Regents Prep Transformation Rules

__Transformations:__

__Transformations:__

- A
__transformation__is a correspondence between one figure, called a__preimage__, and a second figure, its__image__. Each point of the preimage is paired with exactly one point of the image, and each point of the image is paired with exactly one point of the preimage.

__Reflections:__

__Reflections:__

One important transformation of the plane is a reflection in a line. A reflection is simply a “mirror image.”

__In the coordinate plane, there are three special line reflections:__

1) The image of P(x,y) under a reflection in the x-axis is the point P^{1}(x,-y).

P(x,y) → P^{1}(x,-y)

2) The image of P(x,y) under a reflection in the y-axis is the point P^{1}(-x,y).

P(x,y) → P^{1}(-x,y)

3) The image of P(x,y) under a reflection in y = x is the point P^{1}(y,x).

P(x,y) → P^{1}(y,x)

A **reflection in a point** is another type of transformation. Usually the point of reflection is the origin. In that case, the coordinates (x,y) of any point in the preimage are reflected to (-x,-y) in the image.

P(x,y) → P^{1}(-x,-y)

__Examples:__

1) What is the point that represents the image of G(4,-3) after a reflection in the x-axis?

2) What is the point that represents the image of X(-3,-7) after a reflection in the y-axis?

3) What is the point that represents the image of V(3,2) after a reflection in the line y = x?

4) What is the point that represents the image of A(9,-3) after a reflection in the origin?

5) The vertices of triangle ABC are A(-3,3), B(6,6), and C(4,1). Sketch the image and give the coordinates of this triangle after a reflection in the x-axis.

6) The vertices of triangle ABC are A(-3,3), B(6,6), and C(4,1). Sketch the image and give the coordinates of this triangle after a reflection in the y-axis.

7) The vertices of triangle ABC are A(-3,3), B(6,6), and C(4,1). Sketch the image and give the coordinates of this triangle after a reflection in the origin.

8) The vertices of quadrilateral EUDP are E(3,3), U(5,3), D(1,-1), and P(-1,-1). Sketch the image and give the coordinates of this quadrilateral under a reflection in the line y = x.

9) The vertices of triangle ABC are A(-6,7), B(4,8), and C(-2,3). Sketch the image and give the coordinates of this triangle after a reflection in the line y = 3.

10) The vertices of quadrilateral ABCD are A(-4,5), B(-4,8), C(-2,3), and D(-2,6). Sketch the image and give the coordinates of this triangle after a reflection in the line x = 1.

11) The vertices of triangle ABC are A(-3,3), B(6,6), and C(4,1). Sketch the image and give the coordinates of this triangle after a reflection in the line y = -x.

__Translations:__

__Translations:__

A translation is a transformation of a figure in which each point is moved the same distance in the same direction. Since you can imagine sliding triangle ABC onto A^{1}B^{1}C^{1}, we usually call a translation a “slide.”

The image of P(x,y) under a translation a units horizontally and b units vertically is given by: P(x,y) → P^{1}(x + a,y + b)

__Characteristics of a Translation:__

- If a line segment is translated, the image is a line segment congruent to the original line segment.
- If an angle is translated, the image is an angle congruent to the original angle.
- The orientation (order of the vertices) of the image is the same as that of the original object.

__Examples:__

1) The coordinates of the vertices of triangle ABC are (A-1,2), B(4,6), and C(3,-2). Find the coordinates of the image, triangle A^{1}B^{1}C^{1}, under a translation to the left 4 units and down 2 units.

2) What are the coordinates of the point C(-5,4) after a translation of 2 units left and 4 units up?

3) Quadrilateral ABCD with vertices A(1,0), B(4,7), C(6,4), and D(5,0). Graph and sketch the coordinates after a translation of 2 units left and 2 units up.

4) What are the coordinates of the image of A(3,5) under a translation 5 units left and 4 units down?

5) A translation maps (-6,-2) onto (-4,-2). Find the image of (3,5) under the same translation.

6) Find the coordinates of the point (-1,3) after the transformation T_{-3,2}.

__Dilations:__

__Dilations:__

A __dilation__ with center O is a transformation in which a given figure is enlarged or reduced.

__Dilations in the Coordinate Plane:__

The image of P(x,y) under a dilation in the coordinate plane with the origin as the center of dilation and scale factor k (k a nonzero real number is P^{1 }(kx,ky).

P(x,y) → P^{1}(kx,ky)

__Characteristics of a Dilation:__

- Let k represent a positive real number.
- If a segment is dilated with a scale factor k, the image is a line segment whose length is k times that of the original.
- If an angle is dilated with a scale factor k, the image is an angle congruent to the original angle.

__Examples:__

1) Under a dilation with center at O(0,0), the image of A(-4,2) is a^{1}(-2,1). What is the scale factor for the dilation?

2) The vertices of triangle ABC are A(-2,1), B(3,1), and C(-2,5). Graph and give the coordinates after a dilation of 3.

3) The vertices of triangle DEF are D(-5,2), E(-3,7), and F(-5,-5). Graph and state the coordinates of triangle D^{1}E^{1}F^{1}, the image of DEF after D_{2}.

4) What are the coordinates of the point (-3,4) under D_{3}?

5) Quadrilateral ABCD has the following coordinates: A(-2,-2), B(-1,-3), C(4,1), and D(4,-2). Graph and state the coordinates of quadrilateral A^{1}B^{1}C^{1}D^{1}, the image of ABCD after D_{-2}.

__Rotations:__

__Rotations:__

An __isometry__ is a transformation under which image and preimage are congruent (size does not change). Reflections, translations, and rotations are all isometries.

When you translate a figure, you slide it to another position in the plane. Under a __rotation__ in the plane, you swing a figure around in the plane using a fixed distance from a fixed point in the plane called the __center of rotation__. The rotation may be clockwise or counterclockwise.

*** With rotations, it is assumed the direction of the rotation is counterclockwise unless it is otherwise stated.

***To determine equivalent rotations in a positive (counterclockwise) and negative (clockwise) direction, subtract the angle measure you are given from 360 degrees.

__Special Rotations:__

1) Rotation of 90 degrees. P(x,y) → P^{1}(-y,x)

2) Rotation of 180 degrees (1/2 turn). P(x,y) → P^{1}(-x,-y)

3) Rotation of 270 degrees. P(x,y) → P^{1}(y,-x)

__Examples:__

1) Sketch the image of the triangle with vertices A(1,2), B(5,3), C(4,6) under a 90 degrees rotation about the origin.

2) Sketch the image of the triangle with vertices X(2,2), Y(5,3), and Z(4,5) under a half-turn about the origin.

3) What are the coordinates of B(3,4) after a 90 degree rotation about the origin?

4) Triangle ABC has coordinates A(1,2), B(6,2), and C(3,6). Graph and state the coordinates of A^{1}B^{1}C^{1}, the image of triangle ABC after R_{-90}.

5) What positive rotation would be equivalent to R_{-60}?

**See also: **