WEBVTT
1
00:00:00.810 --> 00:00:04.440 A:middle L:90%
We want to describe how the craft of F of
2
00:00:04.440 --> 00:00:08.599 A:middle L:90%
X is equal to X squared plus C. That's
3
00:00:09.480 --> 00:00:13.679 A:middle L:90%
Berries as C Berries. And we're going to then
4
00:00:14.220 --> 00:00:17.980 A:middle L:90%
graft several members of the family to illustrate trends that
5
00:00:17.980 --> 00:00:21.850 A:middle L:90%
we discovered. And in particular, we want to
6
00:00:21.850 --> 00:00:27.079 A:middle L:90%
investigate how our maximum minimum and points of inflection move
7
00:00:27.300 --> 00:00:31.050 A:middle L:90%
as sea changes. We also want to identify any
8
00:00:31.050 --> 00:00:34.899 A:middle L:90%
transitional values of C at which the basics shape of
9
00:00:34.899 --> 00:00:41.750 A:middle L:90%
the curve will change. So here you can see
10
00:00:41.750 --> 00:00:46.049 A:middle L:90%
I have the functions listed for seizing with a negative
11
00:00:46.049 --> 00:00:50.460 A:middle L:90%
one. Negative too negative. 1012 But before we
12
00:00:50.460 --> 00:00:54.570 A:middle L:90%
look at these graphs more, let's just do a
13
00:00:54.570 --> 00:01:00.310 A:middle L:90%
little bit of stuff with the general function. So
14
00:01:00.310 --> 00:01:02.479 A:middle L:90%
the first thing you might know this is we could
15
00:01:02.479 --> 00:01:07.500 A:middle L:90%
go ahead and factor this because we can pull out
16
00:01:07.500 --> 00:01:12.140 A:middle L:90%
a X so we get X is equal to,
17
00:01:12.150 --> 00:01:15.549 A:middle L:90%
or this is equal to X Times X plus C
18
00:01:17.739 --> 00:01:19.370 A:middle L:90%
. And if we were to go ahead and set
19
00:01:19.379 --> 00:01:23.569 A:middle L:90%
this equal zero, that's going to imply that either
20
00:01:23.719 --> 00:01:26.159 A:middle L:90%
X is equal zero or X is equal to negative
21
00:01:26.480 --> 00:01:32.549 A:middle L:90%
. Seen this using the zero product property, so
22
00:01:32.939 --> 00:01:34.379 A:middle L:90%
we know that no matter what, we will always
23
00:01:34.379 --> 00:01:38.030 A:middle L:90%
have an X intercept, that zero. And for
24
00:01:38.030 --> 00:01:41.900 A:middle L:90%
whatever value we choose for C, the negative of
25
00:01:41.909 --> 00:01:44.750 A:middle L:90%
that will be our other ex Anderson. All right
26
00:01:44.750 --> 00:01:47.879 A:middle L:90%
, so that gives us a little bit of information
27
00:01:48.540 --> 00:01:51.959 A:middle L:90%
. Now let's go ahead and look at the derivative
28
00:01:52.230 --> 00:01:55.370 A:middle L:90%
of this so f prime of X. Because remember
29
00:01:55.370 --> 00:01:57.620 A:middle L:90%
to find maximum or minimum points, we can look
30
00:01:57.629 --> 00:02:01.549 A:middle L:90%
at critical values, which is where our directive is
31
00:02:01.549 --> 00:02:06.629 A:middle L:90%
equal to Israel or undefined. So taking the derivative
32
00:02:06.629 --> 00:02:07.860 A:middle L:90%
of X squared plus C x we would use power
33
00:02:07.860 --> 00:02:14.419 A:middle L:90%
rule so X squared becomes to X and see X
34
00:02:14.430 --> 00:02:15.330 A:middle L:90%
will see is a constant. So the derivative of
35
00:02:15.330 --> 00:02:17.460 A:middle L:90%
X is just one. So we just end up
36
00:02:17.460 --> 00:02:22.080 A:middle L:90%
with two explosive now to find our critical values.
