To find the corresponding coordinate on the left parabola \(y = (-x-2)^2\) that reflects to the right parabola \(f(x) = (x-2)^2\), we first rewrite the equations in a more usable form.
-
Left parabola:
- \(y = (-x - 2)^2\) can be rewritten as \(y = (-(x + 2))^2 = (x + 2)^2\) reflected over the y-axis.
- The vertex of the left parabola is at \( (-2, 0) \).
-
Right parabola:
- \(f(x) = (x - 2)^2\) has a vertex at \( (2, 0) \).
To find the coordinate point correspondence:
- We reflect points over the vertical line \(x = 0\) (the y-axis).
- For any point on the left parabola \( (x_1, y_1) \), the corresponding point becomes \( (-x_1, y_1) \) on the right parabola.
Let's take various values of \( x_1 \) in the left parabola and find the corresponding points, and see if they match any \( x \) values in the options.
For \(x_1 = -1\):
- Left: \((-1, (-(-1)-2)^2)\) = \((-1, 1^2)\) = \((-1, 1)\)
- Reflection: \((1, f(1)) = (1, (1-2)^2) = (1, 1)\)
For \(x_1 = -3\):
- Left: \((-3, (-(-3)-2)^2)\) = \((-3, (1)^2) = (-3, 1)\)
- Reflection: \((3, f(3)) = (3, (3-2)^2) = (3, 1)\)
For \(x_1 = -2\):
- Left: \((-2, (-(-2)-2)^2)\) = \((-2, 0)\)
- Reflection: \( (2, f(2)) = (2, (2-2)^2) = (2, 0)\)
By looking at the points, we have the reflection pairs:
- Left: \((-1, 1) \rightarrow\) Right: \((1, 1)\)
- Left: \((-3, 1) \rightarrow\) Right: \((3, 1)\)
- Left: \((-2, 0) \rightarrow\) Right: \((2, 0)\)
From this analysis, the coordinates that correspond correctly to the right parabola's vertex or point \( (2, 0) \) is the coordinate from the left parabola \((-2, 0)\).
Thus, the coordinate on the left parabola that corresponds to the identified coordinate on the right parabola \( (2, 0) \) is:
\((-2, 0)\)
So the final answer is:
\((-2, 0)\)