To determine the vertical displacement for the graph of [tex]\( y = -2 - \cos(x - \pi) \)[/tex], we start by analyzing the given equation and comparing it to the basic form of a cosine function.
### Step-by-Step Solution:
1. Understand the Basic Function:
The basic form of a cosine function is [tex]\( y = \cos(x) \)[/tex].
2. Transformations Applied:
The given function is [tex]\( y = -2 - \cos(x - \pi) \)[/tex]. There are several transformations applied to the basic cosine function:
- Horizontal shift by [tex]\( \pi \)[/tex] units to the right, indicated by [tex]\( (x - \pi) \)[/tex].
- Reflection across the x-axis, indicated by the negative sign before the cosine function.
- Vertical shift downward.
3. Identify Vertical Shift:
The term affecting the vertical shift is the constant term outside of the cosine function.
In the given function [tex]\( y = -2 - \cos(x - \pi) \)[/tex], the constant term is [tex]\(-2\)[/tex]. This term represents a vertical displacement.
4. Conclusion:
The constant term [tex]\(-2\)[/tex] indicates that the entire graph of the function is shifted downward by 2 units from its original position.
So, the vertical displacement is:
[tex]\[
\text{The graph is shifted 2 units downward.}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{2 \text{ units down}}
\][/tex]
Given the options:
A. [tex]\( \pi \)[/tex] units down
B. [tex]\( \pi \)[/tex] units up
C. 2 units down
D. 3 units down
The correct choice is:
[tex]\[
\text{C. 2 units down}
\][/tex]
