1
00:00:00,900 --> 00:00:02,730
Robert McKeown: The slides have some tips on how
2
00:00:02,730 --> 00:00:05,790
to transform graphs. Essentially, you want to
3
00:00:05,790 --> 00:00:08,100
think about having a function in its general
4
00:00:08,100 --> 00:00:10,770
form like that. And then there's two
5
00:00:10,800 --> 00:00:14,910
manipulations that might be made. One is to
6
00:00:15,060 --> 00:00:18,900
double the function itself. In English, what
7
00:00:18,900 --> 00:00:21,840
does that mean? Well, if you have the same
8
00:00:21,840 --> 00:00:27,330
amount of f of x, right, if x is the same value,
9
00:00:27,480 --> 00:00:31,860
you're going to get twice as much y. Now, what
10
00:00:31,860 --> 00:00:34,560
happens if we put that now notice that the two
11
00:00:34,800 --> 00:00:37,440
here is outside the function?
12
00:00:44,940 --> 00:00:48,720
Looking at the next example, we're inside, it's
13
00:00:48,720 --> 00:00:52,650
inside the function. Well, what does that mean?
14
00:00:52,650 --> 00:00:57,960
Well, that means that you can have the same f of
15
00:00:57,960 --> 00:01:05,850
x the same y with half as much x half as much x.
16
00:01:07,230 --> 00:01:09,090
So we're trying to do that to give you a little
17
00:01:09,090 --> 00:01:16,620
bit of intuition. So here we are on ALEKS, we've
18
00:01:16,620 --> 00:01:21,030
got a question. And it says that we've got one
19
00:01:21,030 --> 00:01:25,410
half of X the function of one half x, this means
20
00:01:25,410 --> 00:01:29,280
that we're going to need twice as much X to get
21
00:01:29,280 --> 00:01:34,860
the same amount of y. So if I want to get y
22
00:01:34,860 --> 00:01:38,970
equal to negative four, and before, that would
23
00:01:39,000 --> 00:01:42,480
require two X, now it's going to take four x, so
24
00:01:42,480 --> 00:01:45,420
I'm going to draw a little blue cross make a
25
00:01:45,420 --> 00:01:50,070
little point there. Now, if we can see by the
26
00:01:50,070 --> 00:01:54,150
original function here, that if x is equal to
27
00:01:54,150 --> 00:02:00,750
zero, y will be equal to zero. So multiplying x,
28
00:02:01,020 --> 00:02:03,030
or sorry, zero by one half is not going to
29
00:02:03,030 --> 00:02:07,620
change that. So the origin stays the same. And
30
00:02:07,620 --> 00:02:09,900
similarly, on the other side of the diagram, if
31
00:02:09,900 --> 00:02:14,940
negative two x got us negative four y, now it's
32
00:02:14,940 --> 00:02:19,110
going to take negative four X to get negative
33
00:02:19,110 --> 00:02:23,670
four y is going to take twice as much X to get
34
00:02:23,670 --> 00:02:27,840
us there. I'm going to just draw the lines, now
35
00:02:27,840 --> 00:02:32,940
I got to draw two lines. And so we stretch the
36
00:02:32,940 --> 00:02:35,730
function out to the left and the right, because
37
00:02:35,730 --> 00:02:39,660
we need more X to get the same y. So let's see
38
00:02:39,660 --> 00:02:46,920
if that was correct. And it was correct. So we
39
00:02:46,920 --> 00:02:49,980
needed twice as much X to get the same amount of
40
00:02:49,980 --> 00:03:00,720
y. So a parabola looks something like this.
41
00:03:05,729 --> 00:03:13,859
Or maybe like this. And it just keeps going off
42
00:03:13,859 --> 00:03:17,939
in that direction. This form here would be
43
00:03:17,939 --> 00:03:22,949
something like y is equal to negative two x
44
00:03:22,949 --> 00:03:33,179
squared plus four x plus four, something like
45
00:03:33,179 --> 00:03:38,639
that. And when this is negative, we're going to
46
00:03:38,639 --> 00:03:48,509
get out max up here. The one on the right, could
47
00:03:48,509 --> 00:03:55,229
be something like two x squared minus four x.
48
00:03:56,279 --> 00:04:01,859
And it looks like maybe negative four. Something
49
00:04:01,859 --> 00:04:05,969
like that. And where if this term is positive,
50
00:04:07,709 --> 00:04:11,729
this is gonna be a min and so this one over here
51
00:04:11,759 --> 00:04:19,079
is decreasing. As the absolute value of x
52
00:04:19,079 --> 00:04:25,139
increases, and over here, y is increasing as the
53
00:04:25,139 --> 00:04:33,029
absolute value of x increases. Here, the max it,
54
00:04:33,179 --> 00:04:37,439
this is the vertex. The vertex is a max here.
55
00:04:38,159 --> 00:04:41,759
The men is also the vertex, the vertex is the
56
00:04:41,759 --> 00:04:44,669
min. So the question is asking us to find the
57
00:04:44,669 --> 00:04:48,569
vertex. There are a few ways to find the vertex.
