Answer :
Sure, let's work through the problem step-by-step to find the standard form of the function [tex]\( f(x) = -9(x+5)^2 + 4 \)[/tex].
1. Expand [tex]\((x+5)^2\)[/tex]:
To expand [tex]\((x+5)^2\)[/tex], use the formula [tex]\((a+b)^2 = a^2 + 2ab + b^2\)[/tex].
[tex]\[
(x+5)^2 = x^2 + 2 \cdot 5 \cdot x + 5^2 = x^2 + 10x + 25
\][/tex]
2. Distribute [tex]\(-9\)[/tex] across the expanded form:
Now, multiply [tex]\(-9\)[/tex] with each term in the expansion:
[tex]\[
-9(x^2 + 10x + 25) = -9x^2 - 90x - 225
\][/tex]
3. Add 4 to the result:
Finally, add the constant term 4 to the expanded and multiplied expression:
[tex]\[
-9x^2 - 90x - 225 + 4 = -9x^2 - 90x - 221
\][/tex]
So, the standard form of the given function is:
[tex]\[
f(x) = -9x^2 - 90x - 221
\][/tex]
This matches the second option in your list:
[tex]\[
f(x) = -9x^2 - 90x - 221
\][/tex]
1. Expand [tex]\((x+5)^2\)[/tex]:
To expand [tex]\((x+5)^2\)[/tex], use the formula [tex]\((a+b)^2 = a^2 + 2ab + b^2\)[/tex].
[tex]\[
(x+5)^2 = x^2 + 2 \cdot 5 \cdot x + 5^2 = x^2 + 10x + 25
\][/tex]
2. Distribute [tex]\(-9\)[/tex] across the expanded form:
Now, multiply [tex]\(-9\)[/tex] with each term in the expansion:
[tex]\[
-9(x^2 + 10x + 25) = -9x^2 - 90x - 225
\][/tex]
3. Add 4 to the result:
Finally, add the constant term 4 to the expanded and multiplied expression:
[tex]\[
-9x^2 - 90x - 225 + 4 = -9x^2 - 90x - 221
\][/tex]
So, the standard form of the given function is:
[tex]\[
f(x) = -9x^2 - 90x - 221
\][/tex]
This matches the second option in your list:
[tex]\[
f(x) = -9x^2 - 90x - 221
\][/tex]
