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COVER
TITLE PAGE
PREFACE
INTRODUCTION
1. ABOUT THIS BOOK
2. ABOUT MATHEMATICAL MODELING
3. ABOUT MATHEMATICAL MODELING TECHNIQUES
4. ABOUT BEING A GOOD MODELER
5. ABOUT THE MODELING PROBLEMS
6. ABOUT GEOGEBRA
7. ABOUT WOLFRAM ALPHA
8. ABOUT TYPOGRAPHY
9. ABOUT THE DIGITAL MATERIAL FOR THE BOOK
10. ABOUT THE AUTHORS
11. REFERENCES
ABOUT THE COMPANION WEBSITE
1 SOME INTRODUCTORY PROBLEMS
1. 1.1 TICKET PRICES
2. 1.2 HOW LONG WILL THE PASTURE LAST IN A FIELD?
3. 1.3 A BIT OF CHEMISTRY
4. 1.4 SYDNEY HARBOR BRIDGE
5. 1.5 PERSPECTIVE
6. 1.6 LAKE ERIE’S AREA
7. 1.7 ZEBRA CROSSING
8. 1.8 THE SECURITY CASE
9. 1.9 PERSONAL MEASUREMENTS
10. 1.10 HEIGHT OF THE BODY
11. 1.11 LAMP POLE
12. 1.12 THE SKYSCRAPER
13. 1.13 THE FENCE
14. 1.14 THE CORRIDOR
15. 1.15 BIRD FEEDERS
16. 1.16 GOLF
2 LINEAR MODELS
1. 2.1 ARE WOMEN FASTER THAN MEN?
2. 2.2 TAXI COMPANIES
3. 2.3 CRIME DEVELOPMENT
4. 2.4 THE METAL WIRE
5. 2.5 OPTIONS TRADING
6. 2.6 FLYING FOXES
7. 2.7 KNOTS ON A ROPE
8. 2.8 THE CANDLE
9. 2.9 HOOKE’S LAW
10. 2.10 RANKING
11. 2.11 DOLBEAR'S LAW
12. 2.12 MAN AT OFFICE
13. 2.13 A STACK OF PAPER
14. 2.14 MILK PRODUCTION IN COWS
3 NONLINEAR EMPIRICAL MODELS I
1. 3.1 GALAXY ROTATION
2. 3.2 OLYMPIC POLE VAULTING
3. 3.3 KEPLER’S THIRD LAW
4. 3.4 DENSITY
5. 3.5 YEAST
6. 3.6 COOLING I
7. 3.7 MODELING THE POPULATION OF IRELAND
8. 3.8 THE RULE OF 72
9. 3.9 THE FISH FARM I
10. 3.10 NEW ORLEANS TEMPERATURES
11. 3.11 THE RECORD MILE
12. 3.12 THE ROCKET
13. 3.13 STOPPING DISTANCES
14. 3.14 A BOTTLE WITH HOLES
15. 3.15 THE PENDULUM
16. 3.16 RADIO RANGE
17. 3.17 RUNNING 400 METERS
18. 3.18 BLUE WHALE
19. 3.19 USED CARS
20. 3.20 TEXTS
4 NONLINEAR EMPIRICAL MODELS II
1. 4.1 COOLING II
2. 4.2 BODY SURFACE AREA
3. 4.3 WARM‐BLOODED ANIMALS
4. 4.4 CONTROL OF INSECT PESTS
5. 4.5 SELLING MAGAZINES FOR CHRISTMAS
6. 4.6 TUMOR
7. 4.7 FREE FALL
8. 4.8 CONCENTRATION
9. 4.9 AIR CURRENT
10. 4.10 TIDES
11. 4.11 FITNESS
12. 4.12 LIFE EXPECTANCY VERSUS AVERAGE INCOME
13. 4.13 STOCKHOLM CENTER
14. 4.14 WORKFORCE
15. 4.15 POPULATION OF SWEDEN
16. 4.16 WHO KILLED THE LION?
17. 4.17 AIDS IN UNITED STATES
18. 4.18 THERMAL COMFORT
19. 4.19 WATTS AND LUMEN
20. 4.20 THE BEAUFORT SCALE
21. 4.21 THE VON BERTALANFFY GROWTH EQUATION
5 MODELING WITH CALCULUS
1. 5.1 THE FISH FARM II
2. 5.2 TITRATION
3. 5.3 THE BOWL
4. 5.4 THE AIRCRAFT WING
5. 5.5 THE GATEWAY ARCH IN ST. LOUIS
6. 5.6 VOLUME OF A PEAR
7. 5.7 STORM FLOOD
8. 5.8 EXERCISE
9. 5.9 BICYCLE REFLECTORS
10. 5.10 CARDIAC OUTPUT
11. 5.11 MEDICATION
12. 5.12 NEW SONG ON SPOTIFY
13. 5.13 TEMPERATURE CHANGE
14. 5.14 TAR
15. 5.15 BICYCLE REFLECTORS REVISITED
16. 5.16 GAS PRESSURE
17. 5.17 AIRBORNE ATTACKS
18. 5.18 RAILROAD TRACKS
19. 5.19 COBB–DOUGLAS PRODUCTION FUNCTIONS
20. 5.20 FUTURE CARBON DIOXIDE EMISSIONS
21. 5.21 OVERTAKING
22. 5.22 POPULATION DYNAMICS OF INDIA
23. 5.23 DRAG RACING
24. 5.24 SUPER EGGS
25. 5.25 MEASURING STICKS
26. 5.26 THE LECTURE HALL
27. 5.27 PROGRESSIVE BRAKING DISTANCES
28. 5.28 CYLINDER IN A CONE
6 USING DIFFERENTIAL EQUATIONS
1. 6.1 COOLING III
2. 6.2 MOOSE HUNTING
3. 6.3 THE WATER CONTAINER
4. 6.4 SKYDIVING
5. 6.5 FLU EPIDEMICS
6. 6.6 USA’S POPULATION
7. 6.7 PREDATORS AND PREY
8. 6.8 SMOKE
9. 6.9 ALCOHOL CONSUMPTION
10. 6.10 WHO KILLED THE MATHEMATICS TEACHER
11. 6.11 RIVER CLAMS
12. 6.12 CONTAMINATION
13. 6.13 DAMPED OSCILLATION
14. 6.14 THE POTASSIUM–ARGON METHOD
15. 6.15 BARIUM, LANTHANUM, AND CERIUM
16. 6.16 IODINE
17. 6.17 ENDEMIC EPIDEMICS
18. 6.18 WAR
19. 6.19 FARMERS, BANDITS, AND RULERS
20. 6.20 EPIDEMICS WITHOUT IMMUNITY
21. 6.21 ZOMBIE APOCALYPSE I
22. 6.22 ZOMBIE APOCALYPSE II
7 GEOMETRICAL MODELS
1. 7.1 THE LOOPING PEN
2. 7.2 COMPARING AREAS
3. 7.3 CROSSING LINES
4. 7.4 POINTS IN A TRIANGLE
5. 7.5 TRISECTED AREA
6. 7.6 SPIROGRAPH
7. 7.7 CONNECTED LP PLAYERS
8. 7.8 FOLDING PAPER
9. 7.9 THE LOCOMOTIVE
10. 7.10 MAXIMUM VOLUME
11. 7.11 PASCAL’S SNAIL OR LIMAÇON
12. 7.12 EQUILATERAL TRIANGLE DISSECTION
13. 7.13 DIVIDING THE SIDES OF A TRIANGLE
14. 7.14 THE PEDAL TRIANGLE
15. 7.15 THE INFINITY DIAGRAM
16. 7.16 DISSECTING A CIRCULAR SEGMENT
17. 7.17 NEUBERG CUBIC ART
18. 7.18 PHASE PLOTS FOR TRIANGLES
19. 7.19 THE JOUKOWSKI AIRFOIL
8 DISCRETE MODELS
1. 8.1 THE CABINETMAKER
2. 8.2 WEATHER
3. 8.3 Squirrels
4. 8.4 CHLORINE
5. 8.5 THE DEER FARM
