Let's consider the function [tex]\( f(x) = -(x-4)^2 + 2 \)[/tex].
To convert this function to its standard form [tex]\( ax^2 + bx + c \)[/tex], we need to expand and simplify the given expression step-by-step.
1. Start with the function:
[tex]\[
f(x) = -(x-4)^2 + 2
\][/tex]
2. Expand the squared term [tex]\((x-4)^2\)[/tex]:
[tex]\[
(x-4)^2 = x^2 - 8x + 16
\][/tex]
3. Substitute this expansion back into the function:
[tex]\[
f(x) = - (x^2 - 8x + 16) + 2
\][/tex]
4. Distribute the negative sign across the terms inside the parentheses:
[tex]\[
f(x) = -x^2 + 8x - 16 + 2
\][/tex]
5. Combine the constant terms:
[tex]\[
-16 + 2 = -14
\][/tex]
6. Therefore, the standard form of the function is:
[tex]\[
f(x) = -x^2 + 8x - 14
\][/tex]
The correct answer is:
[tex]\[
\boxed{D. f(x) = -x^2 + 8x - 14}
\][/tex]
