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Inmathematics, acubic function is afunction of the form that is, apolynomial function of degree three. In many texts, thecoefficientsa,b,c, andd are supposed to bereal numbers, and the function is considered as areal function that maps real numbers to real numbers or as a complex function that mapscomplex numbers 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 itscodomain, even when thedomain is restricted to the real numbers.
Settingf(x) = 0 produces acubic equation of the form
whose solutions are calledroots of the function. Thederivative of a cubic function is aquadratic function.
A cubic function with real coefficients has either one or three real roots (which may not be distinct);[1] all odd-degree polynomials with real coefficients have at least one real root.
Thegraph of a cubic function always has a singleinflection point. It may have twocritical points, a local minimum and a local maximum. Otherwise, a cubic function ismonotonic. 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 anaffine transformation, there are only three possible graphs for cubic functions.
Cubic functions are fundamental forcubic interpolation.
Thecritical points of a cubic function are itsstationary points, that is the points where the slope of the function is zero.[2] Thus the critical points of a cubic functionf defined by
occur at values ofx such that thederivative
of the cubic function is zero.
The solutions of this equation are thex-values of the critical points and are given, using thequadratic formula, by
The sign of the expressionΔ0 =b2 − 3ac 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. Ifb2 − 3ac = 0, then there is only one critical point, which is aninflection point. Ifb2 − 3ac < 0, then there are no (real) critical points. In the two latter cases, that is, ifb2 − 3ac is nonpositive, the cubic function is strictlymonotonic. See the figure for an example of the caseΔ0 > 0.
The inflection point of a function is where that function changesconcavity.[3] An inflection point occurs when thesecond derivative is zero, and the third derivative is nonzero. Thus a cubic function has always a single inflection point, which occurs at
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Thegraph of a cubic function is acubic 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 alwayssimilar to the graph of a function of the form
This similarity can be built as the composition oftranslations parallel to the coordinates axes, ahomothecy (uniform scaling), and, possibly, areflection (mirror image) with respect to they-axis. A furthernon-uniform scaling can transform the graph into the graph of one among the three cubic functions
This means that there are only three graphs of cubic functionsup to anaffine transformation.
The abovegeometric transformations can be built in the following way, when starting from a general cubic function
Firstly, ifa < 0, thechange of variablex → −x allows supposinga > 0. After this change of variable, the new graph is the mirror image of the previous one, with respect of they-axis.
Then, the change of variablex =x1 −b/3a provides a function of the form
This corresponds to a translation parallel to thex-axis.
The change of variabley =y1 +q corresponds to a translation with respect to they-axis, and gives a function of the form
The change of variable corresponds to a uniform scaling, and give, after multiplication by a function of the form
which is the simplest form that can be obtained by a similarity.
Then, ifp ≠ 0, the non-uniform scaling gives, after division by
where has the value 1 or −1, depending on the sign ofp. If one defines the latter form of the function applies to all cases (with and).
For a cubic function of the form the inflection point is thus the origin. As such a function is anodd 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 bysimilarity, 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.
The tangent lines to the graph of a cubic function at threecollinear points intercept the cubic again at collinear points.[4] This can be seen as follows.
As this property is invariant under arigid motion, one may suppose that the function has the form
Ifα is a real number, then the tangent to the graph off at the point(α,f(α)) is the line
So, the intersection point between this line and the graph off can be obtained solving the equationf(x) =f(α) + (x −α)f ′(α), that is
which can be rewritten
and factorized as
So, the tangent intercepts the cubic at
So, the function that maps a point(x,y) of the graph to the other point where the tangent intercepts the graph is
This is anaffine transformation that transforms collinear points into collinear points. This proves the claimed result.
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 acubic 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 caninterpolate the function with acontinuously differentiable function, which is apiecewise cubic function.
If the value of a function is known at several points,cubic interpolation consists in approximating the function by acontinuously differentiable function, which ispiecewise 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 zerocurvature at the endpoints.
Thus a cubic equation has either three real roots... or one real root...
A point at which the graph of the function f changes concavity is called an inflection point of f
