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The Process of Naming Transformations on a Graph

Instructor: Melanie Olczak

Melanie has taught high school Mathematics courses for the past ten years and has a master's degree in Mathematics Education.

This lesson will show that by determining coordinates on a graph we can discover what transformation has taken place. There are four transformations that will be discussed, translation, dilation, rotation and reflection.

Transformations in the Coordinate Plane

Have you ever wondered if you could take a two dimensional object and move it around the coordinate plane? Well, you can! In fact, learning just how to do so can be instrumental in software engineering, biological modeling, as well as of course, mathematics.

We can change the location, shape and size of a figure by completing a transformation. There are four transformations that can be done in a coordinate plane to two dimensional figures.

Translations

A translation is a slide that moves a two dimensional figure up or down and left or right. Determining if a figure has been translated just requires looking at the two figures.

The original figure below is rectangle ABCD. The image, which is the translated figure, is rectangle A'B'C'D'. We use the apostrophe to denote that those are the corresponding coordinates of the new figure.


Translation ex 1


As you can see, the rectangle did not change size, but only the location. Point A was at (1, 4) and is now at (-2, 0). This means that the original figure was shifted to the left 2, and down 4. If we looked at each point, B(4, 4) and B'(2,0), C(4,2) and C'(2, -2); and D(1, 2) and D'(-1, -2) the same transformation has occurred.

Rotations

A rotation turns a two dimensional figure in the coordinate plane a certain number of degrees around a certain point. The most common rotations are turns about the origin and 90 degrees, 180 degrees and 270 degrees. Each of these turns can be clockwise or counterclockwise.

There are certain rules for each rotation.

  • Turning a figure 90 degrees counterclockwise and 270 degrees clockwise about the origin are the same rotation, and the coordinates go from (x, y) to (-y, x).
  • Turning a figure 270 degrees counterclockwise and 90 degrees clockwise about the origin are the same rotation, and the coordinates go from (x, y) to (y, -x).
  • Turning a figure 180 degrees both clockwise and counterclockwise will result in the same figure, where the coordinates go from (x, y) to (-x, -y).


Rotation Ex 2


The same triangle has been rotated around the origin (0,0) 90 degrees counterclockwise, 180 degrees counterclockwise, 270 degrees counterclockwise. The coordinates follow the rules listed above. The original point B(1, 3) changes to B'(-3, 1) after a 90 degree counterclockwise rotation. Then B(1, 3) changes to B'(-1, - 3) after a 180 degree rotation. Then B(1, 3) changes to B''(3, -1) after a 270 degree counterclockwise rotation.

Reflections

A reflection flips a two dimensional figure in the coordinate plane over a line or a point. Line reflections are the most common reflections. A two dimensional figure can be reflected over the x-axis, the y-axis or a specific line in the coordinate plane.

The orientation of the two dimensional figure changes when a figure is reflected. The orientation of a figure is the order in which the points are labeled.

In the example below, the triangle ABC is reflected over the x-axis. In the original triangle (purple) the vertices are labeled ABC if we read them clockwise. In the reflection (blue) triangle, the vertices are labeled A'C'B' if we read them in the same clockwise direction.


Reflection


A reflection can be identified in the coordinate plane by looking at the orientation of the vertices. The letters will appear in a different order if the figure has been reflected.

Dilations

A dilation is a change in the size and shape of a figure. A dilation can be:

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