1. Transformations on the coordinate plane

2. 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.

3. 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’.

4. 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’.

5. 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)

6. 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)

7. 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).

8. 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)

9. 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’.

10. 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)

11. 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)

12. 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).

13. 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)