What is the period of the function y equals 2 sine left parenthesis 3 x minus Start Fraction pi over 6 End Fraction right parenthesis plus 4?

Question

(1 point)
Responses

Start Fraction pi over 6 End Fraction
Image with alt text: Start Fraction pi over 6 End Fraction

one half
Image with alt text: one half

Start Fraction 2 pi over 3 End Fraction
Image with alt text: Start Fraction 2 pi over 3 End Fraction

π
π

2π

Answer 1

The period of a sine function \( y = a \sin(bx - c) + d \) is given by the formula:

\[ \text{Period} = \frac{2\pi}{|b|} \]

In the function you provided, \( y = 2 \sin(3x - \frac{\pi}{6}) + 4 \), the coefficient \( b \) is 3. Therefore, we can calculate the period as follows:

\[ \text{Period} = \frac{2\pi}{|3|} = \frac{2\pi}{3} \]

Thus, the period of the function is:

\[ \frac{2\pi}{3} \]

So the correct response is:

\(\frac{2\pi}{3}\)