Graph the following function.

y equals cosine left parenthesis x plus StartFraction pi Over 12 EndFraction right parenthesisy=cosx+π12

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Part 1

Use the graphing tool to graph the function. 

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graph

​(For any answer boxes shown with the​ grapher, type an exact answer. Type the word pi to insert the symbol 

piπ 

as​ needed.) give me phase shift as a numbe

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.

Graph the following function. y = cos(x + π/12). Use the graphing tool to graph the function. Give me the phase shift as a number.

Learn how to graph the function y = cos(x + π/12) and find its phase shift. This problem covers key trigonometric concepts, including phase shifts, graphing cosine functions, and understanding function transformations. Suitable for students in Grades 10-12.

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