Question

1. reflection across the x-axis. reflection across the y-axis.

Explanation:

Step1: Identify Original Coordinates

First, find the coordinates of the original triangle's vertices. From the graph, let's assume the grid has integer coordinates. Let's say \( V(1,1) \), \( U(2,3) \), \( T(4,1) \) (adjusting for the grid; need to check the actual grid. Wait, looking at the graph, the y-axis is at the middle, so let's re - check. Let's assume the original triangle: \( V \) is at \( (1,1) \), \( U \) at \( (2,3) \), \( T \) at \( (4,1) \) (since from the y - axis, \( V \) is 1 unit right on x, 1 unit up on y; \( U \) is 2 units right, 3 units up; \( T \) is 4 units right, 1 unit up).

Step2: Reflect across x - axis (First Reflection)

The rule for reflection across the x - axis is \( (x,y)\to(x, - y) \).

• For \( V(1,1) \): After reflection, \( V_1(1, - 1) \)
• For \( U(2,3) \): After reflection, \( U_1(2, - 3) \)
• For \( T(4,1) \): After reflection, \( T_1(4, - 1) \)

Step3: Reflect across y - axis (Second Reflection)

The rule for reflection across the y - axis is \( (x,y)\to(-x,y) \). Now we take the points from the first reflection (\( V_1(1, - 1) \), \( U_1(2, - 3) \), \( T_1(4, - 1) \)) and reflect them across the y - axis.

• For \( V_1(1, - 1) \): After reflection, \( V_2(-1, - 1) \)
• For \( U_1(2, - 3) \): After reflection, \( U_2(-2, - 3) \)
• For \( T_1(4, - 1) \): After reflection, \( T_2(-4, - 1) \)

(If we first reflect across y - axis then x - axis, the order matters? Wait, the problem says "Reflection across the x - axis. Reflection across the y - axis." So first x - axis, then y - axis.

Alternatively, if the original coordinates are, let's re - examine the graph. Let's assume the original triangle has vertices: Let's count the grid squares. The y - axis is the vertical line in the middle. Let's say \( V \) is at \( (1,1) \), \( U \) at \( (2,3) \), \( T \) at \( (4,1) \) (x - coordinates are positive, to the right of y - axis).

First reflection (x - axis): \( (x,y)\to(x, - y) \)

• \( V(1,1)\to(1, - 1) \)
• \( U(2,3)\to(2, - 3) \)
• \( T(4,1)\to(4, - 1) \)

Second reflection (y - axis): \( (x,y)\to(-x,y) \)

• \( (1, - 1)\to(-1, - 1) \)
• \( (2, - 3)\to(-2, - 3) \)
• \( (4, - 1)\to(-4, - 1) \)

If we were to write the coordinates for the "First" (after x - axis reflection) and "Second" (after y - axis reflection) boxes:

For "First" (x - axis reflection):

• \( V(1,1)\to(1, - 1) \)
• \( U(2,3)\to(2, - 3) \)
• \( T(4,1)\to(4, - 1) \)

For "Second" (y - axis reflection of the first):

• \( (1, - 1)\to(-1, - 1) \)
• \( (2, - 3)\to(-2, - 3) \)
• \( (4, - 1)\to(-4, - 1) \)

(Note: The actual coordinates may vary slightly depending on the exact grid lines, but the process of reflection is based on the rules \( (x,y)\to(x, - y) \) for x - axis and \( (x,y)\to(-x,y) \) for y - axis.)

Answer:

To fill the boxes:

• Original (assuming the original vertices): Let's say \( V(1,1) \), \( U(2,3) \), \( T(4,1) \) (or adjust based on precise grid, but the key is the reflection rules)
• First (after x - axis reflection): \( V(1, - 1) \), \( U(2, - 3) \), \( T(4, - 1) \)
• Second (after y - axis reflection of the first): \( V(-1, - 1) \), \( U(-2, - 3) \), \( T(-4, - 1) \)

(If the question was to find the coordinates after each reflection, these are the steps. If it was to draw, the process is to plot the reflected points.)