Adding to 10

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:02.947
- [Voiceover] We've
got three bananas here.

00:00:02.947 --> 00:00:04.373
One, two, three.

00:00:04.677 --> 00:00:06.430
So I'm starting with three bananas

00:00:06.430 --> 00:00:10.057
and I want to add some number of bananas

00:00:10.057 --> 00:00:12.687
so that I end up with 10 bananas.

00:00:12.687 --> 00:00:13.848
So this is what I'm starting with.

00:00:13.848 --> 00:00:15.090
I'm starting with three

00:00:15.090 --> 00:00:17.157
and I want to end up with 10.

00:00:17.157 --> 00:00:19.153
One, two, three, four, five, six, seven,

00:00:19.153 --> 00:00:20.649
eight, nine, ten.

00:00:20.953 --> 00:00:24.134
So what do I have to add to the three

00:00:24.134 --> 00:00:25.979
to get to 10?

00:00:25.979 --> 00:00:27.779
I encourage you to pause this video

00:00:27.779 --> 00:00:30.133
and try this out on your own.

00:00:31.180 --> 00:00:32.617
So let's work through this.

00:00:32.956 --> 00:00:35.615
So let's just start adding bananas.

00:00:35.615 --> 00:00:36.666
So that's three.

00:00:36.666 --> 00:00:41.279
So now let me add a fourth.

00:00:41.617 --> 00:00:43.298
That would be my fourth banana.

00:00:43.997 --> 00:00:45.701
Five bananas.

00:00:46.400 --> 00:00:48.244
Six bananas.

00:00:48.803 --> 00:00:50.461
Seven bananas.

00:00:50.997 --> 00:00:52.400
Eight bananas.

00:00:53.365 --> 00:00:54.977
Nine bananas.

00:00:55.766 --> 00:00:57.659
And ten bananas.

00:00:57.997 --> 00:00:59.715
So how many bananas did I add?

00:00:59.715 --> 00:01:02.258
So now I added some
number of bananas here.

00:01:02.258 --> 00:01:04.835
So in total I have one,
two, three, four, five,

00:01:04.835 --> 00:01:07.282
six, seven, eight, nine, ten.

00:01:07.876 --> 00:01:10.377
So how many did I have to add to the three

00:01:10.377 --> 00:01:14.575
in order to get to the total of 10

00:01:14.575 --> 00:01:16.500
in order to get this 10 right over here?

00:01:16.804 --> 00:01:18.218
We could just count these.

00:01:18.278 --> 00:01:22.155
This is one, this is two, this is three,

00:01:22.155 --> 00:01:26.983
four, five, six, seven.

00:01:27.182 --> 00:01:29.694
I added seven bananas.

00:01:30.032 --> 00:01:34.906
I added seven bananas to
the original three bananas

00:01:34.908 --> 00:01:36.276
to get to 10.

00:01:36.381 --> 00:01:38.791
So three plus seven is equal to 10.

00:01:38.791 --> 00:01:41.914
And you can see that what
we have on this left side

00:01:41.914 --> 00:01:44.561
is indeed equal to what
we have on the right side.

00:01:44.561 --> 00:01:47.161
On the left side we have
one, two, three, four,

00:01:47.161 --> 00:01:49.750
five, six, seven, eight, nine, ten bananas

00:01:49.750 --> 00:01:52.780
and on the right hand side
we have one, two, three,

00:01:52.780 --> 00:01:55.889
four, five, six, seven, eight, nine, ten.

00:01:55.938 --> 00:01:58.954
So they are indeed equal.


Understanding place value when adding ones

WEBVTT
Kind: captions
Language: en

00:00:00.966 --> 00:00:03.166
- So, this right over here is one way

00:00:03.166 --> 00:00:08.166
to represent the number 37.

00:00:08.200 --> 00:00:13.200
The number 37 has a three right
over here in the tens place.

00:00:13.533 --> 00:00:18.000
So, this three literally
represents three tens.

00:00:18.000 --> 00:00:21.366
Notice, this is a ten
here, this is a ten here,

00:00:21.366 --> 00:00:25.300
and this is a ten here, three tens or 30.

00:00:25.300 --> 00:00:28.133
You see each of these tens,
there's one, two, three, four,

00:00:28.133 --> 00:00:31.667
five, six, seven, eight, nine, ten.

00:00:31.667 --> 00:00:33.666
Now, the seven is in the ones place.

00:00:33.666 --> 00:00:36.700
It literally represents seven ones.

00:00:36.700 --> 00:00:39.800
You see that right over here,
one, two, three, four, five,

00:00:39.800 --> 00:00:44.800
six, seven, so you can say that 37

00:00:47.900 --> 00:00:52.666
is equal to three tens,

00:00:52.666 --> 00:00:57.633
do this in the purple color,
is equal to three tens

00:00:57.633 --> 00:01:02.633
plus seven ones.

00:01:06.567 --> 00:01:10.000
Now, what if I wanted
to add something to 37?

00:01:10.000 --> 00:01:15.000
What would 37

00:01:18.266 --> 00:01:23.266
plus two be equal to?

00:01:23.466 --> 00:01:26.367
How many tens and how many ones?

00:01:26.367 --> 00:01:26.933
Well, let's think about it.

00:01:26.933 --> 00:01:31.833
We're now going to add two,
so we're adding two ones,

00:01:31.833 --> 00:01:36.000
so we're going to add one and two.

00:01:36.000 --> 00:01:40.035
So, now we still have three
tens, so let's write that down.

00:01:40.035 --> 00:01:44.333
So we still have three
tens, but now we have

00:01:44.333 --> 00:01:47.866
a different number of ones,
we now have one, two, three,

00:01:47.866 --> 00:01:51.900
four, five, six, seven, eight,
nine, we now have nine ones.

00:01:51.900 --> 00:01:52.800
Another way to think about it is

00:01:52.800 --> 00:01:57.800
seven ones plus two ones is nine ones.

00:01:57.933 --> 00:01:59.567
So let me write that down.

00:01:59.567 --> 00:02:04.567
So, plus nine ones, or another
way to think about it is

00:02:05.033 --> 00:02:09.633
37, let me just write it the same way,

00:02:09.633 --> 00:02:14.633
37 plus 2 is equal to three tens,

00:02:17.200 --> 00:02:19.933
so I can write this as 30

00:02:19.933 --> 00:02:24.933
and then nine ones, is equal to 39,

00:02:25.000 --> 00:02:26.300
and that's exactly what we have here.

00:02:26.300 --> 00:02:28.566
We have 30, this was the three tens,

00:02:28.566 --> 00:02:33.566
and then we have nine
ones, which is 39 squares.


Subtracting ten or one hundred

WEBVTT
Kind: captions
Language: en

00:00:00.232 --> 00:00:04.132
- [Voiceover] I have here the number 324,

00:00:04.132 --> 00:00:05.298
which we can view as,

00:00:05.298 --> 00:00:06.765
since I have a three
in the hundreds place,

00:00:06.765 --> 00:00:09.360
we could say this is three hundreds,

00:00:10.299 --> 00:00:12.097
I have a two in the tens place,

00:00:12.098 --> 00:00:15.781
so we could say, plus two tens,

00:00:15.781 --> 00:00:17.569
and then I have a four in the ones place,

00:00:17.569 --> 00:00:21.326
so I can say, plus four ones,

00:00:21.665 --> 00:00:23.564
and if you want to think
about what that looks like,

00:00:23.564 --> 00:00:25.533
you could look at all these blocks here.

00:00:25.533 --> 00:00:30.415
Three groups of 100, three
groups of 100, one, two, three,

00:00:30.415 --> 00:00:32.847
two groups of 10, two groups of 10,

00:00:32.847 --> 00:00:35.432
I have two groups of 10
right over here in yellow,

00:00:35.432 --> 00:00:37.613
four ones, I have four ones here,

00:00:37.613 --> 00:00:41.069
so this is 324 blocks.

00:00:41.069 --> 00:00:45.068
So now let's get some
practice subtracting from 324,

00:00:45.068 --> 00:00:49.799
and the first thing I want
to subtract from 324 is,

00:00:49.799 --> 00:00:54.064
let's subtract a 10.

00:00:54.064 --> 00:00:57.083
So, I want to subtract a 10 here.

00:00:57.083 --> 00:01:01.419
So a 10 is one 10 and zero ones.

00:01:01.419 --> 00:01:03.797
Let me do that in the blue color.

00:01:03.798 --> 00:01:05.648
Zero ones.

00:01:05.648 --> 00:01:08.548
So, I'm not subtracting any hundreds,

00:01:08.549 --> 00:01:12.269
I AM subtracting one 10,
you see it right over there,

00:01:12.269 --> 00:01:17.133
so minus one 10, and I'm
not subtracting any ones,

00:01:17.133 --> 00:01:19.332
so what is that going to get me?

00:01:19.332 --> 00:01:22.181
What is that going to get me?

00:01:22.181 --> 00:01:24.134
Well, my hundreds don't change.

00:01:24.134 --> 00:01:25.466
I'm not subtracting any hundreds,

00:01:25.466 --> 00:01:27.097
so I still have three hundreds,

00:01:27.097 --> 00:01:29.131
so I could write that over here,

00:01:29.132 --> 00:01:31.202
I still have three hundreds,

00:01:31.202 --> 00:01:34.798
I had two tens, I'm
taking one of them away,

00:01:34.798 --> 00:01:37.687
so now I have two minus one is,

00:01:37.687 --> 00:01:41.652
two minus one is one 10.

00:01:41.868 --> 00:01:45.082
I had two tens, now I take away one 10,

00:01:45.082 --> 00:01:48.314
I should just call that
a ten instead of a tens,

00:01:48.314 --> 00:01:52.178
I take away one 10, and
now I'm going to have

00:01:53.087 --> 00:01:56.322
plus one 10, if I had
two, I take away one,

00:01:56.322 --> 00:01:59.605
I'm now going to have
one, and I had four ones,

00:01:59.605 --> 00:02:01.154
and I'm not taking any ones away,

00:02:01.154 --> 00:02:04.737
so I still have four ones,

00:02:04.737 --> 00:02:08.702
four ones minus zero ones, four ones.

00:02:08.702 --> 00:02:11.068
So I'm left with 314,

00:02:11.068 --> 00:02:14.655
three hundreds, plus
one 10, plus four ones,

00:02:14.655 --> 00:02:16.237
and if you look at it
over here, what did we do?

00:02:16.237 --> 00:02:20.918
We had 324, and then we
take away one of the tens,

00:02:20.919 --> 00:02:24.024
so we took away one of the
tens, and we are left with

00:02:24.024 --> 00:02:27.142
three hundred, one 10, and four ones,

00:02:27.142 --> 00:02:30.280
or 314.

00:02:30.280 --> 00:02:30.988
That was interesting.

00:02:30.988 --> 00:02:35.128
Now let's subtract 100 from 324

00:02:35.128 --> 00:02:36.912
instead of a 10,

00:02:37.375 --> 00:02:40.719
so, let's now subtract,

00:02:40.719 --> 00:02:44.661
let's now subtract 100,

00:02:44.661 --> 00:02:46.300
so that's one hundred,

00:02:46.300 --> 00:02:50.126
so I'm now gonna subtract one hundred,

00:02:51.188 --> 00:02:54.147
and zero tens,

00:02:54.147 --> 00:02:57.367
and zero ones,

00:02:57.367 --> 00:02:59.712
so let's subtract that now.

00:02:59.866 --> 00:03:01.899
What's that going to be?

00:03:01.899 --> 00:03:03.167
Well, I had three hundreds,

00:03:03.167 --> 00:03:04.006
I take one of them away,

00:03:04.006 --> 00:03:06.050
now I'm going to have two hundreds.

00:03:06.050 --> 00:03:07.983
I had two tens, I take away no tens,

00:03:07.983 --> 00:03:10.148
I still have two tens.

00:03:10.148 --> 00:03:13.700
I had four ones, I take away zero ones,

00:03:13.700 --> 00:03:15.948
I'm still going to have four ones.

00:03:15.949 --> 00:03:17.016
If you look at it over here,

00:03:17.016 --> 00:03:18.700
what's that going to be?

00:03:18.700 --> 00:03:20.916
Let me write an equal sign.

00:03:20.916 --> 00:03:23.799
I had three hundreds, I take away 100,

00:03:23.799 --> 00:03:26.107
I'm going to have two hundreds.

00:03:27.200 --> 00:03:30.034
I had two tens, I'm not
taking away any tens,

00:03:30.034 --> 00:03:32.671
so I still have two tens,

00:03:33.133 --> 00:03:35.501
and I had four ones, and I'm
not taking away any ones,

00:03:35.501 --> 00:03:37.835
so I still have four ones,

00:03:37.835 --> 00:03:39.700
and if you think about
it in this picture here,

00:03:39.700 --> 00:03:41.215
what did I just do?

