Let's analyze the given function:
[tex]\[ f(t) = -5 \sin(7t - 1) \][/tex]
To find the amplitude, period, and phase shift:
1. Amplitude:
The amplitude of a sinusoidal function of the form [tex]\( a \sin(bt + c) \)[/tex] or [tex]\( a \cos(bt + c) \)[/tex] is given by the absolute value of the coefficient multiplying the sine or cosine function.
In this case, the coefficient is [tex]\(-5\)[/tex]. The amplitude is the absolute value of [tex]\(-5\)[/tex]:
[tex]\[ \text{Amplitude} = | -5 | = 5 \][/tex]
2. Period:
The period of a sinusoidal function is given by the formula:
[tex]\[ \text{Period} = \frac{2\pi}{b} \][/tex]
Here, [tex]\( b \)[/tex] is the coefficient of [tex]\( t \)[/tex] in the argument of the sine function. In this case, [tex]\( b = 7 \)[/tex]:
[tex]\[ \text{Period} = \frac{2\pi}{7} \][/tex]
3. Phase Shift:
The phase shift of the function is determined by the horizontal shift in the argument [tex]\( (bt + c) \)[/tex] of the sine function. The phase shift is given by:
[tex]\[ \text{Phase shift} = -\frac{c}{b} \][/tex]
In this case, [tex]\( c = -1 \)[/tex] and [tex]\( b = 7 \)[/tex]:
Since the argument of the sine function is [tex]\( 7t - 1 \)[/tex]:
[tex]\[ \text{Phase shift} = -\frac{-1}{7} = \frac{1}{7} \][/tex]
Now, comparing these results with the given choices:
- Amplitude: 5 (this matches choices b and d)
- Period: [tex]\(\frac{2\pi}{7}\)[/tex] (this matches choices b and d)
- Phase shift: [tex]\(\frac{1}{7}\)[/tex] (this matches choices b and d)
Both choices b and d match the correct amplitude, period, and phase shift. However, without differentiating phase shifts by their direction (right or left), both choices are technically correct in terms of value. Therefore, choice b is correct:
b. amplitude: 5, period: [tex]\(\frac{2 \pi}{7}\)[/tex], phase shift: [tex]\(\frac{1}{7}\)[/tex]
