write a function rule for the illustrated transformation of f(x)=|x|

To write a function rule for a transformation of \( f(x) = |x| \), we first need to understand the types of transformations involved. Common transformations include:

1. **Vertical translation**: Shifts the graph up or down.
   \[
   g(x) = |x| + c \quad \text{(upward by \(c\))} \quad \text{or} \quad g(x) = |x| - c \quad \text{(downward by \(c\))}.
   \]

2. **Horizontal translation**: Shifts the graph left or right.
   \[
   g(x) = |x - h| \quad \text{(right by \(h\))} \quad \text{or} \quad g(x) = |x + h| \quad \text{(left by \(h\))}.
   \]

3. **Vertical stretch/compression**: Stretches or compresses the graph vertically.
   \[
   g(x) = a|x| \quad \text{(stretched if \(a > 1\))} \quad \text{or} \quad g(x) = a|x| \quad \text{(compressed if \(0 < a < 1\))}.
   \]

4. **Horizontal stretch/compression**: Stretches or compresses the graph horizontally.
   \[
   g(x) = |bx| \quad \text{(compressed if \(b > 1\))} \quad \text{or} \quad g(x) = |bx| \quad \text{(stretched if \(0 < b < 1\))}.
   \]

5. **Reflection**: Reflects the graph over the x-axis or y-axis.
   \[
   g(x) = -|x| \quad \text{(reflection over the x-axis)} \quad \text{or} \quad g(x) = |-x| \quad \text{(reflection over the y-axis)}.
   \]

If you provide the specific transformation (such as shifting, stretching, reflecting, etc.), I can give you the exact function rule. Please describe the transformation in more detail!

Would you like to provide more details or ask about a specific transformation?

### Follow-up questions:
1. Is the transformation you're referring to a vertical shift or horizontal shift?
2. Are there any reflections involved in the transformation?
3. Is there any stretching or compression applied to the graph?
4. Does the graph appear to be stretched vertically or horizontally?
5. Are there multiple transformations combined, like reflection and shift?

**Tip**: Always start by identifying the key points (such as vertex) of the graph after transformation; this helps in determining shifts and reflections!

Explore how to write a function rule for the transformation of the absolute value function f(x) = |x|. This includes vertical and horizontal shifts, stretches, compressions, and reflections. Suitable for students in Grades 9-11, this guide covers key transformation concepts.

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