To determine the transformation that turns [tex]\( f(x) = x^2 + 4 \)[/tex] into [tex]\( g(x) = (x-2)^2 + 4 \)[/tex], we need to carefully analyze the transformations involved.
Given:
[tex]\[ f(x) = x^2 + 4 \][/tex]
[tex]\[ g(x) = (x-2)^2 + 4 \][/tex]
We can see that [tex]\( g(x) \)[/tex] is related to [tex]\( f(x) \)[/tex]. Observe the expression inside the quadratic term of [tex]\( g(x) \)[/tex] and compare it with [tex]\( f(x) \)[/tex].
Starting with [tex]\( f(x) = x^2 + 4 \)[/tex], if we replace [tex]\( x \)[/tex] with [tex]\( x - 2 \)[/tex], we get:
[tex]\[ f(x - 2) = (x - 2)^2 + 4 \][/tex]
Therefore, substituting [tex]\( x - 2 \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(x - 2) = (x - 2)^2 + 4 \][/tex]
By comparing this with [tex]\( g(x) = (x-2)^2 + 4 \)[/tex], we see that:
[tex]\[ g(x) = f(x - 2) \][/tex]
Therefore, the transformation from [tex]\( f(x) \)[/tex] to [tex]\( g(x) \)[/tex] is a horizontal shift to the right by 2 units.
So, the correct answer is:
[tex]\[ \boxed{E. \, g(x) = f(x - 2)} \][/tex]
