1. Copyright © 2007 Pearson Education, Inc. Slide 4-1

2. Copyright © 2007 Pearson Education, Inc. Slide 4-2 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 4.4Functions Defined by Powers and Roots 4.5 Equations, Inequalities, and Applications Involving Root Functions

3. Copyright © 2007 Pearson Education, Inc. Slide 4-3 4.2 More on Graphs of Rational Functions Vertical and Horizontal Asymptotes

4. Copyright © 2007 Pearson Education, Inc. Slide 4-4 4.2 Finding Asymptotes: Example 1 Example 1Find the asymptotes of the graph of SolutionVertical asymptotes: set denominator equal to 0 and solve.

5. Copyright © 2007 Pearson Education, Inc. Slide 4-5 4.2 Finding Asymptotes: Example 1 Horizontal asymptote: divide each term by the variable factor of greatest degree, in this case x 2. Therefore, the line y = 0 is the horizontal asymptote.

6. Copyright © 2007 Pearson Education, Inc. Slide 4-6 4.2 Finding Asymptotes: Example 2 Example 2Find the asymptotes of the graph of Solution Vertical asymptote: solve the equation x – 3 = 0. Horizontal asymptote: divide each term by x.

7. Copyright © 2007 Pearson Education, Inc. Slide 4-7 Example 3Find the asymptotes of the graph of Solution Vertical asymptote: Horizontal asymptote: 4.2 Finding Asymptotes: Example 3

8. Copyright © 2007 Pearson Education, Inc. Slide 4-8 4.2 Finding Asymptotes: Example 3 Rewrite f using synthetic division as follows: For very large values of is close to 0, and the graph approaches the line y = x +2. This line is an oblique asymptote (neither vertical nor horizontal) for the graph of the function.

9. Copyright © 2007 Pearson Education, Inc. Slide 4-9 4.2 Determining Asymptotes To find asymptotes of a rational function defined by a rational expression in lowest terms, use the following procedures. 1.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. 2.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

10. Copyright © 2007 Pearson Education, Inc. Slide 4-10 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)The graph of a rational function may have more than one vertical asymptote, but can not intersect them. ii)The graph of a rational function may have only one other non- vertical asymptote, and may intersect it.

11. Copyright © 2007 Pearson Education, Inc. Slide 4-11 4.2 Graphing Rational Functions Let define a rational expression in lowest terms. To sketch its graph, follow these steps. 1.Find all asymptotes. 2.Find the x- and y-intercepts. 3.Determine whether the graph will intersect its non- vertical 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. 4.Plot a few selected points, as necessary. Choose an x- value between the vertical asymptotes and x-intercepts. 5.Complete the sketch.

12. Copyright © 2007 Pearson Education, Inc. Slide 4-12 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 non- vertical 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).

13. Copyright © 2007 Pearson Education, Inc. Slide 4-13 4.2 Graphing a Rational Function ExampleGraph Solution Step 1 Step 2x-intercept: solve f (x) = 0

14. Copyright © 2007 Pearson Education, Inc. Slide 4-14 4.2 Graphing a Rational Function y-intercept: evaluate f (0) Step 3To determine if the graph intersects the horizontal asymptote, solve Since the horizontal asymptote is the x-axis, the graph intersects it at the point (–1,0).

15. Copyright © 2007 Pearson Education, Inc. Slide 4-15 4.2 Graphing a Rational Function Step 4Plot a point in each of the intervals determined by the x-intercepts and vertical asymptotes, to get an idea of how the graph behaves in each region. Step 5Complete the sketch. The graph approaches its asymptotes as the points become farther away from the origin.

16. Copyright © 2007 Pearson Education, Inc. Slide 4-16 4.2 Graphing a Rational Function That Does Not Intersect Its Horizontal Asymptote ExampleGraph SolutionVertical Asymptote: Horizontal Asymptote: x-intercept: y-intercept: Does the graph intersect the horizontal asymptote?

17. Copyright © 2007 Pearson Education, Inc. Slide 4-17 4.2 Graphing a Rational Function That Does Not Intersect Its Horizontal Asymptote To complete the graph of choose points (–4,1) and.

18. Copyright © 2007 Pearson Education, Inc. Slide 4-18 4.2 Graphing a Rational Function with an Oblique Asymptote ExampleGraph SolutionVertical asymptote: Oblique asymptote: x-intercept: None since x 2 + 1 has no real solutions. y-intercept:

19. Copyright © 2007 Pearson Education, Inc. Slide 4-19 4.2 Graphing a Rational Function with an Oblique Asymptote Does the graph intersect the oblique asymptote? To complete the graph, choose the points

20. Copyright © 2007 Pearson Education, Inc. Slide 4-20 4.2 Graphing a Rational Function with a Hole Example Graph SolutionNotice the domain of the function cannot include 2. Rewrite f in lowest terms by factoring the numerator. The graph of f is the graph of the line y = x + 2 with the exception of the point with x-value 2.