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**Transformations on the coordinate plane**

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**Transformations Review**

Type Diagram A translation moves a figure left, right, up, or down A reflection moves a figure across its line of reflection to create its mirror image. A rotation moves a figure around a given point.

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Now we will look at how each transformation looks on a coordinate plane. The transformed figure is often named with the same letters, but adding an apostrophe. The transformation of ABC is A’B’C’.

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**Translation Find point A and Translate ABC 6 units to the right.**

Find point B and Find point C and 6 Units A A’ count 6 units to the right. Plot point A’. B’ B C C’ count 6 units to the right. Plot point B’. count 6 units to the right. Plot point C’.

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Translation Rules To translate a figure a units to the right, increase the x-coordinate of each point by a amount. To translate a figure a units to the left, decrease the x-coordinate of each point by a amount. Translate point P (3, 2) 9 units to the right. Since we are going to the right, we add 9 to the x-coordinate = 12, so the new coordinates of P’ are (12, 2) Translate point P (3, 2) 6 units to the left. Since we are going up, we subtract 6 to the x-coordinate = -3, so the new coordinates of P’ are (-3, 2)

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Translation Rules To translate a figure a units up, increase the y-coordinate of each point by a amount. Translate point P (3, 2) 9 units up. To translate a figure a units down, decrease the y-coordinate of each point by a amount. Since we are going up, we add 9 to the y-coordinate = 11, so the new coordinates of P’ are (3, 11) Translate point P (3, 2) 6 units down. Since we are going down, we subtract 6 to the y-coordinate = -4, so the new coordinates of P’ are (3, -4)

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Translation Example 2 The coordinates of point A are (-5, 4) Since we are moving to the right we increase the x-coordinate by 6. The coordinates of point B are (-2, 3) The coordinates of point C are (-3, 1) = 1, so the new coordinates of A’ are (1, 4). = 4, so the new coordinates of B’ are (4, 3). = 3, so the new coordinates of C’ are (3, 1).

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**Practice Point P (5, 8). Translate 2 to the left and 6 up. P’ (3, 14)**

Point Z (-3, -6). Translate 5 to the right and 9 down. Translate LMN, whose coordinates are (3, 6), (5, 9), and (7, 12), 9 units left and 14 units up. P’ (3, 14) Z’ (2, -15) L’M’N’ (-6, 20), (-4, 23), (-2, 26)

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**Reflection Reflect ABC across the y-axis.**

Count the number of units point A is from the line of reflection. 5 Units 5 Units A A’ Count the same number of units on the other side and plot point A’. 2 Units 2 Units B B’ 3 Units 3 Units Count the number of units point B is from the line of reflection. C C’ Count the same number of units on the other side and plot point B’. Count the number of units point C is from the line of reflection. Count the same number of units on the other side and plot point C’.

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Reflection Rules To reflect point (a, b) across the y-axis use the opposite of the x-coordinate and keep the y coordinate the same. Reflect point P (3, 2) across the y-axis. To reflect point (a, b) across the x-axis keep the x-coordinate the same and use the opposite of the y-coordinate Since we reflecting across the y-axis. Keep the y the same and use the opposite of the x. (-3, 2) Reflect point P (3, 2) across the x-axis. Since we reflecting across the x-axis. Keep the x the same and use the opposite of the y. (3, -2)

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Practice The coordinates of ABC are: (-5, 4), (-2, 3), (-3, 1) Reflect ABC across the y-axis and then reflect it across the x-axis. To reflect it across the y-axis keep the y the same and use the opposite x. The new coordinates are: (5, 4), (2, 3), (3, 1) To reflect it across the x-axis keep the x the same and use the opposite y. The new coordinates are: (5, -4), (2, -3), (3, -1)

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Rotation Rules To rotate a point 90° clockwise, switch the coordinates, and then multiply the new y-coordinate by -1. Rotate point P (3, 2) clockwise about the origin. To rotate a point 180°, just multiply each coordinate by -1. Since we are rotating it clockwise, we switch the coordinates (2, 3) and multiply the new y by -1, so the new coordinates are (2, -3) Rotate point P (3, 2) clockwise about the origin. Since we are rotating it 180°, we simply multiply the coordinates by -1, so the new coordinates are (-3, -2).

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**Practice Point P (5, 8). Rotate 90° clockwise about the origin.**

Point Z (-3, -6). Rotate 180° about the origin. Rotate LMN, whose coordinates are (3, 6), (5, 9), and (7, 12), 90° clockwise about the origin. P’ (8, -5) Z’ (-3, -6) L’M’N’ (6, -3), (9, -5), (7, -12)

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