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].
