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Chapter 4: Rational, Power, and Root Functions 4.1 Rational Functions and Graphs 4.2 More on Graphs of Rational Functions 4.3 Rational Equations, Inequalities, Applications, and Models 4.4 Functions Defined by Powers and Roots 4.5 Equations, Inequalities, and Applications Involving Root Functions Copyright © 2007 Pearson Education, Inc. Slide 4-2 4.2 More on Graphs of Rational Functions Vertical and Horizontal Asymptotes p( x) For the rational function f ( x) q ( x ) , written in lowest terms, if f ( x) as x a, then the line x a is a vertical asymptote; if f ( x) a as x , then the line y a is a horizontal asymptote. Copyright © 2007 Pearson Education, Inc. Slide 4-3 4.2 Finding Asymptotes: Example 1 Example 1 Find the asymptotes of the graph of x 1 f ( x) . 2 x 5x 3 2 Solution Vertical asymptotes: set denominator equal to 0 and solve. 2 x 5x 3 0 1 (2 x 1)( x 3) 0 x or x 3 2 The equations of the vertical asymptotesare 2 x 1 and x 3. 2 Copyright © 2007 Pearson Education, Inc. Slide 4-4 4.2 Finding Asymptotes: Example 1 Horizontal asymptote: divide each term by the variable factor of greatest degree, in this case x2. x 1 1 1 2 2 2 f ( x) 2x x x x 2 x 5x 3 2 5 3 2 2 2 x x2 x x x As x gets larger and larger, 1 1 5 3 , 2 , , 2 all approach 0. x x x x 00 0 0 200 2 Therefore, the line y = 0 is the horizontal asymptote. Copyright © 2007 Pearson Education, Inc. Slide 4-5 4.2 Finding Asymptotes: Example 2 Example 2 Find the asymptotes of the graph of 2x 1 f ( x) . x3 Solution Vertical asymptote: solve the equation x – 3 = 0. The line x 3 is the vertical asymptote. Horizontal asymptote: divide each term by x. 2x 1 1 2 x 20 2 2 f ( x) x x x 3 3 1 0 1 1 x x x as x . Therefore, the horizontal asymptoteis the line y 2. Copyright © 2007 Pearson Education, Inc. Slide 4-6 4.2 Finding Asymptotes: Example 3 Example 3 Find the asymptotes of the graph of x 1 f ( x) . x2 2 Solution Vertical asymptote: x 2 0 Horizontal asymptote: x2 x2 1 1 1 2 2 2 x x x f ( x) x 2 1 2 2 2 2 x x x x 1 does not approach any real number as x since is undefined . 0 Therefore, there is no horizontal asymptote . This occurs when the degree of the numerator is greater th an the degree in the denominato r. Copyright © 2007 Pearson Education, Inc. Slide 4-7 4.2 Finding Asymptotes: Example 3 Rewrite f using synthetic division as follows: x 1 5 f ( x) x2 x2 x2 2 5 For very large values of x , is close to 0, and x2 the graph approaches the line y = x +2. This line is an oblique asymptote (neither vertical nor horizontal) for the graph of the function. Copyright © 2007 Pearson Education, Inc. Slide 4-8 4.2 Determining Asymptotes To find asymptotes of a rational function defined by a rational expression in lowest terms, use the following procedures. 1. 2. Vertical Asymptotes Set the denominator equal to 0 and solve for x. If a is a zero of the denominator but not the numerator, then the line x = a is a vertical asymptote. Other Asymptotes Consider three possibilities: (a) If the numerator has lower degree than the denominator, there is a horizontal asymptote, y = 0 (x-axis). (b) If the numerator and denominator have the same degree, and f is an x n a0 f ( x) , where bn 0, n bn x b0 the horizontal asymptoteis the line y Copyright © 2007 Pearson Education, Inc. an . bn Slide 4-9 4.2 Determining Asymptotes 2. Other Asymptotes (continued) (c) If the numerator is of degree exactly one greater than the denominator, there may be an oblique asymptote. To find it, divide the numerator by the denominator and disregard any remainder. Set the rest of the quotient equal to y to get the equation of the asymptote. Notes: i) ii) The graph of a rational function may have more than one vertical asymptote, but can not intersect them. The graph of a rational function may have only one other nonvertical asymptote, and may intersect it. Copyright © 2007 Pearson Education, Inc. Slide 4-10 4.2 Graphing Rational Functions Let f ( x) p ( x) define a rational expression in lowest terms. q ( x) To sketch its graph, follow these steps. 