https://file.aiquickdraw.com/mathbot/file/1726333642688dbefhhy6.jpg

The graph shown in the image is a function that displays symmetry about the y-axis and several oscillatory features, which suggests that the function depends on the absolute value of \(x\) and involves exponential decay or growth. Let's analyze the options based on the visual clues:

### Key features from the graph:
1. The function appears to exhibit exponential decay as \(x\) increases in both directions from the origin.
2. The graph is symmetric about the y-axis, suggesting the involvement of \(|x|\).
3. There are dips and peaks near \(x = 0\), indicating oscillatory behavior likely modulated by an exponential term.

### Evaluation of each option:
- **Option (A): \( f(x) = x^2 \cdot 2^{-|x|} \)**
   - This function includes \(x^2\), which is symmetric about the y-axis. 
   - The factor \(2^{-|x|}\) introduces exponential decay for increasing \(|x|\).
   - This matches the symmetry and decaying pattern in the graph, along with the dip near \(x = 0\).

- **Option (B): \( f(x) = x \cdot 2^{-|x|} \)**
   - This function has the term \(x\), which would result in a non-symmetric graph about the y-axis, since \(x\) changes sign on different sides of the y-axis.
   - This does not match the observed symmetry in the graph.

- **Option (C): \( f(x) = |x| \cdot 2^{-|x|} \)**
   - This function involves \(|x|\), maintaining the symmetry about the y-axis.
   - However, the graph does not seem to have a simple linear behavior for small values of \(|x|\); instead, there are peaks and dips.
   - This option seems close but does not fully explain the oscillations.

- **Option (D): \( f(x) = x \cdot 2^{-x} \)**
   - This function is neither symmetric about the y-axis nor does it account for oscillatory behavior.
   - It can be immediately ruled out based on the lack of symmetry.

### Conclusion:
The graph best matches **Option (A)**, \( f(x) = x^2 \cdot 2^{-|x|} \), as it explains the symmetry, the exponential decay, and the oscillatory dips near \(x = 0\).

Would you like further clarification or details?

### Related Questions:
1. How does the behavior of \( f(x) = x^2 \cdot 2^{-|x|} \) change as \(x\) approaches infinity?
2. How does the absolute value function affect the symmetry of functions like \( f(x) \)?
3. What is the role of exponential decay in functions involving \( 2^{-|x|} \)?
4. How does squaring the \(x\) term influence the shape of the graph near \(x = 0\)?
5. How do functions with different bases for exponential terms (like 2 or \(e\)) affect the graph's shape?

### Tip:
When analyzing graphs for symmetry, always check if the function involves absolute values or even powers of \(x\). These ensure the graph is symmetric about the y-axis.

Which one of the following options represents the given graph?

This problem involves analyzing the graph of an exponential function with symmetry about the y-axis. The function features both exponential decay and oscillatory behavior, requiring knowledge of absolute value functions and exponential growth and decay patterns. Suitable for undergraduate-level students.

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