1. **State the problem:** We need to write the equation of a sine function in the form $$y = A \sin(Bx + C) + D$$ based on the given graph. 2. **Identify key features from the graph:** - The wave oscillates between -4 and 4, so the amplitude $$A = 4$$. - The midline (vertical shift) is at $$y = 0$$, so $$D = 0$$. - The wave passes through the origin (0,0), which is a zero crossing point of the sine function. - The period can be found by observing the distance between two consecutive peaks or troughs. 3. **Calculate the period:** - Peaks occur near $$x = -\frac{3\pi}{2}$$ and $$x = \frac{\pi}{2}$$. - The distance between these peaks is $$\frac{\pi}{2} - \left(-\frac{3\pi}{2}\right) = \frac{\pi}{2} + \frac{3\pi}{2} = 2\pi$$. - So, the period $$T = 2\pi$$. 4. **Calculate $$B$$:** - The formula for period is $$T = \frac{2\pi}{B}$$. - Substitute $$T = 2\pi$$: $$2\pi = \frac{2\pi}{B}$$ - Multiply both sides by $$B$$: $$2\pi B = 2\pi$$ - Divide both sides by $$2\pi$$: $$\cancel{2\pi} B = \cancel{2\pi}$$ $$B = 1$$ 5. **Calculate phase shift $$C$$:** - Since the sine wave passes through the origin (0,0) and matches the basic sine function shape, the phase shift $$C = 0$$. 6. **Write the final equation:** $$y = 4 \sin(1 \cdot x + 0) + 0 = 4 \sin x$$ **Final answer:** $$\boxed{y = 4 \sin x}$$