1. Definitions
   • Polynomials

     functions of the form

     

     where n is a non-negative integer, and the are real numbers called coefficients. If the leading coefficient , then f is a polynomial of degree n.

   • Power function

     A power function is of the form

     

     where n is a non-negative integer.

   • cubic polynomial

     a third degree polynomial

   • quartic polynomial

     a fourth degree polynomial

   • quintic polynomial

     a fifth degree polynomial

   • rational function

     a function defined as the ratio of two polynomials:

     

   • Vertical asymptote

     if values of function f increases in magnitude without bound as x approaches a value h, then the line x=h is a vertical asymptote of f.

   • Horizontal asymptote

     if values of function f approach a number k as |x| gets larger and larger, the line line y=k is a horizontal asymptote of f.

   • roots

     values of the independent variable for which the function is zero.

2. Theorems/Formulas

3. Properties/Tricks/Hints/Etc.

   Properties of Polynomial functions:

   1. A polynomial function of degree n can have at most n roots (that is, places where the function value is zero).
   2. A polynomial function of degree n can have at most n-1 peaks or pits; conversely, if the graph of a polynomial function has n-1 peaks or pits, then it must be of degree (at least) n.
   3. In the graph of a polynomial function of even degree, each extreme of the function heads to positive infinity (up), or each heads to negative infinity (down). If the degree is odd, then one end goes to positive infinity and the other heads to negative infinity.
   4. If the graph goes up to positive infinity as x becomes large, then the leading coefficient must be positive; if it goes down to negative infinity then the leading coefficient is negative.

4. Summary

   This section focuses on some very important functions: polynomials, and rational functions (which are simply ratios of polynomials). In particular, properties of polynomials such as numbers of pits and peaks, numbers of roots, behavior at plus and minus infinity, etc. are discussed.

   Rational functions have a few additional properties, including asymptotes (both vertical and horizontal). These are important modeling properties: many functions ``asymptote'' to a fixed value - that is, tend to a fixed value - as the independent variable increases without bound. For example, no matter how much effort you put into preparing for an exam, the best you can do it 100%: if we look at performance versus effort, we expect to see a horizontal asymptote.