Sure! Let's go through the problem step-by-step.
First, we'll understand the given transformation rule: the rule is [tex]\( r_y \)[/tex], which transforms a point [tex]\( (x, y) \)[/tex] to [tex]\((-x, y)\)[/tex]. This rule reflects each point across the y-axis.
To find the pre-images of the given points under this transformation, we apply the inverse of this rule.
### Original Points and Their Reflections:
1. Point [tex]\(A(-4, 2)\)[/tex]:
- According to the transformation rule, the point [tex]\( (-4, 2) \)[/tex] becomes [tex]\( (4, 2) \)[/tex].
2. Point [tex]\(A(-2, -4)\)[/tex]:
- According to the transformation rule, the point [tex]\( (-2, -4) \)[/tex] becomes [tex]\( (2, -4) \)[/tex].
3. Point [tex]\(A(2, 4)\)[/tex]:
- According to the transformation rule, the point [tex]\( (2, 4) \)[/tex] becomes [tex]\((-2, 4)\)[/tex].
4. Point [tex]\(A(4, -2)\)[/tex]:
- According to the transformation rule, the point [tex]\( (4, -2) \)[/tex] becomes [tex]\((-4, -2)\)[/tex].
So, the pre-images of the given vertices before the transformation [tex]\( r_y \)[/tex] are:
- For [tex]\( (4, 2) \)[/tex] the pre-image is [tex]\( A(-4, 2) \)[/tex].
- For [tex]\( (2, -4) \)[/tex] the pre-image is [tex]\( A(-2, -4) \)[/tex].
- For [tex]\( (-2, 4) \)[/tex] the pre-image is [tex]\( A(2, 4) \)[/tex].
- For [tex]\( (-4, -2) \)[/tex] the pre-image is [tex]\( A(4, -2) \)[/tex].
This gives us the set of pre-image points:
[tex]\[ \{(4, 2), (2, -4), (-2, 4), (-4, -2)\} \][/tex]
Therefore, the transformed images have pre-images:
- [tex]\( (4, 2) \)[/tex] originating from [tex]\( A(-4, 2) \)[/tex]
- [tex]\( (2, -4) \)[/tex] originating from [tex]\( A(-2, -4) \)[/tex]
- [tex]\( (-2, 4) \)[/tex] originating from [tex]\( A(2, 4) \)[/tex]
- [tex]\( (-4, -2) \)[/tex] originating from [tex]\( A(4, -2) \)[/tex]
In summary, the pre-images of the vertices under the transformation rule [tex]\( r_y \)[/tex] are:
[tex]\[ \{(4, 2), (2, -4), (-2, 4), (-4, -2)\} \][/tex]