6. 8.6 ANALYZING A NUMBER SEQUENCE
7. 8.7 INNER AREAS IN A SQUARE
8. 8.8 INNER AREAS IN A TRIANGLE
9. 8.9 A CLIMATE MODEL BASED ON ALBEDO
10. 8.10 TRAFFIC JAM
11. 8.11 WILDFIRE
12. 8.12 A MODERN CARPENTER
13. 8.13 CONWAY'S GAME OF LIFE
14. 8.14 MATRIX TAXIS
15. 8.15 THE CAR PARK
16. 8.16 SELECTING A COLLAGE
17. 8.17 APPORTIONMENT
18. 8.18 STEINER TREES FOR REGULAR POLYGONS
19. 8.19 HUGS AND HIGH FIVES
20. 8.20 PYTHAGOREAN TRIPLES
21. 8.21 CREDITS
22. 8.22 THE PIANO
9 MODELING IN THE CLASSROOM
1. 9.1 THE TEACHER CREATING DIAGRAMS
2. 9.2 STUDENT’S LAB REPORTS
3. 9.3 MAKING SCREENCAST INSTRUCTIONS
4. 9.4 DEMONSTRATIONS
5. 9.5 STUDENTS INVESTIGATING CONSTRUCTIONS
6. 9.6 WORKING IN GROUPS
7. 9.7 STUDENTS CONSTRUCTING MODELS
8. 9.8 BROADER ASSIGNMENTS
9. 9.9 THE SAME OR DIFFERENT ASSIGNMENTS
10. 9.10 PREVIOUS ASSIGNMENTS
11. 9.11 THE CONSULTANCY BUREAU
10 ASSESSING MODELING
1. 10.1 TO EVALUATE MATHEMATICAL MODELING ASSIGNMENTS
2. 10.2 CONCRETIZING GRADING CRITERIA
3. 10.3 EVALUATING STUDENTS’ WORK
11 ASSESSING MODELS
1. 11.1 RELATIVE ERROR
2. 11.2 CORRELATION
3. 11.3 SUM OF SQUARED ERRORS
4. 11.4 SIMPLE LINEAR REGRESSION
5. 11.5 MULTIPLE REGRESSION ANALYSIS
6. 11.6 NONLINEAR REGRESSION
7. 11.7 CONFIDENCE INTERVALS
8. 11.8 2D CONFIDENCE INTERVAL TOOLS
12 INTERPRETING MODELS
1. 12.1 MATHEMATICAL REPRESENTATIONS
2. 12.2 GRAPHICAL REPRESENTATIONS
3. 12.3 A SAMPLE MODEL INTERPRETED
4. 12.4 CREATING THE MODEL
APPENDIX A: INTRODUCTION TO GEOGEBRA
1. A.1 GEOGEBRATUBE AND THE ECOSYSTEM
2. A.2 GEOGEBRABOOKS
3. A.3 GEOGEBRA ON DIFFERENT DEVICES
4. A.4 THE USER INTERFACE
5. A.5 CUSTOMIZING GEOGEBRA
6. A.6 SLIDERS
7. A.7 BASIC SKILLS AND EXERCISES
8. A.8 WRITING INSTRUCTIONS
9. A.9 REMEMBER
APPENDIX B: FUNCTION LIBRARY
1. B.1 DIFFERENT FUNCTIONS AND THEIR PARAMETRIZATIONS
2. B.2 LINEAR TRANSFORMATIONS IN GENERAL
3. B.3 DRAGGING A FUNCTION IN GEOGEBRA
4. B.4 GEOGEBRA'S GENERIC FITTING COMMANDS
5. B.5 EXAMPLE OF A GENERIC FIT
INTEGER PROPERTIES
INDEX
LIST OF PROBLEMS BY NAME
END USER LICENSE AGREEMENT

List of Tables

Chapter 01
1. TABLE 1.1 Linear Relationship between the Number of Sold Air Seats and the Price for a Seat
2. TABLE 1.2 Organizing Your Data in a Table
3. TABLE 1.3 Two by Two Table for Organizing Proportional Data
4. TABLE 1.4 Proportionality Is Invariant under Rotation and Mirroring Operations
Chapter 02
1. TABLE 2.1 Olympic Winners in the 200 Meter Sprint
2. TABLE 2.2 Number of Reported Crimes in Sweden, 1950 to 1990
3. TABLE 2.3 Thermal Expansion of a Wire
4. TABLE 2.4 Points versus Rank
5. TABLE 2.5 Cricket Chirps at Different Temperatures
6. TABLE 2.6 Heights and Weights for Male Office Workers
Chapter 03
1. TABLE 3.1 Galaxy Rotation Values
2. TABLE 3.2 Olympic Records in Pole Vaulting for Men, 1900 to 1996
3. TABLE 3.3 Planetary Average Distances from the Sun and Orbital Times in Our Solar System
4. TABLE 3.4 Density of Water at Different Temperatures
5. TABLE 3.5 Number of Yeast Cells Does Not Grow without Bounds
6. TABLE 3.6 Cooling Temperatures
7. TABLE 3.7 Population of Ireland, 1672 to 1966
8. TABLE 3.8 New Orleans Temperature Averages, 1951 to 1980
9. TABLE 3.9 Rocket elevations
Chapter 04
1. TABLE 4.1 Cooling a Cupper Sphere
2. TABLE 4.2 Physical Exam Data for 16 Different Patients
3. TABLE 4.3 Heat Production of Some Animals
4. TABLE 4.4 Trapped Male Moths as a Function of the Odorant Amount
5. TABLE 4.5 Points from Sales
6. TABLE 4.6 Earnings from Points
7. TABLE 4.7 Calculating the Size of the Errors of the Model
8. TABLE 4.8 “Price” of an iPhone
9. TABLE 4.9 Coordinates of Tumor Edge
10. TABLE 4.10 Distance Fallen as a Function of Time
11. TABLE 4.11 Concentration of Dissolved Agent over Time
12. TABLE 4.12 Jackknife Resampling Results
13. TABLE 4.13 Varying the Initial Concentration to Get a Range of Possible Values
14. TABLE 4.14 Air Current Data
15. TABLE 4.15 Tidal Data for Nova Scotia
16. TABLE 4.16 An Increasing Workforce
17. TABLE 4.17 Population of Sweden
18. TABLE 4.18 AIDS Cases in the United States
19. TABLE 4.19 Watts to Lumen
Chapter 05
1. TABLE 5.1 Titration Results
2. TABLE 5.2 Titration Points by Different Models
3. TABLE 5.3 Results So Far
4. TABLE 5.4 Coordinates for a Future Bowl
5. TABLE 5.5 Coordinates for a Wing
6. TABLE 5.6 Coordinates for the Gateway Arch
7. TABLE 5.7 Treadmill Data for Increasing Load
