1. The original curve is given by $y = f(x)$, which has a minimum point at the coordinates $(5, -4)$. This means the function $f(x)$ reaches its minimum value of $-4$ when $x = 5$. 2. The problem asks for the minimum point of the new curve $y = f(x + 7)$. 3. To understand this, note that substituting $x + 7$ into $f$ shifts the graph horizontally by $-7$ units. Specifically, the input $x$ in the new function corresponds to input $x + 7$ in the original function. 4. Since the minimum of $f(x)$ occurs at $x = 5$, in the new function $f(x + 7)$, the minimum occurs where $x + 7 = 5$. 5. Solve for $x$: $$x + 7 = 5 \\ x = 5 - 7 = -2$$ 6. The $y$-coordinate of the minimum point remains the same because the function value at the minimum does not change by horizontal shifts. Thus, $y = -4$. 7. Therefore, the minimum point of the curve $y = f(x + 7)$ is at $oxed{(-2, -4)}$.