https://file.aiquickdraw.com/mathbot/file/1726014912901kxjhdqw4.jpg

The graph provided contains two curves: a solid black graph and a dotted orange graph. Based on the structure of the question, we are trying to determine a function that represents the dotted graph based on the given function \( y = f(x) \), which is represented by the solid black graph.

### Observing the Graph:
1. **Solid Black Graph ( \( f(x) \) ):**
   - The solid black graph seems to start from the origin and increase in the positive direction for both \( x \) and \( y \). This suggests that \( f(x) \) could be a logarithmic or square root function. The behavior resembles a square root function, typically in the form \( y = \sqrt{x} \).

2. **Dotted Orange Graph:**
   - The dotted orange graph appears to be a reflection of the solid black graph over the \( x \)-axis. This indicates that the dotted graph could be the negative version of the solid graph. If the solid graph is \( y = f(x) \), the dotted graph likely represents \( y = -f(x) \).

### Conclusion:
If \( f(x) = \sqrt{x} \), then the dotted graph represents the function:
\[
y = -f(x) = -\sqrt{x}
\]

Would you like further details or clarification on this explanation?

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Here are 5 questions that expand on this topic:
1. What transformations would occur if the graph was reflected over the \( y \)-axis instead of the \( x \)-axis?
2. How would the graph change if we applied a vertical stretch to \( f(x) \)?
3. What happens to the graph if we add or subtract a constant to \( f(x) \)?
4. How would the graph behave if \( f(x) \) was squared (i.e., \( f(x)^2 \))?
5. Can the dotted graph be shifted horizontally or vertically to coincide with the solid black graph?

**Tip:** Reflections across the axes change the sign of either \( x \) or \( y \), depending on which axis the reflection occurs over.

The graph of y = f(x) is the solid black graph below. Which function represents the dotted graph?

Explore how the graph of y = f(x) reflects over the x-axis. Understand the transformations and identify the corresponding function for the dotted graph, suitable for students in Grades 9-12.

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