No like add the coordinates of every point for each one

Let's go through each transformation and add specific coordinates for notable points on the graph.

### Part a: \( f(x) = 2\log_2(x) - 3 \)

**Parent Function Key Points:**
- \( (1, 0) \) since \( \log_2(1) = 0 \)
- \( (2, 1) \) since \( \log_2(2) = 1 \)
- \( (4, 2) \) since \( \log_2(4) = 2 \)
- \( \left(\frac{1}{2}, -1\right) \) since \( \log_2\left(\frac{1}{2}\right) = -1 \)

**Transformation Steps and Resulting Points:**

1. **Vertical Stretch by 2:**
   - \( (1, 0) \to (1, 0 \cdot 2) = (1, 0) \)
   - \( (2, 1) \to (2, 1 \cdot 2) = (2, 2) \)
   - \( (4, 2) \to (4, 2 \cdot 2) = (4, 4) \)
   - \( \left(\frac{1}{2}, -1\right) \to \left(\frac{1}{2}, -1 \cdot 2\right) = \left(\frac{1}{2}, -2\right) \)

2. **Vertical Translation Downward by 3:**
   - \( (1, 0) \to (1, 0 - 3) = (1, -3) \)
   - \( (2, 2) \to (2, 2 - 3) = (2, -1) \)
   - \( (4, 4) \to (4, 4 - 3) = (4, 1) \)
   - \( \left(\frac{1}{2}, -2\right) \to \left(\frac{1}{2}, -2 - 3\right) = \left(\frac{1}{2}, -5\right) \)

**Final Key Points:**
- \( (1, -3) \)
- \( (2, -1) \)
- \( (4, 1) \)
- \( \left(\frac{1}{2}, -5\right) \)

### Part b: \( f(x) = -3\log_2(x - 2) + 1 \)

**Parent Function Key Points:**
- \( (1, 0) \)
- \( (2, 1) \)
- \( (4, 2) \)
- \( \left(\frac{1}{2}, -1\right) \)

**Transformation Steps and Resulting Points:**

1. **Horizontal Shift Right by 2:**
   - \( (1, 0) \to (1 + 2, 0) = (3, 0) \)
   - \( (2, 1) \to (2 + 2, 1) = (4, 1) \)
   - \( (4, 2) \to (4 + 2, 2) = (6, 2) \)
   - \( \left(\frac{1}{2}, -1\right) \to \left(\frac{1}{2} + 2, -1\right) = \left(\frac{5}{2}, -1\right) \)

2. **Vertical Reflection and Stretch by -3:**
   - \( (3, 0) \to (3, 0 \cdot -3) = (3, 0) \)
   - \( (4, 1) \to (4, 1 \cdot -3) = (4, -3) \)
   - \( (6, 2) \to (6, 2 \cdot -3) = (6, -6) \)
   - \( \left(\frac{5}{2}, -1\right) \to \left(\frac{5}{2}, -1 \cdot -3\right) = \left(\frac{5}{2}, 3\right) \)

3. **Vertical Translation Upward by 1:**
   - \( (3, 0) \to (3, 0 + 1) = (3, 1) \)
   - \( (4, -3) \to (4, -3 + 1) = (4, -2) \)
   - \( (6, -6) \to (6, -6 + 1) = (6, -5) \)
   - \( \left(\frac{5}{2}, 3\right) \to \left(\frac{5}{2}, 3 + 1\right) = \left(\frac{5}{2}, 4\right) \)

**Final Key Points:**
- \( (3, 1) \)
- \( (4, -2) \)
- \( (6, -5) \)
- \( \left(\frac{5}{2}, 4\right) \)

**Summary:**

1. **Function a:**
   - **Final Points:** \( (1, -3), (2, -1), (4, 1), \left(\frac{1}{2}, -5\right) \)

2. **Function b:**
   - **Final Points:** \( (3, 1), (4, -2), (6, -5), \left(\frac{5}{2}, 4\right) \)

Let me know if you need further details or clarifications, or if you would like to see the graph plotted.

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### Additional Questions:

1. How do the transformations affect the position of the vertical asymptote?
2. What happens to the graph when a logarithmic function is vertically stretched by a negative factor?
3. Why does the graph of a logarithmic function not touch the y-axis?
4. What is the impact of a horizontal shift on the domain of a logarithmic function?
5. How does the base of the logarithm influence the steepness of the curve?
6. What does a vertical translation do to the range of the function?
7. Can a logarithmic function have more than one vertical asymptote? Why or why not?
8. How do you determine the intersection points with the axes after transformations?

### Tip:

When dealing with logarithmic transformations, carefully apply horizontal shifts before vertical reflections and stretches, as these affect the graph's position and orientation.

Explore how logarithmic functions are transformed through vertical stretches, translations, horizontal shifts, and reflections. Detailed examples include step-by-step transformations and resulting key points for functions such as f(x) = 2log_2(x) - 3 and f(x) = -3log_2(x - 2) + 1. Suitable for advanced high school students and beyond.

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