Question

what is the rule for the reflection?
$r_{y=x}(x,y) \to (-y,-x)$
$r_{y=-x}(x,y) \to (-y,-x)$
$r_{y=x}(x,y) \to (y,x)$
$r_{y=-x}(x,y) \to (y,x)$

Explanation:

Step1: Identify coordinates of a point

Take point \( A(-2, 1) \) from the original triangle. After reflection, \( A' \) should be at \( ( - 1, 2) \)? Wait, no, let's check the graph again. Wait, original points: Let's find coordinates of \( A \), \( B \), \( C \). From the grid, \( A(-2,1) \), \( B(-3,4) \), \( C(-4,1) \). Reflected points \( A'(1, - 2) \)? Wait, no, looking at the lower triangle, \( A' \) seems to be \( (1, - 2) \)? Wait, no, maybe I misread. Wait, the line of reflection is \( y = -x \). The rule for reflection over \( y=-x \) is \( (x,y)\to(-y,-x) \). Let's test with point \( A(-2,1) \). Applying \( (-y,-x) \), we get \( (-1, 2) \)? Wait, no, maybe I got the coordinates wrong. Wait, original triangle: let's take point \( B(-3,4) \). Reflect over \( y=-x \): \( (x,y)\to(-y,-x) \), so \( (-4, 3) \)? Wait, no, the reflected triangle \( B' \) seems to be at \( (4, - 3) \)? Wait, maybe I messed up the coordinates. Wait, the correct rule for reflection over \( y = -x \) is \( (x,y)\to(-y,-x) \). Let's check the options. The options have \( r_{y = -x}(x,y)\to(-y,-x) \) as one of them. So that's the correct rule.

Answer:

$r_{y=-x}(x, y) \to (-y, -x)$