To determine the pre-image of the given points under the transformation rule [tex]\( r_{y\text{-axis}}: (x, y) \rightarrow (-x, y) \)[/tex], we need to apply the reverse transformation to each point. The reverse transformation of [tex]\( (-x, y) \)[/tex] is [tex]\( (x, y) \)[/tex].
Let's find the pre-image of each point step by step:
1. For the point [tex]\( (-4, 2) \)[/tex]:
The transformation rule [tex]\( r_{y\text{-axis}} \)[/tex] sends [tex]\( (x, y) \)[/tex] to [tex]\( (-x, y) \)[/tex].
We perform the reverse transformation:
[tex]\[
(-x, y) = (-4, 2) \implies x = 4, y = 2
\][/tex]
Therefore, the pre-image of [tex]\( (-4, 2) \)[/tex] is [tex]\( (4, 2) \)[/tex].
2. For the point [tex]\( (-2, -4) \)[/tex]:
The reverse transformation:
[tex]\[
(-x, y) = (-2, -4) \implies x = 2, y = -4
\][/tex]
Thus, the pre-image of [tex]\( (-2, -4) \)[/tex] is [tex]\( (2, -4) \)[/tex].
3. For the point [tex]\( (2, 4) \)[/tex]:
The reverse transformation:
[tex]\[
(-x, y) = (2, 4) \implies x = -2, y = 4
\][/tex]
Hence, the pre-image of [tex]\( (2, 4) \)[/tex] is [tex]\( (-2, 4) \)[/tex].
To summarize, the pre-images of the given points are:
[tex]\[
A(-4, 2) \rightarrow (4, 2)
\][/tex]
[tex]\[
A(-2, -4) \rightarrow (2, -4)
\][/tex]
[tex]\[
A(2, 4) \rightarrow (-2, 4)
\][/tex]
