Answer :
Sure! Let's find the standard form of the given function by expanding it step-by-step.
We start with the function:
[tex]\[ f(x) = -9(x+5)^2 + 4 \][/tex]
### Step 1: Expand the Square
First, we need to expand the expression [tex]\((x+5)^2\)[/tex]. Using the formula [tex]\((a+b)^2 = a^2 + 2ab + b^2\)[/tex], we get:
[tex]\[
(x+5)^2 = x^2 + 2 \cdot x \cdot 5 + 5^2 = x^2 + 10x + 25
\][/tex]
### Step 2: Multiply by [tex]\(-9\)[/tex]
Now, multiply each term in the expanded expression by [tex]\(-9\)[/tex]:
[tex]\[
-9(x^2 + 10x + 25) = -9x^2 - 90x - 225
\][/tex]
### Step 3: Add the Constant
Finally, add the constant [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]
This matches the option:
[tex]\[ f(x) = -9x^2 - 90x - 221 \][/tex]
Therefore, this is the standard form of the function.
We start with the function:
[tex]\[ f(x) = -9(x+5)^2 + 4 \][/tex]
### Step 1: Expand the Square
First, we need to expand the expression [tex]\((x+5)^2\)[/tex]. Using the formula [tex]\((a+b)^2 = a^2 + 2ab + b^2\)[/tex], we get:
[tex]\[
(x+5)^2 = x^2 + 2 \cdot x \cdot 5 + 5^2 = x^2 + 10x + 25
\][/tex]
### Step 2: Multiply by [tex]\(-9\)[/tex]
Now, multiply each term in the expanded expression by [tex]\(-9\)[/tex]:
[tex]\[
-9(x^2 + 10x + 25) = -9x^2 - 90x - 225
\][/tex]
### Step 3: Add the Constant
Finally, add the constant [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]
This matches the option:
[tex]\[ f(x) = -9x^2 - 90x - 221 \][/tex]
Therefore, this is the standard form of the function.
