Here's the graph of the function $y = 4 \sin \left(\frac{3}{4}x - \frac{3\pi}{16}\right)$ over one complete period. It starts from a shift to the right by $\frac{\pi}{4}$ and continues for $\frac{8\pi}{3}$, showing how the function progresses through its maximum, minimum, and back to the midpoint.

Would you like more details or do you have any questions about how these calculations were done or about the graph?

Further Exploration

1. What is the significance of amplitude in real-world applications of sine waves?
2. How does changing the coefficient $B$ affect the frequency of the sine wave?
3. What is the effect of a horizontal shift on the phase of a wave in physics?
4. Can you determine the vertical shift if the function was $y = 4 \sin\left(\frac{3}{4}x - \frac{3\pi}{16}\right) + C$?
5. How would the graph change if the function was $y = -4 \sin\left(\frac{3}{4}x - \frac{3\pi}{16}\right)$?

Tip: Always check the units of your x-axis when graphing trigonometric functions, especially if the context involves real-world measurements (like time or distance), to ensure accurate interpretation.