubic 
y

y_{0}
= a_{3}(x

x_{0})^{3}
+
a_{1}(x

x_{0}), 

after
expanding and reducing we obtain 
y
=
a_{3}x^{3}
+ a_{2}x^{2}
+ a_{1}x
+ a_{0}
the cubic function
in the general form. 
Thus,
y
=
a_{3}x^{3}
+ a_{2}x^{2}
+ a_{1}x
+ a_{0} _{
}or_{ }
y

y_{0}
= a_{3}(x

x_{0})^{3}
+
a_{1}(x

x_{0}), 

by
setting x_{0}
=
0 and y_{0}
= 0 we get
the source cubic function
y
= a_{3}x^{3}
+
a_{1}x
where a_{1}=
tana_{t}_{
}. 



Coordinates
of the point of inflection coincide with the coordinates of
translations, i.e., I
(x_{0},
y_{0}).


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)
= a_{3}x^{3}
+
a_{1}x
then f
(x)
= [a_{3}(x)^{3}
+
a_{1}(x)]
= a_{3}x^{3}
+
a_{1}x
=
f (x). 
That
is, change of the sign of the independent variable of a function
reflects the graph of the function about the yaxis,
while change of the sign of a function reflects the graph of the
function about the xaxis. 
The
graphs of
the translated cubic functions are symmetric about its
point of inflection. 

There are three types
(shapes) of cubic
functions whose graphs of the source functions are shown in the figure below: 
type
1 
y
=
a_{3}x^{3}
+ a_{2}x^{2}
+ a_{1}x
+ a_{0
}or_{ }_{ }y

y_{0}
= a_{3}(x

x_{0})^{3},^{ }

a_{2}^{2}
+ 3a_{3}a_{1}
= 0 or a_{1}
= 0. 

therefore,
its source function y
=
a_{3}x^{3},^{
}and the tangent line through the point of
inflection is horizontal. 
type
2/1 
y
=
a_{3}x^{3}
+ a_{2}x^{2}
+ a_{1}x
+ a_{0
}or_{ }
y

y_{0}
= a_{3}(x

x_{0})^{3}
+
a_{1}(x

x_{0}),
where a_{3}a_{1}>
0 

whose
slope of the tangent line through the point of inflection is
positive and equals a_{1}. 
type
2/2 
y
=
a_{3}x^{3}
+ a_{2}x^{2}
+ a_{1}x
+ a_{0
}or_{ }
y

y_{0}
= a_{3}(x

x_{0})^{3}
+
a_{1}(x

x_{0}),
where a_{3}a_{1}<
0 

whose
slope of the tangent line through the point of inflection is
negative and is equal a_{1}. 
The
graph of its source function has three zeros or roots at 


and two turning
points at 



The
graphs
of the source cubic functions  the classification criteria
diagram 











Functions
contents C 



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