244 CHAPTER 4 Polynomial and Rational Functions Figure 42 ( ) = − + − G x x x x 2 2 1 5 3 3 Solution Since the degree of the numerator, 5, is greater than the degree of the denominator, 3, by more than one, the rational function G has no horizontal or oblique asymptote. The end behavior of the graph resembles the power function = = − y x x 2 2 . 5 3 2 To see why this is the case, investigate the behavior of G as →−∞ x or as →∞ x . When x is very large, the numerator of G can be approximated by the power function = y x2 ,5 and the denominator can be approximated by the power function = y x .3 This means as →−∞ x or as →∞ x , ( ) = − + − ≈ = = − G x x x x x x x x 2 2 1 2 2 2 5 3 3 5 3 5 3 2 Since this is not linear, the graph of G has no horizontal or oblique asymptote. Figure 42 shows the graph of ( ) = − + − G x x x x 2 2 1 5 3 3 and = y x2 2 using Desmos. Now Work PROBLEMS 45, 47, AND 49 (FIND THE HORIZONTAL OR OBLIQUE ASYMPTOTES, IF ONE EXISTS.) ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 4.5 Assess Your Understanding 1. True or False The quotient of two polynomial expressions is a rational expression. (pp. A35–A36) 2. What are the quotient and remainder when − x x 3 4 2 is divided by − + x x 1. 3 2 (pp. A25–A27) 3. Graph = y x 1 . (pp. 25–26) 4. Graph ( ) = + − y x 2 1 3 2 using transformations. (pp. 112–120) Concepts and Vocabulary 5. Interactive Figure Exercise Exploring Vertical Asymptotes Open the “Multiplicity and Vertical”Asymptotes interactive figure, which is available in the Video & Resource Library of MyLab Math (under Sullivan Interactive Figures). (a) In the interactive figure, a represents the exponent on the factor −x 1 in the denominator, while b represents the exponent on the factor +x 2 in the denominator. What are the equations of the two vertical asymptotes on the graph of R ? (b) Use the slider to change the value of a to 1 and the value of b to 1. Observe the behavior of the graph of the rational function near each vertical asymptote. Change the value of a to 2. Observe the behavior of the graph of the rational function near the vertical asymptote = − x 2. Change the value of b to 2. Observe the behavior of the graph of the rational function near the vertical asymptote = x 1. Continue to experiment with other values of a and b while observing the behavior of the graph of the rational function near each vertical asymptote. Then, answer the following: (i) If the multiplicity of the zero that gives rise to a vertical asymptote is _______ (even/odd), the graph approaches ∞ on one side of the vertical asymptote and approaches −∞ on the other side. (ii) If the multiplicity of the zero that gives rise to a vertical asymptote is _______ (even/odd), the graph approaches either ∞ or −∞ on both sides of the vertical asymptote. (c) Consult the figure. x 5 6 7 3 1 2 24 21 23 y 3 4 5 2 1 21 x 5 4 x 5 22 22 (i) The multiplicity of the zero that gives rise to the vertical asymptote = − x 2 is _______ (even/odd). (ii) The multiplicity of the zero that gives rise to the vertical asymptote = x 4 is _______(even/odd). 6. True or False The domain of every rational function is the set of all real numbers. 7. If, as →−∞ x or as →∞ x , the values of ( ) R x approach some fixed number L, then the line = y L is a ____________ ____________ of the graph of R. 8. If, as x approaches some number c, the values of ( ) →∞ R x , then the line = x c is a ____________ ____________ of the graph of R. 9. True or False The graph of a rational function may intersect a horizontal asymptote. 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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