8. TABLE 5.8 Oxygen Uptake
9. TABLE 5.9 Power during a Race
10. TABLE 5.10 Reading of Dye Concentration in Blood
11. TABLE 5.11 Amount of Medicine in Blood over Time
12. TABLE 5.12 Temperature Change in Gothenburg
13. TABLE 5.13 Amount of Tar Inhaled in Different Sections of the Cigarette
14. TABLE 5.14 Comparison of Measured and Calculated Values of Tar Amounts
15. TABLE 5.15 Equations from Wolfram Alpha
16. TABLE 5.16 Comparing Different Solutions to Estimate Errors
17. TABLE 5.17 Predicted Carbon Dioxide Emissions
Chapter 06
1. TABLE 6.1 Cooling Data
2. TABLE 6.2 US Population over Time
3. TABLE 6.3 Comparing the Models
4. TABLE 6.4 Comparing Polynomial Models
5. TABLE 6.5 Are Antelopes Endangered?
6. TABLE 6.6 Four Different Solution Attempts
Chapter 07
1. TABLE 7.1 Number of “Flower Petals” as r Varies
Chapter 08
1. TABLE 8.1 Transition Probabilities of Three Weather Conditions
2. TABLE 8.2 Number of Areas Overrun by Different Squirrel Types from Year to Year
3. TABLE 8.3 Instructions for Command Buttons
Chapter 10
1. TABLE 10.1 Summary of Criteria for Grading in Mathematics
2. TABLE 10.2 Summary of Grading Criteria Adjusted to Mathematical Modeling
Chapter 11
1. TABLE 11.1 Measured Times for 10 Swings of a Pendulum

List of Illustrations

Introduction
1. FIGURE 0.1 Example of a model presenting Ohms law as a (near) proportionality.
2. FIGURE 0.2 Main stages in modeling (from Mason, 1988, p. 209).
Chapter 01
1. FIGURE 1.1 How to copy a formula using the fill handle.
2. FIGURE 1.2 Table with values and corresponding points of a graphical representation.
3. FIGURE 1.3 The dock button that allows you to dock a free window.
4. FIGURE 1.4 Graphic Window 2 showing another graph with a different scale.
5. FIGURE 1.5 Income as it decreases dramatically after the first 50 empty seats.
6. FIGURE 1.6 Cow in a dandelion field.
7. FIGURE 1.7 Linear model.
8. FIGURE 1.8 Nonlinear model based on the concept of cow days.
9. FIGURE 1.9 Solutions.
10. FIGURE 1.10 Dynamic graphic model of the chemical mixture.
11. FIGURE 1.11 Solving a problem automatically.
12. FIGURE 1.12 Input box dialogue.
13. FIGURE 1.13 A simple calculator.
14. FIGURE 1.14 Button dialogue.
15. FIGURE 1.15 CAS window used to help you solve for variables in a formula.
16. FIGURE 1.16 Where to find the layout settings.
17. FIGURE 1.17 A “Solver” tool in GeoGebra.
18. FIGURE 1.18 Sydney Harbor Bridge. By Yun Huang Yong via Flickr, CC BY‐SA 2.0. https://www.flickr.com/photos/goosmurf/3001997390/sizes/o/.
19. FIGURE 1.19 One of the bridge heads stands prominent in the mist behind a ferry.
20. FIGURE 1.20 Digitize It, a program for digitalization of photos.
21. FIGURE 1.21 Perspective run.
22. FIGURE 1.22 Lock the image in the coordinate system.
23. FIGURE 1.23 Measuring the girl’s height in GeoGebra.
24. FIGURE 1.24 Lake Erie, Google Earth. Image Data from Landsat and NOAA.
25. FIGURE 1.25 Estimating the area of the lake by a polygon.
26. FIGURE 1.26 A segment used to measure a distance.
27. FIGURE 1.27 Wolfram Alpha can provide you with geographical data.
28. FIGURE 1.28 Lorain Lighthouse.
29. FIGURE 1.29 Zebra crossing.
30. FIGURE 1.30 Camera photographing people strolling over Zebra crossing.
31. FIGURE 1.31 Triangles KCB and POB are uniform.
32. FIGURE 1.32 Distances from the bottom of the image to the upper edge of the white lines as shown in the spreadsheet.
33. FIGURE 1.33 Measuring a man’s length to find the scale.
34. FIGURE 1.34 Window to change the graphics area you want the object to be visible in.
35. FIGURE 1.35 Regression used to find answers.
36. FIGURE 1.36 Type of security locker case.
37. FIGURE 1.37 Dynamic model of the locker case.
38. FIGURE 1.38 Plot of cost c against side s.
Chapter 02
1. FIGURE 2.1 Information stored in the GeoGebra spreadsheet.
2. FIGURE 2.2 Women’s record times as seen to decrease faster than men’s.
3. FIGURE 2.3 Taxis G7's Paris fares 2015.
4. FIGURE 2.4 Using the spreadsheet in GeoGebra.
5. FIGURE 2.5 Comparing two taxi companies.
6. FIGURE 2.6 Style Bars for the Graphics and Algebra windows.
7. FIGURE 2.7 How to define a slider.
8. FIGURE 2.8 Objects may be redefined by a double‐click on them.
9. FIGURE 2.9 Any object may be visible in both graphic areas at the same time, in one of them at a time, or be completely hidden.
