Example 14 (1) Let the parabola be $y=2 x^{2}$. If it is shifted $p$ units to the right, or $q$ units downward, it can make the parabola intersect the line $y=x-4$ at exactly one point. Find the values of $p, q$;
(2) If the parabola $y=2 x^{2}$ is shifted $p$ units to the left and $q$ units upward, the resulting parabola passes through the points $(1,3)$ and $(4,9)$. Find the values of $p, q$.
(3) If the parabola $y=a x^{2}+b x+c$ is shifted 3 units to the left and 2 units downward, the resulting graph is the parabola $y=a x^{2}$ passing through the point $\left(-1,-\frac{1}{2}\right)$. Find the original quadratic function.