1. **State the problem:** Find the amplitude, period, phase shift, and vertical shift of the function $$y = 3 + 4 \sin\left(x - \frac{\pi}{2}\right)$$ and sketch the graph for $$0 \leq x \leq 2\pi$$. 2. **Recall the general form of a sine function:** $$y = A \sin(B(x - C)) + D$$ where: - Amplitude = $$|A|$$ - Period = $$\frac{2\pi}{|B|}$$ - Phase shift = $$C$$ - Vertical shift = $$D$$ 3. **Identify parameters from the given function:** - $$A = 4$$ - $$B = 1$$ (coefficient of $$x$$ inside sine) - $$C = \frac{\pi}{2}$$ - $$D = 3$$ 4. **Calculate amplitude:** $$\text{Amplitude} = |4| = 4$$ 5. **Calculate period:** $$\text{Period} = \frac{2\pi}{1} = 2\pi$$ 6. **Calculate phase shift:** $$\text{Phase shift} = \frac{\pi}{2}$$ to the right (since it is $$x - \frac{\pi}{2}$$) 7. **Calculate vertical shift:** $$\text{Vertical shift} = 3$$ (the whole graph is shifted up by 3 units) 8. **Summary:** - Amplitude = 4 - Period = $$2\pi$$ - Phase shift = $$\frac{\pi}{2}$$ right - Vertical shift = 3 up 9. **Sketching the graph:** - Start with the basic sine curve. - Shift it right by $$\frac{\pi}{2}$$. - Stretch vertically by 4. - Shift up by 3. - The graph oscillates between $$3 - 4 = -1$$ and $$3 + 4 = 7$$. - One full cycle occurs over $$0 \leq x \leq 2\pi$$. This completes the analysis of the function.