Determine the function whose graph has a vertex at $(2,-7)$ and passes through the point $(5,-4)$.

Step 1 ：We are given a function with a vertex at (2, -7). The general form of such a function is \(f(x) = a(x-h)^2 + k\), where (h, k) is the vertex of the function.

Step 2 ：Substituting the given vertex into the equation, we get \(f(x) = a(x-2)^2 - 7\).

Step 3 ：We also know that the function passes through the point (5, -4). Substituting these values into the equation, we get \(-4 = a(5-2)^2 - 7\).

Step 4 ：Solving this equation for a, we find that \(a = \frac{1}{3}\).

Step 5 ：Substituting a back into the equation, we get the final function \(f(x) = \frac{1}{3}(x-2)^2 - 7\).

Step 6 ：\(\boxed{f(x) = \frac{1}{3}(x-2)^2 - 7}\) is the function whose graph has a vertex at (2,-7) and passes through the point (5,-4).