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Finding a Pre-image

What is the pre-image of vertex [tex]\( A^{\prime} \)[/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 the vertex [tex]\( A' \)[/tex] given the rule of reflection across the y-axis [tex]\( r_{\text{y-axis}}(x, y) \rightarrow (-x, y) \)[/tex]:

1. Understand the Reflection Rule: When reflecting a point across the y-axis, the x-coordinate changes sign while the y-coordinate remains the same. Thus, for a point [tex]\((x, y)\)[/tex] under reflection, the image becomes [tex]\((-x, y)\)[/tex].

2. Identify the Given Image [tex]\( A' \)[/tex]:
Suppose the coordinates of the image vertex [tex]\( A' \)[/tex] are [tex]\( (4, 2) \)[/tex].

3. Use the Rule to Find Pre-image:
- According to [tex]\( r_{\text{y-axis}}(x, y) \rightarrow (-x, y) \)[/tex], given the image [tex]\( A' = (4, 2) \)[/tex]:
- Let the pre-image be [tex]\((x, y)\)[/tex]. For the point [tex]\((4, 2)\)[/tex] to be the reflected image, we reverse the sign change transformation:
[tex]\[ -x = 4 \implies x = -4 \][/tex]
The y-coordinate remains unchanged:
[tex]\[ y = 2 \][/tex]

4. Combine the Coordinates and Verify:
Thus, the pre-image [tex]\( A \)[/tex] is [tex]\((-4, 2)\)[/tex].

Therefore, the pre-image of the vertex [tex]\( A' \)[/tex] is [tex]\( \boxed{(-4, 2)} \)[/tex].

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