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]
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]
