Cubic function

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 is restricted to the real numbers.

Setting f(x) = 0 produces a cubic equation of the form whose solutions are called roots of the function.

The graph of a cubic function always has a single inflection point.

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.

[2] Thus the critical points of a cubic function f defined by occur at values of x such that the derivative of the cubic function is zero.

The solutions of this equation are the x-values of the critical points and are given, using the quadratic formula, by The sign of the expression Δ0 = b2 – 3ac inside the square root determines the number of critical points.

In the two latter cases, that is, if b2 – 3ac is nonpositive, the cubic function is strictly monotonic.

[3] An inflection point occurs when the second derivative

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 similar to the graph of a function of the form This similarity can be built as the composition of translations parallel to the coordinates axes, a homothecy (uniform scaling), and, possibly, a reflection (mirror image) with respect to the y-axis.

A further non-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 functions up to an affine transformation.

The above geometric transformations can be built in the following way, when starting from a general cubic function

Firstly, if a < 0, the change of variable x → –x allows supposing a > 0.

After this change of variable, the new graph is the mirror image of the previous one, with respect of the y-axis.

Then, the change of variable x = x1 – ⁠b/3a⁠ provides a function of the form This corresponds to a translation parallel to the x-axis.

The change of variable y = y1 + q corresponds to a translation with respect to the y-axis, and gives a function of the form The change of variable

corresponds to a uniform scaling, and give, after multiplication by

has the value 1 or –1, depending on the sign of p. If one defines

the latter form of the function applies to all cases (with

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, 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 three collinear points intercept the cubic again at collinear points.

As this property is invariant under a rigid motion, one may suppose that the function has the form If α is a real number, then the tangent to the graph of f at the point (α, f(α)) is the line So, the intersection point between this line and the graph of f can be obtained solving the equation f(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 an affine transformation that transforms collinear points into collinear points.

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.

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.

Graph of a cubic function with 3 real roots (where the curve crosses the horizontal axis—where y = 0 ). The case shown has two critical points . Here the function is f ( x ) = ( x 3 + 3 x 2 − 6 x − 8)/4 .
The roots , stationary points , inflection point and concavity of a cubic polynomial x 3 − 6 x 2 + 9 x − 4 (solid black curve) and its first (dashed red) and second (dotted orange) derivatives .
Cubic functions of the form
The graph of any cubic function is similar to such a curve.
The points P 1 , P 2 , and P 3 (in blue) are collinear and belong to the graph of x 3 + ⁠ 3 / 2 ⁠ x 2 − ⁠ 5 / 2 ⁠ x + ⁠ 5 / 4 ⁠ . The points T 1 , T 2 , and T 3 (in red) are the intersections of the (dotted) tangent lines to the graph at these points with the graph itself. They are collinear too.