TEXTEAMS Part 1: Algebra II and Precalculus Institute

TEXTEAMS Part 1: Algebra II and Precalculus Institute i

TEXTEAMS Algebra II/Precalculus Institute

Acknowledgements

The Algebra II/ Precalculus Institute was developed under the direction and assistance ofthe following:

Writers:Pam Harris ConsultantSusan May ConsultantDiane McGowan Charles A. Dana Center.

Writers who contributed material that was used from the previous AlgebraII/Precalculus Institute:

Pam Chandler, Consultant; Susan Cinque, Fort Bend ISD; Maggie Dement, Consultant;Tommy Eads, North Lamar ISD; Lois Moseley, Consultant; Richard Parr, RiceUniversity; Rachel Pinkston, Alief ISD; Diane Reed, Ysleta ISD; Rozanne Rubin, AliefISD; GT Springer, Texas Instruments; Susan Williams, University of Houston

TEXTEAMS Part 1: Algebra II and Precalculus Institute ii

Advisory Team:

Beverly Anderson Region XVII: Educational ServiceCecilia Avendano Brownsville ISDTommy Bryan Baylor UniversityNick Carter Independent ConsultantOscar Chavarria Pasadena ISDLibby Chaskin Northside ISD Kathi Cook Charles A. Dana CenterBeth Glassman Leander ISDGaye Glenn Region II Educational Service CenterBasia Hall Houston ISDPam Harris ConsultantSusan Hull Charles A. Dana CenterHee Joon Kim Charles A. Dana CenterLaurie Mathis Charles A. Dana CenterSusan May ConsultantDiane McGowan Charles A. Dana CenterPaul Mlakar Region IV Educational Service CenterBarbara Montalto Texas Education AgencyRichard Parr Rice UniversityErika Pierce Charles A. Dana CenterTim Pope Hays ISDDiane Reed Ysleta ISDRozanne Rubin Alief ISDCindy Schimek Katy ISDSusan Thomas Alamo Heights ISDDick Stanley University of California at BerkeleySusan Williams University of HoustonJeanne Womack Region I Educational Service CenterBeverly Weldon Region X Educational Service CenterWard Roberts Witchita Falls: ISD

TEXTEAMS Part 1: Algebra II and Precalculus Institute iii

Table of ContentsPart I: Algebra II/Precalculus

Foundations of Functions

I. Foundations of FunctionsIntroduction: Ball Drop ………………………………………………………….1

1. Foundations1.1 Transformations ………………………………………………….. 6

Activity 1: Explore Transformations………………………. 17Activity 2: Domains ………………………………………… 22Activity 3: Transformations with Technology ……………..….…. 23Activity 4: Summarize Your Findings …………………….. 29Reflect and Apply …………………………………………… 30Student Activity 1: Move the Monster Algebra II …………….…. 31Student Activity 2: Move the Monster Precalculus …………. 35Student Activity 3: Combinations of Transformations: ……. 40Student Activity 4: Transformations on Generic Graphs... …. 44Student Activity 5: Transformation Practice Algebra II ……. 48Student Activity 6: Transformation Practice Precalculus .. ….…. 54

1.2 What Is the Difference? ………………………………………… 64

2.Applying Transformations2.1 Applications ………………………………………………………. 77

Activity 1:Good Viewing Windows ………………………… 85Activity 2:Friendly Viewing Windows ……………………... 90Reflect and Apply …………………………………………… 91Student Activity: Matching Parent Functions …………….….…. 92

2.2 Airlines ……………………………………………………………. 102Activity 1:Trend Lines ………………………………………. 105Activity 2:Another Trend Line ……………………………… 106Reflect and Apply …………………………………………… 107

3. Making Connections3.1 Inverses …………………………………………………………… 108

Activity 1:Parametric Equations …………………………… 113Activity 2:Restricted Domains ………………………………….….114Reflect and Apply …………………………………………… 115Student Activity: MIRA Reflections ………………………. 116

TEXTEAMS Part 1: Algebra II and Precalculus Institute iv

3.2 Stays the Same ………………………………………………….. 128Activity: Solving Equations Step by Step Graphically …... ….…. 133Reflect and Apply……………………………………………. 134

II. Exponential and Logarithmic Functions

Introduction: Linear and Exponential Models ……………………………….. 135

1.Exponential Growth and Decay1.1 Bacteria Growth...………………………………………………… 142

Activity 1:E .Coli Growth Rate …………………………..… 150Activity 2:The Flu Epidemic: A Logistic Growth Model ……. 153Reflect and Apply …………………………………………… 156Student Activity 1: Recursion and Bacteria Cell Division ….…. 157Student Activity 2: Space Debris in Earth ’s Orbit …………. 166

1.2 Exponential Decay ……………………………………………… 171Activity: Cooling Down……………………………………… 175Reflect and Apply

2. Logarithmic Functions2.1 The Energy of Earthquakes ………………………………………. 180

Activity 1:The Richter Scale ……………………………….… 187Activity 2:Graphing the Magnitude………………………….… 189Reflect and Apply …………………………………………… 191Student Activity 1: Exploring Logs ………………………… 192Student Activity 2: Change of Base ……………………….… 196Appendix: Bad Bad Bugs ………………………………….… 204

III. Power Functions and Polynomials

Introductory Activity: Spread of Aids …….….….….….….….….….….…. 209

I. Power Functions and Polynomials1.1 Power Functions ………………………………………………… 214

Activity 1: Introduction to Power Functions ……………….… 223Activity 2: Comparing Power Functions…………………… 224Activity 3: Comparing Functions…………………………… 228Reflect and Apply …………………………………………… 229Student Activity 1: Truth or Consequences ……………….… 230Student Activity 2: Classifying Functions ……………….…… 232

1.2 Compounding Interest …………………………………………… 236Activity 1: Money Talk ……………………………………… 242Activity 2: Compounding Continuously …………………… 244Reflect and Apply …………………………………………… 246

TEXTEAMS Part 1: Algebra II and Precalculus Institute v

About TEXTEAMS Institutes

TEXTEAMS Philosophy

• Teachers at all levels benefit from extending their own mathematical knowledge andunderstanding to include new content and new ways of conceptualizing the content they alreadypossess.

• Professional development experiences, much like the school mathematics curriculum itself,should focus on few activities in great depth.

• Professional development experiences must provide opportunities for teachers to connect andapply what they have learned to their day-to-day teaching.

Features of TEXTEAMS Institute Materials

Multiple representations (verbal, concrete, pictorial, tabular, symbolic, graphical)Mathematical ideas will be represented in many different formats. This helps both teachers and studentsunderstand mathematical relationships in different ways.

Integration of manipulative materials and graphing technologyThe emphasis of TEXTEAMS Institutes is on mathematics, not on learning about particular manipulativematerials or calculator keystrokes. However, such tools are used in various ways throughout the institutes.

Rich Connections within and outside mathematicsInstitutes focus on using important mathematical ideas to connect various mathematical topics and onmaking connections to content areas and applications outside of mathematics.

Questioning strategiesA variety of questions are developed within each activity that help illicit deep levels of mathematicalunderstanding and proficiency.

Hands-on approach with “get-up-and-move” activitiesInstitutes are designed to balance intense thinking with hands-on experiences.

Math Notes and Reflect and ApplyA feature called Math Notes includes short discussions of mathematical concepts accompanying thelearning activities. Similarly, the Reflect and Apply feature is designed to extend and apply participants’understanding of the mathematical concepts.

TEXTEAMS Part 1: Algebra II and Precalculus Institute vi

About the Algebra II/Precalculus Institute

The Algebra II / Precalculus Institute is a professional development experience designedto stretch and extend participants mathematical knowledge. The activities in this instituteare designed as a professional development for teachers, rather than simply as anopportunity to go through classroom tasks. The topics address in this institute extendwell beyond what might typically be covered in the Algebra II or Precalculus classroom.This institute begins by developing foundations for functions and exploring functions as aunifying theme for this institute and the two additional institutes, Part 2: Algebra II andPart 2: Precalculus. The three institutes include materials from the previous TEXTEAMSAlgebra II/Precalculus Institute.

As with other TEXTEAMS Institutes, this session does not represent a completeprofessional development curriculum for Algebra II or Precalculus teachers. Additionalprofessional development resources for secondary teachers are available throughTEXTEAMS. Course specific TEXTEAMS institutes include Algebra I: 2000 andBeyond, Practiced-Based Professional Development: Algebra I Assessments, RethinkingSecondary Mathematics: Algebraic and Geometric Modeling, High School Geometry:Supporting the TEKS and TAKS, Practiced-Based Professional Development: GeometryAssessments, Rethinking Secondary Mathematics: Statistical Reasoning Across the TEKS,Rethinking Secondary Mathematics: In-depth Secondary Mathematics (IDSM, and,Practiced-Based Professional Development: Algebra II Assessments (in development).

TEXTEAMS Part 1: Algebra II and Precalculus Institute vii

References:

Connally, Hughes-Hallett, Gleason, et al. (2000) Functions Modeling Change: APreparation for Calculus, New York, John Wiley & Sons, Inc.

Demana, Waits, Clemens. (1994). Precalculus Mathematics: A Graphing Appproach,Menlo Park, CA: Addison-Wesley.

Finney, Thomas, Demana, Waits (1994). Calculus: Graphical, Numerical, Algebraic,Readin, MA: Addison-Wesley.

Forester, Paul, Calculus: Concepts and Applications (2000), Emeryville, California, KeyCurriculum Press,

Murdoch, Kamischke, Kamischke, (1998). Advanced Algebra Through DataExploration, Berkekly, CA: Key Curriculum Press.

The University of Chicago School Mathematics Project, (1992). Precalculus andDiscrete Mathematics. Glenview, IL: Scott, Foresman.

TEXTEAMS Part 1: Algebra II and Precalculus Institute viii

OverviewActivity Overview Materials

I. Foundations of FunctionsIntroduction

Ball DropParticipants use data collection tomotivate the need for transformations offunctions.

Motion detector anddata collection device,overhead graphingcalculator, balls

1. Foundations

1.1 IntroducingTransformations

Participants transform a functionnumerically and graphically and makegeneralizations.

Transparency pens, 1”grid paper, markers

1.2 What is theDifference?

Participants will become aware of theTEKS for each course and clarify thesimilarities and differences in the twocourses.

Chart paper, markers,copies of TEKS forAlgebra II andPrecalculus

2. Applying Transformations

2.1 Applications Participants use their knowledge oftransformations to determine appropriateviewing windows on graphing calculators fortransformed parent functions.

Graphing calculator,set of activity cards foreach group

2.2 Airlines Participants graph real data on a scatter plotand find trend lines. They use transformationsto explain aspects of the trend lines.Participants find a real application forhorizontal shifts.

Graphing calculator

3. Making Connections

3.1 Inverses Participants use parametric equations to graphinverse relations and inverse functions and toinvestigate domain and range.

Graphing calculator

3.2 Stays the Same Participants use graphing calculators to modelthe solutions to equations graphically andmake connections between transformations offunctions, inverse functions, and algebraicsolutions.

Graphing calculator,1” grid paper,markers, string,washers, timers, tapemeasures or metersticks, compasses

I. Foundations Introductory Activity: Trainers' Notes

TEXTEAMS Part 1: Algebra II and Precalculus Institute 1DRAFT 3-29-03

Ball Drop IntroductionOverview: In this introductory activity, we use data collection to motivate the need for

transformations of functions.

Objective: Algebra II TEKS(b.1.B) In solving problems, the student collects data and records results,organizes the data, makes scatter plots, fits the curves to the appropriate parentfunction, interprets the results, and proceeds to model, predict, and makedecisions and critical judgments.(c.1.A) The student identifies and sketches graphs of parent functions, including… inverse (

y =

1x

) … functions.

(c.1.B) The student extends parent functions with parameters such as m in y = mx and describes parameter changes on the graph of parent functions.Precalculus TEKS(c.2.A) The student is expected to apply basic transformations, includinga • f x( ), f x( ) + d, f x − c( ), f b • x( ), f x( ) , f x( ) , to the parent functions.(c.2.B) The student is expected to perform operations including compositionson functions, find inverses, and describe these procedures and results verbally,numerically, symbolically, and graphically.

Terms: Transformation

Materials: Motion detector and data collection device, overhead graphing calculator, ball(Racket balls, golf balls, and basketballs work well.)

Procedures: Introduce this initial activity by explaining the focus of the Foundationsection. The intent of the Foundation section is to tie together commonalitiesacross many of the topics found in the Algebra II and Precalculus TEKS. Thisinstitute builds on the functions-based approach from the Algebra I Institute.Understanding transformations is an essential part of this approach. Thisactivity uses data to drive the need for transformation of functions.

Activity: Ball DropHold up a ball and then let it bounce in front of whole group. Next, introducethe motion detector that collects data about the distance from an object to themotion detector.Explain that the motion detector sends out an ultrasonic pulse. The pulsebounces off the ball, and the motion detector records the distance from the ballto the motion detector at that time. The calculator displays the data as a graphwith the distance measured in meters or feet and the time measured inseconds.



Hold up the motion detector and drop the ball under it.1. Ask participants to individually sketch their guess for a graph of the

relationship between the distance of the ball from the motion detector overtime. Have them compare their graph in small groups and arrive at agroup consensus.

2. Collect distance data using the motion detector for about 4 seconds. (Donot use a ball bounce program; simply use a distance program.) Repeat ifnecessary for good data. Have participants sketch the graph of thecollected data.

Sample Data:

• How does your sketch compare with the graph from our collecteddata? If maximum and minimum points are not mentioned, askquestions about them.

• What do the peak points on the graph represent? The ball’s maximumdistance from the motion detector. This distance should be close to thedistance between the motion detector and the floor.

Trace to a point on the scatter plot; have a participant hold up the ball at thedistance away from the motion detector represented by the y-coordinate of thetraced point. (Do not choose a point halfway in between the floor and themotion detector because participants may confuse distance from the motiondetector and distance from the floor.)

• Write a sentence using the two numbers from the ordered pair in thecontext of the ball drop. For example, for the ordered pair (0.84, 2.7),the sentence could be “At 0.84 seconds after the ball was dropped, thedistance between the ball and the motion detector was 2.7 units.” Thisis important because it is easy to confuse the path of the ball with thegraph of the relationship of the distance from the motion detector overtime.

• Did the volunteer hold the ball at the correct distance from the motiondetector? Why or why not?

• Is this a picture of the ball’s path? No, it is a graph of the relationshipof distance from the motion detector over time.

Discuss with participants where the data from the motion detector is stored onthe calculator and how the calculator creates a scatter plot from the data. Savethe data in 2 named lists to use in Exercise 4.



3. Tell participants that for this second experiment we will use a programthat graphs the data differently. Use a ball bounce program that reflectsand shifts the data to repeat the experiment. The resulting graphrepresents the relationship of the distance from the floor and the ball.Have participants sketch the resulting graph.

Sample Graph:

• What is the difference between this graph and the previous graph? Thefirst graph is the graph of the ordered pair (elapsed time, distancefrom the motion detector); the second graph is the graph of theordered pair (elapsed time, distance from the floor).

• What did the program do to the collected data? It shifted and reflectedthe data and graphed the transformed data.

• Is this a picture of the ball’s path? No, it is a graph of the relationshipof distance from the floor over time.

4. Ask participants to discuss in small groups how the program transformedthe data in the lists to obtain the graph of (elapsed time, distance from thefloor). As a whole group, elicit suggestions and try them one at a time onthe data from the first experiment until you get the desired results.

Sample Data:

• What is this process called? Transformation of functions



Focus on the TEKS:What Algebra II and Precalculus TEKS were introduced, reinforced, orextended in the activities? Justify your answer.In these activities participants collect data (A-b.1.B), and explore and applytransformations, numerically, algebraically, and graphically (A-c.1.A, A-c.1.B, P-c.2.A, P-c.2.B).

How do these activities support subsequent courses?A solid understanding of parent functions and how transformations act onparent functions is a good basis for all of the work that students will do withfunctions in calculus.

Can you take this activity to the classroom? If so, how would you adapt it?This activity is appropriate for the classroom to motivate the need for learningtransformations. Algebra II students may need to experiment more to find thenecessary transformations.

Summary: The big idea here is to introduce transformations in a context in whichparticipants have some intuitive notions. Transformations of data are used asa precursor to transformations of functions.

I. Foundations Introductory Activity: Activity


Activity: Ball Drop

1. Sketch a prediction for the graph of the distance of the ball from themotion detector over time.

2. Sketch the results of the motion detector experiment.

3. Sketch the results using a bouncing ball program.

4. How could you transform the data collected from the motion detectorin the first experiment to get it to look like the data displayed usingthe ball bouncing program?

I. Foundations 1.1 Transformations: Trainers' Notes


1.1 TransformationsOverview: Participants follow the transformations of a function numerically and

graphically. Participants investigate the effects of transformations of parentfunctions using graphing calculators and make generalizations.

Objective: Algebra II TEKS(c.1) The student connects algebraic and geometric representations of functions.(c.1.A) The student identifies and sketches graphs of parent functions, includinglinear ( y = x ), quadratic ( y = x2 ), square root ( y = x ), inverse (

y =

1x

),

exponential ( y = ax ), and logarithmic ( y = loga x ) functions.(c.1.B) The student extends parent functions with parameters such as m in y = mx and describes parameter changes on the graph of parent functions.Precalculus TEKS(c.1) The student defines functions, describes characteristics of functions, andtranslates among verbal, numerical, graphical, and symbolic representationsof functions, including polynomial, rational, radical, exponential, logarithmic,trigonometric, piecewise-defined functions.(c.2) The student interprets the meaning of the symbolic representations offunctions and operations on functions within a context.(c.2.A) The student is expected to apply basic transformations, includinga • f x( ), f x( ) + d, f x − c( ), f b • x( ), f x( ) , f x( ) , to the parent functions.(c.2.B) The student is expected to perform operations including compositionson functions, find inverses, and describe these procedures and results verbally,numerically, symbolically, and graphically.(c.3.c) The student is expected to use properties of functions to analyze andsolve problems and make predictions.

Terms: Transformation, function notation, translation, reflection, stretch,compression, domain, range

Materials: Graphing calculator, transparency pens, big graph paper, markers, wall-chartpaper

Procedures:Unfortunately it has been all too common that algebra II and precalculus havebeen taught as a collection of discrete topics, unconnected and separate. Lineswere lines, parabolas were parabolas, and trigonometry might be the first timethat transformations were really addressed. Students learned about eachfamily of functions, but not in relation to the other parent functions and not bylooking at the similarities of transformations on the parent functions. In thisactivity, we take a transformational approach to functions. By looking at theeffects of transforming each parent function, formulas like y = a x − h( )2 + k



become less of a formula to memorize and more of a specific example oftransforming any function, this one just being a quadratic function that hasbeen shifted over, shifted up or down, and stretched or compressed vertically.

When facilitating this activity, consider the following suggestions for order.Begin by having participants complete Activity 1: Explore Transformationstogether and then discuss as a whole group. Next have participants look atStudent Activity 1 and Student Activity 2 and compare the way the activitieshave been written to focus on either Algebra II or Precalculus. Then haveparticipants work together on Activity 2: Domains and then discuss. Activity3: Transformations with Technology is rather lengthy, so you may find itadvantageous to have participants do half of the suggested examples for eachtransformation and then discuss the rest with their partners. Then participantsshould work together on Activity 4: Summarize Your Findings. Next haveparticipants complete Student Activity 3 and Student Activity 4. Lastly,have participants discuss the differences between Student Activity 5 andStudent Activity 6, discussing how each is appropriate for its suggestedcourse.

Activity 1: Explore TransformationsInstruct participants to fill in the tables and graph the corresponding points.Circulate and ask guiding questions.

• What does f 1( ) refer to? The y-value of the function f at x =1 .f 1( ) =1 .

• To fill in the table for Exercise 1 where y = f x + 2( ) , the first x-valueis −4 . What is −4 + 2? Well, −4 + 2 = −2 , thereforef −4 + 2( ) = f −2( ) .

• What is f −2( )? From the table or graph f −2( ) = 1 . Thus, whenx = −4 , f x + 2( ) =1 . So, graph the point −4, 1( ) .

Give different participants transparencies of the Activity and a transparencypen and ask them to each present one of the Exercises. Overlay thetransparency of f onto the transparency of their drawn transformation of f tocompare the original and the transformed function.• What is the effect of the transformation?Questions may arise concerning transformations where participants are askedto find the transformed function evaluated at a value for which f x( ) is notdefined. For example, Exercise 1 asks for f x + 2( ) , evaluated at x = −4 .However, participants may note that in the table of values for f x( ) , f x( ) isnot defined for –4. Help participants to realize that f x( ) does not need to bedefined at –4, because they are not evaluating f x( ) at –4, rather, they are toevaluate f x + 2( ) at x = −4 . Note that f x + 2( ) at x = −4 isf −4 + 2( ) = f −2( ) and f −2( ) is defined.



Activity 2: Domains gets at this idea more in depth as teachers must findappropriate input values for the various transformations.

Answers:1. 2.

3. 4.

5. 6.



7. 8.

9. 10.

11. It is important to note thatparticipants must estimate the x-intercepts of the function to correctlysketch y = f x( ) .

12.

Talk participants through the Student Activities: Move the Monster (AlgebraII) and Move the Monster (Precalculus). Have them compare the twoactivities and discuss how the two are similar and different.

Activity 2: Domains

1. We chose values to evaluate in the transformed functions such that theoriginal function would be defined. For example, Exercise 1 asks fory = f x + 2( ) and provides the x-values of –4, –3, –2, –1, 0, 1. Adding 2 to



these x-values gives –2, –1, 0, 1, 2, 3, and these are the domain values forwhich f is defined. Also, Exercise 2 asks for y = f x − 2( ) and providesthe x-values of 0, 1, 2, 3, 4, 5. Subtracting 2 from these x-values againgives –2, –1, 0, 1, 2, 3, and these are the domain values for which f isdefined.

2. Sample:Given: Graph of g Table for g

g −3( ) = 0g 0( ) = 2g 1( ) = 0g 2( ) = −3g 3( ) =1

y = g x + 3( )x y

–6 0–3 2–2 0–1 –30 1

Activity 3: Transformations with TechnologyIn Activity 1: Explore Transformations, we looked at the effect oftransformations of a piecewise defined function. Now we will investigatefurther by using technology to explore the effects of transformations of parentfunctions.

Reason and Communicate:• If f (x) = x2 , then what is f x + 2( )? x + 2( ) 2

• How does your graphing calculator denote function notation? Helpparticipants with individual calculators.

• What are some attributes of functions that you should keep in mindwhen you describe the effect of the transformation on the parentfunctions? Domain, range, shape

Explain to participants that they are going to use composition of functions andtheir graphing calculators as tools to explore transformations. They will enterparent functions one at a time into y1 . They will enter y2 as a transformationof y1 . This composition approach gives participants a sort of transformation



exploring machine. Illustrate by doing Exercise 1a in Activity 1 withparticipants as follows. Enter y1 = x and y2 = y1 x −1( ) . Explain that this isanalogous to writing f x( ) = x and g x( ) = f x −1( ) . Show thaty2 = y1 x −1( ) = x −1 , so the graph of y2 should be a transformation of y1 .Instruct participants to work in pairs to sketch the given functions andrespective transformations in Activity 1. Suggest to participants that they usedifferent graph styles to help distinguish the graph of the parent function fromthe graph of the transformed function. Also instruct participants to graph a – cof each set and then guess the graph of d before checking with the calculator.Give transparency pens and a transparency of the activity sheets for thedifferent types of transformations I – IV to some of the partners. Ask thepartners to prepare to present their findings. Use the following questions toelicit notions about the transformation or to emphasize points they make.

I. Explore Vertical Translations3. What type of transformation occurs when you add a to the function f x( ) ?

Adding a to the function creates a vertical translation (slide, shift) up ifa>0 and down if a<0.• How does this transformation affect the domain of the original

function? It has no effect on the domain.• How does this transformation affect the range of the original function?

If the range of the original function is all real numbers, then it has noeffect because the range is a set of numbers. If not, then it shifts thevalues of the range up a units if a>0 and shifts the values down a unitsif a<0.

• Give an example of a function for which its range would not beaffected. y = x3 , y = x , y = log x , y = tan x .

• Give an example of a function for which its range would be affected.y = x2 , y = cos x , y = x .

• How does it affect the shape of the graph? It does not change theshape of the graph. It only slides the graph up or down.

II. Explore Horizontal Translations3. What type of transformation occurs when you replace x with x − a in the

function, f x( ) ? Replacing x with x – a in the function creates ahorizontal slide (shift) right if a>0 and left if a<0.• How does this transformation affect the domain? It shifts the values of

the domain right k units if a>0 and shifts the values left k units if a<0.If the domain is all real numbers, then the set is not affected.

• How does this transformation affect the range? It has no effect).• How does it affect the shape of the graph? It does not change the

shape of the graph. It only slides the graph left or right.



