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9.4

?

Algebraic

Representations of

Transformations

ESSENTIAL QUESTION

8.G.1.3

Describe the effect of ¡­,

translations, rotations,

and reflections on twodimensional figures using

coordinates.

YOUR TURN

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Math Trainer

Online Assessment

and Intervention

my.

1.

A rectangle has vertices at (0, -2), (0, 3), (3, -2), and (3, 3). What are the

coordinates of the vertices of the image after the translation

(x, y) ¡ú (x - 6, y - 3)? Describe the translation.

(-6, -5), (-6, 0), (-3, -5), and (-3, 0); the rectangle

is translated 6 units to the left and 3 units down.

How can you describe the effect of a translation, rotation, or

reflection on coordinates using an algebraic representation?

Algebraic Representations

of Reflections

Algebraic Representations

of Translations

The rules shown in the table describe how coordinates change when a figure is

translated up, down, right, and left on the coordinate plane.

Math On the Spot

Math On the Spot

my.

my.

The signs of the coordinates of a figure change when the figure is reflected

across the x-axis and y-axis. The table shows the rules for changing the signs

of the coordinates after a reflection.

Translations

Reflections

Right a units

Add a to the x-coordinate: (x, y) ¡ú (x + a, y)

Left a units

Subtract a from the x-coordinate: (x, y) ¡ú (x - a, y)

Up b units

Add b to the y-coordinate: (x, y) ¡ú (x, y + b)

Down b units

Subtract b from the y-coordinate: (x, y) ¡ú (x, y - b)

My Notes

8.G.1.3

? Houghton Mifflin Harcourt Publishing Company

Triangle XYZ has vertices X(0, 0), Y(2, 3), and Z(4, -1). Find the vertices of

triangle X¡äY¡äZ¡ä after a translation of 3 units to the right and 1 unit down.

Then graph the triangle and its image.

Add 3 to the x-coordinate of each

STEP 2

Apply the rule to find the vertices of the image.

vertex and subtract 1 from the

y-coordinate of each vertex.

Vertices of ¡÷ XYZ

Rule: (x + 3, y - 1)

Vertices of ¡÷ X¡ä Y¡ä Z¡ä

X(0, 0)

(0 + 3, 0 - 1)

X¡ä(3, -1)

Y(2, 3)

(2 + 3, 3 - 1)

Y¡ä(5, 2)

Z(4, -1)

(4 + 3, -1 - 1)

Z¡ä(7, -2)

Graph triangle XYZ and its image.

y

3

X

O

Across the y-axis

Multiply each x-coordinate by -1: (x, y) ¡ú (-x, y)

You add to

or subtract

from the

x-coordinate.

8.G.1.3

Rectangle RSTU has vertices R(-4, -1), S(-1, -1), T(-1, -3), and U(-4, -3).

Find the vertices of rectangle R¡äS¡äT¡äU¡ä after a reflection across

the y-axis. Then graph the rectangle and its image.

Multiply the

x-coordinate of

each vertex by -1.

STEP 1 Apply the rule to find the vertices of the image.

STEP 2

Vertices of RSTU

Rule: (-1 ¡¤ x, y)

Vertices of R¡äS¡äT¡äU¡ä

R(-4, -1)

(-1 ¡¤ (-4), - 1)

R¡ä(4, -1)

S(-1, -1)

(-1 ¡¤ (-1), - 1)

S¡ä(1, -1)

T(-1, -3)

(-1 ¡¤ (-1), - 3)

T¡ä(1, -3)

U(-4, -3)

(-1 ¡¤ (-4), - 3)

U¡ä(4, -3)

Graph rectangle RSTU and its image.

y

3

Y

Y¡ä

x

X¡ä

Multiply each y-coordinate by -1: (x, y) ¡ú (x, -y)

EXAMPLE 2

EXAMPL 1

EXAMPLE

STEP 1

Across the x-axis

Z

-3

7

Z¡ä

Math Talk

8_MFLESE056718_U4M09L4.indd 297

x

-5

Mathematical Practices

When you translate a

figure to the left or

right, which coordinate

do you change?

Lesson 9.4

? Houghton Mifflin Harcourt Publishing Company

LESSON

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297

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R

S

U

T

O

S¡ä

5

R¡ä

T¡ä

U¡ä

-5

298

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PROFESSIONAL DEVELOPMENT

Integrate Mathematical

Practices MP.3.1

This lesson provides an opportunity to address

this Mathematical Practices standard. It calls for

students to use logic to analyze situations.

Students use the rules for translations, reflections,

and rotations to find the vertices of the image

using an algebraic representation instead of

graphs. Also, students use an algebraic rule to

create a graph of an image, then use the graph

to describe the transformation.

