2.2 Polynomial Functions of a Higher Degree           

The Graph of a polynomial Function is a countinuous..

              1. No Holes

              2. No Breaks

              3. No Gaps

              4. No Sharp Turns

         Continuous                 Not Continuous

     >If n is even: resembles a parabola

     >If n is odd: resembles x^3

     >As n gets bigger, graph gets flatter

      example.  Sketch    f(x)= -x^5                                                                      G(x)= x^4+1

End Behavior--->   (right side, left side behavior)

                 as x--> -infinity, y--> infinity                                 x--> -infinity, y--> infinity

                      x--> infinity , y--> -infinity                                x--> infinity, y--> infinity

  y= x^3-2x^2+1

        = x--> -infinity, y--> -infinity                    

           x--> infinity, y--> inifinity

   f(x)= x^3

       x--> -infinity       y--> - infinity

       x--> infinity        y--> infinity   

For a Polynomial Function of a degree n,

       1). The graph has at most n zeros (x-int)

       2). The function has at most n-1 extrema (max/min)

example. Find all real zeros and relative extrema of f(x)= -2x^4+2x^2

                    -2x^4+2x^2=0

                    -2x^2(x^2-1)=0

                    -2x^2(x+1)(x-1)=0

                        x= 1, -1, 0                                        

                   Min: (0,0) Max: (+/-.707, .5)

 example. write a Polynomial with zeros at -2,-1,1,2

          f(x)= (x+2)(x+1)(x-1)(x-2)

                     (x^2-1)(x^2-4)

                     =x^4-5x^2+4

example. Sketch a graph of f(x)= 3x^4-4x^3

1. Shape                                                      3x^4-4x^3=0                         

2. Zeros                                                       x^3(3x-4)=0

3. Pick additional points                                 x= 0, 4/3 

        _x_l_y_

          1  l  -1

         .5  l  -.3125