00:03:41.216 --> 00:03:43.326
Well, I took away one group of 100.

00:03:43.326 --> 00:03:45.367
I took away one group of 100,

00:03:45.367 --> 00:03:48.715
so I could take away that
group of 100 right over there,

00:03:48.716 --> 00:03:50.033
and what am I left with?

00:03:50.033 --> 00:03:54.466
I'm left with two hundreds, two tens,

00:03:54.467 --> 00:03:57.167
that's the two tens there, and four ones.

00:03:57.167 --> 00:04:00.482
Two hundreds,

00:04:00.482 --> 00:04:02.135
let me write it this way,

00:04:02.135 --> 00:04:06.003
two hundreds, two tens, four ones,

00:04:06.003 --> 00:04:09.880
or 224.


Fence posts for horses

WEBVTT
Kind: captions
Language: en

00:00:00.533 --> 00:00:03.651
- [Voiceover] A farmer used
fence posts to build a fence

00:00:03.651 --> 00:00:06.666
for Hazel Horse and Pauly Pony.

00:00:06.666 --> 00:00:10.300
The farmer used 26 fence posts

00:00:10.300 --> 00:00:13.201
to build Hazel Horse's fence

00:00:13.201 --> 00:00:16.667
and 19 fence posts

00:00:16.667 --> 00:00:19.733
to build Pauly Pony's fence.

00:00:19.733 --> 00:00:24.133
How many fence posts did
the farmer use in all?

00:00:24.133 --> 00:00:27.104
If you like, I encourage
you to pause the video

00:00:27.104 --> 00:00:29.032
and try to figure it out.

00:00:29.633 --> 00:00:32.266
I'm assuming you have tried it,

00:00:32.267 --> 00:00:33.702
so let's think it through together.

00:00:33.702 --> 00:00:35.466
Whenever you think about
a word problem like this,

00:00:35.466 --> 00:00:36.633
what are they asking?

00:00:36.633 --> 00:00:38.333
What's the story in our head?

00:00:38.333 --> 00:00:41.067
A farmer is building fence
posts for two different horses,

00:00:41.067 --> 00:00:42.458
Hazel Horse and Pauly Pony.

00:00:42.458 --> 00:00:45.184
There's even a picture of Hazel
Horse here and Pauly Pony.

00:00:45.184 --> 00:00:46.766
They look very similar.

00:00:46.766 --> 00:00:50.333
The farmer used 26 fence posts
to build Hazel Horse's fence,

00:00:50.333 --> 00:00:54.933
so that's these 26 fence
posts right over here.

00:00:55.640 --> 00:00:58.469
These are the 26 fence posts

00:00:58.469 --> 00:01:01.222
for Hazel Horse's fence.

00:01:01.222 --> 00:01:02.966
I haven't counted it, but I'll
just take their word for it

00:01:02.966 --> 00:01:05.701
that there are 26 here,
and then 19 fence posts

00:01:05.701 --> 00:01:07.500
to build Pauly Pony's fence.

00:01:07.500 --> 00:01:11.769
These are 19 fence posts right over here.

00:01:12.133 --> 00:01:16.276
How many fence posts did
the farmer use in all?

00:01:16.276 --> 00:01:18.067
In all, so what do they want us to do?

00:01:18.067 --> 00:01:20.907
Do they want us to add or
subtract these numbers?

00:01:20.907 --> 00:01:22.612
Well, they're looking at "in all."

00:01:22.612 --> 00:01:26.466
How many total fence
posts did the farmer use?

00:01:26.804 --> 00:01:28.466
Well, the total is how many used

00:01:28.466 --> 00:01:31.999
in Hazel's fence and Pauly's fence,

00:01:31.999 --> 00:01:34.300
so we should add these two numbers.

00:01:34.300 --> 00:01:39.076
We're going to add 26 plus
the 19, or let me see.

00:01:39.076 --> 00:01:43.113
The 26 is in that brownish color, 26.

00:01:43.113 --> 00:01:45.367
That's the 26 fence posts to
build Hazel Horse's fence,

00:01:45.367 --> 00:01:47.833
and we're going to add that to 19.

00:01:47.833 --> 00:01:50.133
We're going to add that to 19,

00:01:50.466 --> 00:01:54.765
so 26 for Hazel Horse

00:01:54.766 --> 00:01:59.033
and then 19 for Pauly Pony.

00:01:59.033 --> 00:02:00.281
We want to add them because we're saying

00:02:00.281 --> 00:02:02.666
how many fence posts were used in all,

00:02:02.666 --> 00:02:06.630
or in Hazel's and Pauly's fences.

00:02:06.630 --> 00:02:08.334
Let's add these two together.

00:02:09.072 --> 00:02:10.699
First, let's go to the ones place.

00:02:10.699 --> 00:02:15.660
Six ones plus nine ones is 15 ones,

00:02:16.123 --> 00:02:18.829
but we only can write one digit here,

00:02:18.830 --> 00:02:20.733
so 15 ones is the same thing

00:02:20.733 --> 00:02:24.340
as five ones and one 10, right?

00:02:24.340 --> 00:02:27.100
15 is just one 10 and five ones,

00:02:27.100 --> 00:02:30.466
so we've essentially
regrouped those 10 ones

00:02:30.467 --> 00:02:33.900
as one 10 right over here,
or as some people say,

00:02:33.900 --> 00:02:35.733
you carry the one, but
that's all it's saying.

00:02:35.733 --> 00:02:37.633
We're saying six plus nine is 15.

00:02:37.633 --> 00:02:40.723
15 is five ones and one 10.

00:02:40.723 --> 00:02:43.333
We wrote that one 10 in the tens place,

00:02:44.102 --> 00:02:47.317
and then one 10 plus two 10s plus one 10.

00:02:47.317 --> 00:02:49.235
Well, the one plus two plus one is four,

00:02:49.235 --> 00:02:52.353
so it's going to be four 10s, so 45.

00:02:52.353 --> 00:02:53.768
How many fence posts used in all?

00:02:53.768 --> 00:02:57.033
45.

00:02:57.033 --> 00:02:59.200
Now, some of you all that
might have somehow tried

00:02:59.200 --> 00:03:02.100
to maybe subtract the 19 from 26,

00:03:02.100 --> 00:03:03.799
that would have been
what you would have done

00:03:03.800 --> 00:03:08.515
if they said, "How many more
fence posts did the farmer use

00:03:08.515 --> 00:03:10.799
"for Hazel's fence than for Pauly's?"

00:03:10.800 --> 00:03:13.667
Then, you would subtract 19
from 26 to find the difference,

00:03:13.667 --> 00:03:17.466
but they didn't ask us how
many more fence posts were used

00:03:17.466 --> 00:03:20.938
for Hazel than for Pauly, or
how many fewer fence posts

00:03:20.938 --> 00:03:22.366
were used for Pauly than Hazel.

00:03:22.366 --> 00:03:23.266
They didn't ask us that.

00:03:23.266 --> 00:03:27.035
They said, "How many fence posts
did the farmer use in all,"

00:03:27.035 --> 00:03:29.171
and that's why we added.


Missing numbers between 0 and 120

WEBVTT
Kind: captions
Language: en

00:00:00.407 --> 00:00:02.783
- [Voiceover] What is the missing number,

00:00:02.783 --> 00:00:07.783
100, 101, 102, 103, 104, 105,

00:00:08.583 --> 00:00:11.450
106 is the missing number,

00:00:11.450 --> 00:00:14.783
and then we go 107, 108, 109, 110.

00:00:14.783 --> 00:00:18.650
So, the missing number is 106,
that's the number after 105

00:00:18.650 --> 00:00:20.983
and before 107.

00:00:20.983 --> 00:00:22.217
Let's keep going.

00:00:22.217 --> 00:00:24.284
What is the missing number?

00:00:24.284 --> 00:00:28.617
Well, if I start at 100, the
next number is going to be 101,

00:00:28.617 --> 00:00:32.233
and then we go to 102,
103, 104, we keep going.

00:00:33.167 --> 00:00:34.817
This is exciting.

00:00:34.817 --> 00:00:39.050
What whole number comes right before 105?

00:00:39.050 --> 00:00:41.617
Well, what whole number
comes right before five?

00:00:41.617 --> 00:00:42.717
Well, four.

00:00:42.717 --> 00:00:45.185
So what number comes right before 105?

00:00:45.185 --> 00:00:47.788
Well, it's going to be 104.

00:00:47.788 --> 00:00:52.283
If you're going after 100 you
get 100, 101, 102, 103, 104,

00:00:52.283 --> 00:00:54.233
and then you go to 105.

00:00:57.083 --> 00:00:58.717
All right, so they want us to figure out

00:00:58.717 --> 00:01:00.483
what the missing number is.

00:01:00.483 --> 00:01:02.117
And we see that they're
counting, one, two three,

00:01:02.117 --> 00:01:03.517
all the way up to 10 and then here

00:01:03.517 --> 00:01:04.950
they're going from 11 to 20.

00:01:04.950 --> 00:01:08.183
They're counting all the way up to 120.

00:01:08.183 --> 00:01:09.850
But there's a missing square here.

00:01:09.850 --> 00:01:13.784
So, this is the number
that is one larger than 118

00:01:13.784 --> 00:01:15.650
and one less than 120.

00:01:15.650 --> 00:01:19.550
Well, if I was counting, I'd
go 111, 112, 113, 114, 115,

00:01:19.550 --> 00:01:24.550
116, 117, 118, 119, and then 120.

00:01:25.285 --> 00:01:28.257
Nineteen is before 20,

00:01:28.257 --> 00:01:32.307
so 119 is gonna be before 120.

00:01:32.307 --> 00:01:36.484
And 19 is after 18 so the number after 118

00:01:36.484 --> 00:01:38.400
is going to be 119.

00:01:39.283 --> 00:01:40.350
Let's check our answer.

00:01:40.350 --> 00:01:41.917
Let's do one more of these.

00:01:41.917 --> 00:01:43.717
Well, we've done these a lot already.

00:01:43.717 --> 00:01:45.083
What number is missing?

00:01:45.083 --> 00:01:48.247
Right after seven comes
eight, so right after 107

00:01:48.247 --> 00:01:52.300
comes 108 and then 109 and 110.


Recognizing shapes

WEBVTT
Kind: captions
Language: en

00:00:00.041 --> 00:00:02.104
I have a bunch of shapes here, let's think
about what they are

00:00:03.004 --> 00:00:07.060
so this first shape here, you probably
recognize it is the shape of the moon in

00:00:07.006 --> 00:00:07.506
the sky

00:00:08.038 --> 00:00:11.040
or atleast on a full moon and we call it

00:00:11.058 --> 00:00:14.077
a circle. Now

00:00:14.599 --> 00:00:17.667
what makes it a circle when you get
further in math we come up with fancier

00:00:18.279 --> 00:00:21.590
ways to define a circle but one with
think about it is if we take the center

00:00:21.059 --> 00:00:21.559
the circle

00:00:22.004 --> 00:00:25.061
and if we go from one side
of the circle to the other side circle

00:00:25.097 --> 00:00:26.146
through the center

00:00:27.019 --> 00:00:30.760
that's always going to be the same
length so that's the same length

00:00:30.076 --> 00:00:35.077
as is this as this is on his way to the
center the circle

00:00:35.086 --> 00:00:38.101
as that now

00:00:39.001 --> 00:00:42.002
if that the circle what would something
like this to be

00:00:42.011 --> 00:00:45.190
what you might call this an oval.

00:00:45.289 --> 00:00:49.440
all circles are ovals but this
oval is not a circle because if you have

00:00:49.044 --> 00:00:50.051
its center

00:00:50.051 --> 00:00:54.068
see right over there this distance going
in that direction

00:00:54.068 --> 00:00:57.069
is going to be much shorter or just
shorter

00:00:57.078 --> 00:01:01.085
than this distance right over here so
this one right over here

00:01:01.085 --> 00:01:06.092
is not a circle, this one is. Now what
about the shape over here that has three

00:01:06.092 --> 00:01:06.592
sides

00:01:07.063 --> 00:01:10.372
one two three

00:01:10.939 --> 00:01:14.430
you could also say it has three corners
one two

00:01:14.043 --> 00:01:18.082
three if you have three sides we call that a
triangle

00:01:18.082 --> 00:01:23.154
Tri angle whenever you see this Tri over
hear that's a good

00:01:24.054 --> 00:01:27.111
sign that you're dealing with the number
three. This is a triangle

00:01:28.011 --> 00:01:31.078
has three sides note about these shapes
over here

00:01:31.078 --> 00:01:34.597
these shapes are interesting how many
shapes. So how many sides

00:01:35.299 --> 00:01:38.345
do these shapes have. Well if you count them

00:01:38.759 --> 00:01:42.560
you see that this one up here in purple

00:01:42.056 --> 00:01:45.795
has one two three four sides

00:01:46.299 --> 00:01:49.670
this one has one two three

00:01:49.067 --> 00:01:52.196
four sides, this has one two

00:01:52.799 --> 00:01:56.490
three four sides, this has one

00:01:56.049 --> 00:02:00.084
two three four sides. So they all have four sides.

