The equation \( y = |2x - 10| \) represents the absolute value function. To better understand this function, we can break it down into two cases based on the definition of absolute value:

### Case 1: When \( 2x - 10 \geq 0 \)
If the expression inside the absolute value is non-negative, the absolute value simply removes the absolute value bars:
\[
y = 2x - 10
\]
This occurs when:
\[
2x - 10 \geq 0 \implies x \geq 5
\]
So, for \( x \geq 5 \), the function is \( y = 2x - 10 \).

### Case 2: When \( 2x - 10 < 0 \)
If the expression inside the absolute value is negative, the absolute value changes the sign of the expression:
\[
y = -(2x - 10) = -2x + 10
\]
This occurs when:
\[
2x - 10 < 0 \implies x < 5
\]
So, for \( x < 5 \), the function is \( y = -2x + 10 \).

### Summary of the piecewise function:
\[
y = 
\begin{cases} 
2x - 10 & \text{if } x \geq 5 \\
-2x + 10 & \text{if } x < 5
\end{cases}
\]

The graph of this function is a V-shape with the vertex at \( (5, 0) \), and the function is symmetric about the vertical line \( x = 5 \).

Would you like a more detailed graph or further explanations?

Here are 5 related questions for deeper exploration:
1. What is the slope of each piece of the function?
2. How do you find the vertex of an absolute value function?
3. Can you calculate the y-intercept of this function?
4. How would the function change if the absolute value were removed?
5. How does the function behave as \( x \to \infty \) and \( x \to -\infty \)?

**Tip**: The vertex of an absolute value function occurs where the expression inside the absolute value equals zero.

Explore how to solve and graph the absolute value function y = |2x - 10|. This problem demonstrates how to break down an absolute value function into two linear equations and analyze its V-shaped graph. Suitable for students in Grades 9-11.

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