**a change in the position, size, or shape of a geometric figure. The given image is called the preimage and the resulting figure is called the image. A transformation maps a figure onto its image.**

__Transformations:__Transformation Vocabulary:

Congruent Dilation Image

Isometry Preimage Reflection

Rigid Motion Rotation Similarity

Translation

The four primary transformations we will be looking at are;

Translations

Reflections

Dilations and

Rotations

What these look like on the coordinate plane, we will see in class, but there are rules to help map these.

__TRANSLATIONS__With translations, we move the preimage horizontally and/or vertically as indicated by the arrow rule.

To move horizontally, (left or right) we add or subtract from the x coordinate.

To move vertically, (up or down) we add or subtract from the y coordinate.

For example, if we wanted to move a point 4 units right (horizontally) and 3 units down (vertically) the arrow rule would look like this (x, y) --> (x+4, y-3).

Moving a point 6 units to the left and 2 units up would look like (x, y) --> (x-6, y+2)

Translations result in an image that is

*congruent*to the preimage.

__DILATIONS__Dilations enlarge or reduce a figure by multiplying the x and y coordinates by a given factor. The shape of the figure doesn't change but the size does.

To enlarge the preimage, the factor would be greater than 1. To reduce the preimage, the factor would be between 0 and 1.

For example, to enlarge a point by a factor of 3, the arrow rule would be (x, y) --> (3x, 3y)

To reduce a point by a factor of 1/2, the arrow rule would be (x, y) --> (1/2x, 1/2y).

Since dilations change the size but not the shape, the image is

*similar*to the preimage.

__REFLECTIONS__Reflections "flip" a figure over a given line. While we can flip over any line, we typically focus on four options.

1) Reflect over the x axis

2) Reflect over the y axis

3) Reflect over the line y = x

4) Reflect over the line y = -x

Each of these reflections has an arrow rule to help find the coordinates of the resulting image.

Reflection over the x axis, (x, y) --> (x, -y). The x value doesn't change but the sign of the y value does.

Reflection over the y axis, (x, y) --> (-x, y). The sign of the x value changes but y stays the same.

Reflection over the line y = x, (x, y) --> (y, x). The x and y coordinates change place, the y becomes the x and the x becomes the y.

Reflection over the line y = -x, (x, y) --> (-y, -x). The x and y changes places (like above) and both signs change.

Since reflected figures do not change size, the resulting image is congruent to the preimage.

We can reflect over others lines as well but in the scope of this course, we do not examine those particular reflections.

__ROTATIONS__Rotations are transformations that spin around a given point called the center of rotation. While rotations can take place around any given point, we will limit ourselves to rotations around the origin.

Indicated rotations are specified by indicating the "two D's." These are direction and degrees. By direction, we mean counter clockwise or clockwise rotation by degrees, we mean 90, 180 or 270 degree rotations.

An important thing to remember is that if no direction of rotation is indicated, it is understood to be

*counterclockwise.*

The arrow rules for rotations are as follows;

90 degrees counter clockwise (x, y) --> (-y, x)

180 degrees counter clockwise (x, y) --> (-x, -y)

270 degrees counter clockwise (x, y) --> (y, -x)

90 degrees clockwise (x, y) --> (y, -x)

180 degrees clockwise (x, y) --> (-x, -y)

270 degrees clockwise (x, y) --> (-y, x)

Since rotations do not change the size or shape of the preimage, the resulting figure is congruent to the preimage.