00:02:00.084 --> 00:02:04.573
So a name for any shape that has four
sides is quadrilateral

00:02:05.329 --> 00:02:06.368
Quad

00:02:06.719 --> 00:02:09.768
dri quadrilateral

00:02:10.209 --> 00:02:15.280
quadrilateral and when you see this quad
at the beginning of the word

00:02:15.028 --> 00:02:18.087
it's a pretty good sign that you're
dealing with the number four

00:02:18.087 --> 00:02:21.706
four sides, four sides

00:02:22.489 --> 00:02:26.110
that's what quadrilateral means but
these are all different special cases of

00:02:26.011 --> 00:02:27.019
quadrilaterals

00:02:27.019 --> 00:02:31.026
and actually you could even have a quadrilateral that
looks like this, could even have a

00:02:31.026 --> 00:02:32.108
quadrilateral that look like this. One

00:02:33.008 --> 00:02:37.777
two three four sides as long as as four sides
it's a quadrilateral

00:02:37.849 --> 00:02:41.940
Let's think about each of these so this
shape right over here

00:02:41.094 --> 00:02:44.122
what you might see is the corners the
corners

00:02:45.022 --> 00:02:48.089
all come in, sometimes is called square
sometimes

00:02:48.089 --> 00:02:51.116
you could call it a right angle so

00:02:52.016 --> 00:02:57.115
and notice I could put little small squares
here. SO if all four corners are square

00:02:58.015 --> 00:03:01.022
or they're all come in at a right angle

00:03:01.022 --> 00:03:04.131
like this then you're dealing with a
rectangle

00:03:04.329 --> 00:03:08.910
you are dealing with the rectangle and so
just to be clear

00:03:08.091 --> 00:03:12.095
this down here is not a rectangle notice
this corner right over if I tried to

00:03:13.031 --> 00:03:14.460
pull all square here

00:03:14.739 --> 00:03:17.790
it would either look like this or
doesn't

00:03:17.079 --> 00:03:20.080
it doesn't the two sides aren't two
sides of a square

00:03:20.089 --> 00:03:23.278
or would look like or it would look like
this

00:03:24.079 --> 00:03:27.650
so this is not a square corner neither
it is this

00:03:27.065 --> 00:03:30.159
or neither is this so this is not a
rectangle

00:03:31.059 --> 00:03:35.548
all four corners need to be square this
one also is not a rectangle

00:03:36.079 --> 00:03:39.620
notice the corners are not

00:03:39.062 --> 00:03:43.081
the corners are not right angles so

00:03:43.081 --> 00:03:46.138
this is a direct rectangle this is a
direct rectangle and what about this

00:03:47.038 --> 00:03:51.081
well I can do that I can put little
squares

00:03:51.081 --> 00:03:54.104
I could put all squares in the corner or

00:03:55.004 --> 00:03:58.020
I could see that they are all right all
the corners

00:03:58.002 --> 00:04:01.011
are right angles so this actually is a
rectangle as well

00:04:02.001 --> 00:04:06.040
so these are both these are both
rectangles

00:04:06.139 --> 00:04:10.000
now you might recognize this purple one
on the left is a special type a

00:04:10.000 --> 00:04:10.078
rectangle

00:04:10.078 --> 00:04:13.078
not only are all the corners square

00:04:13.078 --> 00:04:14.107
not all do they have

00:04:15.007 --> 00:04:18.050
these right angles but all the sides are
the same length

00:04:18.005 --> 00:04:21.005
at least it looks like it from the
picture this side is the same length as this

00:04:21.051 --> 00:04:22.115
side is the same length as that side

00:04:23.015 --> 00:04:27.093
same length as that side. So if the sides are
of same length and all the corners are

00:04:27.093 --> 00:04:31.096
square where they have right angles,
then we call it a square

00:04:31.096 --> 00:04:34.134
so this right over here is a, this

00:04:35.034 --> 00:04:38.112
right over here is a square special type
a rectangle

00:04:39.012 --> 00:04:42.096
this one not all the side to the same
this one all the sides are the same

00:04:42.096 --> 00:04:46.122
out if we go down here looks like all
the sides are the same one

00:04:47.022 --> 00:04:51.031
two three four

00:04:51.031 --> 00:04:54.069
but it doesn't have

00:04:54.069 --> 00:04:57.161
square corners. Notice if I tried to put
a little square here

00:04:58.061 --> 00:05:02.074
it wouldn't fit perfectly, if I tried
square here doesn't fit perfectly these

00:05:02.074 --> 00:05:03.091
aren't right angles

00:05:03.091 --> 00:05:06.132
So if all four sides are the same and
that's all we know about it

00:05:07.032 --> 00:05:11.124
we call it a rhombus. We call it a
rhombus

00:05:12.024 --> 00:05:16.028
so you might be saying, the square all
four sides are the same, is that a rhombus?

00:05:16.028 --> 00:05:19.075
yes it is. So the square is a rectangle

00:05:19.075 --> 00:05:22.133
it's a rectangle and

00:05:23.033 --> 00:05:29.088
is a rhombus so this last shape here
this last quadrilateral

00:05:29.088 --> 00:05:32.093
what's interesting about it is you have

00:05:32.093 --> 00:05:35.108
two of the side's going in the same
direction

00:05:36.008 --> 00:05:39.043
so this one right over here knows what I
mean by the same direction it sometimes

00:05:39.043 --> 00:05:40.075
is called parallel

00:05:40.075 --> 00:05:44.157
if I draw two lines here these two lines
are never going to cross each other

00:05:45.057 --> 00:05:49.111
so these two lines are parallel if you
have opposite one pair of opposite sides

00:05:50.011 --> 00:05:50.511
that are parallel

00:05:51.004 --> 00:05:56.053
we call them trapezoids we call them
trapezoids.

00:05:56.053 --> 00:05:59.096
Notice the other blinds the other
opposite sides

00:05:59.096 --> 00:06:02.100
these will cross, these will cross

00:06:03.036 --> 00:06:06.119
eventually to that right over there is
called a trapezoid.


Halves and fourths

WEBVTT
Kind: captions
Language: en

00:00:00.512 --> 00:00:04.087
- [Voiceover] Which picture
divides a rectangle in half?

00:00:04.087 --> 00:00:06.827
So to divide something in half we need to

00:00:06.827 --> 00:00:10.612
divide it into two equal sections.

00:00:10.612 --> 00:00:13.027
Over here we're divided into two sections

00:00:13.027 --> 00:00:14.240
but these aren't equal.

00:00:14.240 --> 00:00:16.034
These don't have the same area.

00:00:16.034 --> 00:00:18.913
This part has a much
larger area than this one.

00:00:18.913 --> 00:00:20.980
So this isn't divided in half.

00:00:20.980 --> 00:00:23.162
This one's also divided into two sections

00:00:23.162 --> 00:00:24.300
but they're not equal.

00:00:24.300 --> 00:00:27.226
This one is much bigger than this one.

00:00:27.226 --> 00:00:29.478
This is divided into equal sections,

00:00:29.478 --> 00:00:30.917
but it's not two equal sections.

00:00:30.917 --> 00:00:33.425
This is divided into three equal sections,

00:00:33.425 --> 00:00:35.283
so this isn't divided in half.

00:00:35.283 --> 00:00:39.067
Here we're divided into
two equal sections.

00:00:39.067 --> 00:00:40.553
One, two sections, and they

00:00:40.553 --> 00:00:43.456
are the same size, the same area.

00:00:43.456 --> 00:00:44.965
So I'd pick that one.

00:00:44.965 --> 00:00:46.822
Let's keep going. This is a lot of fun.

00:00:46.822 --> 00:00:49.771
Which picture divides
a circle into halves?

00:00:49.771 --> 00:00:51.791
Once again, two equal sections.

00:00:51.791 --> 00:00:53.300
This one, this one, and this one divide

00:00:53.300 --> 00:00:56.413
into three sections,
and actually these two

00:00:56.413 --> 00:00:58.848
right over here, the three
sections aren't even equal.

00:00:58.848 --> 00:01:01.124
Then here, this is three equal sections

00:01:01.124 --> 00:01:03.538
but that would be thirds not halves.

00:01:03.538 --> 00:01:06.069
Halves would be two equal sections,

00:01:06.069 --> 00:01:09.134
so it'll be that, right over there.

00:01:09.134 --> 00:01:11.688
Which picture divides
a square into fourths

00:01:11.688 --> 00:01:12.989
or sometimes people might say

00:01:12.989 --> 00:01:14.946
divides a square into quarters.

00:01:14.946 --> 00:01:19.474
Well fourths or quarters
would be four equal sections.

00:01:19.474 --> 00:01:21.110
Well this is two equal sections.

00:01:21.110 --> 00:01:24.036
This is one section,
this is just one whole.

00:01:24.036 --> 00:01:27.612
This is two equal
sections, right over here.

00:01:27.612 --> 00:01:28.830
Or I guess you could say this one

00:01:28.830 --> 00:01:30.583
actually is just not divided at all.

00:01:30.583 --> 00:01:34.112
This right over here
is two equal sections.

00:01:34.112 --> 00:01:37.200
This is four equal sections,
so this is fourths,

00:01:37.200 --> 00:01:40.195
or you could say this is
divided into quarters.

00:01:40.195 --> 00:01:41.612
Let's keep going.

00:01:41.612 --> 00:01:43.330
Let's do a few more of these.

00:01:43.330 --> 00:01:46.627
Which picture divides a circle in half?

00:01:46.627 --> 00:01:48.624
So this circle isn't divided.

00:01:48.624 --> 00:01:52.176
This is a circle divided into
thirds, three equal sections.

00:01:52.176 --> 00:01:54.661
This is a circle divided into two pieces,

00:01:54.661 --> 00:01:57.656
but they're not equal, so this
isn't divided into halves.

00:01:57.656 --> 00:01:59.769
This is two equal
sections, right over here,

00:01:59.769 --> 00:02:01.975
so that one is halves.

00:02:01.975 --> 00:02:03.832
Let's do one more.

00:02:03.832 --> 00:02:07.315
Which picture divides a
rectangle into halves?

00:02:07.315 --> 00:02:09.242
Well, it has to be two equal sections.

00:02:09.242 --> 00:02:11.100
This is three equal sections.

00:02:11.100 --> 00:02:13.283
This is three not even equal sections.

00:02:13.283 --> 00:02:16.069
This is three equal sections,
so this would be thirds,

00:02:16.069 --> 00:02:18.042
just as this one would be thirds.

00:02:18.042 --> 00:02:20.550
This one over here is two equal sections,

00:02:20.550 --> 00:02:23.161
so that is halves.


Estimating lengths

WEBVTT
Kind: captions
Language: en

00:00:00.361 --> 00:00:03.492
- [Voiceover] Lets get some
practice estimating lengths.

00:00:03.492 --> 00:00:06.143
So they tell us this line is one inch.

00:00:06.143 --> 00:00:08.639
So that distance right
over there is one inch.

00:00:08.639 --> 00:00:11.280
About how long is a toy car?

00:00:11.280 --> 00:00:13.840
Well, this is one of those
lines that they just did

00:00:13.840 --> 00:00:15.956
so that's one inch, but let's just see

00:00:15.956 --> 00:00:17.205
how many of those we need

00:00:17.205 --> 00:00:19.529
to go the whole length of the car.

00:00:19.529 --> 00:00:23.304
So if I were to draw another one

00:00:23.304 --> 00:00:24.407
that's about to that,

00:00:24.407 --> 00:00:26.172
and I'm estimating, it's not exact,

00:00:26.172 --> 00:00:27.403
that would be two inches.

00:00:27.403 --> 00:00:29.849
That's one inch, that's two inches.

00:00:29.849 --> 00:00:33.068
Then that would be three inches.

00:00:33.068 --> 00:00:37.001
And then that would be four inches.

00:00:37.001 --> 00:00:41.776
So it looks like the car
is about four inches.

00:00:41.776 --> 00:00:43.174
Let's do another one of these

00:00:43.174 --> 00:00:45.855
estimating length problems.

00:00:45.855 --> 00:00:48.107
So I have a picture of two sunflowers

00:00:48.107 --> 00:00:49.703
right over here, and they tell us that

00:00:49.703 --> 00:00:53.799
this first sunflower
is 40 centimeters tall.

