1. **State the problem:** Find the equations of the sinusoidal graph as translations of $y=\sin(x)$ and $y=\cos(x)$ with the closest right shift. 2. **Analyze the graph characteristics:** - Midline: $y=1$ - Maximum: $y=2$ - Minimum: $y=-4$ - Period: $2\pi$ - Domain shown: $[-2\pi, 2\pi]$ 3. **Determine amplitude and vertical shift:** Amplitude $A = \frac{\text{max} - \text{min}}{2} = \frac{2 - (-4)}{2} = \frac{6}{2} = 3$ Vertical shift $D = \frac{\text{max} + \text{min}}{2} = \frac{2 + (-4)}{2} = \frac{-2}{2} = -1$ 4. **Check period and frequency:** Given period $P = 2\pi$, so frequency $B = \frac{2\pi}{P} = 1$ 5. **Form general sinusoidal equations:** - For sine: $y = A \sin(B(x - C)) + D$ - For cosine: $y = A \cos(B(x - C)) + D$ 6. **Find phase shift $C$ for sine:** Sine normally starts at midline going upward at $x=0$. Here, the midline crossing going upward is at $x = -\frac{\pi}{2}$ (closest right shift). So, phase shift for sine: $C = -\frac{\pi}{2} \approx -1.57$ Equation for sine: $$y = 3 \sin\left(x - \left(-\frac{\pi}{2}\right)\right) - 1 = 3 \sin\left(x + 1.57\right) - 1$$ 7. **Find phase shift $C$ for cosine:** Cosine normally starts at maximum at $x=0$. Here, maximum is at $x = \pi$ (closest right shift). So, phase shift for cosine: $C = \pi \approx 3.14$ Equation for cosine: $$y = 3 \cos(x - 3.14) - 1$$ **Final answers:** - $y = 3 \sin(x + 1.57) - 1$ - $y = 3 \cos(x - 3.14) - 1$