III. Explore Vertical Stretches and Compressions (Dilations)3. What happens to the graph of a function when you multiply a function by a?

When |a| > 1 and a is the leading coefficient, the graph is stretched verticallyand appears thinner than the original parent function. In effect, the functionvalues (y values) get bigger faster. When 0 < |a| < 1, the graph is shrunk orcompressed vertically and appears wider than the original parent function.In effect, the function values (y values) get bigger slower. When a < 0, thegraph is reflected across the x axis.

Have participants trace to any point on y1. Use the down arrow key tomove the cursor to y2. Notice that for a given x-coordinate, the y-coordinate of y2 is three times the value of the y-coordinate of y1.

Also examine the effect a has on the y-values by looking at the table onthe graphing calculator. Notice again that for a given x-coordinate, the y-coordinate of y2 is three times the value of the y-coordinate of y1.

• Do these transformations affect the domain? No.• Do these transformations affect the range? Sometimes.• When? When the leading coefficient is negative, the range of the

following functions changes from y ≥ 0 to y ≤ 0: y = x2 , y = x ,

y = x2 , y = 1x2

, y = ex , y = x .

• How do these transformations affect the shape of the graph? Verticalstretches occur when a > 1, and vertical compressions occur when0 < a < 1 .

IV. Explore Horizontal Stretches and Compressions (Dilations)Trigonometric functions:

• Does this transformation affect the domain? Horizontaltransformations may affect the domain of a function, although they donot in the case of sine, cosine, and tangent.

• Does this transformation affect the range? No.• How does it affect the shape of the graph? For a periodic function, it

changes the period of the function. When |k| > 1 as in sin (kx), theperiod of the function is shrunk, and this transformation is referred toas a horizontal compression. When 0 < k < 1, the period of thefunction is stretched, and this transformation is referred to as ahorizontal stretch.

For y = x2 :• Is the y1 (2x) transformation a horizontal compression by a factor of 2, or

is it a vertical stretch by a factor of 4?



For y = x3

• Is the y112x

transformation a horizontal stretch by a factor of 0.5, or is

it a vertical compression by a factor of 0.125? In each case both answersare correct! Stress that it is a characteristic of quadratic and cubicfunctions that a horizontal compression will look the same as a verticalstretch. Likewise with quadratics and cubics, a horizontal stretch willlook the same as a vertical compression. Not all parent functions sharethis characteristic. Note that for f x( ) = x 2 , f 2x( ) = 2x( ) 2 = 4x2 = 4 f x( )and for f x( ) = x3 , f 2x( ) = 2x( ) 3 = 8x3 = 8 f x( ) .

For y1 = 9 − x 2 :• Do these transformations affect the domain? Yes, the domain of the

original function is –3 ≤ x ≤ 3, the domain of y2 is horizontallycompressed to –1.5 ≤ x ≤ 1.5, and the domain of y3 is horizontallystretched to –6 ≤ x ≤ 6.)

• Do these transformations affect the range? (no, the range is 0 ≤ y ≤ 3for all three functions)

• Do they affect the shape of the graph? Yes.• How is the shape of the graph affected? The first transformation

horizontally compresses the function. The second transformationhorizontally stretches the function. There is no vertical stretching orcompressing.

Curriculum Issues.When should the terms “translation”, “dilation”, “stretch”, and “compression”be introduced? In Algebra I? in Algebra II? in Precalculus? Before?

Should the horizontal stretches and compressions be reserved untiltrigonometric functions are introduced? Or should horizontal stretches andcompressions be introduced early and then revisited during the study oftrigonometric functions?We often use slang terms such as slide, flip, and turn, but the 4th –6th gradeTEKS use correct mathematical vocabulary.

V. Explore Reflections2. What happens to the graph of a function when you multiply the function by

−1? It is a “flip” or a reflection across the x-axis.• Does the transformation affect the domain of the function? (no)• Does it affect the range of the function? Yes, the range is affected for

some of the parent functions. Parent functions whose ranges areaffected are as follows: y = x2 , y = x , y = x2 , y = 1

x2, y = ex ,

y = x .



4. How do you describe the transformation f −x( )? It is a reflection acrossthe y-axis.• Does the transformation affect the domain of the function? Yes, the

domain is affected for some of the parent functions. Parent functionswhose domains are affected are as follows: y = x , y = ln x( ) .

• Does it affect the range of the function? No.• For which functions does the graph of f(x) look the same as the graph

of f −x( )? y = x2 , y = x2 , y = 1x2

, y = x , and y = cos x( ) .

• What common feature do these graphs share? They all are symmetricwith respect to the y-axis. They are all “even” functions.

VI. Explore Absolute ValueNote that y = x is not a transformation of y = x , but if f x( ) = x , thenh x( ) = x is the composition of g x( ) = x and f x( ) = x , g o f .2. What happens when you graph the absolute value of a function? One

possible description is that positive y-values of the original functionremain positive, while any negative y-values are made positive thusreflecting those points across the x-axis.• Does this transformation affect the domain of the function? No.• Does this transformation affect the range of the function? In some

cases it does. Whenever the range of the parent function includesvalues less than 0, then it changes to include only positive values.

• Does this transformation affect the shape of the function? If the rangeof the parent function includes no negative values, then the shape ofthe transformed function is the same as the parent function. If therange of the parent function includes negative values, then the shapeof the transformed function is different from the original parentfunction. All points that have negative y-values are reflected acrossthe x-axis.

• Identify a parent function such that y1 = | y1|. Any function whoserange is y ≥ 0 fits this criteria, e.g., y = x2 , y = x , y = 1

x2, y = x .

4. What happens to the graph of a function when you replace x with x ?

f x( ) =f x( ), if x ≥ 0f −x( ), if x < 0

• If the parent function is symmetric with the y-axis, then no change isapparent looking at the graph. However, if the parent function is notsymmetric with the y-axis, then any piece of the original parent graphlocated in the 2nd or 3rd quadrants is eliminated and each point locatedin the 1st and 4th quadrants is reflected across the y-axis. This



transformation “forces” the function to be symmetric with respect tothe y-axis.

• Does this transformation affect the domain of the function? In somecases it does. For example when the domain of the parent function isx≥0, the domain of the transformed function becomes all real numbers.

• Does this transformation affect the range of the function? In somecases it does. For example the range of y = x3 is all real numbers.The range of y = | x| 3is y≥0.

• Does this transformation affect the shape of the graph? If the graph ofthe parent function is symmetric with respect to the y-axis, then theshape of the transformed function is not affected. If the graph of theparent function is not symmetric with respect to the y-axis, then theshape of the transformed function is affected.

Talk participants through the Student Activities: Combinations ofTransformations and Transformations on Generic Graphs. Depending on thelevel of your participants, you may want them to complete the activities.

Activity 4: Summarize your findings

Have participants work in small groups to create a summary sheet on wallchart paper. Hang the summary sheets about the room for reference.A possible wall chart:

Summary:f x( ) + a results in a graph that is a vertical translation of f x( ) up (a>0)

or down (a<0).f x − a( ) results in a graph that is a horizontal translation of f x( ) right

(a>0) or left (a<0).af x( ) results in a graph that is a vertical stretch ( a > 1) or a vertical

shrink (0 < a <1) and a reflection about the x-axis (a<0).f ax( ) results in a graph that is a horizontal stretch (0 < a <1) or shrink

( a > 1) or a reflection about the y-axis (a<0).− f x( ) results in a graph that is a reflection about the x-axis.f −x( ) results in a graph that is a reflection about the y-axis.f x( ) results in a graph so that all points that have negative y-values in

the graph of f x( ) are reflected across the x-axis and all of the pointswith positive values remain the same.

f x( ) results in a graph such that f x( ) =f x( ), if x ≥ 0f −x( ), if x < 0

.



Talk participants through the Student Activities: Transformation Practice(Algebra II) and Transformation Practice (Precalculus). Have them comparethe two activities and discuss how the two are similar and different.

Answers to Reflect and Apply:

1. g x( ) = f x + 2( ) + 1, g x( ) =x + 2( )2

x + 2( )2 +1+1

2. g x( ) = − f x − 2( ) , g x( ) = −x − 2( )2

x − 2( ) 2 +1

3. g x( ) = 2 f x( ) − 1 , g x( ) =2x2

x2 +1−1

Focus on the TEKS:What Algebra II and Precalculus TEKS were introduced, reinforced, orextended in the activities? Justify your answer.In these activities participants explore and apply transformations,numerically, algebraically, and graphically (A-c.1.A-c.1.B, P-c.2.A, P-c.2.B).

How do these activities support subsequent courses?A solid understanding of parent functions and how transformations act onparent functions is a good basis for all of the work that students will do withfunctions in calculus.

Can you take this activity to the classroom? If so, how would you adapt it?All of these activities are appropriate for the classroom. To adapt theactivities for Algebra II students, cut the trigonometric functions and restrictthe transformations to those listed in the Algebra II TEKS. Have participantsdiscuss the similarities and differences between the teacher activities and thestudent activities.

Summary: The big idea here is to have participants explore the effects of transformationson a piecewise-defined function. By using small numbers and a few points,participants can concentrate on what happens to the function numerically andgraphically for various transformations. This activity also reinforces functionnotation. Then participants can use technology to quickly generate manyexamples, to explore ideas, and to make conjectures about transformations ofparent functions. They can then test their conjectures and performtransformations without technology. Participants learn the effects thatdifferent transformations have on the domain, range, and shape of functions

I. Foundations 1.1 Transformations: Activity 1


Activity 1: Explore Transformations

Given: Graph of f Table for ff −2( ) =1f −1( ) = −1f 0( ) = 2f 1( ) =1f 2( ) = 3f 3( ) = 0

Complete tables and sketch graphs of:

1. y = f x + 2( )x y-4-3-2-101



2. y = f x − 2( )x y012345

3. y = f x( ) +2x y-2-10123

4. y = f x( ) −2x y-2-10123



5. y = f 2x( )x y-1

-1/20

1/21

3/2

6. y = f 12x

x y-4-20246

7. y = 2 f x( )x y-2-10123



8. y = 12f x( )

x y-2-10123

9. y = f − x( )x y-3-2-1012

10. y = − f x( )x y-2-10123



11. y = f x( )x y-2-10123

12. y = f x( )x y-3-2-10123



Activity 2: Domains

1. Look back at Activity 1. Notice that input values, the x-values,change for some of the transformations. The function is defined forthe x-values: –2, –1, 0, 1, 2, 3. But, for example, Exercise 1, y = f x +2( ) , provides the x-values of –4, –3, –2, –1, 0, 1 andExercise 2, y = f x − 2( ) , provides the x-values of 0, 1, 2, 3, 4, 5.Why? Use an example in your explanation.

2. Design a simple function g that is similar to f. Show a table andgraph that would allow your students to fill in the table and graph thetransformation g x +3( ) .

Given: Graph of g Table for gg( ) =g( ) =g( ) =g( ) =g( ) =g( ) =

Complete the table and sketch the graph of:

y = g x +3( )x y



Activity 3: Transformations with Technology

I. Explore Vertical Translations

1. Enter into y2 : y2 = y1 +1Enter the following functions, one at a time, into y1 .Use a friendly window.Sketch the graph of y1 and y2 .

a. y1 = x b. y1 = ln x c. y1 = cos x d. y1 = −x + 2

2. Now enter y2 = y1 − 2

a. y1 = 0.5x b. y1 = sin x c. y1 = 2x d. y1 = − x

3. Generalize: What happens to the graph of a function when youadd a constant to the function rule?



II. Explore Horizontal Shifts

1. Enter into y2 : y2 = y1 x −1( )Enter the following functions, one at a time, into y1 .Use a friendly window.Sketch the graphs of y1 and y2 .

a. y1 = x b. y1= x2 c.

y1 =

1x

d. y1 = x +1

2. Now enter y2 = y1 x + 2( )

a. y1 = 9 − x2 b. y1 = sin x c. y1 = x3 d. y1 = x

3. Generalize: What happens to the graph of a function when youreplace x with x −a?



III. Explore Vertical Stretches and Compressions

1. Enter into y2 : y2 = 3y1Enter the following functions, one at a time, into y1 .Use a friendly window.Sketch the graphs of y1 and y2 .

a. y1 = e x b. y1 = x 3 c. y1 = cos x d. y1 = x

2. Now enter y2 =13y1 .

a. y1 = x b. y1 = sin x c. y1 = e x d. y1 = x

3. Generalize: What happens to the graph of a function when youmultiply the function rule by a?

I. Foundations 1.1 Transformations: Student Activity 1

TEXTEAMS Part I: Algebra II and Precalculus Institute 31DRAFT 3-29-03

Student Activity 1: Move the Monster Algebra IIOverview: Students sketch the graphs of transformations of a piece-wise defined

function.

Objective: Algebra II TEKS(b.1.A) For a variety of situations, the student identifies the mathematicaldomains and ranges and determines reasonable domain and range values forgiven situations.(b.1.B) In solving problems, the student collects data and records results,organizes the data, makes scatter plots, fits the curves to the appropriate parentfunction, interprets the results, and proceeds to model, predict, and makedecisions and critical judgments.(c.1.B) The student extends parent functions with parameters such as m in y = mx and describes parameter changes on the graph of parent functions.

Terms: translation, reflection, dilation

Materials:

Procedures: This activity is intended to reinforce students’ understanding after studentshave completed an activity similar to 1.1.1 Introducing Transformations. In1.1.1, students use a table to graph the transformations of a function. In thisActivity, students practice a more visual approach. Many students will “see”a table of values in their minds to help graph the transformations. The ideahere is to apply the “rules” they wrote in their summaries of transformations.Have students complete the activity and share graphing strategies.

• What is the domain of h ? .• What is the range of h ?• Compare the domain and range of f x( ) with the domains and ranges

of af x( ) , f x( ) + b , f ax( ) , f −x( ) , − f x( ) , f x + h( ) .

Summary: The big idea here is that students can use the general summary statements theymade about transformations to graph the transformations of a function withoutdoing it numerically first.



Activity: Move the Monster

Given h x( ):

Sketch the graph of:

1. −h x( ) 2. h −x( )



3. h x( )+ 2 4. h x( )− 2

5. h x + 2( ) 6. h x − 2( )

I. Foundations 1.1 Introducing Transformations: Student Activity Precalculus


7. 2h x( ) 8. 12h x( )

9. h 2x( )10. h 1

2x



Student Activity 2: Move the Monster PrecalculusOverview: Students sketch the graphs of transformations of a piece-wise defined

function.

Objective: Precalculus TEKS (c.2.A) The student is expected to apply basic transformations, includinga • f x( ), f x( ) + d, f x − c( ), f b • x( ), f x( ) , f x( ) , to the parent functions.(c.2.B) The student is expected to perform operations including compositionson functions, find inverses, and describe these procedures and results verbally,numerically, symbolically, and graphically.(c.3.c) The student is expected to use properties of functions to analyze andsolve problems and make predictions.


Materials:

Procedures: This activity is intended to reinforce students’ understanding after studentshave completed an activity similar to 1.1.1 Introducing Transformations. In1.1.1, students use a table to graph the transformations of a function. In thisActivity, students practice a more visual approach. Many students will “see”a table of values in their minds to help graph the transformations. The ideahere is to apply the “rules” they wrote in their summaries of transformations.Have students complete the activity and share graphing strategies.

• What is the domain of h ?• What is the range of h ?• Compare the domain and range of f x( ) with the domains and ranges

of af x( ) , f x( ) + b , f ax( ) , f −x( ) , − f x( ) , f x + h( ) .

Summary: The big idea here is that students can use the general summary statements theymade about transformations to graph the transformations of a function withoutdoing it numerically first.



Activity: Move the Monster

Given h x( ):

Sketch the graph of:

1. −h x( ) 2. h −x( )



3. h x( )+ 2 4. h x( )− 2

5. h x + 2( ) 6. h x − 2( )



7. 2h x( ) 8. 12h x( )

9. h 2x( )10. h 1

2x



11. h x( ) 12. h x( )

13. Design a function, f , so that f x( ) = f − x( )

14. Design a function, g , so that g −x( ) = −g x( )



Student Activity 3: Combinations of TransformationsOverview: Students sketch the graphs of function built from transformations of a parent

function.

Objective: Algebra II TEKS(b.1.A) For a variety of situations, the student identifies the mathematicaldomains and ranges and determines reasonable domain and range values forgiven situations. (c.1.B) The student extends parent functions with parameters such as m in y = mx and describes parameter changes on the graph of parent functions.


Materials: Graphing calculators

Procedures:Activity: Combinations of Transformations

A hint about “order of transformations”—there are rules similar to “order ofoperations”. When you are trying to sketch the graph of a new function, it isgenerally helpful to do any stretches, shrinks, or reflections before you do thehorizontal or vertical shifts. In the same sense that multiplication takesprecedence over addition, choose to do the stretches, shrinks, or flips first.

Solution: 1. Begin with the parent function y = x2. The transformations are:

Flip or reflect across the x-axis to get y = -x2.Vertical stretch by a factor of 2 (y-values are doubled) to get y = -2x2.Horizontal slide left 3 units to get y = -2(x + 3)2.Vertical slide down 1 unit to get y = -2(x + 3)2 – 1.The result will be a parabola that opens downward. It will be “narrower”than normal, since y-coordinates are doubled.

2. All points, including the vertex, will then be moved 3 units to the left anddown 1 unit. The vertex of this parabola will be (-3, -1). The domain ofthe function will still be all real numbers; the range becomes y ≤ -1.

4. A possible solution: Let f x( ) = x 2 , g x( ) = x + 3 , h x( ) = −2x , andj x( ) = x −1. Describe y = −2 x + 3( ) 2 −1 as a composition of functions.

Solution: y = −2 x + 3( ) 2 −1 can be expressed as y = j h f g( )( )( ) x( ) .



To understand better the “order of transformations,” use the graphingcalculator to do the steps listed in 1. A non-example and an example follow.Note that in the non-example, the vertical shift is done first and then thereflection across the x-axis. In the example the reflection is done first,followed by the vertical shift. If possible, use two overhead calculators toshow both the non-example and the example step by step at the same time.

Non-example (vertical shift before reflection) Example (reflection before vertical shift)



• Why does the order of transformations matter in this example? Show algebraically.Non-example example

y1 = x2

y2 = y1 x + 3( ) = x +3( )2

y3 = 2y2 = 2 x + 3( )2

y4 = y3 +1 = 2 x + 3( )2 +1 y4 = −y3 = −2 x + 3( )2

y5 = −y4 = − 2 x + 3( )2 +1( )= −2 x + 3( )2 −1

y5 = y4 +1 = −2 x + 3( ) 2 +1

Note that in the last step in the non-example, the –1 is multiplied throughthe entire expression producing a shift down instead of the correct shift up.

Summary: The big idea here is that students can combine transformations to graph thetransformations of a parent function but they must be done in a certain order.



Activity: Combinations of Transformations

1. List the transformations applied to y = x2 to get y = −2 x +3( )2 −1.

2. Describe the changes to domain and range.

3. Sketch the graphs of y = x2 and y = −2 x +3( )2 −1.

4. Describe y = −2 x +3( )2 −1 as a composition of functions.

5. Write your own function with at least 3 transformations. Sketch thegraph. Verify with a grapher.



Student Activity 4: Transformations on Generic GraphsOverview: Students sketch the graphs of transformations of a generic function.

Objective: Algebra II TEKS(b.1.A) For a variety of situations, the student identifies the mathematicaldomains and ranges and determines reasonable domain and range values forgiven situations. (c.1.B) The student extends parent functions with parameters such as m in y = mx and describes parameter changes on the graph of parent functions.


Materials:

Procedures:Activity: Transformations on Generic GraphsA good assessment of whether someone understands the notation and thedescriptions of these transformations is determined by using a “generic”graph. This can be any function, such as the one graphed below, where only agraph is given, with no analytical formula. Now the graphing calculatorcannot provide the answers for the student. The calculator has been used asan exploration tool; now the student is applying mathematical concepts.Our “generic” graph is defined as follows:



1. y = − f x( ) 2. y = f x + 2( )

3. y =12f x( )

− 3 4. y = f −x( )

5. y = f x( ) 6. y = 2 f x( )

Summary: The big idea here is that after students have learned the affects oftransformations on parent functions, they can apply this knowledge to findtransformations of a generic graph.



Activity: Transformations on Generic Graphs

Given: the generic graph shown below.

1. Sketch the graphs of the following transformations and describe thetransformations in words.

1. y = − f x( ) 2. y = f x + 2( )



3. y =12f x( )

−3

4. y = f −x( )

5. y = f x( ) 6. y = 2 f x( )


TEXTEAMS Part 1: Algebra II/Precalculus Institute 48DRAFT 3-29-03

Student Activity 5: Transformation Practice Algebra IIOverview: Students graph the effects of transformations of parent functions, using a

graphing calculator to check.

Objective: Algebra II TEKS(b.1.A) For a variety of situations, the student identifies the mathematicaldomains and ranges and determines reasonable domain and range values forgiven situations.(c.1.B) The student extends parent functions with parameters such as m in y=mxand describes parameter changes on the graph of parent functions.(d.2.B) The student uses the parent function to investigate, describe, and predictthe effects of changes in a, h, and k on the graphs of y=a(x-h)2+k form of afunction in applied and purely mathematical situations.

Terms: Parent functions, transformation, domain, range, slide, shift, stretch,reflection, symmetry

Materials: Graphing calculator

Procedures:Students are to:(a) predict what the graph of each of the following transformations will look

like and(b) identify the effects of the transformation on the domain and range of the

parent function.Then they are to use their graphing calculators to verify their predictions.

Function: y1 = x2 − 4x + 3

Viewing window: (–4.7, 4.7, 1, –2, 6, 1)Domain: all real numbersRange: y ≥ −1 or [−1, ∞)Vertex: (2, –1)


Viewing window: (–3, 6.4, 1, –4, 6, 1)Domain: all real numbersRange: y ≥ −2 or [−2, ∞)Vertex: (3, –2)

Function:

€

y = x + 4Viewing window: (–4.7, 4.7, 1, –2, 5, 1)Domain: x≥ –4 or [–4, ∞)Range: y≥0 or [0, ∞)



To check using a graphing calculator, you can use the following:

y = f x( ) − 3 Use y2 = y1 − 3y = f x − 3( ) Use y2 = y1 x − 3( )y = 2 f x( ) Use y2 = 2y1y = − f x( ) Use y2 = −y1

Answers for transformations to y1 = x2 − 4x + 3 .

Function Transformation Domain/Rangey=f(x)+2 graph is shifted up 2;

vertex becomes (2,1)range is affected;range becomes [1, ∞)

y=f(x+2) graph is shifted left 2;vertex becomes (0, -1)

neither is affected

y = 2 f x( ) vertex becomes (2,-2);graph is steeper

range is affected;range becomes [-2, ∞)

y = − f x( ) graph is reflected aboutthe x axis;vertex becomes (2, 1)

range is affected;range becomes (-∞, 1]


Function Transformation Domain/Rangey = f x( ) − 3 graph is shifted down 3;

vertex becomes (3, -5)range is affected;range becomes [-5, ∞)

y = f x − 3( ) graph is shifted right 3;vertex becomes (6, -2)

neither is affected



y = − f x( ) graph is reflected aboutthe x-axis;vertex becomes (3, 2)




Answers for transformations to

€

y = x + 4 .

Function Transformation Domain/Rangey = f x( ) − 3 graph is shifted down 3 domain is not affected;


y = f x − 3( ) graph is shifted right 3 domain is affected;domain becomes [-1, ∞)

y = 2 f x( ) graph is vertically stretched neither is affected;

y = − f x( ) graph is reflected aboutthe x-axis

domain is not affected;range is affected;range becomes (-∞, 0]

Summary: Students graph the effects of transformations of parent functions, using agraphing calculator to check.



Transformation Practice 1

Function: y = x2 − 4x + 3Window:Domain:

Range:

Vertex:

a. Predict the graph of each transformation.b. Describe the transformation in words.c. Identify effects on the domain and range.

1. y = f x( ) +2 2. y = f x + 2( )

a.

b.

c.

a.

b.

c.

3. y = 2 f x( ) 4. y = − f x( )

a.

b.

c.

a.

b.

c.




Function: y = x2 −6x + 7Window:Domain:

Range:

Vertex:


1. y = f x( ) −3 2. y = f x − 3( )

a.

b.

c.

a.

b.

c.

3. y = 2 f x( ) 4. y = − f x( )

a.

b.

c.

a.

b.

c.




Function:

€

y = x+ 4Window:Domain:

Range:


1. y = f x( ) −3 2. y = f x − 3( )

a.

b.

c.

a.

b.

c.

3. y = 2 f x( ) 4. y = − f x( )

a.

b.

c.

a.

b.

c.


TEXTEAMS Part 1: Algebra II/Precalculus Institute Draft 54DRAFT 3-29-03

Student Activity 6: Transformation Practice Precalculus

Overview: Students graph the effects of transformations of parent functions, using agraphing calculator to check.

