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The image shows a mathematical equation written on graph paper: \( y = |x| \).

This equation represents the **absolute value function**. The absolute value of a number \( x \), denoted by \( |x| \), is the non-negative value of \( x \). This means:

\[
y = |x| = 
\begin{cases} 
x & \text{if } x \geq 0 \\
-x & \text{if } x < 0 
\end{cases}
\]

### Characteristics of the Function \( y = |x| \):

1. **Domain**: The domain of \( y = |x| \) is all real numbers (\( x \in \mathbb{R} \)).
2. **Range**: The range is all non-negative real numbers (\( y \geq 0 \)).
3. **Graph**: The graph of \( y = |x| \) is a V-shaped curve that intersects the y-axis at the origin (0,0).
4. **Symmetry**: The function is symmetric with respect to the y-axis, making it an **even function**.
5. **Vertex**: The vertex of the graph is at the point (0,0).

Would you like further details on any specific part of this function or have any other questions?

Here are some related questions you might find interesting:

1. How do you graph the function \( y = |x| \) on a coordinate plane?
2. What transformations occur if the function is modified to \( y = |x - 2| + 3 \)?
3. Can you explain the concept of an even function and how it relates to symmetry?
4. How would the equation change if we wanted to reflect the graph across the x-axis?
5. What is the difference between \( y = |x| \) and \( y = x^2 \) in terms of their graphs and properties?

**Tip**: When graphing absolute value functions, identifying the vertex and understanding how the graph opens (upward or downward) helps in sketching the correct shape.

What is the mathematical expression for y = |x|?

Explore the properties of the absolute value function y = |x|, including its domain, range, symmetry, and graph. Suitable for students in Grades 8-10, this lesson covers key mathematical concepts like function symmetry and graph transformations.

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