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Answer :
To find the standard form of the function [tex]\( f(x) = -9(x+5)^2 + 4 \)[/tex], you need to expand it step by step. Here's how:
1. Expand the Squared Binomial:
The expression inside the function is [tex]\((x + 5)^2\)[/tex]. Using the formula [tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex], we get:
[tex]\((x + 5)^2 = x^2 + 10x + 25\)[/tex].
2. Distribute the [tex]\(-9\)[/tex]:
Next, multiply each term of the expanded binomial by [tex]\(-9\)[/tex]:
[tex]\(-9(x^2 + 10x + 25) = -9x^2 - 90x - 225\)[/tex].
3. Add the Constant Term:
Add the [tex]\(+4\)[/tex] from the original function to the result from step 2:
[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]
Thus, the correct answer is: [tex]\(-9x^2 - 90x - 221\)[/tex].
1. Expand the Squared Binomial:
The expression inside the function is [tex]\((x + 5)^2\)[/tex]. Using the formula [tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex], we get:
[tex]\((x + 5)^2 = x^2 + 10x + 25\)[/tex].
2. Distribute the [tex]\(-9\)[/tex]:
Next, multiply each term of the expanded binomial by [tex]\(-9\)[/tex]:
[tex]\(-9(x^2 + 10x + 25) = -9x^2 - 90x - 225\)[/tex].
3. Add the Constant Term:
Add the [tex]\(+4\)[/tex] from the original function to the result from step 2:
[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]
Thus, the correct answer is: [tex]\(-9x^2 - 90x - 221\)[/tex].
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