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Let's analyze the function \( f(x) = -2 \log_3 (x + 4) \) and fill in the blanks in the statement provided.

1. **Type of Function**: This is a logarithmic function because it involves a logarithm, specifically \( \log_3(x + 4) \).

2. **Asymptote**: Logarithmic functions generally have a vertical asymptote where the argument of the logarithm is zero. In this case:
   \[
   x + 4 = 0 \Rightarrow x = -4
   \]
   So, the vertical asymptote is at \( x = -4 \).

3. **Range**: Since the logarithmic function here is scaled and reflected vertically (due to the \(-2\) coefficient), it will produce all real values as \( f(x) \). Thus, the range is \( (-\infty, \infty) \).

4. **Monotonicity**: The function has a negative coefficient in front of the logarithmic term, which means it will be **decreasing** over its domain. The function decreases as \( x \) increases from left to right.

5. **Domain**: The domain of the function is where the argument of the logarithm is positive:
   \[
   x + 4 > 0 \Rightarrow x > -4
   \]
   So, the domain is \( (-4, \infty) \).

6. **End Behavior**:
   - **Left Side (approaching \( x = -4 \) from the right)**: As \( x \) approaches \(-4\) from the right, \( f(x) \to \infty \).
   - **Right Side (as \( x \to \infty \))**: As \( x \to \infty \), \( f(x) \to -\infty \) because of the negative scaling.

### Filling in the Blanks
Using this information, we can fill in the answers:

- **Function Type**: Logarithmic
- **Asymptote**: Vertical, \( x = -4 \)
- **Range**: \( (-\infty, \infty) \)
- **Monotonicity**: Decreasing on \( (-4, \infty) \)
- **Left End Behavior**: As \( x \to -4^+ \), \( f(x) \to \infty \)
- **Right End Behavior**: As \( x \to \infty \), \( f(x) \to -\infty \)

Would you like further details on any of these points? 

Here are some additional questions for practice:
1. How would the graph change if the coefficient were positive instead of negative?
2. What would be the effect on the domain if we had \( f(x) = -2 \log_3 (x - 4) \) instead?
3. How does the base of the logarithm (\(3\)) affect the graph's shape?
4. Can you find the x-intercept of this function?
5. What happens to the asymptote if we add a constant outside the logarithm?

**Tip**: Remember that for a logarithmic function \( f(x) = \log_b(x + c) \), the asymptote is always at \( x = -c \).

What are the features of the function f(x) = -2 log_3 (x + 4) graphed below?

Analyze the features of the logarithmic function f(x) = -2 log_3 (x + 4), including its asymptote, range, domain, end behavior, and monotonicity. This problem covers important logarithmic concepts suitable for high school students in Grades 11-12.

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