3 points define a parabola, so the regression will be unique. You want to find a quadratic polynomial [tex]P_2(x)=\beta_0+\beta_1x+\beta_2x^2[/tex] such that
[tex]\begin{cases}\beta_0-\beta_1+\beta_2=1\\\beta_0=0\\\beta_0+\beta_1+\beta_2=1\end{cases}[/tex]
where the system above is generated by setting [tex]P_2(-1)=1[/tex], [tex]P_2(0)=0[/tex], and [tex]P_2(1)=1[/tex].
Since [tex]\beta_0=0[/tex], we have
[tex]\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[/tex]
So the regression for the given data points is [tex]P_2(x)=x^2[/tex].
