Transformations of Geometric Figures Dr. Shildneck Fall, 2015

Writing a Matrix that represents a Geometric Figure The way we will write the matrix representing a figure for the purposes of transformations is as follows. 1. Determine the coordinates for each of the vertices of the figure. 2. Write each point in a column of a matrix

Example 1 Write a matrix that represents the quadrilateral with vertices (-4, -5), (0, -8), (4, 0), and (2, 6).

Translating a Figure To translate (or slide) a figure, we want to change the value of the x’s and/or y’s by the same amount for each vertex. This means we want to add the same number(s) to each coordinate of the vertex. In order to add matrices, remember that the matrices must be the same dimensions. Thus, we must create a translation matrix to add to the figure matrix that has the same dimensions.

Example 2 Translate the quadrilateral with vertices (-4, -5), (0, -8), (4, 0), and (2, 6) four units up and 7 units to the left.

Dilation of a Figure Dilating a figure means that we are enlarging or shrinking the size of a figure by a specific factor. To dilate a geometric figure, we simply need to multiply the figure matrix by a scalar that represents the factor by which we wish to scale the figure.

Example 3 Dilate the quadrilateral with vertices (-4, -5), (0, -8), (4, 0), and (2, 6) by a factor of ½.

Reflecting a Figure There are several ways to reflect a figure. The three primary reflections that we will use are - reflection across the x-axis - reflection across the y-axis - reflection across the line y=x In order to use matrices to determine the coordinates for a reflected figure we must multiply by a matrix that will change the coordinates in an appropriate way.

Reflecting a Figure Reflecting across the x-axis - we want to change the sign of the y-coordinate Reflecting across the y-axis - we want to change the sign of the x-coordinate Reflecting across the line y=x - we want to switch the x and y coordinate

Example 4 Reflect the quadrilateral with vertices (-4, -5), (0, -8), (4, 0), and (2, 6) across the x-axis.

Example 5 Reflect the quadrilateral with vertices (-4, -5), (0, -8), (4, 0), and (2, 6) across the y-axis.

Example 6 Reflect the quadrilateral with vertices (-4, -5), (0, -8), (4, 0), and (2, 6) across the line y = x.

Rotating a Figure

Example 7 Rotate the quadrilateral with vertices (-4, -5), (0, -8), (4, 0), and (2, 6) 90 degrees clockwise.

Example 8 Rotate the quadrilateral with vertices (-4, -5), (0, -8), (4, 0), and (2, 6) 180 degrees.

Example 7 Rotate the quadrilateral with vertices (-4, -5), (0, -8), (4, 0), and (2, 6) 90 degrees counter-clockwise.