Topics
6. 2D Geometry
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United States of AmericaVirginia
Grade 8

6.06 Reflections in the coordinate plane

Lesson
Practice

Reflections in the coordinate plane

A reflection is a transformation in which an image is formed by reflecting the preimage over a line called the line of reflection.

Exploration

  • Change the shape of the preimage triangle by dragging the blue points on its vertices.
  • Change the position of the line of reflection by dragging the blue points on the line.
  • When you're ready, check the Flip box to see the image created by reflecting across the line.
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  1. Start with a vertical line of reflection. What do you notice about the shape of preimage and image?

  2. What do you notice about the size of the preimage and image?

  3. What do you notice about the distance between the image and preimage to the line of reflection?

  4. Change the line of reflection so it is horizontal? Are your observations still true?

A reflection is a transformation that flips a preimage over a line called the line of reflection, forming a mirror image. Each point in the image is the same distance from the line of reflection as its corresponding point in the preimage. Reflections also reverse the orientation of the image.

Reflecting over the y-axis

A triangle reflected over the y axis. Ask your teacher for more information.

Here a figure has been reflected over the y-axis.

Notice how the point (-2,\,1) becomes (2,\,1). The y-value has stayed the same while the x-value has changed signs.

Similarly the point (-6,\,3) becomes (6,\,3). For any corresponding points you choose on these figures, the y-values will stay the same and the x-values will have opposite signs.

Reflecting over the x-axis

A triangle reflected over the x axis. Ask your teacher for more information.

Here a figure has been reflected over the x-axis.

Notice how the point (4,\,3) becomes (4,\,-3). The x-value has stayed the same while the y-value has changed signs.

Similarly the point (0,\,5) becomes (0,\,-5). For any corresponding points you choose on these figures, the x-values will stay the same and the y-values will have opposite signs.

For any figure reflected horizontally across the y-axis, the y-values of the coordinates will stay the same and the x-values will have opposite signs.

For any figure reflected vertically across the x-axis, the x-values of the coordinates will stay the same and the y-values will have opposite signs.

Examples

Example 1

Which of the following does not show a reflection?

A
An image of two identical musical note facing opposite each other.
B
An image of two triangle. Ask your teacher for more information.
C
An image of 2 identical ogres. One is standing straight facing front and the other one is upside down facing front.
D
An image of two identical elephants facing each other.
Worked Solution
Create a strategy

Remember, a reflection is when the object flipped or reflected across a line.

Apply the idea

Which of object has not been flipped?

The correct answer is B.

An image of two triangle. Ask your teacher for more information.

Example 2

Trapezoid ABCD is plotted on the coordinate plane below.

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Plot trapezoid A'B'C'D', as a reflection of trapezoid ABCD over the x-axis.

Worked Solution
Create a strategy

A reflection over the x-axis creates an image on the other side of the x-axis where corresponding points on the pre-image and image are the same distance away from the x-axis.

Apply the idea
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Point A is located 2 units above the x-axis, therefore A' will be 2 units below the x-axis.

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Point B is located 4 units above the x-axis, therefore B' will be 4 units below the x-axis.

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Point C is located 4 units above the x-axis, therefore C' will be 4 units below the x-axis.

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Point D is located 2 units above the x-axis, therefore D' will be 2 units below the x-axis.

Reflect and check

We can also look at how the signs of the x and y-coordinates change for a reflection over the x-axis.

When reflecting a point across the x-axis, the x-coordinate stays the same and the sign of the y-coordinate changes.

A(4,2)'s reflection is A'(4,-2).

B(2,4)'s reflection is B'(2,-4).

C(-1,4)'s reflection is C'(-1,-4).

D(-1,2)'s reflection is D'(-1,-2).

Example 3

Plot the following.

a

Plot the polygon PQRS, where the vertices are P(-3,\,-1),\,Q(-2,\,2),\,R(0,\,2),\, and S(0,\,0).

