Question

what is the rule for the reflection?
t_y-axis (x, y) → (-x, y)
t_x-axis (x, y) → (-x, y)
t_x-axis (x, y) → (x, -y)
t_y-axis (x, y) → (x, -y)

Explanation:

Step1: Analyze Coordinates

Take a point, e.g., \( A(1, -2) \) and its image \( A'(1, 2) \). The \( x \)-coordinate remains \( 1 \), \( y \)-coordinate changes from \( -2 \) to \( 2 \) (i.e., \( -y \)).

Step2: Match Reflection Rule

Reflection over \( x \)-axis rule is \( (x, y) \to (x, -y) \)? Wait, no—wait, \( A(1, -2) \) to \( A'(1, 2) \): \( y \)-coordinate flips sign. Wait, original \( y \) is negative, image \( y \) is positive. So \( (x, y) \to (x, -y) \) when reflecting over \( x \)-axis? Wait, no: reflection over \( x \)-axis: \( (x, y) \to (x, -y) \). Let's check another point: \( B(1, -4) \) to \( B'(1, 4) \): same, \( y \) flips. So the rule is reflection over \( x \)-axis: \( (x, y) \to (x, -y) \), which matches the second option (labeled as \( T_{x\text{-axis}} (x, y) \to (x, -y) \)). Wait, the options: let's recheck. The second option (from left, second) is \( T_{x\text{-axis}} (x, y) \to (x, -y) \). Wait, the first option (leftmost) is \( T_{y\text{-axis}} (x, y) \to (x, -y) \) (no, that's wrong). Wait, no—wait the labels: the second option (second from left) is \( T_{x\text{-axis}} (x, y) \to (x, -y) \). Let's confirm coordinates:

Original points (top figure): \( A(1, -2) \), \( B(1, -4) \), \( D(4, -2) \), \( C(5, -4) \).

Image points (bottom figure): \( A'(1, 2) \), \( B'(1, 4) \), \( D'(4, 2) \), \( C'(5, 4) \).

For each point, \( x \)-coordinate same, \( y \)-coordinate becomes \( -y \) (since \( -(-2)=2 \), \( -(-4)=4 \), etc.). So the reflection is over \( x \)-axis, rule \( (x, y) \to (x, -y) \), which is the second option (labeled as \( T_{x\text{-axis}} (x, y) \to (x, -y) \)). Wait, the options: let's list them (from left to right):

1. \( T_{y\text{-axis}} (x, y) \to (x, -y) \) (wrong, \( y \)-axis reflection is \( (-x, y) \))
2. \( T_{x\text{-axis}} (x, y) \to (x, -y) \) (correct, since \( x \) same, \( y \) flips)
3. \( T_{x\text{-axis}} (x, y) \to (-x, y) \) (wrong)
4. \( T_{y\text{-axis}} (x, y) \to (-x, y) \) (wrong)

So the correct option is the second one: \( T_{x\text{-axis}} (x, y) \to (x, -y) \).

Answer:

The second option (from the left) labeled as \( T_{x\text{-axis}} (x, y) \to (x, -y) \) (or the option with text " \( T_{x\text{-axis}} (x, y) \to (x, -y) \) " depending on exact labeling—assuming the second rectangular option from the left).