1. **State the problem:** We need to find the equations of the sine and cosine functions that match the given graph, expressed as translations of $y=\sin(x)$ and $y=\cos(x)$ with a horizontal shift (right shift) and vertical shift. 2. **Analyze the graph:** - The midline is at $y = -5$ (since the range is from -6 to -4, midpoint is $\frac{-6 + (-4)}{2} = -5$). - Amplitude $A = 1$ (since max is -4 and min is -6, amplitude is half the distance: $\frac{-4 - (-6)}{2} = 1$). - The graph crosses the midline with positive slope at $x = \frac{\pi}{4}$. - The maximum occurs at $x = \frac{3\pi}{4}$. 3. **General forms:** - For sine: $$y = A \sin(x - c) + d$$ - For cosine: $$y = A \cos(x - c) + d$$ where $c$ is the horizontal shift (phase shift), $d$ is the vertical shift, and $A$ is amplitude. 4. **Vertical shift and amplitude:** - From the graph, $A = 1$, $d = -5$. 5. **Find horizontal shift for sine:** - The sine function crosses the midline going upward at $x = c$ (since $\sin(0) = 0$ and slope positive at 0). - Given the graph crosses midline with positive slope at $x = \frac{\pi}{4}$, so for sine: $$x - c = 0 \implies c = \frac{\pi}{4}$$ - So sine equation is: $$y = \sin\left(x - \frac{\pi}{4}\right) - 5$$ 6. **Find horizontal shift for cosine:** - Cosine has maximum at $x = c$ (since $\cos(0) = 1$ max at 0). - Given max at $x = \frac{3\pi}{4}$, so for cosine: $$x - c = 0 \implies c = \frac{3\pi}{4}$$ - So cosine equation is: $$y = \cos\left(x - \frac{3\pi}{4}\right) - 5$$ 7. **Round shifts to two decimals:** - $\frac{\pi}{4} \approx 0.79$ - $\frac{3\pi}{4} \approx 2.36$ **Final answers:** $$y = \sin(x - 0.79) - 5$$ $$y = \cos(x - 2.36) - 5$$