Building Functions

Tim Brzezinski, Dec 28, 2016

Table of Contents

  1. Building (Authoring) Functions
    1. Open Middle Linear Functions Exercise
    2. Quiz: Writing Exponential Growth & Decay Functions
    3. Box with Open Top
    4. Relating SLOPE and Y-INTERCEPT to the REAL WORLD
    5. Equation of a Line: Dynamic Illustrator
    6. Writing the Equation of a Line (I-VA)
    7. Quiz: Modeling with Linear Functions
    8. Writing the Equation of a Line Given a Slope & Point (V1)
    9. Composition of Functions: Algebraic (Quiz)
    10. Writing Linear Equations Given Slope & Point (V2)
    11. Writing Linear Equations Given 2 Points (y-int = integer)
    12. Writing Linear Equations Given 2 Points (y-int not necessarily an integer)
    13. Quiz: Application Problems - Writing Linear Equations Given 2 Points
    14. Writing the Equation of a Line Given Both Intercepts
    15. Writing Equations of Parallel and Perpendicular Lines
  2. Function Operations
    1. Building Functions Graphically
    2. Create Your Own: Building Functions Graphically
    3. Adding Functions Graphically
    4. Quiz: Adding Functions Graphically
    5. Function Composition: Dynamic Illustrator (1)
    6. Function Composition: Dynamic Illustrator (2)
    7. Animation 157
    8. Composition of Functions: Algebraic (Quiz)
    9. Quiz: Composition of Functions (Graph & Table)
    10. Composition of Functions: Graphical (Quiz) - V1
    11. Quiz: Modeling with Linear Functions
    12. Application Problems: Solving Linear Equations (1)
    13. Perimeter of a Rectangle: Solving Linear Equations
    14. Application Problems: Solving Linear Equations (3)
    15. Friends and Money: Solving Linear Equations
    16. Application Problems: Solving Linear Equations (5)
  3. Sequences
    1. AQR Section 21: Arithmetic or Geometric?
  4. Function Transformations
    1. Function Transformations
    2. Exploring Function Transformations in 3D
    3. Transformations of Graphs (a, h, k)
    4. What Makes a Function Even?
    5. Animation 143
    6. What Makes a Function Odd?
    7. Animation 144
    8. Odd Functions (Another Look)
    9. Odd Functions: Another Look (Take 2)!
  5. Inverse Functions
    1. Building Functions with Inverses
    2. Domain-Range Swap: Quick Graphical Exploration
    3. Inverse Relations: Graphs
    4. Writing Inverse Functions: Quiz (1)
    5. Writing Inverse Functions: Quiz (2)
    6. Writing Inverse Functions: Quiz (3)
    7. Graphing Inverse Functions: Quiz (1)
  6. Logarithms & Properties
    1. What is a Logarithm?
    2. Logarithmic Functions Anatomy
    3. Quiz: Graphing Logarithmic Functions (Includes Transformations)
    4. Logarithmic Action!
    5. Logarithmic Action (2)!
    6. Logarithmic Action (3)!
    7. Logarithmic Action (3: V2)
    8. Logarithmic Action (4)!
    9. Logarithmic Action (4)! V2
    10. Solving Exponential Equations (I)
  7. Other Functions Resources
    1. Functions Resources

1.

Building (Authoring) Functions

  • 1. Open Middle Linear Functions Exercise
  • 2. Quiz: Writing Exponential Growth & Decay Functions
  • 3. Box with Open Top
  • 4. Relating SLOPE and Y-INTERCEPT to the REAL WORLD
  • 5. Equation of a Line: Dynamic Illustrator
  • 6. Writing the Equation of a Line (I-VA)
  • 7. Quiz: Modeling with Linear Functions
  • 8. Writing the Equation of a Line Given a Slope & Point (V1)
  • 9. Composition of Functions: Algebraic (Quiz)
  • 10. Writing Linear Equations Given Slope & Point (V2)
  • 11. Writing Linear Equations Given 2 Points (y-int = integer)
  • 12. Writing Linear Equations Given 2 Points (y-int not necessarily an integer)
  • 13. Quiz: Application Problems - Writing Linear Equations Given 2 Points
  • 14. Writing the Equation of a Line Given Both Intercepts
  • 15. Writing Equations of Parallel and Perpendicular Lines

1.1.

Open Middle Linear Functions Exercise

Using the digits 0-9 at most 1 time each, fill in the missing boxes to make all these statements true. How many solutions can you find?
– Tim Brzezinski

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1.6.

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1.7.

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1.8.

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1.9.

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1.10.

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1.11.

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1.12.

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1.13.

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1.14.

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1.15.

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2.

Function Operations

  • 1. Building Functions Graphically
  • 2. Create Your Own: Building Functions Graphically
  • 3. Adding Functions Graphically
  • 4. Quiz: Adding Functions Graphically
  • 5. Function Composition: Dynamic Illustrator (1)
  • 6. Function Composition: Dynamic Illustrator (2)
  • 7. Animation 157
  • 8. Composition of Functions: Algebraic (Quiz)
  • 9. Quiz: Composition of Functions (Graph & Table)
  • 10. Composition of Functions: Graphical (Quiz) - V1
  • 11. Quiz: Modeling with Linear Functions
  • 12. Application Problems: Solving Linear Equations (1)
  • 13. Perimeter of a Rectangle: Solving Linear Equations
  • 14. Application Problems: Solving Linear Equations (3)
  • 15. Friends and Money: Solving Linear Equations
  • 16. Application Problems: Solving Linear Equations (5)

2.1.

