Jonathan Choate Groton School jchoate@groton.org www.zebragraph.com Polynomiography Jonathan Choate Groton School jchoate@groton.org www.zebragraph.com

VCTAL

Iteration Pick a function f[x] Pick a seed x(0) Create a sequence {x(0) , x(1) = f[x(0]], x(2)=f[(x(1)], … ,x(n) = f[x(n-1)], ….} This sequence is the ORBIT of x(0) under iteration of f[x]

An Example Iterate f[x] = (1/2)x + 4 for different seeds. Two Interesting Questions 1) For a given seed does the orbit converge to a value S within a given accuracy? 2) If it does converge, how many iterations does it take? The set of seeds that converge to S is called the BASIN OF ATTRACTION for S and will be denoted by B(S)

Solving Polynomial Equations Formulas exist for Quadratics, Cubics and Quartics Have to use numerical methods for polynomials of degree higher than 4.

The Babylonian Method for Finding Square Roots Iterate f[x] =(1/2)(x + a/x) If a>0 orbits converge to and What is B( )? B( )? What happens if a < 0? What happens if a is complex? Babylonian Method

Newton’s Method Given a function f(x) to solve f(x) =0 iterate the function N[x] = x – f(x)/f’(x) Use the spreadsheet Newton’s Method for Quadratics - Doesn’t handle the complex roots - Doesn’t give much info about the Basins of attraction for the roots. Newton’s Quadratic Analysis

Cayley’s Discovery Use a spreadsheet to apply Newton’s Method to Cubics Basins of Attraction get complicated!! Newton’s Cubic Analysis

Polynomiography Polynomioigraphy

The Fundamental Theorem of Algebra: Version 1 An n-th degree polynomial equation with real co-efficients has n roots with complex roots occuring in pairs. Good Exercise: What are the possible solutions for a quadratic equation with real co-efficients? What are the possible solutions for a cubic equation with real co-efficients?

Good Group Activity Create Poly Images for all the possible 5th degree equations?

Challenge What is the equation of the 6th degree polynomial whose polynomiograph is shown below?

Solutions Real Roots z = 3 , z =4 Complex Roots z = 3 + 3i , z = 3 – 3i , z = 1 + I , z = 1 – I (z -3)(z-4)( z- [3+3i] )(z –[ 3 – 3i ]} )( z- [1+i] )(z –[ 1 – i ]} = 0

DeMoivre’s Theorem: A Visual Approach

Play z12 + z2 – 4z + 54

A Final Linear Thought We teach about arithmetic sequences x(n+1) = x(n) + d / x(0) = c Or x(n) = c + n*d We teach about geometric sequences x(n+1) = r x(n) / x(0) = c x(n) = c rn

How about teaching about mixed sequences? A mixed sequence x(n+1) = r*x(n) + d / x(0) = c Or x(n) = c rn[ c – (d/(1-r) ] + (d/(1-r)

Use the Language of Dynamical Systems Let f(x) = ax + b. F has a fixed point F = b/(1—a) The orbit of x(0) under iteration of f has an n-th iterate equal to x(n) = an( x(0) – F) + F