Final answer:
The trigonometric function y = -\(\frac{3}{2}\) sin(2x) represents a reflection across the x-axis, a vertical stretch by a factor of \(\frac{3}{2}\), and a horizontal compression by a factor of 2. There is no phase shift or vertical shift present.
Step-by-step explanation:
The trigonometric function in question is y = -\(\frac{3}{2}\) sin(2x). When we analyze its transformations, we can identify the following characteristics:
- A) Reflection across the x-axis: This occurs because of the negative sign in front of the \(\frac{3}{2}\). It means that the graph of sin(2x) is flipped over the x-axis.
- D) Vertical stretch by a factor of \(\frac{3}{2}\): Here, the absolute value of \(\frac{3}{2}\), which is greater than 1, indicates that the graph is stretched vertically compared to the basic sin(x) function.
- Compression by a factor of 2: The '2' inside the sine function (in sin(2x)) causes the function to complete its cycle in half the time of the standard sine function, indicating a horizontal compression by a factor of 2.
There is no phase shift like \(\pi/2\) present, as there is no horizontal translation included in the function y = -\(\frac{3}{2}\) sin(2x). Additionally, the no vertical shift is indicated by the lack of a constant term added or subtracted from the function.