Defining Rotations, Reflections, and Translations ~ Adapted from Walch Education
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.
Translations A translation is an isometry where all points in the preimage are moved parallel to a given line. No matter which direction or distance the translation moves the preimage, the image will have the same orientation as the preimage. Because the orientation does not change, a translation is also called a slide.
Translations Translations are described in the coordinate plane by the distance each point is moved with respect to the x-axis and y-axis. If we assign h to be the change in x and k to be the change in y, we can define the translation function T such that T h, k (x, y) = (x + h, y + k).
Reflections A reflection is an isometry in which a figure is moved along a line perpendicular to a given line called the line of reflection. Each point in the figure will move a distance determined by its distance to the line of reflection. A reflection is the mirror image of the original figure; therefore, a reflection is also called a flip.
Reflections Reflections can be complicated to describe as a function, so we will only consider the following three reflections (for now): through the x-axis: r x-axis (x, y) = (x, –y) through the y-axis: r y-axis (x, y) = (–x, y) through the line y = x: r y = x (x, y) = (y, x)
Rotations A rotation is an isometry where all points in the preimage are moved along circular arcs determined by the center of rotation and the angle of rotation. A rotation may also be called a turn. This transformation can be more complex than a translation or reflection because the image is determined by circular arcs instead of parallel or perpendicular lines.
Similar to a reflection, a rotation will not move a set of points a uniform distance. When a rotation is applied to a figure, each point in the figure will move a distance determined by its distance from the point of rotation. A figure may be rotated clockwise, in the direction that the hands on a clock move, or counterclockwise, in the opposite direction that the hands on a clock move.
The figure below shows a 90° counterclockwise rotation around the point R. Comparing the arc lengths in the figure, we see that point B moves farther than points A and C. This is because point B is farther from the center of rotation, R.
Rotations Depending on the point and angle of rotation, the function describing a rotation can be complex. Thus, we will consider the following counterclockwise rotations, which can be easily defined. 90° rotation about the origin: R 90 (x, y) = (–y, x) 180° rotation about the origin: R 180 (x, y) = (–x, –y) 270° rotation about the origin: R 270 (x, y) = (y, –x)
Practice # 1 How far and in what direction does the point P (x, y) move when translated by the function T 24, 10 ? Each point translated by T 24,10 will be moved right 24 units, parallel to the x-axis. The point will then be moved up 10 units, parallel to the y-axis. Therefore, T 24,10 (P) = = (x + 24, y + 10)
Your Turn… Using the definitions described earlier, write the translation T 5, 3 of the rotation R 180 in terms of a function F on (x, y).