To determine which graph represents the function $f(x) = -3 - \ln(x+3)$, let's analyze the properties and characteristics of this function.

Steps to analyze the function:

1. Vertical Asymptote: The natural logarithm function $\ln(x+3)$ is undefined when $x+3 \leq 0$. Therefore, $x = -3$ is a vertical asymptote.
2. Behavior as $x \to -3^+$: As $x$ approaches $-3$ from the right ($x \to -3^+$), $\ln(x+3) \to -\infty$. Therefore, $f(x) \to -3 - (-\infty) = \infty$.
3. Behavior as $x \to \infty$: As $x$ approaches $\infty$, $\ln(x+3) \to \infty$. Therefore, $f(x) \to -3 - \infty = -\infty$.
4. Intercepts: To find the $y$-intercept, set $x = 0$: $f(0) = -3 - \ln(3)$ Since $\ln(3)$ is a positive number, $f(0)$ will be less than $-3$.

Checking the graphs:

• Graph A: Does not match the vertical asymptote at $x = -3$.
• Graph B: Has a vertical asymptote at $x = -3$, but the behavior as $x \to \infty$ is not correct.
• Graph C: Does not match the vertical asymptote at $x = -3$.
• Graph D: Matches the vertical asymptote at $x = -3$ and the behavior as $x \to \infty$.

Conclusion:

The graph that represents the function $f(x) = -3 - \ln(x+3)$ is:

Graph D

Would you like more details or have any questions? Here are some other questions you might want to ask:

1. How to graph a natural logarithmic function with transformations?
2. How to determine vertical asymptotes of logarithmic functions?
3. How to find the domain and range of a logarithmic function?
4. What are common properties of logarithmic functions?
5. How to calculate intercepts of logarithmic functions?

Tip: Remember that transformations like translations and reflections significantly alter the graph's shape and position, but vertical asymptotes remain key indicators of logarithmic behavior.