Math 2325
Project 5 (Chapter 14) - Iteration of Quadratic Functions

Helmut Knaust, Department of Mathematical Sciences, UTEP, El Paso TX 79968
hknaust@utep.edu
10/31/2012.Last edit 11/04//2020.

The graph of f(x)=a x (1-x) for values of a between 2 and 4:

f[x_]=a x (1-x)
Manipulate[Plot[{a x(1-x),x},{x,0,1},ImageSize->600,PlotStyle->{{Red,Thick},{Dashed}}],{a,2,4,.1}]

The additional blue graph shows the composition of the function f with itself :

f[f[x]]//Expand

Manipulate[Plot[{a (1-x) x,-(a^3) (x^4)+2 (a^3) (x^3)-(a^3) (x^2)-(a^2) (x^2)+(a^2) x,x},{x,0,1},ImageSize->600,PlotStyle->{{Red,Thick},{Blue,Thick},{Dashed}}],{a,2,4,.1}]

The additional green graph shows the 4-fold composition of the function f with itself.

f[f[f[f[x]]]]//Expand

Manipulate[Plot[{a (1-x) x,-(a^3) (x^4)+2 (a^3) (x^3)-(a^3) (x^2)-(a^2) (x^2)+(a^2) x,-(a^15) (x^16)+8 (a^15) (x^15)-28 (a^15) (x^14)-4 (a^14) (x^14)+56 (a^15) (x^13)+28 (a^14) (x^13)-70 (a^15) (x^12)-84 (a^14) (x^12)-6 (a^13) (x^12)-2 (a^12) (x^12)+56 (a^15) (x^11)+140 (a^14) (x^11)+36 (a^13) (x^11)+12 (a^12) (x^11)-28 (a^15) (x^10)-140 (a^14) (x^10)-90 (a^13) (x^10)-34 (a^12) (x^10)-6 (a^11) (x^10)+8 (a^15) (x^9)+84 (a^14) (x^9)+120 (a^13) (x^9)+60 (a^12) (x^9)+30 (a^11) (x^9)-(a^15) (x^8)-28 (a^14) (x^8)-90 (a^13) (x^8)-70 (a^12) (x^8)-61 (a^11) (x^8)-6 (a^10) (x^8)-(a^9) (x^8)-(a^8) (x^8)+4 (a^14) (x^7)+36 (a^13) (x^7)+52 (a^12) (x^7)+64 (a^11) (x^7)+24 (a^10) (x^7)+4 (a^9) (x^7)+4 (a^8) (x^7)-6 (a^13) (x^6)-22 (a^12) (x^6)-36 (a^11) (x^6)-36 (a^10) (x^6)-8 (a^9) (x^6)-8 (a^8) (x^6)-2 (a^7) (x^6)+4 (a^12) (x^5)+10 (a^11) (x^5)+24 (a^10) (x^5)+10 (a^9) (x^5)+10 (a^8) (x^5)+6 (a^7) (x^5)-(a^11) (x^4)-6 (a^10) (x^4)-7 (a^9) (x^4)-7 (a^8) (x^4)-7 (a^7) (x^4)-(a^6) (x^4)-(a^5) (x^4)+2 (a^9) (x^3)+2 (a^8) (x^3)+4 (a^7) (x^3)+2 (a^6) (x^3)+2 (a^5) (x^3)-(a^7) (x^2)-(a^6) (x^2)-(a^5) (x^2)-(a^4) (x^2)+(a^4) x,x},{x,0,1},ImageSize->600,PlotStyle->{{Red,Thick},{Blue,Thick},{Green,Thick},{Dashed}}],{a,2,4,.1}]