1. The problem states that the curve $y=f(x)$ has a minimum point at $(5,-4)$. 2. We need to find the coordinates of the minimum point on the curve $y=f(x+7)$. 3. The function $y=f(x+7)$ represents a horizontal shift of the original function $f(x)$ by 7 units to the left. 4. Horizontal shifts affect the $x$-coordinate of points on the graph but do not change the $y$-coordinate. 5. Since the original minimum point is at $x=5$, shifting the graph 7 units left means the new $x$-coordinate is: $$5 - 7 = -2$$ 6. The $y$-coordinate remains the same, so it is still $-4$. 7. Therefore, the coordinates of the minimum point on $y=f(x+7)$ are $(-2,-4)$. Final answer: $\boxed{(-2,-4)}$