Ex: Quadratic Function Review — Transcript

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1. 0:00IN THIS VIDEO WE'LL REVIEW
2. 0:02HOW TO FIND ALL THE MAJOR COMPONENTS
3. 0:04OF A QUADRATIC FUNCTION
4. 0:05AS WELL AS HOW TO GRAPH A QUADRATIC FUNCTION.
5. 0:08HERE WE'RE GIVEN F OF X = -2/SQUARED + 5X + 3.
6. 0:13OUR FIRST QUESTION IS, DOES THE PARABOLA OPEN UP OR DOWN.
7. 0:16THIS IS DETERMINED BY THE SIGN OF "A"
8. 0:19OR THE SIGN OF THE LEADING COEFFICIENT.
9. 0:21HERE "A" EQUAL TO -2
10. 0:24AND SINCE "A" IS NEGATIVE THE PARABOLA OPENS DOWN.
11. 0:31NEXT, WE'RE ASKED TO DETERMINE THE EQUATION
12. 0:32OF THE AXIS OF SYMMETRY.
13. 0:34FOR THIS WE'LL USE THE EQUATION X = -B DIVIDED BY 2A.
14. 0:39AGAIN, "A" IS EQUAL TO -2 AND B IS EQUAL TO +5.
15. 0:46SO OUR EQUATION WILL BE X = -B DIVIDED BY 2A.
16. 0:52WELL B IS EQUAL TO 5 AND "A" EQUAL TO -2 SO WE HAVE 2 x -2.
17. 1:00SO HERE WE HAVE -5 DIVIDED BY -4 WHICH WILL BE +5/4.
18. 1:06SO THE EQUATION OF THE AXIS OF SYMMETRY IS X = 5/4.
19. 1:13NEXT, WE'RE ASKED TO DETERMINE THE COORDINATES OF THE VERTEX.
20. 1:16REMEMBER THE VERTEX LIES ON THE AXIS OF SYMMETRY
21. 1:19SO THE X COORDINATE OF THE VERTEX WILL BE 5/4
22. 1:24AND THEN TO FIND THE Y COORDINATE
23. 1:27WE NEED TO DETERMINE F OF 5/4
24. 1:29OR SUBSTITUTE 5/4 FOR X INTO OUR FUNCTION.
25. 1:33SO F OF 5/4 IS EQUAL TO -2 x 5/4/SQUARED + 5 x 5/4 + 3.
26. 1:51WELL 5/4/SQUARED WOULD BE 25/16
27. 1:54SO WE'LL HAVE -2/1 x 25/16 + 5/1 x 5/4 + 3.
28. 2:06HERE OUR FACTOR OF 2 SIMPLIFIES.
29. 2:08THIS SIMPLIFIES TO 1, THIS SIMPLIFIES TO 8.
30. 2:12SO NOW WE HAVE -25/8 + THIS IS GOING TO BE 25/4
31. 2:20BUT TO HAVE OUR COMMON DENOMINATOR OF 8
32. 2:22THIS IS EQUAL TO 50/8 + 3 = 24/8.
33. 2:31SO NOW WE HAVE A COMMON DENOMINATOR OF 8.
34. 2:34WE'LL ADD THE NUMERATORS.
35. 2:35-25 + 50 IS 25 + 24 = 49.
36. 2:41SO THE Y COORDINATE WOULD BE 49/8.
37. 2:44WE WILL CONVERT THIS TO A DECIMAL
38. 2:45WHEN IT COMES TIME TO PLOT THE POINT.
39. 2:49NOW WE WANT TO FIND THE X AND Y INTERCEPTS OF THE GRAPH.
40. 2:52SO TO DETERMINE THE X INTERCEPTS OF THE FUNCTION
41. 2:55WE'RE GOING TO SET Y OR F OF X = 0 AND THEN SOLVE FOR X.
42. 3:00SO WE WANT TO SOLVE THE EQUATION
43. 3:010 = -2X/SQUARED + 5X + 3.
44. 3:12LET'S GO AHEAD AND FACTOR OUT THE NEGATIVE,
45. 3:14AND THEN SEE IF IT FACTORS.
