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.