math lessons - Cubic function

In mathematics, a cubic function is a function of the form

$f(x)=ax^3+bx^2+cx+d,\,$

where a is nonzero; or in other words, a polynomial of degree three. The derivative of a cubic function is a quadratic function. The integral of a cubic function is a quartic function.

Bipartite cubics

The graph of

$y^2 = x(x-a)(x-b)\,$

where $0 < a < b$ is called a bipartite cubic. This is from the theory of elliptic curves.

You can graph a bipartite cubic on a graphing device by graphing the function

$f(x) = \sqrt{x(x-a)(x-b)}\,$

corresponding to the upper half of the bipartite cubic. It is defined on

$(0,a) \cup (b,+\infty).\,$

Root-finding Formula

The formula for finding the roots of a cubic function is fairly complicated. Therefore, it is common for some students to use the rational roots test or a numerical solution instead.

If we have

$f(x) = ax^3 + bx^2 + cx + d = (x - x_1)(x - x_2)(x - x_3)\,$