

The
cubic
function 
General or translated form of cubic function 
Graphs
of the source cubic functions 





The
cubic
function 
General
or translated form of cubic function 
The
general cubic function 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}_{
}. 


There are three types
(shapes of graphs) 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 


Graphs
of the source cubic functions 


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

The source cubic functions are
odd functions. 
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. 










Precalculus contents
D 



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