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Answer :
To convert the given function into standard form, we start with the function given in vertex form:
[tex]\[ f(x) = -9(x+5)^2 + 4 \][/tex]
Our goal is to expand this expression into the standard form of a quadratic function, which is:
[tex]\[ ax^2 + bx + c \][/tex]
Let's expand the expression step by step:
1. Expand the squared term:
[tex]\((x+5)^2 = x^2 + 10x + 25\)[/tex]
2. Multiply by -9:
[tex]\(-9(x^2 + 10x + 25)\)[/tex]
Distribute the -9 across each term inside the parentheses:
[tex]\(-9 \cdot x^2 = -9x^2\)[/tex]
[tex]\(-9 \cdot 10x = -90x\)[/tex]
[tex]\(-9 \cdot 25 = -225\)[/tex]
So, after distribution, we have:
[tex]\[-9x^2 - 90x - 225\][/tex]
3. Add the constant term 4:
Combine [tex]\(-225\)[/tex] and [tex]\(4\)[/tex]:
[tex]\(-225 + 4 = -221\)[/tex]
Putting it all together, the expanded form of the function is:
[tex]\[ f(x) = -9x^2 - 90x - 221 \][/tex]
Therefore, the function in standard form is:
[tex]\[ f(x) = -9x^2 - 90x - 221 \][/tex]
So, the correct answer is:
[tex]\[ f(x) = -9x^2 - 90x - 221 \][/tex]
[tex]\[ f(x) = -9(x+5)^2 + 4 \][/tex]
Our goal is to expand this expression into the standard form of a quadratic function, which is:
[tex]\[ ax^2 + bx + c \][/tex]
Let's expand the expression step by step:
1. Expand the squared term:
[tex]\((x+5)^2 = x^2 + 10x + 25\)[/tex]
2. Multiply by -9:
[tex]\(-9(x^2 + 10x + 25)\)[/tex]
Distribute the -9 across each term inside the parentheses:
[tex]\(-9 \cdot x^2 = -9x^2\)[/tex]
[tex]\(-9 \cdot 10x = -90x\)[/tex]
[tex]\(-9 \cdot 25 = -225\)[/tex]
So, after distribution, we have:
[tex]\[-9x^2 - 90x - 225\][/tex]
3. Add the constant term 4:
Combine [tex]\(-225\)[/tex] and [tex]\(4\)[/tex]:
[tex]\(-225 + 4 = -221\)[/tex]
Putting it all together, the expanded form of the function is:
[tex]\[ f(x) = -9x^2 - 90x - 221 \][/tex]
Therefore, the function in standard form is:
[tex]\[ f(x) = -9x^2 - 90x - 221 \][/tex]
So, the correct answer is:
[tex]\[ f(x) = -9x^2 - 90x - 221 \][/tex]
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