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1. Page 597 Page 597 Page 597 8.4 - Graphs of Rational Functions [Day 2]

2. Graphs of Rational Functions/Day 2/ Section 8-4 8.4 - Graphs of Rational Functions [Day 2]

3. Asymptotes Horizontal Asymptotes are found by looking at the degrees of p(x) and q(x): To Graph: • Putting the numerator equal to zero to determine its roots • Making the denominator equal to zero to find the vertical asymptotes • Follow the rules for horizontal asymptotes • Drawing a dotted line onto the graph to represent the vertical and horizontal asymptote • Use the graphing calculator to determine the accuracy of the graphs 8.4 - Graphs of Rational Functions [Day 2]

4. Horizontal Asymptotes Horizontal Asymptotes are found by looking at the degrees of p(x) and q(x): 8.4 - Graphs of Rational Functions [Day 2]

5. Your Turn Your Turn Your Turn Graph . Determine the zeros, vertical and horizontal asymptote(s), domain, and y-intercept(s) 8-4: Graphs of Rational Functions [Day 1]

6. Example 5 Graph . Determine the zeros, vertical and horizontal asymptote(s), domain, range, and y-intercept(s) x = –2 x = 2 y = 2 8-4: Graphs of Rational Functions [Day 1]

7. Example 5 Given x = –2 x = 2 y = 2 8-4: Graphs of Rational Functions [Day 1]

8. Your Turn Your Turn Your Turn Graph . Determine the zeros, vertical and horizontal asymptote(s), domain, and y-intercept(s) 8-4: Graphs of Rational Functions [Day 1]

9. Example 6 Example 6 Example 6 Given, Determine the zeros, vertical and horizontal asymptote(s) 8-4: Graphs of Rational Functions [Day 1]

10. 1. f(x) = 2. f(x) = 3. f(x) = 4x –12 (x + 2)(x – 3) x –2 x2+ x x–1 x+1 Review Identify the zeros and determine all of the asymptotes to the following: 8.4 - Graphs of Rational Functions [Day 2]

11. Holes Holes Holes Holesare omitted points in a graph. In some cases, both the numerator and the denominator of a rational function will equal 0 for a particular value of x. The difference between an asymptote and hole is: • Holes are applied to cancelled expressions which cause the graph to be discontinuous • For asymptote graphs, they generally approach the asymptote • A hole CANNOT be a vertical asymptote • A hole CANNOT be a zero HOLES TAKE PRECEDENCE OVER ASYMPTOTES 8.4 - Graphs of Rational Functions [Day 2]

12. f(x) = (x –1)(x + 1) For x ≠ 1, f(x) = = x + 1 (x – 1)(x + 1) x–1 (x – 1) Example 1 Example 1 Example 1 Given graph, determine the zeros, all asymptotes/holes, and domain Factor the numerator. Divide out common factors. Solve for x for the hole. The expression x – 1 is a factor of both the numerator and the denominator. There is a hole in the graph at x = 1. 8.4 - Graphs of Rational Functions [Day 2]

13. Example 2 Given , graph, determine the zeros, all asymptotes/holes, and domain 8.4 - Graphs of Rational Functions [Day 2]

14. Hole at x = 3 Example 3 Example 3 Example 3 Given, graph, determine the zeros and all asymptotes/holes 8.4 - Graphs of Rational Functions [Day 2]

15. When Graphing… Put the original equation into the calculator Deciding zeros and asymptotes, use the simplified form Domain’s restrictions include the HOLES and vertical asymptotes 8.4 - Graphs of Rational Functions [Day 2]

16. Example 4 Given graph, determine the zeros, all asymptotes/holes, and domain 8.4 - Graphs of Rational Functions [Day 2]

17. Example 5 Example 5 Example 5 Given, graph, determine the zeros, all asymptotes/holes, and domain 8.4 - Graphs of Rational Functions [Day 2]

18. Given graph, determine the zeros, all asymptotes/holes, and domain Your Turn 10/23/2014 10:23 AM 8.4 - Graphs of Rational Functions [Day 2] 18

19. Given graph, determine the zeros, all asymptotes/holes, and domain Example 6 10/23/2014 10:23 AM 8.4 - Graphs of Rational Functions [Day 2] 19

20. Adiscontinuous functionis a function whose graph has one or more gaps or breaks. The hyperbola graphed in Example 2 and many other rational functions are discontinuous functions. A continuous functionis a function whose graph has no gaps or breaks. The functions you have studied before this, including linear, quadratic, polynomial, exponential, and logarithmic functions, are continuous functions. Definitions Definitions Definitions 10/23/2014 10:23 AM 8.4 - Graphs of Rational Functions [Day 2] 20

21. Write an equation where there is a hole at x = 2, zero is at x = –3, vertical asymptote is at x = –1, and horizontal asymptote is at y = 1. Example 7 10/23/2014 10:23 AM 8.4 - Graphs of Rational Functions [Day 2] 21

22. Write an equation where there is a hole at x = 2, zero is at x = –3, vertical asymptote is at x = –1, and horizontal asymptote is at y = 1. Example 7 10/23/2014 10:23 AM 8.4 - Graphs of Rational Functions [Day 2] 22

23. Write an equation where there is a hole at x = 2, zero is at x = –3, vertical asymptote is at x = –1, and horizontal asymptote is at y = 1. Example 7 10/23/2014 10:23 AM 8.4 - Graphs of Rational Functions [Day 2] 23

24. Write an equation where there is a hole at x = 2, zero is at x = –3, vertical asymptote is at x = –1, and horizontal asymptote is at y = 1. Example 7 10/23/2014 10:23 AM 8.4 - Graphs of Rational Functions [Day 2] 24

25. Write an equation where there is a hole at x = –2/3, zero is at x = 1/2, vertical asymptote is at x = –5, and horizontal asymptote is at y = 1. Your Turn 10/23/2014 10:23 AM 8.4 - Graphs of Rational Functions [Day 2] 25

26. Example 8 Little Suzie was calculating her grade to see what grade she needs in order to pass a certain class. Right now, her test grades are 65, 68, and 71 out of 100 points for each test. Here is a given equation, Write the equation of her current situation is and determine the horizontal asymptote, if there is any. 8.4 - Graphs of Rational Functions [Day 2]

27. Holes Worksheet 8.4 - Graphs of Rational Functions [Day 2]

28. Assignment Finish Worksheet 8.4 - Graphs of Rational Functions [Day 2]

29. Pg 59829, 31, 33-38 all, 39, 41 8.4 - Graphs of Rational Functions [Day 2]