Suppose we want to interpolate a quadratic polynomial between the five points (x[k], y[k]) of Exhibit 2.5.

Exhibit 2.5: Five points through which we wish to interpolate a quadratic polynomial.

The general form for a quadratic polynomial from 2 to is

[2.40]

Solving for the six constants βj based upon the five points (x[k], y[k]) of Exhibit 2.5 would entail a system of five equations in six unknowns. Such a system is likely to have infinitely many solutions. To obtain a unique solution, we may consider a less general form of quadratic polynomial than [2.40]. We might require β5 = β4 or set β1 = 0, etc. For this example, let’s interpolate a quadratic polynomial with zero cross term, β6 = 0. Our polynomial then has form

[2.41]

We require that this polynomial intercept each of our five points. This renders a system of five equations in five unknowns, which we express in matrix notation as

[2.42]

Solving, we obtain

[2.43]

This and the five points are graphed in Exhibit 2.6.

Exhibit 2.6: A quadratic polynomial of form [2.41] is interpolated between the points of Exhibit 2.5.