00:00:53.799 --> 00:00:56.727
About how long is this
second sunflower going to be?

00:00:57.264 --> 00:00:58.069
Lets think about it.

00:00:58.069 --> 00:00:59.725
If this entire distance,

00:00:59.725 --> 00:01:04.725
if this entire height is 40 centimeters.

00:01:05.625 --> 00:01:09.344
And this sunflower, if
you look at its height,

00:01:09.344 --> 00:01:10.881
I'll draw a dotted line here.

00:01:10.881 --> 00:01:12.664
It looks like it's about half of

00:01:12.664 --> 00:01:16.155
the height of the taller sunflower.

00:01:16.416 --> 00:01:18.887
So what is half of 40 going to be?

00:01:18.887 --> 00:01:20.250
Well, 40 is four tens.

00:01:20.250 --> 00:01:22.333
Half of four tens is going to be two tens.

00:01:22.333 --> 00:01:26.867
So this looks like it's
about 20 centimeters,

00:01:26.867 --> 00:01:28.157
let me write it this way,

00:01:28.157 --> 00:01:32.693
this looks like it's about 20 centimeters.

00:01:32.693 --> 00:01:33.909
Another way to think about it is

00:01:33.909 --> 00:01:38.909
if this whole thing is 40
centimeters then half would be 20,

00:01:39.147 --> 00:01:40.740
then this would be 10 centimeters.

00:01:40.740 --> 00:01:43.693
Let me write this, this
would be 10 centimeters.

00:01:43.693 --> 00:01:45.814
This would be 20 centimeters.

00:01:46.152 --> 00:01:48.846
Then this would be 30 centimeters.

00:01:49.108 --> 00:01:51.918
Then this would be 40 centimeters.

00:01:52.287 --> 00:01:56.908
So we already saw that the
sunflower is 40 centimeters tall.

00:01:56.908 --> 00:01:58.764
And then this one would be half of that.

00:01:58.764 --> 00:02:01.790
It would be 20 centimeters tall.


Addition word problems within 10

WEBVTT
Kind: captions
Language: en

00:00:00.736 --> 00:00:03.399
3 flowers bloomed yesterday

00:00:03.399 --> 00:00:05.411
and they draw the flowers 1, 2, 3

00:00:05.431 --> 00:00:09.174
6 more bloomed today. 1, 2, 3, 4, 5, 6.

00:00:09.814 --> 00:00:11.942
How many flowers bloomed?

00:00:12.402 --> 00:00:15.680
Well 3 yesterday, 6 more today. So you're going to have

00:00:15.680 --> 00:00:18.946
the 3 from yesterday, plus the 6 more today.

00:00:19.726 --> 00:00:22.146
Well what's 3 plus 6 equal to?

00:00:22.996 --> 00:00:28.031
Well 3 plus 6 is the same thing as 6 plus 3 and that's going to be 9.

00:00:29.121 --> 00:00:32.374
6 + 1 is 7, 6 + 2 is 8, 6 + 3 is 9.

00:00:33.654 --> 00:00:37.523
And it's good to know, just in your head, 6 + 3 is 9. 3 + 6 is 9

00:00:37.523 --> 00:00:39.442
or you could even count them......

00:00:39.442 --> 00:00:42.663
1, 2, 3, 4, 5, 6, 7, 8, 9.

00:00:43.483 --> 00:00:47.199
Let's do another one.

00:00:47.849 --> 00:00:50.012
There were 2 sharks. 1, 2.

00:00:50.712 --> 00:00:52.936
3 more sharks came. 1, 2, 3.

00:00:53.576 --> 00:00:55.725
How many sharks are there now?

00:00:55.855 --> 00:00:58.834
Well there's the 2 that were there before.

00:00:58.994 --> 00:01:01.707
Plus the 3 that came.

00:01:01.957 --> 00:01:04.035
So, there's 2 plus 3 sharks.

00:01:04.255 --> 00:01:06.156
Well, what is 2 plus 3?

00:01:06.156 --> 00:01:08.092
Well, 2 + 3 is 5.

00:01:08.092 --> 00:01:10.779
2 + 3 is 5 and you could count it. 1,2, 3, 4, 5.

00:01:12.799 --> 00:01:14.712
Let's do one more.

00:01:14.982 --> 00:01:16.845
A squirrel picked up 7 acorns.

00:01:17.095 --> 00:01:19.525
(counting) 1, 2, 3, 4, 5, 6, 7.

00:01:20.235 --> 00:01:22.400
She picked up one more acorn.

00:01:22.400 --> 00:01:25.218
Find out how many acorns she picked up.

00:01:25.528 --> 00:01:27.158
Well she picked up the 7.

00:01:27.458 --> 00:01:30.533
Plus 1 acorn. So it's going to be 7 plus 1 acorns.

00:01:31.473 --> 00:01:33.727
And that is equal to 8.

00:01:33.727 --> 00:01:35.793
(counting) 1, 2, 3, 4, 5, 6, 7, 8.

00:01:38.553 --> 00:01:40.481
You got it right.


How to evaluate a piecewise function (example)

WEBVTT
Kind: captions
Language: en

00:00:00.574 --> 00:00:03.480
- [Instructor] Consider the
following piecewise function

00:00:03.480 --> 00:00:05.918
and we say f(t) is equal to

00:00:05.918 --> 00:00:07.515
and they tell us what it's equal to

00:00:07.515 --> 00:00:09.401
based on what t is,

00:00:09.401 --> 00:00:10.924
so if t is less than or equal to -10,

00:00:10.924 --> 00:00:11.963
we use this case.

00:00:11.963 --> 00:00:13.969
If t is between -10 and -2,

00:00:13.969 --> 00:00:15.006
we use this case.

00:00:15.006 --> 00:00:17.083
And if t is greater than or equal to -2,

00:00:17.083 --> 00:00:18.737
we use this case.

00:00:18.737 --> 00:00:19.570
And then they ask us

00:00:19.570 --> 00:00:21.809
what is the value of f(-10)?

00:00:21.809 --> 00:00:23.766
So t is going to be equal to -10,

00:00:23.766 --> 00:00:25.810
so which case do we use?

00:00:25.810 --> 00:00:26.643
So let's see.

00:00:26.643 --> 00:00:29.345
If t is less than or equal to -10,

00:00:29.345 --> 00:00:31.374
we use this top case, right over here

00:00:31.374 --> 00:00:33.218
and t is equal to -10,

00:00:33.218 --> 00:00:34.839
that's the one that
we're trying to evaluate.

00:00:34.839 --> 00:00:38.334
So we wanna use this case right over here.

00:00:38.334 --> 00:00:42.501
So f(-10) is going to be equal to -10,

00:00:44.289 --> 00:00:45.674
everywhere we see a t here,

00:00:45.674 --> 00:00:48.267
we substitute it with a -10.

00:00:48.267 --> 00:00:50.434
- 10 squared minus 5 times,

00:00:51.396 --> 00:00:52.942
actually I don't have a denominator there,

00:00:52.942 --> 00:00:54.498
I don't know why I wrote it so high.

00:00:54.498 --> 00:00:58.415
So it's gonna be -10
squared minus 5 times -10.

00:01:00.498 --> 00:01:01.331
So let's see.

00:01:01.331 --> 00:01:04.257
- 10 squared, that's positive 100

00:01:04.257 --> 00:01:05.295
and then negative,

00:01:05.295 --> 00:01:07.545
or subtracting 5 times -10,

00:01:08.746 --> 00:01:11.376
this is going to be subtracting -50

00:01:11.376 --> 00:01:13.575
or you're going to add 50,

00:01:13.575 --> 00:01:16.232
so this is going to be equal to 150.

00:01:16.232 --> 00:01:18.942
f(10) is 150,

00:01:18.942 --> 00:01:20.317
'cause we used this case up here,

00:01:20.317 --> 00:01:21.650
'cause t is -10.

00:01:22.769 --> 00:01:25.174
Let's do another one of these examples.

00:01:25.174 --> 00:01:28.115
So, here we have

00:01:28.115 --> 00:01:30.592
consider the following
piecewise function, alright.

00:01:30.592 --> 00:01:32.944
What is the value of h(-3)?

00:01:32.944 --> 00:01:34.455
See when h is -3,

00:01:34.455 --> 00:01:36.304
which case do we use?

00:01:36.304 --> 00:01:41.175
We use this case if our x
is between negative infinity

00:01:41.175 --> 00:01:42.643
and zero.

00:01:42.643 --> 00:01:46.288
And -3 is in between
negative infinity and zero,

00:01:46.288 --> 00:01:48.783
so we're gonna use this
case right over here.

00:01:48.783 --> 00:01:50.153
If it was positive three,

00:01:50.153 --> 00:01:51.525
we would use this case.

00:01:51.525 --> 00:01:52.602
If it was positive 30,

00:01:52.602 --> 00:01:54.228
we would use this case.

00:01:54.228 --> 00:01:57.124
So we're going to use the first case again

00:01:57.124 --> 00:01:59.957
and so for h(-3),

00:02:00.794 --> 00:02:03.018
we're gonna take -3 to the third power.

00:02:03.018 --> 00:02:03.851
So let's see.

00:02:03.851 --> 00:02:08.445
h(-3) is going to be -3 to the third power

00:02:08.445 --> 00:02:09.528
which is -27.

00:02:11.828 --> 00:02:12.661
And we're done.

00:02:12.661 --> 00:02:14.602
That's h(-3).

00:02:14.602 --> 00:02:15.599
Because we're using this case,

00:02:15.599 --> 00:02:17.779
you could almost ignore
these second two cases

00:02:17.779 --> 00:02:19.161
right over here.

00:02:19.161 --> 00:02:20.805
Let's do one more example.

00:02:20.805 --> 00:02:22.981
This one's a little bit different.

00:02:22.981 --> 00:02:26.819
Below is a graph of the step function g(x)

00:02:26.819 --> 00:02:28.964
so we can see g(x) right over here.

00:02:28.964 --> 00:02:31.504
It starts when x equals -9, it's at 3,

00:02:31.504 --> 00:02:32.342
and then it jumps up,

00:02:32.342 --> 00:02:33.667
and then it jumps down.

00:02:33.667 --> 00:02:36.409
Match each expression with its value.

00:02:36.409 --> 00:02:38.992
So g(-3.0001),

00:02:40.436 --> 00:02:42.936
so -3.0001,

00:02:44.017 --> 00:02:45.854
so that's right over here

00:02:45.854 --> 00:02:47.177
and g of that,

00:02:47.177 --> 00:02:48.927
we see is equal to 3.

00:02:50.251 --> 00:02:54.073
So this is going to be
equal to 3 right over here.

00:02:54.073 --> 00:02:57.113
g(3.99999)

00:02:57.113 --> 00:03:01.676
3.99999, almost 4,

00:03:01.676 --> 00:03:04.473
so let's draw a dotted line right here,

00:03:04.473 --> 00:03:06.312
it's gonna be almost 4,

00:03:06.312 --> 00:03:10.394
well g(3.99999) is going to be 7.

00:03:10.394 --> 00:03:12.417
We see that right over there.

00:03:12.417 --> 00:03:14.437
So that is equal to 7.

00:03:14.437 --> 00:03:17.303
g(4.00001).

00:03:17.303 --> 00:03:19.887
So g(4) is still 7,

00:03:19.887 --> 00:03:22.656
but as soon as we go above 4,

00:03:22.656 --> 00:03:24.114
we drop down over here,

00:03:24.114 --> 00:03:28.197
so g(4.00001) is going to be -3.

00:03:29.961 --> 00:03:32.402
I wanna, actually, let's focus
on that a little bit more.

00:03:32.402 --> 00:03:33.922
How did I know that?

00:03:33.922 --> 00:03:37.303
Well I know that g(4) is 7 and not -3

00:03:37.303 --> 00:03:40.280
because we have this dot
is circled in up here

00:03:40.280 --> 00:03:41.919
and it's hollow down here.

00:03:41.919 --> 00:03:44.873
But as soon as we get
any amount larger than 4,

00:03:44.873 --> 00:03:46.854
then the function drops down to this.

00:03:46.854 --> 00:03:49.183
So 4.0000, as many,

00:03:49.183 --> 00:03:50.690
just slightly above 4,

00:03:50.690 --> 00:03:53.844
the value of our function
is going to be -3.

00:03:53.844 --> 00:03:55.991
Now let's do g(9).

00:03:55.991 --> 00:03:58.287
So g(9), that's when x is 9

00:03:58.287 --> 00:03:59.423
and we go down here.

00:03:59.423 --> 00:04:01.289
You might be tempted to say it's -3,

00:04:01.289 --> 00:04:03.036
but you see, at this
point right over here,

00:04:03.036 --> 00:04:05.058
we have an open circle.

