1.2 Characteristics of Polynomial Functions

Today, we learned that there are 3 key features of a polynomial function.

#1: The degree of a polynomial function can be used to determine the finite differences. How?

General Rule: A polynomial function of degree “n” (“n” is a positive integer, the “nth” differences are constant and have the same sign as the leading coefficient.

Example #1: f(x)=2 x^3 +5

• The degree is 3, which means the 3rd finite differences will be constant.
  
• The leading coefficient (2) is positive, which means the 3rd finite differences are positive.

Example #2: f(x)= -5 x^24 +2

• The degree is 24, which means the 24th differences will be constant.
  
• The leading coefficient (-5) is negative, which means the 24th differences are negative.

* That is, for example, if a function has a degree of 123, you do not need to calculate 123 finite differences using the table of values! All you need to do is look at the degree and the leading coefficient to determine when the differences are constant (in this case in the 123rd differences) and if it is positive or negative.

#2: You can classify a function as increasing or decreasing.

a. The graph rises from left to right - increasing
b. The graph falls from left to right - decreasing

For example:

*When writing in interval notation, make sure you include "xE" in front of the brackets!

#3: A turning point is used to describe the point where the function changes from increasing to decreasing (or vice-versa).

Some new terminologies: