1. Splash Screen

2. Lesson Menu Five-Minute Check (over Lesson 8–3) CCSS Then/Now New Vocabulary Key Concept: Vertical and Horizontal Asymptotes Example 1:Graph with No Horizontal Asymptote Example 2:Real-World Example: Use Graphs of Rational Functions Key Concept: Oblique Asymptotes Example 3:Determine Oblique Asymptotes Key Concept: Point Discontinuity Example 4:Graph with Point Discontinuity

3. Over Lesson 8–3 5-Minute Check 1 A.x = –6 B.f(x) = 0 C.x = 2 D.f(x) = 4 Which is not an asymptote of the function

4. Over Lesson 8–3 5-Minute Check 2 A.x = –4 B.x = 7 C.x = 4 D.f(x) = 0 Which is not an asymptote of the function

5. Over Lesson 8–3 5-Minute Check 3 A.D = {x | x ≠ –6} R = {f(x) | f(x) ≠ 0} B.D = {x | x ≠ 0} R = {f(x) | f(x) ≠ 0} C.D = {x | x ≠ –6, 0} R = {f(x) | f(x) ≠ 0} D.D = {x | x ≠ 0} R = {all real numbers} State the domain and range of

6. Over Lesson 8–3 5-Minute Check 4 A. B. C. D.

7. Over Lesson 8–3 5-Minute Check 5 A.–7 and –2 B.7 and –2 C.–7 and 2 D.7 and 2 For what value of x is undefined?

8. CCSS Content Standards A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Mathematical Practices 7 Look for and make use of structure.

9. Then/Now You graphed reciprocal functions. Graph rational functions with vertical and horizontal asymptotes. Graph rational functions with oblique asymptotes and point discontinuity.

10. Vocabulary rational function vertical asymptote horizontal asymptote oblique asymptote point discontinuity

11. Concept

12. Example 1 Graph with No Horizontal Asymptote Step 1Find the zeros. x 3 =0Set a(x) = 0. x=0Take the cube root of each side. There is a zero at x = 0.

13. Example 1 Graph with No Horizontal Asymptote Step 2Draw the asymptotes. Find the vertical asymptote. x + 1=0Set b(x) = 0. x=–1Subtract 1 from each side. There is a vertical asymptote at x = –1. The degree of the numerator is greater than the degree of the denominator. So, there is no horizontal asymptote.

14. Example 1 Graph with No Horizontal Asymptote Step 3Draw the graph. Use a table to find ordered pairs on the graph. Then connect the points.

15. Example 1 Graph with No Horizontal Asymptote Answer:

16. Example 1 A.B. C.D.

17. Example 2A Use Graphs of Rational Functions A. AVERAGE SPEED A boat traveled upstream at r 1 miles per hour. During the return trip to its original starting point, the boat traveled at r 2 miles per hour. The average speed for the entire trip R is given by the formula Draw the graph if r 2 = 15 miles per hour.

18. Example 2A Use Graphs of Rational Functions Simplify. The vertical asymptote is r 1 = –15. Graph the vertical asymptote and function. Notice the horizontal asymptote is R = 30. Original equation r 2 = 15

19. Example 2A Use Graphs of Rational Functions Answer:

20. Example 2B Use Graphs of Rational Functions B. What is the R-intercept of the graph? Answer: The R-intercept is 0.

21. Example 2C Use Graphs of Rational Functions C. What domain and range values are meaningful in the context of the problem? Answer: Values of r 1 greater than or equal to 0 and values of R between 0 and 30 are meaningful.

22. Example 2 A. TRANSPORTATION A train travels at one velocity V 1 for a given amount of time t 1 and then another velocity V 2 for a different amount of time t 2. The average velocity is given by. Let t 1 be the independent variable and let V be the dependent variable. Which graph is represented if V 1 = 60 miles per hour, V 2 = 30 miles per hour, and t 2 = 1 hour?

23. Example 2 A.B. C.D.

24. Example 2 B. What is the V-intercept of the graph? A.30 B.1 C.–1 D.–30

25. Example 2 C. What values of t 1 and V are meaningful in the context of the problem? A.t 1 is negative and V is between 30 and 60. B.t 1 is positive and V is between 30 and 60. C.t 1 is positive and V is is less than 60. D.t 1 and V are positive.

26. Concept

27. Example 3 Determine Oblique Asymptotes Step 1Find the zeros. x 2 =0Set a(x) = 0. x=0Take the square root of each side. There is a zero at x = 0. Graph

28. Example 3 Determine Oblique Asymptotes Step 2Find the asymptotes. x + 1=0Set b(x) = 0. x=–1Subtract 1 from each side. There is a vertical asymptote at x = –1. The degree of the numerator is greater than the degree of the denominator, so there is no horizontal asymptote. The difference between the degree of the numerator and the degree of the denominator is 1, so there is an oblique asymptote.

29. Example 3 Determine Oblique Asymptotes Divide the numerator by the denominator to determine the equation of the oblique asymptote. The equation of the asymptote is the quotient excluding any remainder. Thus, the oblique asymptote is the line y = x – 1. (–) – 1 1

30. Example 3 Determine Oblique Asymptotes Answer: Step 3Draw the asymptotes, and then use a table of values to graph the function.

31. Example 3 Graph A.B. C.D.

32. Concept

33. Example 4 Graph with Point Discontinuity Graph. Notice that or x + 2. Therefore, the graph of is the graph of f(x) = x + 2 with a hole at x = 2.

34. Example 4 Graph with Point Discontinuity Answer:

35. Example 4 A.B. C.D. Which graph below is the graph of ?

36. End of the Lesson