00:04:05.058 --> 00:04:06.792
So that means that while it's not,

00:04:06.792 --> 00:04:10.147
you can't say that the
function is -3 right over there

00:04:10.147 --> 00:04:12.779
and there's no other place
where we have a filled-in circle

00:04:12.779 --> 00:04:14.549
for x equals 9

00:04:14.549 --> 00:04:18.895
so the function g actually
isn't defined at x equals 9.

00:04:18.895 --> 00:04:22.645
So I'm gonna put undefined
right over there.


How to determine the domain of a modeling function

WEBVTT
Kind: captions
Language: en

00:00:00.240 --> 00:00:03.070
Thomas has 400 candy bars in his shop

00:00:03.070 --> 00:00:06.290
and each cost 50 cents. Let p of b

00:00:06.290 --> 00:00:09.889
denote the price, p, measured in
dollars

00:00:09.889 --> 00:00:13.210
of a purchase of b candy bars. Alright

00:00:13.210 --> 00:00:16.379
I input b, the number candy bars I wanna buy,

00:00:16.379 --> 00:00:19.630
and p(b) will tell me what's the
purchase price is really

00:00:19.630 --> 00:00:22.990
just taking the number of candy bars multiplied by 50 cents, but we won't have to worry

00:00:22.990 --> 00:00:23.939
about that just yet.

00:00:23.939 --> 00:00:28.519
Which number type is more appropriate
for the domain of the function?

00:00:28.519 --> 00:00:32.450
So just to remind ourselves, what is the domain of a function? A domain is a set

00:00:32.450 --> 00:00:35.280
of all inputs over which the function is
defined.

00:00:35.280 --> 00:00:38.670
So it is the set of all b's. It is the set of
all inputs

00:00:38.670 --> 00:00:42.489
over which p of b will produce a
defined response

00:00:42.489 --> 00:00:45.520
So let's think about it. Is it integers or
real numbers?

00:00:45.520 --> 00:00:51.000
So I could buy -- b could be 0 candy bars, 1 candy bars, 2 candy bars, all up to

00:00:51.000 --> 00:00:52.440
400 candy bars.

00:00:52.440 --> 00:00:55.879
Could I -- Could I have a fractional can--
Could be b

00:00:55.879 --> 00:00:59.480
0.372 of a candy bar?

00:00:59.480 --> 00:01:02.690
Well, this is a normal candy shop.

00:01:02.690 --> 00:01:06.790
It's -- each candy bar is gonna be in its own packet. It's going to be in a discrete

00:01:06.790 --> 00:01:10.020
chunk. You're not going to be able to buy
0.372

00:01:10.020 --> 00:01:15.189
of a candy bar. You can either buy a one
more or none more, so you buy your 1, 2, 3

00:01:15.189 --> 00:01:16.520
all the way up to 400.

00:01:16.520 --> 00:01:21.439
So I would say integers -- that the domain of this function is going to be

00:01:21.439 --> 00:01:25.680
is going to be a subset of integers. It's
not -- you not, you can't have

00:01:25.680 --> 00:01:29.860
a real, all real number, but integers
are obviously a subset real numbers.

00:01:29.860 --> 00:01:33.860
But you can't say, hey, I'm gonna buy pi candy
bars, or I'm gonna buy

00:01:33.860 --> 00:01:38.350
the square root of two candy bars. You're gonna buy integer number candy bars.

00:01:38.350 --> 00:01:43.189
Now they say, define the interval of the
domain. So the fewest candy bars I could

00:01:43.189 --> 00:01:43.610
buy

00:01:43.610 --> 00:01:49.070
are 0 candy bars, and I have to decide whether I put a
bracket or I put a parenthesis.

00:01:49.070 --> 00:01:52.649
I can actually buy 0 candy bars so I'm
gonna put a bracket.

00:01:52.649 --> 00:01:56.530
If I put a parentheses, that means I
could have values above zero but not

00:01:56.530 --> 00:01:58.850
including 0, but I want to include 0

00:01:58.850 --> 00:02:01.890
so I'm gonna put the bracket there. So the
least I could buy

00:02:01.890 --> 00:02:06.390
is 0, and in the most I could buy, the store
has 400 candy bars so that's the most I

00:02:06.390 --> 00:02:07.040
can buy.

00:02:07.040 --> 00:02:11.190
The most I could buy are 400
candy bars, and I can buy 400.

00:02:11.190 --> 00:02:13.050
So I would put

00:02:13.050 --> 00:02:15.980
brackets there as well. So the interval
of the domain,

00:02:15.980 --> 00:02:19.319
I would want to select integers. So b

00:02:19.319 --> 00:02:24.010
is a member of integers such that b is
also a member of this interval. It

00:02:24.010 --> 00:02:25.870
could be as low as 0

00:02:25.870 --> 00:02:29.080
including 0, and as high as 400
including

00:02:29.080 --> 00:02:31.900
400. Got it right.


How to determine the domain of algebraic functions (examples)

WEBVTT
Kind: captions
Language: en

00:00:00.299 --> 00:00:03.490
Let's do some more examples finding do
mains of functions.

00:00:03.490 --> 00:00:08.590
So let's say we have a function g of x. So this is our
function definition here tells us, look,

00:00:08.590 --> 00:00:13.690
if we have an input x, the output g of x is going to
be equal to 1

00:00:13.690 --> 00:00:17.400
over the square root of 6

00:00:17.400 --> 00:00:20.699
minus -- we write this little bit neater, 1

00:00:20.699 --> 00:00:24.080
over the square root

00:00:24.080 --> 00:00:28.650
of 6 minus the absolute value of x

00:00:28.650 --> 00:00:31.960
So like always, pause this video and see if
you can figure out what

00:00:31.960 --> 00:00:36.000
what is the domain of this function.
Based on this function definition what

00:00:36.000 --> 00:00:37.219
is the domain of g?

00:00:37.219 --> 00:00:41.780
What is the set of all inputs for which
this function is defined?

00:00:41.780 --> 00:00:46.730
Alright. So, to think about all of the
inputs that would allow this function to

00:00:46.730 --> 00:00:49.090
be defined, it may be easier to state when
is this function

00:00:49.090 --> 00:00:53.510
not get defined. Well if we divide by 0

00:00:53.510 --> 00:00:56.910
then we're not going to be defined. Or if
we have a negative

00:00:56.910 --> 00:01:01.940
under the radical. So if you think about
it -- if if what we have under the radical

00:01:01.940 --> 00:01:07.049
is 0 or negative -- if it is 0 or
negative. If it's

00:01:07.049 --> 00:01:11.070
0, you can take the, you can take the
principal root of 0. It's going to be 0.

00:01:11.070 --> 00:01:14.130
But then you are going to divide by 0.
That's going to be undefined.

00:01:14.130 --> 00:01:17.810
And if what you have another radical is
negative, the principal root isn't

00:01:17.810 --> 00:01:22.170
defined for native number, at least the
classic principal root is a defined for

00:01:22.170 --> 00:01:27.100
a negative number. So if 6 minus the absolute
value of x is zero or negative,

00:01:27.100 --> 00:01:30.770
this thing is not going to be defined.
Or another way to think about it is

00:01:30.770 --> 00:01:34.740
it's going to be defined -- so g is

00:01:34.740 --> 00:01:38.799
is defined -- is defined

00:01:38.799 --> 00:01:42.240
if -- g is defined

00:01:42.240 --> 00:01:47.280
if 6 minus, maybe I can write if and only if.

00:01:47.280 --> 00:01:51.770
Sometimes people write if and only if
with two f's right there, iff.

00:01:51.770 --> 00:01:54.570
g is defined if and only if --

00:01:54.570 --> 00:01:59.810
this is kind of mathy way of saying if and
only if -- 6 minus the absolute value

00:01:59.810 --> 00:02:03.250
of x is greater than 0

00:02:03.250 --> 00:02:06.979
it has to be, it has to be positive.
If it's zero,

00:02:06.979 --> 00:02:11.020
we're going take the square root of 0 is 0.
Then you divide by 0. That's undefined. And if it's

00:02:11.020 --> 00:02:14.319
less than zero, then you're taking, you're
trying to find the principal root of a negative

00:02:14.319 --> 00:02:15.709
number, that's not defined.

00:02:15.709 --> 00:02:19.569
So let's see, it's g is defined if and only
if this is true.

00:02:19.569 --> 00:02:23.230
And let's see. We could add the absolute
value

00:02:23.230 --> 00:02:27.800
of x to both sides. We could add the
absolute value of

00:02:27.800 --> 00:02:32.040
x to both sides, then that would give us 6

00:02:32.040 --> 00:02:37.140
is greater than the absolute value of x,
or that the absolute value of x

00:02:37.140 --> 00:02:40.650
is less than 6. Or we could say that,

00:02:40.650 --> 00:02:43.720
you know, let me write that way.
The absolute value of x is less than 6.

00:02:43.720 --> 00:02:44.950
Another way of saying that is

00:02:44.950 --> 00:02:49.860
x would have to be less than 6 and
greater than negative 6.

00:02:49.860 --> 00:02:52.920
or x is between negative 6 and 6.
These two things --

00:02:52.920 --> 00:02:57.799
these two things are equivalent. If the
magnitude of x is less than 6

00:02:57.799 --> 00:03:02.860
then x is greater than negative 6 and
less than positive 6.

00:03:02.860 --> 00:03:07.100
So if we wanted to write the domain in
kinda fancy domain-set notation,

00:03:07.100 --> 00:03:10.430
we could write the domain of

00:03:10.430 --> 00:03:15.709
g is going to be x, all the x's that are
a member of

00:03:15.709 --> 00:03:19.250
real numbers such that negative 6

00:03:19.250 --> 00:03:22.430
is less than x, which is less than

00:03:22.430 --> 00:03:26.900
6, and we're done. Let's do another one.
And this was gonna get even a little bit

00:03:26.900 --> 00:03:30.540
a little bit hairier, just for, just for
kicks. Alright.

00:03:30.540 --> 00:03:32.050
So let's say that I have --

00:03:32.050 --> 00:03:35.910
let's say that I have -- h of x

00:03:35.910 --> 00:03:40.590
is equal to --  and I have a kind of
composite definition here.

00:03:40.590 --> 00:03:43.940
So let's say it's x plus 10

00:03:43.940 --> 00:03:47.040
over x plus

00:03:47.040 --> 00:03:50.470
10 times x minus 9

00:03:50.470 --> 00:03:54.470
times x minus 5

00:03:54.470 --> 00:03:57.520
times x minus 5 and it's this if it --

00:03:57.520 --> 00:04:00.760
h of x is this if x does not equal

00:04:00.760 --> 00:04:04.900
5 and it's equal to -- it's equal to

00:04:04.900 --> 00:04:08.270
pi if x is equal

00:04:08.270 --> 00:04:12.490
if x is equal to 5. So once again, pause
this video.

00:04:12.490 --> 00:04:16.970
Think about what is the domain of h, or
another way to think about is what made h

00:04:16.970 --> 00:04:19.989
not defined? So let's think about it.

00:04:19.989 --> 00:04:23.000
So what would make h not defined?

00:04:23.000 --> 00:04:28.700
So if some -- if x is anything other than 5,
we go this clause. If it's 5, we go this clause.

00:04:28.720 --> 00:04:32.290
So in this clause up here, what would make
this thing undefined?

00:04:32.290 --> 00:04:35.910
Well the most obvious thing is if we
divide by 0.

00:04:35.910 --> 00:04:39.700
So what's going to cause us to divide by
0? So

00:04:39.700 --> 00:04:43.169
if -- if x is equal to

00:04:43.169 --> 00:04:47.169
if -- let me write it here -- so we're going
to divide by 0 --

00:04:47.169 --> 00:04:50.600
divide by 0.

00:04:50.600 --> 00:04:54.120
That would happen if x is equal to 9.

00:04:54.120 --> 00:04:57.520
That would happen if x is equal to negative 10.

00:04:57.520 --> 00:05:01.669
x is equal to negative 10. Now we have to be
careful, would that happen

00:05:01.669 --> 00:05:05.890
if this was the only definition here, it
would happen when x equals to 5.

00:05:05.890 --> 00:05:09.100
But remember when x is equal 5, we don't
look at this part

00:05:09.100 --> 00:05:13.030
of the compound definition. We look at this
part. So it's true that up here you would

00:05:13.030 --> 00:05:17.030
be dividing by 0 if x is equaling 5
but x equaling 5, you wouldn't even look at --

00:05:17.030 --> 00:05:18.650
look there. For input is 5,

00:05:18.650 --> 00:05:21.690
you use this part of the definition. So you'd

00:05:21.690 --> 00:05:25.110
divide by 0. Maybe I should write it this way,
divide by 0

00:05:25.110 --> 00:05:28.540
on, I guess you could say the top -- the top

00:05:28.540 --> 00:05:31.870
clause or the top part a definition. Part

00:05:31.870 --> 00:05:35.180
of the definition.

