https://file.aiquickdraw.com/mathbot/file/1727032697385t8p2jp1q.jpg

Let's break down the given function:

The function is \( f(x) = \log_2(-x) - 2 \).

### Key features:
1. **Logarithmic Nature**: The logarithmic function \( \log_2(-x) \) is defined only when \( -x > 0 \), i.e., when \( x < 0 \). This restricts the domain to negative values of \( x \).
2. **Vertical Shift**: The "-2" in \( \log_2(-x) - 2 \) means that the entire graph is shifted down by 2 units.
3. **Asymptotes**: The function \( \log_2(-x) \) has a vertical asymptote at \( x = 0 \) (since \( \log_2(-x) \) becomes undefined at \( x = 0 \)).

### Behavior:
- The graph is only defined for \( x < 0 \).
- There is a vertical asymptote at \( x = 0 \), meaning the graph will approach negative infinity as \( x \) approaches 0 from the left.
- The entire graph is shifted downward by 2 units because of the "-2".

### Graph Selection:
- The correct graph should:
  - Only exist on the negative side of the x-axis.
  - Have a vertical asymptote at \( x = 0 \).
  - Be shifted downward by 2 units.

Looking at the graphs provided, the **third graph** (from top to bottom) fits this description: it is on the negative side, has an asymptote at \( x = 0 \), and the shift downwards by 2 is visible.

Let me know if you would like any further explanation!

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Here are 5 relative questions:
1. How would the graph change if we had \( f(x) = \log_2(x) - 2 \)?
2. What happens if we change the base of the logarithm, such as \( f(x) = \log_3(-x) - 2 \)?
3. How would the graph behave if the vertical shift was positive, like \( f(x) = \log_2(-x) + 2 \)?
4. What is the domain and range of \( f(x) = \log_2(-x) - 2 \)?
5. How can we determine the asymptotes of a logarithmic function?

**Tip**: When analyzing transformations of functions, always consider shifts (vertical and horizontal) and stretches to understand the overall change in the graph.

Find the graph of the function: f(x) = log_2(-x) - 2

Learn how to identify the correct graph for the function f(x) = log_2(-x) - 2. This problem covers logarithmic functions, graph transformations, and asymptotes. Suitable for students in Grades 9-12.

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