To determine the amplitude, period, and phase shift of the trigonometric function given by the equation [tex]\( y = -8\sin(7x) \)[/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 in front of the sine function, which is [tex]\( |a| \)[/tex].
Here, [tex]\( a = -8 \)[/tex]. Therefore, the amplitude is:
[tex]\[ \text{Amplitude} = | -8 | = 8 \][/tex]
### 2. Period
The period of a sine function [tex]\( y = a \sin(bx + c) \)[/tex] is determined by the coefficient [tex]\( b \)[/tex] in front of the [tex]\( x \)[/tex]-term inside the sine function. The period [tex]\( T \)[/tex] is calculated using the formula:
[tex]\[ T = \frac{2\pi}{b} \][/tex]
In this function, [tex]\( b = 7 \)[/tex]. Therefore, the period is:
[tex]\[ \text{Period} = \frac{2\pi}{7} \approx 0.8975979010256552 \][/tex]
### 3. Phase Shift
The phase shift of a sine function [tex]\( y = a \sin(bx + c) \)[/tex] is given by the value [tex]\( \frac{-c}{b} \)[/tex].
In the given equation [tex]\( y = -8 \sin(7x) \)[/tex], the [tex]\( c \)[/tex] value is 0 because there is no horizontal shift indicated within the argument of the sine function. Hence, the phase shift is:
[tex]\[ \text{Phase Shift} = \frac{0}{7} = 0 \][/tex]
This means the function is not shifted to the left or the right.
Summary:
- Amplitude: 8
- Period: approximately [tex]\( 0.8975979010256552 \)[/tex]
- Phase Shift: 0 (indicating no phase shift)
