Question

the image of a point is given by the rule $r_{y=-x}(x, y) \to (-4, 9)$. what are the coordinates of its pre-image?
$\circ$ $(-9, 4)$
$\circ$ $(-4, -9)$
$\circ$ $(4, 9)$
$\circ$ $(9, -4)$

Explanation:

Step1: Recall the reflection rule over \( y = -x \)

The rule for reflecting a point \((x, y)\) over the line \( y=-x \) is \( r_{y = -x}(x, y)=(-y, -x) \). Let the pre - image be \((x, y)\) and the image be \((- 4,9)\). So we have the equations:
\(-y=-4\) and \(-x = 9\)

Step2: Solve for \(x\) and \(y\)

From \(-y=-4\), multiply both sides by \(- 1\) to get \(y = 4\).
From \(-x=9\), multiply both sides by \(-1\) to get \(x=-9\).
So the pre - image \((x, y)\) is \((-9,4)\).

Answer:

\((-9, 4)\) (corresponding to the option \((-9,4)\))