Determining Vertical and Horizontal Asymptotes of Rational Functions — Transcript

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1. 0:00Welcome to a lesson on determining the horizontal  and vertical asymptotes of a rational function. A
2. 0:06rational function is a function that can be  described as a quotient of two polynomials,
3. 0:11where q of x is not the zero polynomial,  or the domain of the rational function
4. 0:16would be all the values of x that don't make  q of x or the denominator equal to zero.
5. 0:23Here's an example of a rational function,  where the numerator is the polynomial x
6. 0:27squared minus four and the denominator is the  polynomial x squared minus two x minus three.
7. 0:32It's often helpful to express a rational  function in factored form, as we see here.
8. 0:37Let's take a look at the graph of this function.  The one thing we should notice right away is that
9. 0:42this graph has two vertical asymptotes, here at  x equals negative one and here at x equals three.
10. 0:49Vertical asymptotes are lines that the  graph approaches but never touches,
11. 0:53and this graph also has a horizontal asymptote  of y equals one. A horizontal asymptote
12. 1:01is a line that a graph approaches, but it may  also cross it, as we see here. One last thing to
13. 1:07notice about the domain of this function is that  since x equals negative one and x equals three
14. 1:14would make the denominator equal to zero, we  must exclude those two values from the domain.
15. 1:25Let's take a look at how we're going to determine  the equations of the vertical asymptotes.
16. 1:29The line x equals a is a vertical asymptote of  the graph of the function if a is a zero of the
17. 1:36denominator and does not come from a common factor  with the numerator. Zeros of the denominator that
18. 1:43are also zeros of the numerator result in a  hole in the graph, not a vertical asymptote.
19. 1:50For example, if we take a look at this  function here, the first thing we want to
20. 1:53do is factor both the numerator and denominator.  So, we'd have x plus three times x minus three,
21. 2:01and the denominator would be the factors  of positive six that add to negative five:
22. 2:06we'd have x minus two and x minus three.  Notice there are two values that make the
23. 2:12denominator equal to zero. The domain of this  function would have to exclude x equals two
24. 2:19and x equals three, but these both  do not result in vertical asymptotes.
25. 2:25Since the zero x equals three is also a zero of  the numerator, there's a hole at x equals three,
26. 2:34not a vertical asymptote. However, x equals  two is a zero of the denominator but not of
27. 2:40the numerator, and therefore there is  a vertical asymptote at x equals two.
28. 2:48If x equals a is a vertical asymptote,  as x approaches the value of a, f of x
29. 2:54or y approaches either positive infinity or  negative infinity, meaning the graph will
30. 2:59go up indefinitely or down indefinitely  as it approaches a vertical asymptote.
31. 3:05Now let's talk about horizontal asymptotes.  Horizontal asymptotes are horizontal lines
32. 3:10that the graph approaches as x approaches  either positive infinity or negative infinity.
33. 3:17For horizontal asymptotes, we're  going to see how the function behaves
34. 3:21as x increases or decreases without bound.
35. 3:26One way to help us determine the horizontal  asymptotes is to think of how the numerator and
36. 3:30denominator behave as they approach infinity.  If we can determine whether the numerator or
37. 3:35denominator wins, it will help us determine  the equation of the horizontal asymptote.
38. 3:40What I mean by that is, if the degree of the  numerator is higher than the degree of the
39. 3:44denominator, as we see here in this first  function, there will not be a horizontal
40. 3:49asymptote. The reason there isn't one is  because the degree of the numerator is three
41. 3:54and the degree of the denominator is equal  to one. So as x approaches positive infinity,
42. 3:59the numerator is increasing faster, and therefore  the function value increases without bound.
43. 4:05We can illustrate this pretty  quickly with a graphing calculator.
44. 4:08I've already typed the function  into y1. If we go to our table
45. 4:13and then use the table feature, we can see  that as x increases, y increases without bound.
46. 4:23Here's an illustration of why there's  no horizontal asymptote. We can also
47. 4:28look at this graphically. If we press the graph
48. 4:31key, you can see that as we move right along  the graph, the graph moves up indefinitely,
49. 4:39or if we move left, it moves down indefinitely.  Therefore, there's no horizontal asymptote.
50. 4:45If the degree of the numerator and denominator are  equal, the horizontal asymptote will be the ratio
51. 4:50of the leading coefficients. So looking at our  second example here, our numerator and denominator
52. 4:56have degree one, and the ratio of the  leading coefficients would be two over one.
53. 5:01Therefore, the equation of the  horizontal asymptote is y equals two.