Objective: Precalculus TEKS(c.1) The student defines functions, describes characteristics of functions, andtranslates among verbal, numerical, graphical, and symbolic representationsof functions, including polynomial, rational, radical, exponential, logarithmic,trigonometric, piecewise-defined functions.(c.2.A) The student is expected to apply basic transformations, includinga•f(x), f(x)+d, f(x-c), f(b•x), |f(x)|, f(|x|), to the parent functions.

Terms: Parent functions, transformation, domain, range, slide, shift, stretch,reflection, symmetry


Procedures:Students are to:(a) predict what the graph of each of the following transformations will look

like and(b) identify the effects of the transformation on the domain and range of the

parent function.Then they are to use their graphing calculators to verify their predictions.


Viewing window: (–4.7, 4.7, 1, –2, 6, 1)Domain: all real numbersRange: y ≥ −1 or [−1, ∞)Vertex: (2, –1)


Viewing window: (–3, 6.4, 1, –4, 6, 1)Domain: all real numbersRange: y ≥ −2 or [−2, ∞)Vertex: (3, –2)

Function: y1 = x3 − 4x

Viewing window: (–4.7, 4.7, 1, –6, 6, 1)Domain: all real numbersRange: all real numbersTurning points: approx. (–1.15, 3.08) (1.15, –3.08)



To check using a graphing calculator, you can use the following:

y = f x( ) − 3 Use y2 = y1 − 3y = f x − 3( ) Use y2 = y1 x − 3( )y = 2 f x( ) Use y2 = 2y1y = f 2x( ) Use y2 = y1 2x( )y = − f x( ) Use y2 = −y1y = f −x( ) Use y2 = y1 −x( )y = f x( ) Use y2 = abs y1( )y = f x( ) Use y2 = y1 abs x( )( )


Function Transformation Domain/Rangey=f(x)+2 graph is shifted up 2;

vertex becomes (2,1)range is affected;range becomes [1, ∞)

y=f(x+2) graph is shifted left 2;vertex becomes (0, -1)

neither is affected



y = f 2x( ) Vertex becomes (1, -1);graph is steeper

Neither is affected

y = − f x( ) graph is reflected aboutthe x axis;vertex becomes (2, 1)


y = f −x( ) graph is reflected aboutthe y axis;vertex becomes (-2, -1)

neither is affected

y = f x( ) the section of the originalgraph that was below thex axis is reflected aboutthe x axis

range is affected;range becomes [0, ∞)

y = f x( ) the section of the originalgraph that was to the rightof the y axis is kept andreflected about the y axis;the graph looks like an “w”

neither is affected





vertex becomes (3, -5)range is affected;range becomes [-5, ∞)

€

y = f (x − 3) graph is shifted right 3;vertex becomes (6, -2)

neither is affected

€

y = 2 f (x) vertex becomes (3,-4);graph is steeper


€

y = f (2x) vertex becomes (1.5, -2);graph is steeper

neither is affected

y = − f x( ) graph is reflected aboutthe x-axis;vertex becomes (3, 2)


y = f −x( ) graph is reflected aboutthe y-axis;vertex becomes (-3, -2)

neither is affected

y = f x( ) the section of the originalgraph that was below thex-axis is reflected aboutthe x-axis


y = f x( ) the section of the originalgraph that was to the rightof the y-axis is kept andreflected about the y-axis;the graph looks like an “w”

neither is affected



Answers for transformations to y1 = x3 − 4x .


turning points (-1.15, 0.08)and (1.15, -6.08)

neither is affected

y = f x − 3( ) graph is shifted right 3;turning points (1.85, 3.08)and (4.15, -3.08)

neither is affected

y = 2 f x( ) graph is vertically stretched;turning points (-1.15, 6.16)and (1.15, -6.16)

neither is affected;

y = f 2x( ) graph is horizontally shrunk;turning points (-0.575, 3.08)and (0.575, -3.08)

neither is affected

y = − f x( ) graph is reflected aboutthe x-axis;turning points (-1.15, -3.08)and (1.15, 3.08)

neither is affected

y = f −x( ) graph is reflected aboutthe y-axis; turning points(-1.15, -3.08) and (1.15, 3.08)

neither is affected

y = f x( ) the section of the originalgraph that was below thex-axis is reflected aboutthe x-axis


y = f x( ) the section of the originalgraph that was to the rightof the y-axis is kept andreflected about the y-axis

range is affected;range becomesapproximately [-3.08, ∞)

Summary: Students graph the effects of transformations of parent functions, using agraphing calculator to check.




Function: y = x2 − 4x + 3Window:Domain:

Range:

Vertex:


1. y = f x( ) +2 2. y = − f x( )

a.

b.

c.

a.

b.

c.

3. y = f x + 2( ) 4. y = f − x( )

a.

b.

c.

a.

b.

c.



5. y = 2 f x( ) 6. y = f x( )

a.

b.

c.

a.

b.

c.

7. y = f 2x( ) 8. y = f x( )

a.

b.

c.

a.

b.

c.




Function: y = x2 −6x + 7Window:Domain:

Range:

Vertex:


1. y = f x( ) −3 2. y = f x − 3( )

a.

b.

c.

a.

b.

c.

3. y = 2 f x( ) 4. y = f 2x( )

a.

b.

c.

a.

b.

c.



5. y = − f x( ) 6. y = f − x( )

a.

b.

c.

a.

b.

c.

7. y = f x( ) 8. y = f x( )

a.

b.

c.

a.

b.

c.




Function: y = x3 − 4xWindow:Domain:

Range:

Turning Points:


1. y = f x( ) −3 2. y = f x − 3( )

a.

b.

c.

a.

b.

c.3. y = 2 f x( ) 4. y = f 2x( )

a.

b.

c.

a.

b.

c.



5. y = − f x( ) 6. y = f − x( )

a.

b.

c.

a.

b.

c.

7. y = f x( ) 8. y = f x( )

a.

b.

c.

a.

b.

c.

I. Foundations 1.2 What's the Difference?: Trainer Notes

TEXTEAMS Part 1: Algebra II/Precalulus Institute 64DRAFT 3-29-03

What's the Difference?

Overview: Participants compare and contrast the Algebra II and PrecalculusTEKS.

Objectives: Participants will become aware of the TEKS for each course andclarify the similarities and differences in the two courses.

Materials: Copy of the TEKS for Algebra II and Precalculus for each participant.

Procedure: Ask participants to work in pairs or table groups on Section I of theactivity.After they have completed the chart, ask each group to share theircomments on one of the functions they listed. Use chart paper torecord their comments.The participants are divided into three groups. Assign Sections II, III,and IV. They should list a TEKS reference for each comment. Oneexample is given for each chart. Groups will share their comments ontheir charts.

To summarize the activity show the transparency with three questions.Ask tables to discuss the questions and share comments with the wholegroup.• Describe the difference in the organization of the TEKS in bothcourses.

In Algebra II, the TEKS are organized, for the most part, around aspecific function type. In Precalculus, although specific functions aredescribed, the organization is around how the functions are used.

• Describe the difference in language in the two courses regardingmodifications on parent functions.

In Alg II the TEKS describe "parameter changes" on the parentfunctions. In PC the TEKS describe "transformations to the parentfunctions". Although the language is different, the action is the same.Teachers should be aware of this language difference and be sure thatstudents are not seeing these two as different concepts. Although thelanguage of parameters is important, Alg II teachers should make theconnection for students that when they change certain parameters, thefunction is transformed in a certain way.

I. Foundations 1.2 What's the Difference?: Trainer Notes

TEXTEAMS Part 1: Algebra II/Precalulus Institute 65DRAFT 3-29-03

• List some broad common threads that exist throughout bothcourses.

Using functions to solve real world problems, connecting graphs withtheir parent functions, modeling the physical world.

I. Foundations 2.1 Applications: Trainers' Notes


2.1 ApplicationsOverview: Participants use their knowledge of transformations to determine appropriate

viewing windows on graphing calculators for transformed parent functions.

Objective: Algebra II TEKS(b.1.A) For a variety of situations, the student identifies the mathematicaldomains and ranges and determines reasonable domain and range values forgiven situations.(c.1.A) The student identifies and sketches graphs of parent functions, includinglinear, quadratic, square root, inverse (rational), exponential, and logarithmicfunctions.

Precalculus TEKS(c.1.A) The student is expected to describe parent functions symbolically andgraphically.(c.1.B) The student is expected to determine the domain and range of functionsusing graphs, tables, and symbols.

Terms: Viewing window, period, frequency


Procedures:Participants have just completed an extensive transformational look atfunctions. An additional way to experience the transformation approach tographing functions is to find good viewing windows for varioustransformations of parent functions. If you know the transformation, then youknow where to look for the window.

Activity 1: Good Viewing WindowsInstruct participants to work in pairs to find a “good” viewing window for thefunctions. Each pair should do at least two of the exercises 1 – 5. Ask somepairs to enter their windows into the overhead calculator for each of thefunctions and justify their window choices.

• What is the parent function in this exercise? In Exercise 3, the parentfunction is a cubic. Therefore, participants should find a viewingwindow that shows a cubic.

• How can you use what you know about transformations to help youfind a good viewing window? In Exercise 2, the parent function,y = x2 is reflected across the x-axis, is vertically stretched, and isshifted up 1.19. Look for an upside down parabola that for a smallchange in x, the y-values decrease quickly.



• What is a “good” viewing window? It shows the important features ofthe graph.

• Is there only one “good” viewing window? It may not be possible tofind one good viewing window that shows all of the importantfeatures; it may be necessary to have more than one window.

• What features of a function are important to show? x-, y- intercepts,characteristics of the parent function, end behavior

• What strategies did you use to help you find a good viewing window?Answers might include: tracing on the function to find some functionvalues even when the curve is off the screen, using the table feature tofind function values, estimating a few function values mentally.

Sample answers: Remember that many viewing windows will be “good.” Ifthe window includes intercepts, characteristics of the parent function, and endbehavior, it is probably acceptable. In the sample answers below, we includenegative window values in some places where they do not make sense in theproblem situation, but they facilitate seeing the relevant portion of the graphwhen tracing on the graph. The first answers below are for the algebraicmodel and the second answers are for the problem situation.1.

Here we chose positive values up to 100 for weight andlength of the whales.

2.

`



Here we chose positive values for velocity and distance.Note that these arteries must be pretty small.

3.

We chose positive values for volume. We chose sidelength values, 0 ≤ x ≤ 23 because after that, you have cutout more than half of the page for only one of the squares.In other words, you have run out of paper and the rest ofthe x-values do not make sense in the problem situation.

4.

We chose positive values for both time and amount ofmoney.

5.



We chose positive values for time, 0 ≤ x ≤ 0.008 , enoughto show two complete periods. We chose both positive andnegative y-values, −0.5 ≤ y ≤ 0.5 , enough to show theamplitude of the curve.

6.

a.

b.

c.

d.

e.

f. You need one window to see the long term behavior and a differentwindow to see the short term behavior.



g.

h.

i.

j.

k.

l.



The following window is NOT appropriate, but participants might trysomething like it. It is a strange effect of the calculator evaluating justthe right points out of many cycles of the curve and then connectingthem on the screen.

To help participants see that the above window is not appropriate,trace on the curve in the next (smaller) window and watch the cursorjump up and down as it traces points from different cycles of thecurve.

Activity 2: Friendly Viewing WindowsThis activity is a short introduction to friendly viewing windows. You maywant to determine ahead of time which graphing calculators your participantswill be using so you can be prepared with friendly windows for them.

1. The relation was graphed with two functions,y = 4 − x 2and y = − 4 − x2 .• Which window is friendlier? The second window.• Why? It makes the circle look like a circle and not an oval. Also,

unlike the first window which show the circle not connected, thesecond window shows a complete circle. The window has the correctratio between the width and height of the screen. This is importantbecause the screen on the graphing calculator is rectangular and,therefore, the window must be set to counteract that so that the graphis not incorrectly stretched. You want things that are round to lookround, etc.

2. The graphs show y1 = 2x + 5 and y2 = 8 in the same windows.



• Which window is friendlier? The middle one.• Why? Because the trace value in the middle window is the correct

solution. Tracing in this window gives “nice” values. It is possible toset windows so that the trace function picks off integers, even integers,decimals rounded to tenths, hundredths, etc. These friendly valueshave to do with the number of pixels on the graphing calculatorscreen. If you set a window that has a multiple or integer quotient ofthe number of pixels than you get these friendly values.

Do not spend too much time here. We will go into setting friendly windowsin depth in section IV. Rational Functions.

3. A friendly window can refer to a width-height ratio friendly window or atrace value friendly window. It shows the correct shape of a graph. It canalso show desired trace values in the trace function.

4a. Answers will vary depending on the graphing calculator being used. Seethe calculator manual to choose an integer quotient of the number of pixelsthat go across the screen. The graph should look like:

4b. Answers will vary depending on the graphing calculator being used. Seethe calculator manual to choose a multiple of the number of pixels that goacross the screen. The graph could look like:

Possible Answers to Reflect and Apply:

2.a. −10 ≤ x ≤ 10

−100 ≤ y ≤ 300b. −5 ≤ x ≤ −3

−100 ≤ y ≤ 200c. −20 ≤ x ≤ 20

−100 ≤ y ≤ 200d. −20 ≤ x ≤ −1

−100 ≤ y ≤ 200e. −3 ≤ x ≤10

−100 ≤ y ≤ 200



3. −10 ≤ x ≤ 4−10 ≤ y ≤ 21

Focus on the TEKS:What Algebra II and Precalculus TEKS were introduced, reinforced, orextended in the activities? Justify your answer.In these activities participants determine reasonable domains and ranges (A-b.1.A, P-c.1.B) , and identify and sketch various parent functions (A-c.1.A, P-c.1.A).

How do these activities support subsequent courses?A solid understanding of how transformations act on parent functions is agood basis for all of the work that students will do with functions in calculus.

Can you take this activity to the classroom? If so, how would you adapt it?This activity could be appropriate for the classroom if students had theprevious experiences with the activities exploring transformations. Anadaptation for Algebra II is to leave out the trigonometric functions.

Summary: The big idea here is that a good viewing window shows the important featuresof a graph, usually including the intercepts and end behavior. A friendlyviewing window shows the correct (not distorted) shape of the graph andallows for “nice” values when tracing. We use the exercise of finding viewingwindows to review the basic characteristics of parent functions and to practicedetermining the effects of transformations on those parent functions.

I. Foundations 2.1 Applications: Activity 1


Activity 1: Good Viewing Windows

Find a good viewing window for the graph of the function rule and finda good viewing window for the graph of the problem situation. Sketchthe graph and record the window:

1. A model for the relationship of the weight and length of a humpbackwhale is w = −42.958+1.705l , where l is the length in feet and w isthe weight in long tons. (A long ton is about 2240 pounds.)

2. The rate at which a blood cell flows depends on the distance of thecell from the center of the artery. Research has determined that amathematical model of this problem situation isv = 1.19− 1.85×104( )r 2 , where r is the distance (in cm) of the bloodcell from the center of the artery and v is the velocity (in cm persecond).



3. The function v = 2666x −210x 2 + 4x 3 represents the volume of acertain box that has been made by removing equal squares of sidelength x from each corner of a rectangular sheet of material and thenfolding up the sides.

4. If you deposit $1000 into a savings account earning 8% interestcompounded annually, the amount of money A you would have aftertime T is given by A =1000 1.08( )T .



5. A student found a model for the data collected from a tuning forkstruck in front of a microphone, y = 0.341cos 1643 x − 0.003( )( ). Finda viewing window that includes 2 complete periods. Can youapproximate the frequency of the tuning fork?



6. Find a good viewing window for the following functions. Sketch thefunction and record the window:

a. y = .005x − 0.02 b. y = 20∗2 x

c. y = −950x2 +950x − 600 d. y =100 x

e. y = − x −10 − 6 f. y = x3 −10x 2 +33x −15

g. y = 1x +12

h. y = ex



i. y = ln x −11( ) j. y = tan x

k. y = 0.02cos(0.2x) l. y = sin(50x)

7. Define a “good” viewing window.

8. Is there a “best” viewing window?

I. Foundations 2.2 Airlines: Trainers' Notes


2.2 AirlinesOverview: Participants use real data to graph a scatter plot and find trend lines. They use

transformations to explain aspects of the trend lines. Participants find a realapplication for horizontal shifts.

Objective: Algebra II TEKS(b.1) The student uses properties and attributes of functions and applies functionsto problem situation.(b.1.B) In solving problems, the student collects data and records results,organizes the data, makes scatter plots, fits the curves to the appropriate parentfunction, interprets the results, and proceeds to model, predict, and makedecisions and critical judgments.(c.1.C) The student recognizes inverse relationships between various functions.

Precalculus TEKS(c.2.B) The student is expected to perform operations including compositions onfunctions, find inverses, and describe these procedures and results verbally,numerically, symbolically, and graphically.(c.3) The student uses functions and their properties to model and solve real-lifeproblems.(c.3.C) The student is expected to use properties of functions to analyze andsolve problems and make predictions.

Terms: Scatter plot, trend line, independent variable, dependent variable


Procedures:Activity 1: Trend LinesInstruct participants to enter the data into their graphing calculators and createa scatter plot.1. Refrain from using regression to find a trend line. Encourage participant to

simply guess and check a trend line by estimating the y-intercept and theslope, graphing that estimated y = b + mx and then adjusting the functionrule until it is a reasonable trend line. This line is not a “line of best fit.”It is a trend line.



2.• What are the units of the slope? Miles per hour.• What does the slope mean? In our example above, the slope means

that on average, the planes are flying about 440 miles per hour.• What is the unit for the y-intercept? Miles• Why is the y-intercept negative? It does not make sense that at time =

0 , the airplane has traveled a negative distance. But consider the x-intercept. This is the time after which the airplane starts to fly andgain distance. Therefore, the negative y-intercept allows for the timespent not gaining distance, ie. time on the sitting on the tarmac, timespent circling in the air waiting for permission to land, etc.

3. Using our trend line above, y = −220 + 440x , we write this as atransformation of y = x as y = 440 x − 0.5( ) . This form shows that thetime spent not gaining any distance is about a half of an hour. What wasyour wait time last time you flew?

Activity 2: Another Trend Line1. To set up the scatter plot of the inverse relation, simply reverse the x- and

y-variables in the scatter plot set-up. Also exchange the x- and y- windowvalues.

2. In our example, the trend line with distance as the independent variable is

y =1440

x +

12

.

3. The two trend lines are inverses of each other.



4.• Now what are the units of slope and the y-intercept? The units for

slope are hours per mile. The units for the y-intercept are hours.• Which variable in the problem situation is dependent? Which is

independent? Does it make more sense to say that the distance you flydepends on the time it takes to get there (Activity 1) or to say that thetime it takes you to fly somewhere depends on the distance that placeis from where you are (Activity 2)?

• With distance as the independent variable, what does the y-intercepttell you? The y-intercept of 0.5 hours means that to fly nowhere it willtake you about one half of an hour. To fly anywhere, you will have toadd in about a half of an hour for wait time.

Focus on the TEKS:What Algebra II and Precalculus TEKS were introduced, reinforced, orextended in the activities? Justify your answer.In these activities participants makes scatter plots of time and distance datafrom airplane flights, fits curve to the data, interprets the results and makespredictions (A-b.1.B, P-c.3.C), and finds inverses (A-c.1.C, P-c.2.B)

How do these activities support subsequent courses?A solid understanding of linear functions, inverses and trend lines is a goodbasis for all of the work that students will do with other functions in calculus.

Can you take this activity to the classroom? If so, how would you adapt it?This activity could be appropriate for the classroom as a review of linearfunctions, stretching students by finding scatter plots, trend lines, and inverseswithout much adaptation.

Summary: The big idea here is that we can use real data to see an application of thehorizontal translation of a function. We also review the slope y-intercept formof a line by estimating trend lines and we review the notion of independenceand dependence. It is an introduction to our next Activity, which is on inversefunctions.

I. Foundations 2.2 Airlines: Activity 1


Activity 1: Trend Lines

The table below shows the United flying time and mileage from Chicagoto indicated cities.

City Time DistanceSt. Louis 1:10 258Phoenix 3:41 1440Salt Lake City 3:29 1249Dallas 2:20 802Los Angeles 4:20 1745Denver 2:27 901Minneapolis 1:20 334Kansas City 1:21 403Memphis 1:41 491

1. Make a scatter plot and find a trend line. Use time for theindependent variable.

2. What is the slope? What is the y-intercept? What do each mean?

3. Write the function as a transformation of the parent function y = x .What information do you note in this form?

I. Foundations 2.2 Airlines: Activity 2


Activity 2: Another Trend Line

Suppose you use distance as the independent variable in the Unitedflight distance-time situation.

1. Set up the scatter plot so that distance is the independent variable.

2. Write the trend line with distance as the independent variable.

3. How does this trend line relate to the trend line in Activity 1?

4. What information do you note when the trend line is written withdistance is the independent variable?

I. Foundations 2.2 Airlines: Reflect and Apply


Reflect and Apply

Using the internet, find your own airline flight time and distance data. Find a trendline and interpret your findings.

I. Foundations 3.1 Inverses: Trainers' Notes


3.1 InversesOverview: Participants use parametric equations to graph inverse relations and inverse

functions.

Objective: Algebra II TEKS(c.1) The student connects algebraic and geometric representations offunctions.(c.1.C) The student recognizes inverse relationships between variousfunctions.

Precalculus TEKS(c.2) The student interprets the meaning of the symbolic representations offunctions and operations on functions within a given context.(c.2.B) The student is expected to perform operations including compositionon functions, find inverses, and describe these procedures and results verbally,numerically, symbolically, and graphically.

Terms: Function, inverse relation, inverse function, one-to-one function, symmetry,reflection, parametric equations


Procedures: A relation is any ordered pair (a, b) in the coordinate plane. Traditionally, wespecify a relation using an equation or inequality in terms of x and y. There isanother way to specify a relation by employing two functions where the inputvariable for both functions is T, called a parameter, and the output values are xand y. The two equations are called parametric equations.Consider the following table of values to graph y = x2 .

x y-2 4-1 10 01 12 4

We can define the same function as:

€

xT = TyT = T 2

, where both x and y are defined

in terms of T.



T x y-2 -2 4-1 -1 10 0 01 1 12 2 4

We can use parametric graphing to graph the inverse of relations and explorethe properties of inverses. We can also get a feel for what the graph of theinverse of a relation looks like by graphing many inverses quickly with thepower of technology.

Activity 1: Parametric EquationsSee the appendix for a friendly window and for the calculator keys forspecific brands of calculators.

1. xT = TyT = T 2

2. See the appendix for a friendly window and for the calculator keys forspecific brands of calculators.

3.xT = T 2

yT = T or if

x1T = Ty1T = T 2

then x2 T = y1Ty2T = x1T

graphs the inverse.

4.

5.xT = TyT = T

.

After you have looked at this graph, change the calculator to graph insimultaneous mode. Point out that you can see the symmetry happen liveas the calculator graphs. Consider switching back and forth fromsimultaneous to sequential mode in the rest of the activity and askparticipants to note the advantages of each mode. Simultaneous modehelps you see the symmetry happen as it graphs. Sequential mode helps



you establish the graph of the original function, mentally estimate theinverse, and then check your mental estimation as the inverse is thengraphed.

6. Relation7. Yes, because we restricted the original function to be one-to-one by

restricting the domain. Therefore its inverse is now a function.

8. Yes, same as above.

Activity 2: Restricted DomainsEncourage participants to try to keep the range the same while restricting thedomain. This is not possible for Exercise 4.

1. Yes the inverse is a function. In fact it is the famous function,y = loge x = ln x .

2.



The inverse is not a function unless you restrict the domain. By definition,

the domain of y = arcsin x is − π2,π2

.

3. The inverse is a function.

4.

The inverse is not a function. One possible restricted domain to make theinverse a function is:

Focus on the TEKS:What Algebra II and Precalculus TEKS were introduced, reinforced, orextended in the activities? Justify your answer.In these activities participants find the different representations of inverses(A-c.1.C, P-c.2.B).

How do these activities support subsequent courses?Inverses will continue to be an area of study in precalculus and calculus.

Can you take this activity to the classroom? If so, how would you adapt it?This activity could be appropriate for the classroom. Students will need helprestricting domains to make the functions one-to-one as they won’t have theprior knowledge that teachers will have coming into the activity.




f 0( ) ≈ 2.5

f −1 0( ) ≈ 2f 3( ) ≈ −5

f −1 3( ) ≈ −1order: f 3( ), f − 1 3( ), 0, f − 1 0( ), f 0( )

Summary: The big idea here is to create inverse functions and relations using parametricequations. Participants gain intuition for symmetry over the line y = x byusing technology to graph an inverse relation simultaneously with the originalfunction. Using window settings, participants restrict domains to makeinverse relations into inverse functions, by forcing the original function to beone-to-one.

