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7.4 - Polynomial Functions & their Graphs

7.4 - Polynomial Functions & their Graphs

Author: Kyle Webb
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Review of the Degree of a Polynomial

If you recall, in the first section of this unit we examined two ways to classify a polynomial:  by number of terms and by degree.   In this section we will see how the degree of the polynomial helps us understand important information about the appearance of the graph of the polynomial function.  The table below reviews the names of polynomials up to degree four. 


The graph of a polynomial function is continuous.  Essentially, this means that the graph of a polynomial function has no breaks, holes, or gaps as shown below. 

Another feature of the graph of a polynomial function is that it has only smooth, rounded turns.  It cannot have a sharp pointed turn. 

Informally, you can say that a function is continuous if its graph can be drawn with a pencil without lifting the pencil from the paper. 

Graphing Polynomial Functions

Graphing Polynomial Functions with Repeated Factors

Leading Coefficient


To summarize, when n is odd, the end behavior to the left and the right is opposite, and when n is even, the end behavior to the left and the right is the same. 

Zeros of Polynomial Functions

A zero of a polynomial function is defined as a point at which the graph intersects the x-axis.  If we are given a graph of a polynomial function, we can count the number of real zeros by counting the number of times the graph touches or crosses the x-axis.

In this example, you answer should be two real zeros.  From examining the graph of the function, is it an even degree function or an odd degree function, and what is the sign of the leading coefficient?


In general, it can be shown that for a polynomial function f of degree n, the following statements are true:

  1)  The graph of f has at most n real zeros. 

  2)  A zero of a function is a number a for which f(a)=0.

  3)  (a, 0) is an x-intercept of the graph of f  if a is a zero of the function.   

  4)  If (x – a) is a factor of the polynomial function, a is a zero of the function.