Worked Solution
Create a strategy

Plot and connect the vertices.

Apply the idea
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Plot the reflection of the polygon over the y-axis.

Worked Solution
Create a strategy

A reflection over the y-axis creates an image on the other side of the y-axis where corresponding points on the pre-image and image are the same distance away from the y-axis.

Apply the idea

Point P is located 3 units to the left of the y-axis, so its reflection, P' will be 3 units to the right of the y-axis.

Point Q is located 2 units to the left of the y-axis, so its reflection, Q' will be 2 units to the right of the y-axis.

Point R on the y-axis, so it is 0 units from the y-axis. So its reflection, R' will be 0 units from the y-axis. This means it will also be on the y-axis.

Point S is located on the y-axis, so it is 0 units from the y-axis. So its reflection, S' will be 0 units from the y-axis. This means it will also be on the y-axis.

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Reflect and check

We can also look at how the signs of the x-coordinates change for a reflection over the y-axis.

When reflecting a point across the y-axis, the y-coordinate stays the same and the sign of the x-coordinate changes.

P(-3,\,-1)'s reflection is P'(3,\,-1).

Q(-2,\,2)'s reflection is Q'(2,\,2).

R(0,\,2)'s reflection is R'(0,\,2).

S(0,\,0)'s reflection is S'(0,\,0).

Example 4

A fabric designer is creating a new pattern for a popular textile brand. The initial pattern is designed on a coordinate plane and consists of several geometric shapes arranged in a unique layout.

The designer plans to reflect the pattern across the y-axis to create a symmetrical design that will be printed on fabric.

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Draw the original pattern and its reflection across the y-axis on the same coordinate plane to visualize the complete design.

Worked Solution
Create a strategy

When reflecting over the y-axis the points should be equidistant to the y-axis on either side.

The y-axis acts as the mirror line of the image and preimage.

Apply the idea

Start with the hexagon. Each point needs to be reflected across the y-axis and be the same distance from the y-axis as the original point.

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Next, reflect the equilateral triangle.

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Then, reflect the right triangle.

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Finally, reflect the right triangle. Each point needs to be reflected to the other side of the y-axis and be equidistant to the y-axis as the original point.

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Reflect and check

We can check that each ordered pair has been reflected correctly by verifying that signs of the y-coordinates have changed while the signs of the y-coordinates have stayed the same.

For the hexagon:

(0,\,0)'s reflection is '(0,\,0).

(2,\,0)'s reflection is (-2,\,0).

(3,\,2)'s reflection is (-3,\,2).

(2,\,4)'s reflection is (-2,\,4).

(0,\,4)'s reflection is (0,\,4).

(-1,\,2)'s reflection is (1,\,2).

For the equilateral triangle:

(-1,\,-1)'s reflection is '(1,\,-1).

(-3,\,-1)'s reflection is (3,\,-1).

(-2,\,-3)'s reflection is (2,\,-3).

For the right triangle:

(1,\,1)'s reflection is '(-1,\,1).

(4,\,1)'s reflection is (-4,\,1).

(3,\,3)'s reflection is (-3,\,3).

Idea summary

A reflection is a transformation in which an image is formed by reflecting the preimage over a line called the line of reflection.

Every point on the object or shape has a corresponding point on the image, and they will both be the same distance from the reflection line.

For any figure reflected horizontally across the y-axis, the y-values of the coordinates will stay the same and the x-values will have opposite signs.

For any figure reflected vertically across the x-axis, the x-values of the coordinates will stay the same and the y-values will have opposite signs.

Outcomes

8.MG.3

The student will apply translations and reflections to polygons in the coordinate plane.

8.MG.3b

Given a preimage in the coordinate plane, identify the coordinates of the image of a polygon that has been reflected over the x- or y-axis

8.MG.3e

Sketch the image of a polygon that has been reflected over the x- or y-axis.

8.MG.3g

Identify and describe transformations in context (e.g., tiling, fabric, wallpaper designs, art).

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