Characteristics of Polynomial Functions

Names of Polynomial Functions

Degree	0	1	2	3	4	5	6	7	8	9	10
Name	constant	linear	quadratic	cubic	quartic	quintic	sextic	septic	octic	nonic	decic

Key Features

Using key features of a polynomial function like its intercepts, end behaviors, extreme points, can be used to help produce an image of a graph.

Feature	Description
Intercepts	The x and y intercepts. The x-intercepts can be found by setting y to zero and the y-intercept can be found by setting x to zero. These points are when the function crosses its respective intercept.
Sign of leading coefficient	This is the sign (+/-) of the term with the highest degree on the polynomial (\(ax^n\)). This can be used to determine the end behavior of the function
End behavior	This is where the function ends when \(x \to +\infty\) and \(x \to -\infty\) A positive even function will alwas be \(Q2 \to Q1\), a negative even function always goes from \(Q3 \to Q4\), a positive odd function will always go from \(Q3 \to Q1\) and a negative odd function always goes from \(Q2 \to Q4\) (these are all \(x\) from \(-\infty \to +\infty\))
Extreme points	These are all maximum and minimum points. A local extreme point is any point that is a vertex (the point changes direction from \(+ \to -\) or \(- \to +\) and a absolute extreme point (also called global extreme point) is the absolute most extreme value on the entire funtion. Finding the absolute extreme is usually an optimization problem. If a function extends to \(+\infty\) it does not have a absolute maxima, the same is true with \(-\infty\) and an absolute minima.

Extreme Points and Turning Points

Polynomial functions with a degree n (\(n \epsilon \mathbb{W}, n > 1\)) will have at most n - 1 extreme points. With the same conditions, a polynomial function will also have at most n - 1 turning points.

Finite Differences

When taking the differences in y values for any constant values of x of a function, this will result in first differences. \(f(x) - f(x + a), f(x + a) - f(x + 2a), f(x + 2a) - f(x + 3a)\)… will result in the first differnces. Subtracting first differences from each other will result in second differences, subtracting second differences result in third differences, and so on. When the differences become a constant, it is called the finite difference.
For all polynomial functions of degree n, the nth differences are the finite difference. \(a(n(n - 1)(n - 2)(n - 3)...(3)(2)(1)) = an!\) where a is the leading coefficient. The finite difference can be expressed as diff \(= an!\)