Reflection

MCC8.G.3Describe the effect of dilations, translations, rotations and reflections on two-dimensional figures using coordinates.

• One type of transformation uses a line that acts like a mirror, with an image reflected across a line is a reflection and the mirror line is the line of reflection.

12.2 Reflection (flip)Example A

Quadrilateral ABCD is being reflected across the y-axis.

1. What are the coordinates for quadrilateral ABCD?

Point A (1,1)

Point B (2,3)

Point C (4,4)

Point D (5,2)

2. How far is each point from the line of reflection?

Point A is 1 unit

Point B is 2 units

Point C is 4 units

Point D is 5 units

12.2 Reflection (flip)Example A

Each vertex should be the same distance from the line of symmetry, just in the opposite position.

REMEMBER:

3. Using the information in question 2, how far should the image be from the line of reflection?

Pre-image

Point A is 1 unit

Point B is 2 units

Point C is 4 units

Point D is 5 units

Image

Point A′ is 1unit

Point B′ is 2 units

Point C′ is 4 units

Point D′ is 5 units

12.2 Reflection (flip)Example A

4. What are coordinates for quadrilateral A’B’C’D’?

A′ (-1,1)

B′ (-2,3)

C′ (-4,4)

D′ (-5,2)

5. Compare and contrast the coordinates for original and the image?

Pre-image

A (1,1)

B (2,3)

C (4,4)

D (5,2)

Image

A′ (-1,1)

B′ (-2,3)

C′ (-4,4)

D′ (-5,2)

12.2 Reflection (flip)Example B

This time the original is being reflected over the x-axis.

Write down the coordinates for the original and the image. Compare and contrast the coordinates?

Pre-Image

F (2,3)

G (4,1)

H (1,0)

Image

F′ (2,-3)

G′ (4,-1)

H′ (1,0)

When you reflect over the x-axis the x-coordinates stay the same and the y-coordinates change to its opposite.

When you reflect over the y-axis the x-coordinates change to its opposite and the y-coordinates stay the same.

What to remember…

• Reflection- the figure is flipped over a line.

Reflection over the x-axis:(x, y) (x, -y)

Reflection over the y-axis:(x, y) (-x, y)

What happens if the line of reflection is not the axis?

Reflections in other lines . . .• Let’s look at what happens if you reflect a figure across the

line y = x or line y = -x

y = x y = -x

Look at corresponding points. Notice that for (x, y), the corresponding image point is (y, x). For (-2, 5), image point is (5, -2).

Look at corresponding points. Notice that for (x, y), the corresponding image point is (-y, -x). For (6, 3), image point is (-3, -6)

What does y=a look like?• a represents any number.Let’s graph y = 3

What type of line did you graph?Horizontal line

y = 3x y (x,y)-2 3 (-2,3)1 3 (1,3)4 3 (4,3)

What does x=c look like?• c represents any number.Let’s graph x = -2

What type of line did you graph?Vertical line

x = -2x y (x,y)-2 5 (-2,5)-2 0 (-2,0)-2 -3 (-2,-3)

Reflections in more lines . . .• What happens if you reflect in a line

y = 3? or x = -2 ?

Each point and corresponding image must be equidistant from the line. Note A (4, 2) and image point A′(4, 4) are each 1 unit from the line y = 3.

Each point and corresponding image must be equidistant from the line. Note B (0, 4) and image point B′(-4, 4) are 2 units from the line x = -2.

y = 3 A

A’

x = -2

BB’

Steps to finding the image coordinates 1. Determine if the figure will reflect

horizontally or vertically. This will tell you which coordinate will change.

Since it Y=3 only the y-coordinate will change.

2. Find the distance between the pre-image coordinate and the line of reflection by subtracting the coordinate from the value of the line.

U (-3, 5)

5 – 3 = 2U is 2 units above y=3

So how far should U′ be the line of reflection?2 units below

Steps to finding the image coordinates 3. Since it should be below, subtract the

distance from the value of the line of reflection.

4. Check by graphing

U (-3,5)U′ should be 2 units belowLine of reflection y=3

3-2 = 1

So U′ should be at (-3,1)

Reteach Video next

ReflectionReflection

Reflect ABC across the x-axis.Reflect ABC across the x-axis.

4 Units3 Units

A

B

C

4 Units

A’

3 Units

B’

1 Unit

1 UnitC’

Count the number of units point A is from the line of reflection.Count the same number of units on the other side and plot point A’.

Count the number of units point B is from the line of reflection.Count the same number of units on the other side and plot point B’.

Count the number of units point C is from the line of reflection.Count the same number of units on the other side and plot point C’.

Reflections in a line• Reflections can be made across the x-axis.

Look at the corresponding points in the figures. The point (-4, 4) corresponds to the image point (-4, -4). The point (2, 4) corresponds to (2, -4).

Notice that in a reflection over the x-axis, the coordinates of the x’s stay the same but the y’s change sign.

In a reflection across the x-axis, the point (x, y) reflects onto image (x, -y).

x-axis

ReflectionReflection

Reflect ABC across the y-axis.Reflect ABC across the y-axis.

5 Units

2 UnitsA

B

C

5 Units A’

2 Units

B’

3 Units 3 UnitsC’

Count the number of units point A is from the line of reflection.Count the same number of units on the other side and plot point A’.

Count the number of units point B is from the line of reflection.Count the same number of units on the other side and plot point B’.

Count the number of units point C is from the line of reflection.Count the same number of units on the other side and plot point C’.

Reflections in the y axis• Reflections can be made across the y-axis.

Check the corresponding points here.

Notice that the point (2, 1) corresponds to (-2, 1). The point (7, 1) corresponds to (-7, 1). The y

values stay the same, but the x values change sign.

In a reflection across the y axis, the point (x, y) reflects onto image (-x, y).

y-axis