Example 1 Suppose the quadratic function $y=a x^{2}+b x+c$ passes through the points $(1,0),(2,0),(3,6)$. Find the analytical expression of this function.
Solve: From the corollary, we know $a=\frac{C}{2!}=\frac{6-0}{2}=3$.
Then, "extract" $3 x^{2}$ (i.e., $y-3 x^{2}$) from the quadratic function, to get the new data points $(1,-3),(2,-12),(3,-21)$.
Thus, $b=\frac{-12-(-3)}{1!}=-9$.
Finally, we get $c=6$.
Therefore, the required function is $y=3 x^{2}-9 x+6$.