1. Sketching the Graphs of Rational Equations

2. Consider the equation below: Solve for the discontinuities.

3. Your Turn: Solve for the discontinuities of problems 1 – 6 on Sketching the Graphs of Rational Equations – Part I

4. Answers: HA: y = 1 VA: x = 2 Holes: DNE HA: y = –½ VA: x = –3 Holes: DNE HA: y = 2 VA: x = 1 Holes: x = –2 HA: y = 0 VA: x = 2 Holes: DNE HA: y = 0 VA: x = 2 Holes: x = –1 HA: y = 2 VA: x = –3 Holes: x = 0

5. Summary – What We Know How To: Identify discontinuities Algebraically solve for discontinuities Tell the difference between vertical asymptotes and removable discontinuities

6. But aren’t we missing something? But discontinuities represent where the graph isn’t… …and not where the graph is. We need points!  y-intercept  x-intercept(s)  Additional points

7. Fold your paper in half!!! Solving for the y-intercept Solving for the x-intercept(s)

8. Solving for the y-intercept Step 1: Rewrite the equation Step 2: Substitute zero for x Step 3: Solve for y  Leave Blank for Now…

9. Example #1

10. Example #2

11. Your Turn: For problems 1 – 6, solve for the y-intercept. Check your answers in your graphing calculator!!!

12. Answers: 1. y-int = –3 2. y-int = –⅔ 3. y-int = –6 4. y-int = –1.5 5. y-int = –½ 6. y-int = DNE

13. HA: y = 2 VA: x = –3 Holes: x = 0 #6

14. Solving for the y-intercept Step 1: Rewrite the equation Step 2: Substitute zero for x Step 3: Solve for y  If the y-intercept is undefined or indeterminate, then the y-int. is DNE!!!

15. HA: y = 0 VA: x = 0 Holes: DNE Additional Example #1:

16. HA: y = 1 VA: x = 5 Holes: x = 0 Additional Example #2:

17. Solving for the x-intercept(s) Step 1: Rewrite the equation Step 2: Substitute zero for y Step 3: Solve for x  Leave Blank for Now… Step 4: Leave Blank for Now…

18. Example #1

19. Example #2

20. Your Turn: For problems 1 – 4, solve for the x- intercept(s). Check your answers in your graphing calculator!!!

21. Answers: 1. x-int = –6 2. x-int = –4 3. x-int = –3 4. x-int = DNE

22. HA: y = 2 VA: x = 1 Holes: x = –2 #3

23. HA: y = 0 VA: x = 2 Holes: DNE #4

24. Solving for the x-intercept(s) Step 1: Rewrite the equation Step 2: Substitute zero for y Step 3: Solve for x  If the answer is impossible, then the x- intercept is DNE Step 4: Check if the x-intercept matches any of the discontinuities. If it does, REJECT that x-intercept!!!!

25. Your Turn: Solve for the x-intercept(s) of problems 5 – 6.

26. HA: y = 0 VA: x = 2 Holes: x = –1 #5

27. HA: y = 2 VA: x = –3 Holes: x = 0 #6

28. Solve for the y-int. and the x-int.

29. Finding Additional Points We can use our graphing calculators to find additional points! Step 1: Make a table that has two points before and after each VA and hole. Step 2: Type the equation into y1 of graphing calculator. Step 3: Use the table function to find points to fill into the table.  Pick points that are easy to graph!!!

30. Example: HA: y = 1 VA: x = 4 Holes: DNE

31. #1 HA: y = 1 VA: x = 2 Holes: DNE

32. Your Turn: On the “Sketching the Graphs of Rational Equations – Part I” handout, make a table of additional points for problems 2 – 6.

33. Sketching – Putting It All Together!!! Step 1: Graph the HAs and VAs  Remember, we use dashed lines to represent asymptotes! Step 2: Graph the y-intercept and the x- intercept(s) (if they exist) Step 3: Graph the points from the table Step 4: Connect the points with lines Step 5: Graph the any holes

34. HA: y = 1 VA: x = 4 Holes: none y-int. = –0.5 x-int. = –2 x-valuesy-values 2–2 3–5 4Error 64 73

35. HA: y = 1 VA: x = 2 Holes: none y-int. = –3 x-int. = –6 x-valuesy-values 0–3 1–7 2Error 39 45 #1

36. Your Turn: On the “Sketching the Graphs of Rational Equations – Part I” handout, sketch the graphs of problems 2 – 6.

37. Homework Finish “Sketching the Graphs of Rational Equations – Part II”.