Ex 1: Key Characteristics of the Graph of a Quadratic Function (Opens Up) — Transcript

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1. 0:01- IN THIS PROBLEM WE'RE GIVEN THE GRAPH
2. 0:02OF A QUADRATIC FUNCTION AND ASKED TO FIND
3. 0:04ALL THE KEY CHARACTERISTICS.
4. 0:06WE FIRST WANT TO START
5. 0:07BY FINDING THE DOMAIN OF THE FUNCTION.
6. 0:10REMEMBER THE DOMAIN IS A SET OF ALL POSSIBLE X VALUES OR INPUTS
7. 0:14AND SINCE THE X AXIS IS THE HORIZONTAL AXIS HERE,
8. 0:17NOTICE HOW--WELL WE CAN SEE THE GRAPH IS MOVING UP VERY QUICKLY.
9. 0:21IT'S ALSO MOVING LEFT FOREVER AND RIGHT FOREVER
10. 0:24WITHOUT ANY HOLES OR BREAKS.
11. 0:26AND THEREFORE, IF WERE TO PROJECT THIS GRAPH
12. 0:28ON TO THE X AXIS IT WOULD MOVE TO THE LEFT FOREVER
13. 0:31AND MOVE TO THE RIGHT FOREVER
14. 0:33AND THEREFORE THE DOMAIN WOULD BE ALL REAL NUMBERS,
15. 0:36WHICH IS A DOMAIN FOR ALL QUADRATIC FUNCTIONS.
16. 0:39WE CAN EXPRESS THIS SEVERAL WAYS.
17. 0:41USING INTERVAL NOTATION WE'D HAVE THE OPEN INTERVAL
18. 0:44FROM NEGATIVE INFINITY TO POSITIVE INFINITY.
19. 0:50OF COURSE, WE COULD ALSO SAY
20. 0:51THAT X IS GREATER THAN NEGATIVE INFINITY
21. 0:55AND LESS THAN POSITIVE INFINITY.
22. 0:58THESE TWO ARE EQUIVALENT.
23. 1:00THE RANGE IS A SET OF ALL POSSIBLE Y VALUES
24. 1:03AND SINCE THE Y VALUES RUN ALONG THE VERTICAL AXIS
25. 1:07WE CAN THINK OF PROJECTING THIS ONTO THE Y AXIS
26. 1:10OR NOTICE HOW THE LOWEST POINT ON THIS GRAPH IS AT THE VERTEX
27. 1:13WHERE THE Y VALUE IS -4.
28. 1:17AND THEN FROM -4 THE Y VALUES INCREASE
29. 1:20WITHOUT BOUND
30. 1:23THEREFORE, THE RANGE WOULD BE THE INTERVAL
31. 1:26FROM -4 TO INFINITY,
32. 1:29CLOSED ON -4 BECAUSE IT DOES INCLUDE -4.
33. 1:33SO AGAIN USING INTERVAL NOTATION
34. 1:34WE CAN SAY INTERVAL THAT'S CLOSED
35. 1:37ON -4 TO POSITIVE INFINITY.
36. 1:41WHERE USING INEQUALITIES WE COULD JUST SAY
37. 1:43Y IS GREATER THEN OR EQUAL TO -4,
38. 1:46AGAIN THESE TWO OUR EQUIVALENT.
39. 1:49NEXT WE'RE ASKED TO DETERMINE
40. 1:50WHEN THE FUNCTION IS INCREASING AND DECREASING.
41. 1:53INFORMALLY, IF WE WALK ON THE GRAPH FROM LEFT TO RIGHT,
42. 1:57IF WE'RE GOING DOWNHILL THE FUNCTION IS DECREASING
43. 2:00AND IF WE'RE WALKING UPHILL THE FUNCTION WOULD BE INCREASING.
44. 2:04AGAIN, WHEN WALKING FROM LEFT TO RIGHT.
