https://file.aiquickdraw.com/mathbot/file/1727394082495k8uq2jmg.jpg

To analyze this transformation between \( f(x) \) and \( g(x) \), we will focus on understanding how the graph of \( f(x) \) has been modified to create \( g(x) \).

### Graphical Analysis:
- The graph of \( f(x) \) is shown in **green**, while \( g(x) \) is in **red**.
- To figure out the transformation, we look for **shifts** (horizontal and vertical) and possibly **reflections** or **scalings**.

From the graph:
- \( g(x) \) appears to be **shifted downward** compared to \( f(x) \).
- There also appears to be a **horizontal shift** as \( g(x) \) is shifted to the right.

### Vertical and Horizontal Shifts:
1. **Vertical Shift**: Comparing the y-values, \( g(x) \) seems to be shifted downward by 4 units compared to \( f(x) \). This indicates a vertical transformation of \( -4 \).

2. **Horizontal Shift**: Comparing the x-values, \( g(x) \) seems to be shifted to the **right** by 2 units compared to \( f(x) \). This suggests a horizontal transformation involving \( (x - 2) \).

### Conclusion:
The correct transformation that defines \( g(x) \) in terms of \( f(x) \) is:
\[
g(x) = f(x - 2) - 4
\]
This corresponds to the first option: 
\[
g(x) = f(x - 2) - 4
\]

Would you like further details or have any questions about this?

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Here are 5 questions to expand on this topic:
1. How do horizontal shifts affect the equation of a function?
2. How can you determine vertical shifts by looking at a graph?
3. What is the effect of a transformation of the form \( f(x + a) \)?
4. Can a function's graph be transformed by reflections or scalings in addition to shifts?
5. How do transformations help in understanding function behavior in real-world applications?

**Tip**: To quickly identify transformations, compare key points of the function (like the intercepts or maxima/minima) before and after transformation!

The function f(x) has been transformed into the function g(x). Which of the following defines g(x) in terms of f(x)?

Analyze how the function f(x) is transformed into g(x) by examining horizontal and vertical shifts. The problem covers concepts related to function transformations and graphical analysis, suitable for students in Grades 8-10.

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