• Question 1
  • What are the fixed points of f(x)=x^2 ?:
    

    p = 0,1,-1.
    
    

  • What is the relation between the fixed points of the square function and the intersection points of the graph of the square function with the special lines y = x and y = -x?
    

    The square function and the special lines y=x and y=-x intersect at the fixed points of the square function.
    
    

• Question 2
  • Consider the square function on your calculator. Pick a point "near" (distance less than 0.1 is near enough) a fixed point. Keep iterating (pushing the squaring button) until you see a pattern for the long term behavior. Try again with other numbers near each of the fixed points. Write down what happens for each choice.
    

    Any number that was greater than 1 or less than -1 tended towards infinity or negative infinity, respectively, for the long term behavior. Any number greater than -1 but less than 1 (that is, between -1 and 1) converged to 0.
    
    

  • Does it matter if the number you pick is less than or greater than the fixed point?
    

    Yes. As stated above, any number less than the fixed point -1 will diverge, as will any number greater than the fixed point 1. Any number in between the open interval of -1 and 1 will converge to the fixed point 0.
    
    

• Question 3
  • Which of the fixed points for the square function are attracting? Which are repelling?

    Attracting: p=0

    Repelling: p=1,-1
    
    

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