1. The problem states that the curve $y = f(x)$ has a single minimum point at $(8, -12)$. 2. For part (i), the new curve is $y = f(x) + 3$. Adding 3 to the function shifts the entire graph vertically upwards by 3 units. 3. Therefore, the minimum point's $x$-coordinate remains the same, but the $y$-coordinate increases by 3: $$ (8, -12 + 3) = (8, -9) $$ 4. For part (ii), the new curve is $y = f(2x)$. This transformation compresses the graph horizontally by a factor of 2. 5. To find the new minimum point, we set $2x = 8$ (since the original minimum is at $x=8$), so: $$ x = \frac{8}{2} = 4 $$ 6. The $y$-coordinate remains the same because the function value at $2x$ is the same as the original function at $x=8$: $$ y = f(8) = -12 $$ 7. Thus, the minimum point for $y = f(2x)$ is: $$ (4, -12) $$