To identify which second-degree polynomial function fits the given conditions, we need to follow these steps:
1. Understand the Conditions:
- The polynomial has a root at [tex]\(x = 4\)[/tex] with multiplicity 2.
- The leading coefficient of the polynomial is -1.
2. Construct the Polynomial Using the Root:
- Since the root [tex]\(x = 4\)[/tex] has multiplicity 2, it implies that the polynomial can be expressed as [tex]\((x - 4)^2\)[/tex].
3. Consider the Leading Coefficient:
- The polynomial needs to have a leading coefficient of -1. This means we should multiply the expanded form of [tex]\((x - 4)^2\)[/tex] by -1.
4. Expand the Polynomial:
- Start with [tex]\( (x - 4)^2 \)[/tex]:
[tex]\[
(x - 4)^2 = (x - 4)(x - 4) = x^2 - 8x + 16
\][/tex]
- Now, multiply the resulting expression by -1:
[tex]\[
-1 \cdot (x^2 - 8x + 16) = -x^2 + 8x - 16
\][/tex]
5. Match the Polynomial to the Given Options:
- Now, compare the constructed polynomial [tex]\(-x^2 + 8x - 16\)[/tex] with the given options.
Analyzing each option:
- [tex]\( f(x) = -x^2 - 8x - 16 \)[/tex]: Incorrect because the coefficient of [tex]\(x\)[/tex] should be [tex]\(+8x\)[/tex].
- [tex]\( f(x) = -x^2 + 8x - 16 \)[/tex]: Correct as it matches exactly with our polynomial.
- [tex]\( f(x) = -x^2 - 8x + 16 \)[/tex]: Incorrect due to the wrong signs in front of [tex]\(8x\)[/tex] and [tex]\(16\)[/tex].
- [tex]\( f(x) = -x^2 + 8x + 16 \)[/tex]: Incorrect because the constant term should be [tex]\(-16\)[/tex].
Given the detailed steps and checking each option, we conclude that the correct polynomial function is:
[tex]\[ f(x) = -x^2 + 8x - 16 \][/tex]
Therefore, the correct option is:
[tex]\[ \boxed{2} \][/tex]
