College

Select the correct answer.

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

A. [tex]f(x) = -9(x + 5)^2 + 4[/tex]
B. [tex]f(x) = 9x^2 - 180x + 221[/tex]
C. [tex]f(x) = 9x^2 - 90x - 221[/tex]
D. [tex]f(x) = -9x^2 - 180x - 221[/tex]
E. [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 work through the process step-by-step:

1. Start with the Original Function:
We have [tex]\( f(x) = -9(x+5)^2 + 4 \)[/tex].

2. Expand the Squared Term:
First, expand the expression [tex]\((x+5)^2\)[/tex]:
[tex]\[
(x+5)^2 = x^2 + 10x + 25
\][/tex]

3. Substitute the Expanded Form:
Replace [tex]\((x+5)^2\)[/tex] in the original function with its expanded form:
[tex]\[
f(x) = -9(x^2 + 10x + 25) + 4
\][/tex]

4. Distribute the [tex]\(-9\)[/tex]:
Distribute [tex]\(-9\)[/tex] through the terms within the parentheses:
[tex]\[
f(x) = -9x^2 - 90x - 225
\][/tex]

5. Combine the Constant Terms:
Now, combine the constant term from the expansion [tex]\(-225\)[/tex] with the [tex]\(+4\)[/tex] outside:
[tex]\[
f(x) = -9x^2 - 90x - 225 + 4 = -9x^2 - 90x - 221
\][/tex]

Therefore, the standard form of the function is [tex]\(-9x^2 - 90x - 221\)[/tex].

The correct answer is:

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