unit2 5.2.1 and 5.2.2

Unit 5.2

Key Concepts•The coordinate plane is separated into four quadrants, or sections:

• In Quadrant I, x and y are positive.

• In Quadrant II, x is negative and y is positive.

• In Quadrant III, x and y are negative.

• In Quadrant IV, x is positive and y is negative.

)

Transformations on the Coordinate Plane

(3, – 2)

III

Q (0, 1)

J (1, 4) & S (1, 0)

(– 3, – 2)

Transformations on the Coordinate Plane

Transformations are movements of geometric figures.

The preimage is the position of the figure before the transformation, and the image is the position of the figure after the transformation.

Transformations on the Coordinate Plane

Transformations on the Coordinate Plane

Reflection: a figure is flipped over a line

Translation: a figure is slid in any direction

Dilation: a figure is enlarged or reduced

Rotation: a figure is turned around a point

Identify the transformation as a reflection, translation, dilation, or rotation.

Answer: The figure has been increased in size.This is a dilation.

Identify Transformations

Identify the transformation as a reflection, translation, dilation, or rotation.

Answer: The figure has been shifted horizontally to the right. This is a translation.

Identify Transformations

Identify the transformation as a reflection, translation, dilation, or rotation.

Answer: The figure has been turned around a point.This is a rotation.

Identify Transformations

Identify the transformation as a reflection, translation, dilation, or rotation.

Answer: The figure has been flipped over a line.This is a reflection.

Identify Transformations

Answer: rotation

Answer: translation

Answer: reflection

Answer: dilation

Identify each transformation as a reflection, translation, dilation, or rotation.

a. b.

c.

d.

Identify Transformations

Transformations on the Coordinate Plane

.• through the x-axis: rx-axis(x, y) = (x, –y) Multiply the y-

coordinate by -1• through the y-axis: ry-axis(x, y) = (–x, y) Multiply the x-

coordinate by -1• through the line y = x: ry = x(x, y) = (y, x) flip the

coordinates

Reflection

A trapezoid has vertices W(–1, 4), X(4, 4), Y(4, 1) and Z(–3, 1).

Trapezoid WXYZ is reflected over the y-axis. Find the coordinates of the vertices of the image.

To reflect the figure over the y-axis, multiply each x-coordinate by –1.

(x, y) (–x, y)

W(–1, 4) (1, 4)

X(4, 4) (–4, 4)

Y(4, 1) (–4, 1)

Z(–3, 1) (3, 1)

Answer: The coordinates of the vertices of the image are W(1, 4), X(–4, 4), Y(–4, 1), and Z(3, 1).

Reflection

A trapezoid has vertices W(–1, 4), X(4, 4), Y(4, 1), and Z(–3, 1).

Graph trapezoid WXYZ and its image W X Y Z.

Graph each vertex of the trapezoid WXYZ.Connect the points.

Answer:

Graph each vertex of the reflected image W X Y Z. Connect the points.

W X

YZ

WX

Y Z

Reflection

A parallelogram has vertices A(–4, 7), B(2, 7), C(0, 4) and D(–2, 4).

a. Parallelogram ABCD is reflected over the x-axis. Find the coordinates of the vertices of the image.

Answer: A(–4, –7), B(2, –7), C(0, –4), D(–2, –4)

Reflection

b. Graph parallelogram ABCD and its image A B C D.

Answer:

Reflection

Transformations on the Coordinate Plane

Triangle ABC has vertices A(–2, 1), B(2, 4), and C(1, 1).

Find the coordinates of the vertices of the image if it is translated 3 units to the right and 5 units down.

To translate the triangle 3 units to the right, add 3 to the x-coordinate of each vertex. To translate the triangle 5 units down, add –5 to the y-coordinate of each vertex.

Answer: The coordinates of the vertices of the image are A(1, –4), B(5, –1), and C(4, –4).

Translation

Graph triangle ABC and its image.

Answer:

The preimage is .

The translated image is

A

B

C

A

B

C

Translation

Triangle JKL has vertices J(2, –3), K(4, 0), and L(6, –3).a. Find the coordinates of the vertices of the image if it is

translated 5 units to the left and 2 units up.

b. Graph triangle JKLand its image.

Answer: J(–3, –1), K(–1, 2), L(1, –1)

Answer:

Translation

Transformations on the Coordinate Plane

• Rotation:

• 90° rotation about the origin: R90(x, y) = (–y, x)

• 180° rotation about the origin: R180(x, y) = (–x, –y)

• 270° rotation about the origin: R270(x, y) = (y, –x)

Triangle ABC has vertices A(1, –3), B(3, 1), and C(5, –2).

