Let's analyze the function [tex]\( y = \frac{1}{2} \sin(2x) \)[/tex] to determine the amplitude and period.
### Amplitude
The general form of a sine function is [tex]\( y = A \sin(Bx) \)[/tex] where:
- [tex]\( A \)[/tex] is the amplitude.
- [tex]\( B \)[/tex] affects the period of the sine wave.
In our case, [tex]\( y = \frac{1}{2} \sin(2x) \)[/tex], we can see that the coefficient [tex]\( \frac{1}{2} \)[/tex] is in front of the sine function. This coefficient [tex]\( \frac{1}{2} \)[/tex] represents the amplitude of the function.
Therefore, the amplitude is:
[tex]\[
\boxed{\frac{1}{2}}
\][/tex]
### Period
The period of the sine function [tex]\( y = A \sin(Bx) \)[/tex] is determined by the value of [tex]\( B \)[/tex]. The period [tex]\( T \)[/tex] of a sine function is given by the formula:
[tex]\[
T = \frac{2\pi}{B}
\][/tex]
In our function, [tex]\( y = \frac{1}{2} \sin(2x) \)[/tex], we have [tex]\( B = 2 \)[/tex].
Substituting [tex]\( B = 2 \)[/tex] into the period formula, we get:
[tex]\[
T = \frac{2\pi}{2} = \pi
\][/tex]
Therefore, the period is:
[tex]\[
\boxed{\pi}
\][/tex]
In summary, for the function [tex]\( y = \frac{1}{2} \sin(2x) \)[/tex]:
- The amplitude is [tex]\( \boxed{\frac{1}{2}} \)[/tex].
- The period is [tex]\( \boxed{\pi} \)[/tex].