10. FIGURE 2.10 A point that leaves a trace indicates the functional relation.
11. FIGURE 2.11 Linear relations vary with respect to the value of k.
12. FIGURE 2.12 Varying k gives rational functions with a vertical asymptote at k = 5.
13. FIGURE 2.13 Crime development in Sweden, 1950 to 1990.
14. FIGURE 2.14 Linear regression of the data from 1970 and forward.
15. FIGURE 2.15 Error limits by manual calculations.
16. FIGURE 2.16 Two variable analyses in GeoGebra.
17. FIGURE 2.17 Calculations in the spreadsheet.
18. FIGURE 2.18 Measured values in a diagram.
19. FIGURE 2.19 Make sure that the x‐axis will intersect the y‐axis at y = 999.
20. FIGURE 2.20 Data points and the calculated regression line.
21. FIGURE 2.21 Trading at the São Paolo stock exchange.
22. FIGURE 2.22 Bull and Bear in front of Frankfurt’s stock exchange.
23. FIGURE 2.23 Profit of buying a call option as a piecewise defined function.
24. FIGURE 2.24 Profit of selling a call option is also a piecewise defined function.
25. FIGURE 2.25 Total profit for a spread with call options having two “knees.”
26. FIGURE 2.26 A spread where the profit never is negative.
27. FIGURE 2.27 Grey‐headed flying fox.
28. FIGURE 2.28 Two data points with large error margins visualized as two sets of four points.
29. FIGURE 2.29 Linear models.
30. FIGURE 2.30 Exponential models can give really high results.
Chapter 03
1. FIGURE 3.1 Messier 88 Galaxy.
2. FIGURE 3.2 A minimum is easiest found by fitting a polynomial to the points closest to the minimum.
3. FIGURE 3.3 Théo Mancheron in the pole vaults completions in the Athletics Games France 2013.
4. FIGURE 3.4 Graphical representation for the gold medal heights in pole vault jumping in the Olympic Games.
5. FIGURE 3.5 Line tool box in GeoGebra.
6. FIGURE 3.6 Simple linear model describing how the height of pole vaults has increased in 1912 to 1996.
7. FIGURE 3.7 A first attempt to fit a logarithmic function to our data.
8. FIGURE 3.8 A function of the expected shape.
9. FIGURE 3.9 Pole vaulting with stiff poles a long time ago.
10. FIGURE 3.10 Planet’s orbital time against their average distance to the Sun represented in GeoGebra.
11. FIGURE 3.11 Style Bar is shown when you click on the small upper triangle.
12. FIGURE 3.12 A power function fits the data well.
13. FIGURE 3.13 A quadratic function fits the data almost perfectly.
14. FIGURE 3.14 Two Variable Regression Analysis dialogue.
15. FIGURE 3.15 Two Variable Regression Analysis with (a) second‐ and (b) fourth‐degree polynomial. Pay attention to the difference between the residual diagrams.
16. FIGURE 3.16 Two Variable Analysis using a logistic model.
17. FIGURE 3.17 Hot water (cropped).
18. FIGURE 3.18 Exponential fit to data not decreasing toward zero.
19. FIGURE 3.19 First six points are selected for removal.
20. FIGURE 3.20 New graph fits the data much better because the first six data points are excluded.
21. FIGURE 3.21 General fit to the data, using the model m(x).
22. FIGURE 3.22 Intersection will give us the answer to our question.
23. FIGURE 3.23 120 Irish passengers were on the Titanic in 1912.
24. FIGURE 3.24 Population of Ireland, 1672 to 1966.
25. FIGURE 3.25 Two different exponential functions fit to different parts of the data.
26. FIGURE 3.26 Money, making more money.
27. FIGURE 3.27 Both functions are very close in a large interval.
28. FIGURE 3.28 Difference between two functions can give more information than the individual functions if they are close.
29. FIGURE 3.29 Finding a better, but as simple, function for low interest rates.
30. FIGURE 3.30 70/p works best in the interval 1 to 5%.
31. FIGURE 3.31 Fish farming in Turkey.
32. FIGURE 3.32 Nearly cubic function fits the data well.
33. FIGURE 3.33 Total mass as a function of time.
34. FIGURE 3.34 Relation between r and the optimal time.
35. FIGURE 3.35 Manually fitted relation between a parameter and its function.
36. FIGURE 3.36 Parameter k’s influence over the optimal time for extracting the fish.
37. FIGURE 3.37 New Orleans town center.
38. FIGURE 3.38 Temperature shown as both a bar chart (data) and a function (model).
Chapter 04
1. FIGURE 4.1 A cooling graph fitted to data.
2. FIGURE 4.2 A sixth‐degree polynomial fits the first few data points well but fails in the region where we are most interested in exact results.
3. FIGURE 4.3 A fourth‐degree polynomial with 5 parameters passes exactly through 5 data points.
4. FIGURE 4.4 Calculating the length of the patients in the spreadsheet.
5. FIGURE 4.5 Calculating the sum of squared errors, the SSE.
6. FIGURE 4.6 Three warm‐blooded animals.
7. FIGURE 4.7 A linear fit seems to work well enough.
8. FIGURE 4.8 Constructing log‐log diagrams in GeoGebra.
9. FIGURE 4.9 Initial and logarithmic values with a power function and corresponding linear function fitted to the data.
10. FIGURE 4.10 Lymantria dispar, the gypsy moth.
11. FIGURE 4.11 Just data points.
12. FIGURE 4.12 Finding a power function to fit data.
13. FIGURE 4.13 A popular phone.
14. FIGURE 4.14 Both tables represented graphically.
15. FIGURE 4.15 Earnings are a linear function of the points received.
16. FIGURE 4.16 A quadratic function is fit to the first part of the table.
17. FIGURE 4.17 Using CAS to calculate f(g(x)).
18. FIGURE 4.18 Table represented as a piecewise defined function.
19. FIGURE 4.19 Converting points to earnings, as a piecewise defined function.
20. FIGURE 4.20 Earnings as a piecewise defined function of sales.
21. FIGURE 4.21 Input Box dialogue.
22. FIGURE 4.22 An “app” to calculate our earnings.
23. FIGURE 4.23 Creating a minimalistic look.
24. FIGURE 4.24 Using the app online.
25. FIGURE 4.25 Possible cancer in left lung.
26. FIGURE 4.26 How we fit a circle to geometrical data points.
27. FIGURE 4.27 Circle in the process of being adjusted to the data points.
28. FIGURE 4.28 Final, best circle.
29. FIGURE 4.29 Fitting an implicit curve.
30. FIGURE 4.30 Letting GeoGebra re‐discover the conic from the curve.
31. FIGURE 4.31 Analyzing the conic.
32. FIGURE 4.32 Sky diving.
33. FIGURE 4.33 Exponential function fitted to the data.
34. FIGURE 4.34 Calculating errors in the spreadsheet.
35. FIGURE 4.35 Docking button allows you to dock or undock a window from the rest.
36. FIGURE 4.36 Residual diagram created manually.
37. FIGURE 4.37 Same construction with another model.
38. FIGURE 4.38 Sum of two exponential functions can model an initial pulse.
39. FIGURE 4.39 Danger of having too many parameters.
40. FIGURE 4.40 More elaborate model—maybe?
41. FIGURE 4.41 Too few parameters fit the data poorly.
42. FIGURE 4.42 Wing profile in a wind tunnel.
43. FIGURE 4.43 Scattered data requires a good theory to fit.
44. FIGURE 4.44 Intuitive way of estimating the error of the fitted parameters.
45. FIGURE 4.45 Hall’s Harbor, Nova Scotia, at low tide.
46. FIGURE 4.46 A trigonometric fit.
47. FIGURE 4.47 Successive regression analysis.
Chapter 05
1. FIGURE 5.1 Loch Ainort fish farm.
2. FIGURE 5.2 Two possible supply functions.
3. FIGURE 5.3 Finding the maximum profit from a linear supply function.
4. FIGURE 5.4 Maximum profit for a cubic supply function.
5. FIGURE 5.5 Using more information to find a better supply function.
6. FIGURE 5.6 Points are created when you type the coordinates within parentheses.
7. FIGURE 5.7 Typical titration curve.
8. FIGURE 5.8 Derivative of the titration curve.
9. FIGURE 5.9 Second‐Degree Polynomial Fit to Three Points for the Derivative.
10. FIGURE 5.10 Pulse function of the type 1/(1 + x²) adjusted to the derivative of the titration curve.
11. FIGURE 5.11 Normal distribution curve adjusted for our purposes.
12. FIGURE 5.12 Linear and cubic regression functions from the points around the half‐titration point.
13. FIGURE 5.13 Exponential function is used to trace the inside of a bowl.
14. FIGURE 5.14 3D rendering of a bowl created from an exponential function.
15. FIGURE 5.15 Mathematical model of a bowl.
16. FIGURE 5.16 Cross‐sectional area as a function of y along the x‐axis.
17. FIGURE 5.17 One liter bowl.
18. FIGURE 5.18 Schematic image of an aircraft wing.
19. FIGURE 5.19 Model of an aircraft wing.
20. FIGURE 5.20 Gateway Arch in St. Louis.
21. FIGURE 5.21 Model of the Gateway Arch.
22. FIGURE 5.22 Wolfram Alpha helps you to calculate the arc length to about 453 m.
23. FIGURE 5.23 Fitting a curve in Curve Expert.
24. FIGURE 5.24 User‐defined model in Curve Expert.
25. FIGURE 5.25 Result from Curve Expert.
26. FIGURE 5.26 Eight varieties of pears.
27. FIGURE 5.27 Williams pear. Public domain/Wikimedia Commons. https://upload.wikimedia.org/wikipedia/commons/0/08/Williams_Bon_Chr%C3%A9tien_1822.png.
