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.