Interpreting Functions

Tim Brzezinski, Dec 28, 2016

Table of Contents

  1. Function Definition and Related Vocab
    1. Functions: Input and Output (I)
    2. Functions: Input & Output (II)
    3. Functions: Input & Output (3)
    4. Functions: Input & Output (4)
    5. Functions: Input & Output (III)
    6. Function Or Not?
  2. Function Notation, Evaluate Outputs for Inputs
    1. Evaluating Functions Using Graphs (Quiz)
    2. Evaluating Functions Algebraically
    3. Quiz: Function Operations (1)
    4. Composition of Functions: Algebraic (Quiz)
    5. Quiz: Composition of Functions (Graph & Table)
    6. Composition of Functions: Graphical (Quiz) - V1
  3. Sequences
    1. Sequences (Dynamic Illustrator)
  4. Function Behavior, Types, etc.
    1. Function Behavior
    2. Quick Intercepts Illustrator
    3. Relative Extrema Illustrator
    4. Polynomial End Behavior (Dance...)
    5. What Makes a Function Even?
    6. Animation 143
    7. What Makes a Function Odd?
    8. Animation 144
    9. Odd Functions (Another Look)
    10. Odd Functions: Another Look (Take 2)!
    11. Periodic Function Action!
    12. Animation 189
    13. Sine & Cosine Period Action (1)!
    14. Animation 190
    15. Animation 191
    16. Sine & Cosine Period Action (1)!
    17. Sine & Cosine Period Action (2)!
    18. Tangent Period Action (1)!
    19. Animation 192
    20. Tangent Period Action (2)!
    21. Dolbear's Law
    22. Half-Life Function
  5. Domain and range
    1. Domain & Range Illustrator
    2. Domain and Range: Quick Quiz Questions and Illustrator
    3. Animation 95
    4. Animation 93
  6. Average rate of change
    1. Average Rate of Change of a Function: Dynamic Illustration
    2. The Difference Quotient (Illustrated)
    3. Quiz: Average Rate of Change of a Function (1)
    4. Quiz: Average Rate of Change (2)
    5. Quiz: Average Rate of Change of a Function (3)
  7. Different types of functions
    1. Sketching Graph Models
    2. Equation of a Line: Dynamic Illustrator
    3. IRS Tax Function (2016) for Tax Payers Filing Single
    4. Quadratic Function
    5. Transforming Absolute Value Function Graphs: 3-Part Exploration with Assessment Qs
    6. Graphing Absolute Value Functions (with Transformations)
    7. Writing Absolute Value Functions from Graphs
    8. Open Middle: Absolute Value Graphs (1)
    9. Open Middle: Absolute Value Graphs (2)
    10. Open Middle: Absolute Value Graphs (3)
    11. Open Middle: Absolute Value Graphs (4)
    12. Toblerone Modeling (Part 2)
    13. Absolute Value Function
    14. Square Root Function
    15. Cube Root Function
    16. Exponential Functions: Graphs
    17. Exponential Functions (Graph & Equation Anatomy)
    18. Logarithmic Functions: Graphs
    19. Logarithmic Functions (Graph & Equation Anatomy)
    20. Quiz: Dividing Polynomial Expressions
    21. Graphing Sine & Cosine Functions (II)
    22. Graphing Sine & Cosine Functions (I)
    23. Graph y = a sin(x) + d (Quiz)
    24. Graph y = a cos(x) + d (Quiz)
    25. Rational Functions Anatomy
    26. Quiz: Graphing Rational Functions (Transformations Included)
    27. Quiz: Writing Rational Functions (Transformations Included)
    28. Horizontal Asymptote Explorer
    29. Open Middle: Rational Functions Graph (1)
    30. Open Middle: Rational Functions Graph (2)
    31. Rational Function Grapher
  8. Completing the Square, Quadratic and Exponential Functions
    1. Factoring Trinomials
    2. Factoring Trinomials: Intro
    3. Completing the Square (1)
    4. Completing the Square (2)
    5. Completing the Square (3)
    6. Quiz: Completing the Square (1)
    7. Quiz: Completing the Square (2)
    8. Exponential Growth vs. Decay (Intro)
    9. Quiz: Writing Exponential Growth & Decay Functions
  9. Other Functions Resources
    1. Functions Resources

1.

