To find the vertex of the given absolute value function, [tex]\( f(x) = |x - 5| + 10 \)[/tex], we can use the vertex form of an absolute value function: [tex]\( f(x) = a|x - h| + k \)[/tex].
Let's break it down step-by-step:
1. Identify the horizontal shift:
- In the expression [tex]\( |x - 5| \)[/tex], the term inside the absolute value, [tex]\( x - 5 \)[/tex], indicates a horizontal shift.
- The horizontal shift [tex]\( h \)[/tex] is obtained by equating the expression inside the absolute value to zero: [tex]\( x - 5 = 0 \)[/tex]. Therefore, [tex]\( h = 5 \)[/tex].
2. Identify the vertical shift:
- The constant term outside the absolute value, [tex]\( +10 \)[/tex], indicates a vertical shift.
- The vertical shift [tex]\( k \)[/tex] is given directly by this constant term: [tex]\( k = 10 \)[/tex].
3. Determine the vertex:
- The vertex [tex]\( (h, k) \)[/tex] combines the horizontal and vertical shifts.
Putting it all together, the vertex of the function [tex]\( f(x) = |x - 5| + 10 \)[/tex] is:
[tex]\[ (h, k) = (5, 10) \][/tex]
So, the vertex is at [tex]\((5, 10)\)[/tex].
