Back To Course

Supplemental Math: Study Aid1 chapters | 19 lessons

Are you a student or a teacher?

Try Study.com, risk-free

As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Try it risk-freeWhat teachers are saying about Study.com

Already registered? Login here for access

Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Jennifer Beddoe*

In geometry, transformation refers to the movement of objects in the coordinate plane. This lesson will define and give examples of each of the four common transformations and end with a quiz to make sure you are moving in the right direction.

Geometric transformations involve taking a **preimage** and transforming it in some way to produce an **image**. There are two different categories of transformations:

- The
**rigid transformation**, which does not change the shape or size of the preimage. - The
**non-rigid transformation**, which will change the size but not the shape of the preimage.

Within the rigid and non-rigid categories, there are four main types of transformations that we'll learn today. Three of them fall in the rigid transformation category, and one is a non-rigid transformation.

**Rotation**: rotating an object about a fixed point without changing its size or shape**Translation**: moving an object in space without changing its size, shape or orientation**Dilation**: expanding or contracting an object without changing its shape or orientation**Reflection**: flipping an object across a line without changing its size or shape

Why is dilation the only non-rigid transformation? Remember that in a non-rigid transformation, the shape will change its size, but it won't change its shape.

1. Which figure represents the translation of the yellow figure?

The answer is Q. It is the only figure that is a translation. Figure P is a reflection, so it is not facing the same direction. Figure R is larger than the original figure; therefore, it is not a translation, but a dilation.

2. Which type of transformation is represented by this figure?

The preimage has been rotated around the origin, so the transformation shown is a rotation.

Most transformations are performed on the coordinate plane, which makes things easier to count and draw. The best way to perform a transformation on an object is to perform the required operations on the vertices of the preimage and then connect the dots to obtain the figure.

A **translation** is performed by moving the preimage the requested number of spaces.

Move the above figure to the right five spaces and down three spaces. If you take each vertex of the rectangle and move the requested number of spaces, then draw the new rectangle. This will be your translated image:

The mathematical way to write a translation is the following: (*x*, *y*) â†’ (*x* + 5, *y* - 3), because you have moved five positive spaces in the *x* direction and three negative spaces in the *y* direction.

**Rotation** of an object involves moving that object about a fixed point. To rotate a preimage, you can use the following rules. To rotate an object 90° the rule is (*x*, *y*) â†’ (-*y*, *x*). You can use this rule to rotate a preimage by taking the points of each vertex, translating them according to the rule and drawing the image. For example, if the points that mark the ends of the preimage are (1, 1) and (3, 3), when you rotate the image using the 90° rule, the end points of the image will be (-1, 1) and (-3, 3).

The rules for the other common degree rotations are:

- For 180°, the rule is (
*x*,*y*) â†’ (-*x*, -*y*) - For 270°, the rule is (
*x*,*y*) â†’ (*y*, -*x*)

Returning to our example, if the preimage were rotated 180°, the end points would be (-1, -1) and (-3, -3). If it were rotated 270°, the end points would be (1, -1) and (3, -3).

Here is what all those rotations would look like on a graph:

**Reflection** of a geometric figure is creating the mirror image of that figure across the line of reflection. To draw a reflection, just draw each point of the preimage on the opposite side of the line of reflection, making sure to draw them the same distance away from the line as the preimage. Here's an example:

In this example, the preimage is a rectangle, and the line of reflection is the *y*-axis. To draw the image, simply plot the rectangle's points on the opposite side of the line of reflection.

Point (-5, 4) reflects to (5, 4)

Point (-5, 2) reflects to (5, 2)

Point (-2, 4) reflects to (2, 4)

Point (-2, 2) reflects to (2, 2)

Then, connect the vertices to get your image.

Images can also be reflected across the *y*-axis and across other lines in the coordinate plane. Every reflection follows the same method for drawing.

The **dilation** of a geometric figure will either expand or contract the figure based on a predetermined **scale factor**. To perform a dilation, just multiply each side of the preimage by the scale factor to get the side lengths of the image, then graph.

In this example, the scale factor is 1.5 (since 2 * 1.5 = 3), so each side of the triangle is increased by 1.5. The angle measures stay the same.

