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The problem asks you to sketch the graph of \( f(x) = -2\log_{0.5}(2 - x) + 1 \) and compare it with its parent function, which would be \( g(x) = \log_{0.5}(x) \). You're also required to show 4 accuracy points on the graph using transformations and then plot the graph.

### Step-by-Step Solution:

#### Step 1: Understand the Parent Function
The parent function is \( g(x) = \log_{0.5}(x) \), which is a logarithmic function with base \( 0.5 \). Since the base is less than 1, this function decreases as \( x \) increases.

#### Step 2: Apply Transformations
The function \( f(x) = -2\log_{0.5}(2 - x) + 1 \) can be derived from the parent function through the following transformations:
1. **Horizontal Shift:** Replace \( x \) with \( 2 - x \), which shifts the graph to the right by 2 units and reflects it across the y-axis.
2. **Vertical Stretch:** Multiply by -2, which vertically stretches the graph by a factor of 2 and reflects it across the x-axis.
3. **Vertical Shift:** Add 1 to shift the entire graph up by 1 unit.

#### Step 3: Identify 4 Key Points
To ensure accuracy, calculate the function's value at specific points that can be easily plotted:

1. **When \( x = 0 \):**
   \[
   f(0) = -2\log_{0.5}(2 - 0) + 1 = -2\log_{0.5}(2) + 1 = -2 \times (-1) + 1 = 3
   \]
   Point: \( (0, 3) \)

2. **When \( x = 1 \):**
   \[
   f(1) = -2\log_{0.5}(2 - 1) + 1 = -2\log_{0.5}(1) + 1 = -2 \times 0 + 1 = 1
   \]
   Point: \( (1, 1) \)

3. **When \( x = -1 \):**
   \[
   f(-1) = -2\log_{0.5}(2 - (-1)) + 1 = -2\log_{0.5}(3) + 1
   \]
   \( \log_{0.5}(3) \) will be a positive number less than \( -1 \), so this will give you a point less than 1.
   Point: \( (-1, \text{approx.}) \)

4. **When \( x = 1.5 \):**
   \[
   f(1.5) = -2\log_{0.5}(2 - 1.5) + 1 = -2\log_{0.5}(0.5) + 1 = -2 \times 1 + 1 = -1
   \]
   Point: \( (1.5, -1) \)

#### Step 4: Plot the Graph
Using the 4 points, plot the transformed function on a graph. Start with the parent function, and then apply the transformations step by step to get the final graph.

#### Step 5: Compare with the Parent Function
- The parent function \( g(x) = \log_{0.5}(x) \) decreases and crosses through points like \( (1, 0) \) and \( (0.5, 1) \).
- The function \( f(x) \) shifts this graph to the right, reflects it, and vertically stretches and shifts it.

Would you like further assistance with graphing or comparing this function?

### Further Exploration Questions:
1. What is the exact value of \( \log_{0.5}(3) \) for precise graphing?
2. How would the graph change if the base was greater than 1?
3. What is the inverse of \( f(x) \)?
4. How does the reflection across the x-axis affect the concavity of the graph?
5. Can you find the derivative of \( f(x) \) to understand the slope at different points?
6. What is the domain and range of the parent function \( g(x) \)?
7. How does the vertical stretch by -2 affect the steepness of the graph?
8. How does shifting the graph vertically by 1 unit affect its intercepts?

#### Tip:
When sketching graphs, transformations are your best tool. Break down each transformation step-by-step to avoid confusion.

Learn how to graph the function f(x) = -2log0.5(2 - x) + 1 with detailed step-by-step transformations and accuracy points. Explore logarithmic functions and their transformations, suitable for high school students studying functions and graphing.

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