1. **Problem Statement:** Given a sine wave oscillating between -2 and 2 with x-axis labeled in multiples of $\frac{\pi}{3}$, and two full cycles from 0 to $\frac{10\pi}{3}$, find the equation of the sine wave. 2. **Formula and Explanation:** The general form of a sine wave is: $$y = A \sin(Bx + C) + D$$ where: - $A$ is the amplitude (half the distance between max and min values), - $B$ affects the period (period $= \frac{2\pi}{B}$), - $C$ is the phase shift, - $D$ is the vertical shift. 3. **Determine Amplitude $A$:** The wave oscillates between -2 and 2, so amplitude is: $$A = \frac{2 - (-2)}{2} = \frac{4}{2} = 2$$ 4. **Determine Vertical Shift $D$:** Since the wave oscillates symmetrically about 0, vertical shift is: $$D = 0$$ 5. **Determine Period and $B$:** Two full cycles occur from 0 to $\frac{10\pi}{3}$, so one full cycle period $T$ is: $$T = \frac{10\pi/3}{2} = \frac{10\pi}{6} = \frac{5\pi}{3}$$ Using the period formula: $$T = \frac{2\pi}{B} \implies B = \frac{2\pi}{T} = \frac{2\pi}{\frac{5\pi}{3}} = \frac{2\pi \times 3}{5\pi} = \frac{6}{5}$$ 6. **Determine Phase Shift $C$:** The graph starts at $x=0$ with $y=0$ and follows a sine wave pattern, so no horizontal shift: $$C = 0$$ 7. **Final Equation:** Substitute values into the general form: $$y = 2 \sin\left(\frac{6}{5}x\right)$$ **Answer:** The equation of the sine wave is: $$y = 2 \sin\left(\frac{6}{5}x\right)$$