00:05:35.180 --> 00:05:38.980
If x equals 9, X equals negative 10
or -- and that's it

00:05:38.980 --> 00:05:42.750
because x equal 5 doesn't apply to this
top part. If this clause wasn't here

00:05:42.750 --> 00:05:45.060
then yes, you would write x equals 5.

00:05:45.060 --> 00:05:48.490
Now we're almost done, but some of you
might say, wait, wait, wait, but look,

00:05:48.490 --> 00:05:52.160
can't I simplify this? I have x plus 10 in
the numerator and x plus 10 in the denominator.

00:05:52.160 --> 00:05:54.450
Can I just simplify this and that will
disappear?

00:05:54.450 --> 00:05:59.630
And you could except, if you did that, you
are now creating a different

00:05:59.630 --> 00:06:03.920
function definition. Because if you just
simplify this, you just said 1 over x minus 9 --

00:06:03.920 --> 00:06:08.910
1 over x minus 9 times x minus 5.
This is now a different function.

00:06:08.910 --> 00:06:12.250
That one actually would be defined at x
equals negative 10. But the one that we

00:06:12.250 --> 00:06:16.840
have started with, this one is not. This
is -- you're gonna end up with 0 over 0.  You

00:06:16.840 --> 00:06:17.470
gonna end up with that

00:06:17.470 --> 00:06:22.010
indeterminate form. So for this function,
exactly the way it's written,

00:06:22.010 --> 00:06:26.200
it's not going to be defined with
x is equal to 9 or x is equal to

00:06:26.200 --> 00:06:29.940
negative 10. So once again if you want a fan --
write in our fancy

00:06:29.940 --> 00:06:33.420
domain set notation. The domain

00:06:33.420 --> 00:06:36.970
is going to be x

00:06:36.970 --> 00:06:40.420
all the x's that are a member of the
real such that

00:06:40.420 --> 00:06:43.800
x does not equal 9 and x

00:06:43.800 --> 00:06:47.780
does not equal negative 10.
Any other real number x --

00:06:47.780 --> 00:06:51.430
it's going to work including 5.
If x equals 5, h of --

00:06:51.430 --> 00:06:54.470
h of 5 is going to be equal to pi,

00:06:54.470 --> 00:06:58.580
because you default to this one over here.
h of 5 so OK x is equal 5,

00:06:58.580 --> 00:07:02.580
we do that one right over there. Now if
you gave x equals 9

00:07:02.580 --> 00:07:05.670
you gonna divide by 0. If x equals negative 10,
you gonna divide

00:07:05.670 --> 00:07:09.190
by 0, but that's gonna work for anything else.

00:07:09.190 --> 00:07:11.300
So that right over there is the domain.


What is the domain of a function?

WEBVTT
Kind: captions
Language: en

00:00:00.290 --> 00:00:03.480
Let's have a little bit of a review of what
a function is before we talk about what

00:00:03.480 --> 00:00:03.919
it means that

00:00:03.919 --> 00:00:07.870
what the domain of a function means.
So function we can view as

00:00:07.870 --> 00:00:11.219
something -- so I put a function in this box
here and it takes

00:00:11.219 --> 00:00:15.019
inputs, and for a given input, it's going to
produce an output which we call

00:00:15.019 --> 00:00:19.740
f of x. So, for example, let's say that
we have the function --

00:00:19.740 --> 00:00:22.949
let's say we have the function f of x

00:00:22.949 --> 00:00:26.439
is equal to 2 over

00:00:26.439 --> 00:00:29.810
x. So in this case if -- let me see --

00:00:29.810 --> 00:00:33.579
that's my function f. If I were to input

00:00:33.579 --> 00:00:37.090
the number 3. Well, f of 3

00:00:37.090 --> 00:00:40.100
that we're going to output -- we have, we
know how to figure that out. We've

00:00:40.100 --> 00:00:42.910
defined it right over here. It's going to be
equal to 2

00:00:42.910 --> 00:00:45.940
over 3. It's going to be equal to 2

00:00:45.940 --> 00:00:49.100
over 3. So we're able, for that input, we're
able to find an output.

00:00:49.100 --> 00:00:53.629
If our input was pi, then

00:00:53.629 --> 00:00:57.030
we input into our function and then
f of pi --

00:00:57.030 --> 00:01:01.090
when x is pi, we're going to output
f of pi,

00:01:01.090 --> 00:01:04.629
which is equal to 2 over pi. So we
could write this as 2

00:01:04.629 --> 00:01:08.630
over pi. We're able to find the output
pretty easily. But I want to do something

00:01:08.630 --> 00:01:12.310
interesting. Let's attempt to input 0

00:01:12.310 --> 00:01:15.380
into the function. If we input 0 then
the function

00:01:15.380 --> 00:01:19.030
tells us what we need to output. Does this
definition tell us what we need to output?

00:01:19.030 --> 00:01:22.939
So if I attempt to put x equal 0, then this
definition would say

00:01:22.939 --> 00:01:28.150
f of 0 be 2 over 0, but 2 over 0 is
undefined.

00:01:28.150 --> 00:01:33.159
Rewrite this -- 2 over 0. This is undefined.
This function definition does not tell us

00:01:33.159 --> 00:01:37.729
what to actually do with 0. It gives us an
undefined answer.

00:01:37.729 --> 00:01:42.060
So this function is not defined here.
It gives a question mark.

00:01:42.060 --> 00:01:45.290
So this gets to the essence of what domain is.

00:01:45.290 --> 00:01:49.119
Domain is the set of all inputs over
which the function

00:01:49.119 --> 00:01:53.579
is defined. So the domain of this
function f would be all real numbers

00:01:53.579 --> 00:01:57.060
except for x equals 0.

00:01:57.060 --> 00:02:00.939
So we write down these, these big ideas.
This is

00:02:00.939 --> 00:02:04.170
the domain -- the domain of a function --

00:02:04.170 --> 00:02:08.020
Actually let me write that out. The domain of a function

00:02:08.020 --> 00:02:11.130
A domain of a function

00:02:11.130 --> 00:02:14.500
is the set of

00:02:14.500 --> 00:02:18.510
all inputs -- inputs

00:02:18.510 --> 00:02:22.130
over which the function is defined -- over

00:02:22.130 --> 00:02:30.370
which the function is defined,
or the function has defined outputs

00:02:30.370 --> 00:02:33.810
over which the function has defined

00:02:33.810 --> 00:02:40.840
outputs. So the domain for this f
in particular --

00:02:40.840 --> 00:02:43.910
so the domain for this one  -- if I want
to say

00:02:43.910 --> 00:02:48.770
its domain, I could say, look, it's going to
be the set of these curly brackets. These are

00:02:48.770 --> 00:02:53.820
kind of typical mathy set notation. I said OK
, it could be the set of -- I gonna put

00:02:53.820 --> 00:02:57.920
curly brackets like that. Well, x can be a
member

00:02:57.920 --> 00:03:02.460
So this little symbol means a
member of the real numbers. But it can't be

00:03:02.460 --> 00:03:03.620
any real number. It could be

00:03:03.620 --> 00:03:06.720
most of the real numbers except it
cannot be 0 because we don't know

00:03:06.720 --> 00:03:09.840
-- this definition is undefined when
you put the input as 0

00:03:09.840 --> 00:03:13.400
So x is a member of the real numbers,
and we write real numbers -- we write it

00:03:13.400 --> 00:03:18.400
with this kind of double stroke right over here.
That's the set of all real numbers

00:03:18.400 --> 00:03:22.950
such that -- we have to put
the exception. 0 is not a --

00:03:22.950 --> 00:03:26.770
x equals to 0 is not a member of that
domain -- such that x

00:03:26.770 --> 00:03:30.020
does not -- does not equal

00:03:30.020 --> 00:03:34.610
0. Now let's make this a little bit
more concrete by do some more examples

00:03:34.610 --> 00:03:36.160
So more examples we do,

00:03:36.160 --> 00:03:40.950
hopefully the clearer this will become.
So let's say we have another function.

00:03:40.950 --> 00:03:42.830
Just be clear, we don't
always have to use f's

00:03:42.830 --> 00:03:46.110
and x's. We could say, let's say we
have g

00:03:46.110 --> 00:03:49.440
of y is equal to the square root

00:03:49.440 --> 00:03:52.530
of y minus

00:03:52.530 --> 00:03:55.820
6. So what is the domain here? What is the set

00:03:55.820 --> 00:03:59.450
of all inputs over which this function g
is defined? So here

00:03:59.450 --> 00:04:02.680
we are in putting a y it to function g

00:04:02.680 --> 00:04:04.300
and we're gonna output

00:04:04.300 --> 00:04:09.490
g of y. Well it's going to be defined
as long as whatever we have under the

00:04:09.490 --> 00:04:10.730
radical right over here

00:04:10.730 --> 00:04:15.080
is non-negative. If this becomes a
negative, our traditional principal

00:04:15.080 --> 00:04:16.280
root operator here

00:04:16.280 --> 00:04:19.690
is not defined. We need something that --
if this was a negative

00:04:19.690 --> 00:04:22.140
number, how would you take the principal
root of a negative number?

00:04:22.140 --> 00:04:25.610
We just think this is kind of the the
traditional principal root operator.

00:04:25.610 --> 00:04:30.270
So y minus 6, y minus 6 needs to be greater than

00:04:30.270 --> 00:04:36.310
or equal to 0, in order for, in order for g
to be defined for that input y.

00:04:36.310 --> 00:04:40.490
Or you could say add six to both sides.
y is to be greater than or equal 6.

00:04:40.490 --> 00:04:45.150
Or you could say g is defined for any inputs
y that are greater than or equal to 6.

00:04:45.150 --> 00:04:49.470
So you could say the domain here, we
could say the domain here

00:04:49.470 --> 00:04:54.850
is the set of all y's that are members of
the real numbers

00:04:54.850 --> 00:04:58.690
such that y, such that they're also
greater than

00:04:58.690 --> 00:05:01.980
or equal, such that they're also

00:05:01.980 --> 00:05:06.990
greater than or equal to 6. So hopefully
this is starting to make some sense --

00:05:06.990 --> 00:05:09.750
You're all used to a function that is
defined this way. You could even see

00:05:09.750 --> 00:05:10.930
functions that are divided

00:05:10.930 --> 00:05:14.650
fairly exotic ways. You could see a
function -- let me say h of x --

00:05:14.650 --> 00:05:20.470
h of x could be defined as -- it literally
could be defined as, well h of x is gonna be 1

00:05:20.470 --> 00:05:24.280
if x is equal to pi

00:05:24.280 --> 00:05:27.950
and it's equal to 0 if, if,

00:05:27.950 --> 00:05:32.550
x is equal to 3. Now what's the domain
here? And I encourage you to pause the video and

00:05:32.550 --> 00:05:33.360
think about it.

00:05:33.360 --> 00:05:37.500
Well, this function is actually only
defined for two input. If you,

00:05:37.500 --> 00:05:41.250
we know h of -- we know h of pi --
if you input pi into it

00:05:41.250 --> 00:05:45.860
we know you're gonna output 1, and
we know that if you input 3 into it

00:05:45.860 --> 00:05:49.200
h of 3, when x equals 3, you're going to
-- you're going to --

00:05:49.200 --> 00:05:54.440
put some commas here. You're gonna get 0.
But if you input anything else, what's

00:05:54.440 --> 00:05:55.710
h of 4 going to be?

00:05:55.710 --> 00:06:00.020
Well, it hasn't defined. It's undefined.
What's h of negative 1 going to be?

00:06:00.020 --> 00:06:03.289
It hasn't defined. So the domain, the domain here,

00:06:03.289 --> 00:06:06.750
the domain of h

00:06:06.750 --> 00:06:11.490
is literally -- it's just literally
going to be the the two valid inputs

00:06:11.490 --> 00:06:12.629
that x can be

00:06:12.629 --> 00:06:15.740
are 3 and pi.

00:06:15.740 --> 00:06:20.569
These are the only valid inputs.
These are the only two numbers over

00:06:20.569 --> 00:06:25.139
which this function is actually defined.
So this hopefully starts to give you a flavor

00:06:25.139 --> 00:06:29.150
of why we care about to the domain.
It's not all functions are defined over

00:06:29.150 --> 00:06:32.330
all real numbers. Some are defined
for only a small subset

00:06:32.330 --> 00:06:36.720
of real numbers, or for some other
thing, or only whole numbers, or natural

00:06:36.720 --> 00:06:39.520
numbers, or positive numbers, and negative
numbers. So they have exceptions.

