1. **State the problem:** We need to write the equation of a sinusoidal function given its graph characteristics. 2. **Identify key features from the graph:** - Maximum value: 5 - Minimum value: -3 - Midline (vertical shift): $\frac{5 + (-3)}{2} = 1$ - Amplitude: $\frac{5 - (-3)}{2} = 4$ - Period: $\pi$ - Peaks at $x = -\frac{3\pi}{4}, \frac{\pi}{4}, \frac{5\pi}{4}$ - Troughs at $x = -\frac{5\pi}{4}, -\frac{\pi}{4}, \frac{3\pi}{4}$ - Passes through $(0,1)$ increasing 3. **Formula for sinusoidal function:** $$y = A \sin(B(x - C)) + D$$ where - $A$ is amplitude - $B$ relates to period by $B = \frac{2\pi}{\text{period}}$ - $C$ is horizontal shift - $D$ is vertical shift (midline) 4. **Calculate $B$ using the period:** $$B = \frac{2\pi}{\pi} = 2$$ 5. **Determine phase shift $C$:** Since the graph passes through $(0,1)$ and is increasing there, and the midline is at $y=1$, this matches the sine function starting at midline going upward with no horizontal shift. So, $C = 0$. 6. **Write the equation:** $$y = 4 \sin(2x) + 1$$ 7. **Summary:** The equation of the sinusoidal function is $$\boxed{y = 4 \sin(2x) + 1}$$