Slide 2 Defining Rotations, Reflections, and Translations ~
Adapted from Walch Education Slide 3 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. Slide 4 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. Slide 5 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).
Slide 6 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. Slide 7 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) Slide 8 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. Slide 9 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. Slide 10 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. Slide 11 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) Slide 12
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) Slide 13 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).
Slide 14 Thanks for Watching ~Ms. Dambreville
