College

Select the correct answer.

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

Given: [tex]f(x) = -9(x+5)^2 + 4[/tex]

A. [tex]f(x) = -9x^2 - 180x - 221[/tex]

B. [tex]f(x) = 9x^2 - 90x - 221[/tex]

C. [tex]f(x) = 9x^2 - 180x + 221[/tex]

D. [tex]f(x) = -9x^2 - 90x - 221[/tex]

Answer :

We are given the function

$$
f(x) = -9(x+5)^2 + 4.
$$

Our goal is to convert it into standard form, that is

$$
f(x) = ax^2 + bx + c.
$$

**Step 1. Expand the square**

First, expand the binomial:

$$
(x+5)^2 = x^2 + 2\cdot5\cdot x + 5^2 = x^2 + 10x + 25.
$$

**Step 2. Distribute the coefficient**

Multiply the expanded form by $-9$:

$$
-9(x^2 + 10x + 25) = -9x^2 - 90x - 225.
$$

**Step 3. Add the constant**

Now, add the constant $+4$:

$$
-9x^2 - 90x - 225 + 4 = -9x^2 - 90x - 221.
$$

Thus, the standard form of the function is:

$$
f(x) = -9x^2 - 90x - 221.
$$

This corresponds to the correct answer.