Unit+9

=Unit 9 Rational Functions=

Vocabulary

 * branch (p. 485) || independent events (p. 519) || rational function (p. 491) ||
 * combined variation (p. 480) || inverse variation (p. 478) || simplest form of a rational expression (p. 499) ||
 * complex fraction (p. 506) || mutually exclusive events (p. 521) ||  ||
 * dependent events (p. 519) || point of discontinuity (p. 491) ||

9-1 Objectives
An equation in two variables of the form //y// = or //xy// = //k//, where //k// ≠ 0 , is an **inverse variation.** **Combined variation** is an extension of direct and inverse variation to more complicated relationships.
 * 1) To use inverse variation (p. 478)
 * 2) To use combined variation (p. 480)

9-2 Objectives
The graph of an inverse variation has two parts called **branches.** The graph of //y// = + //c// is a translation of //y// = by //b// units horizontally and //c// units vertically. It has a vertical asymptote at //x// = //b// and a horizontal asymptote at //y// = //c//.
 * 1) To graph inverse variations (p. 485)
 * 2) To graph translations of inverse variations (p. 487)

9-3 Objectives
The rational function //f//(//x//) = has a **point of discontinuity** for each real zero of //Q//(//x//). If //P//(//x//) and //Q//(//x//) have no common factors, then the graph of //f//(//x//) has a vertical asymptote when //Q//(//x//) = 0. If //P//(//x//) and //Q//(//x//) have a common real zero //a//, then there is a hole or a vertical asymptote at //x// = //a//. If the degree of //Q//(//x//) is greater than the degree of //P//(//x//), then the graph of //f//(//x//) has a horizontal asymptote at //y// = 0. If //P//(//x//) and //Q//(//x//) have equal degrees, then there is a horizontal asymptote at //y// =, where //a// and //b// are the coefficients of the terms of greatest degree in //P//(//x//) and //Q//(//x//). If the degree of //P//(//x//) is greater than the degree of //Q//(//x//), then there is no horizontal asymptote.
 * 1) To identify properties of rational functions (p. 491)
 * 2) To graph rational functions (p. 494)

9-4 and 9-5 Objectives
A rational expression is in **simplest form** when its numerator and denominator are polynomials that have no common divisors. To add or subtract rational expressions with different denominators, write each expression with the least common denominator. A fraction that has a fraction in its numerator or denominator or in both is called a **complex fraction.** You can simplify a complex fraction by multiplying the numerator and denominator by the LCD of all the rational expressions.
 * 1) To simplify rational expressions (p. 499)
 * 2) To multiply and divide rational expressions (p. 500)
 * 3) To add and subtract rational expressions (p. 504)
 * 4) To simplify complex fractions (p. 506)

9-6 Objectives
Solving a rational equation often requires multiplying both sides by an algebraic expression. This may introduce an **extraneous solution**—a solution of a derived equation but not of the original equation. Check all possible solutions in the original equation.
 * 1) To solve rational equations (p. 512)
 * 2) To use rational equations in solving problems (p. 513)