To graph the function $f(x) = \log_{\frac{1}{2}}(x + 1) - 2$, follow these steps:

1. Understand the Components:

   • $\log_{\frac{1}{2}}(x + 1)$: This is a logarithmic function with base $\frac{1}{2}$. The logarithm with a base less than 1 is a decreasing function.
   • $-2$: This is a vertical shift downward by 2 units.

2. Identify Key Points:

   • The domain of the function is $x + 1 > 0 \Rightarrow x > -1$.
   • The vertical asymptote is at $x = -1$.
   • When $x = 0$, $f(x) = \log_{\frac{1}{2}}(1) - 2 = 0 - 2 = -2$.
   • When $x = 1$, $f(x) = \log_{\frac{1}{2}}(2) - 2 = -1 - 2 = -3$.

3. Graphing Steps:

   • Draw the vertical asymptote at $x = -1$.
   • Plot the point (0, -2).
   • Plot the point (1, -3).
   • As $x$ approaches $-1$ from the right, $f(x)$ approaches $\infty$.
   • As $x$ increases, $f(x)$ decreases without bound, but at a decreasing rate.

4. Sketch the Graph:

   • Start near the vertical asymptote at $x = -1$ and draw the curve approaching infinity as $x$ approaches $-1$ from the right.
   • Pass through the points (0, -2) and (1, -3).
   • Continue the curve downward as $x$ increases, showing the decreasing nature of the logarithmic function.

Let's create a graph based on these details:

Would you like further details or have any questions? Here are some relative questions for deeper understanding:

1. What is the effect of the base of a logarithmic function on its graph?
2. How do vertical shifts affect the graph of a function?
3. What are vertical asymptotes, and how are they determined for logarithmic functions?
4. How do you find the domain of a logarithmic function?
5. What are the general characteristics of logarithmic functions with bases less than 1?

Tip: Always check the domain of the logarithmic function before graphing, as it defines where the function is valid.