ccm6+/7+ - unit 13 - page 1 unit...

CCM6+/7+ - Unit 13 -

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UNIT 13

Transformations

CCM6+/7+ Name: _________________

Math Teacher: ___________

Projected Test Date: ____________________

Main Idea Pages

Unit 9 Vocabulary 2

Translations 3 – 10

Rotations 11 – 17

Reflections 18 – 22

Transformations Notes & Rules 23

Dilations 24 – 34

Composition of Transformations 35 - 39

Unit 9 Study Guide 40 – 44

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Common Core Math 7 Plus Unit 9 Vocabulary

Definitions of Critical Vocabulary and Underlying Concepts

coordinate plane the plane formed by two lines intersecting at their zero points-the

horizontal line is the "x-axis" and the vertical line is the "y-axis”

transformation when a figure or point is changed in size and position on a coordinate plane

rigid transformation when a figure changes position on the coordinate plane but maintains the

same size and shape

translation a transformation that moves points right-left or up-down or a combination

of these (slide)

reflection a transformation that flips a figure over a given line of reflection-each point

moves an equal distance from the line of reflection but on the opposite side

rotation a transformation where a given figure rotates around a given point

dilation a transformation which enlarges or reduces a figure using a given scale

factor

scale factor the factor by which you multiply original numbers to increase or decrease

size

prime coordinates the coordinates that come from applying a scale factor of original points

coordinates the pair of numbers used to describe points on a coordinate plane

composition of

transformation

A composition of two transformations is a transformation in which a

second transformation is performed on the image of a first transformation

glide reflection A composition of a translation and a reflection in a line parallel to the

direction of the translation

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TRANSLATIONS

When working with any TRANSFORMATIONS the original points create the PRE-IMAGE. You

can name the points using letters.

For example A(4, 5) tells you that “point A is located at position 4, 5 on the graph”.

Once the point is moved to its new position it is called a “prime point” and named like this:

A’ - read this as “A prime” the figure is now called the “IMAGE”

TRANSLATIONS involve moves that are either right-left, up-down, or a combination of these.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Create a pre-image by graphing and labeling the following points:

A(-3, 2), B(-3, 6), C(-7, 2)

Now take each point and move it 8 units right and then label the new points as “primes”.

You have modeled a TRANSLATION. Name the new prime coordinates below: A’(___,___), B’(___,___), C’(___,___)

Did the shape or size of the figure change?

Look at the new “x” numbers. What do you notice happened to the “x” part of each ordered

pair?

Why do you think it was the x affected and not the y?

What type of move do you think would affect the y?

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Look at the graphed image below. Write in the coordinates for the pre-image and the image.

Make sure that you label the coordinates that you list.

How did you determine which was the

pre-image and which is the image?

_______________________________

_______________________________

What happened to the y part of the coordinates?_________________________________

Why?____________________________________________________________________

If you are moving UP or DOWN the _____ part of your ordered pair will change.

If you go up you will ________ the number of units to the original y. If you go

down you will ____________ the number of units from the original y.

If you are moving RIGHT or LEFT the _____ part of your ordered pair will change.

If you go right you will _________ the number of units to the original x. If you go

left you will ___________ the number of units to the original x.

A B

C D

A’ B’

C’ D’

Pre-Image Coordinates Image Coordinates

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Graph and label the following points and then translate 3 units left and 2 units up.

Label and list your new prime points.

M(5, 8) M’(____,____)

A(0, 6) A’(____,____)

P(-3, -2) P’(____,____)

Now describe what happened to

each part of the ordered pairs:

(x _______, y________)

Describe the translation that you see below.

Can you give the new prime points without creating the graphs for these two translations.

Example 1: Translate 2 units left and Example 2: Translate 5 units right and 3 units down 4 units down

A(5, -2) (5_____, -2______) A’(____,_____)

M(2, 6) (2_____, 6_______) M’(____,____)

B(0, -3) (0_____,-3_______) B’(____,____)

C(3, 5) (3_____, -2______) C’(____,____)

W(-2,5) (-2_____, 6______) W’(____,____)

T(1, -7) (1_____,-3______) T’(____,____)

2

1

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Translation Practice

Plot the coordinates given for each pre-image.

Translate the figure as instructed.

Draw the image on the coordinate plane.

List the coordinates of the new image.

