High School

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------------------------------------------------ Select the correct answer.

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

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

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

B. [tex]\[ f(x) = -9x^2 - 180x - 221 \][/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 function [tex]\( f(x) = -9(x+5)^2 + 4 \)[/tex], let's go through the steps of expanding and simplifying it.

1. Expand the Square: Start by expanding the square term [tex]\((x+5)^2\)[/tex].
[tex]\((x+5)^2 = x^2 + 10x + 25\)[/tex].

2. Multiply by -9: Next, multiply each term inside the parenthesis by [tex]\(-9\)[/tex].
[tex]\(-9(x^2 + 10x + 25) = -9x^2 - 90x - 225\)[/tex].

3. Add the Constant Term: Finally, add the constant term 4 from the original equation.
[tex]\(-9x^2 - 90x - 225 + 4 = -9x^2 - 90x - 221\)[/tex].

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

The correct answer from the given options is:
[tex]\[ f(x) = -9x^2 - 90x - 221 \][/tex]