Cubic functions
There are three types of the cubic functions - the classification criteria diagram
The graphs of the source cubic functions
Translated cubic functions
Translated cubic function the type 2/2
Graphing a cubic function type 2/2 example
The graphs of the source cubic functions - the classification criteria diagram
Translated cubic function type 2/2
 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).
Graphing a cubic function type 2/2 example
Example:  Given is cubic function y = (-1/3)x3 - 4x2 - 12x - 25/3, find its source or original function and calculate the coordinates of translations, the zero points, the turning points and the point of inflection.
Draw graphs of the source and the given cubic.
Solution:  1)  Calculate the coordinates of translations
y0 f (x0)   =>   y0 = f (- 4) = (-1/3) · (- 4)3 - 4 · (- 4)2 - 12 · (- 4) - 25/3  = - 3,     y0 = - 3.
Therefore, the point of inflection I(- 4, - 3).
2)  To get the source cubic function, plug the coordinates of translations into the general form of the cubic,
y + y0 =  a3(x + x0)3 + a2(x + x0)2 + a1(x + x0) + a0   thus,
y - 3 = (-1/3) · (x - 4)3 - 4(x - 4)2 - 12(x - 4) - 25/3,    y = (-1/3)x3 + 4the source function.
As,  a3 < 0  and  a3a1 < 0  given cubic is the type 2/2 whose graph is reflected around the x-axis.
 the function has three real zeroes,
By plugging the coefficients of the source function and the coordinates of translations x0 and y0 we calculate the turning points,
Functions contents C