https://file.aiquickdraw.com/mathbot/file/1733510627787zna83qh9.jpg

The problem involves determining the function that represents the dotted graph, based on the transformations applied to the original function \( y = f(x) \), represented by the solid black graph.

### Analysis:
The dotted graph appears to be a horizontally shifted and reflected version of the original graph.

1. **Original function** (\( y = f(x) \)):
   - The black graph resembles the quadratic function \( y = x^2 \), but we don't explicitly need the form of \( f(x) \). It's sufficient to compare the transformations between the solid and dotted graphs.

2. **Transformation applied**:
   - The dotted graph has been **reflected horizontally** across the \( y \)-axis.
   - It has also been **shifted horizontally** to the left by 3 units.

### Transformed function:
A horizontal reflection involves replacing \( x \) with \( -x \). A horizontal shift to the left by 3 involves replacing \( x \) with \( x + 3 \). Combining these gives the transformation:
\[
y = f(-(x + 3)).
\]

Thus, the function representing the dotted graph is:
\[
y = f(-x - 3).
\]

Let me know if you need further details! Here are some follow-up questions to deepen understanding:

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### Relative Questions:
1. What does a horizontal reflection do to the graph of any function?
2. How does a shift of \( x \) to the left differ from a shift to the right?
3. Can you derive the general rule for combining multiple transformations of functions?
4. How does replacing \( x \) with \( -x \) affect the symmetry of the graph?
5. Could vertical transformations also apply to this graph? How would they look?

### Tip:
Always apply transformations step-by-step and in the correct order: reflections, scaling, shifts. This avoids errors when combining multiple transformations.

Determine the function that represents the dotted graph, based on the transformations applied to the original function y = f(x), represented by the solid black graph.

Learn how to determine the function representing a graph after applying transformations, including a horizontal reflection and a shift to the left by 3 units. This solution focuses on the transformations of quadratic functions, suitable for students in Grades 8-10.

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