consider the function $y = \\cos(x+\frac{\\pi}{6})$. which of the following is true? there is a phase shift…

consider the function $y = \\cos(x+\frac{\\pi}{6})$. which of the following is true? there is a phase shift to the left. there is a phase shift to the right. the graph is reflected across the x - axis. the graph is stretched horizontally. the graph is compressed horizontally.

consider the function $y = \\cos(x+\frac{\\pi}{6})$. which of the following is true? there is a phase shift to the left. there is a phase shift to the right. the graph is reflected across the x - axis. the graph is stretched horizontally. the graph is compressed horizontally.

Answer

Explanation:

Step1: Recall cosine function form

The general form of a cosine function is $y = A\cos(B(x - C))+D$. In the given function $y=\cos(x+\frac{\pi}{6})$, we can rewrite it as $y=\cos(x - (-\frac{\pi}{6}))$, where $C =-\frac{\pi}{6}$.

Step2: Analyze phase - shift

The value of $C$ determines the phase - shift of the cosine function. If $C>0$, the graph shifts to the right by $C$ units, and if $C < 0$, the graph shifts to the left by $|C|$ units. Since $C=-\frac{\pi}{6}<0$, the graph of $y = \cos(x)$ shifts to the left by $\frac{\pi}{6}$ units.

Answer:

There is a phase shift to the left.