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The image of a point is given by the rule [tex]r_{y=-x}(x, y) = (-4, 9)[/tex]. What are the coordinates of its pre-image?

A. [tex]\((-9, 4)\)[/tex]
B. [tex]\((-4, -9)\)[/tex]
C. [tex]\((4, 9)\)[/tex]
D. [tex]\((9, -4)\)[/tex]



Answer :

GinnyAnswer
Let's explore the rule for reflection across the line [tex]\( y = -x \)[/tex]. When a point [tex]\( (x, y) \)[/tex] is reflected across this line, its coordinates are swapped and their signs are inverted. Specifically, the point [tex]\( (x, y) \)[/tex] becomes [tex]\( (-y, -x) \)[/tex].

Given the reflected point is [tex]\((-4, 9)\)[/tex], we need to find the original point, or pre-image, that results in this coordinate after reflection.

Let's denote the original point by [tex]\( (a, b) \)[/tex]. According to the reflection rule:

1. Swap the coordinates [tex]\( (a, b) \)[/tex] and change their signs.
2. The point [tex]\( (a, b) \)[/tex] becomes [tex]\( (-b, -a) \)[/tex].

Given:
- After reflection, the coordinates are [tex]\((-4, 9)\)[/tex].

From the reflection rule:
- The original x-coordinate [tex]\( -b \)[/tex] must equal [tex]\(-4\)[/tex].
- The original y-coordinate [tex]\( -a \)[/tex] must equal [tex]\(9\)[/tex].

To find the original coordinates:
1. Solve for [tex]\( b \)[/tex]:
[tex]\[ -b = -4 \implies b = 4 \][/tex]
2. Solve for [tex]\( a \)[/tex]:
[tex]\[ -a = 9 \implies a = -9 \][/tex]

Therefore, the original point, or the pre-image, is:
[tex]\[ (a, b) = (-9, 4) \][/tex]

So the correct answer is:
[tex]\[ (-9, 4) \][/tex]

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