Let's break down the problem based on the graph and the function given:

Problem Details:

• You are asked to use the graph of $y = f(x)$ (on the left side) to graph the function $g(x) = \frac{1}{2} f(x - 2)$.
• The operation $g(x) = \frac{1}{2} f(x - 2)$ applies two transformations to the function $f(x)$:
  1. Horizontal Shift: $x - 2$ shifts the graph 2 units to the right.
  2. Vertical Compression: $\frac{1}{2}$ compresses the graph vertically by a factor of 2 (all y-values are halved).

Applying Transformations:

1. Horizontal Shift: Move the entire graph of $f(x)$ 2 units to the right.
2. Vertical Compression: For each y-value of the shifted graph, multiply it by $\frac{1}{2}$.

Observing the Graphs:

From the provided options (A, B, C, D), the graph that correctly applies both transformations is B. It shows:

• A shift 2 units to the right.
• All y-values compressed to half of their original values.

Answer: The correct graph is B.

Would you like further clarification on the transformations or the solution process?

Here are five follow-up questions for more practice:

1. How would the graph of $g(x) = 2f(x + 1)$ transform the graph of $f(x)$?
2. What effect does multiplying a function by a negative constant have on its graph?
3. How would $g(x) = \frac{1}{3}f(x + 3)$ change the graph of $f(x)$?
4. What happens if you add a constant to $f(x)$, like $g(x) = f(x) + 3$?
5. Can you describe the transformation for $g(x) = -f(x - 1)$?

Tip: Always apply horizontal shifts before vertical stretches or compressions when analyzing transformations.