Back To Course

High School Precalculus: Help and Review32 chapters | 297 lessons

Start Your Free Trial To Continue Watching

As a member, you'll also get unlimited access to over 70,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Free 5-day trial
Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Let's see how to graph 1 - cos(x). We'll do this using transformations of the function cos(x). After graphing 1 - cos(x), let's dig a little deeper into transformations of functions, and see how useful they are in graphing seemingly complex functions.

To graph 1 - cos(*x*), we need to be familiar with two things. The first is that we need to know what the graph of *y* = cos(*x*) looks like.

We see that the graph of *y* = cos(*x*) looks like a series of hills and valleys, with maximum values of 1 and minimum values of -1. There are some key points labeled as well.

Okay, that's one thing down. The next thing we need to be familiar with is transformations of functions. In mathematics, **transformations of functions** are algebraic manipulations of a function that correspond to transformations of the graph of the function. Notice that to get from cos(*x*) to 1 - cos(*x*), we multiply the function by a negative to get -cos(*x*), then we add 1 to get 1 - cos(*x*). Each of these manipulations correspond to a transformation of the graph of cos(*x*).

There are four different types of transformations of functions, but we will wait to talk a bit more about this until after we graph 1 - cos(*x*). For now, we are only interested in two of the transformations: reflections and vertical shifts.

A **reflection** is a transformation that reflects a function over the *x* or *y* axis. This type of transformation is represented through multiplication by a negative. When we multiply the whole function by a negative, we reflect the graph of that function over the *x*-axis. When we only multiply the *x* variable by a negative, we reflect the function over the *y*-axis. Recall, that to get from cos(*x*) to 1 - cos(*x*), we said that we first multiply cos(*x*) by a negative. We are multiplying the whole function by a negative, so this corresponds to reflecting the graph of cos(*x*) over the *x*-axis.

We now have the graph of -cos(*x*). The next thing we do to 1 - cos(*x*) is add 1. This corresponds to a vertical shift. A **vertical shift** is a transformation that shifts the graph of a function up or down. Algebraically, if we add *c* to a function, we shift the graph of that function up *c* units. Similarly, if we subtract *c* from a function, we shift the graph of that function down *c* units. Since we are adding 1 to -cos(*x*), we will shift the graph up 1 unit.

Alright! All together, we have that to graph 1 - cos(*x*), we start with the graph of cos(*x*), reflect it over the *y*-axis, and shift that graph up 1 unit.

So now that it's solved, take a look at the graph of 1 - cos(*x*), which is shown on your screen now.

As we just saw, transformations of functions come in extremely handy when trying to graph variations of a well-known function. We just saw reflections and vertical shifts in action. Let's take a look at the two other types of transformations of functions. Those are horizontal shifts and stretching/shrinking.

A **horizontal shift** is a transformation that shifts the graph of a function to the right or left. These types of transformations correspond to adding or subtracting a number to or from *x* in a function. If we add *c* to *x* in a function, then we shift the graph of the function left *c* units, and if we subtract *c* from *x* in a function, then we shift the graph of the function right *c* units. That may sound backwards to you, but that is how horizontal shifts work. Just remember, when dealing with horizontal shifts, think 'opposites' (right = subtraction, left = addition).

That's a lot of words! Putting things into practice is always better, so to illustrate this, consider the function *y* = cos(*x* + 5). Since we are adding 5 to *x* in the function cos(*x*), we shift the graph of cos(*x*) five units to the left to obtain the graph of cos(*x* + 5).

**Stretching** and **shrinking** are transformations that either stretch or shrink (compress) a function. Algebraically, this transformation corresponds to multiplying a function or the *x* variable of a function by a number. If we multiply a whole function by *c* or 1/*c*, we stretch or shrink the function vertically, and if we multiply just the *x*-variable in a function by *c* or 1/*c*, we stretch or shrink the function horizontally.

When it comes to vertical stretching and shrinking, if we multiply by *c*, then we are stretching the function vertically by a factor of *c*. If we multiply by 1/*c*, then we are shrinking the function vertically by a factor of *c*.

On the other hand, when it come to horizontal stretching and shrinking, if we multiply *x* by *c*, then we are shrinking the function horizontally by a factor of *c*. If we multiply *x* by 1/*c*, then we are stretching the function horizontally by a factor of *c*.

Oh boy, that's a lot of words again! Let's look at an example again to better understand this concept. Consider the function 2cos(*x*). Since we are multiplying the whole function of cos(*x*) by 2, we are stretching the function cos(*x*) by a factor of 2.

