Transcript Defining Rotations, Reflections, and Translations ~ Adapted from Walch Education • The coordinate plane is separated into four quadrants, or sections:  In Quadrant I, x and.

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
Th, 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: 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)
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: 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)
Practice # 1
• How 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)
Your Turn…
Using the definitions described earlier,
write the translation T5, 3 of the rotation
R180 in terms of a function F on (x, y).
Thanks for Watching
~Ms. Dambreville