Back To Course

Math 105: Precalculus Algebra14 chapters | 124 lessons | 12 flashcard sets

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:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

All exponential functions follow a basic graph. But when you make changes to the function, you will see the graph shift and make changes. Watch this video lesson to learn how to easily identify these changes or transformations.

An **exponential function** is any function where the variable is the exponent of a constant. The basic exponential function is *f*(*x*) = *b*^*x*, where the *b* is your constant, also called base for these types of functions. Keep in mind that this base is always positive for exponential functions. All other exponential functions are based off of the basic exponential function.

You will see that different exponential functions will add numbers to the basic exponential function in various locations, and these changes will produce changes in the graph as well. When you graph any exponential function though, they will all have the same general look, based off of the basic exponential function graph. Depending on the change, or **transformation**, the graph can be shifted up or down, left or right, or even reversed. Keep watching, and we will identify what causes these changes.

First, let's see what the basic exponential function looks like graphed. Before I show you how it looks, I want you to think about exponentials for a moment. What happens as you increase the exponent? Doesn't the value of the function get increasingly bigger at an increasingly faster rate? It does, so you will see the graph curve upwards quickly. And looking at our graph, we see that it does.

This particular graph shows the graph of *f*(*x*) = 2^*x*. Also notice that the graph crosses the *y*-axis when the exponent equals 0. For the basic function, it is when the *x* = 0 since our exponent is *x* by itself. Now let's see what happens to the graph when we change the function slightly.

What can we change? Well, we can change the exponent to a negative so our function becomes *f*(*x*) = 2^(-*x*). What happens then? Think about what is happening to the exponent. The negative sign essentially reverses our variable. As you decrease the *x*-value and as it gets more negative, the function gets larger and larger more quickly. Why? Well, if we plug in a -2 for our *x*, the function becomes 2^-(-2) = 2^2 since the negative changes the sign of the exponent. If we plug in a -3, the function becomes 2^-(-3) = 2^3. The more negative we get, the bigger our function becomes. So your graph flips or reverses itself.

When you add or subtract a number from the basic function, we get vertical shifts. Think about this. What can you do to the graph to make it go up or down? Wouldn't you just have to add or subtract some numbers? If we added a 3 to our function to get *f*(*x*) = 2^*x* + 3, we would be shifting our graph 3 points upwards. You can think of this as adding 3 to every single point on our basic graph.

If we subtract 4 from the function, what do you think will happen? Our graph will shift down by 4 points. We are essentially subtracting 4 from every single point of our basic graph.

Also, both graphs cross the *y*-axis when *x* = 0 since the exponent is only *x*.

When we add or subtract from the exponent, the graph moves sideways. If we add a 2 to the exponent, we see the graph shifts 2 points to the left. Why? When we change the exponent, we are changing where the graph crosses the *y*-axis. Remember that the graph crosses the *y*-axis when the exponent is 0. So, if our exponent has an added 2, we need to subtract 2 to get back to 0. Going left on the *x*-axis will give us that minus 2 that we need.

If we subtracted 5 from the exponent, our graph would shift to the right by 5 points. Again, it is because the graph crosses when the exponent equals 0. When we subtract 5 from the exponent, we need to add 5 to get it back to where it normally equals 0, hence the shift of 5 to the right.

Wow! Look at all we've learned! We've learned that an **exponential function** is any function where the variable is the exponent of a constant. Our basic exponential function is *f*(*x*) = *b*^*x*, where *b* is our base, which is a positive constant. All other exponential functions are modifications to this basic form.

**Transformations** are changes to the graph. Transformations include vertical shifts, horizontal shifts, and graph reversals. Changing the sign of the exponent will result in a graph reversal or flip. A positive exponent has the graph heading to infinity as *x* gets bigger. A negative exponent has the graph heading to infinity as *x* gets smaller.

