Question

what is the rule for the reflection?
$t_{y\text{-axis}} (x, y) \to (-x, y)$
$t_{x\text{-axis}} (x, y) \to (x, -y)$
$t_{x\text{-axis}} (x, y) \to (-x, y)$
$t_{y\text{-axis}} (x, y) \to (x, -y)$

Explanation:

Step1: Identify a point and its image

Take point \( A \) and \( A' \). From the graph, \( A \) has coordinates (let's assume) \( (1, 2) \) and \( A' \) has coordinates \( (1, -2) \)? Wait, no, looking at the grid, maybe better to check the y - axis reflection or x - axis. Wait, the original figure is above the x - axis, the reflected is below? Wait, no, the coordinates: Let's take point \( B \) and \( B' \). Suppose \( B \) is \( (1, 4) \) and \( B' \) is \( (1, - 4) \)? Wait, no, the rule for reflection over x - axis is \( (x,y)\to(x, - y) \), over y - axis is \( (x,y)\to(-x,y) \). Let's check the coordinates. Let's take a vertex, say \( A \): original \( A \) (from the top figure) has coordinates (let's see the grid) maybe \( (1, 2) \), and \( A' \) (bottom figure) has \( (1, - 2) \)? Wait, no, the x - coordinate stays the same, y - coordinate flips sign. So the reflection is over the x - axis, rule \( T_{x\text{-axis}}(x,y)\to(x, - y) \). Wait, let's check the options. The second option is \( T_{x\text{-axis}}(x,y)\to(x, - y) \). Wait, let's confirm with a point. Let's take point \( B \) in the top figure: if \( B \) is \( (1, 4) \), then \( B' \) in the bottom figure is \( (1, - 4) \), which is \( (x, - y) \) where \( x = 1 \), \( y = 4 \), so \( (1, - 4) \), which matches \( (x, - y) \). So the rule is reflection over x - axis, \( (x,y)\to(x, - y) \), which is the second option (the one with \( T_{x\text{-axis}}(x,y)\to(x, - y) \)).

Step2: Match with the options

The options: the second option (from left, second) is \( T_{x\text{-axis}}(x,y)\to(x, - y) \), which is the rule for reflection over the x - axis.

Answer:

The correct option is the one labeled \( T_{x\text{-axis}}(x,y)\to(x, - y) \) (the second option among the four, with the rule \( (x,y)\to(x, - y) \) for reflection over x - axis).