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!

---

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.