## Graphing Complex Transformations

#### Aligned To Common Core Standard:

**High School Congruence** - HSG-CO.A.3

What are Complex Transformations in Geometry?
Many students struggle with the complex transformations that take place in geometry. For functions that include real variable such as f(x) = sin x, g(x) = x^{2} + 2 etc. you are used to illustrating these transformations geometrically and usually on a cartesian coordinate system. If the functions that are involved have complex variables such as w = sin z or w = z^{2} +2, it is not possible to make a cartesian graph because z cannot be represented on an x, y plane system.
It would be difficult to visualize this on a cartesian system, and therefore, we need to make two copies of the complex plane and then look at the points that are made on the z-plane and see how they are being transformed into the points on w-plane.
Example: w = f(z) = z + 2. This simply shifts every single point two units in the direction of the real axis - which is called a translation. These worksheets and lessons help students learn and experiment with the movement of common transformations.

### Printable Worksheets And Lessons

- Translation and
Rotation Step-by-step Lesson - We throw a triangle in a different
direction and have you document it.

- Guided Lesson
- We use notation to move the figures around in these translations.

- Guided Lesson Explanation
- I like to examine each point closely to see where it is going.

- Practice Worksheet
- This one is spread over four pages. The graphs are a bit oversized,
but you can see them clearly.

- Matching Worksheet
- Find the graph that fits the translation that occurs.

- Working with
Translations 5 Pack - Find the shape that was transformed and
then determine if a vertical or horizontal transformation occurred.

- Rotations Worksheet Five Pack
- This pulls geometric shapes into the real world. You need to assume
that all the shapes are geometrically perfected.

#### Homework Sheets

Understanding the vocabulary for this section is paramount.

- Homework 1 - Write the coordinates of the vertices after a translation and rotation. Graph the image of ABC after the following transformations: Translation (x, y) (x + 5, y + 4) Rotation 270° counterclockwise around the origin.
- Homework 2 - Graph the image of PQRS after the following transformations: Translation (x, y) (x - 3, y + 3)
- Homework 3 - Use the transformation rule (x, y) (x-3, y+5) to find the image of each of its three vertices.

#### Practice Worksheets

I would go over the transformation rules and scales with students first before working on these.

- Practice 1 - Graph the image of PQRS after the following transformations: Translation (x, y) ----> (x – 1, y + 2).
- Practice 2 - Graph the image of XYZ after the translation.
- Practice 3 - Graph the image of STV after the following transformations: Rotation 270° counterclockwise around the origin.

#### Math Skill Quizzes

Quiz 1 and 3 are very traditional. The second quiz is me trying to adapt the skill to the real world.