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