46. 3:16FACTORING A NEGATIVE OUT
47. 3:17IS GOING TO CHANGE THE SIGN OF ALL THE TERMS.
48. 3:19SO WE'LL HAVE 2X/SQUARED - 5X - 3.
49. 3:23I THINK THIS WILL FACTOR.
50. 3:27FACTORS OF 2X/SQUARED WOULD BE 2X AND X.
51. 3:31NOW WE WANT TO PLACE THE FACTORS OF -3 IN HERE
52. 3:34SO THAT THE SUM OF THE INNER PRODUCT AND THE OUTER PRODUCT
53. 3:36WOULD BE -5X.
54. 3:39SO WE'LL PUT - 3 HERE, THAT'S -6X.
55. 3:42AND +1 HERE + 1X DOES = -5X.
56. 3:48SO THIS PRODUCT IS EQUAL TO 0 WHEN 2X + 1 = 0
57. 3:55OR WHEN X - 3 = 0.
58. 3:59SO HERE WE'D HAVE X = -1/2 AND HERE WE'D HAVE X = 3.
59. 4:05THESE ARE THE X INTERCEPTS.
60. 4:07WE DO WANT TO EXPRESS THESE AS ORDERED PAIRS.
61. 4:11SO IF THERE ARE TWO X INTERCEPTS,
62. 4:12ONE OF THEM IS -1/2, 0 AND THE OTHER ONE IS 3, 0.
63. 4:21NOW TO DETERMINE THE Y INTERCEPT WE WANT TO SET X = 0
64. 4:25AND THEN SOLVE FOR Y OR SOLVE FOR F OF X.
65. 4:27WELL IF WE SET X = 0 NOTICE HOW WE WOULD HAVE 0 + 0 + 3
66. 4:33SO THE Y INTERCEPT IS JUST THE POINT 0, 3.
67. 4:39SO WHEN WE HAVE A QUADRATIC FUNCTION IN THIS FORM
68. 4:42THE Y INTERCEPT IS ALWAYS JUST GOING TO BE C.
69. 4:45NOW LET'S TAKE ALL THIS INFORMATION
70. 4:46ONTO THE NEXT SLIDE AND CONTINUE.
71. 4:49NOW WE WANT TO GRAPH THIS FUNCTION.
72. 4:51LET'S TAKE A LOOK AT THE INFORMATION
73. 4:53THAT I PROVIDED HERE FROM THE PREVIOUS SLIDE.
74. 4:55NOTICE FOR THE EQUATION OF THE AXIS, X = 5/4.
75. 4:59I CONVERTED THIS TO A DECIMAL.
76. 5:015 DIVIDED BY 4 IS 1.25
77. 5:04WHICH AGAIN IS THE X COORDINATE OF THE VERTEX
78. 5:07AND 6.125 IS EQUAL TO 49/8
79. 5:11WHICH WAS THE Y COORDINATE OF THE VERTEX.
80. 5:15LET'S START BY SKETCHING THE EQUATION
81. 5:16OF THE AXIS OF SYMMETRY WHICH IS X = 1.25,
82. 5:21SOME WHERE IN HERE.
83. 5:30NEXT LET'S PLOT THE VERTEX WHICH IS ON THIS AXIS,
84. 5:34HAS THE COORDINATES 1.25, 6.125, SO SOME WHERE IN HERE.
85. 5:42NEXT, WE'LL PLOT THE INTERCEPTS
86. 5:44SO WE HAVE THE POINT 3, 0 HERE
87. 5:47AND WE HAVE THE POINT -1/2, 0 WHICH WOULD BE HERE.
88. 5:52NOTICE THESE TWO POINTS ARE EQUAL DISTANCE
89. 5:54TO THE AXIS OF SYMMETRY
90. 5:57AND THEN WE KNOW THAT THE Y INTERCEPT IS THE POINT 0, 3,
91. 6:01WHICH WOULD BE THIS POINT HERE.
