1. **Problem statement:** We need to find the vertical translation of a sine function that has a minimum value of $-2$ and a maximum value of $10$. 2. **Recall the general sine function:** The standard sine function is $y = \sin(x)$, which oscillates between $-1$ and $1$. 3. **Amplitude and vertical translation:** The amplitude $A$ is half the distance between the max and min values, and the vertical translation $D$ is the midpoint between the max and min values. 4. **Calculate amplitude:** $$A = \frac{\text{max} - \text{min}}{2} = \frac{10 - (-2)}{2} = \frac{12}{2} = 6$$ 5. **Calculate vertical translation:** $$D = \frac{\text{max} + \text{min}}{2} = \frac{10 + (-2)}{2} = \frac{8}{2} = 4$$ 6. **Write the translated sine function:** $$y = 6 \sin(x) + 4$$ This function oscillates between $4 - 6 = -2$ and $4 + 6 = 10$, matching the given min and max values.