Question

graph the image of rectangle qrst after a reflection over the line y = x.

Explanation:

Step1: Identify coordinates of Q, R, S, T

First, find the coordinates of the vertices of rectangle QRST. From the graph:

• \( Q(4, -8) \)
• \( R(9, -8) \) (assuming S is at \( (9, -5) \) and T at \( (4, -5) \)? Wait, no, looking at the grid: T is at (4, -5)? Wait, no, the y-coordinate for T: let's check the grid. The rectangle has T and S on the top, Q and R on the bottom. So T: x=4, y=-5? Wait, no, the vertical lines: T is at (4, -5)? Wait, the grid lines: each square is 1 unit. Let's re-express:

Looking at the graph:

• T: (4, -5) (since it's 4 on x, -5 on y? Wait, no, the bottom vertices Q and R are at y=-8. So Q: (4, -8), R: (9, -8), S: (9, -5), T: (4, -5). Yes, that makes a rectangle.

Step2: Apply reflection over \( y = x \)

The rule for reflection over the line \( y = x \) is \( (x, y) \to (y, x) \). So we swap the x and y coordinates of each vertex.

• For \( Q(4, -8) \): reflected point \( Q'(-8, 4) \)
• For \( R(9, -8) \): reflected point \( R'(-8, 9) \)
• For \( S(9, -5) \): reflected point \( S'(-5, 9) \)
• For \( T(4, -5) \): reflected point \( T'(-5, 4) \)

Step3: Plot the reflected points

Now, plot \( Q'(-8, 4) \), \( R'(-8, 9) \), \( S'(-5, 9) \), \( T'(-5, 4) \) on the coordinate plane and connect them to form the reflected rectangle.

Answer:

The image of rectangle QRST after reflection over \( y = x \) has vertices at \( Q'(-8, 4) \), \( R'(-8, 9) \), \( S'(-5, 9) \), \( T'(-5, 4) \). (To graph, plot these points and connect them in order.)