As we saw with our initial problem of graphing 1 - cos(*x*), we can have more than one transformation happening at a time. Consider the function *y* = (1/3)cos(-*x* - 2) - 4. At first glance, this looks very complicated to graph, but really it's just a matter of taking the graph of *y* = cos(*x*), shifting it 2 units to the right, shrinking vertically by a factor of 3, reflecting over the *y* axis, and shifting down 4 units.

Pretty neat, huh?

Let's review what we've learned. In this lesson, we looked at the function 1- cos(x), which is an example of **transformations of functions**, or algebraic manipulations of a function that correspond to transformations of the graph of the function. We looked at four types of transformations, which included **reflections**, which are transformations that reflect a function over the *x* or *y* axis; **vertical shifts**, which are transformations that shift the graph of a function up or down; **horizontal shifts**, which are transformations that shift the graph of a function to the right or left; and **stretching** and **shrinking**, which are transformations that either stretch out or compress a function.

It is easy to see that transformations of functions are extremely useful when trying to graph complex functions. These transformations can take a seemingly difficult problem and transform it into a problem that is quite easy!

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 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
7 in chapter 21 of the course:

Back To Course

High School Precalculus: Help and Review32 chapters | 297 lessons

- Graphing Sine and Cosine Transformations 8:39
- Graphing the Tangent Function: Amplitude, Period, Phase Shift & Vertical Shift 9:42
- Graphing the Cosecant, Secant & Cotangent Functions 7:10
- Using Graphs to Determine Trigonometric Identity 5:02
- Solving a Trigonometric Equation Graphically 5:45
- How to Graph cos(x)
- How to Graph 1-cos(x) 6:15
- How to Find the Phase Shift of a Trig Function
- How to Find the Vertical Shift of a Trig Function
- How to Find the Frequency of a Trig Function 4:59
- Go to Trigonometric Graphs: Help and Review

- College 101: College Prep & Hospitality Business Lite
- College 101: College Prep & Retail Business Lite
- Recruiting & Managing the Multigenerational Workforce
- CTEL 1, 2, 3 Combined Exam (031/032/033): Study Guide & Practice
- MoGEA Science & Social Studies Subtest: Study Guide & Practice
- Business Management Theory
- ICAS Science - Paper J Flashcards
- PSAT - Reading Test Flashcards
- NES Family & Consumer Sciences (310) Flashcards
- UExcel Bioethics - Philosophical Issues Flashcards
- How To Pass The Elementary Algebra Accuplacer
- Accuplacer Tips
- How to Study for the Accuplacer
- HESI Test Cost
- Study.com ASWB Scholarship: Application Form & Information
- ACCUPLACER Prep Product Comparison
- Accuplacer Test Locations

- Strategies for Teaching Semantics to ESOL Students
- Applying Morphology to ESOL Instruction
- Jim Crow Laws in To Kill a Mockingbird
- Teaching ELL Students Narrative Writing
- Sorting Algorithm Comparison: Strengths & Weaknesses
- Practical Application: Writing Job Interview Questions
- Calculate the Intrinsic Value of a Firm
- College 101 Course Orientation & Objective
- Quiz & Worksheet - Language Objective for ESL Students
- Quiz & Worksheet - Victorian Architecture
- Quiz & Worksheet - Origins of the Muslim-Hindu Conflict
- Quiz & Worksheet - Choosing Grade-Appropriate Texts
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies

- ILTS Social Science - Geography: Test Practice and Study Guide
- Post-Civil War U.S. History: Help and Review
- Introduction to Astronomy: Certificate Program
- Mentoring in the Workplace
- Principles of Macroeconomics: Certificate Program
- FTCE Social Science: African History
- Harcourt Social Studies - World History Chapter 17: Times of Rapid Change
- Quiz & Worksheet - Finding the Frequency of a Trig Function
- Quiz & Worksheet - A Modell of Christian Charity
- Quiz & Worksheet - Pedestrian & Automotive Safety Issues
- Quiz & Worksheet - Characteristics of Osmosis
- Quiz & Worksheet - Baroque Painting Characteristics & Artists

- How Religion Developed in the Stone Age and Bronze Age
- The Bronze Age in Mesopotamia: Civilization, Architecture & Weapons
- What is the TACHS Exam?
- Do Private Schools Take Standardized Tests?
- Do Homeschoolers Have to take Standardized Tests?
- SAT Subject Test Registration Information
- 504 Plans in Missouri
- How to Pass the Series 7 Exam
- How To Create SAT Vocabulary Flashcards
- Homeschooling in Michigan
- Middle School Summer Reading List
- What is the EPT Test?

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

Browse by subject