To determine the period of the sinusoidal function [tex]\( y = -4 \sin \left( \frac{2\pi}{3} x \right) \)[/tex], we need to understand the general form of a sinusoidal function and how to calculate its period.
The general form of a sinusoidal function is given by:
[tex]\[ y = A \sin(Bx + C) + D \][/tex]
where:
- [tex]\( A \)[/tex] is the amplitude,
- [tex]\( B \)[/tex] affects the period of the function,
- [tex]\( C \)[/tex] is the phase shift,
- and [tex]\( D \)[/tex] is the vertical shift.
For a sine function, the period is determined by the coefficient [tex]\( B \)[/tex] in the argument of the sine function. The period [tex]\( T \)[/tex] of a sinusoidal function [tex]\( y = A \sin(Bx + C) + D \)[/tex] is calculated using the formula:
[tex]\[ T = \frac{2\pi}{B} \][/tex]
In your given function:
[tex]\[ y = -4 \sin \left( \frac{2\pi}{3} x \right) \][/tex]
we can identify the value of [tex]\( B \)[/tex] as [tex]\( \frac{2\pi}{3} \)[/tex].
To find the period, we use the formula:
[tex]\[ T = \frac{2\pi}{B} \][/tex]
Substitute [tex]\( B = \frac{2\pi}{3} \)[/tex] into the formula:
[tex]\[ T = \frac{2\pi}{\frac{2\pi}{3}} \][/tex]
Simplify the fraction:
[tex]\[ T = \frac{2\pi \cdot 3}{2\pi} = 3 \][/tex]
Thus, the period of the sinusoidal function [tex]\( y = -4 \sin \left( \frac{2\pi}{3} x \right) \)[/tex] is:
[tex]\[ T = 3 \][/tex]
So, the period of the given sinusoidal function is [tex]\(\boxed{3}\)[/tex].