Describe what happened to both the x and y

1. Slide the triangle 5 units left and 3 units up

A (3, 7) A’ ( , )

B (3, 2) B’ ( , )

C (7, 2) C’ ( , )

(x_______, y________)

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2. Translate the parallelogram 10 units left and 2 units down.

H (3, -2) H’ ( , )

I (1, -5) I’ ( , )

J (6, -5) J’ ( , )

K (8, -2) K’ ( , )

(x_______, y________)

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ROTATIONS

You will be exploring a TRANSFORMATION called a ROTATION.

A ROTATION is a movement of a figure that involves rotating in 90 degree increments around the origin. The new prime points will be in the quadrant that is the given number of

degrees clockwise or counterclockwise from the original figure.

The following activities will help you discover what happens when a

point, line, or figure is rotated a given number of degrees.

In a rotation, the original shape does not change in size or shape but does

move to a new position on the coordinate plane.

You will need to remember the names of the quadrants:

I

III

I

II

IV

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EXAMPLE #1:

STEP 1: The following is a 900 clockwise rotation:

STEP 2: List the pre-image points and the image points below.

A(____,____) A’ (____,____)

B(____,____) B’ (____,____)

C(____,____) C’ (____,____)

In what quadrant is the

Pre-Image?______ ; Image?______

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - EXAMPLE #2:



A(____,____) A’ (____,____)

B(____,____) B’ (____,____)

C(____,____) C’ (____,____)



- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Using the rule you have discovered, find the prime coordinates for a line with pre-image points at (2, -6) (____,____) (7, -1) (____,____)

Compare each set of PRE-IMAGE coordinates with the IMAGE coordinates. What is the relationship between the pre-image coordinates and the image coordinates? ______________________________________________________________________________________________

A

B

C

B’ A’

C’

Compare each set of PRE-IMAGE coordinates with the IMAGE coordinates . What is the relationship between the pre-image coordinates and the image coordinates? ______________________________________________________________________________________________

A B

C

B’

A’

C’

Using the two examples above, describe what happens to the coordinates in a 900 clockwise rotation? ___________________________________________________________________________

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EXAMPLE #3:

STEP 1: The following is a 900 counter-clockwise rotation:


A(____,____) A’ (____,____)

B(____,____) B’ (____,____)

C(____,____) C’ (____,____)



- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - EXAMPLE #4:

STEP 1: The following is a 900 counter-clockwise rotation:


A(____,____) A’ (____,____)

B(____,____) B’ (____,____)

C(____,____) C’ (____,____)



- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -



A

B

C B’ A’

C’


A B

C B’

A’

C’

Using the two examples above, describe what happens to the coordinates in a 900 counter-clockwise rotation? ___________________________________________________________________________

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EXAMPLE #5:

STEP 1: The following is a 1800 rotation:


A(____,____) A’ (____,____)

B(____,____) B’ (____,____)

C(____,____) C’ (____,____)



- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - EXAMPLE #6:



A(____,____) A’ (____,____)

B(____,____) B’ (____,____)

C(____,____) C’ (____,____)



- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -



A

B

C

B’

A’ C’


A B

C

B’

A’

C’

Using the two examples above, describe what happens to the coordinates in a 1800 rotation? ___________________________________________________________________________

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Translations and Rotations Practice

Translations:

1. Translate ΔQRS (x+4, y-2). List the new

coordinates below.

2. Translate ΔQRS (x-3, y+1). List the new

coordinates below.

3. ΔABC has vertices at A (-1, -1), B (4,-1),

C(1, 3). Where are the new vertices located

after a move 3 units up and 5 units left?

4. Create your own translation. List the pre-

image coordinates, the translation

movements, and the image coordinates.

Rotations:

1. Translate ΔQRS 180 degrees. List the new

coordinates below.

2. Translate ΔQRS 90 degrees

counterclockwise. List the new coordinates

below.

3. ΔABC has vertices at A (-1, -1), B (-4,-1),

C(-2, -3). Where are the new vertices

located after a 90 degree clockwise

rotation?

4. Create your own rotation. List the pre-

image coordinates, the direction and

degrees of rotation, and the image

coordinates.

Q

R S

Q

R S

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REFLECTIONS

You will need: a straight edge, pencil, and several pieces of patty paper

You will be exploring a TRANSFORMATION called a REFLECTION.