54. 5:05Let's take a look at this on the graphing  calculator. I'm going to go ahead and turn off y1
55. 5:13and turn on y2. Let's go to  our table, and notice that as
56. 5:19x increases, we can see that the y value  is approaching the value of positive two.
57. 5:30We can also take a look at the graph  and see that as we move to the right,
58. 5:38or as x approaches positive infinity,  the y value is approaching positive two.
59. 5:46And if we approach negative infinity, you can  see we'd be approaching positive two as well.
60. 5:56Lastly, if the degree of the denominator  is higher than the degree of the numerator,
61. 6:00the horizontal asymptote will be y equals  zero. In this case, the numerator has degree
62. 6:06zero and the degree of the denominator is  degree one. As x increases without bound,
63. 6:12the numerator stays at two and the denominator  increases without bound. Therefore,
64. 6:17the function value is approaching zero, so  y equals zero is our horizontal asymptote.
65. 6:23Let's go back to the calculator one more time.  Let's turn off y2 and turn on y3. Let's look at
66. 6:30the table first. As x increases without bound, we  can see the y values are quickly approaching zero.
67. 6:40We can see as x approaches positive infinity to  the right, the function values approach zero,
68. 6:50and they do the same if we approach  negative infinity to the left.
69. 7:02Let's go and take a look at our examples now. Here  we want to determine the vertical and horizontal
70. 7:07asymptotes and then graph the function. Notice  the numerator and denominator cannot be factored.
71. 7:13So for the vertical asymptote, if  we set x plus one equal to zero,
72. 7:18we would have x equals negative one as a vertical  asymptote. Let's go ahead and sketch that.
73. 7:29Then for the horizontal asymptote, the degree  of the numerator is zero, and the degree of the
74. 7:34denominator is one. Therefore, our horizontal  asymptote would be y equals zero. Or again, you
75. 7:42can think of what happens to the function value as  x increases without bound. Our numerator stays at
76. 7:47four, and our denominator gets larger and larger,  so the function values would approach zero.
77. 7:54Now, these asymptotes make it a lot easier to  graph this rational function because we know
78. 7:58the pieces of the graph will approach these lines.  So let's go ahead and determine a couple of points
79. 8:03on this function, and then we should be  able to make a nice graph of the function.
80. 8:10We're going to pick values that are close to the  vertical asymptote. So we'll select x equals zero
81. 8:17and x equals one to the right of the vertical  asymptote, and then we'll select x equals negative
82. 8:23two and x equals negative three. When x is zero,  we'd have four divided by one, that's four.
83. 8:31When x is one, we'll have four divided  by one plus one, that'll be two.
84. 8:35When x is negative two, we'll have four divided  by negative one, that'll be negative four.
85. 8:40And then when x is negative three,  we'll have four divided by negative two.
86. 8:44Let's go ahead and plot these four points: zero,  four; one, two; negative two, negative four;
87. 8:52and negative three, negative two. Notice  without these vertical asymptotes,
88. 8:58we probably couldn't make a nice graph, but since  we do know the function approaches these lines,
89. 9:04we can make a pretty good graph with just these  four points. It would look something like this.
90. 9:13And here's the graph done on some  software. Let's take a look at another
91. 9:18example. The first step is to factor  both the numerator and denominator.
92. 9:25x squared would be x times x, and the  denominator also has a common factor of x.
93. 9:34So even though zero and two are zeros of the  denominator, they both do not result in vertical
94. 9:41asymptotes because x equals zero is also a zero  of the numerator. There's a hole at x equals zero,
95. 9:50and there would be a vertical  asymptote at x equals two.
96. 9:55For the horizontal asymptote, since the degree  of the denominator is equal to two and so is the
97. 10:01degree of the numerator, the horizontal asymptote  would be the ratio of the leading coefficients,
98. 10:06or the horizontal asymptote would be y equals one.
99. 10:10Let's go ahead and sketch our two asymptotes.  We have x equals two and we have y equals one.
100. 10:21Let's go and select a couple values of x and then  determine y. Let's go and select x equals three
101. 10:27and x equals one. f of three would be nine divided  by nine minus six, that would give us three. When
102. 10:38x is equal to one, we'd have one divided by one  minus two, that's going to give us negative one.
103. 10:45So we have the point three, three and  then we have the point one, negative one.
104. 10:53Even though we have two points, we know the  graph is going to approach these asymptotes.
105. 10:58So we could sketch something like this. This  piece would be approaching these asymptotes.
106. 11:06The only thing to be careful about is there's  supposed to be a hole at x equals zero.
107. 11:11So what we should do is go over to our graph at  x equals zero and make sure we have a hole there.
108. 11:18Looking at some software, the graph would  look like this. I hope you found this helpful.