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The problem asks you to use the graph of \( y = f(x) \) and transform it according to the function \( g(x) = 2f\left(\frac{1}{2}x\right) \), and then choose the correct graph.

### Step 1: Analyze the transformation

1. **Horizontal scaling**: The expression \( \frac{1}{2}x \) inside the function indicates a horizontal stretch by a factor of 2. This means that every x-value on the graph will be doubled, stretching the graph horizontally.
   
2. **Vertical scaling**: The coefficient 2 outside the function \( 2f(\cdot) \) means a vertical stretch by a factor of 2. This stretches the graph vertically, doubling the y-values.

### Step 2: Apply the transformation

- **Horizontal stretch**: Every x-coordinate on the original graph will be multiplied by 2. For example, if a point on the original graph was at \( (1, y) \), it will now be at \( (2, y) \).
  
- **Vertical stretch**: Every y-coordinate will be multiplied by 2. For example, if a point had a y-value of 2, it will now have a y-value of 4.

### Step 3: Find the correct graph

- In the original graph, key points are approximately:
  - At \( (-4, 1) \), \( (-2, 3) \), \( (0, 2) \), \( (2, 2) \), \( (4, 1) \).
  
- After transformation, the new points (after applying horizontal and vertical stretches) will be:
  - \( (-8, 2) \), \( (-4, 6) \), \( (0, 4) \), \( (4, 4) \), \( (8, 2) \).

### Step 4: Compare to the choices

- The correct graph is **Option B**, which reflects both the horizontal and vertical stretches correctly.

Would you like more details or have any questions about this transformation?

### Related Questions:
1. What is the general effect of a horizontal stretch on a function's graph?
2. How does vertical scaling affect the shape of a graph?
3. What happens to the x-intercepts and y-intercepts after a horizontal and vertical stretch?
4. How do you determine if a transformation is a stretch or compression?
5. What is the effect of combining multiple transformations on a graph?

**Tip:** Always apply horizontal transformations before vertical ones when analyzing composite transformations.

Use the graph of y = f(x) to graph the function g(x) = 2f(1/2 x) and choose the correct graph of g.

Learn how to use the graph of y = f(x) to find the graph of g(x) = 2f(1/2 x) by applying horizontal and vertical stretches. This example provides step-by-step transformations suitable for high school students in Grades 10-12.

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