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What is the standard form of this function?

[tex]\[ f(x)=-(x-4)^2+2 \][/tex]

A. [tex]\( f(x)=-x^2+4x-30 \)[/tex]
B. [tex]\( f(x)=x^2+4x-30 \)[/tex]
C. [tex]\( f(x)=x^2+8x-14 \)[/tex]
D. [tex]\( f(x)=-x^2+8x-14 \)[/tex]


Sagot :

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]