Polynomial and Rational Functions

    Polynomial and rational functions are defined and analyzed, including graphing, determining intervals of increasing and decreasing, domain and range, intercepts, and vertical and horizontal asymptotes.

Definition (Polynomial Functions) Let be a non-negative integer. Functions of the form  



for real numbers are called polynomial functions. The numbers are called the coefficients. The are called the terms, the is called the leading term, the is called the leading coefficient, and is called the degree of

Example (Polynomial Functions) Constant functions, linear functions, and quadratic functions are examples of polynomial functions. All of the following are examples of polynomial functions, and All of the following functions are not polynomial functions, and

Definition (Roots of a Polynomial) Let be a polynomial function. Any real number such that is called a root of also called a zero of

    Polynomial functions are easy to work with if they are given in a factored form. For example, given



we can see that the solutions to the equation are and This means that which means that the zeros of are and Notice that the roots of correspond to the intercepts and of Here's the graph of Notice that has exponent 2 and there is a bounce at



The intercept is determined by

Example (Graphing Polynomial Functions) Polynomial functions are examples of continuous functions (functions having no holes or breaks). Intutively, continuous functions are functions that can be drawn with a pencil without removing the pencil from the paper. All polynomials are continuous on the whole real line.

Continuous function:



Discontinuous function:

Definition (Increasing and Decreasing) A function is increasing on the interval if whenever and a function is decreasing on if whenever

Example (Increasing and Decreasing) Determine where the following functions are increasing and decreasing:

(a) The quadratic function is decreasing on and is increasing on



(b) The cubic function is increasing on and is decreasing on

Definition (Rational Functions) Rational functions are functions of the form where and are polynomial functions and are defined whenever

Example (Rational Functions) All of the following functions are rational functions: All of the following are not rational functions: and

Definition (Vertical and Horizontal Asymptotes) Let be a rational function. The vertical asymptotes of are the vertical lines defined by for every where The horizontal asymptotes is a horizontal line which is defined by either if the degree of is greater than the degree of or is defined as where is the coefficient of the leading term of and is the coefficient of the leading term of if the degrees of and are the same.

Example (Graphing Rational Functions) The rational function has a vertical asymptote at since and Also since the degree of and the degree of are both 1, the horizontal asymptote is The asymptotes should be used to help sketch a graph of the function.

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Polynomial And Rational Functions
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/polynomial-and-rational-functions.html