Objective:Students will be able to represent translations, dilations, reflections and rotations with matrices.

*Geometric Transformations with Matrices

*Translations*A transformation is a change made to a

figure. *The original figure is called the preimage (A), while the transformed figure is called the image (A’).  *When we slide a figure without changing the

size or shape of the figure, it is said to be a translation.  *By using matrix addition, we can translate

the vertices of a figure.

Given triangle ABC where A (–4, 1), B (– 2, 5) and C (0, 2),  translate the preimage 5 units right and 3 units down.  Then, sketch the image.

 

Quadrilateral ABCD has vertices A(0,0), B(-2,5), C(2,3) and D(4.1). Use a matrix to find the coordinates of the vertices of the image translated 5 units left and 2 units up. Then graph ABCD and A’B’C’D’.

*Dilation*A dilation is a transformation that changes

the size of a figure.EXAMPLE:  Given triangle ABC where A (–5,0), B (8,-1) and C (4,5)  Find the coordinates of each image under the following dilations. Then graph the images.:a.) 4 b.) 1/5 c.) -1.5

* Reflections and Rotations with Matrices

*A reflection, or flip, is a transformation that creates symmetry on the coordinate plane.  *You can use matrix multiplication to graph reflections

in the coordinate plane.

*A rotation is a transformation that turns a figure about a fixed point called a center of rotation. *You can rotate a figure as much as 360 degrees. In

this text, all rotations are counterclockwise about the origin.

Matrices for Reflections in the Coordinate Plane

Reflection in the y-axis

Reflection in the x-axis

Reflection in the line y = x 

Reflection in the line y = –x

*RotationsMatrices for Rotations in the Coordinate Plane

Rotation of 90° Rotation of

180°Rotation of

270°Rotation of

360°

*EXAMPLE: Given triangle ABC where A (-3,0), B (– 4,4) and C (1,1). Reflect the triangle across the y-axis, x-axis, y=x and y = -x. Then, sketch the image.

*For example…*EXAMPLE: Given quadrilateral ABCD where A (1, 1), B (3,1), C (6,4),and D(1,3). Rotate the quadrilateral:

a.) 90 °b.) 180°c.) 270°d.) 360°

Then, sketch the image.