1. **State the problem:** Graph one cycle of the function $$y = 3 \cos\left(x + \frac{\pi}{2}\right)$$ and explain its characteristics. 2. **Formula and rules:** The general cosine function is $$y = A \cos(Bx - C) + D$$ where: - Amplitude = $$|A|$$ - Period = $$\frac{2\pi}{|B|}$$ - Phase shift = $$\frac{C}{B}$$ - Vertical shift = $$D$$ 3. **Identify parameters:** For $$y = 3 \cos\left(x + \frac{\pi}{2}\right)$$: - $$A = 3$$ (amplitude) - $$B = 1$$ (coefficient of $$x$$) - Inside is $$x + \frac{\pi}{2}$$, so $$C = -\frac{\pi}{2}$$ (because $$Bx - C = x - (-\frac{\pi}{2})$$) - $$D = 0$$ (no vertical shift) 4. **Calculate period:** $$\text{Period} = \frac{2\pi}{|1|} = 2\pi$$ 5. **Calculate phase shift:** $$\text{Phase shift} = \frac{C}{B} = \frac{-\frac{\pi}{2}}{1} = -\frac{\pi}{2}$$ This means the graph shifts left by $$\frac{\pi}{2}$$. 6. **Graph one cycle:** One cycle spans from $$x = -\frac{\pi}{2}$$ to $$x = -\frac{\pi}{2} + 2\pi = \frac{3\pi}{2}$$. 7. **Key points:** - At $$x = -\frac{\pi}{2}$$, $$y = 3 \cos(0) = 3$$ - At $$x = \frac{\pi}{2}$$, $$y = 3 \cos(\pi) = -3$$ - At $$x = \frac{3\pi}{2}$$, $$y = 3 \cos(2\pi) = 3$$ 8. **Final answer:** The graph is a cosine wave with amplitude 3, period $$2\pi$$, shifted left by $$\frac{\pi}{2}$$, and no vertical shift.