Question

Find the quadratic regression for (-1,1) , (0,0) , (1,1)

Answer 1

3 points define a parabola, so the regression will be unique. You want to find a quadratic polynomial
`P_2(x)=\beta_0+\beta_1x+\beta_2x^2` such that

`\begin{cases}\beta_0-\beta_1+\beta_2=1\\\beta_0=0\\\beta_0+\beta_1+\beta_2=1\end{cases}`

where the system above is generated by setting
`P_2(-1)=1`,
`P_2(0)=0`, and
`P_2(1)=1`.

Since
`\beta_0=0`, we have

`\begin{cases}-\beta_1+\beta_2=1\\\beta_1+\beta_2=1\end{cases}\implies2\beta_2=2\implies\beta_2=1\implies\beta_1=0`

So the regression for the given data points is
`P_2(x)=x^2`.