The given equation is:

\[
y = |x - 1| + 4
\]

This represents an absolute value function. Here's how we can interpret and understand it:

1. **Absolute value function**: The absolute value function, \( |x - 1| \), means that for any value of \( x \), the expression inside the absolute value will always be non-negative. So, if \( x - 1 \) is positive, the function will be just \( x - 1 \). If \( x - 1 \) is negative, the function will flip the sign, becoming \( -(x - 1) \).

2. **Transformation**: The graph of \( y = |x - 1| \) would typically form a "V" shape with its vertex at \( (1, 0) \). However, since we have \( +4 \) outside the absolute value, this shifts the entire graph **upward by 4 units**. So, the vertex of the graph will now be at \( (1, 4) \).

3. **Graph characteristics**:
    - For \( x \geq 1 \), the function behaves as \( y = (x - 1) + 4 = x + 3 \).
    - For \( x < 1 \), the function behaves as \( y = -(x - 1) + 4 = -x + 5 \).

### Summary:
- The vertex of the graph is at \( (1, 4) \).
- The graph is "V"-shaped, opening upwards.
- The slope on the right side of the vertex (for \( x \geq 1 \)) is 1, and the slope on the left side (for \( x < 1 \)) is -1.

Would you like a more detailed explanation, or do you have any questions about graphing this function?

Here are some related questions:
1. How does the graph change if we replace the "+4" with a different constant?
2. What would the graph of \( y = |x - 1| - 4 \) look like?
3. How do absolute value equations behave when multiplied by a constant?
4. What would happen if the equation was \( y = 2|x - 1| + 4 \)?
5. How do transformations affect the graph of absolute value functions?

Tip: Always pay attention to transformations when working with absolute value functions—shifting, stretching, or reflecting can drastically change the graph!

Learn how to graph the absolute value function y = |x - 1| + 4 with a detailed step-by-step explanation. This includes transformations like vertical shifts and the behavior of the graph, suitable for students in Grades 8-10.

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