To find the vertex of the function [tex]\( f(x) = |x - 9| + 2 \)[/tex], let’s break it down step by step.
1. Understand the structure of the function:
- The given function is [tex]\( f(x) = |x - 9| + 2 \)[/tex].
- This is an absolute value function in the form [tex]\( f(x) = |x - h| + k \)[/tex], where [tex]\( h \)[/tex] and [tex]\( k \)[/tex] represent the horizontal and vertical shifts, respectively.
2. Identify the values of [tex]\( h \)[/tex] and [tex]\( k \)[/tex]:
- The expression inside the absolute value is [tex]\( x - 9 \)[/tex].
- The value of [tex]\( x \)[/tex] that makes the expression [tex]\( x - 9 \)[/tex] equal to zero is [tex]\( x = 9 \)[/tex]. Therefore, [tex]\( h = 9 \)[/tex].
- The constant term added outside the absolute value is [tex]\( +2 \)[/tex]. So, [tex]\( k = 2 \)[/tex].
3. Determine the vertex of the function:
- In an absolute value function [tex]\( f(x) = |x - h| + k \)[/tex], the vertex of the graph is at the point [tex]\( (h, k) \)[/tex].
- Substituting our values of [tex]\( h \)[/tex] and [tex]\( k \)[/tex], we get the vertex at [tex]\( (9, 2) \)[/tex].
Therefore, the correct answer is:
A. [tex]\((9, 2)\)[/tex]
