To determine the amplitude, period, and phase shift of the trigonometric equation [tex]\( 4y = -2 \sin (7x + 6\pi) \)[/tex], let's rewrite it first to analyze the function.
We can start by isolating [tex]\( y \)[/tex]:
[tex]\[ y = -\frac{1}{2} \sin(7x + 6\pi) \][/tex]
Now, we can identify the amplitude, period, and phase shift from the general form of a sine function:
[tex]\[ y = A \sin(Bx + C) \][/tex]
### Amplitude
The amplitude is the coefficient in front of the sine function, which tells us how far the maximum and minimum values of the function are from the midline (in this case, the horizontal axis). For our function:
[tex]\[ y = -\frac{1}{2} \sin(7x + 6\pi) \][/tex]
The amplitude is the absolute value of the coefficient [tex]\(-\frac{1}{2} \)[/tex] in front of the sine function. Therefore, the amplitude is:
[tex]\[ \text{Amplitude} = |\frac{-2}{4}| = |-\frac{1}{2}| = \frac{1}{2} \][/tex]
### Period
The period of a sine function [tex]\( y = A \sin(Bx + C) \)[/tex] is given by the formula:
[tex]\[ \text{Period} = \frac{2\pi}{B} \][/tex]
In our function, [tex]\( B = 7 \)[/tex]. Thus, the period is:
[tex]\[ \text{Period} = \frac{2\pi}{7} \approx 0.8976 \][/tex]
### Phase Shift
The phase shift of the sine function [tex]\( y = A \sin(Bx + C) \)[/tex] is calculated as:
[tex]\[ \text{Phase shift} = -\frac{C}{B} \][/tex]
In our function, [tex]\( C = 6\pi \)[/tex] and [tex]\( B = 7 \)[/tex]. Therefore, the phase shift is:
[tex]\[ \text{Phase shift} = -\frac{6\pi}{7} \approx -2.6928 \][/tex]
Since the phase shift is negative, it indicates a shift to the left.
### Summary
- Amplitude: 2
- Period: 0.8976
- Phase Shift: shifted to the left by -2.6928 units
Therefore, the precise answers are:
- Amplitude: 2
- Period: 0.8975979010256552
- Phase Shift: shifted to the left by 2.6927937030769655 units
