Ex: Find the Intercepts, Asymptotes, and Hole of a Rational Function — Transcript

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1. 0:01- WE'RE GIVEN A RATIONAL FUNCTION
2. 0:02AND ASKED TO DETERMINE THE Y-INTERCEPT, X-INTERCEPTS,
3. 0:06VERTICAL ASYMPTOTES, HORIZONTAL ASYMPTOTES,
4. 0:08AND ANY POSSIBLE HOLES IN THE FUNCTION.
5. 0:11WHEN WORKING WITH RATIONAL FUNCTIONS,
6. 0:13IT'S OFTEN BEST TO HAVE IT IN FACTORED FORM.
7. 0:15LET'S GO AHEAD AND START BY FACTORING
8. 0:17THE NUMERATOR AND DENOMINATOR.
9. 0:19LOOKING AT THE NUMERATOR,
10. 0:20WE WOULD HAVE A FACTOR OF X AND A FACTOR OF X HERE
11. 0:24AND THE FACTORS OF -12 THAT ADD TO -1 ARE -4 AND 3.
12. 0:31LOOKING AT THE DENOMINATOR,
13. 0:32NOTICE HOW WE FIRST HAVE A COMMON FACTOR OF 2.
14. 0:34LET'S FACTOR 2 OUT.
15. 0:36THAT WOULD GIVE US X SQUARED - 2X - 8,
16. 0:42BUT THIS DOES FACTOR AGAIN.
17. 0:45WE STILL HAVE OUR FACTOR OF 2,
18. 0:47AND THEN WE'LL HAVE TWO BINOMIAL FACTORS.
19. 0:49THE FACTORS OF X SQUARED ARE X AND X.
20. 0:52THE FACTORS OF -8 THAT ADD TO -2 ARE -4 AND 2.
21. 0:58ONE OF THE REASONS IT'S IMPORTANT
22. 0:59TO FACTOR OUR RATIONAL FUNCTIONS
23. 1:01IS NOTICE IN THIS CASE WE HAVE A COMMON FACTOR
24. 1:04OF X - 4 BETWEEN THE NUMERATOR AND DENOMINATOR.
25. 1:07THE REASON THIS IS IMPORTANT IS THAT THE VALUES OF X
26. 1:10THAT MAKE THIS COMMON FACTOR EQUAL TO 0
27. 1:12WILL NOT GIVE US AN X-INTERCEPT OR A VERTICAL ASYMPTOTE.
28. 1:16IT'S ACTUALLY GOING TO GIVE US A HOLE.
29. 1:18AGAIN, SINCE 4 MAKES THIS FACTOR EQUAL TO 0,
30. 1:22THAT MEANS WE'RE GOING TO HAVE A HOLE AT THE POINT
31. 1:25WHERE THE X COORDINATE IS EQUAL TO 4.
32. 1:29ONCE WE KNOW THIS, WE CAN GO AHEAD AND SIMPLIFY THE FUNCTION,
33. 1:32MEANING SIMPLIFY OUT THIS COMMON FACTOR,
34. 1:35LEAVING US WITH THE FUNCTION,
35. 1:37THE QUANTITY (X + 3) DIVIDED BY 2 x THE QUANTITY (X + 2).
36. 1:45THE GRAPH OF THIS SIMPLIFIED FUNCTION WILL BE EXACTLY
37. 1:48THE SAME AS THE GRAPH OF THE ORIGINAL FUNCTION
38. 1:51EXCEPT THERE WILL A HOLE AT X = 4.
39. 1:54NOW TO DETERMINE THE Y COORDINATE
40. 1:56WHERE THIS HOLE WILL BE,
41. 1:58WE CAN NOW EVALUATE THE SIMPLIFIED FUNCTION AT X = 4.
42. 2:03SO F(4) OF THE SIMPLIFIED FUNCTION
43. 2:06WOULD BE 4 + 3 DIVIDED BY WE'D HAVE 2 X 4 + 2.
44. 2:15THIS IS GOING TO BE 7.
45. 2:16THIS WILL BE 6 X 2 OR 12.
46. 2:19SO THERE WILL BE A HOLE IN THE GRAPH AT THE POINT (4, 7/12).
47. 2:25NOW THAT WE HAVE A HOLE HERE,
48. 2:26WE CAN USE THE SIMPLIFIED FUNCTION HERE
49. 2:28TO DETERMINE THE REST OF THE INFORMATION.
50. 2:32SO TO FIND THE Y-INTERCEPT OF OUR FUNCTION,
51. 2:34WE'LL SET X EQUAL TO 0 WHICH MEANS WE NEED TO EVALUATE F(0),
52. 2:41AND WE CAN GO AHEAD AND USE THE SIMPLIFIED FUNCTION.
