Certainly! Let's solve the problem step-by-step to find the original point whose reflection over the line [tex]\( y = -x \)[/tex] results in the point [tex]\((-4, 9)\)[/tex].
The rule [tex]\( r_{y = -x}(x, y) \)[/tex] tells us how to transform a point [tex]\((x, y)\)[/tex] by reflecting it across the line [tex]\( y = -x \)[/tex]. This means that if you have a point [tex]\((x, y)\)[/tex], after reflection, it will become [tex]\((-y, -x)\)[/tex].
We are given the image of the reflected point as [tex]\((-4, 9)\)[/tex]. We need to determine the original point [tex]\((x, y)\)[/tex] before it was reflected.
1. Start with the reflection rule: [tex]\( r_{y = -x}(x, y) \rightarrow (-y, -x) \)[/tex].
2. According to the problem, after applying the reflection rule, the coordinates of the image are [tex]\((-4, 9)\)[/tex]. Therefore, we set up the following equations based on the rule:
[tex]\[
-y = -4 \quad \text{and} \quad -x = 9
\][/tex]
3. Solve the equations for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[
-y = -4 \implies y = 4
\][/tex]
[tex]\[
-x = 9 \implies x = -9
\][/tex]
4. Thus, the original point before reflection is [tex]\((9, -4)\)[/tex].
To summarize, by reflecting the point [tex]\((9, -4)\)[/tex] across the line [tex]\( y = -x \)[/tex], we get the point [tex]\((-4, 9)\)[/tex]. Therefore, the original point whose image is [tex]\((-4, 9)\)[/tex] after reflection is:
[tex]\[
(9, -4)
\][/tex]
The correct answer is: [tex]\((9, -4)\)[/tex].
