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Write the transformation of f(x) to g(x) then write the equation g(x) in terms of f(x)=x

To analyze the transformation of \( f(x) \) to \( g(x) \) from the provided graph, we need to compare the two functions visually.

### Step-by-Step Analysis:

1. **\( f(x) \)** is a V-shaped function, suggesting it is an absolute value function. From the graph, \( f(x) \) seems to be \( f(x) = |x| \), which creates the characteristic "V" shape.

2. **\( g(x) \)** appears to be a transformation of \( f(x) \), specifically a reflection and a translation:
   - It is reflected across the x-axis (since the slope of \( g(x) \) is downward).
   - There is a vertical translation downwards, and the function seems to be shifted down by 6 units.

### General Transformation:
The transformation from \( f(x) \) to \( g(x) \) involves:
- A reflection across the x-axis.
- A vertical translation downward by 6 units.

Thus, the equation for \( g(x) \) in terms of \( f(x) \) is:
\[
g(x) = -f(x) - 6
\]
Since \( f(x) = |x| \), the equation of \( g(x) \) becomes:
\[
g(x) = -|x| - 6
\]

This means that:
- The reflection is represented by the negative sign.
- The shift downward by 6 units is captured by subtracting 6.

### Summary:
- The function \( f(x) = |x| \) has been reflected over the x-axis and translated downward by 6 units to form \( g(x) \).
- The equation for \( g(x) \) in terms of \( f(x) = |x| \) is \( g(x) = -|x| - 6 \).

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Do you need any further details or have any questions?

Here are 5 related questions to further explore:
1. What is the general form of a vertical translation and reflection in function transformations?
2. How would the graph change if \( g(x) \) had been shifted upward instead of downward?
3. What happens to the function if \( f(x) = |x| \) is shifted horizontally?
4. How can you derive the vertex of an absolute value function from its equation?
5. How would you reflect \( f(x) \) over the y-axis instead of the x-axis?

**Tip:** When analyzing transformations, always look for reflections, translations, stretches, and compressions to see how they affect the graph visually.

Learn how to transform the absolute value function f(x) = |x| into g(x) with a reflection across the x-axis and a vertical translation downward by 6 units. This example demonstrates key concepts in function transformations, including reflections and translations. Suitable for Grades 8-10.

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