The function you're working with is:

$y = \cos\left(x + \frac{\pi}{12}\right)$

Step-by-step breakdown:

1. Standard Cosine Function: The basic form of the cosine function is $y = \cos(x)$, which has a period of $2\pi$, an amplitude of 1, and no phase shift.

2. Phase Shift: The phase shift is determined by the horizontal translation of the function. The phase shift formula for a cosine function in the form $y = \cos(x + C)$ is given by: $\text{Phase shift} = -\frac{C}{B}$ where $C$ is the constant added inside the cosine function, and $B$ is the coefficient of $x$. For your function $y = \cos\left(x + \frac{\pi}{12}\right)$, $C = \frac{\pi}{12}$ and $B = 1$.

   So, the phase shift is: $\text{Phase shift} = -\frac{\frac{\pi}{12}}{1} = -\frac{\pi}{12}$

   Therefore, the phase shift is $-\frac{\pi}{12}$, meaning the graph is shifted to the left by $\frac{\pi}{12}$.

Would you like me to provide a graph or more details about the graph, such as amplitude, period, and vertical shift? Let me know if you want additional information or have further questions!

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Related Questions:

1. How does changing the amplitude affect the graph of a cosine function?
2. What is the period of a cosine function, and how can it be altered?
3. How do you find the vertical shift of a transformed trigonometric function?
4. How does the phase shift impact the starting point of a cosine function?
5. Can a cosine function have a negative amplitude, and how does that affect the graph?

Tip:

For trigonometric functions, always examine the coefficients and constants inside the function to determine key transformations like amplitude, period, and phase shift.