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Answer :
To find the standard form of the given function [tex]\( f(x) = -9(x+5)^2 + 4 \)[/tex], we need to expand and simplify it. Here's how you can do it:
1. Expand the squared term:
The expression [tex]\( (x+5)^2 \)[/tex] needs to be expanded. Remember the formula for squaring a binomial:
[tex]\[
(a+b)^2 = a^2 + 2ab + b^2
\][/tex]
Applying this formula, we get:
[tex]\[
(x+5)^2 = x^2 + 2 \cdot x \cdot 5 + 5^2 = x^2 + 10x + 25
\][/tex]
2. Multiply by [tex]\(-9\)[/tex]:
Now, take the expanded form and multiply each term 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 [tex]\(+4\)[/tex] to the expression:
[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 is:
[tex]\[
f(x) = -9x^2 - 90x - 221
\][/tex]
1. Expand the squared term:
The expression [tex]\( (x+5)^2 \)[/tex] needs to be expanded. Remember the formula for squaring a binomial:
[tex]\[
(a+b)^2 = a^2 + 2ab + b^2
\][/tex]
Applying this formula, we get:
[tex]\[
(x+5)^2 = x^2 + 2 \cdot x \cdot 5 + 5^2 = x^2 + 10x + 25
\][/tex]
2. Multiply by [tex]\(-9\)[/tex]:
Now, take the expanded form and multiply each term 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 [tex]\(+4\)[/tex] to the expression:
[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 is:
[tex]\[
f(x) = -9x^2 - 90x - 221
\][/tex]
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