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Determining the Vertex of an Absolute Value Function

The vertex of the graph of [tex]$f(x)=|x-3|+6$[/tex] is located at [tex](\square, \square)[/tex].

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To determine the vertex of the absolute value function [tex]\( f(x) = |x - 3| + 6 \)[/tex], follow these steps:

1. Identify the Standard Form: The standard form of an absolute value function is [tex]\( f(x) = |x - h| + k \)[/tex], where [tex]\( (h, k) \)[/tex] is the vertex of the function.

2. Compare with the Given Function:
- Given function: [tex]\( f(x) = |x - 3| + 6 \)[/tex]
- Here, [tex]\( h = 3 \)[/tex] and [tex]\( k = 6 \)[/tex]

3. Determine the Vertex:
- [tex]\( h = 3 \)[/tex] corresponds to the horizontal shift. This means the graph of the function is shifted 3 units to the right from the origin.
- [tex]\( k = 6 \)[/tex] corresponds to the vertical shift. This means the graph of the function is shifted 6 units up from the origin.

4. Vertex: Therefore, combining these horizontal and vertical shifts, the vertex of the absolute value function [tex]\( f(x) = |x - 3| + 6 \)[/tex] is at the point [tex]\( (3, 6) \)[/tex].

Therefore, the vertex of the graph of [tex]\( f(x) = |x - 3| + 6 \)[/tex] is located at [tex]\((3, 6)\)[/tex].
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