Section 3.2 - Basic Rational & Power Functions and More Transformations

Section Objectives

  1. Develop a familiarity with the whole-number power functions and basic rational functions.
  2. Apply transformations to basic functions.



Intro to Polynomials and Rational Functions

An -th-degree polynomial in the variable is a function of the form

,

where the coefficients , , , are real or complex numbers with . We will be almost exclusively interested in polynomials with real coefficients.


Here are some examples of polynomial functions...



A rational function is a ratio of two polynomials. That is, a rational function is a function of the form

where and are polynomials and .


The domain of a rational function is the set of all real numbers for which the denominator is nonzero.


Here are some examples of rational functions...



Reciprocal functions

One of the simplest rational functions is the reciprocal function .




Another of the simpler rational functions is the reciprocal square function .




Examples










Even-number Power Functions: ,

The graphs of , , , ... are U-shaped curves opening upward with the turning point at the origin.


The bigger the exponent, the flatter the curve is on the interval and the steeper the curve is outside that interval.




Odd-number Power Functions: ,

The graphs of , , , ... have the following shape. Each has a flat spot at the origin.


The bigger the exponent, the flatter the curve is on the interval and the steeper the curve is outside that interval.




Examples