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.

This activity might not be viewable on your mobile device. Interactive exercise. Assistance may be required.

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

Click to seeInteractive popup. Assistance may be required. additional instructions Close Pop Up 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

 

 

Yellow

 

 

Cyan

 

 

Black

 

 

Green

 

 

Yellow

 

 

Cyan

 

 

Black

 

 

Interactive popup. Assistance may be required.

See a Sample Answer

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)

Close Pop Up

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. Close Pop Up

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) Close Pop Up

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

 

 

Yellow

 

 

Cyan

 

 

Black

 

 

Green

 

 

Yellow

 

 

Cyan

 

 

Black

 

 

Interactive popup. Assistance may be required.

See a Sample Answer

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)

Close Pop Up

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. Close Pop Up

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) Close Pop Up

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

 

 

Yellow

 

 

Cyan

 

 

Black

 

 

Green

 

 

Yellow

 

 

Cyan

 

 

Black

 

 

Interactive popup. Assistance may be required.

See a Sample Answer

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)

Close Pop Up

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. Close Pop Up

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) Close Pop Up

Practice

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

Pentagon P E N T A graphed on a coordinate plane.

  1. What are the coordinates of each vertex after the pentagon is reflected across the x-axis?
  2. Interactive popup. Assistance may be required. Hint A reflection across the x-axis follows the rule (x, y) → (x, −y). Close Pop Up 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) Close Pop Up

  3. What are the coordinates of the reflection of pentagon PENTA after it is reflected across the y-axis?
  4. Interactive popup. Assistance may be required. Hint A reflection across the x-axis follows the rule (x, y) → (-x, y).Close Pop Up 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)Close Pop Up

  5. What are the coordinates of the reflection of pentagon PENTA after it is reflected across the line y = x?
  6. Interactive popup. Assistance may be required. Hint A reflection across the x-axis follows the rule (x, y) → (y, x).Close Pop Up 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) Close Pop Up