Section 3.5 Rational Functions and Their Graphs Rational Functions Rational Functions are quotients of polynomial functions.
Download ReportTranscript Section 3.5 Rational Functions and Their Graphs Rational Functions Rational Functions are quotients of polynomial functions.
Section 3.5 Rational Functions and Their Graphs Rational Functions Rational Functions are quotients of polynomial functions. This means that rational functions can p ( x) be expressed as f(x)= where p and q are q ( x) polynomial functions and q(x) 0. The domain of a rational function is the set of all real numbers except the x-values that make the denominator zero. Example Find the domain of the rational function. x 16 f ( x) x4 2 Example Find the domain of the rational function. x f ( x) 2 x 36 Vertical Asymptotes of Rational Functions 1 The equation f(x)= x Vertical Asymptote on the y-axis. y x 1 x2 Vertical Asymptote on the y-axis. The equation f(x)= Two Graphs with Vertical Asymptotes, one without x f(x)= 2 x 9 Graphing Calculator Input the equation as you see at left. The first graph is Connected Mode. In connected mode, the graphing calculator plots many points and connects the points with "curves." In dot mode, the graphing calculator plots the Connected mode same points but does not connect them. To change the mode on the calculator press the MODE key then scroll down to the line that says Connected Dot you want to use. Dot Mode and choose the one that Example Find the vertical asymptote, if any, of the graph of the rational function. x f ( x) 2 x 36 Example Find the vertical asymptote, if any, of the graph of the rational function. x f ( x) 2 x 36 Example Find the vertical asymptote, if any, of the graph of the rational function. x6 f ( x) 2 x 36 x2 4 Consider the function f(x)= . Because the denominator is zero when x2 x=2, the function's domain is all real numbers except 2. However, there is a reduced form of the equation in which 2 does not cause the denominator to be zero. A graph with a hole corresponding to the denominator’s zero. Your calculator will not show the hole. Horizontal Asymptotes of Rational Functions Two Graphs with Horizontal Asymptotes, one without Notice how the horizontal asymptote intersects the graph. Example Find the horizontal asymptote, if any, of the graph of the rational function. 3x f ( x) 2 x 1 Example Find the horizontal asymptote, if any, of the graph of the rational function. 6x2 f ( x) 2 x 1 Using Transformations to Graph Rational Functions Graphs of Common Rational Functions Transformations of Rational Functions Example 1 1 Use the graph of f(x)= to graph g(x)= 4 x x 3 y x Example 1 1 Use the graph of f(x)= 2 to graph g(x)= 2 2 x x 4 y x Graphing Rational Functions Example 5x Graph f(x)= using the 7 step strategy from x2 the previous slide. y x Example 2x2 Graph f(x)= 2 using the 7 step strategy. x 25 y x Slant Asymptotes The graph of a rational function has a slant asymptote if the degree of the numerator is one more than the degree of denominator. The equation of the slant asymptote can be found by division. It is the equation of the dividend with the term containing the remainder dropped. Example x2 6 x 2 Find the slant asymptote of the function f(x)= . x Example x3 1 Find the slant asymptote of the function f(x)= 2 . x 2x 2 Applications The cost function, C, for a business is the sum of its fixed and variable costs. The average cost per unit for a company to produce x units is the sum of its fixed and variable costs divided by the number of units produced. The average cost function is a rational function that is denoted by C. Thus Example The Fort Myers Fishing Company discovered a better material for making fishing reels. The fixed monthly cost is $10,000 for the cost of rental of space, manufacturing equipment, as well as wages and benefits for it’s employees. It costs $10 for materials to make each fishing reel. a.Write the cost function, C, of producing x fishing poles. b. Write the average cost function, C, of producing x fishing poles. c. Find and interpret C (1,000), C (100,000) d. What is the horizontal asymptote for the graph of the average cost function C. Describe what this represents for the company. Find the vertical asymptote(s) for the graph of 4 x2 the rational function f(x)= 2 . x 16 (a) y 4 x4 (c) x 6, 6 (d) y 4, 4 (b) Find the horizontal asymptote(s) for the graph of 4 x2 the rational function f(x)= 2 . x 16 (a) y 4 x4 (c) x 6, 6 (d) y 4, 4 (b) 3 3x Find the horizontal asymptote for f(x)= 2 . x 36 (a) y 3 (b) y 6, 6 (c) x 6, 6 (d) none