Cubic function
From Academic Kids
Polynomialdeg3.png
In mathematics, a cubic function is a function of the form
- <math>f(x)=ax^3+bx^2+cx+d,\,<math>
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
- <math>y^2 = x(x-a)(x-b)\,<math>
where <math>0 < a < b<math> 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
- <math>f(x) = \sqrt{x(x-a)(x-b)}\,<math>
corresponding to the upper half of the bipartite cubic. It is defined on
- <math>(0,a) \cup (b,+\infty).\,<math>
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 root test or a numerical solution instead.
If we have
- <math>f(x) = ax^3 + bx^2 + cx + d = a(x - x_1)(x - x_2)(x - x_3)\,<math>
Let
- <math>q = \frac{{3c-b^2}}{{9}}<math> and
- <math>r = \frac{{9bc - 27d - 2b^3}}{{54}}<math>
Now, let
- <math>s = \sqrt[3]{{r + \sqrt{{q^3+r^2}}}}<math> and
- <math>t=\sqrt[3]{{r-\sqrt{{q^3+r^2}}}}<math>
The solutions are
- <math>x_1 = s+t-\frac{{b}}{{3}}<math>
- <math>x_2=-\frac{{1}}{{2}}(s+t)-\frac{{b}}{{3}}+\frac{{\sqrt{{3}}}}{{2}}(s-t)i<math>
- <math>x_3=-\frac{{1}}{{2}}(s+t)-\frac{{b}}{{3}}-\frac{{\sqrt{{3}}}}{{2}}(s-t)i<math>
See also: cubic equation, spline.
