Editing Cubic function (section)

==Collinearities==
[[File:Cubica colinear.png|thumb|The points {{math|''P''<sub>1</sub>}}, {{math|''P''<sub>2</sub>}}, and {{math|''P''<sub>3</sub>}} (in blue) are collinear and belong to the graph of {{math|''x''<sup>3</sup> + {{sfrac|3|2}}''x''<sup>2</sup> − {{sfrac|5|2}}''x'' + {{sfrac|5|4}}}}. The points {{math|''T''<sub>1</sub>}}, {{math|''T''<sub>2</sub>}}, and {{math|''T''<sub>3</sub>}} (in red) are the intersections of the (dotted) tangent lines to the graph at these points with the graph itself. They are collinear too.]]

The tangent lines to the graph of a cubic function at three [[collinear points]] intercept the cubic again at collinear points.<ref>{{Citation|last = Whitworth|first = William Allen|author-link = William Allen Whitworth|title = Trilinear Coordinates and Other Methods of Modern Analytical Geometry of Two Dimensions|publisher = Deighton, Bell, and Co.|year = 1866|place = Cambridge|page = 425|url = https://archive.org/details/trilinearcoordin00whit|chapter = Equations of the third degree|access-date = June 17, 2016}}</ref> This can be seen as follows.

As this property is invariant under a [[rigid motion]], one may suppose that the function has the form 
:<math>f(x)=x^3+px.</math>

If {{mvar|α}} is a real number, then the tangent to the graph of {{mvar|f}} at the point {{math|(''α'', ''f''(''α''))}} is the line
:{{math|{(''x'', ''f''(''α'') + (''x'' − ''α'')''f''&thinsp;′(''α'')) : ''x'' ∈ '''R'''}<nowiki/>}}.
So, the intersection point between this line and the graph of {{mvar|f}} can be obtained solving the equation {{math|''f''(''x'') {{=}} ''f''(''α'') + (''x'' − ''α'')''f''&thinsp;′(''α'')}}, that is
:<math>x^3+px=\alpha^3+p\alpha+ (x-\alpha)(3\alpha^2+p),</math>
which can be rewritten
:<math>x^3 - 3\alpha^2 x +2\alpha^3=0,</math>
and factorized as
:<math>(x-\alpha)^2(x+2\alpha)=0.</math>
So, the tangent intercepts the cubic at
:<math>(-2\alpha, -8\alpha^3-2p\alpha)=(-2\alpha, -8f(\alpha)+6p\alpha).</math>

So, the function that maps a point {{math|(''x'', ''y'')}} of the graph to the other point where the tangent intercepts the graph is 
:<math>(x,y)\mapsto (-2x, -8y+6px).</math>
This is an [[affine transformation]] that transforms collinear points into collinear points. This proves the claimed result.