
Polynomial and/or Polynomial
Functions and Equations 
Quartic
function 
Transformation of the quartic
polynomial from the general to source form and vice versa 
The coordinates of translations formulas 
The values of the
coefficients, a_{2}
and a_{1}
of the source quartic function y
= a_{4}x^{4} + a_{2}x^{2}
+ a_{1}x 
The
basic classification criteria diagram 





Quartic
function
y = a_{4}x^{4}
+ a_{3}x^{3} + a_{2}x^{2}
+ a_{1}x + a_{0} 
Transformation of the quartic
polynomial function from the general to source form and vice versa 
1)
Calculate the
coordinates of translations by plugging
n
= 4
into 

2)
To
get the source quartic function we should plug the coordinates
of translations (with changed signs) 
into the general form
of the quartic,
i.e., 
y
+ y_{0}
= a_{4}(x
+ x_{0})^{4}
+
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
quartic function 

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

y_{0}
=
a_{4}(x

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

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

x_{0}), 

after
expanding and reducing we obtain 
y
= a_{4}x^{4}
+ a_{3}x^{3}
+
a_{2}x^{2}
+
a_{1}x + a_{0}
the
quartic function
in the general form. 

The
coordinates of translations formulas and the values of the
coefficients of the source quartic function 
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. 



The
basic classification criteria diagram for the quartic function 
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. 

Thus,
there are
ten types (different shapes of the graphs) of quartic polynomial
functions. 









Precalculus contents
E 



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