https://file.aiquickdraw.com/mathbot/file/17287947364409hbgnlnv.jpg

The contour plot in the image represents a 3D surface using contour lines that connect points of equal value (height). The shading also gives an indication of the surface's height, where lighter colors correspond to higher values and darker colors to lower values. The contour levels are marked with values on the color bar on the right side of the plot.

You are asked to identify which graph corresponds to the cross-section of this surface at \( y = -2 \).

### Steps to Approach:
1. **Locate the line \( y = -2 \)**: This is a horizontal slice of the plot at \( y = -2 \). You can visually follow this line across the \( x \)-axis to observe the contour behavior along it.
  
2. **Examine the contours at \( y = -2 \)**: 
   - The contours around \( y = -2 \) are fairly symmetrical, and as you go from left to right across the plot (along the \( x \)-axis), you observe how the values change.
   - In the range \( x = -4 \) to \( x = 4 \), the contour lines at \( y = -2 \) are spread out, indicating a smoother gradient of change, with deeper (more negative) values in the center and higher values near the edges.

3. **Cross-section characteristics**: 
   - At \( x = 0 \), the contour value is near its minimum, suggesting the surface is at its lowest point (dark blue region).
   - Moving outward from \( x = 0 \), the values increase as indicated by the contour lines moving toward lighter colors, indicating a rising surface.

This information should help identify the graph corresponding to this cross-section. 

Would you like more detailed analysis or have any specific graphs in mind to compare?

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**Here are 5 follow-up questions you could explore**:
1. How do contour plots reflect gradient changes across a surface?
2. What is the significance of contour line spacing on such plots?
3. How can you interpret 3D surface shapes based on contour patterns?
4. What are the general methods for slicing 3D surfaces to obtain 2D cross-sections?
5. How does the color gradient relate to the values on the surface?

**Tip**: Contour plots are powerful tools to visualize 3D surfaces in 2D. The closer the lines are, the steeper the surface is at that region!

Which one of the following graphs corresponds to the cross-section of this surface through y = -2?

Learn how to identify the cross-section of a 3D surface represented by a contour plot. This problem explores the process of analyzing contour lines to find corresponding values along a horizontal slice at y = -2, suitable for high school math students.

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