Editing Cubic function (section)

==Classification==
[[File:Cubic function (different c).svg|thumb|Cubic functions of the form <math>y=x^3+cx.</math><br/>The graph of any cubic function is [[similarity (geometry)|similar]] to such a curve.]]

The [[graph of a function|graph]] of a cubic function is a [[cubic curve]], though many cubic curves are not graphs of functions.

Although cubic functions depend on four parameters, their graph can have only very few shapes. In fact, the graph of a cubic function is always [[similarity (geometry)|similar]] to the graph of a function of the form
:<math>y=x^3+px.</math> 
This similarity can be built as the composition of [[translation]]s parallel to the coordinates axes, a [[homothecy]] ([[uniform scaling]]), and, possibly, a [[reflection (mathematics)|reflection]] ([[mirror image]]) with respect to the {{mvar|y}}-axis. A further [[uniform scaling|non-uniform scaling]] can transform the graph into the graph of one among the three cubic functions
:<math>\begin{align}
y&=x^3+x\\
y&=x^3\\
y&=x^3-x.
\end{align}
</math>

This means that there are only three graphs of cubic functions [[up to]] an [[affine transformation]].

The above [[geometric transformation]]s can be built in the following way, when starting from a general cubic function 
<math>y=ax^3+bx^2+cx+d.</math>

Firstly, if {{math|''a'' < 0}}, the [[change of variable]] {{math|''x'' → −''x''}} allows supposing {{math|''a'' > 0}}. After this change of variable, the new graph is the mirror image of the previous one, with respect of the {{mvar|y}}-axis.

Then, the change of variable {{math|1=''x'' = ''x''{{sub|1}} − {{sfrac|''b''|3''a''}}}} provides a function of the form
:<math>y=ax_1^3+px_1+q.</math>
This corresponds to a translation parallel to the {{mvar|x}}-axis.

The change of variable {{math|1=''y'' = ''y''{{sub|1}} + ''q''}} corresponds to a translation with respect to the {{mvar|y}}-axis, and gives a function of the form
:<math>y_1=ax_1^3+px_1.</math>

The change of variable <math>\textstyle x_1=\frac {x_2}\sqrt a, y_1=\frac {y_2}\sqrt a</math> corresponds to a uniform scaling, and give, after multiplication by <math>\sqrt a,</math> a function of the form 
:<math>y_2=x_2^3+px_2,</math>
which is the simplest form that can be obtained by a similarity.

Then, if {{math|''p'' ≠ 0}}, the non-uniform scaling <math>\textstyle x_2=x_3\sqrt{|p|},\quad y_2=y_3\sqrt{|p|^3}</math> gives, after division by <math>\textstyle \sqrt{|p|^3},</math>
:<math>y_3 =x_3^3 + x_3\sgn(p),</math>
where <math>\sgn(p)</math> has the value 1 or −1, depending on the sign of {{mvar|p}}. If one defines <math>\sgn(0)=0,</math> the latter form of the function applies to all cases (with <math>x_2 = x_3</math> and <math>y_2 = y_3</math>).