Phantom Graphs

Doug Kuhlmann, Jul 15, 2018

Phantom graphs use the 3D graphing capability of GeoGebra to show all solutions, both real and complex, to many equations. In general, they will show all solutions to f(x)=c where c ranges over all reals. Explore to find that by extending to complex numbers we can graph the solutions to x^2=-4 and even see that cos(z)=7 actually has a complex solution (an infinite number of them). In all the applets in this book if the point A(a, b, c) is on the phantom graph, then letting x=a+ic and y=b will give values that satisfy the original equation. Note that y=b is the middle value since we are using the x-axis as real and the z-axis as the imaginary axis.

Table of Contents

  1. Background and low degree polynomials
    1. Phantom Graphs Background. Notes on creating phantom graphs.
    2. Phantom Graph for f(x)=x^2
    3. Phantom Graphs for Ax^2+Bx+C
    4. Phantom Graph for A(x-H)^2+K
    5. Phantom Graph General Cubic
    6. Phantom Graph for f(x)=x^4
    7. Phantom Graph for Ax^4+Bx^2+C
    8. Phantom Graph 4th Degree General
    9. Phantom Graph for f(x)=x^5
  2. Duals: Functions that are PGs of each other when rotated
    1. Phantom Graph of ellipse is hyperbola (& v.v.)
    2. Phantom Graph of hyperbola
    3. Exp, cosine, cosh, sin & sinh are all connected
    4. Cosine & cosh are PG duals
    5. Cosh (hyperbolic) and Cos are dual PGs.
    6. Phantom Graph sine(x)
    7. Phantom Graph of 1/(1+x^2)
    8. Phantom Graph of 1/(1-x^2)
    9. Phantom Graph of 2x^2/(x^2+1)
    10. Phantom Graph 2x^2/(x^2-1)
    11. Phantom Graph x^4/(1+x^2)
    12. Phantom Graph x^4/(1-x^2)
    13. Phantom Graph (x^2-1)^2
    14. Phantom Graph (x^2+1)^2
    15. Phantom Graph cos(x^2/2)
    16. Phantom Graph cosh(x^2/2)
  3. Exp and sinh: similar but not duals
    1. Phantom Graph of exp(x)
    2. Phantom Graph of f(x)=sinh(x)
  4. Howard K.Fehr (1951)!!
    1. Howard K. Fehr's notes
    2. Fehr problem 1 in notes
    3. Fehr problem 2 in notes
    4. Fehr problem 6&7 in notes
    5. Fehr problem 9 in notes
    6. Fehr problem 10 in notes
    7. Fehr problem 11 complex intersection of disjoint circles
  5. Philip Lloyd of New Zealand. All phantom graphs in this chapter come from his notes.
    1. Philip Lloyd's notes and explanations
    2. Phillips Lloyd's notes part 3
    3. Phantom Graph of f(x)=(x^2-1)(x^2+1)
    4. Phantom Graph of (x^2+1)^2
    5. Phantom Graph of x(x-3)^2
    6. Phantom Graph y^2=x(x-3)^2
    7. Phantom Graph of y=x^2/(x-1)
    8. Phantom Graph of y=x^2/((x-1)(x-4))
    9. Phillips Lloyd's notes part 3
  6. Some Famous Curves
    1. Phantom Graph Witch of Agnesi
    2. Phantom Graph Cissoid of Diocles
    3. Phantom Graph Devil's Curve
    4. Phantom Graph Tschirnhaus's Cubic
    5. Phantom Graph Conchoid de Sluze
    6. Phantom Graph Figure Eight
    7. Phantom Graph Trisectrix of Maclaurin
    8. Phantom Graph Serpentine
    9. Phantom Graph Pear-shaped Quartic
    10. Phantom Graphs Newton's Diverging Parabolas
    11. Phantom Graph Kampyle of Eudoxus
    12. Phantom Graph Strophoid

1.

Background and low degree polynomials

Some background information on how to created these and applets showing phantom graphs for polynomials of degree 2, 3 and 4.

  • 1. Phantom Graphs Background. Notes on creating phantom graphs.
  • 2. Phantom Graph for f(x)=x^2
  • 3. Phantom Graphs for Ax^2+Bx+C
  • 4. Phantom Graph for A(x-H)^2+K
  • 5. Phantom Graph General Cubic
  • 6. Phantom Graph for f(x)=x^4
  • 7. Phantom Graph for Ax^4+Bx^2+C
  • 8. Phantom Graph 4th Degree General
  • 9. Phantom Graph for f(x)=x^5

1.1.

Phantom Graphs Background. Notes on creating phantom graphs.

Phantom Graphs Background. Notes on creating phantom graphs.
– Doug Kuhlmann

1.2.

