Summary of End Behavior or Long Run Behavior of Polynomial Functions — Transcript

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1. 0:00- IF ASKED TO DETERMINE THE END BEHAVIOR OR LONG RUN BEHAVIOR
2. 0:04OF A POLYNOMIAL FUNCTION,
3. 0:06IT'S POSSIBLE TO DETERMINE THE END BEHAVIOR
4. 0:08JUST BY KNOWING THE DEGREE OF THE POLYNOMIAL
5. 0:11AND THE SIGN OF THE LEADING COEFFICIENT.
6. 0:13BUT BEFORE WE START,
7. 0:14LET'S REVIEW WHAT END BEHAVIOR OR LONG RUN BEHAVIOR IS.
8. 0:18THE END BEHAVIOR DESCRIBES THE VALUE OF THE FUNCTION
9. 0:21OR THE Y VALUE
10. 0:22AS X APPROACHES THE LEFT OR APPROACHES NEGATIVE INFINITY
11. 0:27AND APPROACHES THE RIGHT OR APPROACHES POSITIVE INFINITY.
12. 0:32FIRST LET'S CONSIDER WHEN THE DEGREE IS EVEN,
13. 0:34AND WE'LL ASSUME THAT THE TERMS OF THE POLYNOMIAL
14. 0:36ARE IN DESCENDING ORDER,
15. 0:38SO OUR POLYNOMIAL WILL HAVE DEGREE N
16. 0:41AND THE LEADING COEFFICIENT WILL BE A.
17. 0:44SO IF N IS GREATER THAN OR EQUAL OR 2, AND EVEN,
18. 0:47AND THE LEADING COEFFICIENT IS POSITIVE OR GREATER THAN 0,
19. 0:51THEN AS X APPROACHES INFINITY, OR AS WE MOVE RIGHT,
20. 0:55F OF X APPROACHES POSITIVE INFINITY
21. 0:58OR THE Y VALUES INCREASE WITHOUT BOUND.
22. 1:01SO LOOKING AT THE GRAPH, AS X APPROACHES POSITIVE INFINITY,
23. 1:05WE'D BE MOVING RIGHT ALONG THE FUNCTION.
24. 1:07AS WE MOVE RIGHT ALONG THE FUNCTION,
25. 1:09NOTICE HOW THE Y VALUES OR FUNCTION VALUES ARE GOING UP
26. 1:13OR INCREASE WITHOUT BOUND,
27. 1:14THEREFORE, F OF X APPROACHES POSITIVE INFINITY.
28. 1:18AS X APPROACHES NEGATIVE INFINITY,
29. 1:21F OF X ALSO APPROACHES POSITIVE INFINITY.
30. 1:24SO AS X APPROACHES NEGATIVE INFINITY,
31. 1:27WHERE WE MOVE LEFT ALONG THE FUNCTION,
32. 1:29NOTICE HOW THE Y VALUES OR FUNCTION VALUES
33. 1:32ALSO INCREASE WITHOUT BOUND APPROACHING POSITIVE INFINITY.
34. 1:36AND THIS WILL ALWAYS BE THE CASE
35. 1:38WHEN THE DEGREE IS GREATER THAN OR EQUAL TO 2, EVEN,
36. 1:40AND THE LEADING COEFFICIENT IS POSITIVE.
37. 1:43SO FOR EXAMPLE, IF WE HAVE A FUNCTION LIKE F OF X
38. 1:47EQUALS 5X TO THE EIGHTH - 6X TO THE FIFTH,
39. 1:54PLUS 3X SQUARED - 2,
40. 1:57THE END BEHAVIOR OF THIS FUNCTION WOULD BE THE SAME
41. 1:59AS THE END BEHAVIOR DESCRIBED HERE,
42. 2:02AND WE CAN DETERMINE THIS
43. 2:03JUST BY LOOKING AT THE DEGREE OF THE POLYNOMIAL
44. 2:06AND THE SIGN OF THE LEADING COEFFICIENT.
45. 2:09NEXT, IF N IS GREATER THAN OR EQUAL OR 2 AND EVEN,
46. 2:13BUT THE LEADING COEFFICIENT IS NEGATIVE OR LESS THAN 0,
47. 2:17THEN AS X APPROACHES POSITIVE INFINITY,
48. 2:19F OF X APPROACHES NEGATIVE INFINITY.
49. 2:22SO AS WE MOVE RIGHT ALONG THE GRAPH,
50. 2:24NOTICE HOW THE Y VALUES OR FUNCTIONS VALUES
51. 2:27DECREASE WITHOUT BOUND APPROACHING NEGATIVE INFINITY,
52. 2:31AND AS X APPROACHES NEGATIVE INFINITY,
53. 2:34F OF X ALSO APPROACHES NEGATIVE INFINITY.
54. 2:38SO AS WE MOVE LEFT, OR AS X APPROACHES NEGATIVE INFINITY,
55. 2:42THE FUNCTION VALUES OR Y VALUES ALSO DECREASE WITHOUT BOUND,
56. 2:46APPROACHING NEGATIVE INFINITY.
