Translation - a translation is simply a sliding of the figure to a different position on the coordinate plane. THe reason why it is called a translation is because you are taking all the x-coordinates of the original figure and moving them left or right by the same number of spaces. The same is true for the y-coordinates.
Example: The coordinates (-7,-2), (-7,-6), and (-4,-6) form a right triangle in Quadrant III. You can translate this triangle to Quandrant I by adding 9 (in this case) to all your x-coordinates and 10 to all your y-coordinates. So your translated points will be (2, 8), (2, 4) and (4, 4). Try it on grid paper to have a look.
To describe a translation you write T(x,y). For the translation below, sliding from the red triangle to the blue, we moved five spaces to the left, so we subtracted 5 (or -5). We also moved up 3 spaces. Because the x axis is left to oright and the y is up and down, the translation can be written as T(-5,3).
Reflection - a reflection is a mirror image across either the x-axis or the y-axis. Since the y-axis measures the positions of the horitzontal lines, reflecting the image across this axis means that all the y-coordinates remain the same but the oppiste of the x-coordinates are needed to reflect the figure. The same is true for reflecting across the x-axis. However, all the x-coordinates remain the same, but the y's change.
For the reflection above, notice that point A is (1,7). However the reflection or A' is (-1,7). Point B is (2,3) so the reflection across the y-axis is (-2,3). If we were to reflect the image across the x-axis, point A (1,7) transforms into point A' (1,-7).
Rotation - in regards to rotation the figure is turned a certain number of degrees around a specific point. This point is call the center of rotation. The center of rotation can be the origin or even a point one the figure.
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