In the previous section, you investigated coordinate dilations. Dilations are transformations that preserve the shape of a polygon or an object, but the size changes. Reflections are congruence transformations. That is, a reflection is a transformation that preserves both size and shape of a polygon or object. In this section of the resource, you will investigate reflections that are performed on the coordinate plane.

Use the interactive link shown below to investigate coordinate reflections. Reflect a square, a parallelogram, and a triangle. Reflect these objects across the x-axis, the y-axis, and the line y = x. Once you have done so, use your experiences to answer the questions that follow.

Click on the sketch below to access the interactive and perform coordinate reflections.

Click to seeInteractive popup. Assistance may be required. additional instructions in using the interactive sketch.

Part 1: Reflections Across the x-axis

Use the interactive sketch to complete the following table. Reset the sketch and place a new parallelogram on the coordinate grid. Make a copy of the table and paste it into your notes. Fill in the columns for Original Coordinates. Translate your parallelogram across the x-axis and then record the reflected coordinates. Repeat a reflection for a second new parallelogram.

Reflect Across x-axis	Original Coordinates	Reflected Coordinates
Green	(1, 6)	(1, −6)
Yellow	(2, 0)	(2, 0)
Cyan	(5, 6)	(5, −6)
Black	(6, 0)	(6, 0)
Green	(−3, −3)	(−3, 3)
Yellow	(−2, −7)	(−2, 7)
Cyan	(3, −3)	(3, 3)
Black	(4, −7)	(4, 7)

Use your completed table to answer the questions that follow.

What patterns do you observe in the coordinates?

Interactive popup. Assistance may be required. Check Your Answer

The x-coordinate of the reflected coordinates remain the same as the x-coordinate of the original coordinates.
The y-coordinate of the reflected coordinates has the opposite sign as the y-coordinate of the original coordinates.

How could you express that relationship using an algebraic rule?

Interactive popup. Assistance may be required. Check Your Answer

If a polygon is reflected across the x-axis, the coordinates of each vertex of the polygon are changed by the following rule:
(x, y) → (x, -y)

Part 2: Reflections Across the y-axis

Use the interactive sketch to complete the following table. Reset the sketch and place a new parallelogram on the coordinate grid. Make a copy of the table and paste it into your notes. Fill in the columns for Original Coordinates. Reflect your parallelogram across the y-axis and then record the reflected coordinates. Repeat a reflection for a second new parallelogram.

Reflect Across y-axis	Original Coordinates	Reflected Coordinates
Green	(1, 6)	(−1, 6)
Yellow	(2, 0)	(−2, 0)
Cyan	(5, 6)	(−5, 6)
Black	(6, 0)	(−6, 0)
Green	(−3, −3)	(3, −3)
Yellow	(−2, −7)	(2, −7)
Cyan	(3, −3)	(−3, −3)
Black	(4, −7)	(−4, −7)

Use your completed table to answer the questions that follow.

What patterns do you observe in the coordinates?

Interactive popup. Assistance may be required. Check Your Answer

The x-coordinate of the reflected coordinates has the opposite sign as the x-coordinate of the original coordinates.
The y-coordinate of the reflected coordinates remain the same as the y-coordinate of the original coordinates.

How could you express that relationship using an algebraic rule?

Interactive popup. Assistance may be required. Check Your Answer

If a polygon is reflected across the y-axis, the coordinates of each vertex of the polygon are changed by the following rule:
(x, y) → (-x, y)

Part 3: Reflections Across the Line y = x

Use the interactive sketch to complete the following table. Reset the sketch and place a new parallelogram on the coordinate grid. Make a copy of the table and paste it into your notes. Fill in the columns for Original Coordinates. Reflect your parallelogram across the line y = x and then record the reflected coordinates. Repeat a reflection for a second new parallelogram.

Reflect Across y = x	Original Coordinates	Reflected Coordinates
Green	(1, 6)	(6, 1)
Yellow	(2, 0)	(0, 2)
Cyan	(5, 6)	(6, 5)
Black	(6, 0)	(0, 6)
Green	(−3, −3)	(−3, −3)
Yellow	(−2, −7)	(−7, −2)
Cyan	(3, −3)	(−3, 3)
Black	(4, −7)	(−7, 4)

Use your completed table to answer the questions that follow.

What patterns do you observe in the coordinates?

Interactive popup. Assistance may be required. Check Your Answer

The x-coordinate and y-coordinate of the reflected coordinates are the reverse from the x-coordinate and y-coordinate of the original coordinates.

How could you express that relationship using an algebraic rule?

Interactive popup. Assistance may be required. Check Your Answer

If a polygon is reflected across the line y = x, the coordinates of each vertex of the polygon are changed by the following rule:
(x, y) → (y, x)

Practice

Pentagon PENTA has coordinates at P (−4, 3), E (0, 7), N (4, 8), T (6, 1), and A (2, −2).

1. What are the coordinates of each vertex after the pentagon is reflected across the x-axis?

Interactive popup. Assistance may be required. Hint A reflection across the x-axis follows the rule (x, y) → (x, −y). Interactive popup. Assistance may be required. Check Your Answer P' (−4, −3), E'(0, −7), N'(4, −8), T' (6, −1), and A' (2, 2)

2. What are the coordinates of the reflection of pentagon PENTA after it is reflected across the y-axis?

Interactive popup. Assistance may be required. Hint A reflection across the x-axis follows the rule (x, y) → (-x, y). Interactive popup. Assistance may be required. Check Your Answer P' (4, 3), E'(0, 7), N'(−4, 8), T' (−6, 1), and A' (−2, −2)

3. What are the coordinates of the reflection of pentagon PENTA after it is reflected across the line y = x?

Interactive popup. Assistance may be required. Hint A reflection across the x-axis follows the rule (x, y) → (y, x). Interactive popup. Assistance may be required. Check Your Answer P' (3, −4,), E'(7, 0), N'(8, 4), T' (1, 6), and A' (−2, 2)