37
00:02:22.930 --> 00:02:24.090 A:middle L:90%
Remember, the thing we want to do said this
38
00:02:24.099 --> 00:02:30.199 A:middle L:90%
equals zero, and that's going to imply that if
39
00:02:30.199 --> 00:02:31.669 A:middle L:90%
we have any, Max's airmen's is going to occur
40
00:02:31.759 --> 00:02:38.990 A:middle L:90%
at negative. See over. And you might recall
41
00:02:39.000 --> 00:02:43.770 A:middle L:90%
that this year is just the Vertex because we're working
42
00:02:43.770 --> 00:02:45.789 A:middle L:90%
with a quant traffic, so we could have did
43
00:02:45.789 --> 00:02:49.699 A:middle L:90%
the whole negative be over two. A plug goes
44
00:02:49.699 --> 00:02:53.259 A:middle L:90%
in to get the same point. So we know
45
00:02:53.539 --> 00:02:55.379 A:middle L:90%
how are in this case is going to be a
46
00:02:55.379 --> 00:02:59.789 A:middle L:90%
minimum, since we know our leading coefficient is larger
47
00:02:59.789 --> 00:03:05.520 A:middle L:90%
than zero positive, we know how are minimum is
48
00:03:05.520 --> 00:03:09.189 A:middle L:90%
going to change because it's just going to shift to
49
00:03:09.189 --> 00:03:14.159 A:middle L:90%
the left or to the right based off this value
50
00:03:14.159 --> 00:03:17.050 A:middle L:90%
are released, the position or long be why access
51
00:03:17.840 --> 00:03:21.449 A:middle L:90%
the X axis is going to shit. I should
52
00:03:21.449 --> 00:03:24.849 A:middle L:90%
say that. And now for our last part,
53
00:03:25.090 --> 00:03:28.430 A:middle L:90%
let's go ahead and look a second derivative and see
54
00:03:28.430 --> 00:03:30.939 A:middle L:90%
if we can find any points of inflection. So
55
00:03:30.949 --> 00:03:34.000 A:middle L:90%
F Double Prime of X is going to be just
56
00:03:34.169 --> 00:03:37.370 A:middle L:90%
to cause a derivative of two x will driven of
57
00:03:37.370 --> 00:03:39.090 A:middle L:90%
X is one. So we're just up with two
58
00:03:39.210 --> 00:03:42.139 A:middle L:90%
. And the derivative of See Sense is a constant
59
00:03:42.139 --> 00:03:45.310 A:middle L:90%
would be zero. And we know that this here
60
00:03:45.310 --> 00:03:49.039 A:middle L:90%
is strictly greater than zero, which is going to
61
00:03:49.039 --> 00:03:53.650 A:middle L:90%
imply that our function is always con cave up.
62
00:03:53.389 --> 00:03:59.050 A:middle L:90%
So there should be no points of inflection. And
63
00:03:59.050 --> 00:04:01.740 A:middle L:90%
if we just go through and look at these values
64
00:04:01.740 --> 00:04:04.110 A:middle L:90%
, we could kind of see what we expect to
65
00:04:04.110 --> 00:04:09.860 A:middle L:90%
occur from this little analysis does happen so starting at
66
00:04:10.500 --> 00:04:12.740 A:middle L:90%
C is equal to zero, which is this middle
67
00:04:12.740 --> 00:04:16.509 A:middle L:90%
graph here. We could see when c 0 to
68
00:04:16.509 --> 00:04:23.870 A:middle L:90%
0. We get that our ex intercepts are only
69
00:04:23.870 --> 00:04:29.579 A:middle L:90%
at zero. And if we were to go ahead
70
00:04:29.589 --> 00:04:33.959 A:middle L:90%
and plug in zero into negative c over to we
71
00:04:33.959 --> 00:04:36.250 A:middle L:90%
get X zero. Sorber, Texas. Also there
72
00:04:39.540 --> 00:04:43.329 A:middle L:90%
. Now, when C is negative, one Arbor
73
00:04:43.329 --> 00:04:47.250 A:middle L:90%
, Tex. Shifts down and celebrate as well as
74
00:04:47.259 --> 00:04:54.740 A:middle L:90%
we get extra X intercept hat, which is exactly
75
00:04:54.740 --> 00:04:57.620 A:middle L:90%
what we would expect again when c zero to negative
76
00:04:57.629 --> 00:05:01.699 A:middle L:90%
, too. It shifts halfway in between those two
77
00:05:01.699 --> 00:05:04.920 A:middle L:90%
values for seeing for Vertex. So it's now at
78
00:05:04.930 --> 00:05:11.050 A:middle L:90%
one and we get our X intercept over here two
79
00:05:12.180 --> 00:05:15.899 A:middle L:90%
and looking at season one, Exeter said. Get
80
00:05:15.899 --> 00:05:20.850 A:middle L:90%
shifted to the left one. And our Vertex is
81
00:05:20.899 --> 00:05:26.379 A:middle L:90%
right in the middle. Those two and sees it
82
00:05:26.389 --> 00:05:30.740 A:middle L:90%
too. Same thing are other ex intercepted ship the
83
00:05:30.740 --> 00:05:35.550 A:middle L:90%
negative, too. And our burr Tex is half
84
00:05:36.110 --> 00:05:38.170 A:middle L:90%
that distance