00:06:39.520 --> 00:06:42.590
So we'll see that as we do
more and more examples.


11 Computing with set members

WEBVTT
Kind: captions
Language: en

00:00:00.775 --> 00:00:05.036
- [Voiceover] If lowercase
x is a member of capital X,

00:00:05.036 --> 00:00:09.931
lowercase y is a member of set capital Y,

00:00:09.931 --> 00:00:13.626
and lowercase z is a
member of set capital Z,

00:00:13.626 --> 00:00:15.898
which of the following
could have a value of 23?

00:00:15.898 --> 00:00:17.717
So let's look at each of these.

00:00:17.717 --> 00:00:19.749
So x times y times z.

00:00:19.749 --> 00:00:21.262
So you take something from this set,

00:00:21.262 --> 00:00:23.160
something from this set,
and something from that set,

00:00:23.160 --> 00:00:25.108
and multiply them together.

00:00:25.108 --> 00:00:26.698
Well, if you did that, you're going

00:00:26.698 --> 00:00:29.077
to get a non-prime number.

00:00:29.077 --> 00:00:32.003
Maybe you pick one from set X,

00:00:32.003 --> 00:00:33.516
but then you're gonna have to pick a two

00:00:33.516 --> 00:00:37.192
from set Y, or three or four,
and then a three, four, five.

00:00:37.192 --> 00:00:41.821
So you're going to get things
like one times two times three

00:00:41.821 --> 00:00:44.696
or one times three times three.

00:00:44.696 --> 00:00:46.100
No matter what, especially when you look

00:00:46.100 --> 00:00:50.424
at these last two sets, you're
going to get the product

00:00:50.424 --> 00:00:53.491
of at least two primes or you
might even take the product

00:00:53.491 --> 00:00:56.652
of a composite and a
prime or two composites.

00:00:56.652 --> 00:00:59.335
So there's no way multiplying those three.

00:00:59.335 --> 00:01:01.432
Especially a member from these last sets,

00:01:01.432 --> 00:01:02.662
they don't have one in them.

00:01:02.662 --> 00:01:05.397
But you're going to get
a prime number like 23.

00:01:05.397 --> 00:01:08.583
23's factors are clearly
only 23 and itself.

00:01:08.583 --> 00:01:10.762
It doesn't have a two or
three or four as a factor.

00:01:10.762 --> 00:01:12.542
It doesn't have a five as a factor.

00:01:12.542 --> 00:01:14.980
And so, we could rule that out.

00:01:14.980 --> 00:01:17.392
Let's think about it, x plus y plus z.

00:01:17.392 --> 00:01:20.323
Can we make that equal to 23?

00:01:20.323 --> 00:01:22.467
So let's see, 23 is a lot
larger than these numbers.

00:01:22.467 --> 00:01:24.275
Let's see if we'd pick the largest number

00:01:24.275 --> 00:01:27.270
in each of these sets and
see if we can get to 23.

00:01:27.270 --> 00:01:31.898
So three plus four plus five.

00:01:31.898 --> 00:01:34.293
These are the largest
members in each of those sets

00:01:34.293 --> 00:01:36.217
and this still only gets us to 12.

00:01:36.217 --> 00:01:39.842
So there's no way we're going
to be able to get to 23.

00:01:39.842 --> 00:01:43.772
Now at C, x plus y times z.

00:01:43.772 --> 00:01:46.030
x plus y times z.

00:01:46.030 --> 00:01:47.899
Well, I see a way of doing that.

00:01:47.899 --> 00:01:49.130
I see a way of doing that.

00:01:49.130 --> 00:01:52.143
x plus y times, let me do
this in a different color,

00:01:52.143 --> 00:01:53.919
x plus y times z.

00:01:53.919 --> 00:01:54.931
What if...

00:01:54.931 --> 00:01:56.954
Let's see, we can get to 20 by multiplying

00:01:56.954 --> 00:02:00.356
four times five and then you add a three.

00:02:00.356 --> 00:02:02.750
So this could be three plus

00:02:02.750 --> 00:02:05.812
four times five.

00:02:05.812 --> 00:02:07.507
And so we'll go with that one.

00:02:07.507 --> 00:02:09.200
Which of the following
could have a value of 23?

00:02:09.200 --> 00:02:11.477
That one, and we can only pick one choice

00:02:11.477 --> 00:02:12.649
so we know we're already done.

00:02:12.649 --> 00:02:14.036
But we could verify that we can't

00:02:14.036 --> 00:02:15.822
get there with these other ones.

00:02:15.822 --> 00:02:19.255
So let's see, x squared plus
y squared plus z squared.

00:02:19.255 --> 00:02:22.215
This one is tempting.

00:02:22.215 --> 00:02:23.202
It's tempting.

00:02:23.202 --> 00:02:25.628
This is because you could
get numbers large enough.

00:02:25.628 --> 00:02:28.345
But if you play around
enough, you would have trouble

00:02:28.345 --> 00:02:33.051
finding a combination that has a 23.

00:02:33.051 --> 00:02:35.988
x times y plus z.

00:02:35.988 --> 00:02:39.096
So, if you wanted to get
as large as possible,

00:02:39.096 --> 00:02:40.748
you do three times four.

00:02:40.748 --> 00:02:44.927
So you could try three times
four and then plus some z.

00:02:44.927 --> 00:02:48.609
But then in this case, this
is going to be 12 plus z.

00:02:48.609 --> 00:02:49.502
12 plus z.

00:02:49.502 --> 00:02:51.093
z would have to be nine.

00:02:51.093 --> 00:02:52.452
You don't find a nine in this third set.

00:02:52.452 --> 00:02:53.737
There's clearly no nine so you could

00:02:53.737 --> 00:02:55.859
rule that one out as well.

00:02:55.859 --> 00:02:57.744
So we go at C.


14 Running distance ratio

WEBVTT
Kind: captions
Language: en

00:00:00.689 --> 00:00:02.969
- [Voiceover] In preparing for a race,

00:00:02.969 --> 00:00:07.969
Jill ran half the distance that Sam ran.

00:00:08.086 --> 00:00:08.907
So let me write that down.

00:00:08.907 --> 00:00:11.479
So let's say J, is the distance Jill ran,

00:00:11.479 --> 00:00:13.846
is equal to half the distance.

00:00:13.846 --> 00:00:16.889
We'll say S is the distance that Sam ran.

00:00:16.889 --> 00:00:21.591
And Sam ran three times the
distance that Aisha ran.

00:00:21.591 --> 00:00:25.735
So Sam ran three times the distance,

00:00:25.735 --> 00:00:29.009
that we'll use A for the
distance that Aisha ran.

00:00:29.009 --> 00:00:30.771
What was the ratio of the distance

00:00:30.771 --> 00:00:35.614
that Jill ran to the
distance that Aisha ran?

00:00:35.614 --> 00:00:36.396
So let's think about it.

00:00:36.396 --> 00:00:40.235
So here, we have how much
Jill ran in terms of S.

00:00:40.235 --> 00:00:43.748
So maybe we could write how
much Aisha ran in terms of S.

00:00:43.748 --> 00:00:44.900
Then we could find the ratio

00:00:44.900 --> 00:00:47.194
and then the S, it should cancel out.

00:00:47.194 --> 00:00:49.265
So let's see, over here
if we wanna solve for,

00:00:49.265 --> 00:00:52.845
if we wanna write this,
so we have how much

00:00:52.845 --> 00:00:55.652
Aisha ran in terms of how much Sam ran,

00:00:55.652 --> 00:00:58.159
we can just divide both sides by three.

00:00:58.159 --> 00:01:00.504
So you divide both sides by three.

00:01:00.504 --> 00:01:03.794
We're left with Aisha ran

00:01:03.794 --> 00:01:06.718
1/3 the distance Sam ran.

00:01:06.718 --> 00:01:08.526
So if we wanna find the ratio,

00:01:08.526 --> 00:01:09.916
the ratio of the distance Jill ran

00:01:09.916 --> 00:01:11.229
to the distance Aisha ran.

00:01:11.229 --> 00:01:14.746
So it's the ratio J over A.

00:01:14.746 --> 00:01:16.885
The ratio J over A.

00:01:16.885 --> 00:01:19.535
This is going to be equal to 1/2 S.

00:01:19.535 --> 00:01:21.266
That's what J is equal to.

00:01:21.266 --> 00:01:23.866
Over 1/3 S.

00:01:23.866 --> 00:01:27.243
That's what A is equal to, over 1/3 S.

00:01:27.243 --> 00:01:29.264
So it's equal to 1/2 divided by 1/3.

00:01:29.264 --> 00:01:32.379
The S is cancel out.

00:01:32.379 --> 00:01:35.909
So this is going to be equal to 1/2.

00:01:35.909 --> 00:01:37.987
You divide by 1/3, that's the same thing

00:01:37.987 --> 00:01:41.214
as multiplying by three over one.

00:01:41.214 --> 00:01:44.664
And so, one times three is three.

00:01:44.664 --> 00:01:46.998
Two times one is two.

00:01:46.998 --> 00:01:49.701
So the ratio is equal to...

00:01:49.701 --> 00:01:51.761
So the ratio the distance
that Jill ran to the distance

00:01:51.761 --> 00:01:56.322
that Aisha ran is three to two or 3/2.


6 Area of small circles

WEBVTT
Kind: captions
Language: en

00:00:00.837 --> 00:00:03.855
- [Voiceover] In the figure
above, the seven small circles,

00:00:03.855 --> 00:00:06.920
which have equal radii,
and the large circle

00:00:06.920 --> 00:00:09.869
are tangent at the indicated points.

00:00:09.869 --> 00:00:13.282
If the area of the large circle is 36 pi,

00:00:13.282 --> 00:00:16.556
so the area of the whole thing
right over here is 36 pi,

00:00:16.556 --> 00:00:21.271
what is the area of one
of the small circles?

00:00:21.271 --> 00:00:23.568
So let's think about how we could do this.

00:00:23.568 --> 00:00:26.656
If we know the area of the large circle,

00:00:26.656 --> 00:00:29.117
we could figure out the
radius of the large circle,

00:00:29.117 --> 00:00:30.789
and if we know the radius
of the large circle,

00:00:30.789 --> 00:00:32.391
we should be able to figure out the radius

00:00:32.391 --> 00:00:34.458
of one of these smaller circles.

00:00:34.458 --> 00:00:35.456
So let's do that.

00:00:35.456 --> 00:00:40.355
So we know that area is
equal to pi r squared,

00:00:40.355 --> 00:00:43.397
and so the area of the
larger circle is 36 pi.

00:00:43.397 --> 00:00:48.180
So we know that 36 pi is
equal to pi r squared.

00:00:48.180 --> 00:00:53.180
We could divide both sides by
pi, divide both sides by pi,

00:00:53.544 --> 00:00:56.121
and we're left with the
radius of the larger circle,

00:00:56.121 --> 00:00:57.770
I'll swap the two sides,

00:00:57.770 --> 00:01:01.624
the radius of the larger
circle squared is equal to 36.

00:01:01.624 --> 00:01:03.133
Now if you were just solving an equation,

00:01:03.133 --> 00:01:05.014
r squared is equal to 36, you would get

00:01:05.014 --> 00:01:07.429
that r is equal to plus or minus 6,

00:01:07.429 --> 00:01:08.645
but we're talking about a length here,

00:01:08.645 --> 00:01:12.583
so we only want to think about
the principal root of 36.

00:01:12.583 --> 00:01:16.716
So we get the radius of
the larger circle is six.

00:01:16.716 --> 00:01:17.568
Or another way to think about it

00:01:17.568 --> 00:01:20.657
is the diameter of the larger
circle is going to be 12.

00:01:20.657 --> 00:01:22.869
Diameter is twice the radius.

00:01:22.869 --> 00:01:25.006
So if we were to start at
this point right over here

00:01:25.006 --> 00:01:28.326
and go all the way across,
go all the way across,

00:01:28.326 --> 00:01:31.414
that whole distance right there is 12.

00:01:31.414 --> 00:01:32.900
So this distance right over here,

00:01:32.900 --> 00:01:36.220
the diameter of the larger circle is 12.

00:01:36.220 --> 00:01:38.426
Now the diameter of the
larger circle is 12,

00:01:38.426 --> 00:01:39.726
what is going to be the diameter

00:01:39.726 --> 00:01:41.445
of one of the smaller circles?

00:01:41.445 --> 00:01:43.674
And they all have the same diameter.

00:01:43.674 --> 00:01:47.180
Well, three of the diameters
of a smaller circle

00:01:47.180 --> 00:01:49.966
are equal to the diameter
of the larger circle.

00:01:49.966 --> 00:01:51.824
So each of these smaller
circles are going to have

00:01:51.824 --> 00:01:54.238
one-third of the diameter
of the larger one.

