describe transformation, domain and range of y=|x|

### Transformation:
The equation \( y = |x| \) represents the absolute value function, which is a piecewise linear function. The transformation involved in this function is as follows:

1. **Reflection:** The function \( y = |x| \) can be thought of as reflecting the part of the graph of \( y = x \) (for \( x \geq 0 \)) across the y-axis for negative values of \( x \). This reflection creates a "V" shape with the vertex at the origin.

2. **Identity for Non-Negative Values:** For \( x \geq 0 \), the function behaves as the identity function \( y = x \).

3. **Reflection for Negative Values:** For \( x < 0 \), the function reflects \( y = -x \) about the y-axis.

### Domain:
The domain of \( y = |x| \) is the set of all real numbers because the absolute value function is defined for every real number:
\[
\text{Domain: } (-\infty, \infty)
\]

### Range:
The range of \( y = |x| \) consists of all non-negative real numbers because the absolute value of any real number is always zero or positive:
\[
\text{Range: } [0, \infty)
\]

### Summary:
- **Transformation:** Reflection and identity, forming a "V" shape.
- **Domain:** \( (-\infty, \infty) \)
- **Range:** \( [0, \infty) \)

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How would the transformation change for \( y = |x - 2| \)?
2. What is the effect of multiplying \( y = |x| \) by a constant, such as \( y = 2|x| \)?
3. How would you describe the graph of \( y = |x| + 3 \)?
4. Can you explain how to determine the vertex of a transformed absolute value function?
5. What is the domain and range of the function \( y = |x| \) if restricted to \( x > 0 \)?

**Tip:** The absolute value function is essential in understanding piecewise functions, which can model real-world situations where different rules apply to different conditions.

Explore the transformation, domain, and range of the absolute value function y = |x|. Understand how the function reflects and identifies non-negative values, its domain, which covers all real numbers, and its range, which includes non-negative real numbers. Suitable for high school students studying functions and transformations.

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