57. 2:49AND AGAIN, THIS WILL ALWAYS BE TRUE
58. 2:51WHEN WE HAVE A POLYNOMIAL FUNCTION WITH AN EVEN DEGREE
59. 2:53AND A NEGATIVE LEADING COEFFICIENT.
60. 2:56SO FOR EXAMPLE, THE FUNCTION F OF X EQUALS -0.25X TO THE FOURTH
61. 3:04+ 5X TO THE THIRD - 200
62. 3:10WILL HAVE THE SAME END BEHAVIOR AS DESCRIBED HERE,
63. 3:13AND WE KNOW THIS JUST BECAUSE THE DEGREE IS EVEN
64. 3:16AND THE LEADING COEFFICIENT IS NEGATIVE.
65. 3:19NOW, LET'S CONSIDER END BEHAVIOR
66. 3:21WHEN WE HAVE A POLYNOMIAL FUNCTION AND THE DEGREE IS ODD.
67. 3:25IF N IS GREATER THAN OR EQUAL TO 1, AND ODD,
68. 3:28WITH A POSITIVE LEADING COEFFICIENT,
69. 3:32THEN AS X APPROACHES POSITIVE INFINITY,
70. 3:34F OF X APPROACHES POSITIVE INFINITY.
71. 3:37SO AS WE MOVE RIGHT ALONG THE GRAPH,
72. 3:40OR AS X APPROACHES POSITIVE INFINITY,
73. 3:42NOTICE HOW THE FUNCTION VALUES OR Y VALUES
74. 3:45ARE INCREASING WITHOUT BOUND,
75. 3:47THEREFORE, F OF X IS APPROACHING POSITIVE INFINITY,
76. 3:51AND AS X APPROACHES NEGATIVE INFINITY,
77. 3:54F OF X APPROACHES NEGATIVE INFINITY.
78. 3:57SO AS WE MOVE LEFT ALONG THE GRAPH,
79. 3:59NOTICE HOW THE Y VALUES OR FUNCTION VALUES
80. 4:01ARE DECREASING WITHOUT BOUND, APPROACHING NEGATIVE INFINITY.
81. 4:05THIS WILL ALWAYS BE THE CASE WHEN THE DEGREE IS ODD
82. 4:08AND THE LEADING COEFFICIENT IS POSITIVE.
83. 4:10SO FOR EXAMPLE, IF WE HAVE THE FUNCTION F OF X EQUALS
84. 4:142X TO THE SEVENTH - 2X TO THE SIXTH
85. 4:19+ 7X TO THE FIFTH - 3X SQUARED - 1,
86. 4:25AGAIN, BECAUSE THE DEGREE IS ODD
87. 4:27AND THE LEADING COEFFICIENT IS POSITIVE,
88. 4:30THE END BEHAVIOR WILL BE THE SAME AS DESCRIBED HERE.
89. 4:34AND THE LAST CASE
90. 4:35IS WHEN N IS GREATER THAN OR EQUAL TO 1 AND ODD
91. 4:38AND A IS NEGATIVE OR LESS THAN 0,
92. 4:41THAN AS X APPROACHES POSITIVE INFINITY,
93. 4:44F OF X APPROACHES NEGATIVE INFINITY,
94. 4:47SO AS WE MOVE RIGHT ALONG THE GRAPH,
95. 4:49NOTICE HOW THE FUNCTION VALUES OR Y VALUES
96. 4:51ARE DECREASING WITHOUT BOUND APPROACHING NEGATIVE INFINITY,
97. 4:55AND AS X APPROACHES NEGATIVE INFINITY,
98. 4:58F OF X APPROACHES POSITIVE INFINITY.
99. 5:01SO AS WE MOVE LEFT ALONG THE GRAPH,
100. 5:03NOTICE HOW THE FUNCTION VALUES OR Y VALUES
101. 5:05ARE INCREASING WITHOUT BOUND APPROACHING POSITIVE INFINITY.
102. 5:08THIS WILL ALWAYS BE THE CASE WHEN THE DEGREE IS ODD
103. 5:11AND THE LEADING COEFFICIENT IS NEGATIVE.
104. 5:13SO IF WE HAVE F OF X = -10X TO THE FIFTH
105. 5:21+ 4X TO THE FOURTH - 3,
106. 5:25THIS FUNCTION WILL HAVE THE SAME END BEHAVIOR
107. 5:27AS DESCRIBED HERE BECAUSE THE DEGREE IS ODD,
108. 5:30THE LEADING COEFFICIENT IS NEGATIVE.
109. 5:34OF COURSE, WHENEVER YOU'RE ASKED TO DESCRIBE THE END BEHAVIOR
110. 5:36AND YOU'RE NOT SURE,
111. 5:37IT'S BEST TO GRAPH THE FUNCTION AND ANALYZE THE GRAPH.
112. 5:42I HOPE YOU FOUND THIS SUMMARY HELPFUL.