Transformations As Functions
~ Adapted from Walch Education
Transformations
• A transformation changes the position,
shape, or size of a figure on a coordinate
plane.
• The original figure, called a preimage, is
changed or moved, and the resulting figure
is called an image.
An isometry is a transformation in which the
preimage and the image are congruent.
An isometry is also referred to as a “rigid
transformation” because the shape still has the
same size, area, angles, and line lengths.
Figures are congruent if they both have the
same shape, size, lines, and angles. The new
image is simply moving to a new location.
•T(x, y) = (x + h, y + k), then
would be:
ONE-TO-ONE
• Transformations are one-to-one, which means each
point in the set of points will be mapped to exactly
one other point and no other point will be mapped to
that point.
More Info…
• The simplest transformation is the identity
function I where I: (x', y' ) = (x, y).
• Transformations can be combined to form a
new transformation that will be a new
function.
• Because the order in which functions are taken can
affect the output, we always take functions in a
specific order, working from the inside out.
• For example, if we are given the set of functions
h(g(f(x))), we would take f(x) first and then g and
finally h.
Three Isometric Transformations
• A translation, or slide, is a transformation
that moves each point of a figure the same
distance in the same direction.
• A reflection, or flip, is a transformation
where a mirror image is created.
• A rotation, or turn, is a transformation that
turns a figure around a point.
Some transformations are not isometric.
Examples of non-isometric transformations
are horizontal stretch and dilation.
• A dilation stretches or contracts both
coordinates.
Practice #1
• Given the point P(5, 3) and T(x, y) = (x + 2, y + 2),
what are the coordinates of T(P)?
• T(P) = (x + 2, y + 2)
• (5 + 2, 3 + 2)
• (7, 5)
T(P) = (7, 5)
Challenge Problem
Given the transformation of a translation
T5, –3, and the points P (–2, 1) and Q (4, 1),
show that the transformation of a translation is
isometric by calculating the distances, or
lengths, of and .
Thanks for Watching!!!
~Dr. Dambreville