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To solve the problem, we need to match each graph to its corresponding equation based on the transformations applied to the function \( f(x) \) to obtain \( g(x) \).

### Step-by-Step Matching:

1. **Graph 1**:
   - **Observation**: The graph \( g(x) \) appears to be a vertical reflection and vertical shift of \( f(x) \).
   - **Possible Transformation**: A reflection over the x-axis can be achieved by multiplying by -1, and a shift down by 3 units.
   - **Equation**: This matches the equation \( g(x) = -f(x + 3) \). However, we need to check available options. The closest match is **C**: \( g(x) = -f(x + 3) \).

2. **Graph 2**:
   - **Observation**: The graph \( g(x) \) appears to be a horizontal stretch and downward shift.
   - **Possible Transformation**: A horizontal stretch by a factor of 2 followed by a downward shift by 1 unit.
   - **Equation**: This corresponds to \( g(x) = f\left(\frac{1}{2}x\right) - 1 \), which matches option **D**.

3. **Graph 3**:
   - **Observation**: The graph \( g(x) \) appears to be a reflection over the y-axis and a vertical stretch.
   - **Possible Transformation**: Reflecting over the y-axis can be achieved by replacing \( x \) with \( -x \) and a vertical stretch by 2 times.
   - **Equation**: This matches \( g(x) = 2f(-x) + 3 \), which is option **F**.

4. **Graph 4**:
   - **Observation**: The graph \( g(x) \) seems to be a horizontal compression and a shift to the right.
   - **Possible Transformation**: A horizontal compression by a factor of 2 followed by a shift to the right by 4 units.
   - **Equation**: This matches \( g(x) = f(2(x - 4)) \), which is option **H**.

### Final Matching:

- **Graph 1**: \( g(x) = -f(x + 3) \) - **C**
- **Graph 2**: \( g(x) = f\left(\frac{1}{2}x\right) - 1 \) - **D**
- **Graph 3**: \( g(x) = 2f(-x) + 3 \) - **F**
- **Graph 4**: \( g(x) = f(2(x - 4)) \) - **H**

The 4-letter code based on these answers is **CDFH**.

Would you like further details on any of the transformations? Here are some related questions you might find interesting:

1. What is the impact of a horizontal stretch or compression on the function's graph?
2. How do vertical reflections affect the graph of a function?
3. Can you explain the effect of shifting a function horizontally or vertically?
4. What are the key differences between horizontal and vertical transformations?
5. How can transformations be combined to achieve a specific graph?

**Tip**: Remember that horizontal transformations affect the x-values (inside the function argument), while vertical transformations affect the y-values (outside the function argument).

Determine the equation that maps f(x) to g(x) for the given graphs.

Solve a puzzle on function transformations by identifying the equation that maps f(x) to g(x) using vertical and horizontal shifts, reflections, and stretches. Suitable for high school students (Grades 9-12).

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