Chapter 2.2- Polynomial Functions of Higher Degree

Graphs of Polynomial Functions

The graph of a polynomial function is:

• continuous: does not contain breaks, holes, or gaps in the graph

• has only smooth, rounded turns

Note: the graphs cannot have sharp pointed turns







Example of polynomial function:

Note: this graph is continuous- has no breaks, holes or gaps- and has smooth,

rounded turns

Monomials of the form: (Note: n must be an integer greater than zero)

n is even n is odd

When n is even, the graph touches When n is odd, the graph crosses

the axis at the x-intercept. the axis at the x-intercept.

The Leading Coefficient Test

As x moves to the left or right,

the graph rises or falls based on these rules:

When n is odd

• leading coefficient = positive, the graph falls to the left and rises to the right

• leading coefficient = negative, the graph rises to the left and falls to the right

leading coefficient is positive leading coefficient is negative



When n is even

• leading coefficient = positive, the graph rises to the left and right

• leading coefficient = negative, the graph falls to the left and right

Example

Determine the left and right behavior of the graphs:

rises to the left and falls to the right

Why? The degree is odd and the leading coefficient is negative

rises to the left and right

Why? The degree is even and the leading coefficient is positive

Zeros of Polynomial Functions

• 
  
  
  the graph f has at most n real zeros. In other words, the highest degree of a polynomial function determines the number of zeros.
  

• 
  
  the function f has at most x-1 relative extrema.

Note: Extrema are the relative minimums and maximums combined

Zero Rules:

• 
  
  
  x=a is a zero of the function f



• 
  
  
  x=a is a solution of the polynomial equation f(x)=0



• 
  
  
  (x-a) is a factor of the polynomial f(x)



• 
  
  
  (a,0) is an x-intercept of the graph f