Back To Course

Math 101: College Algebra12 chapters | 92 lessons | 11 flashcard sets

Watch short & fun videos
**Start Your Free Trial Today**

Start Your Free Trial To Continue Watching

As a member, you'll also get unlimited access to over

Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Erin Monagan*

Erin has been writing and editing for several years and has a degree in fiction writing.

Although a basic absolute value graph isn't complicated, transformations can make them sufficiently confusing! In this lesson, you'll practice different transformations of absolute value graphs.

We'll start this lesson by recalling that absolute value graphs look like the absolute coolest guitar there is, the Flying V. Also, we can slide these graphs all around by doing the transformation that is called a **translation**.

For example, by taking our parent graph *y=|x|* and changing it to *y=|x-2|+3*, the V gets translated two to the right and up three so that we end up with our vertex at the coordinates (2, 3). Also, remember that the left and right shift does the opposite of what you would expect. So, a -2 on the inside the absolute value actually shifts it to the right because I need to put in a +2 for *x* to turn the absolute value part into 0, which is where my vertex is going to be.

This is not the only kind of transformation that we often see with absolute value graphs. The second most common one is called a **dilation** (or a stretch or a shrink). In absolute value graphs, a dilation makes the V either wider or thinner. We accomplish this by putting a value in front of the absolute value (for example, *y=2|x|* or *y=1/3|x|*). Just like the *m* in *y=mx+b*, this value tells us the new slope of the lines in our V. Meaning that *y=2|x|* will start at the origin (because there is nothing being added or subtracted on the inside or the outside), but then go up two and over one each step of the way. This means it's going to be a steeper V than the normal one. If we did *y=(1/3)|x|*, we'd only go up one and over three each step of the way (and in both directions). That means this V is going to be wider than all the other ones we've looked at.

So, then the question becomes - what happens when we put a negative number in front of the absolute value (say, *y=-4|x|*)? Well, since the -4 is directly in front of the absolute value, which means multiplication. And, because the absolute value will always be positive no matter what we substitute in for *x*, (we don't multiply the positive by a negative, because it gives us a negative number) all our *y* values are going to be negative. So, now instead of our entire graph being above the x-axis, it will be completely below it. We call this kind of transformation a reflection because the graph of y=|x| gets reflected (like a mirror) over the x-axis to end up with *y=-|x|*.

Quickly coming backing to the example mentioned a second ago, graphing *y=-4|x|* is as simple as starting at the origin (because there's no numbers being added or subtracted on the inside or the outside), going down four and then over one in each direction to find how steep or thin the graph is. Again, whether it's positive or negative, the number in front just acts as the slope - so down four over one when it's negative. We could also do *y=-(5/2)|x|*. We start at the origin again and go down five and over two in both directions to figure out how thin or wide our graph is.

When we combine translations, dilations and reflections into one, we end up with the standard form absolute value equation, which is *y=a|x-h|+k*. The *a* tells us the slope, *h* tells us how far to shift the graph left and right, and *k* tells us how far to shift the graph up and down.

Using this equation, we can graph things like *y=-2|x-1|+3*. We start by finding our vertex and going over one to the right and going up three. Then, we use the *a* value of -2 (the slope), which tells us to go down two and over one in both directions, and we end up with this graph. We could also do *y=(3/2)|x|-5*. Again, we start by finding *h* and *k*. There's nothing being added or subtracted from the *x* - kind of like there's a plus 0 - so we don't shift it left and right at all. We go down five to find the vertex at the coordinates (0, -5), then I use the *a* value as the slope. Since it's positive, we go up three and then over two in both directions to find the graph.

To review, translations slide the graph around on the coordinate plane without changing its size. Dilations make the graph wider or thinner. Reflections flip the graph like a mirror. In the standard form absolute value equation, *y=a|x-h|+k*, *h* and *k* give us the coordinates of the vertex, while *a* tells us the slope of the V. Lastly, a negative *a* value makes the V of the absolute value point down instead of how it normally points up.

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

Create your account

Already a member? Log In

BackDid you know… We have over 49 college courses that prepare you to earn credit by exam that is accepted by over 2,000 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.

Back To Course

Math 101: College Algebra12 chapters | 92 lessons | 11 flashcard sets

- What is a Matrix? 5:39
- How to Take a Determinant of a Matrix 7:02
- What is an Absolute Value? 4:42
- How to Evaluate Absolute Value Expressions 5:28
- How to Solve an Absolute Value Equation 5:26
- Solving Absolute Value Practice Problems 7:09
- How to Graph an Absolute Value and Do Transformations 8:14
- Graphing Absolute Value Equations: Dilations & Reflections 6:08
- Go to Matrices and Absolute Value

- Go to Inequalities

- Go to Functions

- BITSAT Exam - Biology: Study Guide & Test Prep
- NDA Exam Preparation & Study Guide
- VCE Physics: Exam Prep & Study Guide
- Basic Nursing Training
- Common Entrance Test (CET): Study Guide & Syllabus
- AQA A-Level Anthropology: Personhood & Identity
- AQA A-Level Anthropology: Human Movement & Migration
- AQA A-Level Anthropology: Defining Boundaries
- AQA A-Level Anthropology: Practicing Anthropology
- Derivative Rules
- What Does Redshirt Mean in Kindergarten?
- Becoming Naomi Leon Lesson Plan
- HRCI Online Recertification & Continuing Education Credit
- Cool Math Puzzles
- Study.com Demo for Workforce College Accelerator
- eBooks vs. Textbooks
- Response to Intervention (RTI) in Illinois

- Mama in A Raisin in the Sun: Character & Quotes
- Conflicts in Brave New World
- How Science & Technology Can Be Used to Solve Society's Problems
- Self-Expression: Definition & Examples
- Reading Music Notes: Symbols & Names
- Using Graphic Organizers for Reading Comprehension
- Almorzar Conjugation: Present Tense & Present Progressive
- Comparing Literary & Informative Nonfiction: Analysis & Methods
- Quiz & Worksheet - Conflicts in Julius Caesar
- Quiz & Worksheet - Questions on The Screwtape Letters Chapter 1
- Quiz & Worksheet - Cell Membrane Location & Characteristics
- Quiz & Worksheet - Jane Bennet in Pride and Prejudice
- Quiz & Worksheet - Lydia Bennet in Pride and Prejudice
- Roman Art History Flashcards
- Cathedral Architecture Terms Flashcards

- History of Major World Religions Study Guide
- Ohio Assessments for Educators - Middle Grades Science: Practice & Study Guide
- Business Math Textbook
- Of Mice and Men Study Guide
- High School Physical Science: Homework Help Resource
- Praxis Earth & Space Sciences: Natural Phenomena & the Earth's Climate
- Prenatal and Human Development
- Quiz & Worksheet - Employer Rights in Unionization & Collective Bargaining
- Quiz & Worksheet - Restoring Balance in Europe After Napoleon
- Quiz & Worksheet - Slant Asymptotes
- Quiz & Worksheet - Building a New Republic in Italy After WWII
- Quiz & Worksheet - Vitamin A Deficiency & Toxicity Symptoms

- How Meiosis & Fertilization Promote Genetic Variation
- Affirmative Action Supreme Court Cases
- Creative Writing Warm-Ups & Exercises
- Election of 1860 Lesson Plan
- What Are General Education Courses?
- Reconstruction Lesson Plan
- Types of Government Lesson Plan
- Alternative Teacher Certification in Maryland
- Romeo and Juliet Prologue Lesson Plan
- 100th Day of School Project Ideas
- ESL Writing Prompts
- Federalism Lesson Plan

Browse by subject