28. FIGURE 5.28 Model for the measuring of the volume of a pear.
29. FIGURE 5.29 Water flow in a river.
30. FIGURE 5.30 Heavy rain falling over the Murrumbidgee river.
31. FIGURE 5.31 Background image in GeoGebra.
32. FIGURE 5.32 Positioning a background image in GeoGebra.
33. FIGURE 5.33 Measuring a given function and recreating it.
34. FIGURE 5.34 Area under the curve is calculated.
35. FIGURE 5.35 Sprint finish.
36. FIGURE 5.36 Power that seems to increase linearly over time.
37. FIGURE 5.37 Oxygen uptake appearing to be nonlinear.
38. FIGURE 5.38 Table created with values for the new function u02(t).
39. FIGURE 5.39 Selecting the graphics areas where the function should be visible.
40. FIGURE 5.40 Calculating the integral that gives the total amount of oxygen taken up by the runner.
41. FIGURE 5.41 Two different running styles.
42. FIGURE 5.42 Let the values start on row 3.
43. FIGURE 5.43 Oxygen uptake values in column I and J.
44. FIGURE 5.44 Jacob will take up more oxygen than Michael.
45. FIGURE 5.45 Bicycle side reflectors.
46. FIGURE 5.46 First attempt with the reflector moved in the wrong direction.
47. FIGURE 5.47 Correct path for the reflector.
48. FIGURE 5.48 Cycloid.
49. FIGURE 5.49 Parametric curve in GeoGebra.
50. FIGURE 5.50 Area calculated by fitting a polygon to data points.
51. FIGURE 5.51 Seventh‐degree polynomial is not a candidate for a good fit but there are worse fits.
52. FIGURE 5.52 Logarithmic diagram. The recirculation part lies above the line.
53. FIGURE 5.53 Diagram adjusted for the recirculation.
54. FIGURE 5.54 Pills containing L‐DOPA.
55. FIGURE 5.55 Typical pulse.
56. FIGURE 5.56 Two different functions adjust to different intervals.
57. FIGURE 5.57 Exponential function without constant term used to model the tail of the pulse.
58. FIGURE 5.58 Fitting a product of a power function and an exponential function.
59. FIGURE 5.59 Model in three parts.
60. FIGURE 5.60 Spotify logotype.
61. FIGURE 5.61 Having determined the period and phase, you now need to determine the amplitude.
62. FIGURE 5.62 Complete model for the playing rate.
63. FIGURE 5.63 Complete model for the total number of played songs.
64. FIGURE 5.64 Parabola as model for the playing rate.
65. FIGURE 5.65 Total number of played songs as a cubic polynomial.
66. FIGURE 5.66 Comparison of the total times the song was played.
67. FIGURE 5.67 Comparison of the playing rate.
68. FIGURE 5.68 Gothenburg panorama.
69. FIGURE 5.69 Differences can be hard to model.
70. FIGURE 5.70 Two sine functions are better than one.
71. FIGURE 5.71 Inhaled tar (mg/cm) as a function of the position of the glow on the cigarette.
72. FIGURE 5.72 Five lines intersect in 10 points.
73. FIGURE 5.73 Drag racing time slip sample.
Chapter 06
1. FIGURE 6.1 Fitting a rational function to cooling data.
2. FIGURE 6.2 New attempt with better—but still not completely satisfying—results.
3. FIGURE 6.3 Rational function with an arbitrary exponent.
4. FIGURE 6.4 Best answer so far.
5. FIGURE 6.5 Moose cow with calves. [CC BY 2.0 (http://creativecommons.org/licenses/by/2.0)], via Wikimedia Commons. https://upload.wikimedia.org/wikipedia/commons/1/1f/Moose_Cow_and_Calves_%286187093400%29.jpg.
6. FIGURE 6.6 Symbolic solution of a differential equation in GeoGebra.
7. FIGURE 6.7 Numerical solution of a differential equation in GeoGebra.
8. FIGURE 6.8 Direct solution of a differential equation in GeoGebra with a slope field.
9. FIGURE 6.9 Possible future for the moose without hunting.
10. FIGURE 6.10 Collapsing moose tribe.
11. FIGURE 6.11 Moderate hunt to ensure a stable population.
12. FIGURE 6.12 Water dispenser. [CC BY‐SA 3.0 (http://creativecommons.org/licenses/by‐sa/3.0)], via Wikimedia Commons. https://upload.wikimedia.org/wikipedia/commons/5/54/Water_dispenser_with_lemons.JPG.
13. FIGURE 6.13 Slope field with a solution through a given point.
14. FIGURE 6.14 Complete solution.
15. FIGURE 6.15 Skydivers quickly reach their terminal velocity. [CC BY‐SA 4.0 (http://creativecommons.org/licenses/by‐sa/4.0)], via Wikimedia Commons. https://upload.wikimedia.org/wikipedia/commons/2/22/Skydiving_16_way_Over_Peterborough.jpg.
16. FIGURE 6.16 Wolfram Alpha used to solve a differential equation based on initial conditions.
17. FIGURE 6.17 vt‐Diagram for a parachute jumper.
18. FIGURE 6.18 Time of flight almost inversely proportional to the square root of the mass.
19. FIGURE 6.19 vt‐Diagram for a skydiving human.
20. FIGURE 6.20 Peregrine in flight. [CC BY 2.0 (http://creativecommons.org/licenses/by/2.0)], via Wikimedia Commons. https://upload.wikimedia.org/wikipedia/commons/1/1e/Peregrine_Falcon_in_flight.jpg.
21. FIGURE 6.21 Swedish Influenza season 2014–2015 summarized in June 2015.
22. FIGURE 6.22 Building a spreadsheet model of epidemics.
23. FIGURE 6.23 Creating points in the spreadsheet.
24. FIGURE 6.24 Visualizing the SIR model.
25. FIGURE 6.25 Connecting the dots.
26. FIGURE 6.26 Adjust the parameters so that the model fits with the epidemic.
27. FIGURE 6.27 Relatively low value for R₀ still demands that more than every fourth person in the population must be vaccinated to stop the epidemic.
