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From the image, you are asked to determine the equation of the function \( f(x) \) in terms of \( g(x) \), based on the given graph and the provided transformations (stretches, reflections, and translations).

### Key Observations:
- **\( g(x) \)** is a piecewise linear function that appears to be composed of two distinct line segments.
- **\( f(x) \)** is also a linear function but seems to be transformed in relation to \( g(x) \).

### Steps to Determine \( f(x) \) in Terms of \( g(x) \):

1. **Identify the equations of the given functions:**
   - The graph shows two lines, \( y = f(x) \) (red) and \( y = g(x) \) (blue).
   - You should first write the explicit equation for both lines. 
   - Based on the graph, it looks like \( g(x) \) has a slope of 1 and passes through \( (0, 2) \), so we hypothesize:
     \[
     g(x) = x + 2
     \]
   - The function \( f(x) \) appears to have a negative slope and crosses the x-axis at \( x = -4 \) and y-axis at \( y = 2 \), so:
     \[
     f(x) = -\frac{1}{2}x + 2
     \]

2. **Look at potential transformations:**
   - **Stretch:** \( f(x) \) seems like it might involve a vertical stretch, as its slope differs from \( g(x) \).
   - **Reflection:** \( f(x) \) has a negative slope, which could indicate a reflection over the x-axis relative to \( g(x) \).
   - **Translation:** Both functions appear to have the same y-intercept, so no vertical translation is present, but there might be a horizontal shift.

3. **Transformations applied to \( g(x) \) to get \( f(x) \):**
   - First, apply a **reflection** over the x-axis, which would change the slope sign: \( g(x) \) becomes \( -g(x) \), so:
     \[
     f(x) = -g(x)
     \]
   - Next, apply a **stretch** by a factor of \( \frac{1}{2} \), affecting the slope, so:
     \[
     f(x) = -\frac{1}{2}g(x)
     \]

### Conclusion:
Thus, the equation of \( f(x) \) in terms of \( g(x) \) is:
\[
f(x) = -\frac{1}{2}g(x)
\]

### Would you like more details, or do you have any questions?

### Expanding Questions:
1. Can you verify the equation of \( g(x) \) from the graph?
2. How would you calculate the slope of \( f(x) \) from the graph?
3. What are the effects of reflecting a function over the x-axis?
4. How do vertical stretches affect the slope of a function?
5. Can you identify horizontal translations between the two functions?

### Tip:
When performing transformations, always apply reflections and stretches first before considering translations.

Determine the equation of the function f(x) in terms of g(x) using transformations (stretches, reflections, and translations) from the given graph.

Learn how to determine the equation of a transformed function f(x) in terms of g(x) using graph transformations such as stretches, reflections, and translations. This step-by-step guide is perfect for students in Grades 10-12 working on linear function transformations.

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