Question

Which of the following correctly describes the phase shift of the function f(x)=2 sin(x+π/6)?

Answer 1

The function \( f(x) = 2 \sin(x + \frac{\pi}{6}) \) can be analyzed for phase shift by examining its argument.

In the standard form of a sine function, \( f(x) = a \sin(bx - c) \), the phase shift is given by \( \frac{c}{b} \).

For the function \( f(x) = 2 \sin(x + \frac{\pi}{6}) \), we can rewrite it as:

\[
f(x) = 2 \sin\left(1 \cdot x + \frac{\pi}{6}\right) = 2 \sin\left(1 \cdot x - \left(-\frac{\pi}{6}\right)\right)
\]

Here, \( b = 1 \) and \( c = -\frac{\pi}{6} \).

To find the phase shift, we calculate:

\[
\text{Phase shift} = \frac{c}{b} = \frac{-\frac{\pi}{6}}{1} = -\frac{\pi}{6}
\]

A negative phase shift means the graph is shifted to the left by \( \frac{\pi}{6} \).

So, the correct phase shift for the function \( f(x) = 2 \sin(x + \frac{\pi}{6}) \) is \( \frac{\pi}{6} \) units to the left.