I. Foundations 3.1 Inverses: Activity 1

Algebra II and Precalculus Institute Draft 3-15-03 113

Activity 1: Parametric Equations

1. Use a graphing calculator to graph y = x2 in Parametric Mode. Writethe equations you use:

xT =yT =

2. Set a square friendly window and note it here:

3. How can you use parametric equations to graph the inverse of y = x2?

4. Use trace to verify that if (a,b) is on the first graph, (b, a) is on thereflection.

5. Add the equation y = x . How?

6. Is the inverse you graphed above, an inverse relation or an inversefunction?

7. Change Tmin = 0 and investigate the graph using trace. Is the inverse

now a function? Why or why not?

8. Go back to your friendly window but make Tmax = 0 and investigate.Is this inverse a function? Why or why not?

I. Foundations 3.1 Inverses: Activity 2


Activity 2: Restricted Domains

Graph each function in parametric mode. Then graph the inverserelation. Is the inverse relation a function? If not, how can you restrictthe domain of the original function to make its inverse relation aninverse function? Graph to confirm.

1. y = ex

2. y = sin x

3. y = x −2

4. y = x3 −2x

I. Foundations 3.1 Inverses: Reflect an d Apply


Reflect and Apply

1. Let f be defined by the following graph:

f

-3

-5

-5

3

Rank the following in order from least to greatest:0, f 0( ) , f −1 0( ) , 3, f 3( ) , f −1 3( )

I. Foundations 3.1 Inverses: Student Activity

TEXTEAMS part 1: Algebra II and Precalculus Institute 116Draft 3-15-03

Student Activity: MIRA Reflections

Overview: Students use a MIRA to investigate the properties of reflections and inverses.They distinguish between an inverse relation and an inverse function.

Objective: Algebra II TEKS(c.1) The student connects algebraic and geometric representations offunctions.(c.1.C) The student recognizes inverse relationships between variousfunctions.


Terms: MIRA, function, inverse, inverse relation, inverse function, domain, range

Materials: MIRAs, Graphing calculators

Procedures: Discuss briefly how to use the MIRA. Give transparencies of the Activities topairs of students and ask them to do each Activity with the MIRA live on theoverhead projector, explaining as they go. Instruct students to completeActivities 1 – 3 along with the presenting pairs. Use the questions below toelaborate on each activity.

Activity 1: MIRA Across the X-AxisPossible answers:2. y = −x2 or − f x( )3. Yes, the reflection is a function because for every x-value there is only one

y-value.4. For every x-value, the y-value of the reflection is the opposite of the y-

value of the function.• How do you know if the reflection is a function or a relation? For the

reflection to be a function, it must pass the vertical line test. For everyx-value, there is only one y-value.

• How did you construct your table?• Compare the domain and range of the original graph with its

reflection. The domains are the same. The ranges are exact oppositesof each other.

6. y = −x3 or − f x( ) .



7. Yes, the reflection is a function because for every x-value there is only oney-value.

8. For every x-value, the y-value of the reflection is the opposite of the y-value of the function.

Activity 2: MIRA Across the Y-AxisPossible answers:2. y = −x( )2 or f −x( )3. Yes, the reflection is a function because for every x-value there is only one

y-value.4. For the opposite of every x-value, the y-value of the reflection is the same

as the y-value of the function.• Does the reflected graph create a new graph? No the reflection is the

same as the original function.• Is y = x2 an even function, an odd function, or neither? Why?

y = x2 = f x( ) is an even function because f −x( ) = f x( ) .

• What are some other even functions? y = cos x , y = x , y = 1x2

6. y = −x3 or − f x( )7. Yes, the reflection is a function because for every x-value there is only one

y-value.8. For the opposite of every x-value, the y-value of the reflection is the same

as the y-value of the function.• Does the reflected graph create a new graph? Yes.• Is y = x3 an even function, an odd function, or neither? Why?

y = x3 = f x( ) is an odd function because f −x( ) = − f x( ) .

• What are some other odd functions? y = 1x

, y = sin x , y = tan x

• Compare the reflections of y = x3 over the x-axis and over the y-axis.What do you find? Why? They are the same graph.

Activity 3: InversesUse a transparency of the activity to identify several points on the graph andtheir reflections, e.g., (1, 2) becomes (2, 1) and (2, 4) becomes (4, 2).Emphasize that the point (a, b) on the original graph becomes (b, a) on theinverse graph.

• What do you call the reflection over the line y = x ? The inverserelation. It may or may not be a function.

2. The point (a, b) on the original graph becomes (b, a) on the inverse graph.3. No, the reflection is not a function. Each input generates two outputs.

For example, there are two y-values when x = 4 .4. You can restrict the domain of the original function to a one-to-one

function. Then its reflection will be a function.6. The point (a, b) on the original graph becomes (b, a) on the inverse graph.



7. Yes, the reflection is a function. Each input generates one output.8 – 9.

• Are the reflections inverses? Yes.• Are the reflections functions? Yes.• How do you know if the reflection is an inverse function or an inverse

relation? For a function’s inverse to be a function, the originalfunction must be one-to-one. Then its inverse will be a function.

• Compare the domain and range of the original with the reflection.They are switched. Just like finding the inverse of a function startswith switching x for y, the domain and range of a function and itsinverse are switched.

Answer to Assessment:

1. 2.

-5

3

-3

5

Summary: Using a MIRA, students get a visual representation of reflections across the x-and y-axis and the line y = x . Students also explore inverse reflections andwhen these inverses are functions.



Activity 1: MIRA Across the X-Axis

1. Graph y = x2 on the grid. Place the MIRA along the x-axis andsketch the reflection of y = x2 .

2. How could we algebraically describe the reflected graph?

3. Is the reflection a function? Why or why not?

4. Make a table that shows values for the original function andcorresponding values for the reflected function. Describe patterns.



Activity 2: MIRA Across the Y-Axis

1. Graph y = x2 on the grid. Place the MIRA along the y-axis andsketch the reflection of y = x2 .





Activity 3: Inverses

1. Graph y = x2 and y = x on the grid. Place the MIRA along the liney = x and sketch the reflection of y = x2 .





4. What could you do to make the inverse a function?



5. Graph y = x3 and y = x on the grid. Place the MIRA along the liney = x and sketch the reflection of y = x3 .





8. Graph y = x2 for x ≥ 0 and reflect over the line y = x using theMIRA.



9. Graph y = x2 for x ≤ 0 and reflect over the line y = x using theMIRA.



Assessment

1. Given the graph of f, sketch f −1. Check your sketch using the MIRA.

f

-3

5

3

-5

2. Given the graph of g, sketch g−1 . Check your sketch using the MIRA.

I. Foundations 3.2 Stays the Same: Trainers' Notes

TEXTEAMS Part 1: Algebra II and Precalculus Institute 128Draft 3-15-03

3.2 Stays the SameOverview: Participants use graphing calculators to model the solutions to equations

graphically and make connections between transformations of functions,inverse functions, and algebraic solutions.

Objective: Algebra II TEKS(b.2.A) The student uses tools including matrices, factoring, and properties ofexponents to simplify expressions and transform and solve equations.(c.1) The student connects algebraic and geometric representations offunctions.


Terms: Inverse function, transformation

Materials: Graphing calculators, big grid paper, markers

Procedures:

Activity: Solving Equations Step by Step GraphicallyTraditionally in Algebra II and Precalculus, students learn to solve equationsusing algebraic transformations. With technology, we can also show thatequations arise from the intersection of two functions and that the solution tothe equation is the intersection of the graphs of the functions. Moreover, thisis true for every algebraic equation-solving step. This activity is designed tohelp participants connect the algebraic and graphical solution of an equation ateach step. In the following problems, as we solve the equation algebraically,we will analyze the transformation on the graphs. At each step, encourageparticipants to predict the change in the graph and to focus on the intersectionpoints. Ask the question, “What stays the same?”As the groups report their work, point out that the transformations behave thesame no matter what the parent function is. Have the participants point outanything unique to their equation.



1.

2.

3.



4.

At this point, we have to reason to find the inverse of the greatest integerfunction. We now have the equation x + 4 =1 . Therefore, 1 ≤ x + 4 < 2and thus, −3 ≤ x < −2 .

5.




xx −1

+ x = 4x − 3x −1

x −1( )x

x −1+ x

= x −1( )4x − 3x −1

Notice the holes in both graphs where they are undefined at x =1 .

x + x −1( )x = 4x − 3Above is the step where the extraneous solution is introduced.

x2 − 4x + 3 = 0

Focus on the TEKS:What Algebra II and Precalculus TEKS were introduced, reinforced, orextended in the activities? Justify your answer.In these activities participants solve equations using transformation,algebraically and graphically (A-b.2.A, P-c.2.B).

How do these activities support subsequent courses?Connecting the graphical and symbolic representations of solving equationswill strengthen students’ ability to solve more complex problems inprecalculus and calculus.



Can you take this activity to the classroom? If so, how would you adapt it?This activity could be appropriate for the classroom. Students may need helpat the point of the solving when you must use the inverse function. You couldadapt the activity for Algebra II students by cutting the trigonometricproblem.

Summary: The big idea here is that each step in the algebraic equation solution has agraphical representation. For each step in the algebraic solution, a differentpair of functions is set equal. Although the functions are very different, theirintersections maintain a common x-value, the solution to all of the equations.

I. Foundations 3.2 Stays the Same: Activity 2


Activity: Solving Equations Step by Step Graphically

For each of the equations below, solve the equation algebraicallyshowing each step in the solution. Use your graphing calculator to grapheach side of the equation as a function and look at the intersection. stepOn a single large sheet of graph paper, create a graphical model of thesolution. Use a different color for each step. Mark the intersectionpoints. Use the large grid paper to summarize your finding by graphingeach step on the same axis.

1. 3cos x + 4( )+ 2 = 5

2. 3 1x +4

+ 2 = 5

3. 3e x+4( ) + 2 = 5.

4. 3[x + 4] + 2 = 5

5. x + 2 = x

I. Foundations 3.2 Stays the Same: Reflect and Apply


Reflect and Apply

1. Shown below are steps in the algebraic solution of the first equation. Grapheach of the equations and show where the extraneous solution appears.

xx −1

+ x = 4x − 3x −1

x − 1( ) xx −1

+ x

= x −1( ) 4x − 3x − 1

x + x − 1( )x = 4x − 3

x2 − 4 x + 3 = 0

x = 1, 3

II. Exponential & Logarithmic Functions Introductory Activity: Trainers' Notes


Linear and Exponential ModelsOverview: Participants fit two sets of median income data with linear models and make

predictions. Participants then examine what makes data exponential. Withthis knowledge, participants fit the income data sets with exponential modelsand make predictions. They compare the linear and exponential models.

Objective: Algebra II TEKS1b. In solving problems, the student collects data and records the results,organizes the data, makes scatterplots, fits the curves to the appropriate parentfunction, interprets the results, and proceeds to model, predict, and makedecisions and critical judgments.

Precalculus TEKS3. The student uses functions and their properties to model and solve real-lifeproblems.

Terms: Linear model, exponential model, quotients, median, percent increase


Procedures: Activity : Median IncomeBriefly discuss the data. Instruct participants to enter the data into theirgraphing calculators and create two scatter plots, men’s median income overtime and women’s median income over time.

1 a. Data is linear if it has a constant rate of change.b. Successive differences do not yield exactly a constant rate of change.Take an average of the differences to find an average rate of change. Anaverage of the differences for men’s income is $995 and for women’sincome is $751. The assumption here is to find first successivedifferences. Later in the institute we find second and third successivedifferences and discuss their implications.c. Using the starting income for the men of $9521 and an average rate ofchange of 995, a linear model is m x( ) = 9521 + 995x . Then we adjustedthe parameters to get about as many points on top of the scatter plot asunderneath the scatter plot. A guess and check trend line ism x( ) = 8750 +1100x . (A calculator regression is m x( ) = 9015 +1084x .)Using the starting income for the women of $5616 and an average rate ofchange of 751, a linear model is w x( ) = 5616 + 751x . Then we adjusted



the parameters to get about as many points on top of the scatter plot asunderneath the scatter plot. A guess and check trend line isw x( ) = 4000 + 850x . (A calculator regression is w x( ) = 803 + 4475x .)

d. Using our models we found predicted earnings for the year 2000,m 30( ) = 41,750 and w 30( ) = 29,500 .

2 a. Have participants fill in the second column by evaluating y = 3• 2 x .b. Have participants fill in the third column with differences. Again herewe assume that participants are finding first differences.• Is the data linear? No, consecutive differences are not constant.It is interesting to note here that the first differences for the exponentialfunction look just like the original function values. Check that the seconddifferences will also be the same. Higher differences for exponentialfunctions are cyclic. Hence, the derivative of ax is ax forever.c. Have participants fill in the fourth column with first consecutivequotients.• What is true about successive quotients for exponential data?

Quotients are constant.d. Compare the data in the table with the model y = 3• 2 x . Where do yousee the 3 from y = 3• 2 x in the table? The value at x=0 is 3.e. Compare the data in the table with the model y = 3• 2 x . Where do yousee the 2 from y = 3• 2 x in the table? The common quotient is 2.

x y = 3• 2 x Differences Quotients0 3 3 21 6 6 22 12 12 23 24 24 24 48 48 25 96

3a. The average quotient for the men’s income data is about 1.056. Theaverage quotient for the women’s income data is about 1.064.



b. Using the starting income for the men of $9521 and an average quotientof 1.056, an exponential model is m x( ) = 9521•1.056 x . A calculatorregression is m x( ) =10498 •1.058x .Using the starting income for the women of $5616 and an averagequotient of 1.064, an exponential model is w x( ) = 5616 •1.064 x . Acalculator regression is w x( ) = 5883•1.068x .

d. Using the calculator regression models we found predicted earningsfor the year 2000, m 30( ) = 57320 and w 30( ) = 42,512 . Compare these tothe predicted earnings with the linear models which were m 30( ) = 41,750and w 30( ) = 29,500 .

4 a.• Compare the men’s and women’s rates of change (slope). Which is

greater? The men’s income slope is higher.• What does that mean for the predictions using the linear models?

Using linear models, men’s incomes are increasing faster thanwomen’s. If the trend continues, women’s incomes will never catchmen’s incomes.

• Compare the quotients for the men’s and women’s incomes. Which isgreater? The quotient of the women’s income is higher.

• What does that mean for the predictions using the exponential models?Using exponential models, women’s incomes are increasing fasterthan men’s. If the trend continues, women’s incomes will eventuallycatch and surpass men’s incomes.

c. Answers will vary. Note that often the government reports data usingpercent increases and decreases.

5. [-2, 22] [-0.1, 1]



A linear model of the data is y = .56 + .0059x .

• What are the units of the slope? The ratio of men’s income to women’sincome per year.

• Based on this model, how long will it take for women’s medianincome to reach men’s median income? 76 years.

Focus on the TEKS:What TEKS (Algebra II/ Precalculus) were introduced, reinforced, orextended in the activities? Justify your answer.This activity reinforces the Algebra II TEKS (b.1.B) and Precalculus TEKS(c.3). Both of these TEKS deal with modeling real data and using the modelsto predict behavior.How do these activities support subsequent courses?A strength of this activity is to look at different mathematical models for thesame data and compare the two. In precalculus, students are expected todecide which model is best for a given situation. The questions in this activitybuild that skill in students.Can you take this activity to the classroom? If so, how would you adapt it?This activity could be taken to the classroom. In Algebra II, the teachershould use this a discussion for looking at similarities and differences inexponential functions and linear functions. In precalculus, the discussionshould be more on comparing the behavior of the graphs and the difference inthe predictions based on each model.

Summary: Linear models have constant differences. Exponential models have constantquotients. Often real data does not clearly exhibit constant differences orquotients and there can be different implications depending on the model youchoose.

II. Exponential & Logarithmic Functions Introductory Activity: Activity

TEXTEAMS Part 1: Algebra II and Precalculus Institute 139

Activity: Median Income

Who will win the income race?

Year Men Women1970 9521 56161971 10038 58721972 11148 63311973 12088 67911974 12786 73701975 13821 81171976 14732 87281977 15726 92571978 16882 101211979 18711 110711980 20297 121561981 21689 132591982 22857 144771983 23891 152921984 25497 161691985 26365 171241986 27335 176751987 28313 185311988 28180 185091989 29556 197521990 29987 205561991 30874 212721992 31408 22141

US Census, 1995



1. Build scatter plots with your calculator, setting the years 1970 – 1992to 0 – 22.a. What makes data linear?

b. Find successive differences.

c. Use the differences to find a reasonable linear model for eachscatter plot.

d. Use your models to predict the earnings in the year 2000.

2. Do the following to answer the question: What makes a functionexponential?

a. Fill in the table.x y = 3• 2x

012345

b. Find successive differences in the third column.c. Find successive quotients in the fourth column.

d. Where does the 3 in y = 3• 2x appear in the table?

e. Where does the 2 from y = 3• 2x appear in the table?



3. Turn off your linear equations for the income data. Do the following.a. Take successive quotients and find the percent increase for each

year.

b. Use the quotients to find reasonable exponential models.

c. Use your models to predict the earnings in the year 2000.

4. Compare the two models: linear and exponential.

a. Using the linear model, will women’s earnings ever catch up withmen’s? Why or why not?

b. Using the exponential models, will women’s earnings ever catchup with men’s? Why or why not?

c. Which do you think makes more sense? Why?

5. Find the percentage of women’s median income to men’s medianincome. Create a scatter plot of this percentage versus years. Basedon this graph, will women’s earnings ever catch up with men’s? Whyor why not?

II. Exponential & Logarithmic Functions 1.1 Bacteria Growth: Trainers’ Notes

TEXTEAMS Part 1: Algebra and Precalculus Institue 142DRAFT 3-29-03

1.1 Bacteria GrowthOverview: In the following biology/mathematics activities, participants are given the task of

exploring bacteria growth as a model of population growth in general. They aregiven real data and are asked to develop a mathematical model that best describes(fits) the data.

Objective: Algebra II TEKS(b.1) The student uses properties and attributes of functions and applies functionsto problem situations.(b.1.f) The student formulates equations and inequalities based on exponentialand logarithmic functions, uses a variety of methods to solve them, and analyzesthe solutions in terms of the situation.

Precalculus TEKS(c.3) The student uses functions and their properties to model and solve real-lifeproblems.

Terms: Exponential function, exponential equation, logarithmic equation,mathematical model

Materials: Graphing calculators, large grid paper, one meter stick

Procedures: Have participants work in groups of 3 to 4 for these activities.

Activity 1: E. coli Growth Rate• How might a scientist measure the number of bacteria in a sample?

Use the following in you discussion.To monitor population growth of bacteria, biologists could examine asmall sample under a powerful microscope and count the populationnumber at different times. This approach, however, takes a great dealof time and is not particularly accurate.A more expedient and accurate approach takes advantage of the factthat as bacteria multiply, they increasingly cloud the solution in whichthey are growing. The cloudiness occurs as a result of the bacteriaabsorbing (or scattering light). The more bacteria, the more that lightis absorbed (or scattered), and hence the cloudier the solutionbecomes.

An instrument, known as a spectrophotometer, measures the amount of lightthat is absorbed by a solution. Absorbance (or optical density) is directlyrelated to the number of light-absorbing cells. The optical density isproportional to the number of cells per unit of solution. This implies that the



optical density (concentration of bacteria in a solution) is equal to a constanttimes the number of cells per unit.

In an experiment, measurements were taken with a Klett-Summersoncolorimeter, which is a simple type of spectrophotometer that measures theoptical density of a culture. Use the transparency of the table in Activity 3 todiscuss the experiment as follows.Two different nutrient solutions were used to grow E. coli bacteria cultures:1. LB, a standard nutrient broth and2. Superbroth, an enriched nutrient broth.

Four flasks, two containing 30 milliliters of LB solution and two containing30 milliliters of Superbroth solution, were each inoculated with 0.3 millilitersof an overnight culture of E. coli bacteria.

Two temperature conditions were used:1. 30˚ C (between room temperature and body temperature)2. 37˚ C (body temperature).

The data table gives Klett readings and number of cells per milliliter asfunctions of elapsed time in minutes for the following 4 situations:1. LB Broth at 30˚ C2. LB Broth at 37˚ C3. Superbroth at 30˚ C4. Superbroth at 37˚ C.

A Klett reading of 1 corresponds to 5 × 106 cells/milliliter.A Klett reading of 10 corresponds to 5 × 107 cells/milliliter.A Klett reading of 100 corresponds to 5 × 108 cells/milliliter.Have participants complete Activity 3 by investigating the growth patterns ineach of the four given situations using graphing utilities and determiningmathematical models for each situation. To save time, assign each group onesituation and then have them present their findings to the whole group.

• What is the constant of proportionality? 5 × 106

• If the optical density were 20, then how many bacteria would there beper milliliter? 20 × 5 × 106 cells/ml or 108 cells/ml

• If the optical density were 57, then how many bacteria would there beper milliliter? 57 × 5 × 106 cells/ml or 2.85 × 108 cells/ml

• What type of function relationship exists between Klett readings andthe corresponding number of bacteria per milliliter? linear orproportional



Each group should:1. use a graphing calculator to make a scatter plot of the E. coli growth data

assigned to their group. [independent variable = time, dependent variable= Klett value]

2. choose a viewing window that will display complete plots.3. make a hard copy of the plot on large grid paper and on the Activity Sheet.4. find an exponential model for the data.5. graph the exponential function on the hard copy.6. repeat the process for one of the other situations if time permits.

A possible calculator window [-10, 250] [-10, 70].

Regression Equations:a. y = 3.558 1.0125( ) x

b. y = 2.308 1.0189( )x

c. y = 2.246 1.0141( ) x

d. y = 2.246 1.0141( ) x

Use the following to help develop a good sense of the size of one bacterium andhow many bacteria it would take to fill a small container, such as a sewingthimble.

• How many bacteria will there be after 6 hours? After graphing theexponential models, you can use the calculator’s table of values todetermine the Klett values. You could also store 360 in x and have thecalculator compute the y values. When x=360 minutes, then the Klettvalues are:

y1≈317.8, y2≈1967, y3≈343.3, y4≈5172.1.The number of bacteria cells is determined by multiplying the Klettvalue by 5 × 10 6. The number of bacteria is:

y1=1,589,000,000; y2=9,835,000,000; y3=1,716,650,000;y4=25,860,500,000

• How many bacteria will there be after 24 hours? When x=1440minutes, then the Klett readings are:

y1=2.27×108, y2=1.2×1012, y3=1.23×109, y4=6.3×101 3.The number of bacteria is:

y1=1.135×1015, y2=6.0×1018, y3=6.13×1015, y4=3.15×1020

• About how long does it take your bacteria culture to double in size?For example in the 30oC LB culture, it takes approximately 55 minutesfor the number of cells to double from 3 × 107 to 6 × 107. Time elapsedwhen the number of cells was 30,000,000 was 42 minutes. Timeelapsed when the number of cells had doubled to 60,000,000 wasapproximately 97 minutes. In the table below, the doubling andtripling times are summarized when (a) the number of cells is doubled



from 30,000,000 to 60,000,000 and (b) the number of cells is tripledfrom 30,000,000 to 90,000,000.In summary, the doubling and tripling rates are much faster in theSuperbroth culture than in the LB culture. In addition, the rate ofgrowth appears to increase as the temperature increases.

Use the transparency of Activity 2 with the table of "Doubling and TriplingTimes" to organize the participants' data.

• About how long does it take your bacteria culture to triple in size? Forexample in the 30oC LB culture, it took approximately 88 minutes forthe number of cells to triple from 3 × 107 to 9 × 10 7. Time elapsedwhen the number of cells was 30,000,000 was 42 minutes. Timeelapsed when the number of cells had tripled to 90,000,000 wasapproximately 130 minutes. See chart.

30°C LB 37°C LB 30°C Super 37°C Super3 x 107

cells42 minutes 51 minutes 70 minutes 46 minutes

6 x 107

cells97 minutes 88 minutes 120 minutes 78 minutes

9 x 107

cells130minutes

110minutes

149 minutes 97 minutes

doublingtime

55 minutes 37 minutes 50 minutes 32 minutes

triplingtime

88 minutes 59 minutes 79 minutes 51 minutes

Extensions1. Use Census Bureau data to investigate human population growth patterns.2. Precalculus students could investigate the exponential model as a semi-log plot.

Activity 2: The Flu Epidemic: A Logistic Growth ModelWhen students begin to study exponential growth and they see a model for therapid growth of, say, E. coli bacteria, they may wonder, “If one E. coli cellcan produce one billion cells in just 10 hours, why haven’t these littlecreatures taken over the earth?”Investigate a possible answer to this question using the following Activity.

Activity 4 was created to simulate the introduction of the flu virus into aclosed environment by means of a single infected individual. As a wholegroup, do the first 4 days to demonstrate the procedure. Using the exampledata, the table would look like the following.



Day Number of initiallyinfected people

Number of newlyinfected people

Total number ofinfected people

1 1 1 22 2 2 43 4 3 74 7

The simulation will probably produce duplications. For example, two peoplemay both infect the same person. Another possibility is that one person maycome in contact with an already infected person. Also, an infected personmay generate his or her own number. These duplications and repetitions are adesired aspect of the simulation, as they cause the exponential-growth stage tochange into the dampened-growth stage.Have participants work in groups of 4 to complete the data collection for Days1 through 10.

