

There
are three types of the cubic functions  the classification
criteria diagram 
The graphs
of the source cubic functions 





The
graphs
of the source cubic functions 
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 









Precalculus contents
E 



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