Answer :
Sure! Let's find the standard form of the given function step-by-step.
We start with the function in its vertex form:
[tex]\[ f(x) = -9(x + 5)^2 + 4 \][/tex]
To convert this function into its standard form (which is of the form [tex]\( ax^2 + bx + c \)[/tex]), we need to expand the expression.
1. First, let's expand the squared term [tex]\((x + 5)^2\)[/tex]:
[tex]\[ (x + 5)^2 = x^2 + 10x + 25 \][/tex]
2. Now, substitute this expansion back into the original function:
[tex]\[ f(x) = -9(x^2 + 10x + 25) + 4 \][/tex]
3. Next, distribute the [tex]\(-9\)[/tex] over each term inside the parentheses:
[tex]\[ f(x) = -9x^2 - 90x - 225 + 4 \][/tex]
4. Finally, combine the constant terms [tex]\(-225\)[/tex] and [tex]\(4\)[/tex]:
[tex]\[ f(x) = -9x^2 - 90x - 221 \][/tex]
Thus, the standard form of the given function is:
[tex]\[ f(x) = -9x^2 - 90x - 221 \][/tex]
So, the correct answer is:
[tex]\[ \boxed{f(x) = -9x^2 - 90x - 221} \][/tex]
We start with the function in its vertex form:
[tex]\[ f(x) = -9(x + 5)^2 + 4 \][/tex]
To convert this function into its standard form (which is of the form [tex]\( ax^2 + bx + c \)[/tex]), we need to expand the expression.
1. First, let's expand the squared term [tex]\((x + 5)^2\)[/tex]:
[tex]\[ (x + 5)^2 = x^2 + 10x + 25 \][/tex]
2. Now, substitute this expansion back into the original function:
[tex]\[ f(x) = -9(x^2 + 10x + 25) + 4 \][/tex]
3. Next, distribute the [tex]\(-9\)[/tex] over each term inside the parentheses:
[tex]\[ f(x) = -9x^2 - 90x - 225 + 4 \][/tex]
4. Finally, combine the constant terms [tex]\(-225\)[/tex] and [tex]\(4\)[/tex]:
[tex]\[ f(x) = -9x^2 - 90x - 221 \][/tex]
Thus, the standard form of the given function is:
[tex]\[ f(x) = -9x^2 - 90x - 221 \][/tex]
So, the correct answer is:
[tex]\[ \boxed{f(x) = -9x^2 - 90x - 221} \][/tex]
