To determine the original coordinates of the point before the transformation, we need to understand the given transformation rule:
[tex]\[ r_y = -x(x, y) \rightarrow (-4, 9) \][/tex]
This rule implies a specific transformation applied to the coordinates [tex]\((x, y)\)[/tex]. Let's break it down:
1. The first part of the transformation, [tex]\( -x \)[/tex], suggests that the original x-coordinate of the point is negated in the transformed point.
2. The second part indicates that the original y-coordinate is preserved to become the x-coordinate in the transformed point.
Given the transformed coordinates [tex]\((-4, 9)\)[/tex]:
- The transformed x-coordinate is [tex]\(-4\)[/tex]. According to the rule, this means the original y-coordinate was [tex]\(-(-4) = 4\)[/tex].
- The transformed y-coordinate is [tex]\(9\)[/tex]. According to the rule, this means the original x-coordinate was [tex]\(9\)[/tex].
Therefore, reversing the transformation, the original coordinates of the point are [tex]\((9, 4)\)[/tex].
So, among the given options:
- [tex]\((-9, 4)\)[/tex]
- [tex]\((-4, -9)\)[/tex]
- [tex]\((4, 9)\)[/tex]
- [tex]\((9, -4)\)[/tex]
The correct pre-image of the point is:
[tex]\[ (9, 4) \][/tex]
