To find the least squares regression parabola y = ax2 + bx + c for a set of points (x1

Chapter 8, Problem 97

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Fitting a Parabola   To find the least squares regression parabola \(y = ax^2 + bx + c\) for a set of points \((x_1,y_1)\), \((x_2,y_2)\), . . ., \((x_n,y_n)\) you can solve the following system of linear equations for a, b, and c.

\(\left\{\begin{array}{r} n c+\left(\sum_{i=1}^{n} x_{i}\right) b+\left(\sum_{i=1}^{n} x_{i}^{2}\right) a=\sum_{i=1}^{n} y_{i} \\ \left(\sum_{i=1}^{n} x_{i}\right) c+\left(\sum_{i=1}^{n} x_{i}^{2}\right) b+\left(\sum_{i=1}^{n} x_{i}^{3}\right) a=\sum_{i=1}^{n} x_{i} y_{i} \\ \left(\sum_{i=1}^{n} x_{i}^{2}\right) c+\left(\sum_{i=1}^{n} x_{i}^{3}\right) b+\left(\sum_{i=1}^{n} x_{i}^{4}\right) a=\sum_{i=1}^{n} x_{i}^{2} y_{i} \end{array}\right. \)

In Exercises 97–100, the sums have been evaluated. Solve the given system for a, b, and c to find the least squares regression parabola for the points. Use a graphing utility to confirm the result.

\(\left\{\begin{aligned} 4 c+40 a &=19 \\ 40 b &=-12 \\ 40 c+544 a &=160 \end{aligned}\right. \)

Text Transcription:

{nc+(sum_{i=1}^{n}x_{i})b+(sum_{i=1}^{n}x_{i}^{2})a=sum_{i=1}^{n} y_{i} _(sum_{i=1}^{n}x_{i})c+(sum_{i=1}^{n}x_{i}^{2})b+(sum_{i=1}^{n}x_{i}^{3})a=sum_{i=1}^{n}x_{i}y_{i} _(sum_{i=1}^{n}x_{i}^{2})c+(sum_{i=1}^{n}x_{i}^{3}\right)b+(sum_{i=1}^{n}x_{i}^{4})a=sum_{i=1}^{n}x_{i}^{2}y_{i}

{4c+40a=19 _40b=-12 _40c+544a=160