Image

The shape that results from a transformation of a figure known as the preimage.

Preimage

The original shape that undergoes a transformation.

Transformation

A mapping of a figure from its original position to a new position.

Vector

A quantity that has both a size and direction.

Isometry

An isometry is a transformation that does not change the shape or size of a figure.  Isometries are also called congruence transformations or rigid motions.  Examples include reflections, translations, and rotations.  In isometry, the image is always congruent to the preimage.

Reflections

A reflection is a transformation across a line, called the line of reflection, so that the line of reflection is the perpendicular bisector of each segment joining each point and its image.

Translations

A translation is a transformation where all the points of a figure are moved the same distance in the same direction.  Since a translation is an isometry, the image of a translated figure is congruent to the preimage.  The translation vector determines the direction and distance of movement.

Transformation of Parent Function y = f(x)

Transformations can be used to graph complicated functions by using the graphs of simpler functions called parent functions.  Here are some examples of parent functions:

For the parent function y = x² , here are two examples of transformations:

Rotations

A rotation is a transformation that turns a figure around a fixed point, called the center of rotation.  Since a rotation is an isometry, the image of the rotated figure is congruent to the preimage.  Each point and its image are the same distance from the point of rotation.

Composition of Transformations

A composition of transformations is one transformation followed by another.

The composition of two reflections across two parallel line is equivalent to a translation.

The translation vector is perpendicular to the lines.

The length of the translation vector is twice the distance between the lines.

The composition of two reflections across two intersecting lines is equivalent to a rotation.

The center of rotation is the intersection of the lines.

The angle of rotation is twice the measure of the angle formed by the lines.

Glide Reflection

A glide reflection is the composition of a translation and a reflection across a line parallel to the translation vector.

Symmetry

Exact matching of size, shape, and relative position (point for point) of parts on opposite sides of a point, line, or plane.

Line Symmetry

A figure has line symmetry (or reflection symmetry) if it can be reflected across a line so that the image coincides with the preimage.

Line of Symmetry

The line of symmetry (also called the axis of symmetry) divides the figure into two congruent halves.

Rotational Symmetry

A figure has rotational symmetry (or radial symmetry) if it can be rotated about a point by an angle greater than 0 degrees and less than 360 degrees so that the image coincides with the preimage.

The angle of rotational symmetry is the smallest angle through which a figure can be rotated to coincide with itself.  The number of times the figure coincides with itself as it rotates through 360 degrees is called the order of the rotational symmetry.

Plane Symmetry

A three-dimensional figure has plane symmetry if a plane can divide the figure into two congruent reflected halves.

Symmetry About an Axis

A three-dimensional figure has symmetry about an axis if there is a line about which the figure can be rotated (by an angle greater than 0 degrees and less than 360 degrees) so that the image coincides with the preimage.

Tessellation

A tessellation is a repeating pattern that completely covers a plane with no gaps or overlaps.  The measures of the angles that meet at each vertex must add up to 360 degrees.

A regular tessellation is formed by congruent regular polygons.  To form a regular tessellation, the angle measures of a regular polygon must be a divisor of 360 degrees.

A semiregular tessellation is formed by two or more different regular polygons, with the same number of each polygon occurring in the same order at every vertex.  The angle measures around a vertex must add up to 360 degrees.

Dilations

A dilation is a transformation that changes the size of a figure but not the shape.  The image and the preimage of a figure under dilation are similar.

In a dilation, the lines connecting points of the image with the corresponding points of the preimage all intersect at the center of dilation.

The scale factor of a dilation is the ratio of the linear measurement of the image to a corresponding measurement of the preimage.  For a dilation with a scale factor of k, if k > 0, then the figure is not turned or flipped.  If k < 0 (negative), then the figure is rotated by 180 degrees.

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