SECTION 4.6 The Graph of a Rational Function 247 Explaining Concepts 64. If the graph of a rational function R has the horizontal asymptote = y 2, the degree of the numerator of R equals the degree of the denominator of R. Explain why. 65. The graph of a rational function cannot have both a horizontal and an oblique asymptote. Explain why. 66. If the graph of a rational function R has the vertical asymptote = x 4, the factor −x 4 must be present in the denominator of R. Explain why. ‘Are You Prepared?’ Answers 1. True 2. Quotient: +x3 3; remainder: − − x x 2 3 3 2 3. x y 22 2 2 22 (1, 1) (21, 21) 4. x y 3 3 23 23 (0, 21) (21, 23) Problems 67–76 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. 67. Find the equation of a vertical line passing through the point ( ) − 5, 3. 68. Solve: ( ) − + = − x x 2 5 3 7 1 4 2 69. Is the graph of the equation − = x xy 2 4 3 2 symmetric with respect to the x -axis, the y -axis, the origin, or none of these? 70. What are the points of intersection of the graphs of the functions ( ) = − + f x x3 2 and ( ) = − − g x x x2 4? 2 71. Find the intercepts of the graph of ( ) = − + f x x x 6 2 . Retain Your Knowledge 72. Use a graphing utility to find the local maximum of ( ) = + − + f x x x x 4 3 1 3 2 73. Where is ( ) = − − < f x x x 5 13 6 0? 2 74. Determine whether the function ( ) = + f x x x 6 3 2 is even, odd, or neither. 75. Simplify: − − + x x 3 9 2 3 2 76. Solve: + − − + − = x x x x x x 9 9 2 3 2 2 4.6 The Graph of a Rational Function Now Work the ‘Are You Prepared?’ problems on page 255. • Finding Intercepts (Section 1.3, pp. 20–21) PREPARING FOR THIS SECTION Before getting started, review the following: 1 Analyze the Graph of a Rational Function Graphing utilities make the task of graphing rational functions less time consuming. However, the results of algebraic analysis must be taken into account before drawing conclusions based on the graph provided by the utility. In the next example we illustrate how to use the information collected in the last section in conjunction with the graphing utility to analyze the graph of a rational function ( ) ( ) ( ) = R x p x q x . OBJECTIVES 1 Analyze the Graph of a Rational Function (p. 247) 2 Solve Applied Problems Involving Rational Functions (p. 254)
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