Answer :
To find the standard form of the function [tex]\( f(x) = -9(x + 5)^2 + 4 \)[/tex], we need to expand and simplify the expression.
Here’s a step-by-step guide:
1. Expand the binomial:
[tex]\((x + 5)^2\)[/tex] means multiplying the binomial by itself:
[tex]\((x + 5)(x + 5)\)[/tex].
2. Use the distributive property (FOIL method) to expand:
[tex]\[
(x + 5)^2 = x^2 + 5x + 5x + 25 = x^2 + 10x + 25
\][/tex]
3. Substitute back into the function:
Replace [tex]\((x + 5)^2\)[/tex] in the original equation:
[tex]\[
f(x) = -9(x^2 + 10x + 25) + 4
\][/tex]
4. Distribute [tex]\(-9\)[/tex] across the trinomial:
[tex]\[
f(x) = -9 \cdot x^2 - 9 \cdot 10x - 9 \cdot 25 + 4
\][/tex]
[tex]\[
f(x) = -9x^2 - 90x - 225 + 4
\][/tex]
5. Combine like terms (constant terms):
Simplify the constants:
[tex]\[
-225 + 4 = -221
\][/tex]
6. Write the final expression in standard form:
[tex]\[
f(x) = -9x^2 - 90x - 221
\][/tex]
Therefore, the correct standard form of the function is:
[tex]\[
f(x) = -9x^2 - 90x - 221
\][/tex]
This matches with the option [tex]\( f(x) = -9 x^2 - 90 x - 221 \)[/tex].
Here’s a step-by-step guide:
1. Expand the binomial:
[tex]\((x + 5)^2\)[/tex] means multiplying the binomial by itself:
[tex]\((x + 5)(x + 5)\)[/tex].
2. Use the distributive property (FOIL method) to expand:
[tex]\[
(x + 5)^2 = x^2 + 5x + 5x + 25 = x^2 + 10x + 25
\][/tex]
3. Substitute back into the function:
Replace [tex]\((x + 5)^2\)[/tex] in the original equation:
[tex]\[
f(x) = -9(x^2 + 10x + 25) + 4
\][/tex]
4. Distribute [tex]\(-9\)[/tex] across the trinomial:
[tex]\[
f(x) = -9 \cdot x^2 - 9 \cdot 10x - 9 \cdot 25 + 4
\][/tex]
[tex]\[
f(x) = -9x^2 - 90x - 225 + 4
\][/tex]
5. Combine like terms (constant terms):
Simplify the constants:
[tex]\[
-225 + 4 = -221
\][/tex]
6. Write the final expression in standard form:
[tex]\[
f(x) = -9x^2 - 90x - 221
\][/tex]
Therefore, the correct standard form of the function is:
[tex]\[
f(x) = -9x^2 - 90x - 221
\][/tex]
This matches with the option [tex]\( f(x) = -9 x^2 - 90 x - 221 \)[/tex].
