 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.