Families of Functions [& Relations]

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] Algebra in the middle and secondary grades is largely the mathematics of function. The topic of Functions is not normally taught early in a student’s algebra career. When the topic of functions is finally taught, it is taught with emphasis on the symbolic representations of functions. Research has shown that approaching algebra from the outset through the study of functions using both symbolic and graphical representations simultaneously and in parallel is a pedagogically powerful teaching and learning strategy. Many people learn better and develop deeper understanding and intuitions about a subject if visual representations of aspects of the subject are prominent in the teaching and learning. All people learn better and develop deeper understanding and intuitions if they have the opportunity and the tools to explore their understanding and intuition by going back and forth between different representations of the a given mathematical object and the procedures for manipulating and transforming it. The Cartesian graph provides a way to express functions with imagery in addition to the symbols we usually use for their representation. The increasingly wide availability of graphing calculators and graphing software for personal computers and mobile devices makes it possible for us as teachers to take advantage of graphical representations in the way we teach the mathematics we teach [i]as well as in the way we ourselves understand the subject[/i].

Table of Contents

  1. Exploring linear & quadratic functions
    1. Linear Functions & Yellow Bowties
    2. Linear functions & quadrilaterals
    3. Parabolas between parabolas
    4. Absolute Value Functions & quadrilaterals
    5. Museum of Mathematics exhibit
    6. the locus of the vertex !
    7. Exploring Factoring & 'FOIL' - I
    8. Exploring Factoring & 'FOIL' - II
    9. Two cones and a plane
  2. Build & Fit...
    1. Build & Fit Linear Functions
    2. Build & Fit - Quadratic Functions
    3. Build & Fit - absolute value f'n I
    4. Build & Fit - absolute value f'n II
    5. Build & Fit - absolute value f'n III
    6. Build & Fit - Ellipse
    7. Build & Fit - Parabola
    8. Build & Fit - hyperbola
  3. Exploring functions through transformations
    1. Constructing F'ns by Sliding/Stretching/Squeezing/Reflecting
    2. CHALLENGE - the sine & cosine challenge
  4. Beyond linear & quadratic functions
    1. Power functions - x^n when n is not an integer
    2. Power functions - when x is complex
    3. spirals
    4. A swirl of spirals
    5. Circular & Hyperbolic Trig F'ns
    6. Approximating Functions with Polynomials
    7. Families of Exponential Functions
    8. E & P have a race - a fable about rate of change
  5. Some curiosities
    1. 'Squaring' the Circle
    2. Was Pythagoras wrong?
    3. Was Pythagoras Wrong? - Related Heresies
    4. 'elliptical' trigonometric functions
    5. A line & an hyperbola
    6. A Bestiary of Curves [degree > 2]
    7. f(x) f'(x) puzzle
    8. f(x) / f'(x) puzzle
    9. f(x) + f'(x) puzzle

1.

Exploring linear & quadratic functions

  • 1. Linear Functions & Yellow Bowties
  • 2. Linear functions & quadrilaterals
  • 3. Parabolas between parabolas
  • 4. Absolute Value Functions & quadrilaterals
  • 5. Museum of Mathematics exhibit
  • 6. the locus of the vertex !
  • 7. Exploring Factoring & 'FOIL' - I
  • 8. Exploring Factoring & 'FOIL' - II
  • 9. Two cones and a plane

1.1.

Linear Functions & Yellow Bowties

Some people think of neckwear when they hear the word "bowtie"; some think of pasta. [br]In this applet "bowtie" refers to the shape formed by two intersecting linear functions and the lines x = a and x = b.[br][br]Use the white dots to position the bowtie and set the values of a and b.[br]Use the blue and green dots to determine the slopes of the two linear functions.[br][br]What linear functions are the blue and green lines?[br]Where do they intersect?[br]What is area of the "bowtie"?[br]How does the area of the "bowtie" depend on the horizontal position of the intersection point of the two functions? on the vertical position?[br]What are some other questions you can ask your students about this construction?
– Judah L Schwartz

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1.9.

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2.

Build & Fit...

  • 1. Build & Fit Linear Functions
  • 2. Build & Fit - Quadratic Functions
  • 3. Build & Fit - absolute value f'n I
  • 4. Build & Fit - absolute value f'n II
  • 5. Build & Fit - absolute value f'n III
  • 6. Build & Fit - Ellipse
  • 7. Build & Fit - Parabola
  • 8. Build & Fit - hyperbola

2.1.

