MAT129 Fall 2025 Notes mathnstuff.com/math/precalc/MAT129fa25/p.9.htm © 2025   A2, Agnes Azzolino (Polynomial)/(polynomial) = (rational fx) Rational Functions, Asymptotes and Graphing. Text 5.6, PG. 414

Welcome to Prealc! (Polynomial)/(polynomial) = (rational fx) Hi,     When one divides polynomials, rational (as in ratio or fraction) functions are created.     Contact me through the pink box on www.mathnstuff.com/math/math.htm or with the phone number on your class CANVAS home page.     Stay safe, A2

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

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   anxn + an-1xn-1 + an-2xn-2 + ... +a2x2 + a1x1 + a0x0 __________________________________________ bnxn + bn-1xn-1 + bn-2xn-2 + ... +b2x2 + b1x1 + b0x0   in which the constants an, an-1, an-2, ... , a2, a1, a0, bn, bn-1, bn-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 To provide a list of things to do when sketching a rational function, To provide two examples of sketching rational functions, To give you a problems to try and their solution, and 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 (Axm)/(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. recipfx.gsp   -- graph of parameter-defined reciprocal function,     * with slide-able tangent line,     * VERTICAL ASYMPTOTE MUST BE POSITIONED BY USER! asymptote definition end behavior   -- definition increasing/decreasing Who Has? Rat'l F(x) Game COMPOSITION -- Composition of Functions polynomial, rational functions pdf compositeFX.gsp  -- build 2 functions using other functions compositeFx.poly.ratl.gsp   -- build functions using other functions 0 - graph of 1 function 1 - graphs of 2 functions 2 - composite of 4 functions 3 - polynomial. links to polynomial, dilations gif, quatratic.htm, equation, discriminant 4 - rational, rational function features & how to find them 5 - end behavior, battle of (Ax^m)/(Bx^n) Spread Sheets -- Dynamic Computation polynomial spreadsheet --     graph a polynomial function written in factored form     complete synthetic division synthetic division spreadsheet -- just complete synthetic division of a polynomial or of a quadratic by a linear binomial rational function spread sheet     Graph and table of (Axm)/(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.