Next, have them generate a scatter plot, using number of days as theindependent variable and total number of people infected with the flu virus asthe dependent variable. This graph will produce a good visual model of thelogistic growth curve.Show the Transparency 3: Logistic Growth Model and discuss thestages below.

Initial-Growth Stage. As the infected person comes in contact with otherpeople, these infected people start to slowly spread the virus, and we canmeasure the spread of the virus by the total number of infected individuals.

Exponential-Growth Stage. For a while, the infection rate grows swiftlybecause during the beginning stages, very few people of the total populationhave yet been infected. The high ratio of people who do not have the diseaseto people who do have the disease produces the stage of rapid exponentialgrowth.

Dampened-Growth Stage. As more and more people become sick, thenumber of possible contacts with new, uninfected hosts decreases. Thisdensity-dependent factor reduces the rate of infection less than exponential.As the ratio of non-infected to infected persons becomes smaller, the chanceof a contact between non-infected and infected individuals becomes smaller.

Equilibrium-Growth Stage. The number of infected people, although stillgrowing, is growing at an ever-decreasing rate. Finally, the curve levels out(approaches a limit) and the infection rate reaches equilibrium.

• What is the limit for this function? 100• What were the environmental conditions that limited the spread of the

flu virus in this simulation? The experiment was run in a closed



environment. There were only 100 people that could be infected. Thevirus has almost no one left to infect, producing a steady-statecondition.

Have them use their calculators to determine a logistic model (if theircalculators have that capability).

If participants are not using calculators that can determine a logistic curve-of-best-fit, tell them that logistic growth patterns can be modeled by a certaintype of equation, one that has the general form

Y =c

1 + a ⋅ e − bx( ) .

For Algebra II classes that have not been introduced to the transcendentalnumber e, a comparable equation can be used that is in the form

Y =c

1 + a ⋅ 2 −bx( ) .

Changing the base from approximately 2.7 to 2 still produces the desiredserpentine nature of the curve. The value of b is used to compensate for thechange in the base.

The data for this experiment can be approximated by the logistic equation

Y =100

1 + 99 ⋅e −7 x( ) or Y =100

1+ 99 ⋅ 2 − .97 x( )

This activity was created to simulate the introduction of the flu virus into aclosed environment by means of a single infected individual. The spread ofthe flu virus is exponential in the beginning, but the pace is dampened as moreand more of the population becomes infected.

Logistic Growth Curve. A representation of this scenario is given by anequation of the form

y = c1+ a e− bx

• List possible reasons for why E. coli or other bacteria and viruses havenot grown without bound. When conditions are ideal, bacteria, suchas E. coli, reproduce exponentially. A realistic environment isrestrictive—scarce food supply or other environmental conditions slowdown or halt cell division.



Mathematics Note:There are four stages of the Logistic Growth Model.1. initial-growth stage2. exponential-growth stage3. dampened-growth stage4. equilibrium-growth stage

The initial-growth stage shows a small but steady increase in numbers.Then, once the specimen adapts to the environmental conditions, itspopulation grows very rapidly. This growth often produces a doubling type ofeffect that is exponential. The third stage is a dampening of growth due tolimiting factors, such as competition for nutrients and safety; predation on theorganism; overcrowding, which can sometimes cause members of a species toeliminate their own kind. Because of these and other conditions, thepopulation of the organism eventually reaches the equilibrium stage. Asteady state in size, known as the carrying capacity of the organism in a closedenvironment, results. This situation does not mean that the size of thepopulation declines. Rather, the average birth and death rate become equal.The growth in the number of organisms reaches a limit.

Logistic Growth Curve. The graphical representation is a serpentine-shapedcurve that starts out with a slow increase; changes into an exponential-growthcurve, as seen by its upward concavity; then starts to slow down, noted by itsdownward concavity, as it approaches the carrying capacity of theenvironment. The carrying capacity is represented by the limit of thisfunction. The point, at which the concavity changes, is called a point ofinflection.

Answers to Reflect and Apply The equation below represents E. coli growth data under realistic conditions,where y represents the Klett Value and x represents the elapsed time inminutes.

y = 215.031+ 248.52e−.03 x( )

1. Using a graphing calculator, sketch a graph of the Klett Value versus Elapsed

Time. Describe the plot.



The graph is always positive and increasing. It has the same behavior as theparent function y=ex.

2. Describe the bacteria growth, based on the plot. Compare to the previous Klettvalue plots.This new graph increases at a faster rate



Focus on the TEKS:What TEKS (Algebra II/ Precalculus) were introduced, reinforced, orextended in the activities? Justify your answer.These activities reinforce the use of exponential functions to model real life asexpressed in Algebra II TEKS (b.1.f). It also follows the spirit of the AlgebraII and Precalculus TEKS in that we analyze real data, create scatter plots, fitfunctions to the data and use our mathematical models to predict futurebehavior.How do these activities support subsequent courses?The modeling context of the problem gives students experience with real data.In the Flu Epidemic problem, students are exposed to setting up a simulationas is done in Statistics.Can you take this activity to the classroom? If so, how would you adapt it?Both of these activities could be taken to the classroom. The teacher wouldneed to lead students through the development of the model in question 3 ofActivity 1, depending on the previous experiences of the class. The FluEpidemic is a nice simulation for the classroom. The calculations becomesomewhat tedious towards the end, but the model requires that data. Studentsmay need some assistance gathering the last pieces of data.

Summary: The big idea of these activities is to give participants experience with variousgrowth situations that can be modeled with exponential models. Participantsbegin with a simple model of exponential growth. Although many growthmodels look as though they grow exponentially, in reality they may level off.Logistic curves may represent a more realistic model.

II. Exponential & Logarithmic Functions 1.1 Bacteria Growth: Activity 1


Activity 1: E. coli Growth Data

LB Cultures Superbroth CulturesTemp = 30° C Temp = 37° C Temp = 30° C Temp = 37° C

Elapsedtime(min)

Klettvalue

Numberof

cells/ml

Klettvalue

Numberof

cells/ml

Klettvalue

Numberof

cells/ml

Klettvalue

Numberof

cells/ml

205080110130165180200230

567172230354057

2.50 x 107

3.00 x 107

3.50 x 107

8.50 x 108

1.10 x 108

1.50 x 108

1.75 x 108

2.00 x 108

2.85 x 108

26112737627587

106

1.00 x 107

3.00 x 107

5.50 x 107

1.35 x 108

1.85 x 108

5.10 x 108

3.75 x 108

4.35 x 108

5.30 x 108

277101620273853

1.00 x 107

3.50 x 107

3.50 x 107

5.00 x 107

8.00 x 107

1.00 x 108

1.35 x 108

1.90 x 108

2.65 x 108

2715305387

125163188

1.00 x 107

3.50 x 107

7.50 x 107

1.50 x 108

2.65 x 108

4.35 x 108

6.25 x 108

8.15 x 108

9.40 x 108

LB Culture(Standard Nutrient Broth)

Superbroth(Enriched Nutrient Broth)

Klett 1 = 5 x 106 cells/ml Klett 10 = 5 x 107 cells/ml Klett 100 = 5 x 108 cells/ml

1. Set up a scatter plot for the E. coli Growth Data using Klett values:a. LB broth at 30°C versus elapsed timeb. LB broth at 37°C versus elapsed timec. Superbroth at 30°C versus elapsed timed. Superbroth at 37°C versus elapsed time.

2. Sketch your plot(s) on large grid paper and on the grid on the next page.Label your graphs.

3. Find a model for your data. Sketch the model for each plot on the largegrid paper and on the next page. Label the model with the equation.



4. Pose two good questions that can be answered using your graphs, tables,or models and answer your questions. What interpretations orpredictions can you make using your graphs, tables, or data?

5. Graph the data and a model for the data. Label. Be creative.

II. Exponential & Logarithmic Functions 1.1 Bacteria Growth: Transparency


Transparency 3Logistic Growth Model

Four Distinct Stages:1. Initial-growth stage2. Exponential-growth stage3. Dampened-growth stage4. Equilibrium-growth stage



Activity 2: The Flu Epidemic: A Logistic Growth Model

This activity will simulate the introduction of the flu virus into a closedenvironment by means of a single infected individual.

For the simulation, imagine a total population of 100 individuals. Eachnumber from 0 through 99 represents an individual, with the number 0used to portray the original host. Record the interactions betweeninfected/non-infected individuals in the table. Use the Hundreds Chartto keep track of the infected individuals by crossing off their number onthe list as they become infected.

Day 1. The original host infects a person represented by a randomlygenerated number. Generate a random integer between (and including)0 and 99 using your graphing calculator.For example, suppose the calculator generates the number {35}. Crossout 0 and 35 on the Hundreds Chart indicating that two people (number0and number35) now have the virus. Record in the chart:

Day Number of initiallyinfected

Number of newlyinfected


1 1 1 2

Day 2. The two infected people from day 1 will now infect two people,so generate two random integers.For example, suppose the numbers generated were {3, 18}.The Hundreds Chart will appear as follows:

0, 1, 2, 3, 4, 5, ….., 17, 18, 19, ….., 34, 35, 36, ….., 98,

Day Number of initiallyinfected

Number of newlyinfected


1 1 1 22 2 2 4



Continue with the rest of the days.



Hundreds Chart

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29

30 31 32 33 34 35 36 37 38 39

40 41 42 43 44 45 46 47 48 49

50 51 52 53 54 55 56 57 58 59

60 61 62 63 64 65 66 67 68 69

70 71 72 73 74 75 76 77 78 79

80 81 82 83 84 85 86 87 88 89

90 91 92 93 94 95 96 97 98 99

Data Collection Summary

Day Number ofinitially

infected people

Number ofnewly infected

people

Total numberof infected

people0123456789



Charting the Spread of the Flu VirusGraph the function that best models the flu virus data. Determine acurve of best fit.

II. Exponential & Logarithmic Functions 1.1 Bacteria Growth: Reflect and Apply


Reflect and Apply

The equation below represents E. coli growth data under realistic conditions, wherey represents the Klet Value and x represents the elapsed time in minutes.

y = 215.031+ 248.52e−.03 x( )

3. Using a graphing calculator, sketch a graph of the Klett Value versus Elapsed

Time. Describe the plot.

4. Describe the bacteria growth, based on the plot. Compare to the previous Klettvalue plots.

II. Exponential & Logarithmic Functions 1.1 Bacteria Growth: Student Activity


Student Activity 1: Recursion and Bacteria Cell Division

Overview: In this activity, students explore a model of cell division using recursion.The students then use exponential functions to model the cell division andanswer questions using their mathematical model.

Objective: Algebra II TEKS(f.4) The student solves exponential and logarithmic equations andinequalities using graphs, tables, and algebraic methods.(f.5) The student analyzes a situation modeled by an exponential function,formulates and equation or inequality, and solves the problem.


Terms: Mathematical model, repeated addition, repeated multiplication, linear model,exponential model.


Procedures: Begin a discussion about E. coli with the whole group.• What are E. coli and where have you heard of them? E. coli, short for

Escherichia coli, is one of the most thoroughly studied of all organisms.Normally, this organism lives in the intestinal tract of humans and aids thedigestive system. Only certain types of E. coli cause problems for thehuman host.

Show Transparency 1: Algebra II Saves Biology Student. Allow time forgroups to consider the questions on Activity 1. Debrief with the whole group.

1. E. coli bacteria are very small – 1 micrometer in length. The followingdiscussion about the size of a micrometer might help to put its size inperspective. You may want to use a meter stick during this part of thediscussion to help participants visualize how tiny a micrometer is.

One micrometer = 10-6 meter or 11,000,000

of a meter. A millimeter is

10-3 meter or 11,000

of a meter. A micrometer, therefore, is 11,000

of a

millimeter. Another way of looking at this situation is as follows: if 1micrometer = 10-6 meters, then 1 meter = ? micrometers. (We wantparticipants as well as students to understand relationships in both



directions. Do students understand that if 1 micrometer = 10-6

or 11,000,000

of a meter, then 1 meter = 106 or 1,000,000 micrometers?)

2. How many micrometers is a five-foot tall person?

Several approaches are shared below.

(1) Using a meter stick, we find that five feet = 60 inches ≈ 152 cm = 1.52m. Knowing that a five-foot person is approximately 1.52 meters tall andthere are 1,000,000 micrometers for each meter, we should expect theperson’s height to be approximately 1.52 million micrometers.

(2) Using dimensional analysis, we get:5 feet × 12 in.

1 ft. × 2.54 cm

1 in. × 1 meter

100 cm = 1.524 m

Because a meter is equal to 1,000,000 micrometers, we can compute afive-foot tall person’s height in micrometers.1.524 m ×

1,000,000 micrometers1 meter

=1,524,000 micrometers

Hence we can conclude that the height of a 5-foot tall person isapproximately equal to the length of 1,524,000 E. coli laid end-to-end.

3. Suppose each E. coli cell divides once a minute and we start with 1bacterium.

• How fast can E. coli grow? Cells of E. coli reproduce by simply splittingin half, so that one cell forms two “daughter cells.” This form of celldivision is called fission. Fission occurs at a rate that depends on thenutrients and conditions (e.g., temperature) that are available. Whenconditions are ideal, E. coli can reproduce very rapidly. On the otherhand, in a restrictive environment, one in which the food supply is scarceor conditions are poor in other ways, cell division may slow down or stopaltogether.



a. Have participants build a table of values like the table below.

NumberMinutes

Cell Division Number E. coli

0 o 11 o o 22 o o o o 43 o o o o o o o o 84 o o o o o o o o o o o o o o o o 16

b. Participants use the recursive feature the home screen of a graphingcalculator to model the E. coli growth.To do this, ask participants to analyze the table above and decide whatoperation can be performed on a value of the number of E. coli to give thenext value. Now, in the home screen of the calculator, enter the initialnumber of E. coli, 1. Then type the operation (for example +2 or *3 orwhatever they decided). When you hit the enter key, the calculatorperforms the operation on the previous entry. Continue to hit the enter keyand see the values for the number of E. coli.

c. Participants create a scatter plot of the data and determine amathematical model that accurately describes this set of data. Participantsmay recognize this table of values as one that satisfies y = 2 x . Others maychoose to use the calculator’s curve-of-best-fit feature to determine themathematical model. Refer back to the Introductory Activity to see thatthe model is:the number of E. coli is the starting amount times the common quotientraised to the time ornumber is starting amount (1) times quotient, 2, raised to time ory = 1 ⋅2 x .

For Algebra II students, a teacher may provide several functions for thestudents to test in order to find a model. For example, the teacher might

provide y = 2x , y = 12x , y = x2 , y = x

12 , y = 2 x , y =

12

x

as possible

choices and allow the students to experiment with the given functions todetermine the best choice.



d. Remind participants about the Table feature or the Calculate Valuefeature of the calculator to determine how many cells there will be after 1hour. Have participants answer the question and describe the method theyused. After 1 hour, there will be approximately 1.15 × 1018 or1,152,921,505,000,000,000 E. coli cells. After 2 hours, there will beapproximately 1.33 × 1036 or 1,329,227,996,000,000,000,000,000,000,000,000,000 E. coli cells.

• Describe in words how the E. coli bacteria in this situation grow?They double every minute.

• Does the value of the number of E. coli increase by repeated additionor repeated multiplication? Repeated multiplication.

• What do we call this type of procedure? Recursion.• What kind of model do we get for the bacteria growth? Exponential.• Would we always get this type of model if growth is modeled by

repeated multiplication? Yes.• What would happen if we looked at a decaying situation modeled by

“repeated division”? Repeated division is the same thing as repeatedmultiplication by a fraction between 0 and 1. In other words, repeateddivision by 2 is the same thing as repeated multiplication by 1

2. We

would still have an exponential model, but the function would be

decreasing. For example, y =12

x

= 2 − x .

e. This question allows us to ask when does y = 1,000,000,000 or when does2x= 1,000,000,000. Use Transparency 2 to help participants see how theneed for equations follows naturally from the study of functions. The solutionis 29.897 minutes or approximately 30 minutes.

Possible Solution Strategies:

(1) Guess and check in the home screen. Guess a value for x that would make2 x =109 .

(2) Use the Calculate Intersection feature of the calculator.Enter Y1 = 2

x and Y2 = 1,000,000,000 , graph, and have the calculatordetermine the point of intersection.

(3) Use the Table feature of the calculator. Enter Y1 = 2x and scan the Y1

column of the Table for 1,000,000,000. The corresponding x-value should beapproximately 30.



(4) Use logarithms to solve 2 x =1,000,000,000 or 109.log2 x = log109

x log2 = 9 log10Because the log 10 is equal to 1, we next get:x log2 = 9

x = 9log2

x = 90.301029957...

x = 29.89735285... or approximately 30 minutes

Note: If students have not yet studied logarithms, then the last approach isobviously not an option. However, a teacher could use this example as anintroduction to logarithms. Before powerful calculators were accessible,computations involving very large and very small numbers were quitetedious because they had to be done by hand. Logarithms were inventedto assist with these types of computations. Rules were established andtables were developed. The tables expressed the logarithm of eachnumber as a power of 10. The tables in essence allowed a person toperform the computation on the exponents of 10 rather than on the originalnumbers. For example, 2 x =1,000,000,000 was approached by looking atthe log 2 and log 1,000,000,000. The log of 2 ≈ 0.3 (because 100.3 ≈ 2) andthe log of 1,000,000,000 = 9 (because 109 = 1,000,000,000). Computationdealing with 0.3 and 9 is much simpler than trying to figure out whatpower of 2 is approximately equal to 1,000,000,000.


Algebra II and Precalculus Institute 164

Transparency 1Algebra II Saves

Biology Students!!!

What are E. coli?

• E. coli are bacteria that live inthe human intestinal tract.

• Most strains are harmless andaid digestion.

• An E. coli bacterium is rod-shaped and 1 micrometer long.

• The growth of E. coli providesa good model for populationgrowth in general.



Transparency 2 Questions Can Lead from

Functions to Equations

Question: How long will it take 1 E. coli cell to divide

and form 1 billion cells?

Functions: y = 2 x and y = 109 .

(where x = number of minutes elapsed and y = number of cells)

Equation: 2 x = 109

The equation forms naturally from the study of the twofunctions:

y1 = 2x and y2 = 10

9 = 1,000,000, 000.



Student Activity 1: Recursion and Bacteria Cell Division

E. coli bacteria are very small – 1 micrometer in length.

1. One micrometer = 10-6 meter or 11,000,000

of a meter. A millimeter

is 10-3 meter or 11,000

of a meter. A micrometer, therefore, is 11,000

of a millimeter. Another way of looking at this situation is asfollows:

If 1 micrometer = 10−6 meters, then 1 meter = _______ micrometers.

2. How many micrometers is a five-foot tall person? Explain how youarrived at your solution.

3. Suppose each E. coli cell divides once a minute and we start with 1bacterium.a. Build a table of values to model the E. coli growth.

Number ofminutes

Cell Division Number of E.coli

0 o 11 o o 22 o o o o34



b. Use the home screen of your graphing calculator to model the E. coli growth.

c. Create a scatter plot of the data and determine a mathematicalmodel that accurately describes this set of data. Sketch yourscatter plot below and write the equation for your model.

d. How many cells will we have after 1 hour? How many after 2hours? Describe how you arrived at your solution.

e. How long will it take 1 E. coli cell to divide so that 1 billion cellsare formed? Describe your solution strategy.



Student Activity 2 : Space Debris in Earth’s Orbit

Overview: Students look at data that describes the amount of debris in the Earth’s orbit.They graph the data, find a mathematical model for the data and use these tomake predictions.

Objective: Algebra II TEKS(f.4) The student solves exponential and logarithmic equations andinequalities using graphs, tables, and algebraic methods.(f.5) The student analyzes a situation modeled by an exponential function,formulates and equation or inequality, and solves the problem.


Terms: Mathematical model, repeated addition, repeated multiplication, linear model,exponential model.


Procedures: Have students work in small groups on this activity.

1. Have students fill in the chart on the activity sheet using the variousgrowth rates.



Debris in Orbit (millions of pounds)Numberof Years

Fixed Amount Increase (1.8) Fixed 10%Increase

Fixed 20%Increase

0 (1990) 4.00 4.00 4.001 (1991) 4 +1.8 = 5.80 4 + 4 • 0.1 = 4 1 + 0.1( )

= 4 1.1( ) = 4.404 1.2( ) = 4.80

2 (1992) 4 +1.8 +1.8 = 4 + 2 1.8( ) = 7.60 4 1.1( ) 1.1( ) = 4 1.1( )2 = 4.84 4 1.2( ) 1.2( ) = 4 1.2( )2

=5.763 (1993) 4 + 3 1.8( ) = 9.40 4 1.1( )3 = 5.32 4 1.2( )3 = 6.914 (1994) 4 + 4 1.8( ) = 11.20 4 1.1( )4 = 5.85 4 1.2( )4 = 8.295 (1995) 4 + 5 1.8( ) = 13.00 4 1.1( )5 = 6.44 4 1.2( )5 = 9.956 (1996) 4 + 6 1.8( ) =14.80 4 1.1( )5 = 7.08 4 1.2( )6 =11.947 (1997) 4 + 7 1.8( ) =16.60 4 1.1( )5 = 7.79 4 1.2( )7 =14.338 (1998) 4 + 8 1.8( ) = 18.40 4 1.1( )5 = 8.57 4 1.2( )8 = 17.209 (1999) 4 + 9 1.8( ) = 20.20 4 1.1( )9 = 9.43 4 1.2( )9 = 20.64

10 (2000) 4 +10 1.8( ) = 22.00 4 1.1( )10 =10.37 4 1.2( )10 = 24.77t 4 + t 1.8( ) = 4 +1.8t 4 •1.1t 4 •1.2t

• Explain how you filled in the chart? The fixed amount increase isrepeated addition of 1.8 to the previous term. The fixed 10% increaseis repeated multiplication of the previous term by 1.10. The fixed 20%increase is repeated multiplication of the previous term by 1.20.

• Based on how the data is generated, what would you expect the graphsto look like? The fixed amount increase will be linear with slope 1.8.The other two graphs will be exponential.

2 – 3. Students should enter the data from the chart into a graphing calculatorand make scatter plots of the number of years versus each of the three growthrates. Below are the three scatter plots: (+ fixed amount increase;fixed 10% increase; • fixed 20% increase); the three models, and themodels graphed with the scatter plots.

4. The mathematical models can be used to predict the amount of debris bysubstituting appropriate values of x into the equations.

Numberof Years

Fixed Amt.Increase (1.8)

Fixed 10%Increase

Fixed 20%Increase

20 (2010) y1(20)=40 y2(20)=26.91 y3(20)=1532530 (2020) y1(30)=58 y2(30)=69.8 y3(30)=949.5140 (2030) y1(40)=76 y2(40)=181.04 y3(40)=5879.09



5 – 6. To find the doubling time, we need to know how long it will take tohave 8 million pounds of debris in our orbit. To find the tripling time weneed to find how long it will take to have 12 million pounds of debris inour orbit. Have students discuss various ways they solved these equations(including graphically, symbolically, or with the graphing calculator).Have students graph y = 8 along with the other three graphs to see agraphical interpretation of the solution to the doubling times. Then havestudents graph y = 12 to see the tripling times.

Fixed Amt. Increase(1.8)

Fixed 10%Increase

Fixed 20%Increase

Doublingtime

2.2 years 7.27 years 3.8 years

Triplingtime

4.4 years 11.53 years 6.03 years

• Could you have solved each of the equations above symbolically? Ifso, how? The linear model requires simple algebraic manipulation.The two exponential models can be solved by taking the logarithm ofboth sides of the equation, then using properties of logarithms.

Summary: This activity gives students the opportunity to look at different models ofgrowth and analyze each one. They generated data to see that repeatedaddition results in a linear model, while repeated multiplication results inexponential models.



Student Activity 2: Space Debris in Earth’s Orbit

In 1990, scientists estimated that a total of 4 millionpounds of debris was in Earth’s orbit. The table belowgives the accumulation of space debris in millions of poundsbased on different growth rates per year.

1. Complete the table.

Debris in Orbit (millions of pounds)

Number ofYears

Fixed AmountIncrease (1.8)

Fixed 10%Increase

Fixed 20%Increase

0 (1990) 4.00 4.00 4.001 (1991) 5.80 4.40 4.802 (1992)3 (1993)4 (1994)5 (1995)6 (1996)7 (1997)8 (1998)9 (1999)

10 (2000)t

2. Graph all three sets of data as a function of time (measured in years).Include labels, legend and title. Use different colors and plottingsymbols for each plot.

3. Find and graph the mathematical model for each plot. Write theequation of the model on the graph in an appropriate place.



4. Use both your graphs and mathematical models to predict the amountof space debris that may be in Earth’s orbit in the years 2010, 2020,2030.