1. 2. 3. 4. 5. Find all asymptotes. Find the x- and y-intercepts. Determine whether the graph will intersect its nonvertical asymptote by solving f (x) = k where y = k is the horizontal asymptote, or f (x) = mx + b where y = mx + b is the equation of the oblique asymptote. Plot a few selected points, as necessary. Choose an xvalue between the vertical asymptotes and x-intercepts. Complete the sketch. Copyright © 2007 Pearson Education, Inc. Slide 4-11 4.2 Comprehensive Graph Criteria for a Rational Function A comprehensive graph of a rational function will exhibits these features: 1. all intercepts, both x and y; 2. location of all asymptotes: vertical, horizontal, and/or oblique; 3. the point at which the graph intersects its nonvertical asymptote (if there is such a point); 4. enough of the graph to exhibit the correct end behavior (i.e. behavior as the graph approaches its nonvertical asymptote). Copyright © 2007 Pearson Education, Inc. Slide 4-12 4.2 Graphing a Rational Function Example x 1 Graph f ( x) . 2 x 5x 3 2 Solution Step 1 From Example1, the vertical asymptotes have equations x 12 and x 3, and the horizontal asymptoteis the x - axis. Step 2 x-intercept: solve f (x) = 0 x 1 0 2 x 5x 3 x 1 0 x 1 2 Copyright © 2007 Pearson Education, Inc. The x - intercept is 1. Slide 4-13 4.2 Graphing a Rational Function y-intercept: evaluate f (0) 0 1 1 f (0) 2 2(0) 5(0) 3 3 Step 3 1 The y - intercept is 3 To determine if the graph intersects the horizontal asymptote, solve f ( x) 0. y - value of horizontal asymptote Since the horizontal asymptote is the x-axis, the graph intersects it at the point (–1,0). Copyright © 2007 Pearson Education, Inc. Slide 4-14 4.2 Graphing a Rational Function Step 4 Plot a point in each of the intervals determined by the x-intercepts and vertical asymptotes, (,3), (3,1), (1, 1 ), and ( 12 , ) to get an 2 idea of how the graph behaves in each region. Step 5 Complete the sketch. The graph approaches its asymptotes as the points become farther away from the origin. Copyright © 2007 Pearson Education, Inc. Slide 4-15 4.2 Graphing a Rational Function That Does Not Intersect Its Horizontal Asymptote 2x 1 Example Graph f ( x) . x3 Solution Vertical Asymptote: x 3 0 Horizontal Asymptote: y 12 x3 y2 x-intercept: 2 x1 0 2 x 1 0 or x 1 x3 2 2(0)1 y-intercept: f (0) 03 13 Does the graph intersect the horizontal asymptote? 2x 1 2 2 x 6 2 x 1 No solution. x3 Copyright © 2007 Pearson Education, Inc. Slide 4-16 4.2 Graphing a Rational Function That Does Not Intersect Its Horizontal Asymptote 2x 1 , choose To complete the graph of f ( x) x3 13 points (–4,1) and 6, 3 . Copyright © 2007 Pearson Education, Inc. Slide 4-17 4.2 Graphing a Rational Function with an Oblique Asymptote x 1 Example Graph f ( x) . x2 2 Solution Vertical asymptote: x 2 0 Oblique asymptote: x 1 2 x2 5 x2 , therefor e the x2 x2 oblique asymptote is the line y x 2. x-intercept: None since x2 + 1 has no real solutions. y-intercept: f (0) 12 , so the y - intercept is 12 . Copyright © 2007 Pearson Education, Inc. Slide 4-18 4.2 Graphing a Rational Function with an Oblique Asymptote Does the graph intersect the oblique asymptote? x 2 1 x2 x 2 x 1 x 4 No solution. 2 2 To complete the graph, choose the points 2. 4, 17 and 1 , 2 3 Copyright © 2007 Pearson Education, Inc. Slide 4-19 4.2 Graphing a Rational Function with a Hole x 4 . Example Graph f ( x) x2 2 Solution Notice the domain of the function cannot include 2. Rewrite f in lowest terms by factoring the numerator. x 2 4 ( x 2)( x 2) f ( x) x 2 ( x 2) x 2 x 2 The graph of f is the graph of the line y = x + 2 with the exception of the point with x-value 2. Copyright © 2007 Pearson Education, Inc. Slide 4-20