45. 2:06MORE FORMALLY, IF AS X INCREASES Y DECREASES
46. 2:10THE FUNCTION IS DECREASING AND IF AS X INCREASES
47. 2:14Y INCREASES THE FUNCTION IS INCREASING.
48. 2:18SO LOOKING AT THE GRAPH OF OUR FUNCTION
49. 2:20NOTICE HOW ON THE LEFT OF THE VERTEX TO THE VERTEX
50. 2:25THE GRAPH WOULD BE DECREASING AND TO THE RIGHT OF THE VERTEX
51. 2:33THE FUNCTION IS GOING UPHILL OR INCREASING.
52. 2:37WHEN GIVING THE INTERVAL
53. 2:38FOR WHICH THE FUNCTION IS INCREASING OR DECREASING
54. 2:40WE WANT TO USE X VALUES.
55. 2:42SO NOTICE HOW THE X VALUE OF THE VERTEX IS -3.
56. 2:47SO THE FUNCTION WOULD BE DECREASING ON THE INTERVAL
57. 2:50FROM NEGATIVE INFINITY TO -3.
58. 2:53AND IT'S INCREASING ON THE INTERVAL FROM -3 TO INFINITY.
59. 2:57AND WE ARE NOT GOING TO INCLUDE -3 IN EITHER OF THESE INTERVALS.
60. 3:02SO THE FUNCTION IS INCREASING TO THE RIGHT OF -3.
61. 3:06SO FROM -3 TO INFINITY OR IF WE WANT WHEN X IS GREATER THAN -3
62. 3:16AND IT'S DECREASING ON THE INTERVAL
63. 3:18FROM NEGATIVE INFINITY TO -3.
64. 3:22AGAIN, BOTH INTERVALS ARE OPEN ON -3
65. 3:24BECAUSE -3 IS NOT INCLUDED.
66. 3:27SO WE COULD SAY THIS WOULD BE 1X IS LESS THAN -3.
67. 3:31BECAUSE THE PARABOLA OPENS UP
68. 3:33THE VERTEX WOULD BE THE LOW POINT ON OUR GRAPH,
69. 3:37WHICH IS HERE WITH AN X COORDINATE OF -3
70. 3:40AND A Y COORDINATE OF -4.
71. 3:43SO THIS WOULD BE THE VERTEX.
72. 3:49REMEMBER THAT THE VERTEX IS ALSO THE ONLY POINT
73. 3:52THAT'S ALSO ON THE AXIS OF SYMMETRY
74. 3:54WHICH WOULD BE THIS VERTICAL LINE HERE X = -3.
75. 4:02NEXT, WE'RE ASKED TO FIND
76. 4:03THE MAXIMUM OR MINIMAL FUNCTION VALUE
77. 4:06AND BECAUSE THIS PARABOLA OPENS UP WE ARE GOING TO HAVE
78. 4:09A MINIMUM VALUE WHICH WOULD OCCUR AT THIS LOW POINT
79. 4:12AND THE MINIMUM VALUE IS ACTUALLY
80. 4:14THE Y COORDINATE OF THE VERTEX.
81. 4:17SO THE MINIMUM VALUE IS Y = -4.
82. 4:24FINALLY, WE'RE ASKED TO FIND THE INTERCEPTS.
83. 4:27THE X INTERCEPTS ARE THE POINTS
84. 4:29WHERE THE GRAPH CROSSES THE X AXIS.
85. 4:31NOTICE HOW THIS PARABOLA HAS TWO X INTERCEPTS,
86. 4:34ONE HERE AT THE POINT -50, AND ONE HERE AT THE POINT -10.
87. 4:44AND THE Y INTERCEPT IS WHERE THE GRAPH CROSSES
88. 4:46THE Y AXIS WHICH OCCURS HERE AT THE POINT 0, 5.
89. 4:58I HOPE YOU FOUND THIS EXPLANATION HELPFUL.