Function Definition and Related Vocab

  • 1. Functions: Input and Output (I)
  • 2. Functions: Input & Output (II)
  • 3. Functions: Input & Output (3)
  • 4. Functions: Input & Output (4)
  • 5. Functions: Input & Output (III)
  • 6. Function Or Not?

1.1.

Functions: Input and Output (I)

– Tim Brzezinski

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1.3.

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1.4.

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1.5.

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1.6.

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2.

Function Notation, Evaluate Outputs for Inputs

  • 1. Evaluating Functions Using Graphs (Quiz)
  • 2. Evaluating Functions Algebraically
  • 3. Quiz: Function Operations (1)
  • 4. Composition of Functions: Algebraic (Quiz)
  • 5. Quiz: Composition of Functions (Graph & Table)
  • 6. Composition of Functions: Graphical (Quiz) - V1

2.1.

Evaluating Functions Using Graphs (Quiz)

– Tim Brzezinski

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3.

Sequences

  • 1. Sequences (Dynamic Illustrator)

3.1.

Sequences (Dynamic Illustrator)

[b]A[/b] [color=#38761d][b]sequence[/b][/color][b] is a function whose[/b] [color=#38761d][b]domain is the set of positive (or sometimes, nonnegative) integers[/b][/color]. [br][br]Recall the domain of a function is the set of all values that can be inputted into the function. [br][br]The following applet dynamically and graphically illustrates the meaning of a [color=#38761d][b]sequence[/b][/color]. [br]Feel free to re-author the [color=#38761d][b]explicit formula for this sequence[/b][/color] in the upper left-hand corner. [br]
– Tim Brzezinski

4.

Function Behavior, Types, etc.

  • 1. Function Behavior
  • 2. Quick Intercepts Illustrator
  • 3. Relative Extrema Illustrator
  • 4. Polynomial End Behavior (Dance...)
  • 5. What Makes a Function Even?
  • 6. Animation 143
  • 7. What Makes a Function Odd?
  • 8. Animation 144
  • 9. Odd Functions (Another Look)
  • 10. Odd Functions: Another Look (Take 2)!
  • 11. Periodic Function Action!
  • 12. Animation 189
  • 13. Sine & Cosine Period Action (1)!
  • 14. Animation 190
  • 15. Animation 191
  • 16. Sine & Cosine Period Action (1)!
  • 17. Sine & Cosine Period Action (2)!
  • 18. Tangent Period Action (1)!
  • 19. Animation 192
  • 20. Tangent Period Action (2)!
  • 21. Dolbear's Law
  • 22. Half-Life Function

4.1.

Function Behavior

Feel free to start by putting the 2 white points anywhere you'd like. [br]Then slide the slider. [br][br]Please answer the questions that follow.
1.
[b][color=#0000ff]How would you describe the pieces of the function that are BLUE? Explain. [/color][/b][br]
2.
[b][color=#ff0000]How would you describe the pieces of the function that are RED? Explain. [/color][br][/b]
3.
[b]How would you describe the pieces of the function that are BLACK? Explain. [/b]
Quick (Silent) Demo
– Tim Brzezinski

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4.18.

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4.19.

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4.20.

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4.21.

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4.22.

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5.

Domain and range

  • 1. Domain & Range Illustrator
  • 2. Domain and Range: Quick Quiz Questions and Illustrator
  • 3. Animation 95
  • 4. Animation 93

5.1.

Domain & Range Illustrator

– Tim Brzezinski

5.2.

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5.3.

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5.4.

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6.

Average rate of change

  • 1. Average Rate of Change of a Function: Dynamic Illustration
  • 2. The Difference Quotient (Illustrated)
  • 3. Quiz: Average Rate of Change of a Function (1)
  • 4. Quiz: Average Rate of Change (2)
  • 5. Quiz: Average Rate of Change of a Function (3)

6.1.

