To determine the amplitude, period, and phase shift of the trigonometric equation [tex]\( y = \frac{1}{2} \sin(x + 6) \)[/tex], let's break it down step by step:
### Amplitude
The amplitude of a sine function [tex]\( y = A \sin(Bx + C) \)[/tex] is given by the absolute value of the coefficient [tex]\( A \)[/tex] in front of the [tex]\( \sin \)[/tex] function.
For the given equation [tex]\( y = \frac{1}{2} \sin(x + 6) \)[/tex]:
- The coefficient [tex]\( A \)[/tex] is [tex]\( \frac{1}{2} \)[/tex].
Therefore, the amplitude is:
[tex]\[
\text{Amplitude} = \left| \frac{1}{2} \right| = 0.5
\][/tex]
### Period
The period of a sine function [tex]\( y = A \sin(Bx + C) \)[/tex] is determined by the coefficient [tex]\( B \)[/tex] in front of [tex]\( x \)[/tex]. The period is calculated using the formula:
[tex]\[
\text{Period} = \frac{2\pi}{B}
\][/tex]
For the given equation [tex]\( y = \frac{1}{2} \sin(x + 6) \)[/tex]:
- The coefficient [tex]\( B \)[/tex] is [tex]\( 1 \)[/tex] since there is no coefficient explicitly written in front of [tex]\( x \)[/tex], so it is assumed to be [tex]\( 1 \)[/tex].
Therefore, the period is:
[tex]\[
\text{Period} = \frac{2\pi}{1} = 2\pi \approx 6.283185307179586
\][/tex]
### Phase Shift
The phase shift of a sine function [tex]\( y = A \sin(Bx + C) \)[/tex] is determined by the constant [tex]\( C \)[/tex] inside the sine function and is calculated using the formula:
[tex]\[
\text{Phase Shift} = -\frac{C}{B}
\][/tex]
For the given equation [tex]\( y = \frac{1}{2} \sin(x + 6) \)[/tex]:
- The constant [tex]\( C \)[/tex] is [tex]\( 6 \)[/tex]
- The coefficient [tex]\( B \)[/tex] is [tex]\( 1 \)[/tex]
Therefore, the phase shift is:
[tex]\[
\text{Phase Shift} = -\frac{6}{1} = -6
\][/tex]
This indicates a shift to the left by 6 units.
### Conclusion
Based on the above calculations, the characteristics of the trigonometric equation [tex]\( y = \frac{1}{2} \sin(x + 6) \)[/tex] are:
- Amplitude: 0.5
- Period: [tex]\( 2\pi \approx 6.283185307179586 \)[/tex]
- Phase Shift: shifted to the left by 6 units
So, the answers are:
Amplitude: [tex]\( 0.5 \)[/tex]
Phase Shift: shifted to the left
