# Graphing Complex Transformations Worksheets

Objects have been known to move from time to time. We have previously learned how to describe the movements of figures in a geometric plane. This can get a bit complicated, at times, especially when two or more transformations are taking place throughout the process. As you read this in symbol form the transformation is usually signified by the use of a left to right arrow (→). As you begin to get more familiar with this material, the arrow will become common place. The worksheets and lessons found below will cover a wide range of material related to this topic. You will at times need to name points, describe transformations with vocabulary words, and draw complete figures.

### Aligned Standard: High School Geometry - HSG-CO.A.3

- 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.

- Answer Keys - These are for all the unlocked materials above.

### 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. Make sure to work off of the starting points.
- 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.

- Quiz 1 - Process a translation and then rotation according to the directions that are given.
- Quiz 2 - What is the measure of the angle tip of the crayons with the line?
- Quiz 3 - Is the rotation counterclockwise? Map out what occurs to determine this.

### What are Complex Transformations in Geometry?

Many students struggle with the complex transformations that take place in geometry. A complex position usually involves, not just one, but several transformations. This often takes time to basically reverse engineer as you compare the starting and ending points of the geometric figure.

For functions that include real variables 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.