To find the pre-image of a point under the reflection rule \( r_{y=-x} \), let's carefully analyze the reflection over the line \( y = -x \).
When a point \((x, y)\) is reflected over the line \( y = -x \):
1. The x-coordinate of the new point will be the negative of the y-coordinate of the original point.
2. The y-coordinate of the new point will be the negative of the x-coordinate of the original point.
Given that the reflected point (image) is \((-4, 9)\), we need to determine the coordinates of the original point (pre-image) that produced this reflection.
Let's solve this step-by-step:
1. Let's denote the pre-image coordinates by \((x, y)\).
2. According to the reflection rule \( r_{y=-x} \), the pre-image point \((x, y)\) will transform as:
[tex]\[
(x, y) \rightarrow (-y, -x)
\][/tex]
3. The given image point is \((-4, 9)\). Therefore, we set up the equations based on the rule:
[tex]\[
(-y, -x) = (-4, 9)
\][/tex]
4. From this, we can extract two equations:
[tex]\[
-y = -4 \quad \Rightarrow \quad y = 4
\][/tex]
[tex]\[
-x = 9 \quad \Rightarrow \quad x = -9
\][/tex]
So, the coordinates of the original point (pre-image) are \((-9, 4)\).
Comparing this with the provided options:
- \((-9, 4)\)
- \((-4, -9)\)
- \((4, 9)\)
- \((9, -4)\)
The correct answer is:
[tex]\[
(-9, 4)
\][/tex]