Find the coordinates of the image of ABC after it is rotated 180° about the origin.

To find the coordinates of the image of ABC after a 180° rotation, multiply both coordinates of each point by –1.

Answer: The coordinates of the vertices of the image are A(–1, 3), B(–3, –1), and C(–5, 2).

Rotation

Graph the preimage and its image.

Answer:

The preimage is .

The translated image is

A

B

C

A

B

C

Rotation

Triangle RST has vertices R(4, 0), S(2, –3), and T(6, –3).

a. Find the coordinates of the image of RST after it is rotated 90° counterclockwise about the origin.

b. Graph the preimage and the image.

Answer: R(0, 4), S(3, 2), T (3, 6)

Answer:

Rotation

Guided Practice

Example 1How far and in what direction does the point P (x, y) move when translated by the function T24, 10?

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5.2.1: Defining Rotations, Reflections, and Translations

Guided Practice: Example 1, continued

1.Each point translated by T24,10 will be moved right 24 units, parallel to the x-axis.

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5.2.1: Defining Rotations, Reflections, and Translations

Guided Practice: Example 1, continued

2.The point will then be moved up 10 units, parallel to the y-axis.

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5.2.1: Defining Rotations, Reflections, and Translations

Guided Practice: Example 1, continuedTherefore, T24,10(P) = = (x + 24, y + 10).

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5.2.1: Defining Rotations, Reflections, and Translations

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Guided Practice

Example 2Using the definitions described earlier, write the translation T5, 3 of the rotation R180 in terms of a

function F on (x, y).

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5.2.1: Defining Rotations, Reflections, and Translations

Guided Practice: Example 2, continued

1.Write the problem symbolically.

F = T5, 3(R180(x, y))

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5.2.1: Defining Rotations, Reflections, and Translations

Guided Practice: Example 2, continued

2.Start from the inside and work outward.

R180(x, y) = (–x, –y)

Therefore, T5, 3(R180(x, y)) = T5, 3(–x, –y).

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5.2.1: Defining Rotations, Reflections, and Translations

Guided Practice: Example 2, continued

3.Now translate the point.

T5, 3(–x, –y) = (–x + 5, –y + 3)

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5.2.1: Defining Rotations, Reflections, and Translations

Guided Practice: Example 2, continued

4.Write the result of both translations.

F = T5, 3(R180(x, y)) = (–x + 5, –y + 3)

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5.2.1: Defining Rotations, Reflections, and Translations

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Guided Practice

Example 2Use the definitions you have learned to graph the reflection of parallelogram ABCD, or , through the y-axis given with the points A (–5, 5), B (–3, 4), C (–4, 1), and D (–6, 2).

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5.2.2: Applying Rotations, Reflections, and Translations

Guided Practice: Example 2, continued

1.Using graph paper, draw the x- and y-axes and graph with A (–5, 5), B (–3, 4), C (–4, 1), and D (–6, 2).

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5.2.2: Applying Rotations, Reflections, and Translations

Guided Practice: Example 2, continued

2.Write the new points.

where

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5.2.2: Applying Rotations, Reflections, and Translations

Guided Practice: Example 2, continued

3.Plot the new points , , , and .

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5.2.2: Applying Rotations, Reflections, and Translations

Guided Practice: Example 2, continued

4.Connect the corners of the points to graph the reflection of .

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5.2.2: Applying Rotations, Reflections, and Translations

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Guided Practice

Example 3Using the definitions you have learned, graph a 90° rotation of with the points A (1, 4), B (6, 3), and C (3, 1).

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5.2.2: Applying Rotations, Reflections, and Translations

Guided Practice: Example 3, continuedUsing graph paper, draw the x- and y-axes and graph with the points A (1, 4), B (6, 3), and C (3, 1).

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5.2.2: Applying Rotations, Reflections, and Translations

Guided Practice: Example 3, continued

2.Write the new points.

where

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5.2.2: Applying Rotations, Reflections, and Translations

Guided Practice: Example 3, continued

3.Plot the new points , , and .

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5.2.2: Applying Rotations, Reflections, and Translations

Guided Practice: Example 3, continued

4.Connect the vertices to graph a 90° rotation of .

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5.2.2: Applying Rotations, Reflections, and Translations

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