Math II Unit 1 Transformations

Transformations A transformation is a change in position, shape, or size of a figure.

Identifying Transformations Figures in a plane can be Reflected Rotated Translated To produce new figures. The new figures is called the IMAGE. The original figures is called the PREIMAGE. The operation that MAPS, or moves the preimage onto the image is called a transformation.

Copy this down Rotation about a point Reflection in a line Translation

Some notation When you name an image, take the corresponding point of the preimage and add a prime symbol. For instance, if the preimage is A, then the image is A’, read as “A prime.”

Terminology In a transformation, the given figure is called the preimage and the transformed figure is called the image. Points on the image that correspond to points on the preimage are labeled similarly but with primes. A transformation is said to map a figure onto its image.

ISOMETRY An ISOMETRY is a transformation that preserves lengths. Isometries also preserve angle measures, parallel lines, and distances between points. Transformations that are isometries are called RIGID TRANSFORMATIONS.

Ex. 2: Identifying Isometries Which of the following appear to be isometries? This transformation appears to be an isometry. The blue parallelogram is reflected in a line to produce a congruent red parallelogram. So reflections are isometric.

Ex. 2: Identifying Isometries Which of the following appear to be isometries? This transformation is not an ISOMETRY because the image is not congruent to the preimage Dilations are not isometric

Ex. 2: Identifying Isometries Which of the following appear to be isometries? This transformation appears to be an isometry. The blue parallelogram is rotated about a point to produce a congruent red parallelogram. Rotations are isometric.

Isometric ARE ISOMETRIC Translations Reflections Rotations IS NOT ISOMETRIC Dilation (because it does not preserve the size)

Rotations

Orientation The orientation of an object refers to the order of its parts as you move around the object in a clockwise or a counter-clockwise direction.

Rotations A rotation is a type transformation that where every point turns around a center.

Rotation Exploration In order to visualize a rotation, try this. Trace the following letter F and its “string” on a piece of tracing paper.

Rotation Exploration Your results will look something like this.

Rotation the Rules Counter wise || clock-clockwise 90/-270 P(x,y) →P’(-y, x) Example P(3,5) to P’(-5,3) 180/-180 P(x,y) →P’(-x,-y) Example P(3,5) to P’(-3,-5) 270/-90 P(x,y) →P’(y, -x) Example P(3,5) to P’(5,-3)

Reflections

The Butterfly Reflect the butterfly.

Solution

A Reflection  A reflection is a transformation in which each point is mapped onto to its image over a line in such a way that the line is the perpendicular bisector of the line segment connecting the point and its image.

A Reflection - a more simple definition A reflection is a type of transformation that acts likes a mirror with an image reflected in the line. If you forget the rules then think of folding your paper.

The Rules of Reflection Over x axis P(x,y) →P’(x,-y) Example P(3,5) to P’(3,-5) Over y axis P(x,y) →P’(-x,y) Example P(3,5) to P’(-3,5) Over y = x P(x,y) →P’(y,x) Example P(3,5) to P’(5, 3) Over y = -x P(x,y) →P’(-y,-x) Example P(3,5) to P’(-5, -3)

Translations  A translation is a transformation that “slides” a figure to a new location. Size and direction are preserved.

Translations In order to translate a figure you need to know two things. How far will it be translated? In what direction will it be translated?

Fact A translation (or slide) preserves size, shape, distance, and orientation.

Application of a translation

Frieze Patterns A frieze pattern is a pattern that repeats itself along a straight line. The pattern may be mapped onto itself with a translation. Wallpaper borders are practical applications of frieze patterns. Frieze patterns can be found around the eaves of some old buildings.

Translations – the rules Example P(3,5) P(x – 3,y + 4) to P’(0,9)

Dilations A dilation is a transformation (notation  ) that produces an image that is the same shape as the original, but is a different size.   A dilation stretches or shrinks the original figure.     The description of a dilation includes the scale factor (or ratio) and the center of the dilation.   The center of dilation is a fixed point in the plane about which all points are expanded or contracted.  It is the only invariant point under a dilation. Copied from http://www.regentsprep.org/regents/math/geometry/gt3/ldilate2.htm

Dilations continued A dilation of scalar factor k whose center of dilation is the origin may be written:   Dk (x, y) = (kx, ky). If the scale factor, k, is greater than 1, the image is an enlargement (a stretch). If the scale factor is between 0 and 1, the image is a reduction (a shrink). (It is possible, but not usual, that the scale factor is 1, thus creating congruent figures.)

Review