When working with any TRANSFORMATIONS the original points create

the PRE-IMAGE. You can name the points using letters.

For example A(4, 5) tells you that “point A is located at position 4, 5 on the graph”.

Once the point is moved to its new position it is called a “prime point”

and named like this:

A’ - read this as “A prime” the figure is now called the “IMAGE”

REFLECTIONS involve moves that “flip” over a given line.

The following activities will help you discover what happens when a

point, line, or figure is reflected over a given “line of reflection”. This can

also be called the LINE OF SYMMETRY.

In a reflection, the original shape does not change in size or shape but

does move to a new position on the coordinate plane.

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EXPLORE:

STEP 1: Create a pre-image by graphing, labeling, and connecting the following points:

A(-3, 2), B(-3, 6), C(-7, 2)

STEP 2: Using a piece of patty paper, and a straight edge, trace the original figure. STEP 3: FLIP the piece of patty paper

over, lining up AB an equal distance from the y-axis but on the opposite side of the y-axis. STEP 4: Record the new coordinates below after the flip and then add this figure to the graph .

A(-3, 2) A’ (____,____)

B(-3, 6) B’ (____,____)

C(-7, 2) C’ (____,____)

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - STEP 1: Create a pre-image by graphing, labeling, and connecting the following points:

R(3, 2), T(3, 6), N(8, 1), Q(8, 8)

STEP 2: Using a piece of patty paper, and a straight edge, trace the original figure.

STEP 3: FLIP the piece of patty paper

over, lining up RT an equal distance from the y-axis but on the opposite side of the y-axis.

STEP 4: Record the new coordinates below after the flip and then add this figure to the graph.

R(3, 2) R’ (____,____)

T(3, 6) T’ (____,____)

N(8, 1) N’ (____,____)

Q(8, 8) Q’ (____,____) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Compare each set of PRE-IMAGE coordinates with the IMAGE coordinates after the move. What do you notice?

_____________________________________________________

_____________________________________________________


_____________________________________________________

_____________________________________________________

DISCOVERY: When a figure is reflected over the y-axis, the y part of each coordinate _______________________ and the x part of each coordinate ______________________.

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STEP 1: Create a pre-image by graphing, labeling, and connecting the following points:

A(-3, 2), B(-3, 6), C(-7, 2)

STEP 2: Using a piece of patty paper, and a straight edge, trace the original figure. STEP 3: FLIP the piece of patty paper

over, lining up AB an equal distance from the x-axis but on the opposite side of the x-axis. STEP 4: Record the new coordinates below after the flip and then add this figure to the graph.

A(-3, 2) A’ (____,____)

B(-3, 6) B’ (____,____)

C(-7, 2) C’ (____,____)

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - STEP 1: Create a pre-image by graphing, labeling, and connecting the following points: R(3, 2), T(3, 6), N(8, 1), Q(8, 8)

STEP 2: Using a piece of patty paper, and a straight edge, trace the original figure.

STEP 3: FLIP the piece of patty paper

over, lining up RT an equal distance from the x-axis but on the opposite side of the x-axis.

STEP 4: Record the new coordinates below after the flip and then add this figure to the graph.

R(3, 2) R’ (____,____)

T(3, 6) T’ (____,____)

N(8, 1) N’ (____,____)

Q(8, 8) Q’ (____,____) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -


_____________________________________________________

_____________________________________________________


_____________________________________________________

_____________________________________________________

DISCOVERY: When a figure is reflected over the x-axis, the x part of each coordinate _______________________ and the y part of each coordinate ______________________.

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Transformations Notes & Rules

Translations Reflection Rotations

Slide Flip Turn If sliding down:

Subtract from y

(y decreases in value)

Over x axis:

Change all y coordinates

to their opposites

( x , oy )

90 degrees clockwise:

(270 degrees counterclockwise)

Flip x and y coordinates

Change x coordinate to

its opposite

( y , ox )

If sliding up:

Add to y

(y increases in value)

Over y axis:

Change all x coordinates

to their opposites

( ox , y )


(180 degrees counterclockwise):

Change x and y

coordinate to their

opposites

( ox , oy )

If sliding left:

Subtract from x

(x decreases in value)


(90 degrees counterclockwise):

Flip x and y coordinates

Change y coordinates to

its opposite

( oy , x )

If sliding right:

Add to x

(x increases in value)

360 degrees/0 degrees clockwise: (360 degrees/0 degrees counterclockwise)

Original position

( x , y )

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DILATIONS

Dilate: ________________________________________________________

_____________________________________________________________

Scale Factor ___________________________________________________

_____________________________________________________________

A scale factor greater than one means ________________________________

_____________________________________________________________

A scale factor less than one means ___________________________________

_____________________________________________________________

A dilated image will always be _______________to its original image.

