To analyze the trigonometric equation [tex]\( y = 3 - 8 \sin(x) \)[/tex], we need to determine its amplitude, period, and phase shift.
### Amplitude
The amplitude of a sine function [tex]\( y = A \sin(Bx - C) + D \)[/tex] is given by the coefficient in front of the sine term, [tex]\( A \)[/tex]. In this equation, [tex]\( y = 3 - 8 \sin(x) \)[/tex], the sine term is [tex]\( -8 \sin(x) \)[/tex], and thus the coefficient is 8. Therefore, the amplitude is:
[tex]\[ \boxed{8} \][/tex]
### Period
The period of a sine function is determined by the coefficient [tex]\( B \)[/tex] in the term [tex]\( Bx \)[/tex] inside the sine function. For a general sine function [tex]\( y = A \sin(Bx - C) + D \)[/tex], the period is calculated as:
[tex]\[ \text{Period} = \frac{2\pi}{B} \][/tex]
In this case, the coefficient [tex]\( B \)[/tex] in front of [tex]\( x \)[/tex] is 1, so the period is:
[tex]\[ \text{Period} = \frac{2\pi}{1} = 2\pi \][/tex]
Therefore, the period of the function is:
[tex]\[ \boxed{6.283185307179586} \][/tex]
### Phase Shift
The phase shift of a sine function [tex]\( y = A \sin(Bx - C) + D \)[/tex] is determined by the term [tex]\( \frac{C}{B} \)[/tex]. If there is no term [tex]\( C \)[/tex] to shift the function horizontally, then there is no phase shift. In this function [tex]\( y = 3 - 8 \sin(x) \)[/tex], there is no horizontal shift term inside the sine function.
Thus, the phase shift is:
[tex]\[ \boxed{\text{no phase shift}} \][/tex]
So, the complete analysis of the function [tex]\( y = 3 - 8 \sin(x) \)[/tex] gives us:
- Amplitude: [tex]\( \boxed{8} \)[/tex]
- Period: [tex]\( \boxed{6.283185307179586} \)[/tex]
- Phase Shift: [tex]\( \boxed{\text{no phase shift}} \)[/tex]
