To determine the period of the function [tex]\( y = \frac{1}{2} \sin(2x) - 3 \)[/tex], it is important to understand the form and properties of the sine function.
The standard form of the sine function is:
[tex]\[ y = A \sin(Bx - C) + D \][/tex]
where:
- [tex]\( A \)[/tex] is the amplitude,
- [tex]\( B \)[/tex] affects the period,
- [tex]\( C \)[/tex] represents a horizontal shift (phase shift),
- [tex]\( D \)[/tex] represents a vertical shift.
The key parameter that affects the period of the sine function is [tex]\( B \)[/tex]. The period of the sine function is given by:
[tex]\[ \text{Period} = \frac{2\pi}{|B|} \][/tex]
For the function [tex]\( y = \frac{1}{2} \sin(2x) - 3 \)[/tex]:
- Here, [tex]\( A = \frac{1}{2} \)[/tex] (affects amplitude, but not the period),
- [tex]\( B = 2 \)[/tex] (this affects the period),
- [tex]\( C = 0 \)[/tex] (no phase shift),
- [tex]\( D = -3 \)[/tex] (shifts the function vertically, but does not affect the period).
We need to calculate the period using the value [tex]\( B = 2 \)[/tex]:
[tex]\[ \text{Period} = \frac{2\pi}{|2|} = \frac{2\pi}{2} = \pi \][/tex]
Therefore, the period of the function [tex]\( y = \frac{1}{2} \sin(2x) - 3 \)[/tex] is [tex]\(\pi\)[/tex].
Hence, the correct answer is B. [tex]\(\pi\)[/tex].