5. Determine the Doubling Times for the three situations.6. Determine the Tripling Times for the three situations.

SPACE DEBRIS IN EARTH’S ORBIT PROJECT

Directions:• Summarize this activity.• Describe the trends and differences in your three plots and their

mathematical models.• Make at least two predictions based on your plots and mathematical

models.• Pose one reasonable “What if” question about the growth of space

debris. Then describe how you would use mathematics to find ananswer to your question.

Final product:Make a three-part poster of this activity for display:

ScenarioDescription

Tableand

Graphs

Predictions,What ifs,

andConclusion

II. Exponential & Logarithmic Functions 1.2 Exponential Decay: Trainers’ Notes

TEXTEAMS part 1: Algebra II and Precalculus Institute 1 7 1DRAFT 3-29-03

1.2 Exponential DecayOverview: Participants investigate the natural process of a substance “cooling down.”

Nature will “cool down” or “heat up” all objects in a room to roomtemperature. For example, a piece of ice sitting on a counter will “heat up” toroom temperature as it melts. A cup of hot chocolate will “cool down” to roomtemperature and be too cold to melt the marshmallows. Participants will collectdata and investigate exponential decay.

Objective: Algebra II TEKS(b.1) The student uses properties and attributes of functions and appliesfunctions to problem situations.(f) The student formulates equations and inequalities based on exponential andlogarithmic functions, uses a variety of methods to solve them, and analyzes thesolutions in terms of the situation.


Terms: Proportional relationships, curve-of-best-fit, regression equation

Materials: Thermometers or data collection device with a temperature probe, pint-sizeplastic containers or 8-ounce Styrofoam cups, hot water (approximately 180° For 82° C), stop watches, graphing calculators, graph paper

Procedure: Introduce the situation presented in the Activity. Ask participants to discusseach of the questions and encourage them to identify which graph representseach situation.

Activity: Cooling Down• Will a cup of hot water retain its heat for a while, and then cool rapidly?

Which graph depicts this situation? Graph 3• Will the water cool rapidly at first and then cool more slowly? Graph 2• Or, will the cup cool at a constant rate? Graph 1

Ask participants to discuss which graph they believe best represents the coolingprocess for this situation. Wait to answer this until after participants havecollected the temperature data below.

Explain that, in this Activity, instead of letting the water cool and recording theresulting temperature, we shorten the time interval necessary to collect the databy simulating water cooling. Thus, the cooling process can be measured inseconds rather than in hours. To do this, we will place the thermometer in a cupof hot water (≈180°F) for 1 minute to heat up. Then we will remove thethermometer from the cup and let it cool to room temperature in the air. Thethermometer cooling in the air simulates the water cooling in the cup.

The temperature reading of the thermometer is a function of time.• What is the independent variable? elapsed time• What is the dependent variable? temperature reading



• How long do you think it will take the thermometer to “cool down” toroom temperature?

Demonstrate the procedure for the experiment below.Have participants conduct the experiment in their groups.

5. Have participants predict the number of seconds that they think it will takethe thermometer to “cool down” to room temperature.

6. Have the groups collect their data

Experiment Procedure without a data collection device. Have oneparticipant from each group handle the thermometer and call out the temperature.Another participant keeps track of the elapsed time. The third participantrecords the data.Although the independent variable is time, the participants will find it easier tocall out temperature decreases of 2° and record the elapsed time. The datarecorder is responsible for recording all data. All participants should be readyfor quick readings initially. If they are surprised by the quick drop intemperature readings, suggest that they start over.

Experiment Procedure using a data collection device.Set up the experiment using a program which measure temperatures over aninterval of time.Have one participant from the group handle the temperature probe. Anotherparticipant operates the graphing calculator. The third participant operates theelectronic data collection device.

8. Once all data are gathered, ask participants to construct a scatter plot of thedata on the grid and/or in their graphing calculators.

• What patterns do you see in the data? As time increases, thetemperature decreases, then levels off.

• What information does the graph give you that was not as obvious fromthe table of values? The rapid drop in temperature in the first severalseconds.

• How are the functions y =12

x

and y = 2− x related? They are the

same.• How can an exponential function model a decaying (or decreasing)

situation? In the form y = abx , b must be between 0 and 1.• What function family has the characteristics seen in the graph of the

data? exponential functions.• What is the long run behavior of the data? The graph will level off at

room temperature.



• What is the long run behavior of an exponential function? As x getslarge, the decreasing exponential function has a horizontal asymptote.In this case, the exponential function modeling the temperature drophas a horizontal asymptote at y = room temperature.

• What is different about the graph of the data and the general function inthis family? A general exponential function has a horizontal asymptoteat y=0. The graph of our data has a horizontal asymptote and y =room temperature.

9. Exponential functions. The graph of the data is approaching the roomtemperature, whereas the general exponential approaches y = 0 .• In order to use the exponential regression formula, we must account for

the vertical shift in our graph. How can we do this? Adjust thetemperature data by subtracting the room temperature from each value.

10. Before entering a list in the calculator, we must subtract the roomtemperature from every value in the table. The graph of this adjusted datawill be a vertical translation, making the graph have a horizontal asymptote aty = 0 . Next have them use their graphing calculators to find an exponentialregression equation of the form y = abx that best fits the data.

• Does this regression equation model the actual data? No, we must addthe room temperature back into the equation to obtain an accuratefunctional model of the data collected.

• What affect did subtracting the room temperature from the data pointshave? It allowed us to use the exponential regression analysis bytranslating the graph to a general exponential function.

• Why did we have to add the room temperature back into the regressionequation? To make our regression equation model the actual data.

• What would happen if we tried to find an exponential regressionequation without compensating for the room temperature shift? Have



participants try it to see that the regression equation would not fit theinitial drop in temperature.

Refer back to the three graphs at the beginning of the Activity. The answer isGraph 2. Have participants reconcile their original answer with their currentanswer, if they are different.

Answers to Reflect and Apply:This is an important part of the activity for it allows the presenter to tie the“Cooling Down” experience to the transformations. The adding of 32 shiftsthe graph up 32 units, but the multiplication by 1.8 stretches the graph. Hence,the two graphs will not run parallel to one another.

Focus on the TEKS:What TEKS (Algebra II/ Precalculus) were introduced, reinforced, or extendedin the activities? Justify your answer.This activity extends on the students work with exponential functions detailed inTEKS (b.1.f) in Algebra II. It gives students a real example of exponentialdecay that they can watch happening. For precalculus students, this is a goodreinforcement of TEKS (c.3). It illustrates the importance that students be ableto use their knowledge of all functions to find a good model for the data.How do these activities support subsequent courses?Exponential decay, and specifically Newton’s Law of Cooling, is exploredextensively in Calculus. Calculus students derive Newton’s Law of Coolingusing differential equations.Can you take this activity to the classroom? If so, how would you adapt it?This activity can easily be taken to the classroom. Depending on the placementin the scope and sequence, students may need more guidance in developingtheir model for the data.

Summary The data from this experiment provide an example of exponential decay and canbe represented by the function y = abx , where 0 < b <1 . Emphasize theimportance of b having a value between 0 and 1 for exponential decay and of bhaving a value greater than 1 for exponential growth. Another big idea in thisexperiment is the need to understand how technology can help modelexperiments mathematically, but the user must be able to apply the technologyappropriately. In this example, a regression equation alone would haveproduced a very inaccurate model.

II. Exponential & Logarithmic Functions 1.2 Exponential Decay: Activity


Activity: Cooling Down

Consider a cup of hot water at approximately 180° F that is placed in aroom where the temperature is 74° F. Think about the cooling effect.

1. How will the water cool over time?

Time (hours)

t

y

60

90

120

150

180

Graph 3

0.5 1.0 1.5 2.0

2. Describe a situation implied by each graph.



Use a thermometer:

We will simulate the water cooling down by recording the temperaturedata of a thermometer cooling down.

Group Roles:Temperature reader – handles the thermometer and calls out thetemperature.Timer – keeps track of the elapsed time.Recorder – records the data.

3. Use the thermometer to determine the room temperature. ________4. Pour hot, not boiling water (≈ 180° F) into a cup. Place the

thermometer in the cup for about a minute to heat up.

5. Predict the number of seconds that you think it will take yourthermometer to “cool down” to room temperature. _________

6. Remove the thermometer from the cup of hot water. (Make sure tomove the thermometer away from the steam of the cup.) Begincollecting data immediately.Although the independent variable is time, you may find it easier to callout temperature decreases of 2° and record the elapsed time. The datarecorder is responsible for recording all data. Be ready for quickreadings initially.

Elapsed Time (seconds) Temperature (°F or °C)

7. What patterns do you see in the data?



Use a graphing calculator and an electronic data collection devicewith a temperature probe.

We will simulate the water cooling down by recording the temperaturedata of a temperature probe cooling down.

Group Roles:Temperature reader - handles the temperature probe.Calculator – handles the graphing calculator.Recorder – handles the electronic data collection device.

3. Use the temperature probe to determine the room temperature. _______

4. Pour hot, not boiling water (≈ 180° F) into a cup. Place the temperatureprobe in the cup for about a minute to heat up.

5. Predict and record the number of seconds that you think it will takeyour thermometer to “cool down” to room temperature.

6. Remove the thermometer from the cup of hot water. (Make sure tomove the thermometer away from the steam of the cup.) Begincollecting data immediately.

7. What patterns do you see in the data?



8. Sketch a scatter plot of the data collected on the grid below.

9. What family of functions has the characteristics seen in the graph of thedata? What is different about the graph of the data and the generalfunction in this family?

10. Adjust a list of data collected to create a scatter plot with a horizontalasymptote at y = 0. Use a graphing calculator to find an exponentialregression equation of the form y = abx that best fits the data.

11. Does this regression equation model the actual data? If not, how canyou adjust it so that it does?

II. Exponential & Logarithmic Functions 1.2 Exponential Decay: Reflect and Apply


Reflect and Apply

1. Convert the °C to °F by creating a new list L4 that is equal to 1.8 * L2 + 32.Record findings.

2. How are the two graphs related to each other. Why?

3. Create a scatter plot of L1, L4.Select a window that will accommodate both the original scatter plot and thenew scatter plot. Sketch the plots below.

4. Compare and contrast the two graphs. Do the two graphs run parallel? If so,why? If not, why not?

II. Exponential & Logarithmic Functions 2.1 The Energy of Earthquakes: Trainers’ Notes


2.1 The Energy of Earthquakes

Overview: In these activities, participants examine the use of logarithms in the Richterscale, which measures the magnitude of earthquakes.

Objective: Algebra II TEKS(f.3) For given contexts, the student determines the reasonable domain andrange values of exponential and logarithmic functions, as well as interpretsand determines the reasonableness of solutions to exponential and logarithmicequations and inequalities.(f.4) The student solves exponential and logarithmic equations andinequalities using graphs, tables, and algebraic methods.

Precalculus TEKS(c.3.A) The student is expected to use functions such as logarithmic,exponential, trigonometric, polynomial, etc. to model real-life data

Terms: Exponential function, properties of logarithms, graphing


Procedure: Activity 1: The Richter ScaleDiscuss with participants the development of the Richter scale as described atthe beginning of Activity 1. The web site referenced has a lot of otherinformation to add to the discussion, including frequency of earthquakes,more formulas and interesting facts about earthquakes. Use the Transparencyto give the participants a feel for the intensity of earthquakes given themagnitude. Have participants work through the Activity in small groups.1. Plugging into the formula and solving for E, we get:

E = 10 11.8 + 1.5( ) 7.4( )( ) ≈ 7.943 ×10222. Again, using the formula we get:

E = 10 11.8 + 1.5( ) 7.1( )( ) ≈ 2.818 ×10223. This question begins to get at the importance of logarithmic formulas.

Although the magnitude differs only by 0.3, the amount of energy releaseddiffers by 5.125 x 1022. The logarithm allows us to take very largenumbers and scale them precisely with numbers in a more comfortablerange.

4. Now we begin to look at the ratios of magnitudes and energy. Althoughthe earthquake magnitude is in a ratio of 2 to 1, the energy released is in aratio of 1000 to 1.

5. Participants should choose any two other magnitudes that are in a ratio of2 to 1 and repeat Exercise 4. The larger the original two magnitudes arethe larger the ratio of energy released will be.



6. We repeat the Exercises 4 and 5 for general magnitudes and see that theratio of energy released is 101.5M, where M is the smaller magnitude. Theanalysis that leads to this follows:1011 .8 +1.5 2 M( )

1011 .8 +1 .5 M=1011 .8 •10 3M

1011 .8 •101.5M= 103 M −1 .5 M =101 .5M .

• Why are logarithms useful in scaling earthquakes? Logarithms allowus to scale very large numbers in a non-linear manner.

• Notice that the doubling factor in this formula is a factor of 10. Whyis this? The base of the logarithm used in the formula is 10.

Activity 2: Graphing the MagnitudesHave the participants work through the Activity in small groups.1. MS =

log x −11.81.5

2. Give the groups time to find a nice graph. Note if they use the zoom fitfeature of a calculator, they may or may not have a useful window. Pointout to the groups that we would like to have a few of the most commonmagnitudes, that is we would like the y-axis to show magnitudes of around1 to 10. Below is a graph with the following window settings: xmin=1011,xmax=1025, xscl=0, ymin= –1, ymax=10, yscl=1.

• What is the domain of the logarithmic function described in question1? x > 0 .

• Does the graph have any asymptotes? If so, describe them. There is avertical asymptote at x=0. There are no horizontal asymptotes.

• Describe the long run behavior of the graph. It continues to increasealthough at a slower and slower rate

3. Notice that the graph rises very, very quickly at first then slows downdramatically.

4. Encourage the groups to look at the graph over various windows to seethe effect on the graph. For example, even with an xmax value of 1018, thesteep beginning of the graph seems unchanged. But, if we change xmax to1014 we see a different picture.

5. We can graph y = 7 along with the magnitude function to see theintersection, but because of the scale of the numbers on the x-axis, it isvery difficult to decide how much energy is released by just looking at thegraph. The window values may be changed. Look at 1022 ≤ x ≤ 1023 anduse the intersect feature. x =1.995 •1022 .



6. Have participants use the table feature of the calculator to help them fillin this chart. They can set the table to ask for the independent variableand enter in each of the values.

Energy Magnitude1011 -.53331012 0.13331013 0.8001014 1.4671015 2.1331016 2.8001017 3.4671018 4.1331019 4.8001020 5.4671021 6.1331022 6.8001023 7.4671024 8.1331025 8.800

7. A scatter plot of the table should look exactly like the graph of thefunction.

8. Below is a scatter plot of the magnitudes versus the power of 10 of theenergy released. Notice the graph is linear.

9. This graph allows us to get a general idea of the amount of energyreleased from the graph.

10. The slope of this line is about 23

. This means that for every power of ten

increase in energy, the magnitude increases by 23

. Remind participants of

the doubling factor from Activity 1. This graph gives a visualunderstanding for why the doubling factor was a power of 10.

• What kind of scale is used in the graph from question 8? Explain.This is a log-scale, because the values on the x-axis represent the log



of the actual data. For example, instead of plotting 1011, we plot 11,which is equivalent to the log1011 .

• Why is a log-scale appropriate for this graph? Because the x-valuesvary to include extremely large numbers, while the y-values fall in avery narrow range.

• When would a log-log-scale (where the log of the data is plotted onboth the x- and y-axis) be appropriate? When both the x-and y-valuesvary from relatively small values to extremely large values.

• What would the graph of y versus x look like on a linear scale if thegraph on a log-scale, where the y-axis is the log-scale, is linear?exponential

Answers to Reflect and Apply: The Rule of 701. Applying the rule, we get:

1%: t=70 years2%: t = 35 years5%: t = 14 years10%: t = 7 years

2. Applying the formula, we get:1%: 100 e0 .01( )t = 200 ; Solve for t to get t=69.3 years.2%: 100 e0 .02( )t = 200; t=34.66 years5%: 100 e0 .05( )t = 200 ; t=13.86 years10%: 100 e0 .1( )t = 200; t=6.93 years

3. Solve Pert = 2P for t.ert = 2ln ert = ln2rt = ln 2t = ln2

rNotice that ln 2 ≈ 0.6931, so suppose r = 7% = 0.07 , we get

ln 20.07

≈0.69310.07

≈707

.

Focus on the TEKS:What TEKS (Algebra II/ Precalculus) were introduced, reinforced, orextended in the activities? Justify your answer.These activities reinforce Algebra II TEKS (f.3 and 4). Participants look atreal-life models using exponential and logarithmic functions. PrecalculusTEKS (c.3.A) is addressed by these activities as well in an introductorymanner.How do these activities support subsequent courses?



These activities give participants experience with exponential and logarithmicfunctions in a real-life situation. In precalculus, students must draw on suchexperiences to determine what function is the best model for a situation.Can you take this activity to the classroom? If so, how would you adapt it?These activities would need guidance and some adaptation to be taken into theclassroom. They provide a good context and motivation for studyinglogarithms for Algebra II students. Activity 2 in particular would need somemodification beginning with question 6. It might be useful to bring in somesemi-log and log-log paper on which the students can plot the data.

Summary: These activities give participants experience in using logarithms to scale largenumbers. Participants also create logarithm scales so that they can seecharacteristics of the graphs.

II. Exponential & Logarithmic Functions 2.1 The Energy of Earthquakes: Transparency


MAGNITUDE VS. INTENSITYFrom the National Earthquake Information Center

Magn. Intensity Description1.0 – 3.0 I I. Not felt except by a very few under

especially favorable conditions.3.0 – 3.9 II – III • Felt only by a few persons at rest,

especially on upper floors ofbuildings.

• Felt quite noticeably by personsindoors, especially on upperfloors of buildings. Many peopledo not recognize it as anearthquake. Standing motor carsmay rock slightly. Vibrationssimilar to the passing of a truck.Duration estimated.

4.0 – 4.9 IV – V • Felt indoors by many, outdoorsby few during the day. At night,some awakened. Dishes,windows, doors disturbed; wallsmake cracking sound. Sensationlike heavy truck striking building.Standing motor cars rockednoticeably.

• Felt by nearly everyone; manyawakened. Some dishes,windows broken. Unstableobjects overturned. Pendulumclocks may stop.

II. Exponential & Logarithmic Functions 2.1 The Energy of Earthquakes: Transparency


5.0 – 5.9 VI - VII • Felt by all, many frightened.Some heavy furniture moved; afew instances of fallen plaster.Damage slight.

• Damage negligible in buildings ofgood design and construction;slight to moderate in well- builtordinary structures; considerabledamage in poorly built or badlydesigned structures; somechimneys broken.

6.0 – 6.9 VIII – IX • Damage slight in speciallydesigned structures; considerabledamage in ordinary substantialbuildings with partial collapse.Damage great in poorly builtstructures. Fall of chimneys,factory stacks, columns,monuments, walls. Heavyfurniture overturned.

• Damage considerable in speciallydesigned structures; well-designed frame structures thrownout of plumb. Damage great insubstantial buildings, with partialcollapse. Buildings shifted offfoundations.

7.0 andhigher

X orhigher

• Some well-built woodenstructures destroyed; mostmasonry and frame structuresdestroyed with foundation. Railsbent.

• Few, if any (masonry) structuresremain standing. Bridgesdestroyed. Rails bent greatly.

• Damage total. Lines of sight andlevel are distorted. Objectsthrown into the air.

II. Exponential & Logarithmic Functions 2.1 The Energy of Earthquakes: Activity 1


Activity 1: The Richter Scale

In the 1930s Charles F. Richter, a California seismologist, developed alogarithmic scale to rate the magnitude of earthquakes. Since then, hisscale has been adapted so that it can be used around the world. Variousmagnitudes are now used to describe different features of theearthquake, but the basis for Richter’s scale is still used today. For moreinformation see the National Earthquake Information Center website:http://gldss7.cr.usgs.govThere are many formulas that relate the various magnitudes of anearthquake to the amplitude, time and depth of the surface and bodywaves. In this activity, we examine one of Richter’s formulas thatrelates the magnitude of surface waves of an earthquake, Ms, and theenergy, E, released in the quake. This formula is:

log E = 11.8 + 1.5Ms, where Ms is the Richter Scale reading.

1. In 1999, Mexico City was hit be a massive earthquake that measureda magnitude of 7.4 on the Richter scale. Calculate the amount ofenergy released by this earthquake.

2. In 1989, the San Francisco Bay Area in California was hit by anearthquake that measured a magnitude of 7.1 on the Richter scale.Calculate the amount of energy released by this earthquake.

3. Compare the amount of energy released in the two earthquakesdescribed above.



4. Suppose one earthquake has a magnitude of 2 and another amagnitude of 4. What is the ratio of energy release between thesecond and first earthquake?

5. Choose two other magnitudes that are in a ratio of 2 and repeatExercise 4.

6. Write a general expression that gives the ratio of energy released interms of magnitude when the two magnitudes are in a ratio 2. Whatinsight does this information give you to the relationship between themagnitude and energy released?



Activity 2: Graphing the Magnitudes

The relationship described in Activity 1, log E = 11.8 + 1.5Ms, can berewritten so that the magnitude is given as a function of the energyreleased.

1. Rewrite the relationship as a function of the energy released.

2. Using a graphing calculator, graph the function. Discuss with yourgroup how to set the window to view the graph. (Note: the largestknown earthquake was in Chile on May 22, 1960. It had a magnitudeof 8.5.)

3. Discuss important features of the graph with your group.

4. Change the window settings to see different parts of the graph andexamine the table values in the calculator.

5. Using the graph, try to find the energy released by an earthquake withmagnitude 7.



6. As a way to try to get a better understanding the values given by thegraph, fill in the table below.

Energy Magnitude1011 -.53331012 0.13331013 0.8001014 1.4671015 2.1331016 2.8001017 3.4671018 4.1331019 4.8001020 5.4671021 6.1331022 6.8001023 7.4671024 8.1331025 8.800

7. What would a scatter plot of the above table look like?

8. Suppose we make scatter plot with the x-axis plotting the exponent often used in the chart above and the y-axis plotting the magnitude.What would this scatter plot look like? Why? Graph the plot on yourcalculator with a line graph.

9. Using the new plot, find the energy released by an earthquake withmagnitude 7. Your answer should be the same as in question 4.Which graph helps you see the answer better?

10. What is the slope of the line plotted in question 7? Describe themeaning of this slope in terms of the situation.

II. Exponential and Logarithmic Functions 2.1 The Energy of Earthquakes: Reflect and Apply

Algebra II/Precalculus Institute Draft 03-11-02 191

Reflect and Apply

The Rule of 70 is a commonly used by investors to see how long it will take fortheir investment to double. The rule gives an easy way to estimate the doublingtime for any exponentially growing quantity. The Rule of 70 says:Doubling time ≈ 70

r, where r % is the constant annual growth rate of the quantity.

1. According to the Rule of 70, how long will it take for a $100 investment todouble if it grows at the following constant annual rates: 1%, 2%, 5%, 10%?

2. Verify the above doubling times using the formula A = Pert , where P is theoriginal investment and r is the annual percent growth rate.

3. Explain why the Rule of 70 works in general.

II. Exponential and Logarithmic Functions 2.1 The Energy of Earthquakes: Student Activity


Student Activity 1: Exploring Logs

Overview: Students explore the definition of a logarithm as the inverse of an exponentialfunction graphically. Then students analyze properties of logarithms using thedefinition and by analyzing graphs.

Objectives: Algebra II TEKS(c.1.C.) The student recognizes inverse relationships between various functions.(f.1.) The student develops the definition of logarithms by exploring anddescribing the relationship between exponential functions and their inverses.

Precalculus TEKS(c.1.A.) The student is expected to describe parent functions symbolically andgraphically, including y=xn, y=ln x, y=logzx, y=1/x, y=ex, y=ax,y=sin x, etc.(c.2.B.) The student is expected to perform operations including composition onfunctions, find inverses, and describe these procedures and results verbally,numerically, symbolically, and graphically.(c.2.C) The student is expected to investigate identities graphically and verifythem symbolically, including logarithmic properties, trigonometric identities, andexponential properties.

Terms: inverse, function, rules of exponents

Materials: graphing calculator

Procedures: Before class begins, set up a demonstration graphing calculator to graph y=10x

and its inverse using parametric equations. Set the graphing calculator toparametric mode and to graph simultaneously. Graph the following parametricequations: x1T=T, y1T=10^T, x2T=10^T, y2T=T. Set the window as follows: Tmin=-2, Tmax=2, Tstep=.05, Xmin==-3, Xmax=3, Xscl=1, Ymin=-2, Ymax=2, Yscl=1. Tellstudents that you are going to graph the function y=2x and its inverse.• What does the graph of y=10x look like? Have students describe the shape,

pointing out that it passes through (0, 1) and approaches the x-axis as x goesto negative infinity.

• What should the graph of the inverse of y=10x look like? It will be areflection across the line y=x.

• What will the relationship between corresponding coordinates on the functionand its inverse be? Every point (x, y) on a graph of a function corresponds tothe point (y, x) on the graph of the inverse.

