To determine the vertex of the absolute value function [tex]\( f(x) = |x - 3| + 6 \)[/tex], let's break down the components and how they affect the graph of the function:
1. Absolute Value Function:
The basic form of an absolute value function is [tex]\( f(x) = |x| \)[/tex]. Its graph is a V-shape with the vertex at the origin (0,0).
2. Horizontal Translation:
When the function takes the form [tex]\( f(x) = |x - h| \)[/tex], it translates the graph horizontally. The vertex moves from (0,0) to (h,0). For our function [tex]\( f(x) = |x - 3| \)[/tex], the graph is translated horizontally 3 units to the right. Therefore, the horizontal coordinate of the vertex is [tex]\( h = 3 \)[/tex].
3. Vertical Translation:
When a constant [tex]\( k \)[/tex] is added to [tex]\( f(x) \)[/tex], as in [tex]\( f(x) = |x - h| + k \)[/tex], it translates the graph vertically. The vertex moves from (h,0) to (h,k). In our function, we have [tex]\( k = 6 \)[/tex]. So, the vertical coordinate of the vertex moves from 0 to [tex]\( k = 6 \)[/tex].
Combining these translations, the vertex of the given function [tex]\( f(x) = |x - 3| + 6 \)[/tex] is located at:
[tex]\[ (3, 6) \][/tex]
So, the vertex of the graph of [tex]\( f(x) = |x - 3| + 6 \)[/tex] is located at [tex]\( (3, 6) \)[/tex].