Math Background

Having a good understanding of transformations

in both their graphical and algebraic representations will be beneficial for students in more

advanced levels of algebra. For example, students

will be using a parent parabola that has a vertex

at the origin, and then translating it while

maintaining its shape. Students will use the

equation for the original parabola and the

translation rule to write the equation of the

translated parabola.

Algebraic Representations of Transformations

298

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Reflect

YOUR TURN

2. Triangle ABC has vertices A(-2, 6), B(0, 5), and C(3, -1). Find the vertices

of triangle A¡äB¡äC¡ä after a reflection across the x-axis.

A¡ä(-2, -6), B¡ä(0, -5), and C¡ä(3, 1)

3.

Personal

Math Trainer

Communicate Mathematical Ideas How would you find the vertices

of an image if a figure were rotated 270¡ã clockwise? Explain.

A 270¡ã clockwise rotation is the same as a 90¡ã

Online Assessment

and Intervention

counterclockwise rotation. Change the sign of the

my.

y-coordinate and switch the coordinates.

Algebraic Representations of Rotations

When points are rotated about the origin, the coordinates of the image can be

found using the rules shown in the table.

90¡ã clockwise

90¡ã

counterclockwise

my.

Multiply each x-coordinate by -1; then switch the x- and

y-coordinates: (x, y) ¡ú (y, -x)

Online Assessment

and Intervention

Multiply each y-coordinate by -1; then switch the x- and

y-coordinates: (x, y) ¡ú (-y, x)

A triangle has vertices at J(-2, -4), K(1, 5), and L(2, 2). What are the

coordinates of the vertices of the image after the triangle is rotated

90¡ã counterclockwise?

J¡ä(4, -2), K¡ä(-5, 1), and L¡ä(-2, 2)

my.

8.G.1.3

Guided Practice

Quadrilateral ABCD has vertices at A(-4, 2), B(-3, 4), C(2, 3), and D(0, 0).

Find the vertices of quadrilateral A¡äB¡äC¡äD¡ä after a 90¡ã clockwise rotation.

Then graph the quadrilateral and its image.

Multiply the x-coordinate of

each vertex by -1, and then

switch the x- and y-coordinates.

STEP 1 Apply the rule to find the vertices of the image.

Vertices of ABCD

Rule: (y, -x)

Vertices of A¡äB¡äC¡äD¡ä

A(-4, 2)

(2, -1 ¡¤ (-4))

A¡ä(2, 4)

B(-3, 4)

(4, -1 ¡¤ (-3))

B¡ä(4, 3)

C(2, 3)

(3, -1 ¡¤ 2)

C¡ä(3, -2)

D(0, 0)

(0, -1 ¡¤ 0)

D¡ä(0, 0)

1. Triangle XYZ has vertices X(-3, -2), Y(-1, 0), and Z(1, -6).

Find the vertices of triangle X¡äY¡äZ¡ä after a translation of

6 units to the right. Then graph the triangle and its image.

(Example 1)

X¡ä(3, -2), Y¡ä(5, 0), and Z¡ä(7, -6)

2. Describe what happens to the x- and y-coordinates after a

point is reflected across the x-axis. (Example 2)

-4

-2

D

-2

X¡ä

Z

Z¡ä

y

5

The triangle is rotated 90¡ã clockwise.

A¡ä

?

?

B¡ä

C

D¡ä

x

7

O

-7

3. Use the rule (x, y) ¡ú (y, -x) to graph the image of the triangle

at right. Then describe the transformation. (Example 3)

2

A

Y¡ä

Y

X

y-coordinate changes sign.

y

4

-3

The x-coordinate remains the same, while the

Graph the quadrilateral and its image.

B

y

3

4. How do the x- and y-coordinates change when a figure is

translated right a units and down b units?

x

2

ESSENTIAL QUESTION CHECK-IN

4

x

-5

The x-coordinates increase by a, and the

C¡ä

O

5

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EXAMPL 3

EXAMPLE

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4.

Personal

Math Trainer

Multiply both coordinates by -1: (x, y) ¡ú (-x, -y)

180¡ã

STEP 2

YOUR TURN

Math On the Spot

Rotations

-3

y-coordinates decrease by b.

-4

Lesson 9.4

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300

Unit 4

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DIFFERENTIATE INSTRUCTION

Communicating Math

Have students work in pairs. One student

provides a transformation. The other describes

how the ordered pairs change using an

algebraic representation.

Student 1: Translate right 3 units and up

2 units.

Student 2: Add 3 to the x-value, and add

2 to the y-value: (x, y) ¡ú (x + 3, y + 2).

Visual Cues

On three separate index cards, have students

write the rules for how to change the coordinates for a figure when it is translated, rotated,

and reflected. Have students use colored pencils

or markers to emphasize the changes made to

the x- and y-values.

Additional Resources

Differentiated Instruction includes:

? Reading Strategies

? Success for English Learners ELL

? Reteach

? Challenge PRE-AP

Algebraic Representations of Transformations

300

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