Ex 1: Graphing Rational Functions — Transcript

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1. 0:00Given the rational function f of x equals four divided by x.
2. 0:04We wanna determine the domain of the function, complete the table of values and then graph the function.
3. 0:10The key to determining the domain of the rational function.
4. 0:13Is to remember that division by zero is undefined.
5. 0:16So here we have four divided by x. So whatever values make the denominator equal to zero.
6. 0:21Can not be in the domain and since we have an x in the denominator.
7. 0:26The domain will be all real numbers except z equals 0.
8. 0:30So we'll say all reals.
9. 0:35Except.
10. 0:37X equals zero.
11. 0:40This is helpful for a variety of reasons. First,
12. 0:43this functional have a vertical asymptote at x equals zero
13. 0:47Which should be this line here which is also the y axis.
14. 0:51Vertical asymptote is a vertical line the graph approaches,
15. 0:55but does not cross. So this will be helpful in making a sketch.
16. 0:59Next when making a table values we're gonna put the value not in the domain or zero.
17. 1:05Right in the middle of the table as we see here. So we know when x equals zero.
18. 1:10The function or y would be undefined.
19. 1:18And now we are going to select x values that are greater than zero and less than zero.
20. 1:24And it is always helpful to pick at least two values that are pretty close to zero.
21. 1:27For example here we selected x equals 1/2 and then increasing values of x.
22. 1:33Then here we selected x equals negative 1/2 and decreasing values of x.
23. 1:38Ofcourse we can select any x values we want.
24. 1:41But since we do have four divided by x.
25. 1:44It is convenient to pick values of x that can divide evenly into four.
26. 1:49Now lets complete our table.
27. 1:51If x is equal to 8, f of x or y would be equal to four divided by 8.
28. 1:58Which simplifies to 1/2 cup
29. 2:01Which means if x is equal to -8, y would be negative 1/2
30. 2:07If x is equal to four, y is equal to four divided by four or one.
31. 2:13If x is negative four y is -1.
32. 2:16If x is equal to 2, y is equal to four divided by two, which is positive two.
33. 2:22So if x is -2, y is equal to -2
34. 2:28If x is equal to one y is equal to four divided by one which is four.
35. 2:35And four divided by -1 would be negative four.
36. 2:39But notice when x is equal to one-half.
37. 2:42We would have f of x or y.
38. 2:46Equal to...
39. 2:48Four divided by 1/2
40. 2:50remember dividing by a fraction is the same as multiplying by the reciprocal.
41. 2:55Should be four times two over one or just four times two.
42. 2:59Which is equal to eight.
43. 3:01So when x is equal to 1/2, y is 8 and when x is equal to -1/2 we would just have -8.
44. 3:09Now lets go ahead and plot these points and see if we can make a nice sketch.
45. 3:13So, 8, 1/2 would be here.
46. 3:174, 1, would be here.
47. 3:212,2 is here.
48. 3:241,4 is here.
49. 3:27And 1/2 8 would be here.
50. 3:32So this piece of the graph would look something like this.
51. 3:40Now we'll plot the values where x is less than zero.
52. 3:43So we have negative 1/2 negative 8, that's here.
53. 3:48-1 -4
54. 3:52-2 -2
55. 3:55And -4 -1
56. 3:59And then -8, -1/2 would be here.
57. 4:05So this piece of the graph would look something like this.
58. 4:15There's one more thing and notice here.
59. 4:17As x increases or decreases with outbound.
60. 4:21Meaning x approaches positive infinity or negative infinity.
61. 4:24Notice how the function value would actually approach 0.
62. 4:28Which means not only do we have a vertical asymptote at x equals 0.
63. 4:33We also have a horizontal asymptote at y equals 0 for the x axis.
64. 4:44We'll take a look at several more examples of graphing rational functions in the next few videos.
65. 4:49I hope you found this helpful!