Question

rotate 90° clockwise around the origin.
reflect across the x-axis.
graph with quadrilateral d, e, f, g on coordinate plane
reflect across the y-axis. rotate 90° counterclockwise around the origin.

Explanation:

To solve the problem of transforming the quadrilateral (let's assume the coordinates of the vertices first, since they are not labeled with coordinates, we can estimate them from the graph. Let's assume the coordinates of the vertices:

• Let's identify the coordinates of \( D \), \( E \), \( F \), \( G \) from the graph. Let's estimate:
• \( D(-4, -1) \) (approximate, since it's near \( x=-4, y=-1 \))
• \( E(0, -2) \) (near \( x=0, y=-2 \))
• \( F(-2, -6) \) (near \( x=-2, y=-6 \))
• \( G(-6, -4) \) (near \( x=-6, y=-4 \))

Part 1: Rotate \( 90^\circ \) clockwise around the origin, then reflect across the \( x \)-axis

Step 1: Rotate \( 90^\circ \) clockwise around the origin

The rule for rotating a point \( (x, y) \) \( 90^\circ \) clockwise around the origin is \( (x, y) \to (y, -x) \)

• For \( D(-4, -1) \): \( (-1, 4) \)
• For \( E(0, -2) \): \( (-2, 0) \)
• For \( F(-2, -6) \): \( (-6, 2) \)
• For \( G(-6, -4) \): \( (-4, 6) \)

Step 2: Reflect across the \( x \)-axis

The rule for reflecting a point \( (x, y) \) across the \( x \)-axis is \( (x, y) \to (x, -y) \)

• For \( D'(-1, 4) \): \( (-1, -4) \)
• For \( E'(-2, 0) \): \( (-2, 0) \)
• For \( F'(-6, 2) \): \( (-6, -2) \)
• For \( G'(-4, 6) \): \( (-4, -6) \)

Part 2: Reflect across the \( y \)-axis, then rotate \( 90^\circ \) counterclockwise around the origin

Step 1: Reflect across the \( y \)-axis

The rule for reflecting a point \( (x, y) \) across the \( y \)-axis is \( (x, y) \to (-x, y) \)

• For \( D(-4, -1) \): \( (4, -1) \)
• For \( E(0, -2) \): \( (0, -2) \)
• For \( F(-2, -6) \): \( (2, -6) \)
• For \( G(-6, -4) \): \( (6, -4) \)

Step 2: Rotate \( 90^\circ \) counterclockwise around the origin

The rule for rotating a point \( (x, y) \) \( 90^\circ \) counterclockwise around the origin is \( (x, y) \to (-y, x) \)

• For \( D''(4, -1) \): \( (1, 4) \)
• For \( E''(0, -2) \): \( (2, 0) \)
• For \( F''(2, -6) \): \( (6, 2) \)
• For \( G''(6, -4) \): \( (4, 6) \)

(Note: Since the exact coordinates were estimated, the transformed coordinates are based on the estimated original coordinates. If the original coordinates were different, the transformed coordinates would change accordingly. The key is to apply the transformation rules correctly: \( 90^\circ \) clockwise rotation: \( (x,y) \to (y, -x) \), reflection over \( x \)-axis: \( (x,y) \to (x, -y) \), reflection over \( y \)-axis: \( (x,y) \to (-x, y) \), \( 90^\circ \) counterclockwise rotation: \( (x,y) \to (-y, x) \))

If you need to plot these points, you can use the transformed coordinates to draw the new quadrilateral.

Answer:

To solve the problem of transforming the quadrilateral (let's assume the coordinates of the vertices first, since they are not labeled with coordinates, we can estimate them from the graph. Let's assume the coordinates of the vertices:

• Let's identify the coordinates of \( D \), \( E \), \( F \), \( G \) from the graph. Let's estimate:
• \( D(-4, -1) \) (approximate, since it's near \( x=-4, y=-1 \))
• \( E(0, -2) \) (near \( x=0, y=-2 \))
• \( F(-2, -6) \) (near \( x=-2, y=-6 \))
• \( G(-6, -4) \) (near \( x=-6, y=-4 \))

Part 1: Rotate \( 90^\circ \) clockwise around the origin, then reflect across the \( x \)-axis

Step 1: Rotate \( 90^\circ \) clockwise around the origin

The rule for rotating a point \( (x, y) \) \( 90^\circ \) clockwise around the origin is \( (x, y) \to (y, -x) \)

• For \( D(-4, -1) \): \( (-1, 4) \)
• For \( E(0, -2) \): \( (-2, 0) \)
• For \( F(-2, -6) \): \( (-6, 2) \)
• For \( G(-6, -4) \): \( (-4, 6) \)

Step 2: Reflect across the \( x \)-axis

The rule for reflecting a point \( (x, y) \) across the \( x \)-axis is \( (x, y) \to (x, -y) \)

• For \( D'(-1, 4) \): \( (-1, -4) \)
• For \( E'(-2, 0) \): \( (-2, 0) \)
• For \( F'(-6, 2) \): \( (-6, -2) \)
• For \( G'(-4, 6) \): \( (-4, -6) \)

Part 2: Reflect across the \( y \)-axis, then rotate \( 90^\circ \) counterclockwise around the origin

Step 1: Reflect across the \( y \)-axis

The rule for reflecting a point \( (x, y) \) across the \( y \)-axis is \( (x, y) \to (-x, y) \)

• For \( D(-4, -1) \): \( (4, -1) \)
• For \( E(0, -2) \): \( (0, -2) \)
• For \( F(-2, -6) \): \( (2, -6) \)
• For \( G(-6, -4) \): \( (6, -4) \)

Step 2: Rotate \( 90^\circ \) counterclockwise around the origin

The rule for rotating a point \( (x, y) \) \( 90^\circ \) counterclockwise around the origin is \( (x, y) \to (-y, x) \)

• For \( D''(4, -1) \): \( (1, 4) \)
• For \( E''(0, -2) \): \( (2, 0) \)
• For \( F''(2, -6) \): \( (6, 2) \)
• For \( G''(6, -4) \): \( (4, 6) \)

(Note: Since the exact coordinates were estimated, the transformed coordinates are based on the estimated original coordinates. If the original coordinates were different, the transformed coordinates would change accordingly. The key is to apply the transformation rules correctly: \( 90^\circ \) clockwise rotation: \( (x,y) \to (y, -x) \), reflection over \( x \)-axis: \( (x,y) \to (x, -y) \), reflection over \( y \)-axis: \( (x,y) \to (-x, y) \), \( 90^\circ \) counterclockwise rotation: \( (x,y) \to (-y, x) \))

If you need to plot these points, you can use the transformed coordinates to draw the new quadrilateral.