Editing Cubic function

{{short description|Polynomial function  of degree 3}}
{{distinguish|Cubic equation}}
{{one source|date=September 2019}}
[[Image:Polynomialdeg3.svg|thumb|right|210px|Graph of a cubic function with 3 [[real number|real]] [[root of a function|roots]] (where the curve crosses the horizontal axis—where {{math|''y'' {{=}} 0}}). The case shown has two [[critical point (mathematics)|critical points]]. Here the function is {{math|''f''(''x'') {{=}} (''x''<sup>3</sup> + 3''x''<sup>2</sup> − 6''x'' − 8)/4}}.]]

In [[mathematics]], a '''cubic function''' is a [[function (mathematics)|function]] of the form <math>f(x)=ax^3+bx^2+cx+d,</math> that is, a [[polynomial function]] of degree three. In many texts, the ''coefficients'' {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, and {{mvar|d}} are supposed to be [[real numbers]], and the function is considered as a [[real function]] that maps real numbers to real numbers or as a complex function that maps [[complex number]]s to complex numbers. In other cases, the coefficients may be complex numbers, and the function is a complex function that has the set of the complex numbers as its [[codomain]], even when the [[domain of a function|domain]] is restricted to the real numbers.

Setting {{math|''f''(''x'') {{=}} 0}} produces a [[cubic equation]] of the form
:<math>ax^3+bx^2+cx+d=0,</math>
whose solutions are called [[root of a function|roots]] of the function. The [[derivative]] of a cubic function is a [[quadratic function]].

A cubic function with real coefficients has either one or three real roots ([[Multiplicity (mathematics)|which may not be distinct]]);<ref>{{Cite book|last1=Bostock|first1=Linda|url=https://books.google.com/books?id=e2C3tFnAR-wC&q=A+cubic+function+has+either+one+or+three+real+roots&pg=PA462|title=Pure Mathematics 2|last2=Chandler|first2=Suzanne|last3=Chandler|first3=F. S.|date=1979|publisher=Nelson Thornes|isbn=978-0-85950-097-5|pages=462|language=en|quote=Thus a cubic equation has either three real roots... or one real root...}}</ref> all odd-degree polynomials with real coefficients have at least one real root.

The [[graph of a function|graph]] of a cubic function always has a single [[inflection point]]. It may have two [[critical point (mathematics)|critical points]], a local minimum and a local maximum. Otherwise, a cubic function is [[monotonic]]. The graph of a cubic function is symmetric with respect to its inflection point; that is, it is invariant under a rotation of a half turn around this point. [[Up to]] an [[affine transformation]], there are only three possible graphs for cubic functions.

Cubic functions are fundamental for [[cubic interpolation]].

==History==
{{main|Cubic equation#History}}

==Critical and inflection points==
{{Cubic_graph_special_points.svg}}
The [[critical point (mathematics)|critical points]] of a cubic function are its [[stationary point]]s, that is the points where the slope of the function is zero.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Stationary Point|url=https://mathworld.wolfram.com/StationaryPoint.html|access-date=2020-07-27|website=mathworld.wolfram.com|language=en}}</ref> Thus the critical points of a cubic function {{math|''f''}} defined by 
:{{math|''f''(''x'') {{=}} ''ax''<sup>3</sup> + ''bx''<sup>2</sup> + ''cx'' + ''d''}}, 
occur at values of {{math|''x''}} such that the [[derivative]]
:<math> 3ax^2 + 2bx + c = 0</math>
of the cubic function is zero.

The solutions of this equation are the {{mvar|x}}-values of the critical points and are given, using the [[quadratic formula]], by <!-- Do not change 3ac into 4ac: here the of the cubic equation coefficients of the quadratic polynomial are not the same as the coefficients generally used for expressing the quadratic formula -->
:<math>x_\text{critical}=\frac{-b \pm \sqrt {b^2-3ac}}{3a}.</math>

The sign of the expression {{math|Δ<sub>0</sub> {{=}} }}{{math|''b''{{sup|2}} − 3''ac''}} inside the square root determines the number of critical points. If it is positive, then there are two critical points, one is a local maximum, and the other is a local minimum. If {{math|''b''{{sup|2}} − 3''ac'' {{=}} 0}}, then there is only one critical point, which is an [[inflection point]]. If {{math|''b''{{sup|2}} − 3''ac'' < 0}}, then there are no (real) critical points. In the two latter cases, that is, if {{math|''b''{{sup|2}} − 3''ac''}} is nonpositive, the cubic function is strictly [[monotonic]]. See the figure for an example of the case {{math|Δ<sub>0</sub> > 0}}.

The inflection point of a function is where that function changes [[Second derivative#Concavity|concavity]].<ref>{{Cite book|last1=Hughes-Hallett|first1=Deborah|url=https://books.google.com/books?id=8CeVDwAAQBAJ&q=inflection+point+of+a+function+is+where+that+function+changes+concavity&pg=PA181|title=Applied Calculus|last2=Lock|first2=Patti Frazer|last3=Gleason|first3=Andrew M.|last4=Flath|first4=Daniel E.|last5=Gordon|first5=Sheldon P.|last6=Lomen|first6=David O.|last7=Lovelock|first7=David|last8=McCallum|first8=William G.|last9=Osgood|first9=Brad G.|date=2017-12-11|publisher=John Wiley & Sons|isbn=978-1-119-27556-5|pages=181|language=en|quote=A point at which the graph of the function f changes concavity is called an inflection point of f}}</ref> An inflection point occurs when the [[second derivative]] <math>f''(x) = 6ax + 2b, </math> is zero, and the third derivative is nonzero. Thus a cubic function has always a single inflection point, which occurs at
:<math>x_\text{inflection} = -\frac{b}{3a}.</math>

==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>).

==Symmetry==
For a cubic function of the form <math>y=x^3+px,</math> the inflection point is thus the origin. As such a function is an [[odd function]], its graph is symmetric with respect to the inflection point, and invariant under a rotation of a half turn around the inflection point. As these properties are invariant by [[similarity (geometry)|similarity]], the following is true for all cubic functions.

''The graph of a cubic function is symmetric with respect to its inflection point, and is invariant under a rotation of a half turn around the inflection point.''

==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.

==Cubic interpolation==
{{main|Spline interpolation}}
Given the values of a function and its derivative at two points, there is exactly one cubic function that has the same four values, which is called a [[cubic Hermite spline]].

There are two standard ways for using this fact. Firstly, if one knows, for example by physical measurement, the values of a function and its derivative at some sampling points, one can ''interpolate'' the function with a [[continuously differentiable function]], which is a [[piecewise]] cubic function.

If the value of a function is known at several points, [[cubic interpolation]] consists in approximating the function by a [[continuously differentiable function]], which is [[piecewise]] cubic. For having a uniquely defined interpolation, two more constraints must be added, such as the values of the derivatives at the endpoints, or a zero [[curvature]] at the endpoints.

==References==
{{Reflist}}

==External links==
{{commons category|Cubic functions}}
* {{springer|title=Cardano formula|id=p/c020350|ref=none}}
*[http://www-history.mcs.st-and.ac.uk/history/HistTopics/Quadratic_etc_equations.html History of quadratic, cubic and quartic equations] on [[MacTutor archive]].

{{Polynomials}}

{{DEFAULTSORT:Cubic Function}}
[[Category:Calculus]]
[[Category:Polynomial functions]]