
The
quartic
function 
The
source
or original quartic
function 
The
quartic
functions classification diagram 
The
graphs of the quartic functions
types 4/1 . . . 4/6 





Quartic
function y
=
a_{4}x^{4}
+ a_{3}x^{3}
+
a_{2}x^{2}
+
a_{1}x + a_{0} 
Thus,
y
= a_{4}x^{4}
+ a_{3}x^{3}
+
a_{2}x^{2}
+
a_{1}x + a_{0}_{
}or_{ }y

y_{0}
=
a_{4}(x

x_{0})^{4}
+
a_{2}(x

x_{0})^{2}
+
a_{1}(x

x_{0}), 


by
setting x_{0}
=
0 and y_{0}
= 0 we get
the source quartic _{ }y
=
a_{4}x^{4}
+
a_{2}x^{2}
+
a_{1}x. 



By
setting the coefficients a_{2}
and a_{1}
of the source quartic to zero interchangeably, obtained is the
basic classification shown in the diagram. 

There are
altogether ten types or shapes of graphs of quartic functions. 

The
graphs of the quartic functions
types 4/1 . . . 4/6 
Remaining
six types of quartic polynomial satisfy the criteria shown in the
diagram below. 


The
roots of the source quartic
y
=
a_{4}x^{4}
+
a_{2}x^{2}
+
a_{1}x
Types,
4/1, 4/2, 4/3 and 4/4 

the
roots of the Types,
4/5 and 4/6 


The
abscissa of the turning point of the Types,
4/1, 4/2 and 4/3 

the
abscissas of the turning points of the Types, 4/4,
4/5 and 4/6 


The
abscissas of the points of inflection of the
source quartic of Types 4/2
to 4/6, 


The
roots of the translated quartic Type 4
we get by adding x_{0}
to the solutions of the equation 
a_{4}x^{4}
+
a_{2}x^{2}
+
a_{1}x
+
y_{0}
= 0. 












Calculus
contents A 



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