By using the method of least squares, find the best parabola through the points:
    $(1, 0)$,   $(-1, -2)$,   $(2, 2)$,   $(0, -1)$  

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Step 1. The general equation of a parabola is $c_0 + c_1x + c_2x^2 = y$. Plugging the data points into this formula gives a matrix equation $\mathrm{A}\mathbf{c} = \mathbf{y}$.

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Step 2. The matrix equation $\mathrm{A}\mathbf{c} = \mathbf{y}$ has no solution, so instead we use the normal equation $\mathrm{A^T} \mathrm{A}\, \hat{\mathbf{c}} = \mathrm{A^T} \mathbf{y}$
    $\mathrm{A^T} \mathrm{A} =$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$

    $\mathrm{A^T} \mathbf{y} =$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$

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Step 3. Solving the normal equation gives the answer
    $\hat{\mathbf{c}} =$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$
which corresponds to the formula
    $y =$