Introduction - DR. D. Dambreville's Math Page · 2018-08-06 · 3. What are the effects on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), ... Quiz - Simplifying ... Shifting

8/5/2018 Properties of Exponents and Graphing Exponential Functions

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Properties of Exponents and Graphing Exponential Functions

IntroductionYou've learned about two types of functions: linear & quadratic; and now it is timefor one more! Exponential Functions model a different type of growth, one thatgrows by a rate, rather than a constant value like a linear function. We are going tostart by learning about properties of exponents and how to graph exponentialfunctions. You will learn to analyze exponential functions so that later on you cancompare them to the other types of functions you've learned about!

Essential Questions1. What are the properties of exponents and how do I use them to simplify

expressions?2. What is an exponential function? How do I graph an exponential function?3. What are the effects on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x +

k) for specific values of k (both positive and negative)?4. How do I find the value of k given the graphs?5. What are the key features of an exponential function, and how do I identify them?6. Why is the concept of a function important, and how do I use function notation to

show a variety of situations modeled by functions?7. What is the average rate of change of an exponential function?

Key Terms

Algebra

Average Rate of Change

Coefficient

Continuous

Discrete

Domain

End Behaviors

Equation

Exponential Function

Expression

Horizontal Translation

Interval Notation

Irrational Number

Natural Numbers

Ordered Pair

Range

Rational Number

Real Numbers

Reflection

Variable

Whole Numbers

x-intercept



y-intercept

What to ExpectIn this module, you will be responsible for completing the following assignments:

Assignment - Simplifying Exponential Expressions Practice WorksheetQuiz - Simplifying Exponential ExpressionsAssignment - Shifting Exponential Functions Practice WorksheetQuiz - Graphing Exponential FunctionsDiscussion - Transformations of FunctionsProject - Paper FoldingTest - Properties of Exponents & Graphing Exponential Functions

Properties of ExponentsIn this module, we are going to explore exponential relationships. Before we do that, it is important that we remember theproperties of exponents you've explored in previous courses.

Properties of Exponents Practice

Simplifying Exponential ExpressionsAn important aspect of learning about exponential relationships is being able to utilize multiple properties in the simplification ofexponential expressions. Watch this video to practice a few problems.

Simplifying Exponential Expressions Practice

Assignment - Simplifying Exponential Expressions Practice WorksheetIt is now time to complete the Simplifying Exponential Expressions Practice Worksheet. Download the worksheet from theBlue Sidebar and submit when completed.

Quiz - Simplifying Exponential ExpressionsIt is now time to complete the Simplifying Exponential Expressions Quiz. You will have a limited amount of time; please planaccordingly.

Graph Imagine that you are offered a job that pays $1 on the first day, then $2 on the second day, $4 on the third, and $8 on the fourth.You will continue to get paid in this manner for as long as you hold the job. How can you determine how much money you willhave on your 32nd day of work? Your 100th day of work? This situation represents an exponential function, and that is what we willlearn about in this module. An exponential function is a nonlinear function in which the independent variable is the exponent.



Let's try graphing one of these functions by making a table:

Plot these points.

x -2 -1 0 1 2f(x) 1/4 1/2 1 2 4

Connect the curve.

This graph represents exponential growth because the base of the function is greater than 1.



Let's try graphing a different function by making a table:

Plot these points.

x -2 -1 0 1 2f(x) 4 2 1 1/2 1/4

Connect the curve.



This graph represents exponential decay because the base of the function is greater than 0 but less than 1.

Both of the functions graphed above have an asymptote. An asymptote is a line that a graph approaches more and more closelybut never touches. For both of those functions, the asymptote is the line y = 0.



Graph PracticeComplete the tables for each of the function below, if the dependent value is a fraction type it in using a / symbol. Example: 1/2

Identify the following functions as exponential growth or exponential decay:

Graph Recall from our lesson on transformations of functions that when a function is multiplied by a value, that value can do a few things.

The transformation effects the vertical component of the graph, so in order to graph one of these functions, we should graph the"base" function first, then apply the transformations.

Let's try graphing .

First graph by making a table.

x



-2 1/9-1 1/30 11 32 9

Now since a = 2, that means the graph is vertically stretched by a factor of 2 so each y-value should be multiplied by 2.

x

-2

-1

0 2(1) = 21 2(3) = 62 2(9) = 18



The graph of this function has an asymptote of y = 0.

Watch this video to try a few more:

Transformations of Exponential Functions PracticeMatch each equation to its graph and each equation to the appropriate transformations:

Shift Exponential FunctionsNow that we have stretched, compressed and reflected the exponential functions, we should practice shifting them. The shifts forexponential functions are described below:

Here is an example:



a = 2 b = 3 h = 4 k = 1From our previous lesson, weknow this stretches the graph

vertically by a factor of 2

The base is greater than1, so this function will be

exponential growth

The value of h is4, so the graphshifts right 4.