TRY THIS:

Looking at the coordinates below, can you identify what scale factor was used?

EXAMPLE 1: EXAMPLE 2:

A (5, 3), B(2, 5) X(-2, 4), Y(2, 8)

A’(10, 6), B’(4, 10) X’(-1, 2), Y’(1, 4)

SCALE FACTOR:________ SCALE FACTOR:_______

Now use the attached graph page to graph the lines formed by both the original

points and the prime points in examples 1 and 2 to see what happens. Can you

guess which would be larger? Which will be smaller?

Now use a straight edge to create a line that goes through A and A’ and another line that goes

through B and B’ in your graph for example 1. Extend your line through the whole graph shown.

Do the lines go through the origin?______ Repeat the process for example 2.

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You have just dilated your first figures in a coordinate plane. Now try the ones listed below and

watch what happens.

EXAMPLE 3:

Plot the following points for a triangle and then draw the triangle. Next apply a scale factor of

3. Write the new prime points below and then plot them on the provided graph.

R(1, 1) R’(____, _____)

S(1, 4) S’(____, _____)

T(4, 1) T’(____, _____)

EXAMPLE 4:

Plot the following points and draw the figure. Next apply a scale factor of 1/3. Write the new

prime points below and then plot and draw them on the provided graph.

M(-3, -3) M’(____,____)

N(0, -9) N’(____,____)

Q(-9, -3) Q’(____,____)

D(-12, -9) D’(____,____)

EXAMPLE 5:

What do you think may happen if the scale factor is negative? _________________________

_____________________________Try it with this figure using a scale factor of -2:

D(-2, 1) D’(____,____)

A(-5, 1) A’(____,____)

B(-2, 6) B’(____,____)

EXAMPLE 6: You create your own figure, apply a scale factor of your choice, and draw the prime figure. You

will trade with another student to see if they can determine your scale factor. Mark blanks on your sheet for

them to fill out points, prime points, and scale factor.

What happens if you go to the graphed figures and draw lines extended from each given point and its prime?

How could you tell that the graphed triangle would get larger or smaller?

What happens if you go to the graphed figures and draw lines extended from each given point and its prime?

How could you tell that the graphed triangle would get larger or smaller?

What happens with this figure if you go to the graphed figures and draw lines extended from each given point and its prime? Does it still go through the origin?

Describe what happened to the figure:

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Dilations Practice

1. Given the following points for an image, list the prime points after a dilation using a scale factor of 3:

W(1, 1), S(2, 1), T(1, 2) G(2, 2) W’(___,___), S’(___,___), T’(___,___), G’(___, ___)

2. For a given dilation, the point (5, 0) has a prime point of (35, 0). What are the coordinates of the prime point using

the same dilation for (10, 2)? (____, ____)

3. What are the coordinates for the prime point given an original point of (4, 6) after a scale factor of -6 is applied?

(____, ____)

4. For the following, identify the scale factor that was used.

D(7, -3) D’(14, -6)

E(-2, -5) E’(-4, -10) SCALE FACTOR:________

M(8, 2) M’(16, 4)

5. Graph the following image then apply a scale factor of ¼ . Draw the new image on the same graph and list the prime

points.

E(4, -2) E’(____,____)

X(4, -8) X’(____,____)

A(12, -4) A’(____,____)

L(12, -8) L’(____,____)

6. How do you know if an image will get larger or smaller based on the scale factor?_____________________________

__________________________________________________________________________________________________

__________________________________________________________________________________________________

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9 10

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Determine if the following scale factor would create an enlargement or a reduction.

11. 3.5 12. 2/5 13. 0.6 14. 1.1 15. 4/3 16. 5/8

Given the point and its image, determine the scale factor.

