The contour plot displayed represents a surface, and the problem asks for a cross-section through $x = -1$. To determine which graph corresponds to this cross-section, we need to analyze the surface at this particular $x$-value.

Here's the general approach:

1. Contour Plot: This plot shows level curves (contours) where the surface has constant height (or value) at different points in the $x, y$-plane. Each contour represents a specific value of the function (seen in the color scale to the right).
2. Cross-section: A cross-section through $x = -1$ means we fix $x = -1$ and examine how the height (or value) of the function changes with $y$. Essentially, this is a 2D slice where the surface is reduced to a line plot of $f(-1, y)$.

Analyzing $x = -1$:

• On the plot, $x = -1$ corresponds to the vertical line at $x = -1$, which cuts through the contours.
• To determine the behavior of the function along this slice, observe how the colors and contour lines change as you move up or down along $y$ at $x = -1$.
• At $y = 0$, near $x = -1$, the contours suggest a minimum, since the contour lines are closely packed, indicating a dip in the surface. Moving up or down in $y$, the surface rises slightly and then falls again (indicated by the contour pattern).

Thus, the graph of this cross-section should reflect this "dip" at $y = 0$ with a symmetric rise and fall around it.

Let me know if you'd like more details, or have any questions about the steps. Here are 5 related questions to expand on this topic:

1. What is the significance of closely spaced contours in a contour plot?
2. How can we compute the function values for specific points on a contour plot?
3. How would the cross-section change if we took $x = 0$ instead of $x = -1$?
4. What types of surfaces produce symmetric cross-sections like this one?
5. How would a saddle point appear in a contour plot and its cross-section?

Tip: In a contour plot, when the lines are closer together, the surface is steeper, while widely spaced lines indicate a flatter region.