Question

What is the period of the function f left parenthesis x right parenthesis equals sine left parenthesis Start Fraction x over 2 End Fraction plus Start Fraction pi over 6 End Fraction right parenthesis plus 1?
(1 point)
Responses

π
π

2π
2 π

4π
4 π

12π

Answer 1

To find the period of the function \( f(x) = \sin\left(\frac{x}{2} + \frac{\pi}{6}\right) + 1 \), we focus on the sine part of the function.

The general form of a sine function is:

\[
\sin(kx)
\]

where the period of \( \sin(kx) \) is given by \( \frac{2\pi}{|k|} \).

In your function, the argument of the sine is \( \frac{x}{2} + \frac{\pi}{6} \). The coefficient of \( x \) here is \( \frac{1}{2} \), so:

\[
k = \frac{1}{2}
\]

Thus, the period \( P \) of the sine function is:

\[
P = \frac{2\pi}{\left|\frac{1}{2}\right|} = \frac{2\pi}{\frac{1}{2}} = 2\pi \times 2 = 4\pi
\]

The \( \frac{\pi}{6} \) added inside the sine function is a phase shift, which does not affect the period.

Since the remaining part of the function, \( +1 \), is a vertical shift and does not affect the period either, the period of the entire function \( f(x) \) is still \( 4\pi \).

Thus, the period of the function \( f(x) \) is:

\[
\boxed{4\pi}
\]