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**Reflections and Translations**

Chapter 9.1 and 9.2 Reflections and Translations

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Reflection A reflection is a flip.

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**Reflect over the x-Axis**

Graph the point A (-2, 4) Graph the image of A’ under a reflection in the x-axis.

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**Reflect over the y-Axis**

Graph the point B (2, -5) Graph the image of B’ under a reflection in the y-axis.

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Concept

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**Reflect over the x-Axis**

Graph the line segment AB A (-6, 0) , B (-2, -6) Graph the image of AB’ under a reflection in the x-axis.

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**Reflect over the y-Axis**

Graph the line segment XY X (2, 2) , Y (4, 5) Graph the image of XY’ under a reflection in the y-axis.

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**Multiply the x-coordinate of each vertex by –1. (x, y) → (–x, y) **

Reflect a Figure in the x- or y-axis B. Graph quadrilateral ABCD with vertices A(1, 1), B(3, 2), C(4, –1), and D(2, –3) and its reflected image in the y-axis. Multiply the x-coordinate of each vertex by –1. (x, y) → (–x, y) A(1, 1) → A'(–1, 1) B(3, 2) → B'(–3, 2) C(4, –1) → C'(–4, –1) D(2, –3) → D'(–2, –3)

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**Multiply the y-coordinate of each vertex by –1. (x, y) → (x, –y) **

Reflect a Figure in the x- or y-axis A. Graph quadrilateral ABCD with vertices A(1, 1), B(3, 2), C(4, –1), and D(2, –3) and its image reflected in the x-axis. Multiply the y-coordinate of each vertex by –1. (x, y) → (x, –y) A(1, 1) → A'(1, –1) B(3, 2) → B'(3, –2) C(4, –1) → C'(4, 1) D(2, –3) → D'(2, 3)

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A. Graph quadrilateral LMNO with vertices L(3, 1), M(5, 2), N(6, –1), and O(4, –3) and its reflected image in the x-axis. Select the correct coordinates for the new quadrilateral L'M'N'O'. A. L'(3, –1), M'(5, –2), N'(6, 1), O'(4, 3) B. L'(–3, 1), M'(–5, 2), N'(–6, –1), O'(–4, –3) C. L'(–3, –1), M'(–5, –2), N'(–6, 1), O'(–4, 3) D. L'(1, 3), M'(2, 5), N'(–1, 6), O'(–3, 4)

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B. Graph quadrilateral LMNO with vertices L(–1, 0), M(1, 1), N(2, –2), and O(0, –4) and its reflected image under the y-axis. Select the correct coordinates for the point M' in the new quadrilateral L'M'N'O'. A. L'(–1, 0), M'(1, –1), N'(2, 2), O'(0, 4) B. L'(1, 0), M'(–1, 1), N'(–2, –2), O'(0, –4) C. L'(1, 0), M'(–1, –1), N'(–2, 2), O'(0, 4) D. L'(0, –1), M'(1, 1), N'(–2, 2), O'(–4, 0)

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Concept

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**Interchange the x- and y-coordinates of each vertex. (x, y) → (y, x) **

Reflect a Figure in the Line y = x Quadrilateral ABCD with vertices A(1, 1), B(3, 2), C(4, –1), and D(2, –3). Graph ABCD and its image under reflection of the line y = x. Interchange the x- and y-coordinates of each vertex. (x, y) → (y, x) A(1, 1) → A'(1, 1) B(3, 2) → B'(2, 3) C(4, –1) → C'(–1, 4) D(2, –3) → D'(–3, 2)

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Example 5 Quadrilateral EFGH has vertices E(–3, 1), F(–1, 3), G(1, 2), and H(–3, –1). Graph EFGH and its image under reflection of the line y = x. Select the correct coordinates for the point H' in the new quadrilateral E'F'G'H'. A. E'(–3, –1), F'(–1, –3), G'(1, –2), H'(–3, 1) B. E'(3, –1), F'(1, –3), G'(–1, 2), H'(3, –1) C. E'(1, –3), F'(3, –1), G'(2, 1), H'(–1, –3) D. E'(–1, 3), F'(–3, 1), G'(–2, –1), H'(1, 3)

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Concept

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**Translations A translation is a slide.**

It moves all points of a figure the same distance in the same direction. Vectors are used to define translations.

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Concept

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**Translations in the Coordinate Plane**

A. Graph ΔTUV with vertices T(–1, –4), U(6, 2), and V(5, –5) along the vector –3, 2.

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**Translations in the Coordinate Plane**

B. Graph pentagon PENTA with vertices P(1, 0), E(2, 2), N(4, 1), T(4, –1), and A(2, –2) along the vector –5, –1.

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A. Graph ΔABC with the vertices A(–3, –2), B(4, 4), C(3, –3) along the vector –1, 3. Choose the correct coordinates for ΔA'B'C'. A. A'(–2, –5), B'(5, 1), C'(4, –6) B. A'(–4, –2), B'(3, 4), C'(2, –3) C. A'(3, 1), B'(–4, 7), C'(1, 0) D. A'(–4, 1), B'(3, 7), C'(2, 0)

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B. Graph ΔGHJK with the vertices G(–4, –2), H(–4, 3), J(1, 3), K(1, –2) along the vector 2, –2. Choose the correct coordinates for ΔG'H'J'K'. A. G'(–6, –4), H'(–6, 1), J'(1, 1), K'(1, –4) B. G'(–2, –4), H'(–2, 1), J'(3, 1), K'(3, –4) C. G'(–2, 0), H'(–2, 5), J'(3, 5), K'(3, 0) D. G'(–8, 4), H'(–8, –6), J'(2, –6), K'(2, 4)

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Example 3 Describing Translations A. ANIMATION The graph shows repeated translations that result in the animation of the raindrop. Describe the translation of the raindrop from position 2 to position 3 in function notation and in words.

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Example 3 A. The graph shows repeated translations that result in the animation of the soccer ball. Choose the correct translation of the soccer ball from position 2 to position 3 in function notation. A. (x, y) → (x + 3, y + 2) B. (x, y) → (x + (–3), y + (–2)) C. (x, y) → (x + (–3), y + 2) D. (x, y) → (x + 3, y + (–2))

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Warm Up (4, –6) (12, 27) (–6, 2) 1. Subtract 3 from the x-coordinate and 2 from the y-coordinate in (7, –4). 2. Multiply each coordinate by 3 in (4, 9).

Warm Up (4, –6) (12, 27) (–6, 2) 1. Subtract 3 from the x-coordinate and 2 from the y-coordinate in (7, –4). 2. Multiply each coordinate by 3 in (4, 9).

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