To determine the coordinates of the zero(s) of the function [tex]\(f(x) = -(x + 3)^2\)[/tex], we need to find the values of [tex]\(x\)[/tex] for which [tex]\(f(x) = 0\)[/tex].
Let's solve the equation [tex]\(f(x) = 0\)[/tex]:
1. Set the function equal to zero:
[tex]\[
-(x + 3)^2 = 0
\][/tex]
2. Eliminate the negative sign (it does not affect the equality):
[tex]\[
(x + 3)^2 = 0
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
[tex]\[
(x + 3)^2 = 0 \implies x + 3 = 0 \implies x = -3
\][/tex]
The value of [tex]\(x\)[/tex] at which the function [tex]\(f(x) = 0\)[/tex] is [tex]\(x = -3\)[/tex].
Now, find the corresponding [tex]\(y\)[/tex]-coordinate by evaluating [tex]\(f(x)\)[/tex] at [tex]\(x = -3\)[/tex]:
[tex]\[
f(-3) = -( (-3) + 3 )^2 = -(0)^2 = 0
\][/tex]
Therefore, the function [tex]\(f(x) = -(x + 3)^2\)[/tex] has a zero at the coordinate [tex]\((-3,0)\)[/tex].
Given the options, the correct answer is:
(A) [tex]\((-3, 0)\)[/tex] only
