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Answer :
To find the standard form of the function [tex]\( f(x) = -9(x+5)^2 + 4 \)[/tex], we need to expand the expression and simplify it. Let's go through the steps:
1. Expansion of the Binomial:
The expression inside the function is [tex]\((x+5)^2\)[/tex]. We expand this using the formula [tex]\((a+b)^2 = a^2 + 2ab + b^2\)[/tex].
- Here, [tex]\(a = x\)[/tex] and [tex]\(b = 5\)[/tex].
- So, [tex]\((x + 5)^2 = x^2 + 2 \cdot x \cdot 5 + 5^2\)[/tex].
- This simplifies to [tex]\(x^2 + 10x + 25\)[/tex].
2. Substitute and Distribute:
Substitute this expanded form back into the function:
[tex]\( f(x) = -9(x^2 + 10x + 25) + 4 \)[/tex].
Next, distribute the [tex]\(-9\)[/tex] across the terms inside the parentheses:
- Multiply [tex]\(-9\)[/tex] by each term: [tex]\(-9 \cdot x^2 = -9x^2\)[/tex],
- [tex]\(-9 \cdot 10x = -90x\)[/tex],
- [tex]\(-9 \cdot 25 = -225\)[/tex].
Now the function looks like this:
[tex]\( f(x) = -9x^2 - 90x - 225 + 4 \)[/tex].
3. Combine Like Terms:
Finally, combine the constant terms [tex]\(-225\)[/tex] and [tex]\(+4\)[/tex]:
- [tex]\(-225 + 4 = -221\)[/tex].
So, the function simplifies to:
[tex]\( f(x) = -9x^2 - 90x - 221 \)[/tex].
Therefore, the standard form of the function is:
[tex]\[ f(x) = -9x^2 - 90x - 221 \][/tex]
Among the given options, this corresponds to:
- [tex]\( f(x) = -9x^2 - 90x - 221 \)[/tex]
This is the correct standard form of the given function.
1. Expansion of the Binomial:
The expression inside the function is [tex]\((x+5)^2\)[/tex]. We expand this using the formula [tex]\((a+b)^2 = a^2 + 2ab + b^2\)[/tex].
- Here, [tex]\(a = x\)[/tex] and [tex]\(b = 5\)[/tex].
- So, [tex]\((x + 5)^2 = x^2 + 2 \cdot x \cdot 5 + 5^2\)[/tex].
- This simplifies to [tex]\(x^2 + 10x + 25\)[/tex].
2. Substitute and Distribute:
Substitute this expanded form back into the function:
[tex]\( f(x) = -9(x^2 + 10x + 25) + 4 \)[/tex].
Next, distribute the [tex]\(-9\)[/tex] across the terms inside the parentheses:
- Multiply [tex]\(-9\)[/tex] by each term: [tex]\(-9 \cdot x^2 = -9x^2\)[/tex],
- [tex]\(-9 \cdot 10x = -90x\)[/tex],
- [tex]\(-9 \cdot 25 = -225\)[/tex].
Now the function looks like this:
[tex]\( f(x) = -9x^2 - 90x - 225 + 4 \)[/tex].
3. Combine Like Terms:
Finally, combine the constant terms [tex]\(-225\)[/tex] and [tex]\(+4\)[/tex]:
- [tex]\(-225 + 4 = -221\)[/tex].
So, the function simplifies to:
[tex]\( f(x) = -9x^2 - 90x - 221 \)[/tex].
Therefore, the standard form of the function is:
[tex]\[ f(x) = -9x^2 - 90x - 221 \][/tex]
Among the given options, this corresponds to:
- [tex]\( f(x) = -9x^2 - 90x - 221 \)[/tex]
This is the correct standard form of the given function.
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