58
00:04:49,529 --> 00:04:53,339
I'm just going to show you how to do it by trial
59
00:04:53,339 --> 00:04:57,059
and error. Essentially. There are other ways but
60
00:04:57,059 --> 00:04:59,099
we don't use them. They don't come up very much
61
00:04:59,099 --> 00:05:02,189
in economics, so I won't spend too much time on
62
00:05:02,189 --> 00:05:05,759
that unless you really want me to. You're more
63
00:05:05,759 --> 00:05:09,149
than welcome to ask me questions. I guess via
64
00:05:09,149 --> 00:05:11,849
email if we're doing distance or after class if
65
00:05:11,849 --> 00:05:14,969
we're in person. So for trial and error, I
66
00:05:14,969 --> 00:05:19,199
usually just, I will just guess that x is equal
67
00:05:19,199 --> 00:05:25,169
to zero. If x is equal to zero, y in this case
68
00:05:25,169 --> 00:05:28,949
is going to be equal to negative two. parabolas
69
00:05:28,949 --> 00:05:30,869
have certain properties that are going to help
70
00:05:30,869 --> 00:05:35,699
me with my gas, I know that this parabola, the
71
00:05:35,699 --> 00:05:39,869
square term is positive. So the vertex is going
72
00:05:39,869 --> 00:05:47,909
to occur at the minimum, so I can plug and play
73
00:05:48,239 --> 00:05:51,449
with some numbers and see if I can find that
74
00:05:51,449 --> 00:05:55,229
minimum. I know that a parabola has a certain
75
00:05:55,229 --> 00:05:58,979
amount of symmetry, if I draw a straight line
76
00:05:59,159 --> 00:06:02,819
coming up from the vertex, then the distance
77
00:06:02,819 --> 00:06:06,689
from that straight line to each of its wings is
78
00:06:06,689 --> 00:06:11,219
going to be the same. Looking back at my guess,
79
00:06:11,249 --> 00:06:13,889
when x was equal to zero, y was equal to
80
00:06:13,889 --> 00:06:16,949
negative two. Well, why don't I guess? Okay,
81
00:06:16,949 --> 00:06:20,099
well, what happens if x is equal to one, well,
82
00:06:20,129 --> 00:06:25,079
then I get three times one squared, plus six
83
00:06:25,109 --> 00:06:30,659
minus two, and that's going to give me seven. So
84
00:06:30,659 --> 00:06:34,799
it's quite a bit higher than negative two. Now,
85
00:06:34,799 --> 00:06:40,619
if I can show that x is at negative one is
86
00:06:40,619 --> 00:06:44,129
greater than y is equal to negative two, then
87
00:06:44,129 --> 00:06:48,689
I'll know that at x zero, y equal negative two,
88
00:06:48,959 --> 00:06:52,199
we've got the vertex. So I'm going to guess x is
89
00:06:52,199 --> 00:06:55,349
equal to negative one, I get three times
90
00:06:55,379 --> 00:06:59,489
negative one squared, plus six times negative
91
00:06:59,489 --> 00:07:07,829
one, minus two, and I end up with minus five.
92
00:07:12,449 --> 00:07:23,129
Now I know that x zero, y negative two is not,
93
00:07:24,659 --> 00:07:26,399
well, maybe I should make it a little more
94
00:07:26,399 --> 00:07:29,399
formal here. I know that the pair's zero,
95
00:07:29,579 --> 00:07:34,829
negative two is not the vertex.
96
00:07:40,350 --> 00:07:47,970
But all hope is not lost. Because the coordinate
97
00:07:47,970 --> 00:07:52,140
negative one, negative five, might be the
98
00:07:52,140 --> 00:07:52,950
vertex.
99
00:07:58,680 --> 00:08:04,320
So I'm gonna guess. x is equal to negative two.
100
00:08:06,180 --> 00:08:13,680
And I've got y is going to be equal to negative
101
00:08:13,680 --> 00:08:17,550
two squared times three, plus six times negative
102
00:08:17,550 --> 00:08:26,640
two, minus two. And I've got 12 minus 12 minus
103
00:08:26,640 --> 00:08:33,000
two, which equals to negative two. So therefore,
104
00:08:33,540 --> 00:08:44,010
negative one, negative five is the vertex. I did
105
00:08:44,010 --> 00:08:46,860
that by guessing. And I know that Cindy has a
106
00:08:46,860 --> 00:08:51,180
problem with the square term, this term right
107
00:08:51,180 --> 00:08:55,380
here being positive. I know that once I find the
108
00:08:55,380 --> 00:08:58,920
minimum point, that's going to be the vertex.
109
00:08:59,220 --> 00:09:03,450
And because of symmetry, any other value of x is
110
00:09:03,450 --> 00:09:08,700
going to give a higher value of y. We can now go
111
00:09:08,700 --> 00:09:12,990
over to ALEKS. And I'll show you how to graph
112
00:09:12,990 --> 00:09:17,490
this parabola and ALEKS, anywhere feature and
113
00:09:17,490 --> 00:09:20,460
ALEKS. And that's that this little diagram here
114
00:09:20,460 --> 00:09:22,770
with the X on it, so I'm going to click on that,
115
00:09:23,070 --> 00:09:26,460
and it says, plot a point anywhere. And I'm
116
00:09:26,460 --> 00:09:30,600
going to go down to here. And I'm just going to
117
00:09:32,880 --> 00:09:38,160
put in all our coordinates
118
00:09:43,410 --> 00:09:50,430
that we calculated before. And notice that every
119
00:09:50,430 --> 00:09:54,630
time I I do that and I click the plot a point
120
00:09:57,540 --> 00:10:01,500
button, a little Little x appears.
121
00:10:07,109 --> 00:10:10,529
Okay, so I've plotted the points of the
122
00:10:10,529 --> 00:10:15,869
parabola. And then I'm good. All I had to do is
123
00:10:15,869 --> 00:10:21,299
press this button right here. And ALEKS has sort
124
00:10:21,299 --> 00:10:23,729
of knows what to do with the plot points. Now
125
00:10:23,729 --> 00:10:31,679
let's see if it's correct. And it was correct.
126
00:10:31,679 --> 00:10:32,939
So we got the right answer.