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 $x = -c$.