The graphs of the elementary 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
      The graphs of the cubic function
The graphs of the elementary functions
Elementary functions are,   Algebraic functions and Transcendental functions
Algebraic functions
  The polynomial function   f (x) =  yanxn + an-1xn-1 + an-2xn-2 + . . . + a2x2 + a1x + a0
                                                    y a1x + a0                                                   - Linear function 
                                                    y = a2x2 + a1x + a0                                                      - Quadratic function 
                                                    y = a3x3 + a2x2 + a1x + a0                                       - Cubic function
                                                    y = a4x4 + a3x3 + a2x2 + a1x + a0                        - Quartic function
                                                    y = a5x5 + a4x4 + a3x3 + a2x2 + a1x + a0         - 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 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
fs(x) = anxn + an-2xn-2 + . . . + a2x2 + a1x
has n - 1 terms lacking second and the constant term, since its coefficients, an-1 = 0 and a0 = 0 while the leading coefficient an, remains unchanged.
Therefore, the source polynomial function passes through the origin.
A coefficient ai 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 x-axis by x0 and in the direction of the y-axis by y0.
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 fs(x), i.e.,
y - y0 = an(x - x0)n + an-2(x - x0)n-2 + . . .  + a2(x - x0)2 + a1(x - x0)
and by expanding above expression we get the polynomial function in the general form 
f(x) =  yanxn + an-1xn-1 + an-2xn-2 + . . . + a2x2 + a1x + a0.
Inversely, by plugging the coordinates of translations into the given polynomial f(x) expressed in the general form, i.e.,
y + y0 = an(x + x0)n + an-1(x + x0)n-1 + . . .  + a1(x + x0) + a0
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.
Cubic function    y = a3x3 + a2x2 + a1x + a0
Applying the same method we can examine the third degree polynomial called cubic function.
1)  Calculate the coordinates of translations
substitute n = 3 in    
and
2)  To get the source cubic function we should plug the coordinates of translations (with changed signs)
     into the general form of the cubic, i.e.,
y + y0 = a3(x + x0)3 + a2(x + x0)2 + a1(x + x0) + a0,
after expanding and reducing obtained is
  the source cubic function.
3)  Inversely, by plugging the coordinates of translations into the source cubic
                                     y - y0 = a3(x - x0)3 + a1(x - x0),
   
after expanding and reducing we obtain
                                     y = a3x3 + a2x2 + a1x + a0   the cubic function in the general form.
Thus,         y = a3x3 + a2x2 + a1x + a0     or      y - y0 = a3(x - x0)3 + a1(x - x0),
by setting  x0 = 0  and  y0 = 0 we get the source cubic function  y = a3x3 + a1x  where  a1= tanat .
Coordinates of the point of inflection coincide with the coordinates of translations, i.e.,  I (x0, y0). 
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) = a3x3 + a1x  then  - f (-x) = - [a3(-x)3 + a1(-x)]  = a3x3 + a1x f (x).
That is, change of the sign of the independent variable of a function reflects the graph of the function about the y-axis, while change of the sign of a function reflects the graph of the function about the x-axis.
The graphs of the translated cubic functions are symmetric about its point of inflection.
There are three types (shapes) of cubic functions whose graphs of the source functions are shown in the figure below:
type 1 y = a3x3 + a2x2 + a1x + a0    or    y - y0 = a3(x - x0)3,    - (a2)2 + 3a3a1 = 0 or a1 = 0.
therefore, its source function  y = a3x3,  and the tangent line through the point of inflection is horizontal.
type 2/1 y = a3x3 + a2x2 + a1x + a0     or      y - y0 = a3(x - x0)3 + a1(x - x0), where  a3a1> 0
whose slope of the tangent line through the point of inflection is positive and equals a1.
type 2/2 y = a3x3 + a2x2 + a1x + a0     or      y - y0 = a3(x - x0)3 + a1(x - x0), where  a3a1< 0
whose slope of the tangent line through the point of inflection is negative and is equal a1
The graph of its source function has three zeros or roots at  
and two turning points at
Graphs of cubic functions
Translated cubic functions
type 1 y = a3x3 + a2x2 + a1x + a0     or      y - y0 = a3(x - x0)3   where,
The root   The point of inflection  I(x0, y0).
type 2/1 y = a3x3 + a2x2 + a1x + a0     or    y - y0 = a3(x - x0)3 + a1(x - x0),    a3  a1 > 0,
  I(x0, y0).
 
type 2/2 y = a3x3 + a2x2 + a1x + a0     or    y - y0 = a3(x - x0)3 + a1(x - x0),    a3  a1 < 0,
If  | y0 | > | yT |
if  | y0 | < | yT |
The turning points The point of inflection  I(x0, y0).
Intermediate algebra contents
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