5.6, MAT129 Notes Fall 2024

MAT129 Fall 2024 Notes

Wednesday  October  9	Topics:	5.6		Notes:	10.09.a, 10.09.b, 10.09.c, 10.09.d, 10.07.d


5.6  - Rational Functions: Asymptotes and Graphing, pg. 414,  Learning Objectives
	* Use arrow notation.
	* Solve applied problems involving rational functions.
	* Find the domains of rational functions.
	* Identify vertical asymptotes.
	* Identify horizontal asymptotes.
	* Graph rational functions.

polynomial, rational functions pdf

Rational Function Definition & Notation

Precalc definition:
an algebraic expression or function, produced by the DIVISION of polynomials, with the use of one variable x -- fraction expressions or fraction functions

a_nxⁿ + a_n-1x^n-1 + a_n-2x^n-2 + ... +a₂x² + a₁x¹ + a₀x⁰
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b_nxⁿ + b_n-1x^n-1 + b_n-2x^n-2 + ... +b₂x² + b₁x¹ + b₀x⁰

in which the constants a_n, a_n-1, a_n-2, ... , a₂, a₁, a₀, b_n, b_n-1, b_n-2, ... , b₂, b₁, b₀ are real,
and serve as coefficients or the constant term,
and in which all exponent of x are counting numbers.

Rational Functions Notes

a list of things to do when sketching a rational function

two examples of sketching rational functions

a problems to try and their solution

Features of a Rational Function

Sketch f(x) = [ (x² - 4)(x + 3)] / [(x+2)(x-1)] animation

Synthetic division & long division notes

to find zeros

to rewrite a rational function to find horizontal and other asymptotes and

to determine the end behavior.

ratl.xls

Graph and table of (Ax^m)/(Bxⁿ)
Watch the power of the top or bottom equal or dominate the other.
Examine hoirzontal asymptotes or infinite increase or decrease.
Think endbehavior and explore.
Graph and table of rational function, polynomial A(x) divided by B(x).
Examine asymptotes.
Examine the role of factors A(x) and 1/B(x).
Graph and table of rational functions written as A(x) / B(x) + C(x)
Really play with C(x), the asymptote
Complete synthetic division and rewrite of quotient and remainder.