Questions: Determine the phase shift of the sine function: (a) sin(x + π/3) The phase shift is and the graph is shifted to the (Choose one).

Transcript text: Determine the phase shift of the sine function: (a) $\sin \left(x+\frac{\pi}{3}\right)$ The phase shift is $\square$ and the graph is shifted to the (Choose one) .

Solution

Solution Steps

To determine the phase shift of the sine function, we need to identify the horizontal shift in the argument of the sine function. The general form of a sine function with a phase shift is $ \sin(x - C) $, where $ C $ is the phase shift. If the argument is $ x + \frac{\pi}{3} $, it can be rewritten as $ x - (-\frac{\pi}{3}) $, indicating a phase shift of $ -\frac{\pi}{3} $. This means the graph is shifted to the left by $ \frac{\pi}{3} $.

Step 1: Identify the Phase Shift

The given function is $ \sin \left(x + \frac{\pi}{3}\right) $. To find the phase shift, we rewrite the argument in the form $ x - C $. This gives us: \[ x + \frac{\pi}{3} = x - \left(-\frac{\pi}{3}\right) \] From this, we can see that the phase shift $ C $ is $ -\frac{\pi}{3} $.

Step 2: Determine the Direction of the Shift

Since the phase shift is negative, it indicates that the graph of the sine function is shifted to the left. Therefore, the direction of the shift is "left".