To solve for the amplitude, period, and phase shift of the trigonometric equation [tex]\( y = -8 - 4 \sin(9x + 6) \)[/tex], follow these steps:
1. Amplitude:
- The amplitude of a sine function [tex]\( y = A \sin(Bx + C) \)[/tex] is given by the absolute value of the coefficient of the sine function, [tex]\( |A| \)[/tex].
- In the equation [tex]\( y = -8 - 4 \sin(9x + 6) \)[/tex], the coefficient of the sine function is [tex]\(-4\)[/tex].
- Therefore, the amplitude is [tex]\( | -4 | = 4 \)[/tex].
2. Period:
- The period of a sine function [tex]\( y = \sin(Bx + C) \)[/tex] is given by [tex]\( \frac{2\pi}{B} \)[/tex], where [tex]\( B \)[/tex] is the coefficient of [tex]\( x \)[/tex] inside the sine function.
- In the equation [tex]\( y = -8 - 4 \sin(9x + 6) \)[/tex], the coefficient [tex]\( B \)[/tex] is [tex]\( 9 \)[/tex].
- Therefore, the period is [tex]\( \frac{2\pi}{9} \approx 0.698 \)[/tex].
3. Phase Shift:
- The phase shift of a sine function [tex]\( y = \sin(Bx + C) \)[/tex] is given by [tex]\( -\frac{C}{B} \)[/tex], where [tex]\( C \)[/tex] is the constant term added to [tex]\( Bx \)[/tex].
- In the equation [tex]\( y = -8 - 4 \sin(9x + 6) \)[/tex], [tex]\( B = 9 \)[/tex] and [tex]\( C = 6 \)[/tex].
- Therefore, the phase shift is [tex]\( -\frac{6}{9} = -\frac{2}{3} \)[/tex].
- A negative phase shift means the graph is shifted to the left.
Summarizing:
- Amplitude: [tex]\( 4 \)[/tex]
- Period: [tex]\( \approx 0.698 \)[/tex] (or [tex]\( \frac{2\pi}{9} \)[/tex])
- Phase Shift: shifted to the left by [tex]\( \frac{2}{3} \)[/tex] units
So the complete answer is:
- Amplitude: [tex]\(4\)[/tex]
- Phase Shift: shifted to the left
