To find the period of the sinusoid given by the equation [tex]\( y = -2 \sin \left(\frac{2 \pi}{7} x\right) \)[/tex], we can follow these steps:
### Step 1: Identify the general form of the sinusoidal function
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,
- [tex]\( D \)[/tex] is the vertical shift.
### Step 2: Determine the value of [tex]\( B \)[/tex]
For the given equation [tex]\( y = -2 \sin \left(\frac{2 \pi}{7} x\right) \)[/tex], we can compare it with the general form to identify [tex]\( B \)[/tex]. Here, we see that:
[tex]\[ B = \frac{2 \pi}{7} \][/tex]
### Step 3: Calculate the period
The period [tex]\( T \)[/tex] of a sinusoidal function [tex]\( y = \sin(Bx) \)[/tex] is given by:
[tex]\[ T = \frac{2 \pi}{B} \][/tex]
### Step 4: Substitute the value of [tex]\( B \)[/tex]
Using the identified value of [tex]\( B \)[/tex]:
[tex]\[ T = \frac{2 \pi}{\frac{2 \pi}{7}} \][/tex]
### Step 5: Simplify the expression
Simplifying the fraction inside the expression for the period:
[tex]\[ T = \frac{2 \pi}{\frac{2 \pi}{7}} = \frac{2 \pi \cdot 7}{2 \pi} = 7 \][/tex]
### Final Answer
The period of the sinusoid [tex]\( y = -2 \sin \left(\frac{2 \pi}{7} x\right) \)[/tex] is [tex]\( \boxed{7} \)[/tex].
