Rational Functions Rational Functions Basics • The graph of a rational function is formed by two symmetrical curves and two asymptotes • Each curve gets infinitely close to each asymptote (but never touches or crosses it) • ]Range: −∞, G[∪] G,+∞[• The inverse of a rational function is another rational function There are two ways to write the rule of a rational function Slant Asymptotes ! Consider a rational function whose denominator is of degree 1 or greater. ! If the degree of the numerator is exactly one more than the degree of the denominator, the graph of the function has a slant (or oblique) asymptote. ! For example, the graph of ! has a slant asymptote . 21 ! To find the equation of a slant asymptote, use long division. ! For instance, by dividing x 2.6 RATIONAL FUNCTIONS In this section you will learn how to: • Find the domain of rational functions • Find horizontal, vertical and slant asymptotes of rational functions • Analyze and sketch the graph of a rational function • Use rational functions to model and solve real-life problems 2.6 Rational Functions January 19, 2011 To Graph a Rational Function: 1. Factor numerator and denominator. 2. Find the domain. 3. Reduce the fraction to lowest terms. 4. Find each vertical asymptote. 5. Find the coordinates of any holes in the function. 6. Find the horizontal asymptote if there is one. Does the graph of the function cross its horizontal asymp- tote? If so, at what point? Vertical Asymptotes of Rational Functions Look again at the graph of in Figure 2.30.The curve approaches,but does not touch, the The or is said to be a vertical asymptoteof the graph.A rational function may have no vertical asymptotes, one vertical asymptote, or several vertical asymptotes. The graph of a rational function never intersects a vertical asymptote.We will use dashed lines to show 2.6 Rational Functions and Asymptotes 193 Applications There are many examples of asymptotic behavior in real life. For instance, Example 5 shows how a vertical asymptote can be used to analyze the cost of removing pollutants from smokestack emissions. EXAMPLE 5 Cost-Benefit Model A utility company burns coal to generate electricity. The cost of removing a certain percent of the pollutants 2.6_bzpc4e - Free download as Powerpoint Presentation (.ppt), PDF File (.pdf), Text File (.txt) or view presentation slides online. Scribd is the world's largest social reading and publishing site. Open navigation menu. Close suggestions Search Search. en Change Language. close menu Language . English Section 2.6 Rational Functions and Asymptotes Objective: In this lesson you learned how to determine the domain and find asymptotes of rational functions. I. Introduction to Rational Functions (Page 189) The domain of a rational functio n of x includes all real numbers except . . . x-values that make the denominator zero. 2.6 Rational Functions August 18, 2010 Vertical Asymptotes are caused by zero values in the denominator. y = ___________2x - 3 (x-1)(x+2) A look at y = ___1 x and some transformations y = ___1 x 1 x+2 y = ___ 1 x-3 y = ___ +1 y = ______________2x(x+2) (x-1)(x+3) The numerator tell us the roots (x-intercepts) of the function. To find equation, divide the function and ignore the remainder. Example 1 : Find all asymptotes of the graph. a) 3 1 2 ( ) 2+ = x x f x b) 1 2 ( ) 2 2 − = x x f x Values (x-values) for which a rational function