Function as Object

Judah L Schwartz, Jan 31, 2014

[b]This is a series of applets drawn from the website mathMINDhabits [[url]sites.google.com/site/mathmindhabits[/url]] that is designed for teachers of mathematics who want to deepen their understanding of the mathematics they teach and that their students are expected to learn.[/b] From a certain perspective the fundamental object of a great deal of mathematics is the function. There are operations such as addition, multiplication and differentiation on functions viewed as objects. Functions are objects that can be transformed by such transformations as translation, dilation and reflection. [b][ [i]For a more extended discussion of "function as object" please read the essay on the mathMINDhabits website entitled "Functions - Objects & Actions"[/i] ][/b]

Table of Contents

  1. operations
    1. Function Table Builder
    2. Functions as Objects - operations on functions
    3. non-invertable operations
    4. non-invertable operations II
  2. transformations
    1. Functions as Objects - transformation on functions
    2. Functions as Objects - how they change - I
    3. Functions as Objects: how they change
    4. The Function Attribute Explorer
    5. an imposter function
  3. composition
    1. Composing with logarithmic & exponential functions
    2. How are the graphs of F & Finverse related?
    3. Composing Transformations
  4. roots
    1. Real & Complex Roots of Quadratic Functions
    2. Conjectures about Complex roots of polynomials
    3. The Three Roots of a Cubic Function
  5. global behavior
    1. Rational F'ns & Asymptotic Behavior
    2. Even & Odd Functions
    3. Relations that Function

1.

operations

  • 1. Function Table Builder
  • 2. Functions as Objects - operations on functions
  • 3. non-invertable operations
  • 4. non-invertable operations II

1.1.

Function Table Builder

Multiplication table for polynomials Make a table that has sketches of constant, linear, quadratic and cubic along the top row and down the left side. Fill in the cells with sketches of the product function corresponding to that row and column. Note that some cells may require more than one entry. Division table for rational functions Make a table that has sketches of constant, linear, quadratic and cubic along the top row and down the left side. Fill in the cells with sketches of the quotient function corresponding to that row and column. Note that some cells may require more than one entry.
– Judah L Schwartz

1.2.

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1.3.

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1.4.

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2.

transformations

  • 1. Functions as Objects - transformation on functions
  • 2. Functions as Objects - how they change - I
  • 3. Functions as Objects: how they change
  • 4. The Function Attribute Explorer
  • 5. an imposter function

2.1.

Functions as Objects - transformation on functions

Challenge: [br]choose a linear function for f(x) and the same linear function for g(x); then dilate and then translate f(x) and translate and then dilate g(x).[br]Are the two resulting functions the same? If yes, why? If not, under what circumstances could they be the same?[br][br]Challenge:[br] Choose two different linear functions for f(x) and g(x). Can you use these transformations to transform one into the other? Do you believe you can do the same for any linear f(x) and linear g(x)?[br] Choose two different quadratic functions for f(x) and g(x). Can you use these transformations to transform one into the other? Do you believe you can do the same for any quadratic f(x) and quadratic g(x)?[br] Can you transform any quadratic function into a linear function? Why or why not? Prove it.[br][br]Challenge:[br] What additional transformation(s), if any, would you need to be able to transform any cubic function into any other cubic function?
– Judah L Schwartz

2.2.

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2.3.

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2.4.

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2.5.

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3.

composition

  • 1. Composing with logarithmic & exponential functions
  • 2. How are the graphs of F & Finverse related?
  • 3. Composing Transformations

3.1.

Composing with logarithmic & exponential functions
– Judah L Schwartz

3.2.

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3.3.

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4.

roots

  • 1. Real & Complex Roots of Quadratic Functions
  • 2. Conjectures about Complex roots of polynomials
  • 3. The Three Roots of a Cubic Function

4.1.

Real & Complex Roots of Quadratic Functions

This environment allows you to explore the following questions:[br][br][br]Given two real numbers, how many quadratic functions are there [br]with those numbers as roots?[br][br][br]How are these functions related to one another?[br][br][br]Given two complex conjugate numbers, how many quadratic functions[br]have these numbers as roots?[br][br][br]How are these functions related to one another?[br][br][br]Challenge – given two quadratic functions with the same roots, can you transform one into the other?
– Judah L Schwartz

4.2.

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4.3.

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5.

global behavior

  • 1. Rational F'ns & Asymptotic Behavior
  • 2. Even & Odd Functions
  • 3. Relations that Function

5.1.

Rational F'ns & Asymptotic Behavior

You can use this environment to explore the behavior of rational functions formed by dividing a constant, linear or quadratic function by a linear or quadratic function. Define the numerator function by dragging the red dots. Define the denominator function by dragging the blue dots. Fit the appropriate asymptote(s) to the rational function you have determined. Alternatively, choose a set of asymptotes and try to find a rational function with those asymptotes.
– Judah L Schwartz

5.2.

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5.3.

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Information