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.