• Will the inverse of y=10x be a function? Why or why not? Yes, the inverse ofy=10 x will be a function because y=10x is a one-to-one function.



• Does the inverse of every function have to be a function? Explain. No, if thefunction is not one-to-one, then the inverse will not be a function. Forexample, the function y=x2 is not one-to-one because the y-value of 4 is aresult of two different x-values, 2 and –2. So, the inverse of y=x2 will haveboth (4, 2) and (4, -2) as points on the graph. This violates the uniquenessdefinition of a function.

Begin the graph of the two parametric functions, pausing the graph periodically tosee that the graphs are reflections of each other across the line y=x. After thegraph is drawn, it might be helpful to go back and add the parametric equation forthe line y=x to the graph (x3T=T, y3T=T). Use the Trace feature of the calculator tohave students go between the two graphs and see that every (x, y) on one graphcorresponds to the point(y, x) on the other graph. Show students the parametric equations used to graphthese functions. Point out that the difference between the two sets of equations issimply the assignment of x and y.• What functional relationship can we write to describe the graph of the inverse

of y=10x? Depending on the students’ previous experiences with inverses,their answers may vary. One possible answer would be to switch the x and yvalues in the original function. Thus, the inverse would be describe by thefunctional relationship x=10y. This is a perfectly fine way to describe theinverse function, but it is not very useful if we need to use function notation.For example, how would we put this function into our graphing calculator?Other students may have learned that logarithms are the inverses ofexponentials and may answer that the graph of the inverse of y=10x can bedescribed by the function y=log10x. No matter which answer students give,have a discussion about the other answer. This is where the definition oflogarithms comes in.

Present students with the definition of a logarithm:For all positive numbers a, with a≠1, and for all positive numbers x,y=logax is equivalent to x=ay.

Have students graph the functions y=10x and y=log10x on their graphingcalculators (in function mode, not parametric mode) using the x-y window usedabove. (Point out that the log button on their calculator is log10. The only otherlog that is on their calculator is the natural log, ln, which is loge. In order to grapha log of any other base, students will first have to learn the change of base rule.)• What are some of the properties of exponents that we might investigate for

logarithms? bn•bm=bm+n,

€

bn

bm= bn−m , b0=1

Have students work through the worksheet in pairs or small groups.



Student Activity 1: Exploring Logs1. Use the definition of logarithms and your knowledge of exponents to fill in each

of the blanks. In order to check yourself, graph both sides of the equation asseparate functions. If the equality is true, the two functions should coincide.

a. log1010 = 1 c. log101 = 0b. logee = 1 d. loge1 = 0

2. Generalize the properties below:a. logaa = 1 b. loga1 = 0

3. Using your graphing calculator, sketch each of the following pairs of functions.Then make a generalization.

a. y1=log(2x); y2=log(2)+log(x)b. y1=log(x2); y2=log(x)+log(x)c. y1=log(x•sin(x)); y2=log(x)+log(sin(x)), for x>0d. y1=ln(5x); y2=ln(5)+ln(x)

Generalization: log(xy)= log(x) + log(y)

4. Based on the above generalization and your knowledge of rules of exponents,what would you guess the log(x)-log(y)=? Use functions like the ones given in a-d of question 3 to support your guess.

€

log xy

= log (x) – log(y)

5. In question 3b, you graphed y2=log(x)+log(x). What would another equivalentform of this function be? Graph the new expression for this function and see if itcoincides with the previous two expression of the function. How else could youwrite y=log(x3)? y=log(x4)? Make a generalization.

log(xc) = clog(x)

Summary: The big idea of this activity is to introduce logarithms to students as the inverse ofexponential functions. Students also connect properties of exponents toanalogous properties of logarithms.



Student Activity 1: Exploring Logs

1. Use the definition of logarithms and your knowledge of exponents to fill in each of theblanks. In order to check yourself, graph both sides of the equation as separate functions. Ifthe equality is true, the two functions should coincide.

a. log1010 =________ c. log101 =__________

b. logee =__________ d. loge1 =__________

2. Generalize the properties below:

a. loga a =___________ b. loga1 =___________

3. Using your graphing calculator, sketch each of the following pairs of functions. Then make ageneralization.

a. y1=log(2x); y2=log(2)+log(x)b. y1=log(x2); y2=log(x)+log(x)c. y1=log(x•sin(x)); y2=log(x)+log(sin(x)), for x>0d. y1=ln(5x); y2=ln(5)+ln(x)

Generalization: log(xy)=___________________________

4. Based on the above generalization and your knowledge of rules of exponents, what wouldyou guess the log(x)-log(y)=? Use functions like the ones given in a-d of question 3 tosupport your guess.

_______________________= log(x)-log(y)

5. In question 3b, you graphed y2=log(x)+log(x). What would another equivalent form of thisfunction be? Graph the new expression for this function and see if it coincides with theprevious two expression of the function. How else could you write y=log(x3)? y=log(x4)?Make a generalization.

log(xc)=__________________



Student Activity 2: Change of Base

Overview: Examining further the connection between exponential and logarithmicfunctions, participants explore the change of base rule to graph logarithmicfunctions.

Objective: Algebra II TEKS(c.1.A) The student identifies and sketches graphs of parent functions,including … exponential ( y = ax ), and logarithmic ( y = loga x ) functions.

(c.1.B) The student extends parent functions with parameter such as in y=mxand describes parameter changes on the graph of parent functions.(c.1.C) The student recognizes inverse relationships between variousfunctions.(f.1) The student develops the definition of logarithms by exploring anddescribing the relationship between exponential functions and their inverses.(f.2) The student uses the parent functions to investigate, describe, and predictthe effects of parameter changes on the graphs of exponential and logarithmicfunctions, describes limitations on the domains and ranges, and examinesasymptotic behavior.(f.4) The student solves exponential and logarithmic equations andinequalities using graphs, tables, and algebraic methods.Precalculus TEKS(c.2.A) The student is expected to apply basic transformations, includinga • f x( ), f x( ) + d, f x − c( ), f b • x( ), f x( ) , f x( ) , to the parent functions.(c.2.B) The student is expected to perform operations including compositionson functions, find inverses, and describe these procedures and results verbally,numerically, symbolically, and graphically.(c.3.C) The student is expected to use properties of functions to analyze andsolve problems and make predictions.

Terms: Exponential functions, logarithmic function, base, exponent, inverse function

Materials: Graphing Calculators

Procedures: In the last activity, we investigated properties of logarithms. This activity willexplore graphing logarithmic functions. Traditionally logarithms were used tosolve equations and simplify computations and one used logarithmic tables todo so. We now take advantage of the power of technology to deepenstudents’ understanding of logarithmic functions by graphing them andcomparing them to the graphs of exponential functions.



• On a graphing calculator, how would you graph y = log10 x ?Demonstrate using the LOG key.

• On a graphing calculator, how would you graph y = loge x ? Demonstrateusing the LN key.

• On a graphing calculator, how would you graph y = logb x , where b is anybase? (Remember that by definition, b > 0 .) Tell participants that thisactivity is to explore using a graphing calculator to graph logarithmicfunctions with any base.

Student Activity: Change of Base Ask participants to complete Exercise 1 with a partner by graphing y = 2x and y = log2 x by hand. Share strategies.Scaffolding questions:

1. What do you know about inverse functions? An inverse function is areflection of the original function over the line y = x . For (a,b) on f , (b,a) ison f

−1 . The inverse of f is a function if and only if f is one-to-one.

• If you know the point (0, 1) is on y = 2x , what point do you know lies on y = log2 x ? The point (1, 0).

• How could the sketch of y = x help? The line y = x is the reflection linefor inverse functions.

2. Work through Exercise 2 together with participants. Emphasize that bydefinition, the inverse of y = 2x is y = log2 x and when you find the inverse of

f = 2 x algebraically, you find f −1 =

log xlog2

(see below.) Therefore,

log2 x =

log xlog2

. This is a specific example of the change of base rule.



y = 2x

x = 2 y

log x = log2 y

log x = y log2

y =log xlog2

3. So in general, logb a =

logalogb

and to graph y = logb x we use logb x =

log xlogb

.

4. Do Exercise 4a live participants, emphasizing that participants should predictthe graph of f

−1 based on the graph of f . Trace to points as shown below.Then have participants complete the rest of Exercise 4 and then Exercises 5 –6. Share strategies.

(Note that you cannot just jump from Y1 to Y2 to get the above screens.Once you find an ordered pair (a, b) on Y1 , then arrow to Y2 and trace tothe (b, a) that you want.)

5-6. Now that participants have seen many examples quickly using the power oftechnology, they should be able to apply their knowledge to graph thefunctions in Exercises 5 – 6 without using calculators.

Scaffolding questions:• To graph y = log5 x , what is its inverse function? How can you graph the

inverse function? How does the graph of the inverse help you graph y = log5 x ?

• Label an ordered pair (a, b) on y = log5 x . How can (a, b) help you find anordered pair on the inverse function?

• Which increases faster in the first quadrant, y = 3 x or y = 2x ? Whatimplications does this have for y = log3 x and y = log2 x ?



• For Exercise 6, which function is in between the other two? How do youknow?

Have participants share strategies for completing Exercise 6.

Assessment:This student used the fact that y=log2x is the inverse of the function y=2x. Thesolution to the equation 2x=5 is the point (x, 5) on the function y=2x. Thecorresponding point on the inverse function would be (5, y) on the functiony=log2x. So the student chose to find the y value on the inverse function andknew that this would be the same value as the x value on the original function.

Summary: The big idea here is that you can use exponential functions to graphlogarithmic functions. In order to graph logarithmic functions on the graphingcalculator, you must use a change of base rule if the base is other than e or 10.

II. Exponential and Logarithmic Functions 2.1 The Energy of Earthquakes: Student Activity 2


Student Activity 2: Change of Base

1. Sketch a graph of y = 2 x below and label of few points.Using what you know about the graphs of inverse functions, add the graph of y = log2 x and label a few points.

x y = 2 x x y = log2 x

2. Find the inverse of y = 2 x algebraically by switching x and y and then solvingfor y by taking the log of both sides.

3. By definition, y = log2 x is the inverse of y = 2 x . Based on your solution in

Exercise 2, log2 x = ____________ .

In general, logb x = ____________ .



4. Use the change of base rule above to graph the below in a square window.

• Trace on f x( ) and label a point (a, b).

• Predict the graph of f−1 x( ) based on the graph of f x( ) .

• Graph f−1 x( ) and confirm that f

−1 x( ) is the inverse of f x( ) by tracingon f

−1 x( ) to (b, a).

a. f x( ) = 4 x , f−1 x( ) = log4 x b. f x( ) = 13x , f

−1 x( ) = log13 x

c. f x( ) = 100x , f−1 x( ) = log100 x d.

f x( ) =

12

x

, f −1 x( ) = log1

2

x



5. Sketch, without using a graphing calculator.a. y = log5 xb. y = log9 xc.

y = log1

3

x

6. Without a graphing calculator, match the following graphs.

____ a. y = 3x

____ b. y = log2 x

____ c. y = log8 x

____ d. y = 2 x

____ e. y = 8x

____ f. y = log3 x

II. Exponential and Logarithmic Functions 2.1 The Energy of Earthquakes: Assessment


Assessment

A student used the following screen to find the solution to 2x = 5.Explain the student’s reasoning.

II. Exponential and Logarithmic Functions Appendix


Appendix: Bad Bad BugsSource: People Magazine, November 9, 1998, pp. 109-111.

Have each group discuss the “Bad Bad Bugs” article that participants were assigned to read theprevious night. Use the following questions on the transparency for disucssion. Once the groupshave had time to discuss the questions, have each group report they answer to one of thequestions until all questions have been addressed by the large group.1. Infectious diseases were the leading cause of death. Tuberculosis, diarrhea diseases and

pneumonia accounted for 30% of all deaths and kept life expectancy at about 47 years.2. Personal hygiene was given a higher priority, food and water supplies were cleaned up,

sanitation services have vastly improved, and just after WWII penicillin, an antibiotic forkilling bacteria without harming the human host, was discovered.

3. Bacteria have evolved and begun to develop resistant strains. Bacterial diseases such astuberculosis, meningitis and salmonella have reappeared. Viral diseases, such as various flustrains, HIV and the newly discovered and potentially deadly hepatitis C, are resistant toantibiotics and pose new challenges.

4. Decisions by pharmaceutical companies provide fewer antibiotic alternatives. Internationaltravel allows bacteria to spread more rapidly. Antibiotics used in agricultural or animalshave had health consequences throughout the food chain. Possible misuse of antibiotics maybe increasing the problem. Rapid growth of bacteria (one single bacterium can producemore than a billion bacteria in one day) allows for resistant mutations.

Have participants discuss the solution in their groups. We originally found that it tookapproximately 30 minutes for 1 E. coli cell to divide and form 1 billion cells. Hence, bychanging the rate of fission from once every 1 minute to once every 20 minutes, it should take 20times longer which is 600 minutes = 10 hours.

II. Exponential and Logarithmic Functions Appendix


Transparency: Bad Bad Bugs

Discuss the key points of the article “BadBad Bugs” in your small groups usingquestions 1 – 4 as a guide.

1. What were the health conditions at the turn of thecentury?

2. Life expectancy is now 76 years. What has happenedto extend life?

3. Have we conquered bacterial-related diseases?

4. Why does Dr. Mitchell Cohen, director of the Divisionof Bacterial and Mycotic Diseases in Atlanta, believethat resistant strains of bacteria have developed?

II. Exponential and Logarithmic Functions

Appendix

Appendix: Bad Bad BugsSource: People Magazine, November 9, 1998, pp. 109-111.

Have each group discuss the “Bad Bad Bugs” article that participants were assigned toread the previous night. Use the following questions on the transparency for discussion.Once the groups have had time to discuss the questions, have each group report theyanswer to one of the questions until all questions have been addressed by the large group.

1. 1. Infectious diseases were the leading cause of death. Tuberculosis,diarrhea diseases and pneumonia accounted for 30% of all deaths and kept lifeexpectancy at about 47 years.2. 2. Personal hygiene was given a higher priority, food and water supplieswere cleaned up, sanitation services have vastly improved, and just after WWII penicillin,an antibiotic for killing bacteria without harming the human host, was discovered.3. 3. Bacteria have evolved and begun to develop resistant strains. Bacterialdiseases such as tuberculosis, meningitis and salmonella have reappeared. Viraldiseases, such as various flu strains, HIV and the newly discovered and potentiallydeadly hepatitis C, are resistant to antibiotics and pose new challenges.4. 4. Decisions by pharmaceutical companies provide fewer antibioticalternatives. International travel allows bacteria to spread more rapidly. Antibiotics usedin agricultural or animals have had health consequences throughout the food chain.Possible misuse of antibiotics may be increasing the problem. Rapid growth of bacteria(one single bacterium can produce more than a billion bacteria in one day) allows forresistant mutations.

Have participants discuss the solution in their groups. We originally found that it tookapproximately 30 minutes for 1 E. coli cell to divide and form 1 billion cells. Hence, bychanging the rate of fission from once every 1 minute to once every 20 minutes, it shouldtake 20 times longer which is 600 minutes = 10 hours.

III. Power Functions and Polynomials Introductory Activity

TEXTEAMS Part1: Algebra II and Precalculus Institute 209DRAFT 3-29-03

Activity: The Spread of AIDS

Overview: Participants investigate an exponential, power, and linear regression equation.Participants examine the different long-term effects of each of these models inrelation to the spread of AIDS.

Objective: Algebra II TEKS(b.1.B) In solving problems, the student … makes scatter plots, fits the curvesto the appropriate parent function, interprets the results, and proceeds tomodel, predict and make decisions and critical judgements.

Precalculus TEKS(c.3.B) The student is expected to use regression to determine a function tomodel real-life data.

Terms: Linear, quadratic, cubic, polynomial, radical, rational, exponential, and powerfunctions; regression equation.


Procedure:Have participants work through the student activity to get them thinking aboutpower functions.

Have participants work in groups of three.1. The graph below shows a scatter plot of the data from 1981-1987.

For demonstration purposes, have all three regression equations on thedemonstration calculator and toggle between them during the discussion.

Linear Exponential Power

When the participants solve for the number of deaths in the year 1998,they should get the following approximate answers:Linear: 102,988



Exponential: 1,588,754,275Power: 525,139These are extremely different predictions and have drastic implications.Some epidemiologists at the time felt that the power function is the bestmodel. But, the answer to which is the best model depends on a lot morethan mathematics. The purpose for the discussion here is to realize thevast difference in prediction factor between linear, power and exponentialmodels that all look to fit data fairly well over a small sample.

• Describe the difference in the long run behavior of the linear,exponential and power models. All three models increase, but at verydifferent rates. The linear model grows steadily. The exponentialmodel increases the most rapidly.

2. With the data given through 1998, participants can check theirpredictions. It turns out of the three regression analyses performed, thepower function does indeed give the best prediction.

3. The first graph below is the scatterplot of the data from 1981 through1998. Next to it is the same graph with the power regression fromquestion 1.

When we look at the scatterplot alone, we notice a slight change inconcavity around 15 and 16. In fact, the Centers for Disease Controlreported that in 1996 the annual death rate from AIDS in the US decreasedfor the first time. The power regression does not reflect this change. Evenif we were to perform another power regression with the extended data,the curve would still not show this change. So, in fact, we are again at apoint in time similar to 1987. It seems as though a different regressionanalysis is needed, perhaps a logistic model.

• What might account for the change in the graph around x =16?Improvements in medical treatments, education and awareness, etc.

• Is the graph increasing or decreasing? The graph is increasing.• How can we describe the change in the graph around x =16

mathematically? The graph is increasing at a decreasing rate; thegraph changes concavity.

• As time goes on, is it possible that the graph might decrease at somepoint? Why or why not? No, this is cumulative data. Even if we



reached a point where there were no more deaths from AIDS, the datawould level off and the graph become a horizontal line.

• What might be the long run behavior of a model based on the new dataset? The new scatter plot indicates the possibility of a horizontalasymptote. Since the data is cumulative, the graph will not decrease.A horizontal asymptote would indicate a victory over AIDS with anegligible increase in the number of deaths from AIDS.

Focus on the TEKS:What TEKS (Algebra II/ Precalculus) were introduced, reinforced, orextended in the activities? Justify your answer.This activity extends Algebra II TEKS (b.1.B) and Precalculus TEKS (c.3.B).In algebra II and precalculus students are expected to make scatter plots, fitthe data with a function and use the model to predict behavior. This activityextends those concepts by asking students to consider a logistic model, whichis new to them. The activity also asks students to consider three differentregression models and compare the three models.How do these activities support subsequent courses?Logistic models are studied using differential equations in calculus. It isimportant that students see this model prior to that course so that they don’tfalsely assume that all growth is exponential.Can you take this activity to the classroom? If so, how would you adapt it?This activity could be taken to the precalculus classroom as is throughquestion 2. The teacher should demonstrate question 3 to a precalculus classwith some exploratory discussion about other growth models. For algebra IIclassrooms, the power regression could be eliminated from question 1.

Summary: In this activity, participants analyze a set data using different mathematicalmodels, including linear, exponential and power functions. The first analysislooks at data from more than 10 years ago. Participants are then given morecurrent data to check their models with. The big idea here is that differentmathematical models produce drastically different predictions and often amodel does not foresee a change in the future. When using mathematicalmodels to predict future behavior, it is important to realize the model mayonly be very good on the domain of the data set.



Activity: The Spread of AIDS

1. The table below gives the number of domestic deaths from AIDSfrom 1981-1987.

t, year since 1980 N, total number ofdeaths to date

1 1592 6193 2,1224 5,6205 12,5926 24,6697 41,027

a. Make a scatter plot of the data.

b. Using a graphing calculator, find a linear function that fits the data.

c. On another graphing calculator, find an exponential function thatfits the data.

d. On a third graphing calculator, find a power function that fits thedata.

e. Make sure that all three calculators have the same viewingwindow. Then decide which functions best fits the data.

f. Use each model to predict the total number of AIDS deaths in theUS by the year 1998, when t =18. Discuss the variance in the threeanswers.

II. Exponential Functions 3.1 Exponential & Power Functions: Student Activity


2. The table below continues the table in Exercise 1. Compare youranswer to question 1f to the actual recorded data below.

t, year since 1980 N, total number ofdeaths to date

8 62,1019 89,817

10 121,25511 157,63712 198,32213 241,78714 288,59715 332,24916 343,00017 N/A18 401,028

3. Make a scatter plot of the data through 1998. (Omit the data point for1997, since the value is not available.) Does the power regressionequation you found in Exercise 1 seem to fit this new data? Whatseems to happen around 1996?

III. Power Functions and Polynomials 1.1 Power Functions: Trainers' Notes


1.1 Power FunctionsOverview: Participants investigate power functions using the graphing calculator.

Objective: Algebra II TEKS(b.1) The student uses properties and attributes of functions and appliesfunctions to problem situations.(c.1) The student connects algebraic and geometric representations offunctions.

Precalculus TEKS(c.1) The student defines functions, describes characteristics of functions, andtranslates among verbal, numerical, graphical, and symbolic representations offunctions, including polynomial, rational, radical, exponential, logarithmic,trigonometric, and piece-wise defined functions.(c.3.A) The student is expected to use functions such as logarithmic,exponential, trigonometric, olynomical, etc. to model real-life data.

Terms: Power function, long run behavior, short run behavior


Procedures: Have a discussion with the participants about what power functions are.Power functions are of the form y=kxp. They are a subset of polynomialfunctions that are not often examined as group of their own. In these activitieswe will analyze some interesting properties of power functions.

Activity 1: Introduction to Power FunctionsDiscuss the scenario. Have participants work on the Activity in groups.Circulate and ask guiding questions.1. P = kd3

2. Solve for k to find k = 3652

93,000,0003. Substitute k back in P = kd3 to get

P = 365 d93,000,000

32.

3.



4.

Activity 2: Comparing Power FunctionsIn these Exercises we will make comparisons between power functions withdifferent exponents and study the end behavior of these functions.Write a general power function on the transparency:

y = kx p , where k and p are any real numbers.

• Can you think of examples of real situations that are modeled by powerfunctions?Possible answers:Area, A, in terms of the radius, r: A = πr2The time it takes to travel 300 miles depends on the rate, r:t = 300

r= 300r −1 .

The side of a square, s, depends upon the area of the square, A:

s = A = A12 .

Work through Exercise 1 with the participants.1. Ask participants to name an appropriate viewing window to graph and

compare the two functions.Below are two possible viewing windows. Both may be necessary tounderstand the behavior of the functions.

The graphs have the common point (0,0).The graph of y = x2 is symmetric to the y-axis while the graph of y = x3 issymmetric to the origin.



To determine the long run behavior one may examine tables as x gets largeand as x decreases.

As x→ +∞ the values for both functions are increasing without bound,but the cubic functions’ values are larger. Both functions grow rapidly,but the cubic function grows large faster than the square function.As x→ −∞ , x2 → +∞ and x3 → −∞ .

To consider the short run behavior zoom in on the point (0,0)

The graphs for both functions appeared flat in the original viewingwindow, but these windows show that for values near 0, it is evident thaty = x3 changes shape. It changes from being concave down to beingconcave up. The graph of y = x2 is always concave up.The point (0,0) is a vertex for y = x2 , but for y = x3 , it is called a point ofinflection, a point where the curve changes its concavity.

Math Note: One might explain concavity by asking the participants toimagine a line which is tangent to a curve. If the curve is above the line, thecurve is concave up at the point of tangency. If the curve is below the line,the curve is concave down at the point of tangency. This may be illustrated bygraphing these two situations:



At this point, the presenter may assign all of the remaining exercises to beworked in groups or assign each group one of the problems 3 – 7.

After these problems have been discussed as a whole group, assign the lastexercise 8 to be discussed in groups.

2.

y = x−1 y = x−2

Tables may be used to help answer some of the questions.

Both functions are discontinuous at x = 0 and have a vertical asymptote atx = 0 . The function y = x−1 is symmetric to the y-axis, while y = x2 issymmetric to the x-axis. There are no zeroes for these functions.

As x becomes very large both functions approach 0. However, y = x−3

approaches 0 faster than y = x−2 .As x becomes very small both functions approach 0, but y = x−2

approaches from the positive direction and y = x−3 approaches from thenegative direction.y = x−2 is concave up and decreasing for positive values of x and concavedown and increasing for negative values of x .y = x−3 is concave up and decreasing for positive values of x and concavedown and decreasing for negative values of x .



• Describe another power function that will be less than y = x−3 forlarge values of x. Any power function that has an integer exponentless than –3 will be smaller than y = x−3 .

3. y = x12( ) y = x

13( )

y = x12( ) has no symmetry. y = x

13( ) is symmetrical to the origin.

y = x12( ) has a restricted domain of all real numbers greater than 0.

y = x13( ) has a domain of all real numbers.

Both functions approach infinity as x gets large but the graph below showsthat y = x

12( ) grows larger faster than y = x

13( ) .

