1. **State the problem:** Find the amplitude, period, phase shift, and vertical shift for the function $$y = 6 \sin x + 2$$ and sketch the graph for $$0^\circ \leq x \leq 360^\circ$$. 2. **Recall the general form of a sine function:** $$y = A \sin(B(x - C)) + D$$ where: - Amplitude = $$|A|$$ - Period = $$\frac{360^\circ}{|B|}$$ (degrees) - Phase shift = $$C$$ (degrees) - Vertical shift = $$D$$ 3. **Identify parameters for $$y = 6 \sin x + 2$$:** - $$A = 6$$ - $$B = 1$$ (since $$\sin x = \sin(1 \cdot x)$$) - $$C = 0$$ (no horizontal shift) - $$D = 2$$ 4. **Calculate amplitude:** $$\text{Amplitude} = |6| = 6$$ 5. **Calculate period:** $$\text{Period} = \frac{360^\circ}{1} = 360^\circ$$ 6. **Calculate phase shift:** $$\text{Phase shift} = 0^\circ$$ 7. **Calculate vertical shift:** $$\text{Vertical shift} = 2$$ 8. **Summary:** - Amplitude: 6 - Period: 360° - Phase shift: 0° - Vertical shift: 2 9. **Graph sketch notes:** - The sine wave oscillates between $$2 - 6 = -4$$ and $$2 + 6 = 8$$. - One full cycle occurs from $$0^\circ$$ to $$360^\circ$$. - The midline is at $$y = 2$$. This completes the solution for the first function.