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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:

![Graph](https://www.desmos.com/calculator/embeddable)

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.

Graph the function f(x) = log_(1/2)(x + 1) - 2.

Learn how to graph the logarithmic function f(x) = log_(1/2)(x + 1) - 2. This solution explains transformations, vertical shifts, and asymptotes for logarithmic functions. Ideal for high school students in Grades 10-12.

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