1. **Stating the problem:** We want to find the equation of a sinusoidal function that fits the given graph data points and characteristics. 2. **Observations from the graph:** - The function oscillates between approximately 2 and -2, so the amplitude $A$ is about 2. - The midline (average value) is around 0, so the vertical shift $D = 0$. - The period $T$ is about $6\pi$ (from the x-axis markings). - The function starts above zero near $y=1.5$ at $x=0$, suggesting a phase shift. 3. **General form of sinusoidal function:** $$y = A \sin(B(x - C)) + D$$ where - $A$ is amplitude, - $B = \frac{2\pi}{T}$ controls the period, - $C$ is the phase shift, - $D$ is vertical shift. 4. **Calculate $B$ using the period:** $$B = \frac{2\pi}{6\pi} = \frac{1}{3}$$ 5. **Determine phase shift $C$:** Since the function starts near $y=1.5$ at $x=0$ and the sine function normally starts at 0, we try to fit a cosine function which starts at maximum. The cosine function form: $$y = A \cos(B(x - C)) + D$$ At $x=0$, $y \approx 1.5$, so: $$1.5 = 2 \cos\left(\frac{1}{3}(0 - C)\right)$$ $$\cos\left(-\frac{C}{3}\right) = \frac{1.5}{2} = 0.75$$ 6. **Solve for $C$:** $$-\frac{C}{3} = \cos^{-1}(0.75)$$ $$\cos^{-1}(0.75) \approx 0.7227$$ $$C = -3 \times 0.7227 = -2.168$$ 7. **Final function:** $$y = 2 \cos\left(\frac{1}{3}(x + 2.168)\right)$$ This matches the amplitude, period, and phase shift observed. **Answer:** $$\boxed{y = 2 \cos\left(\frac{x + 2.168}{3}\right)}$$