Transformations

In the following coordinate planes, the preimage is shown in black. Use the transparency to help with the transformations.

For each of the transformations (translation, reflection, rotation, dilation) below, please do the following:

  1. Label preimage vertices A, B, C.
  2. Draw the image based on the directions.
    Tip: On the transparency, also mark the axes to help line up the shapes.
  3. Label corresponding image vertices A', B', C'.
  4. Provide a general mapping (x, y) → (?, ?) to depict the geometric transformation.
    Tip: It may help to write a mapping of each pair of corresponding points. For example, (1, 1) → (-4, 2) in the first "translations" example.

Translations

Translations move points along the same arrow (formally called a "vector"). Orientation of the preimage does not change. Also notice that "the mathematics of translation is addition" (Pixar).

Translate left 5 and up 1

Canvas is not supported

Translate right 1 and down 5

Canvas is not supported

Reflections (or flips)

A reflection creates a "mirror" image.

Reflect over x-axis

Canvas is not supported

Reflect over y-axis

Canvas is not supported

Rotations

Although rotations can be done around any point, we will focus on rotating around the origin, (0, 0).

Rotate 90° counter-clockwise

Canvas is not supported

Rotate 180°

Canvas is not supported

Rotate 90° clockwise

Canvas is not supported

Dilation

Dilation (or scaling) is the mathematics of multiplication (Pixar). Again, we can use many points for the center of dilation, but we will focus on the origin, (0, 0).

Dilate by 1/2

Canvas is not supported

Dilate by 2

Canvas is not supported

This not a rigid transformation. Why not?


Sequence of transformations

Transformations can build on each other by first applying a transformation to preimage ABC to create image A'B'C', then applying a transformation to image A'B'C' to create image A''B''C'', and so on.

  1. Label the preimage ABC.
  2. Apply the transformations in order.
  3. Label corresponding vertices appropriately.

Reflect on y-axis
Translate down by 4

Canvas is not supported

Rotate 90° clockwise
Reflect on x-axis

Canvas is not supported

Rotate 180°
Translate up by 2

Canvas is not supported

Translate up by 2
Rotate 180°

Canvas is not supported

Reflect y-axis
Translate 1 right
Reflect x-axis

Canvas is not supported

Translate left 1 and down 1
Rotate 90° counter-clockwise
Dilate by 1.5

Canvas is not supported

Congruent transformations

Preimage
Image

Follow the transformations below to unscramble the picture.

  1. Reflect segment f across x-axis
  2. Translate segment e down 7 and right 12
  3. Translate segment f down 12 and right 7
  4. Rotate trapezoid d counter-clockwise 90° about the origin (0, 0)
  5. Rotate ellipse a 180° about the origin (0, 0)
  6. Reflect segment c across y-axis
  7. Reflect triangle g across x-axis
  8. Rotate circle b 180° about the the point (0, 6)
  9. Reflect triangle g across x-axis, then translate left 7
  10. Translate segment c left 0.5 and up 1, then reflect across y-axis