Mathematical **transformations** involve changing an image in some prescribed manner. There are four main types of transformations: **translation**, **rotation**, **reflection** and **dilation**. These transformations fall into two categories: rigid transformations that do not change the shape or size of the preimage and non-rigid transformations that change the size but not the shape of the preimage.

After you've completed this lesson, you should have the ability to:

- Define mathematical transformations and identify the two categories
- Describe the four types of transformations
- Explain how to create each of the four types of transformations

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackWhat teachers are saying about Study.com

Already registered? Login here for access

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
14 in chapter 1 of the course:

Back To Course

Supplemental Math: Study Aid1 chapters | 19 lessons

- Less Than Symbol in Math: Problems & Applications 4:10
- What are 2D Shapes? - Definition & Examples 4:35
- Trapezoid: Definition, Properties & Formulas 3:58
- What is Surface Area? - Definition & Formulas 5:56
- Using Parentheses in Math: Rules & Examples 3:58
- Universal Set in Math: Definition, Example & Symbol 6:03
- Complement of a Set in Math: Definition & Examples 5:59
- Zero Exponent: Rule, Definition & Examples 4:32
- Quotient Of Powers: Property & Examples 4:58
- What is Simplest Form? - Definition & How to Write Fractions in Simplest Form 5:49
- What is Slope? - Definition & Formulas 7:10
- Skewed Distribution: Examples & Definition 5:09
- Change Of Base Formula: Logarithms & Proof 4:54
- Transformations in Math: Definition & Graph 6:27
- Fixed Interval: Examples & Definition 4:00
- Scatterplot and Correlation: Definition, Example & Analysis 7:48
- Dilation in Math: Definition & Meaning 5:30
- Simplifying Fractions: Examples & Explanation 4:44
- Go to Overview of Math Concepts

- Computer Science 109: Introduction to Programming
- Introduction to HTML & CSS
- Introduction to JavaScript
- Computer Science 332: Cybersecurity Policies and Management
- Introduction to SQL
- Algorithmic Analysis, Sorting & Searching
- Computer Programming Basics
- Stacks & Queues for Data Structures
- Functions & Modules in Programming
- Built-In Data Types for Programming
- CEOE Test Cost
- PHR Exam Registration Information
- Claiming a Tax Deduction for Your Study.com Teacher Edition
- What is the PHR Exam?
- Anti-Bullying Survey Finds Teachers Lack the Support They Need
- What is the ASCP Exam?
- ASCPI vs ASCP

- The Lion of the Tribe of Judah: Symbol & Meaning
- Partial Fractions: Rules, Formula & Examples
- Risk Identification: Process & Analysis
- Design Constraints in Engineering: Definition & Example
- Filter Function in R Programming
- What is a VLAN? - Definition, Function & Implementation
- Room 101 Lesson Plan
- Quiz & Worksheet - Difference Between Temple & Synagogue
- Quiz & Worksheet - Narrative of the Life of Frederick Douglass
- Quiz & Worksheet - A Midsummer Night's Dream Play Within a Play
- Quiz & Worksheet - Dorsiflexion vs. Plantar Flexion
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies
- ESL Teaching Strategies for ESL Students
- Punctuation Worksheets

- DSST Technical Writing: Study Guide & Test Prep
- Physical Geology for Teachers: Professional Development
- 6th Grade Language Arts: Lessons & Help
- MTTC Psychology (011): Practice & Study Guide
- McDougal Littell The Americans: Online Textbook Help
- MEGA Middle School Math: Monomials & Polynomials
- Basic Communication Skills
- Quiz & Worksheet - Respiration in Roundworms
- Quiz & Worksheet - The Character of Macbeth
- Quiz & Worksheet - Breaching Experiments
- Quiz & Worksheet - Goneril, Regan & Cordelia in King Lear
- Quiz & Worksheet - Emperor Hadrian of Rome

- Reproduction of Planaria Worms
- What is Enterprise Architecture? - Definition & Examples
- Plant Experiments for Kids
- What Will I Learn in an SAT Class?
- DNA Structure Lesson Plan
- Adult Literacy Resources
- When to Take the GMAT
- Equivalent Fractions Lesson Plan
- Average ELM Test Scores
- AP Macroeconomics Exam: Tips for Long Free-Response Questions
- 4th Grade Georgia Science Standards
- Best Psychology Books for Beginners

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject