1. **Problem Statement:** Write the equations of the sinusoidal functions for the given graph. The graph has the following characteristics: - Amplitude $a = 3$ (since it peaks at 3 and troughs at -3) - Period $T = 4\pi$ (since one full cycle spans from 0 to $4\pi$) - Vertical shift $d = 0$ (wave oscillates symmetrically about the x-axis) - Phase shift $c = 0$ (wave starts at origin for sine) 2. **Formula for sinusoidal functions:** - For sine: $y = a \sin b(x - c) + d$ - For cosine: $y = a \cos b(x - c) + d$ Where: - $a$ is amplitude - $b = \frac{2\pi}{T}$ is the angular frequency - $c$ is phase shift - $d$ is vertical shift 3. **Calculate $b$:** $$b = \frac{2\pi}{T} = \frac{2\pi}{4\pi} = \frac{1}{2}$$ 4. **Write positive sinusoidal equation (sine):** $$y = 3 \sin \frac{1}{2} (x - 0) + 0 = 3 \sin \frac{x}{2}$$ 5. **Write negative sinusoidal equation (cosine):** $$y = -3 \cos \frac{1}{2} (x - 0) + 0 = -3 \cos \frac{x}{2}$$ **Final answers:** - Positive sinusoidal equation: $y = 3 \sin \frac{x}{2}$ - Negative sinusoidal equation: $y = -3 \cos \frac{x}{2}$