


The
graphs of the elementary functions 
The
graphs of algebraic and transcendental functions 
The
graphs of algebraic functions 
The
graphs of the
polynomial functions 
The
source or original polynomial function 
Translating
(parallel shifting) of the polynomial function 
Coordinates
of translations and their role in the polynomial expression 
Translated
power function 






The
graphs
of algebraic
and transcendental functions 
Elementary
functions are, Algebraic functions
and Transcendental functions 
Algebraic
functions 
· The
polynomial function
f (x) =
y = a_{n}x^{n}
+ a_{n}_{1}x^{n}^{}^{1}
+ a_{n}_{2}x^{n}^{}^{2}
+
.
. . +
a_{2}x^{2}
+
a_{1}x + a_{0} 
y
=
a_{1}x
+ a_{0}
 Linear
function 
y
=
a_{2}x^{2}
+
a_{1}x + a_{0}_{
}_{
}
Quadratic
function 
y
=
a_{3}x^{3}
+
a_{2}x^{2}
+
a_{1}x + a_{0}_{
}
Cubic
function 
y
=
a_{4}x^{4}
+
a_{3}x^{3}
+
a_{2}x^{2}
+
a_{1}x + a_{0}_{
}
Quartic
function 
y
=
a_{5}x^{5}
+ a_{4}x^{4}
+
a_{3}x^{3}
+
a_{2}x^{2}
+
a_{1}x + a_{0}_{
}
Quintic
function 
   
        
       
        
·
Rational
functions  a ratio of two polynomials 



Reciprocal function 



Translation of the reciprocal function,
called linear rational function. 



The
graphs of the
polynomial functions 
The
graph of a function f
is drawing on the Cartesian plane, plotted with respect to
coordinate axes, that shows functional relationship between
variables. The points (x,
f
(x)) lying on the curve
satisfy this relation. 

The
source
or original polynomial function 
Any
polynomial f (x)
of degree n >
1 in the general form, consisting
of n
+ 1 terms, shown graphically, represents translation of its
source (original) function in the direction of the coordinate
axes. 
The source polynomial function 

f_{s}(x)
= a_{n}x^{n}
+ a_{n}_{}_{2}x^{n}^{}^{2}
+
.
. . +
a_{2}x^{2}
+ a_{1}x



has
n
 1 terms
lacking second and the constant term, since its coefficients, a_{n}_{}_{1
}=
0
and a_{0
}=
0
while
the leading coefficient a_{n},
remains unchanged. 
Therefore,
the source polynomial function passes through the
origin. 
A
coefficient a_{i
}of
the source function is expressed by the coefficients of the general
form. 

Translating
(parallel shifting) of the polynomial function 
Thus,
to obtain the graph of a given polynomial function f
(x)
we translate (parallel shift)
the
graph of its source function in the direction of the xaxis
by x_{0}
and in the direction of the yaxis
by y_{0}. 
Inversely,
to put a given graph of the polynomial function beck to the
origin, we translate it in the opposite direction, by taking the
values of the
coordinates of translations with opposite sign. 

Coordinates of translations
and their role in the polynomial expression 
The
coordinates of translations we calculate using the formulas, 

Hence,
by plugging the coordinates of translations into
the source polynomial function f_{s}(x),
i.e., 

y
 y_{0}
= a_{n}(x
 x_{0})^{n}
+ a_{n}_{}_{2}(x
 x_{0})^{n}^{}^{2}
+
.
. .
+
a_{2}(x
 x_{0})^{2}
+ a_{1}(x
 x_{0}) 


and
by expanding above expression we get the polynomial function in
the general form 
f
(x)
=
y = a_{n}x^{n}
+ a_{n}_{1}x^{n}^{}^{1}
+ a_{n}_{}_{2}x^{n}^{}^{2}
+
.
. . +
a_{2}x^{2}
+
a_{1}x + a_{0}. 
Inversely, by plugging the coordinates of translations into
the given polynomial f
(x)
expressed in the general form,
i.e., 

y
+ y_{0}
= a_{n}(x
+ x_{0})^{n}
+ a_{n}_{}_{1}(x
+ x_{0})^{n}^{}^{1}
+
.
. .
+ a_{1}(x
+ x_{0})
+ a_{0} 


and
after expanding and reducing above expression we get its source polynomial function. 
Note
that in the above expression the signs of
the coordinates of translations are already changed. 

According
to mathematical induction we can examine any
ndegree
polynomial function using shown method. 
Therefore,
the polynomial f
(x)
=
y = a_{n}x^{n}
+ a_{n}_{1}x^{n}^{}^{1}
+ a_{n}_{}_{2}x^{n}^{}^{2}
+
.
. .
+
a_{2}x^{2}
+
a_{1}x + a_{0} 
we can
write as



while, for k
= 0, a_{n}
=
a_{n}, 

and from which, for k
=
n,
a_{0}
=
f
(x_{0})
= y_{0}. 

Thus,
expanded form of the above
sum is 
y
 y_{0}
= a_{n}(x
 x_{0})^{n}
+ a_{n}_{}_{2}(x
 x_{0})^{n}^{}^{2}
+
.
. . +
a_{2}(x
 x_{0})^{2}
+ a_{1}(x
 x_{0}) 
where
x_{0}
and y_{0}
are coordinates of translations
of the graph of the source polynomial 
f_{s}(x)
= a_{n}x^{n}
+ a_{n}_{}_{2}x^{n}^{}^{2}
+
.
. . +
a_{2}x^{2}
+ a_{1}x

in
the direction of the xaxis
and the yaxis
of a Cartesian coordinate system. 

Therefore,
every given polynomial written in the general form can be
transformed into translatable form by calculating the
coordinates of translations x_{0}
and y_{0 }
and the coefficients a
of its source function. 

Translated
power function 
If
we set all coefficients a in the above expanded form of the
polynomial to zero, we get 
y
 y_{0}
= a_{n}(x
 x_{0})^{n},
where x_{0}
= 
a_{n}_{1}/(
n · a_{n})
and y_{0}
=
f
(x_{0}). 
translated
power (or monomial) function, the exponent of which is an odd or
an even positive integer. 
When
the exponent is even, i.e., of the form n
= 2m,
m Î N,
the graph of the source power function is symmetric about the yaxis,
that is f ( x)
=
f (x). 
When
the exponent is odd, i.e., of the form n
= 2m
+ 1,
m Î N,
the graph of the source power function is symmetric about the origin, that
is f ( x)
=
 f
(x). 









Precalculus contents
D 



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