A function term from conditions

In a «Steckbrief» task a function term with unknown coefficients is wanted. Each property gives an equation: a point P(x₀ | y₀) on the graph gives f(x₀) = y₀, an extreme point at x₀ gives f′(x₀) = 0, a point of inflection at x₁ gives f″(x₁) = 0.

A cubic with four coefficients needs four conditions. They lead to a system of linear equations.

If the term contains a parameter t (a family f_t), each condition fixes t. The extreme points of all curves of the family lie on one curve, the locus: eliminate t from the coordinates.