To determine the amplitude, period, and phase shift of the trigonometric equation [tex]\( y = -2 - 8 \sin(x) \)[/tex], we need to analyze each part of the equation separately.
### Amplitude:
The amplitude of a sine function of the form [tex]\( y = a \sin(x) \)[/tex] is given by the absolute value of the coefficient in front of the sine function. Here, the equation is [tex]\( y = -2 - 8 \sin(x) \)[/tex]. The coefficient in front of [tex]\( \sin(x) \)[/tex] is -8. Therefore, the amplitude is the absolute value of -8, which is 8.
### Period:
The period of the sine function [tex]\( y = \sin(bx) \)[/tex] is calculated using the formula [tex]\( \frac{2\pi}{|b|} \)[/tex]. In the given function [tex]\( y = -2 - 8 \sin(x) \)[/tex], there is no coefficient in front of [tex]\( x \)[/tex] inside the sine function, meaning [tex]\( b = 1 \)[/tex]. Therefore, the period is [tex]\( \frac{2\pi}{1} = 2\pi \)[/tex].
### Phase Shift:
The phase shift of the sine function [tex]\( y = \sin(x - c) \)[/tex] is determined by the horizontal shift applied within the argument of the sine function. In the given equation [tex]\( y = -2 - 8 \sin(x) \)[/tex], there is no horizontal shift indicated within the sine function. Thus, the phase shift is zero, or we say there is no phase shift.
### Summary:
- Amplitude: 8
- Period: [tex]\( 2\pi \approx 6.283185307179586 \)[/tex]
- Phase Shift: No phase shift
So, we have:
- Amplitude: 8
- Period: [tex]\( 6.283185307179586 \)[/tex]
- Phase Shift: No phase shift
Correctly, we should select the following:
Amplitude: 8
Period: [tex]\( 6.283185307179586 \)[/tex]
Phase Shift: No phase shift