17. A(3,6) A’(4.5, 9) 18. G’(3,6) G(1.5,3) 19. B(2,5) B’(1,2.5)

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Composition of Transformations

Composition of Transformation: A composition of two transformations is a transformation in which a second

transformation is performed on the image of a first transformation

Glide reflection: a composition of a translation and a reflection in a line parallel to the direction of the translation

Guided Examples:

Given DEF with D (3, 1), E (-3, 2), and F (-2, -2). Find the image points after:

a. A reflection over the x-axis, then a dilation of 3

1 Complete one transformation at time—IN ORDER.

b. A translation of (x, y) → (x - 5, y + 2), then c. A reflection over the y-axis, then a

a rotation of 90° counter clockwise translation of (x, y) → (x + 1, y – 4)

d. Triangle DEF has vertices D (3, -4), E (2, -2), and F (0, 1). Find the coordinates after a glide reflection composed

of the translation (x, y) → (x, y - 2) and a reflection in the y-axis.

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“You Try”

Given DEF with D (-5, 7), E (-3, 2), and F (-4, 8). Find the image points after:

e. A rotation of 180° counter clockwise, then a dilation of 2.

f. A reflection over the x-axis, then a rotation of 270° counter clockwise

g. Triangle ABC has vertices A (3, 2), B (-1, -3), and C (2, -1). Find the coordinates after a glide reflection of (x, y) →

(x + 3, y) and a reflection over the y-axis.

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TRANSFORMATIONS practice Show me that you can:

Translate a figure.

Reflect a figure.

Rotate a figure.

The original figure:

A(3, -1) B(5, -1) C(4, -3)

Translate it 3 left and 4 up.

A’( , ) B’( , ) C’( , )

Reflect it across the x-axis.

A’’( , ) B’’( , ) C’’( , )

Reflect it across the y-axis.

A’’’( , ) B’’’( , ) C’’’( , )

Now, let’s start with a new shape:

D(1, 1), E(2, 2), F(3, 2), G(4, 1)

Rotate it 90° clockwise:

D’( , ), E’( , ), F’( , ), G’( , )

Rotate it 180° either direction:

D’’( , ), E’’( , ), F’’( , ), G’’( , )

Rotate it 90° counterclockwise:

D’’’( , ), E’’’( , ), F’’’( , ), G’’’( , )

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Unit 13 Study Guide

I. Matching: Match the terms in the left column with the correct definitions or examples in the right

column.

1. _____ Reflection a. (x, y)

2. _____ Translation b. where the x and y axes intersect (0 ,0)

3. _____ Rotation c. a turn that moves 1 quadrant

4. _____ X axis d. the same direction as a clock

5. _____ Y axis e. moving a figure by flipping it in a coordinate grid

6. _____ Origin f. the vertical axis (up and down)

7. _____ Coordinate plane g. a numbered grid with x and y axes

8. _____ 90 degree rotation h. moving a figure by sliding it in a coordinate grid

9. _____ Clockwise i. the horizontal axis (across)

10. _____ Ordered Pair j. moving a figure by turning it in a coordinate grid

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II. Application:

On the coordinate grids provided, transform the figures as directed.

Use prime notation to label each point on the coordinate grid.

Write the ordered pairs for the coordinates of the new image below for each problem.

Plane 1 - Translate triangle ABC x-4, y+1.

A’ __________ B’ __________ C’ __________

Plane 2 - Reflect trapezoid DEFG over the x axis.

D’ __________ E’ __________ F’ __________ G’ __________

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Plane 3 - Rotate parallelogram HIJK over the 180 degrees.

H’ __________ I’ __________ J’ __________ K’ __________

Plane 4 - Dilate square LMNO by a scale factor of 2.

L’ __________ M’ __________ N’ __________ O’ __________

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Plane 5 - Rotate rectangle PQRS 90 degrees clockwise about the origin.

P’ __________ Q’ __________ R’ __________ S’ __________

Plane 6 - Dilate square TUVW 180 by a scale factor of ½ .

T’ __________ U’ __________ V’ __________ W’ __________

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Plane 7 - Plot triangle XYZ on the coordinate grid using the following coordinates:

X (-4, 4) Y (-4, -2) Z (-1, -2)

Reflect the figure over the y-axis, then translate x-2, y+1.

Plane 8 – The pre-image and image have been graphed. Explain the transformations that were applied to get to

the image.

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