To graph the function $f(x) = -3 \log_2(-x)$, let's break down the requirements step-by-step:

1. Understanding the Domain:

   • Since $\log_2(-x)$ only takes real values for $-x > 0$, we require $x < 0$. Thus, the function is defined for $x < 0$ only.

2. Asymptote:

   • As $x$ approaches 0 from the left (i.e., $x \to 0^-$), $-x \to 0^+$ and $\log_2(-x) \to -\infty$.
   • This causes $f(x)$ to go to positive infinity. Hence, there is a vertical asymptote at $x = 0$.

3. Selecting Points to Plot:

   • Choose values of $x$ that are negative, since $f(x)$ is only defined for $x < 0$.
   • For example, try:
     • $x = -1$: $f(-1) = -3 \log_2(1) = 0$
     • $x = -2$: $f(-2) = -3 \log_2(2) = -3$
     • $x = -4$: $f(-4) = -3 \log_2(4) = -6$
   • These points provide integer coordinates for easy plotting.

4. Graph Characteristics:

   • The graph should start near infinity on the left side of the $y$-axis and decrease as $x$ becomes more negative.
   • The points $(-1, 0)$, $(-2, -3)$, and $(-4, -6)$ are good reference points.

Would you like more details on the calculations, or have questions about any specific part?

Related Questions

1. What is the general behavior of logarithmic functions and their transformations?
2. How does the base of a logarithmic function affect its growth rate?
3. How does changing the sign in front of the logarithmic function affect its graph?
4. What is the effect of different vertical stretches/compressions on logarithmic graphs?
5. How do you determine asymptotes for logarithmic functions with transformations?

Tip

For logarithmic functions, vertical transformations often alter the range without affecting the domain.