• How would the function y = x25( ) be related to these two functions?

The graph would be between these two functions for positive values ofx, because the exponent, 2

5, is between 1

3 and 1

2.

4. Functions from least to greatest value for 0 ≤ x <1 .y = x4 , y = x3 , y = x2 , y = x , y = x

12( ) , y = x

13( )

Graphs:



Trace on the graphs to determine the order.• Describe the difference in the concavities of these functions in this

window. The functions with exponent greater than 1 are concave up.The functions with exponent less than 1 are concave down.

• Which of these functions contain the point (1, 1)? All power functionsof the form y = xp contain the point (1, 1).

5. Functions from least to greatest value for x > 1 .y = x

13( ) , y = x

12( ) , y = x , y = x2 , y = x3 , y = x4

• Describe another way to compare these functions. Tables may be usedto compare the function values.

• Which of these functions grows faster? The function y = x4 growsfaster.

6. Functions from least to greatest value for 0 ≤ x <1 .y = x , y = x−1 , y = x−2 , y = x−3

7. Functions from least to greatest value for x > 1 .y = x−3 , y = x−2 , y = x−1 , y = x



• Describe the concavity. Each of these functions is concave up forx > 1 except y = x which is neither concave up nor concave down.

8. The functions should be arranged by their exponents. In the interval from0 to k, the greatest function is the one with the least exponent.a. 3 b. 1 c. 2 d. 4

Activity 3: Comparing FunctionsAssign each group one of the four exercises. Ask them to justify theirreasoning using large grid paper and to present their findings to the group.

1. The two functions are equal when x =100 . This is the point after whichy = x4 overtakes y = 100x3

• What is the behavior of these functions for very small values of x?For very small negative values, the even exponent makes y = x4

always positive and the odd exponent means y = 100x3 is negative.• What has more effect, the coefficient, k, or the exponent p in the long

run?• When you found where f and g intersect, what equation(s) were you

solving? Connect 100x3 = x4 and x4 −100x3 = 0 , x3 x −100( ) = 0 .

2. y = x4 is less than y = 2 x for larger values. To determine when thechange occurs the table could be examined or the intersect feature on thecalculator may be used.

For all x greater than 16, y = x4 is less than y = 2 x .• How did you decide on what graphing window to use?• Will all exponential functions eventually overtake the power function

y = x4 ?



• What grows faster, an exponential or a power function in the long run?If the base is a number greater than 1, the power function wouldeventually overtake y = x4 .

• Try this: graph y = x2 , y = x3 , y = x4 , y = 2 x in the window [0, 20] by[0, 200,000]. Guess first, then confirm. Where is y = x2 ?

3. y = x−3 is greater than y = 0.5x for all values of x.

“Wins” here means which function decreases faster.• Add in f x( ) = x −2 .• What decrease faster, an exponential function or a power function in

the long run? An exponential function.

4. y = x12 is greater than y = log x for all values of x.

“Wins” here means which function increases faster.Try looking at the inverse functions, compare and then reflect over y = x tocompare originals.

• Add in f x( ) = x13 , f x( ) = x

15 .

• What decrease faster, an exponential or a power function in the longrun?

• How could you numerically show that y = x12 is increasing faster?

You could examine the difference in the functions. The gap betweenthe two functions is increasing as x increases.

Reflect and Apply Answers:1. B2. C3. A4 a. y = 15.75x +16.5

b. y = 3 4( )x



c. y = 0.75x 6

Focus on the TEKS:What TEKS (Algebra II/ Precalculus) were introduced, reinforced, orextended in the activities? Justify your answer.These activities extend on the Algebra II TEKS dealing with parent functionsin (c.1). The activities examine parameter changes on the power parentfunction. Activity 1 is an example of Precalculus TEKS (c.3.A) in whichfunctions are used to model real-life.How do these activities support subsequent courses?These activities encourage teachers to think about the family of powerfunctions and how their graphs change as the parameter change. This is acommon activity for linear functions and even quadratic functions, but powerfunctions are rarely considered. Students need to be able to compare graphsand look at the long run and short run behavior of graphs. This will behelpful as students learn about limits formally in calculus.Can you take this activity to the classroom? If so, how would you adapt it?These activities could be taken into the precalculus classroom as a teacher leddiscussion with the use of technology. It provides a good review of conceptsrelated to graphing and analyzing functions, such as domain, symmetry,increasing/decreasing, etc.

Summary: The big idea here is that power functions with the same coefficient may becompared using their exponents and that the comparison depends uponwhether or not the interval is greater than or less than 1. Functions may havethe same long run behavior, but one increases faster than the other does.

Note that the Student Activity is included in this section to provide moreexperience in graphing power functions and other radical functions, toexamine roots, and to consider extraneous roots.

III. Power Functions and Polynomials 1.1 Power Functions: Activity 1


Activity 1: Introduction to Power Functions

One of Johanes Kepler’s three laws of planetary motion states that thesquare of the period, P, of a body orbiting the sun is directlyproportional to the cube of its average distance, d, from the sun.Symbolically, P2 = kd3 .

1. Solve for P.

2. For earth, use a period of 365 days and a distance from the sun ofapproximately 93,000,000 miles. Write the function for the period Pin terms of the average distance, d, from the sun.

3. Find a good viewing window for the situation. Sketch the graph.

4. Given that the planet Jupiter has an average distance from the sun of483,000,000 miles, how long in Earth days is a Jupiter year?



Activity 2: Comparing Power Functions

If k and p are any constants, then y = kx p is a power function of x.

The parameter k is a stretch factor, so we let k =1, so we canconcentrate on the parameter p. Consider the following special powerfunctions:

y = x2 , y = x3, y = x−1 , y = x−2 , y = x12 , y = x

13

1. Compare the graphs of y = x2 and y = x3. Consider the following:

• What points do they have in common?• Symmetry?• What is the long run or end behavior of each?• What is the short run behavior of each?• Increasing and/or decreasing?• Concavity?• Special points of interest?• Which grows faster where?



2. Compare the graphs of y = x−1 and y = x−2 . Consider the following:

• Discontinuities, asymptotes?• Symmetry?• What is the long run or end behavior of each?• What is the short run behavior of each?• Increasing and/or decreasing?• Concavity?• Special points of interest?• Which grows faster where?

3. Compare the graphs of y = x12 and y = x

13 . Consider the following:

• Symmetry?• Domain?• What is the long run or end behavior of each?• Increasing and/or decreasing?• Concavity?• Special points of interest?



Use graphs and tables to order the following:4. Order from least to greatest for 0 ≤ x ≤ 1, y = x

13 , y = x2 , y = x4 , y = x

12 ,

y = x3, y = x . Sketch the graph.

5. Order from least to greatest for x > 1, y = x13 , y = x2 , y = x4 , y = x

12 ,

y = x3, y = x . Sketch the graph.

6. Order from least to greatest for 0 ≤ x ≤ 0.5, y = x−2 , y = x−1 , y = x−3 ,y = x . Sketch the graph.

7. Order from least to greatest for x > 0.5, y = x−2 , y = x−1 , y = x−3 , y = x .Sketch the graph.



8. Match the functions with the graphs:

____ a. y = kx916

____ b. y = kx311

____ c. y = kx38

____ d. y = kx57



Activity 3: Comparing FunctionsLet’s Race. Who wins in the long run? Support your answernumerically and graphically.

1. f x( ) = 100x 3 versus g x( ) = x4 .

2. f x( ) = x 4 versus g x( ) = 2x .

3. f x( ) = x −1 versus g x( ) = 0.5x .

4. f x( ) = x12 versus g x( ) = log x .

III. Power Functions and Polynomials 1.1 Power Functions: Reflect and Apply


Reflect and Apply

Match:_____ 1. y = x p , p >1

_____ 2. y = x p , 0 < p <1

_____ 3. y = x p , p < 0

4. Suppose that f −1( ) = 34

and f 2( ) = 48.

a. Assume that f is a linear function. Find a possible formula for f x( ) .b. Assume that f is an exponential function. Find a possible formula for

f x( ) .c. Assume that f is a power function. Find a possible formula for f x( ) .

III. Power Functions and Polynomials 1.1 Power Functions: Student Activity


Student Activity: Truth or ConsequencesOverview: Students investigate the square root function by investigating the validity of

equations.

Objective: Algebra II TEKS(d.4) The student formulates equations and inequalities based on square rootfunctions, uses a variety of methods to solve them, and analyzes the solutionsin terms of the situation.

Precalculus TEKS(c.1) The student defines functions, describes characteristics of functions, andtranslates among verbal, numerical, graphical, and symbolic representations offunctions, including polynomial, rational, radical, exponential, logarithmic,trigonometric, and piecewise-defined functions.

Terms:


Procedures:For the equations given on the activity sheet, students graph the left and rightsides as separate functions. Then they decide whether each statement isalways true, sometimes true of never true. Students should be told to use thetable feature of the calculator to reinforce their conjectures. Some discussionmay be needed on what is means to prove something is always true. Studentsshould give algebraic support for the conjectures.

Answers:1. S ( x = 0 ) 2. N3. S ( x ≥ 0 ) 4. S ( x ≥ 0 )5. A 6. S ( x = 0 )7. A 8. A9. S ( x = 2 ) 10. S ( x = 0 )

III. Power functions and Polynomials 1.1 Power Functions: Student Activity


Student Activity: Truth or Consequences

Graph each side of the following equations separately and decide if eachstatement is Always true, Sometimes true, or Never true. If the equationis sometimes true, state when this occurs.

Use the table feature of you calculator to support your conjecture andgive algebraic support as well.

1. x2 + 9 = x +3

2. x − 2 = x −2

3. x2 = x

4. x( )2 = x

5. 16x2 = 4 x

6. −x = − x

7. x4 = x 2

8. x9=

x3

9. 3 x −2 = 3x −6

10. x +1= x +1



Student Activity: Classifying FunctionsOverview: Students compare and contrast exponential, polynomial and power functions.

Objective: Algebra II TEKS (c.1) The student connects algebraic and geometricrepresentations of functions.

Precalculus TEKS (c.1) The student defines functions, describescharacteristics of functions, and translates among verbal, numerical, graphical,and symbolic representations of functions, including polynomial, rational,radical, exponential, logarithmic, trigonometric, and piecewise-defined functions.

Terms: Linear, quadratic, cubic, polynomial, rational, radical, exponential, and powerfunctions


Procedures: Have students work in pairs to classify the functions listed on the ActivitySheet. Use the questions below to guide a whole group discussion.

• What is the difference between a power function and an exponentialfunction? In a power function, the exponent is a constant, while in anexponential function, the base is a constant. Power function: f(x) =axb, where a and b are constants; exponential function: g(x) = abx

where a and b are constants.• Are polynomial functions power functions? Only if they are

monomials of the form f(x) = axb.• Are rational functions power functions? When functions are in the

form f(x) = axb where a is a constant and b is a negative integer, weget rational functions that are power functions.

• Are radical functions power functions? When functions are in theform f(x) = axb where a is a constant and b is a non-integer rationalexponent, we get radical functions that are power functions.

• Are exponential functions power functions? No.• Which functions from the classifying student activity sheet are not

power functions? y=x2 + 3x, y=4x3 – 3x2 + 2, and all the exponentialfunctions.



Results of classifying activity:

LinearFunctions

QuadraticFunctions

Cubic Functions Polynomial Functions

f x( ) = x1 f x( ) = x 2

f x( ) = x 2 + 3xf x( ) = x3

f x( ) = 4x3 − 3x2 + 2f x( ) = x1

f x( ) = x 2

f x( ) = x3

f x( ) = x10

f x( ) = x 2 + 3xf x( ) = 4x3 − 3x2 + 2

RationalFunctions

Radical Functions Exponential Functions Power Functions

f x( ) = x1

f x( ) = x 2

f x( ) = x3

f x( ) = x 4

f x( ) = 2x3

f x( ) = x12

f x( ) = x13

f x( ) = x14

f x( ) = x110

f x( ) = 3x12

f x( ) = 0.5( )x13

f x( ) = 2 x

f x( ) = 2 x −1

f x( ) =12

x

f x( ) = ex

f x( ) =1e

x

f x( ) = 4 x

f x( ) = 10 x

f x( ) = x1

f x( ) = x 2

f x( ) = x3

f x( ) = x10

f x( ) = x −1

f x( ) = x −2

f x( ) = x −3

f x( ) = x −4

f x( ) = 2x−3

f x( ) = x12

f x( ) = x13

f x( ) = x14

f x( ) = x110

f x( ) = 3x12

f x( ) = 0.5( )x13



Student Activity: Classifying Functions

Sort the functions into the following categories: linear, quadratic, cubic,polynomial, rational, radical, exponential, and power functions. Somefunctions will fit into more than one category.

f x( ) = x2 f x( ) = x12 f x( ) = 2x

f x( ) = x2 + 3x f x( ) = 3x12 f x( ) = 2x −1

f x( ) =12

x f x( ) = x−2 f x( ) = x1

f x( ) = exf x( ) =

1e

x f x( ) = x−1

f x( ) = x−3 f x( ) = x13 f x( ) = x3

f x( ) = 2x −3 f x( ) = 0.5( )x13 f x( ) = 4x 3 − 3x 2 + 2

f x( ) = x14 f x( ) = 4x f x( ) = x−4

f x( ) =10 x f x( ) = x10 f x( ) = x110



LinearFunctions

QuadraticFunctions

Cubic Functions PolynomialFunctions

RationalFunctions

RadicalFunctions

ExponentialFunctions

PowerFunctions

III. Power Functions and Polynomials 1.2 Compounding Interest: Trainers' Notes


1.2 Compounding InterestOverview: Participants investigate various scenarios of interest bearing bank accounts.

The balances are modeled using power, polynomial and exponential functions.The relationship between the number e and its definition from a polynomialfunction is also investigated.

Objective: Algebra II TEKS(f.5) The student analyzes a situation modeled by an exponential functionformulates an equation or inequality, and solves the problem.

Precalculus TEKS(c.3.A) The student is expected to use functions such as logarithmic,exponential, trigonometric, polynomial, etc. to model real-life data.

Terms: Compounding interest, e


Procedures: We often use exponential functions to describe rapid growth functions.Continuously compounding interest is an example of one such situation. Inthese activities we look at different types of compounding interest and build tothe exponential model through power and polynomial function models.

Activity 1: Money TalkThis activity investigates the growth of a bank account balance when aconstant yearly deposit is made and annual interest is earned. Participants willinvestigate the account numerically and develop a general formula for thegrowth.1.

No. of yearselapsed

Start of yearbalance

End of yeardeposit

End of yearinterest

0 $2,000.00 $1,000 $80.001 $3,080.00 $1,000 $123.202 $4,203.20 $1,000 $168.133 $5,371.33 $1,000 $214.854 $6,586.18 $1,000 $263.455 $7,849.63 $1,000 $313.996 $9,163.61 $1,000 $366.54

2. Neither. It is not growing linearly because the change in balance fromyear to year is not constant. It is not growing exponentially because theratio of successive balances is not constant.

3. Balance after one year = 2000 + 2000r + 1000 = 2000(1+r) + 1000



After participants find the first expression above, suggest rewriting theexpression using (1+r) as shown above. Then have participants fill in thetable in question 4, continuing to use the (1+r) quantity.

4.In year n Balance

0 20001 2000 + 2000r + 1000 = 2000(1 + r) + 10002 [2000(1 + r)+1000] + [2000(1 + r)+1000]r + 1000

=[2000(1 + r)+1000](1 + r) + 1000=2000(1 + r)2 + 1000(1 + r) + 1000

3 [2000(1 + r)2+1000(1 + r)+1000]+ [2000(1 + r)2+1000(1 + r) +1000]r+1000=[2000(1+r)2+1000(1+r)+1000](1+r)+1000=2000(1+r)3 + 1000(1+r)2 + 1000(1+r) + 1000

n 2000(1+r)n + 1000(1+r)n-1 + …+1000(1+r) + 1000

The underlined expressions above may help participants see the commonfactors in the expressions. You may wish to discuss the (1+r) expression.This is the annual growth factor. For example, if the account earns 4%interest, (1+r) = (1+.04) or 100% + 4%. This represents 100% of thebalance plus the interest, 4% of the balance. We could let x=(1+r) to get acleaner polynomial expression.

5. pn = 2000 1+ r( )n +1000 1 + r( )n −1 + ... +1000 1 + r( ) +10006. Graph the following functions on the graphing calculator:

y1=2000(1+x)5 +1000(1+x)4 +1000(1+x)3 +1000(1+x)2 +1000(1+x)+1000y2=9000Find the intersection.

The x-value, .088, represents the APR. The account must earn at least 8.8%

Activity 2: Compounding ContinuouslyThis activity looks at accounts with various compounding time periods andthe effects on the balance of the accounts. This process leads to the discoveryof the number e. Have participants work in small groups on exercises 1 – 4.

1. Assuming no other deposits or withdrawals, the 12% annual interestwould result in:



$1000 + $1000(.12) = $1000(1+.12) = $1000(1.12) = $1120.

The account paying 1% interest every month, again assuming no otherwithdrawals or deposits can be explored using the chart below.

Month Amount1 $1000(1.01)2 ($1000(1.01))(1.01)3 $1000(1.01)(1.01)(1.01)12 $1000(1.01)(1.01)(1.01)…(1.01)

12 months of 1% monthly interest

So the result of the account paying 1% interest monthly is$1000(1.01)12 = $1126.83• Why does the account that breaks the 12% interest into monthly

payments end up with a larger balance at the end of a year? Theinterest earned in the first month is added to the balance that theinterest for the second month is calculated. This happens each month.This is what compounding means. The annual interest would alsocompound over the course of years, but the effect of the compoundingis not seen in a single year.

2.a $1000(1.08)(1.08)(1.08) = $1000(1.08)3 = $1259.71b The annual rate of 8% is split into 12 months, so each month the

interest rate is .0812

. At the end of 3 years, 36 interest payments will

have been made. Therefore, the balance can be calculated as$1000(1+ .08

12)36 = $1270.24

c Now the annual rate is split into 52 weeks, so the interest rate per weekis .0852

. After three years, 52 x 3 = 156 payments will have been made,

resulting in a balance of: $1000(1+ .0852

)156 = $1271.01

d The balance in the account if the interest was paid daily would be:$1000(1+ .08

365)3•365 = $1271.22

• Look at the change in the balance amounts of the variouscompounding methods above. What do you notice? The change fromannual to monthly interest is fairly large, but from then on the changelessens. It seems to approach a limit of some sort.

• How could you continue the list in question two in order to compoundmore frequently? What would (e), (f) or (g) be? We could break thetime down to hourly, by the minute or eventually to the limiting factoror continuously compounding interest.



3. Have the participants share their formulas and discuss any differences.This formula should emerge from the work of the groups during Exercise

2: A = P 1+rn

nt

. Another form of this equation is A = P 1+1nr

nr

rt

.

Discuss with the group why these two equations are equivalent.

4. The initial deposit, P, and the annual interest rate, r, are constant and thecompounding period, described by n, and the time, t change.

As a group, consider the part of the formula above, 1 +1nr

nr

. Let

x = nr

. Then the expression becomes 1 +1x

x

.

5. Encourage participants to investigate the function in various manners. Astarting window for the graph might be 0≤x≤100, 0≤y≤3. From this graph,the participants should notice that the values level off somewhere between2 and 3. From there, the ymin and ymax of the window might benarrowed. The table feature can be used to see numerically the valuesapproaching a limit. Allow participants to decide on a limiting value.

6. Most likely, participants will have decided that f x( ) approaches somelimiting value of at least 2.6. The calculator will estimate e1≈2.718281828. Some participants may look at this value and assume that eis a repeating decimal. In fact, e does not repeat the pattern we see here.e, like π, is an irrational mathematical constant that appears in manyapplications, such as compounding interest.

7. The table start entries below will vary from group to group. They aregiven here as a point of reference. The interesting point to make here ishow slowly the values grow as x gets very large. Finding the x values forthe last two estimates may surprise some participants.

Estimate of e Table Start Value x=2.7 60 74

2.71 150 1632.718 4050 4095

2.7182 10300 10309



8. Questions 5 – 7 lead us to a definition of e as limx→ ∞

1 +1x

x

. So, as n

approaches infinity, the formula A = P 1+1nr

nr

rt

approaches A = Pert.

This formula gives the amount in an account over time that is compoundedcontinuously.

• What is the amount in the account if $1,000 is invested at 8%compounded continuously? 1000e 3( ) 0.08( ) = $1,271.25 1000e(3x.08) =$1271.25


B = 300 1.2( )t This investment earned 12% annually, compounded annually.B = 300 1.12( )tThis investment earned, on average, more that 1% each month.

B = 300 1.06( )2t This investment earned 12% annually, compounded semi-annually.

B = 300 1.06( )t2 This investment earned, on average, less that 3% each quarter.

B = 300 1.03( )4t This investment earned, on average, more that 6% every 6months.

Focus on the TEKS:What TEKS (Algebra II/ Precalculus) were introduced, reinforced, orextended in the activities? Justify your answer.Activity 1 is a good example of Precalculus TEKS (c.3.A) where a polynomialfunction is used to model a real situation. Activity 2 extends the ideaspresented in Algebra II TEKS (f.5) in that it explores what the number erepresents.How do these activities support subsequent courses?Activity 2 is a great lead into the formal definition of e in calculus usinglimits.Can you take this activity to the classroom? If so, how would you adapt it?Activity 1 could be appropriate for precalculus students in small groups withguidance from the teacher. Students may need help organizing their table ofdata and seeing the polynomial pattern appear. Activity 2 is a niceintroduction to the concept of a limit that students will see later. This activitywould need to be done as a demonstration for students.



Summary: These activities give participants opportunities to explore themathematics behind various types of banking situations. These can bemodeled with modified power functions, polynomial functions andexponential functions.

III. Power Functions and Polynomials 1.2 Compounding Interest: Activity 1


Activity 1: Money Talk

Suppose you open a bank account with $1000. Every year on the samedate you deposit another $1000.1. Suppose the account earns 4% annual interest, compounded annually.

Fill in the table below.Number of

years elapsedStart of year

balanceEnd of year

depositEnd of year

interest0 $2000.00 $1000 $80.001 $3080.00 $1000 $123.202 $4203.20 $10003 $10004 $10005 $10006 $1000

2. Describe how the balance in the account grows.

3. Suppose the account earns R% annual interest, compounded annually.Let r be the decimal form of this percent. Write an expression for thebalance in the account after one year.



4. Fill in the chart below assuming the variable interest rate given inExercise 3.

In year n Balance0 20001 2000 + 2000r +1000 = 2000(1+ r) +100023

n

5. Using the chart on the previous page write the equation which showsthe relationship between the balance in the account and the interestrate.

6. Using the formula from Exercise 5 and a graphing calculator,determine the APR of the account in order for an investor to have abalance of $9000 in 5 years.



Activity 2: Compounding Continuously

1. What is the difference between a bank account that pays 12% interestonce per year and one that pays 1% interest every month?

2. Suppose $1000 is deposited into an account paying interest at a rateof 8% per year. Find the balance after 3 years if the interest iscompoundeda. annuallyb. monthlyc. weeklyd. daily.

3. Write a general formula that describes the relationship between theending balance and the starting balance in the account.

4. In Exercise 2, which of the parameters described in Exercise 3 areconstant and which are variable?

5. Investigate the following function using a graphing calculator.

f (x) = 1+1x

x

In this function, we can think of x as the compounding frequencydescribed by n in our formula. If we move toward continuouscompounding, the value of x would get larger and larger. Investigatethe value of this function as x approaches infinity. Use the trace and



table features of your calculator to help you with your investigation.Describe what happens to the value of f x( ) as x grows very large.



6. The limit of the function f (x) = 1+1x

x as x approaches infinity

defines a mathematical constant, e. Use a calculator to give anestimate of e by calculating e1 . What value do you get? How doesthis value compare to your conclusions in Exercise 5?

7. Using the table feature of your graphing calculator, explore whatinteger value of x first gives the following estimates of e.

Estimate of e Table Start Value x =2.72.71

2.7182.7182

8. How can our investigation of the number e be used in a formula forcalculating compounding interest?

III. Power Functions and Polynomials 1.2 Compounding Interest: Reflect and Apply


Reflect and Apply

Suppose that $300 was deposited into one of five different bank accounts. Each ofthe equations on the left gives the balance of an account in dollars, as a function ofthe number of years elapsed, t. On the right are verbal descriptions of fivedifferent situations. Match each situation with the equation or equations it couldpossibly describe.

B = 300 1.2( )t This investment earned 12% annually, compoundedannually.

B = 300 1.12( )t This investment earned, on average, more that 1% eachmonth.

B = 300 1.06( )2t This investment earned 12% annually, compoundedsemi-annually.

B = 300 1.06( )t2 This investment earned, on average, less that 3% each

quarter.

B = 300 1.03( )4t This investment earned, on average, more that 6%every 6 months.

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TEXTEAMS Part 1: Algebra II and Precalculus Institute

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