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Select the correct answer.

Which of these is the standard form of the following function?

1. [tex]f(x) = -9(x+5)^2 + 4[/tex]
2. [tex]f(x) = -9x^2 - 90x - 221[/tex]
3. [tex]f(x) = 9x^2 - 90x - 221[/tex]
4. [tex]f(x) = -9x^2 - 180x - 221[/tex]
5. [tex]f(x) = 9x^2 - 180x + 221[/tex]

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.

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