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Identify the transformations for the following transformed cos function that is reflected over the x-axis:

Period =

Answer 2 Question 2
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Vertical Shift =

Answer 3 Question 2
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Phase Shift =

Answer 4 Question 2
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Let's analyze the provided graph to identify the transformations applied to the basic cosine function \( y = \cos(x) \).

### 1. **Reflection over the x-axis:**
   - The graph appears to be reflected over the x-axis, so the base function \( y = \cos(x) \) changes to \( y = -\cos(x) \).

### 2. **Period:**
   - The period of the cosine function is the length of one complete cycle.
   - Normally, the period of \( y = \cos(x) \) is \( 2\pi \).
   - Here, the graph completes a full cycle from \( 0 \) to \( 2\pi \), so the period remains \( 2\pi \).

### 3. **Vertical Shift:**
   - The midline of the graph is at \( y = -1 \), instead of \( y = 0 \), indicating a vertical shift.
   - The vertical shift is downward by 1 unit.

### 4. **Phase Shift:**
   - The phase shift can be determined by where the graph starts relative to \( x = 0 \).
   - The graph appears to start a complete cycle at \( \frac{\pi}{2} \) to the right.
   - So, there is a phase shift of \( \frac{\pi}{2} \) to the right.

### Summary of Transformations:
- **Period:** \( 2\pi \)
- **Vertical Shift:** Down by 1 unit
- **Phase Shift:** \( \frac{\pi}{2} \) to the right

If you have any further questions or need additional details, feel free to ask!

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**Related Questions:**

1. How would the equation of the function change if the graph were shifted upward by 2 units instead of downward by 1?
2. What would be the period if the graph completed one cycle from 0 to \( \pi \)?
3. How would the graph look if the phase shift was \( \pi \) to the left instead of \( \frac{\pi}{2} \) to the right?
4. If the function was reflected over the y-axis instead of the x-axis, how would the graph change?
5. How would the equation be modified if the graph were stretched vertically by a factor of 2?

**Tip:** Always check the amplitude, period, and phase shift when analyzing trigonometric graphs, as these are key to understanding the transformations applied.

Learn how to identify transformations of a reflected cosine function over the x-axis. Understand concepts such as period, vertical shift, and phase shift through detailed analysis and examples. Suitable for high school students studying trigonometry and function transformations.

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