Let,$\,\,\,\, \phi=A \xi(\xi-1)(\xi+1)$

At node 1,$\ (\xi+1)\,\,vanishes,$
$\therefore \phi_1=A \xi(\xi-1)$
Now, $\phi_1=1\,\,\,\xi=-1$
$\therefore 1=A(-1)(-1-1)$
$\therefore A=\frac{1}{2}$

$\therefore \phi_1=\frac{1}{2}\xi(\xi-1)$

At node 2,$\ (\xi-1)\,\,vanishes,$
$\therefore \phi_2=A \xi(\xi+1)$
Now, $\phi_2=1\,\,\,\xi=1$
$\therefore 1=A(1)(1+1)$
$\therefore A=\frac{1}{2}$

$\therefore \phi_2=\frac{1}{2}\xi(\xi+1)$

At node 3,$\ \xi\,\,vanishes,$
$\therefore \phi_3=A (\xi+1)(\xi-1)$
Now, $\phi_3=1\,\,\,\xi=0$
$\therefore 1=A(-1)(1)$
$\therefore A=-1$

$\therefore \phi_3=-1(\xi-1)(\xi+1)=(1-\xi)(1+\xi)$