92. 6:03THERE MUST A MIRROR IMAGE
93. 6:04ON THE RIGHT SIDE OF THE AXIS OF SYMMETRY
94. 6:06SO THERE MUST BE A POINT SOMEWHERE IN HERE
95. 6:10AND THIS IS MORE THAN ENOUGH INFORMATION
96. 6:11TO SKETCH A NICE GRAPH OF OUR FUNCTION.
97. 6:14IT WILL LOOK SOMETHING LIKE THIS.
98. 6:22NOW WE WANT TO DETERMINE THE DOMAIN AND RANGE
99. 6:23OF THE FUNCTION.
100. 6:24REMEMBER THE DOMAIN IS A SET OF ALL POSSIBLE X VALUES
101. 6:28AND THE RANGE IS THE SET OF ALL POSSIBLE Y VALUES.
102. 6:33SO AGAIN, DOMAIN MEANS X.
103. 6:35RANGE MEANS Y.
104. 6:37WELL THE DOMAIN OF ANY QUADRATIC FUNCTION
105. 6:39WILL ALWAYS BE ALL REAL NUMBERS,
106. 6:41WHICH EXPRESSED USING INTERVAL NOTATION
107. 6:43WOULD BE FROM NEGATIVE INFINITY TO POSITIVE INFINITY.
108. 6:47LOOKING AT THE GRAPH IT IS MOVING DOWN QUICKLY
109. 6:50BUT IT'S ALSO MOVING LEFT AND RIGHT FOREVER
110. 6:52VERIFYING OUR DOMAIN.
111. 6:54AND THEN FOR THE RANGE OF THE POSSIBLE Y VALUES
112. 6:56NOTICE HOW THE LARGEST Y VALUE
113. 6:58IS THE Y COORDINATE OF THE VERTEX WHICH IS 6.125
114. 7:03AND THEN FROM THERE ALL OF THE Y VALUES ARE LESS THAN THAT.
115. 7:07SO WE CAN EXPRESS THE RANGE USING INTERVAL NOTATION
116. 7:11AS THE INTERVAL FROM NEGATIVE INFINITY TO 6.125 OR 49/8
117. 7:17AND IT DOES INCLUDE THIS VALUE
118. 7:20SO WE HAVE A CLOSED INTERVAL HERE.
119. 7:22WE COULD ALSO EXPRESS THIS USING INEQUALITIES
120. 7:24AS Y IS LESS THAN OR EQUAL TO 49/8.
121. 7:30AND FOR THE LAST QUESTION
122. 7:31WE WANT TO DETERMINE THE INTERVALS
123. 7:32FOR WHICH THE FUNCTION IS INCREASING AND DECREASING.
124. 7:36SO IF YOU COULD IMAGINE YOURSELF AS A SMALL PERSON
125. 7:38WALKING ALONG THIS FUNCTION FROM LEFT TO RIGHT,
126. 7:41IF YOU'RE WALKING UPHILL THE FUNCTION IS INCREASING.
127. 7:44IF YOU'RE WALKING DOWNHILL THE FUNCTION IS DECREASING.
128. 7:49BUT IT IS IMPORTANT THAT WE'D BE WALKING FROM LEFT TO RIGHT.
129. 7:51SO THE FUNCTION IS INCREASING ON THIS INTERVAL
130. 7:54AND DECREASING ON THIS INTERVAL
131. 7:56AND IT CHANGES FROM INCREASING TO DECREASING AT THE VERTEX.
132. 8:01SO THE FUNCTION IS INCREASING ON THE INTERVAL
133. 8:06FROM NEGATIVE INFINITY
134. 8:08TO 1.25 THE X COORDINATE OF THE VERTEX.
135. 8:14REMEMBER 1.25 WAS 5/4 AND IT'S GOING TO BE OPEN ON 5/4
136. 8:20AND IT'S GOING TO BE DECREASING TO THE RIGHT OF 5/4
137. 8:23OR THE OPEN INTERVAL FROM 5/4 TO POSITIVE INFINITY.
138. 8:30OKAY. SO I HOPE YOU FOUND THIS REVIEW HELPFUL.
139. 8:32THANK YOU FOR WATCHING.