The value of k is1, so the graph

shifts up 1

Let's try graphing

1. We will start by graphing the base function .

2. Now we will stretch vertically by a factor of 2.


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3. Now we will shift right 4 and up 1.



Notice that the asymptote of y = 0 has been shifted with the function up 1. So the new asymptote is y =1.

Watch this video to try a few more.

More Transformations of Exponential Functions Practice

Assignment - Shifting Exponential Functions Practice WorksheetIt is now time to complete the Shifting Exponential Functions Practice Worksheet. Download the worksheet from the BlueSidebar and submit when completed.

Characteristics of Exponential FunctionsAfter graphing exponential functions, it is important we are able to recognize the key features. Below is a table outlining the keyfeatures you should be able to identify and state.

Feature Definition Picture and ExampleDomain - The set of all

possible values toinput for x.

- For allexponentialfunctions thedomain is all realnumbers becauseyou are allowed toput any value in forx.

- To represent allreal numbers, wesay:



Asymptote - The line the curve

approaches butnever touches.

- For exponentialfunctions, theasymptote isalways y = k.

Range - The set of allpossible outputvalues.

- The range forexponentialfunctions will eitherbe all values abovethe asymptote orbelow theasymptote. Therange will neverinclude theasymptote becausethe curve nevertouches that value!

y-intercept - The point wherethe curve crossesthe y-axis.

- Set x=0 and solvefor y.

Intervalsofincreaseordecrease

- The set of all x-values for whichthe function isincreasing ordecreasing.

EndBehavior

- A statement thattells us what the"ends" of the curveare doing



Watch this video to practice finding these characteristics.

Characteristics of Exponential FunctionsRollover each cell to learn more.

#1 #2

Graph

Equation Solution SolutionAsymptote Solution Solutiony-intercept Solution SolutionDomain Solution SolutionRange Solution SolutionInterval of Increase Solution SolutionInterval of Decrease Solution SolutionEnd Behavior Solution Solution

Average Rate of ChangeIn earlier lessons, we have learned that the rate of change is the slope between two points.

Given two points, (2, 10) and (1, 4), calculate the slope between them: .

If you were to draw a line between the points, the slope of the line would be 6.

If you are not given both the x- and y-parts of the points, you may need to plug in the x-part to find the y-part.

Calculate the rate of change for the function over the interval .



You are given the x-parts of each point.

So first you should find the y-parts.

Now calculate the slope between those points. .

So the rate of change over that interval is 3.

Average Rate of Change Practice

1. Calculate the rate of change for the function over the interval . Solution



When given a table of values, you can recognize an exponential function by the rate of change. The output values in anexponential function increase or decrease by a factor.

x f(x)0 11 32 93 274 81

Notice that each of the y-values is tripled, or increased by a factor of 3!

Discussion - Transformations of FunctionsIt is now time to complete the Transformations of Functions Discussion. Here is an opportunity for you to challenge yourclassmates. Create two exponential functions with at least one of each of the following:

a vertical stretch, compression or reflectiona horizontal shifta vertical shift

Ex:

Number those equations in your post but do not state the transformations, be sure to use the equation editor in your post(you can get to equation editor by pressing the button that looks like an E in the toolbar). Be sure to write your functionsdown on your own paper and list the transformations for yourself.

Quiz - Graphing Exponential FunctionsIt is now time to complete the Graphing Exponential Functions Quiz. You will have a limited amount of time; please planaccordingly.

Properties of Exponents and Graphing Exponential Functions Wrap Up



ChecklistIn this module, you were responsible for completing the following assignments:

Assignment - Simplifying Exponential Expressions Practice Worksheet

Quiz - Simplifying Exponential Expressions

Assignment - Shifting Exponential Functions Practice Worksheet

Quiz - Graphing Exponential Functions

Discussion - Transformations of Functions

Properties of Exponents and GraphingExponential Functions Final Assessments

Paper Folding ProjectIt is now time to complete the Paper Folding Project. Download the project instructions from the Blue Sidebar. Submit yourproject when completed.

Properties of Exponents and Graphing Exponential Functions TestIt is now time to complete the Properties of Exponents and Graphing Exponential Functions Test. Once you havecompleted all self-assessments, assignments, and the review items and feel confident in your understanding of thismaterial, you may begin. You will have a limited amount of time to complete your test and once you begin, you will not be

allowed to restart your test. Please plan accordingly.

Documents

Introduction - DR. D. Dambreville's Math Page · 2018-08-06 · 3. What are the effects on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), ... Quiz - Simplifying ... Shifting