What is the pre-image of vertex [tex]A'[/tex] if the rule that created the image is [tex]r_{y \text{-axis}}(x, y) \rightarrow (-x, y)[/tex]?

A. [tex]A(-4, 2)[/tex]
B. [tex]A(-2, -4)[/tex]
C. [tex]A(2, 4)[/tex]
D. [tex]A(4, -2)[/tex]



Answer :

To determine the pre-image of a vertex \( A' \) given the rule \( r_{y \text{-axis}} (x, y) \rightarrow (-x, y) \), we need to understand that this rule reflects points over the y-axis. This reflection changes the sign of the x-coordinate while leaving the y-coordinate unchanged.

Now, let's determine the pre-image of each of the possible given points:

1. Given point \( A(-4, 2) \):
- If this is a reflection of a point over the y-axis, then we start with the point before reflection, which would be \((x, y) = (4, 2)\).

2. Given point \( A(-2, -4) \):
- Applying the reflection rule in reverse, we change the sign of the x-coordinate, so the pre-image would be \((x, y) = (2, -4)\).

3. Given point \( A(2, 4) \):
- Applying the reflection rule in reverse, we change the sign of the x-coordinate, so the pre-image would be \((x, y) = (-2, 4)\).

4. Given point \( A(4, -2) \):
- Applying the reflection rule in reverse, we change the sign of the x-coordinate, so the pre-image would be \((x, y) = (-4, -2)\).

Thus, the pre-images for the given points are:
- For \( A(-4, 2) \), the pre-image is \( (4, 2) \).
- For \( A(-2, -4) \), the pre-image is \( (2, -4) \).
- For \( A(2, 4) \), the pre-image is \( (-2, 4) \).
- For \( A(4, -2) \), the pre-image is \( (-4, -2) \).

So the pre-images are:
[tex]\[ \begin{aligned} & (4, 2), \\ & (2, -4), \\ & (-2, 4), \\ & (-4, -2). \end{aligned} \][/tex]