Average Rate of Change of a Function: Dynamic Illustration

The average rate of change of a continuous function [i]f[/i] from input value [i]x[/i] = [i]a[/i] to input value [i]x[/i] = [i]b[/i] is given by [math]f_{ave}=\frac{f\left(b\right)-f\left(a\right)}{b-a}[/math]. [br][br]The applet below provides a dynamic visual interpretation of this expression. [br][br]Interact with this applet for a few minutes. [br][color=#444444][b]You can input different functions in the input box in the upper right-hand corner. [/b][/color][b]You can also adjust the input values [i]a[/i] and [i]b[/i] using the sliders or their respective input boxes. [/b]
– Tim Brzezinski

6.2.

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6.3.

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7.

Different types of functions

  • 1. Sketching Graph Models
  • 2. Equation of a Line: Dynamic Illustrator
  • 3. IRS Tax Function (2016) for Tax Payers Filing Single
  • 4. Quadratic Function
  • 5. Transforming Absolute Value Function Graphs: 3-Part Exploration with Assessment Qs
  • 6. Graphing Absolute Value Functions (with Transformations)
  • 7. Writing Absolute Value Functions from Graphs
  • 8. Open Middle: Absolute Value Graphs (1)
  • 9. Open Middle: Absolute Value Graphs (2)
  • 10. Open Middle: Absolute Value Graphs (3)
  • 11. Open Middle: Absolute Value Graphs (4)
  • 12. Toblerone Modeling (Part 2)
  • 13. Absolute Value Function
  • 14. Square Root Function
  • 15. Cube Root Function
  • 16. Exponential Functions: Graphs
  • 17. Exponential Functions (Graph & Equation Anatomy)
  • 18. Logarithmic Functions: Graphs
  • 19. Logarithmic Functions (Graph & Equation Anatomy)
  • 20. Quiz: Dividing Polynomial Expressions
  • 21. Graphing Sine & Cosine Functions (II)
  • 22. Graphing Sine & Cosine Functions (I)
  • 23. Graph y = a sin(x) + d (Quiz)
  • 24. Graph y = a cos(x) + d (Quiz)
  • 25. Rational Functions Anatomy
  • 26. Quiz: Graphing Rational Functions (Transformations Included)
  • 27. Quiz: Writing Rational Functions (Transformations Included)
  • 28. Horizontal Asymptote Explorer
  • 29. Open Middle: Rational Functions Graph (1)
  • 30. Open Middle: Rational Functions Graph (2)
  • 31. Rational Function Grapher

7.1.

Sketching Graph Models

[b][color=#0000ff]Students:[/color][/b][br][br]Use these templates to draw general sketches of each scenario listed on the scenarios page given to you at the beginning of class. Then submit either via our GeoGebra Group or Google Classroom.
SCENARIO A
SCANARIO B
– Tim Brzezinski

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7.27.

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7.28.

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7.29.

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8.

Completing the Square, Quadratic and Exponential Functions

  • 1. Factoring Trinomials
  • 2. Factoring Trinomials: Intro
  • 3. Completing the Square (1)
  • 4. Completing the Square (2)
  • 5. Completing the Square (3)
  • 6. Quiz: Completing the Square (1)
  • 7. Quiz: Completing the Square (2)
  • 8. Exponential Growth vs. Decay (Intro)
  • 9. Quiz: Writing Exponential Growth & Decay Functions

8.1.

Factoring Trinomials

Algebra Tiles: Each shape represents an algebraic expression.
Remember the area of a rectangle = (LENGTH)(WIDTH). Complete the following:
Remember the area of a rectangle = (LENGTH)(WIDTH). Complete the following:
Remember the area of a rectangle = (LENGTH)(WIDTH). Complete the following:
Remember the area of a rectangle = (LENGTH)(WIDTH). Complete the following:
Quick (Silent) Demo
– Tim Brzezinski

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9.

Other Functions Resources

  • 1. Functions Resources

9.1.

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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