1. **State the problem:** We are given the function $$f(x) = -3 \sin\left(\frac{1}{2}(x + \pi)\right) - 1$$ and need to find its amplitude, period, vertical shift (V.S.), and phase shift (P.S.). 2. **Recall the general sine function form:** $$f(x) = A \sin(B(x - C)) + D$$ where: - Amplitude = $$|A|$$ - Period = $$\frac{2\pi}{|B|}$$ - Phase shift = $$C$$ - Vertical shift = $$D$$ 3. **Identify parameters from the given function:** - $$A = -3$$ so amplitude = $$|-3| = 3$$ - Inside the sine, the coefficient of $$x$$ is $$\frac{1}{2}$$, so $$B = \frac{1}{2}$$ - The expression inside sine is $$\frac{1}{2}(x + \pi) = \frac{1}{2}x + \frac{1}{2}\pi$$, which can be rewritten as $$\frac{1}{2}(x - (-\pi))$$, so phase shift $$C = -\pi$$ - Vertical shift $$D = -1$$ 4. **Calculate the period:** $$$\text{Period} = \frac{2\pi}{|B|} = \frac{2\pi}{\frac{1}{2}} = 2\pi \times 2 = 4\pi$$$ 5. **Summarize results:** - Amplitude = 3 - Period = $$4\pi$$ - Vertical shift = -1 - Phase shift = $$-\pi$$ (shift left by $$\pi$$ units) 6. **Graph shape explanation:** - The negative amplitude flips the sine wave vertically. - The amplitude 3 stretches the wave vertically by a factor of 3. - The period is stretched horizontally to $$4\pi$$, making the wave longer. - The phase shift moves the graph left by $$\pi$$. - The vertical shift moves the entire graph down by 1 unit. This fully describes the graph of the function.