Build & Fit Linear Functions

You can make a linear function by dragging the two RED points almost anywhere on the screen (why almost?). The function will have the form - linear(x) = Ux + C, where U is a real number and the value of C is a real number. Now try to build the same linear function by combining the appropriate amounts of linear term and constant term using the BLUE sliders. Try to do this without displaying the functional form of the RED linear function you have made. You can check your work by showing the expressions for the functions on the screen. CHALLENGE - A student asks "How do you know that you can always make a linear function given two points?" What do you respond? Can two points define any other kind of function? Would rise/slant or run/slant be as good a measure of slope as rise/run? Why or why not?
– Judah L Schwartz

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2.8.

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3.

Exploring functions through transformations

  • 1. Constructing F'ns by Sliding/Stretching/Squeezing/Reflecting
  • 2. CHALLENGE - the sine & cosine challenge

3.1.

Constructing F'ns by Sliding/Stretching/Squeezing/Reflecting

Starting with the function [math]x[/math], is it possible to build any possible linear function of the form [math]mx + b[/math] using these transformations? [i][b]If yes, can you prove it? If no, can you find a counterexample?[/b][/i][br][br]Starting with the function [math]x^2[/math], is it possible to build any possible quadratic function of the form [math]ax^2 + bx + c[/math] using these transformations? [i][b]If yes, can you prove it? If no, can you find a counterexample?[/b][/i][br][br]Starting with the function [math]|x|[/math], is it possible to build any possible absolute value function of the form[br] [math]a|x – b| + c[/math] using these transformations? [i][b]If yes, can you prove it? If no, can you find a counterexample?[/b][/i][br][br][i][b]Can you build a constant function with this environment? Why or why not?[/b][/i][br][br]Think about the following questions. For the cases examined in this environment[br][br]• Vertical sliding of f(x) leads to f(x) + a. [Which way does the function slide if a > 0? a < 0?][br][br]• Horizontal sliding of f(x) leads to f(x+a). [Which way does the function slide if a > 0? a < 0?][br][br]• Vertical stretching & squeezing of f(x) leads to af(x). [What happens to the function if a < 0? 0 < a < 1? a > 1? ][br][br]• Horizontal stretching & squeezing of f(x) leads to af(x). [What happens to the function if a < 0? 0< a < 1? a > 1? ][br]Do you believe these statements are true for any function of one variable? If so, can you prove it? If not, can you find a counterexample?[br][br]Can you build a cubic function with this environment? Why or why not?
– Judah L Schwartz

3.2.

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4.

Beyond linear & quadratic functions

  • 1. Power functions - x^n when n is not an integer
  • 2. Power functions - when x is complex
  • 3. spirals
  • 4. A swirl of spirals
  • 5. Circular & Hyperbolic Trig F'ns
  • 6. Approximating Functions with Polynomials
  • 7. Families of Exponential Functions
  • 8. E & P have a race - a fable about rate of change

4.1.

Power functions - x^n when n is not an integer

What happens if the n in the function x^n is not an integer? We know from the rules of combining exponents that x^½ times x^½ is x. This means that x^½ is the same as the square root of x.[br][br]As you slide the slider, you will see that some graphs are plotted for both positive and negative values of x and some graphs for positive values only. Why? Are there any functions of the form x^n where n is not an even integer that have values for negative x? [br][br]Challenge - For each power of x you explore check and uncheck the absolute value functions checkbox. How would you characterize the difference between[br][br] f(x) and absolute value of f(x) – which is written as | f(x) |[br][br]In general, do you believe that |f(x)| and f(|x|) are the same or different? Why?[br][br][[b][i]This applet is better viewed as a Java applet[/i][/b]]
– Judah L Schwartz

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4.8.

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5.

Some curiosities

  • 1. 'Squaring' the Circle
  • 2. Was Pythagoras wrong?
  • 3. Was Pythagoras Wrong? - Related Heresies
  • 4. 'elliptical' trigonometric functions
  • 5. A line & an hyperbola
  • 6. A Bestiary of Curves [degree > 2]
  • 7. f(x) f'(x) puzzle
  • 8. f(x) / f'(x) puzzle
  • 9. f(x) + f'(x) puzzle

5.1.

'Squaring' the Circle

Drag the GOLD dot to 'square' the circle geometrically -[br]when you do this two sets of parallel segments that are[br]perpendicular to one another are connected at the corners[br]by quarter circles.[br][br]Drag the slider to 'square' the circle algebraically -[br]when you do this a generalization of the circle relation [br]is plotted.[br][br]What conjectures do you have in each case about ratio of the area[br]of the 'squared circle' to that of the limiting square?
'Squaring' the Circle
– Judah L Schwartz

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