To solve this problem, we are tasked with using the graph of $y = f(x)$ to graph the function $g(x) = -f(x) - 1$.

Step-by-Step Explanation:

1. Understanding the given graph:

   • The graph of $y = f(x)$ is a horizontal line segment from $x = -3$ to $x = 3$, lying on the line $y = -3$.

2. Transformation for $g(x) = -f(x) - 1$:

   • The function $g(x) = -f(x) - 1$ is obtained from $f(x)$ by applying two transformations:
     • Reflection across the x-axis: This changes the sign of $f(x)$, so the values of $y$ will become positive where they were negative, and vice versa. For the given graph, since $f(x) = -3$, $-f(x) = 3$ for the same $x$-values.
     • Vertical translation by -1: After reflecting, we shift the graph down by 1 unit. This changes the value of $y = 3$ to $y = 2$ for the entire segment.

3. Applying the transformation:

   • The original line segment from $x = -3$ to $x = 3$, which was at $y = -3$, will first reflect to $y = 3$, and then shift down by 1 unit to $y = 2$.

Matching the correct graph:

• Looking at the options, the correct graph will have a horizontal line segment from $x = -3$ to $x = 3$, positioned at $y = 2$.

• Correct answer: Option D is the graph that corresponds to this transformation.

Would you like more details, or have any questions?

Here are 5 related questions:

1. What other transformations can we apply to $f(x)$ besides reflection and translation?
2. How does vertical stretching or shrinking affect the graph of a function?
3. Can you explain horizontal translations and how they would modify $f(x)$?
4. What happens to the graph if we change the function to $g(x) = f(x) + 2$?
5. How would the graph look if we applied the transformation $g(x) = -f(x) + 1$?

Tip: When reflecting across the x-axis, just change the sign of all the $y$-coordinates without altering the $x$-coordinates.