Using Matrices to Transform
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Using Matrices to Transform Geometric Figures. 4-3. Warm Up. Lesson Presentation. Lesson Quiz. Holt Algebra 2. Warm Up Perform the indicated operation. 1. 2. 3. Objective. Use matrices to transform a plane figure. Vocabulary. t ranslation m atrix r eflection m atrix

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4 3

Using Matrices to Transform

Geometric Figures

4-3

Warm Up

Lesson Presentation

Lesson Quiz

Holt Algebra 2


4 3

Warm Up

Perform the indicated operation.

1.

2.

3.


4 3

Objective

Use matrices to transform a plane figure.


4 3

Vocabulary

translation matrix

reflection matrix

rotation matrix


4 3

You can describe the position, shape, and size of a polygon on a coordinate plane by naming the ordered pairs that define its vertices.

The coordinates of ΔABC below are A (–2, –1),

B (0, 3), and C (1, –2) .

You can also define ΔABC by a matrix:

x-coordinates

y-coordinates


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A translation matrix is a matrix used to translate coordinates on the coordinate plane. The matrix sum of a preimage and a translation matrix gives the coordinates of the translated image.


4 3

Reading Math

The prefix pre- means “before,” so the preimage is the original figure before any transformations are applied. The image is the resulting figure after a transformation.


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Example 1: Using Matrices to Translate a Figure

Translate ΔABC with coordinates A(–2, 1), B(3, 2), and C(0, –3), 3 units left and 4 units up. Find the coordinates of the vertices of the image, and graph.

The translation matrix will have –3 in all entries in row 1 and 4 in all entries in row 2.

x-coordinates

y-coordinates


4 3

Example 1 Continued

A'B'C', the image of ABC, has coordinates A'(–5, 5), B'(0, 6), and C'(–3, 1).


4 3

Check It Out! Example 1

Translate ΔGHJ with coordinates G(2, 4), H(3, 1), and J(1, –1) 3 units right and 1 unit down. Find the coordinates of the vertices of the image and graph.

The translation matrix will have 3 in all entries in row 1 and –1 in all entries in row 2.

x-coordinates

y-coordinates


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Check It Out! Example 1 Continued

G'H'J', the image of GHJ, has coordinates G'(5, 3), H'(6, 0), and J'(4, –2).


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A dilation is a transformation that scales—enlarges or reduces—the preimage, resulting in similar figures. Remember that for similar figures, the shape is the same but the size may be different. Angles are congruent, and side lengths are

proportional.

When the center of dilation is the origin, multiplying the coordinate matrix by a scalar gives the coordinates of the dilated image. In this lesson, all dilations assume that the origin is the center of dilation.


4 3

Example 2: Using Matrices to Enlarge a Figure

Enlarge ΔABC with coordinates A(2, 3), B(1, –2), and C(–3, 1), by a factor of 2. Find the coordinates of the vertices of the image, and graph.

Multiply each coordinate by 2 by multiplying each entry by 2.

x-coordinates

y-coordinates


4 3

Example 2 Continued

A'B'C', the image of ABC, has coordinates A'(4, 6), B'(2, –4), and C'(–6, 2).


4 3

Enlarge ΔDEF with coordinates D(2, 3), E(5, 1), and F(–2, –7) a factor of . Find the coordinates of the vertices of the image, and graph.

Multiply each coordinate by by multiplying each entry by .

Check It Out! Example 2


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D'E'F', the image of DEF, has coordinates

Check It Out! Example 2 Continued


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A reflection matrix is a matrix that creates a mirror image by reflecting each vertex over a specified line of symmetry. To reflect a figure across the y-axis, multiply

by the coordinate matrix. This reverses the x-coordinates and keeps the y-coordinates unchanged.


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Caution

Matrix multiplication is not commutative. So be sure to keep the transformation matrix on the left!


4 3

Example 3: Using Matrices to Reflect a Figure

Reflect ΔPQR with coordinates P(2, 2), Q(2, –1), and R(4, 3) across the y-axis. Find the coordinates of the vertices of the image, and graph.

Each x-coordinate is multiplied by –1.

Each y-coordinate is multiplied by 1.


4 3

Example 3 Continued

The coordinates of the vertices of the image are P'(–2, 2), Q'(–2, –1), and R'(–4, 3).


4 3

To reflect a figure across the x-axis, multiply by

.

Reflect ΔJKL with coordinates J(3, 4), K(4, 2), and L(1, –2) across the x-axis. Find the coordinates of the vertices of the image and graph.

Check It Out! Example 3


4 3

Check It Out! Example 3

The coordinates of the vertices of the image are J'(3, –4), K'(4, –2), L'(1, 2).


4 3

A rotation matrix is a matrix used to rotate a figure. Example 4 gives several types of rotation matrices.


4 3

Example 4: Using Matrices to Rotate a Figure

Use each matrix to rotate polygon ABCD with coordinates A(0, 1), B(2, –4), C(5, 1), and D(2, 3) about the origin. Graph and describe the image.

A.

The image A'B'C'D' is rotated 90° counterclockwise.

B.

The image A''B''C''D'' is rotated 90° clockwise.


4 3

Example 4 Continued


4 3

Check It Out! Example 4

Use

Rotate ΔABC with coordinates A(0, 0), B(4, 0), and C(0, –3) about the origin. Graph and describe the image.

A'(0, 0), B'(-4, 0), C'(0, 3); the image is rotated 180°.


4 3

Check It Out! Example 4 Continued


4 3

Lesson Quiz

Transform triangle PQR with vertices

P(–1, –1), Q(3, 1), R(0, 3). For each, show the matrix transformation and state the vertices of the image.

1. Translation 3 units to the left and 2 units up.

2. Dilation by a factor of 1.5.

3. Reflection across the x-axis.

4. 90° rotation, clockwise.


4 3

Lesson Quiz

1.

2.

3. 4.


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