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 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.


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


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



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



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


Answer: rotation

Answer: translation

Answer: reflection

Answer: dilation

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

a. b.

c.

d.



.• 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

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


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

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

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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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1.Write the problem symbolically.

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

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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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3.Now translate the point.

T5, 3(–x, –y) = (–x + 5, –y + 3)

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4.Write the result of both translations.

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

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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


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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2.Write the new points.

where

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3.Plot the new points , , , and .

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

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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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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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2.Write the new points.

where

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3.Plot the new points , , and .

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

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Unit2 5.2.1 and 5.2.2

Documents

Transcript of Unit2 5.2.1 and 5.2.2