[G.CO.2] Congruence #2

Objective

Common Core Text:

  • [G.CO.2] Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

Said Differently:

  • Perform transformations using

    • transparencies

    • software

  • Use functions to represent transformations

  • Compare rigid and non-rigid transformations

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Explanation

Transformations using transparencies

We'll discuss this more in [G.CO.5]

Transformations using software

The simplest way to begin performing transformations with software is to download GeoGebra. It's free. Click here for instructions on downloading.

For the rest of the explanation of how to perform transformations using transparency paper and GeoGebra, check out this link to [G.CO.5].

Transformations using functions

Transformations can also be thought of as functions.

Example 1

(x,y) → (x - 3, y + 2)

This function says that for every point,

  • To find the new x-coordinate, subtract 3 from its original x-coordinate

  • To find the new y-coordinate, add 2 to its original y-coordinate

Lets make a table

Plotting our new points, we get the following for our answer

We can see this was a translation. How can we know just by looking at the function that our transformation would be a translation? Four clues:

  1. There's some addition or subtraction (or both)

  2. New x' is calculated from old x, new y' is calculated from old y.

  3. No coefficients.

  4. No negatives in front of x or y.

Example 2

(x,y) → (y, -x)

This function says that for every point, (careful, notice that the x's and y's have switched)

  • To find the new x-coordinate, simply use the original y-coordinate

  • To find the new y-coordinate, take the negative of the original x-coordinate

Lets make a table

Plotting our new points, we get the following for our answer

We can see this was a 90° rotation clockwise about the origin. How can we know just by looking at the function that our transformation would be a 90° rotation about the origin? Four clues:

  • New x' is calculated from old y, new y' is calculated from old x.

  • Either the x or the y is negative

  • No coefficients.

    • No addition or subtraction

Example 3

(x,y) → (-x, y)

This function says that for every point

  • To find the new x-coordinate, take the negative of the original x-coordinate

  • To find the new y-coordinate, simply use the original y-coordinate

Lets make a table

Plotting our new points, we get the following for our answer

We can see this was a reflection. How can we know just by looking at the function that our transformation would be a reflection? Three clues:

  • New x' is calculated from old x, new y' is calculated from old y.

  • Either the x or the y is negative

  • No coefficients.

Compare rigid and non-rigid transformations

Translations, rotations, and reflections are the three transformations known as rigid. What does rigid mean?

    • Rigid: The position of the shape changes, but the shape and size do not change.

Let's look at some transformations that aren't rigid.

Example 4

(x,y) → (2x, y)

This function says that for every point,

  • To find the new x-coordinate, multiply 2 by its original x-coordinate

  • To find the new y-coordinate, simply use its original y-coordinate

Lets make a table

Plotting our new points, we get the following for our answer

We can see this was a a different transformation, called a horizontal stretch. The shape of the transformed shape is not the same. It's all stretched out sideways.

How can we know just by looking at the function that our transformation would not be a rigid motion? 1 clue:

  1. Either the x or the y (or both) has a coefficient.