Building Functions Graphically

[size=100][size=150]In each app below, functions[b][i][color=#ff7700] f [/color][/i][/b]and [b][color=#1e84cc][i]g [/i][/color][/b]are shown as graphs. [br][br]For each exercise:[br][list=1][*]Plot several points that lie on the graph you're asked to construct. To do this, you can use the POINT tool [icon]/images/ggb/toolbar/mode_point.png[/icon] or simply use the Input... line (at bottom) [/*][*]Use the PEN tool [icon]/images/ggb/toolbar/mode_pen.png[/icon] to hand-sketch the graph as best you can. [/*][/list][/size][/size]
Sketch the graph of (f + g).
Sketch the graph of (g - f).
Sketch the graph of f(g(x)).
Sketch the graph of g(f(x)).
– Tim Brzezinski

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2.5.

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2.6.

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2.7.

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2.8.

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2.9.

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2.10.

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2.11.

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2.12.

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2.13.

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2.14.

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2.15.

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2.16.

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3.

Sequences

  • 1. AQR Section 21: Arithmetic or Geometric?

3.1.

AQR Section 21: Arithmetic or Geometric?

Is the sequence an [b]arithmetic sequence[/b] or a [b]geometric sequence[/b]?[br]Classify the sequence correctly.[br]You will then be prompted for the common difference or common ratio.
– Steve Phelps

4.

Function Transformations

  • 1. Function Transformations
  • 2. Exploring Function Transformations in 3D
  • 3. Transformations of Graphs (a, h, k)
  • 4. What Makes a Function Even?
  • 5. Animation 143
  • 6. What Makes a Function Odd?
  • 7. Animation 144
  • 8. Odd Functions (Another Look)
  • 9. Odd Functions: Another Look (Take 2)!

4.1.

Function Transformations

[left][color=#000000]This applet accompanies the [/color][i][color=#1e84cc][b]Transformations of Functions[/b][/color][/i][color=#000000] packet you received at the beginning of class. [/color][/left]
– Tim Brzezinski

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4.4.

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4.5.

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4.6.

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4.7.

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4.8.

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4.9.

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5.

Inverse Functions

  • 1. Building Functions with Inverses
  • 2. Domain-Range Swap: Quick Graphical Exploration
  • 3. Inverse Relations: Graphs
  • 4. Writing Inverse Functions: Quiz (1)
  • 5. Writing Inverse Functions: Quiz (2)
  • 6. Writing Inverse Functions: Quiz (3)
  • 7. Graphing Inverse Functions: Quiz (1)

5.1.

Building Functions with Inverses

Interact with this app for a few minutes. LARGE POINTS are moveable. Then answer the questions that follow.
What do you notice? What do you wonder?
What does it mean for a relation to be a [b]function[/b]? Describe. [i]Make sure to use the terms "input" and "output" in your description.[/i]
In the app above, reposition the [b]3 LARGE POINTS[/b] of the [b]function[/b] so that the [b][color=#0000ff]graph of the inverse relation[/color] [/b]also becomes a function.
Explain what you did to the [b]original function[/b] to cause the [b][color=#0000ff]graph of the inverse relation[/color][/b] to be a function.
Use this app to help you answer the questions that follow.
In the app above, reposition the [b]3 LARGE POINTS[/b] of the [b]function[/b] so that the [b][color=#0000ff]graph of the inverse relation[/color] [i]is not[/i] [/b]a function.
Explain what you did to the [b]original function[/b] to cause the [b][color=#0000ff]graph of the inverse relation[/color][/b] to [b][i]not be[/i][/b] a function.
Click on the [b][color=#0000ff]TEST INVERSE[/color][/b] checkbox. Drag the point that appears. How does this help illustrate the graph of the inverse relation is not a function? Explain. ([i]In your explanation, avoid using the phrase "vertical line test". Rather, describe using the terms "input" and "output".[/i])
Quick silent demo
– Tim Brzezinski

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5.3.

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5.4.

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5.5.

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5.6.

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5.7.

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6.

Logarithms & Properties

  • 1. What is a Logarithm?
  • 2. Logarithmic Functions Anatomy
  • 3. Quiz: Graphing Logarithmic Functions (Includes Transformations)
  • 4. Logarithmic Action!
  • 5. Logarithmic Action (2)!
  • 6. Logarithmic Action (3)!
  • 7. Logarithmic Action (3: V2)
  • 8. Logarithmic Action (4)!
  • 9. Logarithmic Action (4)! V2
  • 10. Solving Exponential Equations (I)

6.1.

What is a Logarithm?

Move the LARGE POINT on the xAxis as far as it will go. You can drag the LARGE POINT that appears behind it anywhere you'd like.
What exactly [i]is[/i] a [b][color=#9900ff]logarithm[/color][/b]? How does/do your observation(s) support your response? Explain.
Move the LARGE POINT on the xAxis as far as it will go. You can drag the LARGE POINT that appears behind it anywhere you'd like.
What exactly [i]is[/i] a [color=#9900ff][b]natural logarithm[/b][/color]? How does/do your observation(s) support your response? Explain.
– Tim Brzezinski

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6.5.

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6.6.

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6.7.

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6.9.

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6.10.

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7.

Other Functions Resources

  • 1. Functions Resources

7.1.

Functions Resources

[list][*][b][url=https://www.geogebra.org/m/k6Dvu9f3]Interpreting Functions[/url][/b][/*][*][b][url=https://www.geogebra.org/m/uTddJKRC]Building Functions[/url][/b][/*][*][b][url=https://www.geogebra.org/m/GMvvpwrm]Linear, Quadratic, and Exponential Functions[/url][/b][/*][*][b][url=https://www.geogebra.org/m/aWuJMDas]Trigonometric Functions[/url][/b][/*][/list]
Half-life function: Quick Exploration. (Large point & slider moveable.)
What does it mean for a function to be odd? (Points moveable.)
– GeoGebra Team, Tim Brzezinski

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