- The transformation is given by the form \(f(x - h) = \sin(x - \frac{3\pi}{2})\), where \(h = \frac{3\pi}{2}\).
- A positive value of \(h\) indicates a horizontal shift to the right by \(h\) units.
- Consequently, the graph of \(y = \sin(x - \frac{3\pi}{2})\) is the graph of \(y = \sin(x)\) shifted by \(\frac{3\pi}{2}\) units to the right.
- The final answer is \(\boxed{\frac{3\pi}{2} \text{ units to the right}}\).
Explanation
- Identifying the Transformation Form
The function \(f(x) = \sin(x)\) serves as our parent function. We are considering the transformed function \(g(x) = \sin\left(x - \frac{3\pi}{2}\right)\). To understand how the graph shifts, we compare this with the general horizontal translation formula \(y = f(x - h)\). In this formula, \(h\) represents the magnitude and direction of the horizontal shift applied to the graph of \(f(x)\).
- Determining the Shift Magnitude and Direction
By comparing \(g(x) = \sin\left(x - \frac{3\pi}{2}\right)\) with the general form \(f(x - h) = \sin(x - h)\), we can clearly identify the value of \(h\). Here, we have \(h = \frac{3\pi}{2}\). The sign of \(h\) is positive, which is the key indicator for the direction of the horizontal translation.
- Conclusion on Horizontal Shift
According to the rules for horizontal shifts, when \(h > 0\), the graph of the parent function \(f(x)\) is shifted \(h\) units to the right. Since \(h = \frac{3\pi}{2}\) and \(\frac{3\pi}{2} > 0\), the graph of \(y = \sin\left(x - \frac{3\pi}{2}\right)\) is obtained by shifting the graph of \(y = \sin(x)\) exactly \(\frac{3\pi}{2}\) units to the right.
- Final Answer
Based on the analysis, the graph is shifted \(\frac{3\pi}{2}\) units to the right. This corresponds to the second option provided in the problem. The final answer is \(\boxed{\frac{3\pi}{2} \text{ units to the right}}\).
Examples
In fields like audio engineering or signal processing, understanding horizontal shifts (phase shifts) is crucial. A sine wave represents a pure tone; shifting this wave horizontally by a phase angle (like \(\frac{3\pi}{2}\)) changes when the peak occurs in time. By mastering these transformations, engineers can synchronize signals, create effects like interference or cancellation, or precisely tune audio systems to ensure sound waves arrive at the listener exactly when intended.