Find the parabola of the form y=ax2+b which best fits the points (−1,0), (5,5), (6,10) by minimizing the sum of squares, s, given by

The given data is (-1, 0), (5, 5) and (6, 10).
That is,
{x} = [-1, 5, 6]
{y} = [0, 5, 10]

Let the curve of the best least squares fit be
y = ax² + b

The sum of the squares of errors between the data and the curve is


To minimize the error requires that

Also,


Therefore the normal equations are
(x₁⁴+x₂⁴+x₃⁴)a + (x₁²+x₂²+x₃²)b = x₁²y₁+x₂²y₂+x₃²y₃            (1)
(x₁²+x₂²+x₃²)a + 3b = y₁+y₂+y₃                                              (2)

From the given data, obtain
1922a + 62b = 485
62a + 3b = 15
The solution of these equations yields
a = 0.2732
b = -0.6452

The required curve is
y = 0.2732x² - 0.6452
A graph of the curve and the given data is shown below.

  y = 0.2732x² - 0.6452