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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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1.7.

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1.8.

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1.9.

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2.

Duals: Functions that are PGs of each other when rotated

I use the term 'duals' for two functions that are (essentially) the phantom graphs of each other if rotated.

  • 1. Phantom Graph of ellipse is hyperbola (& v.v.)
  • 2. Phantom Graph of hyperbola
  • 3. Exp, cosine, cosh, sin & sinh are all connected
  • 4. Cosine & cosh are PG duals
  • 5. Cosh (hyperbolic) and Cos are dual PGs.
  • 6. Phantom Graph sine(x)
  • 7. Phantom Graph of 1/(1+x^2)
  • 8. Phantom Graph of 1/(1-x^2)
  • 9. Phantom Graph of 2x^2/(x^2+1)
  • 10. Phantom Graph 2x^2/(x^2-1)
  • 11. Phantom Graph x^4/(1+x^2)
  • 12. Phantom Graph x^4/(1-x^2)
  • 13. Phantom Graph (x^2-1)^2
  • 14. Phantom Graph (x^2+1)^2
  • 15. Phantom Graph cos(x^2/2)
  • 16. Phantom Graph cosh(x^2/2)

2.1.

Phantom Graph of ellipse is hyperbola (& v.v.)

Best viewed in app or downloaded
ellipse-hyperbola
– Doug Kuhlmann

2.2.

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2.3.

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2.4.

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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.

Exp and sinh: similar but not duals

Both exp and sinh have phantom graphs that are copies of themselves repeated infinitely often.

  • 1. Phantom Graph of exp(x)
  • 2. Phantom Graph of f(x)=sinh(x)

3.1.

Phantom Graph of exp(x)

Best viewed in app or downloaded
– Doug Kuhlmann

3.2.

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4.

Howard K.Fehr (1951)!!

Howard K. Fehr while a professor at Teachers College Columbia, wrote a book for secondary school teachers that had a section on phantom graphs but did not use the term. That was 67 years ago. Moral: Read old books.

  • 1. Howard K. Fehr's notes
  • 2. Fehr problem 1 in notes
  • 3. Fehr problem 2 in notes
  • 4. Fehr problem 6&7 in notes
  • 5. Fehr problem 9 in notes
  • 6. Fehr problem 10 in notes
  • 7. Fehr problem 11 complex intersection of disjoint circles

4.1.

Howard K. Fehr's notes

A readable if not perfect pdf copy of the section from Fehr that describes this technique. (1951) Fehr was head of the Teaching of Mathematics dept in Teacher's College Columbia.
– Doug Kuhlmann

4.2.

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4.3.

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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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5.

Philip Lloyd of New Zealand. All phantom graphs in this chapter come from his notes.

The first person I found who discovered phantom graphs independently and gave them the name of phantom graphs.

  • 1. Philip Lloyd's notes and explanations
  • 2. Phillips Lloyd's notes part 3
  • 3. Phantom Graph of f(x)=(x^2-1)(x^2+1)
  • 4. Phantom Graph of (x^2+1)^2
  • 5. Phantom Graph of x(x-3)^2
  • 6. Phantom Graph y^2=x(x-3)^2
  • 7. Phantom Graph of y=x^2/(x-1)
  • 8. Phantom Graph of y=x^2/((x-1)(x-4))
  • 9. Phillips Lloyd's notes part 3

5.1.

Philip Lloyd's notes and explanations

PHANTOM GRAPHS - Home1
PHANTOM GRAPHS - Home2
– Doug Kuhlmann

5.2.

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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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5.8.

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5.9.

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6.

Some Famous Curves

Taken from 50 Famous Curves available on-line from the Bloomsburg University Math Dept, Bloomsburg, PA.

  • 1. Phantom Graph Witch of Agnesi
  • 2. Phantom Graph Cissoid of Diocles
  • 3. Phantom Graph Devil's Curve
  • 4. Phantom Graph Tschirnhaus's Cubic
  • 5. Phantom Graph Conchoid de Sluze
  • 6. Phantom Graph Figure Eight
  • 7. Phantom Graph Trisectrix of Maclaurin
  • 8. Phantom Graph Serpentine
  • 9. Phantom Graph Pear-shaped Quartic
  • 10. Phantom Graphs Newton's Diverging Parabolas
  • 11. Phantom Graph Kampyle of Eudoxus
  • 12. Phantom Graph Strophoid

6.1.

Phantom Graph Witch of Agnesi

Best viewed in app or downloaded.
– Doug Kuhlmann

6.2.

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6.3.

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6.4.

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6.5.

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6.6.

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6.7.

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6.8.

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6.9.

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6.10.

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6.11.

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6.12.

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