28. FIGURE 6.28 GeoGebra’s own numerical solution.
29. FIGURE 6.29 People—How many will we become?
30. FIGURE 6.30 US population growth over the last 200 years.
31. FIGURE 6.31 Logistic function that fits the data well.
32. FIGURE 6.32 Spreadsheet so far.
33. FIGURE 6.33 Graphics Area 2 showing r(t) decreasing over time.
34. FIGURE 6.34 New solution g(x) compared with the logistic f(x) (dashed).
35. FIGURE 6.35 Long‐term prognoses differences.
36. FIGURE 6.36 r(t) as an exponentially decreasing function.
37. FIGURE 6.37 Another function that can help you understand the data.
38. FIGURE 6.38 Human population will pass the billion level around year 2260.
39. FIGURE 6.39 Two Variable Regression Analysis tool gives quick information when comparing different standard models.
40. FIGURE 6.40 Table values in a phase diagram.
41. FIGURE 6.41 Model under construction.
42. FIGURE 6.42 Graphics 2 showing the evolution for individuals over time.
43. FIGURE 6.43 Typical phase‐and‐time diagram for Lotka–Volterras equations.
44. FIGURE 6.44 Function for the food intake relation. Maximum value K = 5. Half this value is reached when x/y = 1.
45. FIGURE 6.45 Catastrophic situation where both spices becomes extinct.
46. FIGURE 6.46 Greek letters accessed by clicking the α‐sign in the input field.
47. FIGURE 6.47 Stochastic model affected by randomness.
48. FIGURE 6.48 Dynamic solution of a differential equation.
49. FIGURE 6.49 Seven‐day solution.
50. FIGURE 6.50 Final value does not change from week to week.
51. FIGURE 6.51 Amount of alcohol in the blood decreases almost linearly over time if taken at the start of the evening.
52. FIGURE 6.52 Alcohol intake for the same person when drinking at a constant rate.
53. FIGURE 6.53 Exponentially decreasing fit to two measured temperatures.
54. FIGURE 6.54 Solid curve showing body temperature, which decreases when the outdoor temperature is falling.
Chapter 07
1. FIGURE 7.1 Supported pens.
2. FIGURE 7.2 Building a model of a sliding pen.
3. FIGURE 7.3 Loop that is not a circle.
4. FIGURE 7.4 Compare the areas.
5. FIGURE 7.5 It is easy to measure and compare areas in GeoGebra.
6. FIGURE 7.6 Special case that might be easier to prove.
7. FIGURE 7.7 Four crossing lines form four triangles.
8. FIGURE 7.8 Bisectors and the inscribed circle of a triangle.
9. FIGURE 7.9 Create New Tool Dialogue.
10. FIGURE 7.10 Remember to write a short help text.
11. FIGURE 7.11 New tool created.
12. FIGURE 7.12 Four inscribed circles.
13. FIGURE 7.13 With four circumscribed circles, all four seem to pass through the same point.
14. FIGURE 7.14 Euler line coincides with several interesting points in a triangle.
15. FIGURE 7.15 Which point P is closest to all vertices?
16. FIGURE 7.16 Creating a dynamic text box.
17. FIGURE 7.17 Model for investigating the point “closest” to all vertices.
18. FIGURE 7.18 Drawing a color map to get a better feeling for how the sum varies.
19. FIGURE 7.19 Angles between PA, PB, and PC that seem to be 120°.
20. FIGURE 7.20 P does not lie on the Euler line.
21. FIGURE 7.21 Constructing the Fermat point.
22. FIGURE 7.22 Dudney’s dissection where a triangle can be rearranged to form a square.
23. FIGURE 7.23 Model for investigating the trisection of a triangle.
24. FIGURE 7.24 What properties for this particular trisection can we discover?
25. FIGURE 7.25 Another trisection of a triangle.
26. FIGURE 7.26 What is special about P, and what are its properties?
27. FIGURE 7.27 One of many different ways to divide a triangle into four parts of equal area.
28. FIGURE 7.28 One example of a spirograph.
29. FIGURE 7.29 Patterns produced with a spirograph.
30. FIGURE 7.30 Spirograph’s construction protocol.
31. FIGURE 7.31 Selecting which columns to show.
32. FIGURE 7.32 Spirograph model halfway finished.
33. FIGURE 7.33 Spirograph in action.
34. FIGURE 7.34 Auto‐spirograph generating a pattern directly when the sliders are updated.
35. FIGURE 7.35 Mechanical harmonograph.
36. FIGURE 7.36 Start the construction of an LP harmonograph.
37. FIGURE 7.37 When something is animated, a pause/play‐button appears in the lower left of the Graphics window.
38. FIGURE 7.38 Point J as equally far away from G and H.
39. FIGURE 7.39 First links finished.
40. FIGURE 7.40 Each object can be hidden or shown by clicking on the visibility buttons in the Algebra window.
41. FIGURE 7.41 Two final links.
42. FIGURE 7.42 Model of the linkage that is almost finished.
43. FIGURE 7.43 One of several possible patterns created by the LP‐harmonograph simulator.
44. FIGURE 7.44 Locus demonstrates the patterns that would occur if one of the LP players were standing still.
45. FIGURE 7.45 Folding paper.
46. FIGURE 7.46 Our model of the folded paper.
47. FIGURE 7.47 Trace of our measurements showing a clear maximum.
48. FIGURE 7.48 How to find the maximum of a locus.
49. FIGURE 7.49 Steam locomotive, or steam engine.
50. FIGURE 7.50 Simple model connecting circular to linear motion in a locomotive.
51. FIGURE 7.51 E’s x‐coordinate generating a cosine function.
52. FIGURE 7.52 Piston not describing a pure cosine function.
53. FIGURE 7.53 Pascal's snail.
54. FIGURE 7.54 Creating the pedal triangle.
55. FIGURE 7.55 Infinity diagram.
56. FIGURE 7.56 Dividing a circular segment in two equal areas with a line.
57. FIGURE 7.57 Art from the Euler line.
58. FIGURE 7.58 Reading a ternary diagram.
Chapter 08
1. FIGURE 8.1 Table upon which Ohio’s first constitution was signed. The table is exhibited in Chillicothe’s (Ohio’s first capital town) Town Hall.
2. FIGURE 8.2 Overlapping inequalities forming an area of interest.
3. FIGURE 8.3 Profit line through the optimal vertex with interesting points highlighted.
4. FIGURE 8.4 Raising the price.
5. FIGURE 8.5 Coding Python online at Codingground.
6. FIGURE 8.6 Beautiful weather at the outer coast line in the archipelago.
7. FIGURE 8.7 GeoGebra’s matrix calculations.
8. FIGURE 8.8a Red Squirrel.
9. FIGURE 8.8b Gray squirrel.
10. FIGURE 8.9 Spreadsheet transforms a table to a matrix.
11. FIGURE 8.10 Matrix2 shows the long‐term distribution of squirrel region types.
12. FIGURE 8.11 Tooting Bec Lido, south London. Nick Cooper at en.wikipedia [CC BY‐SA 3.0 (http://creativecommons.org/licenses/by‐sa/3.0) or GFDL (http://www.gnu.org/copyleft/fdl.html)], from Wikimedia Commons. https://upload.wikimedia.org/wikipedia/commons/5/5b/Tooting_Bec_Lido_20080724.JPG.
13. FIGURE 8.12 Decreasing exponential functionmodeled on the data.
14. FIGURE 8.13 Model where all parameters can be varied.
15. FIGURE 8.14 Fallow deer in field. Av Johann‐Nikolaus Andreae (originally posted to Flickr as p9036717.jpg) [CC BY‐SA 2.0 (http://creativecommons.org/licenses/by‐sa/2.0)], via Wikimedia Commons. https://upload.wikimedia.org/wikipedia/commons/f/f3/Fallow_deer_in_field.jpg.
