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!