###### 2.4.2 Example: Quadratic Interpolation

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

The general form for a quadratic polynomial from ^{2} to is

[2.40]

Solving for the six constants β_{j} 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 β_{5} = β_{4} or set β_{1} = 0, etc. For this example, let’s interpolate a quadratic polynomial with zero cross term, β_{6} = 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.