Adding or subtracting numbers to the function will result in vertical, or up and down, shifts. Adding numbers shifts the graph up. Subtracting numbers shifts the graph down. Adding or subtracting numbers to the exponent will result in horizontal, or sideways, shifts. Adding numbers to the exponent shifts the graph to the left, and subtracting numbers to the exponent shifts the graph to the right.

Study and reference this lesson if you'd like to:

- Recall the meaning of a basic exponential function
- Interpret a graph shift along the x- or y-axis
- Understand the transformation of a graph based on the modification to the original function
- Note the correlation between a negative sign and the reversal of a variable
- Distinguish between horizontal and vertical shifts

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
3 in chapter 12 of the course:

Back To Course

Math 105: Precalculus Algebra14 chapters | 124 lessons | 12 flashcard sets

- What Is an Exponential Function? 7:24
- Exponential Growth vs. Decay 8:41
- Transformation of Exponential Functions: Examples & Summary 5:51
- What is a Logarithm? 5:23
- How to Evaluate Logarithms 6:45
- Writing the Inverse of Logarithmic Functions 7:09
- Exponentials, Logarithms & the Natural Log 8:36
- Basic Graphs & Shifted Graphs of Logarithmic Functions: Definition & Examples 8:08
- Logarithmic Properties 5:11
- Practice Problems for Logarithmic Properties 6:44
- How to Solve Logarithmic Equations 6:50
- Using the Change-of-Base Formula for Logarithms: Definition & Example 4:56
- How to Solve Exponential Equations 6:18
- Go to Exponential and Logarithmic Functions

- Computer Science 335: Mobile Forensics
- Electricity, Physics & Engineering Lesson Plans
- Teaching Economics Lesson Plans
- U.S. Politics & Civics Lesson Plans
- US History - Civil War: Lesson Plans & Resources
- iOS Data Analysis & Recovery
- Acquiring Data from iOS Devices
- Foundations of Digital Forensics
- Introduction to Mobile Forensics
- Examination of iOS Devices
- CNE Prep Product Comparison
- IAAP CAP Prep Product Comparison
- TACHS Prep Product Comparison
- Top 50 Blended Learning High Schools
- EPPP Prep Product Comparison
- NMTA Prep Product Comparison
- Study.com NMTA Scholarship: Application Form & Information

- History of Sparta
- Realistic vs Optimistic Thinking
- How Language Reflects Culture & Affects Meaning
- Logical Thinking & Reasoning Questions: Lesson for Kids
- Human Geography Project Ideas
- Asian Heritage Month Activities
- Types of Visualization in Python
- Quiz & Worksheet - Frontalis Muscle
- Octopus Diet: Quiz & Worksheet for Kids
- Quiz & Worksheet - Fezziwig in A Christmas Carol
- Quiz & Worksheet - Dolphin Mating & Reproduction
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies
- Bullying in Schools | Types & Effects of Bullying
- ESL Conversation Questions & Topics for ESL Students

- Political Science 102: American Government
- Praxis English Language Arts - Content & Analysis (5039): Practice & Study Guide
- High School Biology: Homework Help Resource
- High School Chemistry: Homework Help Resource
- Effective Communication in the Workplace: Certificate Program
- The Roman World in Transformation
- Working with Quadratic Functions Lesson Plans
- Quiz & Worksheet - Five Methods of Neutralization
- Quiz & Worksheet - Personification in The Necklace
- Quiz & Worksheet - Role of the Nucleus in a Plant Cell
- Quiz & Worksheet - Othello Character Analysis
- Quiz & Worksheet - Mordred in King Arthur

- Glycine: Formula, Structure & Benefits
- How to Solve Fractional Exponents
- Electricity Experiments for Kids
- Romeo and Juliet Act 5 Lesson Plan
- Sunnyvale, CA Adult Education
- Homeschooling in New York
- Ethos, Pathos & Logos Lesson Plan
- Frindle Lesson Plan
- Letter Writing Lesson Plan
- Solar System Project Ideas
- Logical Reasoning Questions on the LSAT
- Counting Money Lesson Plan

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

Browse by subject