Question

write the rule for the reflection.
$(x,y)\to(\square,\square)$

Explanation:

Step1: Identify Original and Reflected Points

First, find coordinates of original segment (e.g., let's say original points: for the blue segment, maybe \( P(-1, 3) \) and \( Q(-4, 2) \)? Wait, no, looking at the graph, maybe original is the blue segment, reflected is the dashed. Let's take a point: suppose original point \( P(-1, 7) \)? Wait, maybe better to take a clear point. Let's say original point (blue) has coordinates, say, \( (-1, 7) \)? No, maybe \( (-1, 6) \)? Wait, maybe the original segment has a point at \( (-1, 6) \) and \( (-4, 3) \), and reflected has \( (1, 6) \) and \( (4, 3) \)? Wait, no, looking at the x-axis reflection? No, y-axis? Wait, if we reflect over y-axis, the rule is \( (x, y) \to (-x, y) \)? Wait, no, wait the dashed segment: let's take a point from blue (original) and dashed (reflected). Suppose original point \( (-1, 6) \), reflected is \( (1, 6) \)? Wait, no, maybe \( (-1, 6) \) and \( (1, 6) \)? Wait, no, the x-coordinate flips sign. Wait, let's check coordinates. Let's assume original point \( P(-1, 6) \) (blue), reflected point \( P'(1, 6) \) (dashed). Similarly, original \( Q(-4, 3) \), reflected \( Q'(4, 3) \). So the x-coordinate changes sign, y stays same. So reflection over y-axis? Wait, no, wait the rule for reflection over y-axis is \( (x, y) \to (-x, y) \)? Wait, no: if original is \( (x, y) \), reflection over y-axis is \( (-x, y) \)? Wait, no, wait: if you have a point \( (a, b) \), reflecting over y-axis gives \( (-a, b) \)? Wait, no, wait: y-axis is vertical, so x-coordinate flips sign. Wait, no, wait: (1,2) reflected over y-axis is (-1,2). Wait, but in the graph, original (blue) is on left, reflected (dashed) on right. So original x is negative, reflected x is positive. So if original is \( (x, y) \), reflected is \( (-x, y) \)? Wait, no: if original is (-1,6), reflected is (1,6), so \( x \) becomes \( -x \)? Wait, no: \( -(-1) = 1 \). So the rule is \( (x, y) \to (-x, y) \)? Wait, no, wait: original x is -1, reflected x is 1, so \( x \to -x \)? Wait, \( -(-1) = 1 \), yes. So reflection over y-axis: the rule is \( (x, y) \to (-x, y) \)? Wait, no, wait: reflection over y-axis: for a point \( (x, y) \), image is \( (-x, y) \). Wait, but in the graph, original is left (negative x), reflected is right (positive x), so yes, x-coordinate is negated, y stays. So the rule is \( (x, y) \to (-x, y) \)? Wait, no, wait: let's check another point. Original \( Q(-4, 3) \), reflected \( Q'(4, 3) \). So \( x = -4 \), becomes \( 4 = -(-4) \), so yes, \( x \to -x \), y same. So the reflection rule is over y-axis, so \( (x, y) \to (-x, y) \)? Wait, no, wait: wait, maybe I got it reversed. Wait, if you reflect over y-axis, the formula is \( (x, y) \mapsto (-x, y) \). So that's the rule.

Step2: Confirm the Reflection Rule

By observing the coordinates of corresponding points (original and reflected), the x-coordinate changes sign (from negative to positive or vice versa) while the y-coordinate remains the same. This indicates a reflection over the y - axis, whose transformation rule is \( (x, y) \to (-x, y) \)? Wait, no, wait: wait, if original is \( (x, y) \), reflection over y-axis is \( (-x, y) \). Wait, but in the example, original \( x = -1 \), reflected \( x = 1 = -(-1) \), so yes, \( x \to -x \), y same. So the rule is \( (x, y) \to (-x, y) \)? Wait, no, wait: no, wait, reflection over y-axis: the formula is \( (x, y) \mapsto (-x, y) \). So that's the rule.

Answer:

\( (x, y) \to (-x, y) \) (Wait, no, wait: wait, maybe I made a mistake. Wait, if original is on the left (negative x), reflected on right (positive x), so original \( x \) is negative, reflected \( x \) is positive. So \( x \to -x \): if original \( x = -1 \), \( -x = 1 \), which matches. So the rule is reflection over y - axis, so \( (x, y) \to (-x, y) \)? Wait, no, wait: no, reflection over y-axis: the transformation is \( (x, y) \mapsto (-x, y) \). So the answer is \( (x, y) \to (-x, y) \)? Wait, but maybe I mixed up. Wait, let's take a point: original \( (-1, 6) \), reflected \( (1, 6) \). So \( x \) becomes \( -x \) (since \( -(-1) = 1 \)), y same. So the rule is \( (x, y) \to (-x, y) \).