53. 2:43SO IF X IS 0 WE WOULD HAVE 3 ALL OVER 2 x 2 WHICH IS 4.
54. 2:49SO F(0) EQUALS 3/4 WHICH MEANS THE Y-INTERCEPT
55. 2:54IS THE POINT (0, 3/4).
56. 2:57NOW TO DETERMINE THE X-INTERCEPTS,
57. 2:59WE SET Y OR F(X) EQUAL TO 0.
58. 3:04AND AGAIN, WE'RE GOING TO USE THE SIMPLIFIED FUNCTION.
59. 3:06AND BECAUSE WE HAVE A RATIONAL FUNCTION,
60. 3:08THIS FUNCTION IS ONLY GOING TO EQUAL 0
61. 3:11WHEN THE NUMERATOR = 0.
62. 3:14NOTICE THE NUMERATOR WILL EQUAL 0 WHEN X = -3
63. 3:18WHICH MEANS OUR X-INTERCEPT IS THE POINT (-3, 0).
64. 3:22NOTICE THAT WE DID NOT USE THE SIMPLIFIED FUNCTION.
65. 3:24WE WOULD MAKE THE MISTAKE
66. 3:26AND ASSUME THERE WOULD BE AN X-INTERCEPT
67. 3:28WHEN X = 4,
68. 3:29BUT THERE'S NOT GOING TO BE
69. 3:30BECAUSE THE FUNCTION IS UNDEFINED AT 4,
70. 3:33AND THERE'S ACTUALLY A HOLE IN THE FUNCTION.
71. 3:34THAT'S WHY WE HAVE TO USE THE SIMPLIFIED FUNCTION.
72. 3:38AND THEN TO DETERMINE THE VERTICAL ASYMPTOTES,
73. 3:40THESE OCCUR WHEN THE DENOMINATOR IS EQUAL TO 0.
74. 3:43SO X + 2 = 0 WHEN X = -2.
75. 3:47THEREFORE, THE EQUATION OF THE VERTICAL ASYMPTOTE IS X = -2.
76. 3:52AND NOW FOR THE LAST QUESTION,
77. 3:54WE WANT TO DETERMINE THE HORIZONTAL ASYMPTOTES
78. 3:56WHICH CAN BE FOUND BY ANALYZING
79. 3:58THE DEGREE OF THE NUMERATOR AND DENOMINATOR.
80. 4:01AND WE CAN USE THE SIMPLIFIED FUNCTION
81. 4:02OR THE ORIGINAL FUNCTION.
82. 4:04NOTICE IN THIS FORM HERE,
83. 4:06THE NUMERATOR AND DENOMINATOR HAVE DEGREE 1.
84. 4:08AND SINCE THEY HAVE THE SAME DEGREE,
85. 4:10THE EQUATION OF THE HORIZONTAL ASYMPTOTE
86. 4:12WILL BE Y EQUALS THE RATIO OF THE LEADING COEFFICIENTS.
87. 4:16NOTICE THE LEADING COEFFICIENT OF THE NUMERATOR WOULD BE 1,
88. 4:18LEADING COEFFICIENT OF THE DENOMINATOR WOULD BE 2,
89. 4:21SO Y = 1/2 OR Y = 1/2 IS THE EQUATION
90. 4:26OF THE HORIZONTAL ASYMPTOTE.
91. 4:29USING THIS SHORTCUT, REMEMBER IF THE DEGREE OF THE NUMERATOR
92. 4:31WAS LESS THAN THE DEGREE OF THE DENOMINATOR,
93. 4:34THEN THE HORIZONTAL ASYMPTOTE WOULD BE Y = 0.
94. 4:37AND IF THE DEGREE OF THE NUMERATOR
95. 4:38WAS GREATER THAN THE DEGREE OF THE DENOMINATOR
96. 4:40THERE WOULD BE NO HORIZONTAL ASYMPTOTE.
97. 4:42LET'S GO AHEAD AND VERIFY OUR WORK BY GRAPHING THE FUNCTION.
98. 4:46HERE IS THE HOLE IN THE FUNCTION AT X = 4.
99. 4:49HERE IS OUR X-INTERCEPT OF -3, OUR Y-INTERCEPT OF 3/4,
100. 4:55OUR VERTICAL ASYMPTOTE OF X = -3,
101. 4:59AND OUR HORIZONTAL ASYMPTOTE OF Y = 1/2.
102. 5:04SO THIS GRAPH DOES VERIFY THAT OUR WORK IS CORRECT.
103. 5:07WE HOPE YOU FOUND THIS HELPFUL.