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Which of these is the standard form of the following function?

1. [tex]\( f(x) = -9(x+5)^2 + 4 \)[/tex]

A. [tex]\( f(x) = 9x^2 - 180x + 221 \)[/tex]

B. [tex]\( f(x) = -9x^2 - 90x - 221 \)[/tex]

C. [tex]\( f(x) = -9x^2 - 180x - 221 \)[/tex]

D. [tex]\( f(x) = 9x^2 - 90x - 221 \)[/tex]

Answer :

To find the standard form of the given function [tex]\(f(x) = -9(x + 5)^2 + 4\)[/tex], we need to convert it from vertex form to standard form, which is [tex]\(ax^2 + bx + c\)[/tex].

Here are the steps:

1. Expand the squared term:
[tex]\((x + 5)^2\)[/tex] can be expanded as:
[tex]\[
(x + 5)(x + 5) = x^2 + 10x + 25
\][/tex]

2. Multiply by -9:
Take the expansion [tex]\(x^2 + 10x + 25\)[/tex] and multiply each term by [tex]\(-9\)[/tex]:
[tex]\[
-9(x^2 + 10x + 25) = -9x^2 - 90x - 225
\][/tex]

3. Add the constant 4:
Now, add 4 to each term in the expression:
[tex]\[
-9x^2 - 90x - 225 + 4
\][/tex]

4. Combine like terms:
Simplify the expression by combining the constant terms [tex]\(-225\)[/tex] and [tex]\(4\)[/tex]:
[tex]\[
-9x^2 - 90x - 221
\][/tex]

Hence, the standard form of the function is:
[tex]\[ f(x) = -9x^2 - 90x - 221 \][/tex]

This matches the option:
[tex]\( f(x) = -9x^2 - 90x - 221 \)[/tex]

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