To identify the vertex of the quadratic function \( y = 4 (x - 25)^2 - 61 \), we can use the standard vertex form of a parabola, which is given by:
\[ y = a(x - h)^2 + k \]
where \((h, k)\) is the vertex of the parabola.
In the given equation:
- \( a = 4 \)
- \( h = 25 \)
- \( k = -61 \)
Thus, the vertex \((h, k)\) is:
\[ (25, -61) \]
So, the vertex of the function \( y = 4 (x - 25)^2 - 61 \) is \((25, -61)\).
Therefore, the correct answer is:
(25, -61).