1. **State the problem:** We need to write sine and cosine functions for a curve with given parameters: midline $-1$, amplitude $2$, and horizontal shift left by $\frac{\pi}{4}$. The cosine function is given as $y=2\cos(x+\pi)-1$. 2. **Recall the general form of sine and cosine functions:** $$y = A\sin(B(x - C)) + D$$ $$y = A\cos(B(x - C)) + D$$ where: - $A$ is the amplitude, - $B$ affects the period ($\text{period} = \frac{2\pi}{B}$), - $C$ is the phase shift (positive $C$ shifts right, negative $C$ shifts left), - $D$ is the midline (vertical shift). 3. **Given values:** - Amplitude $A = 2$ - Midline $D = -1$ - $B = 1$ (since $b = \frac{2\pi}{2\pi} = 1$) - Phase shift left $\frac{\pi}{4}$ means $C = -\frac{\pi}{4}$ 4. **Write the sine function:** Using the sine form with phase shift left $\frac{\pi}{4}$: $$y = 2\sin\left(x - \left(-\frac{\pi}{4}\right)\right) - 1 = 2\sin\left(x + \frac{\pi}{4}\right) - 1$$ 5. **Check the cosine function given:** $$y = 2\cos(x + \pi) - 1$$ This matches the amplitude and midline, with a phase shift of $-\pi$ (left $\pi$). 6. **Summary:** - Sine function: $$y = 2\sin\left(x + \frac{\pi}{4}\right) - 1$$ - Cosine function: $$y = 2\cos(x + \pi) - 1$$ These functions describe the same wave shifted differently horizontally but with the same amplitude and midline. **Final answer:** Sine equation: $y = 2\sin\left(x + \frac{\pi}{4}\right) - 1$ Cosine equation: $y = 2\cos(x + \pi) - 1$