To determine the horizontal translation from the parent function [tex]\( f(x) = x^2 \)[/tex] to the function [tex]\( g(x) = (x-4)^2 + 2 \)[/tex], we need to focus on the expression inside the parentheses.
1. The given function is [tex]\( g(x) = (x-4)^2 + 2 \)[/tex].
2. For horizontal translation, we examine the term involving [tex]\( x \)[/tex] inside the squared term, which is [tex]\( (x-4) \)[/tex].
The general form for a horizontally translated function is [tex]\( f(x-h) \)[/tex], which represents a shift of the graph of [tex]\( f(x) \)[/tex] horizontally by [tex]\( h \)[/tex] units.
3. In this case, the parent function [tex]\( f(x) = x^2 \)[/tex] is modified to [tex]\( f(x-4) = (x-4)^2 \)[/tex].
4. The term [tex]\( (x-4) \)[/tex] indicates a horizontal shift to the right by 4 units because [tex]\( h = 4 \)[/tex].
Therefore, the value representing the horizontal translation from the graph of the parent function [tex]\( f(x) = x^2 \)[/tex] to [tex]\( g(x) = (x-4)^2 + 2 \)[/tex] is:
[tex]\[ \boxed{4} \][/tex]
