College

Select the correct answer.

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

[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 :

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]