16. FIGURE 8.15 Parameters and the spreadsheet model.
17. FIGURE 8.16 Hunt starts after six years.
18. FIGURE 8.17 Birthrate will decrease with the number of animals when the resources are limited.
19. FIGURE 8.18 Deer population living with limited resources but without being hunted.
20. FIGURE 8.19 Polynomial generating a number sequence.
21. FIGURE 8.20 Difference schema for analyzing a number sequence.
22. FIGURE 8.21 Wolfram Alpha suggests a rational function.
23. FIGURE 8.22 OEIS only finds well‐known series—why not try with 1, 3, 4, 7, 11….
24. FIGURE 8.23 Inner shapes of a square.
25. FIGURE 8.24 How to divide a segment in equal parts.
26. FIGURE 8.25 How to construct points, dividing a segment in different parts.
27. FIGURE 8.26 Ratio of the inner area to the whole area for differently positioned points.
28. FIGURE 8.27 Creating a new tool.
29. FIGURE 8.28 Measured data from the geometrical model.
30. FIGURE 8.29 Inverted value is a better representation for analyzing patterns.
31. FIGURE 8.30 Difference analysis of a number sequence generated by 2m² – 2m + 1 (column B).
32. FIGURE 8.31 n = 1 and m = 2 gives an inner area that is one‐fifth of the outerarea.
33. FIGURE 8.32 Inner shapes of a triangle.
34. FIGURE 8.33 Inner area of a triangle.
35. FIGURE 8.34 Regression analysis can be used to find generating polynomials to number sequences.
36. FIGURE 8.35 Nonagon numbers.
37. FIGURE 8.36 Sunrise over Haleakala.
38. FIGURE 8.37 Freehand function that defines the relation between the average temperature and albedo.
39. FIGURE 8.38 Spreadsheet model for the feedback loop.
40. FIGURE 8.39 Climate model exhibiting a global temperature collapse.
41. FIGURE 8.40 Spreadsheet part of the simulation of traffic jam.
42. FIGURE 8.41 Cars ready to go.
43. FIGURE 8.42 Creating a command button is easy.
44. FIGURE 8.43 Longer script executing several commands at once.
45. FIGURE 8.44 Update script for the time variable where positions and velocities are updated.
46. FIGURE 8.45 Center of the congestion at the gray car.
47. FIGURE 8.46 Hollow symbol car runs one turn but the density maximum (gray) has only moved a little.
48. FIGURE 8.47 Bugaboo forest fire.
49. FIGURE 8.48 Representation of the map in a spreadsheet.
50. FIGURE 8.49 So far no wildfire.
51. FIGURE 8.50 Simple GUI.
52. FIGURE 8.51 Writing a program to write another program ….
53. FIGURE 8.52 Wildfire when the wind is coming from the north.
54. FIGURE 8.53 Steiner Tree for a square.
55. FIGURE 8.54 Folding Pascal's triangle.
Chapter 09
1. FIGURE 9.1 Logotype of the Consultancy Bureau that all students were told to use in their reports and on the posters.
Chapter 10
1. FIGURE 10.1 Poster by Sebastian Genas.
Chapter 11
1. FIGURE 11.1 Geometrical interpretation of the method of least squares.
2. FIGURE 11.2 Two 2D confidence ellipses.
Chapter 12
1. FIGURE 12.1 Moving train.
2. FIGURE 12.2 Train’s movement and velocity in the same graphical representation.
Appendix A
1. FIGURE A.1 GeoGebra forum.
2. FIGURE A.2 Open from GeoGebraTube… dialogue.
3. FIGURE A.3 Shift‐click Open to open constructions on any website.
4. FIGURE A.4 GeoGebraBook with chapters.
5. FIGURE A.5 Starting GeoGebra for the first time.
6. FIGURE A.6 Spreadsheet window.
7. FIGURE A.7 Login for easier uploads.
8. FIGURE A.8 Dropdown toolbar.
9. FIGURE A.9 Selection and Point tools menus.
10. FIGURE A.10 Line and Construction tools menus.
11. FIGURE A.11 Polygon tools menu.
12. FIGURE A.12 Circle tools menu.
13. FIGURE A.13 Conics and Measurement tools menus.
14. FIGURE A.14 Transformation and Special tools menus.
15. FIGURE A.15 Action and Window tools menus.
16. FIGURE A.16 Input bar where you can type commands.
17. FIGURE A.17 Special characters input.
18. FIGURE A.18 GeoGebra has an extensive help system.
19. FIGURE A.19 GeoGebra will autocomplete commands for you.
20. FIGURE A.20 Algebra window and the graphics window show different representations of the objects.
21. FIGURE A.21 Options menu.
22. FIGURE A.22 Style Bars allow immediate adjustments to object appearances.
23. FIGURE A.23 Quick way to the Properties dialogue.
24. FIGURE A.24 Right‐click menu for the graphics window.
25. FIGURE A.25 Different tabs are represented as icons.
26. FIGURE A.26 Layout tab.
27. FIGURE A.27 Tabs in the Graphics window settings.
28. FIGURE A.28 Several different ways to zoom.
29. FIGURE A.29 Object properties is a tab in the Properties dialogue.
30. FIGURE A.30 Graphic representation of a number.
31. FIGURE A.31 Slider dialogue.
32. FIGURE A.32 Blue/white buttons control the object’s visibility.
33. FIGURE A.33 Automatic slider creation.
34. FIGURE A.34 Geometric construction without axes or grid.
35. FIGURE A.35 With the lines selected, choose a dashed line in the Style Bar.
36. FIGURE A.36 Finished construction, complete with angle measurements.
37. FIGURE A.37 GeoGebra spreadsheet. Notice the button Auxiliary Objects.
38. FIGURE A.38 Rectangle controlled by point B and C.
39. FIGURE A.39 2D drawing of a cube in GeoGebra.
40. FIGURE A.40 Creating three sliders to control the box.
41. FIGURE A.41 Creating the back plane.
42. FIGURE A.42 Dynamic box.
43. FIGURE A.43 Width and depth now both depend on the height of the box.
44. FIGURE A.44 Your model goes in one window, your analysis in the other.
45. FIGURE A.45 Recording P:s coordinates to the spreadsheet.
46. FIGURE A.46 Record to Spreadsheet dialogue.
47. FIGURE A.47 Creating a list of points from data.
48. FIGURE A.48 Set your labels to your preferences.
49. FIGURE A.49 Finding the minimum value.
50. FIGURE A.50 CAS will return exact algebraic solutions whenever possible.
51. FIGURE A.51 Least surface value in exact form.
52. FIGURE A.52 Creating a dynamic text.
Appendix B
1. FIGURE B.1a and b Linear functions.
2. FIGURE B.2a and b Quadratic functions.
3. FIGURE B.3a and b Increasing exponential functions with y' > 0.
4. FIGURE B.4a and b Decreasing exponential functions with y' < 0.
5. FIGURE B.5a and b Increasing exponential functions with y' < 0.
6. FIGURE B.6a and b Logistic functions.
7. FIGURE B.7a and b Symmetric single pulses.
8. FIGURE B.8a and b Two kinds of asymmetric pulses.
9. FIGURE B.9a and b Examples of double pulse functions.
10. FIGURE B.9c Example of a double pulse with a distinct start.
11. FIGURE B.10a and b Power functions with positive exponents.
12. FIGURE B.10c Power function with a negative exponent.
13. FIGURE B.11a, b, and c Rational functions are characterized by one or several asymptotes.
14. FIGURE B.12a and b Simple rational functions tending to an asymptotic value.
15. FIGURE B.13a and b Logarithmic functions.
16. FIGURE B.14a and b Periodic functions.
17. FIGURE B.15a and b Examples of asymmetric extreme points.
18. FIGURE B.16a and b Examples of linear combinations of basic functions.
19. FIGURE B.17a and b Examples of products of functions.
20. FIGURE B.18a and b Gradual transitions between functions.
21. FIGURE B.19a, b, and c Examples of linear transforms.
22. FIGURE B.20 f(x) = x³ translated with the vector (2, 3).
23. FIGURE B.21 Sending an image to the background.
24. FIGURE B.22 Sydney Harbor Bridge image in GeoGebra with sliders.
25. FIGURE B.23 Modeling the bridge.
26. FIGURE B.24 The fitted function.

Applications with GeoGebra™