To solve this question, we need to understand the transformation rule [tex]\( r_{y=-x} \)[/tex]. This rule reflects a point across the line [tex]\( y = -x \)[/tex]. When reflecting a point [tex]\((x, y)\)[/tex] across this line, we swap the coordinates and change their signs.
Given the image of the point is [tex]\((-4, 9)\)[/tex], we need to determine the coordinates of its pre-image. Following the transformation rule [tex]\( r_{y=-x} \)[/tex]:
1. Swap the coordinates: Start from the image coordinates [tex]\((-4, 9)\)[/tex].
- The x-coordinate of the image becomes the y-coordinate of the pre-image.
- The y-coordinate of the image becomes the x-coordinate of the pre-image.
2. Change the signs of the swapped coordinates: The value that was the x-coordinate of the image (which is now the y-coordinate of the pre-image) has its sign changed. Similarly, the value that was the y-coordinate of the image (which is now the x-coordinate of the pre-image) also has its sign changed.
Let's apply these steps:
1. Swap the coordinates [tex]\((-4, 9)\)[/tex]:
- The swapped coordinates are [tex]\((9, -4)\)[/tex].
2. Change the signs of the swapped coordinates:
- The x-coordinate [tex]\(9\)[/tex] becomes [tex]\(-9\)[/tex],
- The y-coordinate [tex]\(-4\)[/tex] becomes [tex]\(4\)[/tex].
Thus, the coordinates of the pre-image are [tex]\((-9, 4)\)[/tex].
Answer: [tex]\((-9, 4)\)[/tex]
