The question asks to select the correct logarithmic function from the graph. Let's break down the graph and analyze the options.

Analysis of the Graph:

1. Vertical Shift: The graph crosses the y-axis around $y = 2$, suggesting a vertical shift upward by 2 units.
2. Horizontal Shift: The logarithmic function seems to shift horizontally, meaning it likely involves a transformation of $\log_2(x - c)$, where $c$ affects the left or right movement.
3. Shape: The graph looks like a typical logarithmic curve, gradually increasing and flattening out as $x$ increases.

Comparing with the Choices:

Let's look at the options:

• Option A: $h(x) = \log_2(x - 1) + 2$
  • A horizontal shift right by 1 and a vertical shift up by 2.
• Option B: $h(x) = \log_2(x + 1) - 2$
  • A horizontal shift left by 1 and a vertical shift down by 2.
• Option C: $h(x) = \log_2(x - 2) + 1$
  • A horizontal shift right by 2 and a vertical shift up by 1.
• Option D: $h(x) = \log_2(x + 2) - 1$
  • A horizontal shift left by 2 and a vertical shift down by 1.

Conclusion:

Based on the graph, the logarithmic curve seems to have a horizontal shift to the right by 1 and a vertical shift upward by 2 units. This matches Option A: $h(x) = \log_2(x - 1) + 2$.

Would you like further details on the graph analysis or transformations?

Here are 5 related questions for further exploration:

1. How does the base of a logarithm affect the shape of its graph?
2. What is the difference between vertical and horizontal shifts in functions?
3. How do transformations of logarithmic functions differ from those of exponential functions?
4. What would happen if the vertical shift were downward instead of upward?
5. How do you graph a logarithmic function step by step?

Tip:

Always check the base of the logarithm in a function, as it can significantly impact the growth rate and overall shape of the graph.