Presentation on theme: "Defining Rotations, Reflections, and Translations"— Presentation transcript:
1Defining Rotations, Reflections, and Translations ~ Adapted from Walch Education
2The 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.
3TranslationsA 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.
4TranslationsTranslations 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 thatTh, k(x, y) = (x + h, y + k).
5ReflectionsA 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.
6ReflectionsReflections can be complicated to describe as a function, so we will only consider the following three reflections (for now):through the x-axis: rx-axis(x, y) = (x, –y)through the y-axis: ry-axis(x, y) = (–x, y)through the line y = x: ry = x(x, y) = (y, x)
7RotationsA 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.
8Similar 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.
9The 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.
10RotationsDepending 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: R90(x, y) = (–y, x)180° rotation about the origin: R180(x, y) = (–x, –y)270° rotation about the origin: R270(x, y) = (y, –x)
11Therefore, T24,10(P) = = (x + 24, y + 10) Practice # 1How far and in what direction does the point P (x, y) move when translated by the function T24, 10?Each point translated by T24,10 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, T24,10(P) = = (x + 24, y + 10)
12Your Turn…Using the definitions described earlier, write the translation T5, 3 of the rotation R180 in terms of a function F on (x, y).