To determine which point among the given options is a solution to the equation [tex]\( y = (x - 2)^2 \)[/tex], we need to substitute each point into the equation and check if it holds true.
Let's consider each point one by one:
1. Point [tex]\((3, 1)\)[/tex]:
- Substitute [tex]\( x = 3 \)[/tex] and [tex]\( y = 1 \)[/tex] into the equation [tex]\( y = (x - 2)^2 \)[/tex]:
[tex]\[
y = (3 - 2)^2 = 1^2 = 1
\][/tex]
This is true since [tex]\( y = 1 \)[/tex]. Thus, the point [tex]\((3, 1)\)[/tex] satisfies the equation.
2. Point [tex]\((1, -1)\)[/tex]:
- Substitute [tex]\( x = 1 \)[/tex] and [tex]\( y = -1 \)[/tex] into the equation [tex]\( y = (x - 2)^2 \)[/tex]:
[tex]\[
y = (1 - 2)^2 = (-1)^2 = 1
\][/tex]
This is not true since [tex]\( y \neq -1 \)[/tex]. Thus, the point [tex]\((1, -1)\)[/tex] does not satisfy the equation.
3. Point [tex]\((4, 0)\)[/tex]:
- Substitute [tex]\( x = 4 \)[/tex] and [tex]\( y = 0 \)[/tex] into the equation [tex]\( y = (x - 2)^2 \)[/tex]:
[tex]\[
y = (4 - 2)^2 = 2^2 = 4
\][/tex]
This is not true since [tex]\( y \neq 0 \)[/tex]. Thus, the point [tex]\((4, 0)\)[/tex] does not satisfy the equation.
4. Point [tex]\((-3, 1)\)[/tex]:
- Substitute [tex]\( x = -3 \)[/tex] and [tex]\( y = 1 \)[/tex] into the equation [tex]\( y = (x - 2)^2 \)[/tex]:
[tex]\[
y = (-3 - 2)^2 = (-5)^2 = 25
\][/tex]
This is not true since [tex]\( y \neq 1 \)[/tex]. Thus, the point [tex]\((-3, 1)\)[/tex] does not satisfy the equation.
Having checked all points, the point [tex]\((3, 1)\)[/tex] is the only one that satisfies the equation [tex]\( y = (x - 2)^2 \)[/tex].
Thus, the answer is:
(A) [tex]\((3, 1)\)[/tex]
