To determine the characteristics of the trigonometric equation [tex]\(y = -7 - 9 \sin(x)\)[/tex], we need to identify the amplitude, period, and phase shift of the sine function involved.
### 1. Amplitude
The amplitude of a trigonometric function of the form [tex]\(y = A \sin(Bx + C) + D\)[/tex] is given by the coefficient [tex]\(A\)[/tex] of the sine term. In our equation, the sine coefficient is [tex]\(-9\)[/tex]. The amplitude is always taken as the absolute value of this coefficient.
[tex]\[
\text{Amplitude} = | -9 | = 9
\][/tex]
### 2. Period
The period of a sine function [tex]\(y = A \sin(Bx + C) + D\)[/tex] is determined by the coefficient [tex]\(B\)[/tex] in front of the [tex]\(x\)[/tex] inside the sine function. For a general sine function, the period is calculated as:
[tex]\[
\text{Period} = \frac{2\pi}{B}
\][/tex]
In our equation [tex]\(y = -7 - 9 \sin(x)\)[/tex], there is no coefficient in front of [tex]\(x\)[/tex] (i.e., [tex]\(B = 1\)[/tex]). Thus, the period is:
[tex]\[
\text{Period} = \frac{2\pi}{1} = 2\pi \approx 6.283185307179586
\][/tex]
### 3. Phase Shift
The phase shift of a sine function [tex]\(y = A \sin(Bx + C) + D\)[/tex] is given by [tex]\(\frac{-C}{B}\)[/tex]. In our equation, since there is no horizontal shift term ([tex]\(C = 0\)[/tex]), the phase shift is:
[tex]\[
\text{Phase Shift} = \frac{0}{1} = 0
\][/tex]
Thus, there is no phase shift in our trigonometric equation.
### Summary
- Amplitude: 9
- Period: approximately 6.283185307179586
- Phase Shift: no phase shift
### Final Answer
- Amplitude: [tex]\(9\)[/tex]
- Period: [tex]\(6.283185307179586\)[/tex]
- Phase Shift: no phase shift