00:01:54.238 --> 00:01:57.552
So they're each going to
have a diameter of four.

00:01:57.552 --> 00:02:01.066
Well, if they each have
a diameter of four,

00:02:01.066 --> 00:02:04.849
that means that they each
have a radius of two.

00:02:04.849 --> 00:02:06.962
So this distance right over here,

00:02:06.962 --> 00:02:09.470
this distance right over here is two.

00:02:09.470 --> 00:02:12.488
So now what is the area of
one of the small circles?

00:02:12.488 --> 00:02:16.854
Well, the area is going to be
pi times its radius squared.

00:02:16.854 --> 00:02:20.127
So pi times two squared,

00:02:20.127 --> 00:02:24.144
which is equal to pi
times four, or four pi.

00:02:24.144 --> 00:02:26.304
So the area of one of the smaller circles,

00:02:26.780 --> 00:02:30.780
or area of one of the
small circles, is four pi.


Example: Amplitude and period transformations

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:03.010
We're asked to graph the function

00:00:03.010 --> 00:00:07.848
y = 2sin(-x) on the interval

00:00:07.848 --> 00:00:10.349
the closed interval so it includes the endpoints

00:00:10.349 --> 00:00:13.648
-2π to 2π

00:00:13.648 --> 00:00:14.681
So to do this

00:00:14.681 --> 00:00:18.507
I'm going to graph the function y = sin(x)

00:00:18.507 --> 00:00:21.402
and then think about how it's changed by the 2

00:00:21.402 --> 00:00:24.519
and the negative in front of the x over here

00:00:24.519 --> 00:00:27.873
So let's look at the sine of x first

00:00:27.873 --> 00:00:32.633
So let me draw our x-axis

00:00:32.633 --> 00:00:36.754
let me draw the y-axis

00:00:36.754 --> 00:00:37.894
pretty straight forward

00:00:37.894 --> 00:00:38.869
and we care about it

00:00:38.869 --> 00:00:40.865
between -2π and 2π

00:00:40.865 --> 00:00:43.913
so let's say this is -2π

00:00:43.913 --> 00:00:48.081
and then this one right over here would be -π

00:00:48.081 --> 00:00:50.151
this of course is 0

00:00:50.151 --> 00:00:54.435
then this is positive π

00:00:54.435 --> 00:01:00.842
and then this right over here is 2π again

00:01:00.842 --> 00:01:04.201
And then this could be 1

00:01:04.201 --> 00:01:05.792
this could be 2

00:01:05.792 --> 00:01:07.730
this could be -1

00:01:07.730 --> 00:01:09.876
and this can be -2

00:01:09.876 --> 00:01:12.665
Now let me copy this thing so I can use it for later

00:01:12.665 --> 00:01:14.627
when I adjust this graph

00:01:14.627 --> 00:01:18.021
So just this graph so let me copy

00:01:18.021 --> 00:01:21.804
Alright so let's think about sin(x)

00:01:21.804 --> 00:01:23.863
So what happens when sine is zero?

00:01:23.863 --> 00:01:25.408
When sine is zero

00:01:25.408 --> 00:01:30.330
Oh sorry, when x is zero, sin(0) is 0

00:01:30.330 --> 00:01:33.158
Ok, I'll draw a little unit circle here for reference

00:01:33.158 --> 00:01:35.072
This is what I like to do in my head

00:01:35.072 --> 00:01:36.664
as I like to figure out the value

00:01:36.664 --> 00:01:39.697
of the trigonometric functions

00:01:39.697 --> 00:01:41.128
so this is x

00:01:41.128 --> 00:01:42.741
this is y

00:01:42.741 --> 00:01:45.724
Draw unit circle

00:01:45.724 --> 00:01:49.613
And remember over here x is refering to the angle

00:01:49.613 --> 00:01:52.038
So that's my unit circle radius 1

00:01:52.038 --> 00:01:55.062
So then the angle is 0

00:01:55.062 --> 00:01:56.453
sin is going to be the y-coordinate here

00:01:56.453 --> 00:01:59.062
so sin(0) is a 0

00:01:59.077 --> 00:02:01.230
when as sine increases

00:02:01.230 --> 00:02:05.389
we go up all the way to sin(π/2) which is 1

00:02:05.389 --> 00:02:11.251
so sin(π/2) is going to get you to 1

00:02:11.251 --> 00:02:14.934
then sin(π) gets you to 0

00:02:14.934 --> 00:02:19.618
sin(3π/2) gets you to -1

00:02:19.618 --> 00:02:22.233
and then sin(2π) gets you back to 0

00:02:22.233 --> 00:02:24.535
So if I were to graph this I'd look something like this

00:02:24.535 --> 00:02:30.666
so this is between 0 and 2π it looks something like that

00:02:30.666 --> 00:02:31.925
And we also want to go in the negative direction

00:02:31.925 --> 00:02:39.402
and so as we go in the negative direction

00:02:39.402 --> 00:02:40.799
as we go in the negative direction

00:02:40.799 --> 00:02:49.182
so sin(-π/2) is -1

00:02:49.182 --> 00:02:52.303
then you go back to -π and you go back to zero

00:02:52.303 --> 00:02:56.276
-3π/2 you're going all the way back around like that

00:02:56.276 --> 00:03:00.070
that gets you all the way back to sine is equal to 1

00:03:00.070 --> 00:03:01.872
so sine is equal to 1

00:03:01.873 --> 00:03:07.096
And then 2π you go back sine is back equalling 0

00:03:07.127 --> 00:03:11.258
So the curve will look something like this

00:03:11.258 --> 00:03:13.616
In the negative so as you go from

00:03:13.616 --> 00:03:17.438
between 0 and -2π

00:03:17.438 --> 00:03:20.431
And this is consistent with everything else we know about sine

00:03:20.431 --> 00:03:23.752
the period of sin(x)

00:03:23.752 --> 00:03:24.823
what you see here

00:03:24.823 --> 00:03:26.398
you have a coefficient of 1 here

00:03:26.398 --> 00:03:29.512
so the period is going to be 2π over the absolute value of 1

00:03:29.512 --> 00:03:31.794
which is a little bit obvious it's just 2π

00:03:31.794 --> 00:03:34.176
or you just see here that the period is 2π

00:03:34.176 --> 00:03:42.500
It took 2π length to do our smallest repeating pattern

00:03:42.500 --> 00:03:44.248
And what is the amplitude?

00:03:44.248 --> 00:03:47.918
Well we vary between 1 and -1

00:03:47.918 --> 00:03:50.448
The total difference between the minimum and maximum is 2

00:03:50.463 --> 00:03:52.062
Half of that is 1

00:03:52.062 --> 00:03:53.549
Or another way of thinking about it is

00:03:53.549 --> 00:03:57.787
we vary 1 from our middle point

00:03:57.787 --> 00:04:01.132
So that was pretty straight forward

00:04:01.132 --> 00:04:02.814
Let's change it up a little bit

00:04:02.814 --> 00:04:14.101
Now let's graph y=2sin(x)

00:04:14.101 --> 00:04:16.726
So let me draw

00:04:16.756 --> 00:04:21.071
let me put my little axis there

00:04:21.071 --> 00:04:23.170
want to do it right under it

00:04:23.170 --> 00:04:25.696
So what is going to happen

00:04:25.696 --> 00:04:30.576
now that we have y = 2sin(x)

00:04:30.576 --> 00:04:34.838
how is the graph going to change?

00:04:34.838 --> 00:04:40.336
Well all we did is multiply this function by 2 so

00:04:40.336 --> 00:04:43.410
whatever this value it takes on is going to be twice as high now

00:04:43.410 --> 00:04:45.658
so 2 times 0 is 0

00:04:45.658 --> 00:04:48.645
2 times 1 is now 2

00:04:48.645 --> 00:04:49.742
2 times 1 is 2

00:04:49.742 --> 00:04:51.156
2 times 0 is

00:04:51.156 --> 00:04:52.464
let me be careful

00:04:52.464 --> 00:04:55.302
2 times 1 is 2

00:04:55.302 --> 00:04:56.884
that's a π/2

00:04:56.884 --> 00:04:59.894
2 times 0 is a 0

00:04:59.894 --> 00:05:04.726
2 times -1 is -2

00:05:04.726 --> 00:05:08.256
2 times 0 is 0

00:05:08.256 --> 00:05:10.700
so it looks something like this

00:05:10.700 --> 00:05:15.932
between 0 and 2π

00:05:15.932 --> 00:05:18.533
so it looks something like that

00:05:18.533 --> 00:05:20.936
and we keep going in the negative direction

00:05:20.936 --> 00:05:25.726
2 times -1 is -2

00:05:25.726 --> 00:05:28.238
2 times 0 is 0

00:05:28.238 --> 00:05:31.517
2 times 1 is 2

00:05:31.517 --> 00:05:32.864
2 times 0 is 0

00:05:32.879 --> 00:05:36.950
so in the negative direction it looks something like that

00:05:36.950 --> 00:05:39.845
my best attempt to draw a relatively smooth curve

00:05:39.845 --> 00:05:41.956
hopefully you get the idea

00:05:41.956 --> 00:05:43.605
so it would look something like that

00:05:43.605 --> 00:05:45.820
So what just happened?

00:05:45.820 --> 00:05:49.875
Well the difference between the minimum

00:05:49.875 --> 00:05:52.151
value and the maximum value just increased by a factor of 2

00:05:52.151 --> 00:05:56.532
the total difference is 4, half of that difference is 2

00:05:56.532 --> 00:05:59.178
So what is the amplitude here?

00:05:59.178 --> 00:06:05.101
Well the amplitude is 2

00:06:05.101 --> 00:06:08.124
you can view it as the absolute value of 2

00:06:08.124 --> 00:06:09.972
well it's common sense

00:06:09.972 --> 00:06:14.953
the amplitude here was 1 but now you're swaying from that middle position twice as far

00:06:14.953 --> 00:06:17.983
because you're multiplying by 2

00:06:17.983 --> 00:06:19.488
Now let's go back to

00:06:19.488 --> 00:06:23.580
sin(x) and let's change it in a different way

00:06:23.580 --> 00:06:27.230
Let's graph sin(-x)

00:06:27.230 --> 00:06:31.546
so now let me once again put some graph paper here

00:06:31.546 --> 00:06:40.736
And now my goal is to graph sin(-x)

00:06:40.736 --> 00:06:43.703
y=sin(-x) so at least for the time being I've got rid of that 2 there

00:06:43.703 --> 00:06:47.118
and I'm just going straight from sin(x) to sin(-x)

00:06:47.118 --> 00:06:50.922
So let's think about how the values are going to work out

00:06:50.922 --> 00:06:56.048
So when x is 0 this is still going to be sin(0) which is 0

00:06:56.048 --> 00:07:01.845
But then what as x increases, what happens when x is π/2

00:07:01.845 --> 00:07:04.017
the angle that we're inputting in to sine

00:07:04.017 --> 00:07:05.270
we're going to have to multiply by this negative

00:07:05.270 --> 00:07:06.681
so when x is π/2

00:07:06.681 --> 00:07:09.948
we're really taking sin(-π/2)

00:07:09.948 --> 00:07:16.740
but what's sin(-π/2) but we can see over here here it's -1

00:07:16.740 --> 00:07:18.705
It's - 1

00:07:18.705 --> 00:07:24.988
and then when x = π well sin(-π) we see this is 0

00:07:24.988 --> 00:07:29.588
When x is 3π/2 well it's going to be sin(-3π/2)

00:07:29.588 --> 00:07:32.851
which is 1

00:07:32.851 --> 00:07:38.515
Once again when x is 2π it's going to be sin(-2π)

00:07:38.515 --> 00:07:39.689
is 0

00:07:39.689 --> 00:07:45.083
So notice what was happening as I was trying to graph between 0 and 2π

00:07:45.083 --> 00:07:48.514
I kept referring to the points in the negative direction

00:07:48.514 --> 00:07:51.740
so you can imagine taking this negative side

00:07:51.740 --> 00:07:56.325
right over here between 0 and -2π

00:07:56.325 --> 00:07:59.855
and then flipping it over to get this one right over here

00:07:59.855 --> 00:08:02.187
that's what that -x seems to do

00:08:02.187 --> 00:08:05.079
and by that same logic when we go in then negative direction

00:08:05.079 --> 00:08:10.198
you say when x = -π/2 where you have the negative in front of it

00:08:10.198 --> 00:08:13.654
so it's going to be sin(π/2) so it's going to be equal to 1

00:08:13.654 --> 00:08:17.677
and you can flip this over the y-axis

00:08:17.677 --> 00:08:19.581
so essentially what we have done

00:08:19.581 --> 00:08:20.672
is we have flipped it we have

00:08:20.672 --> 00:08:26.983
reflected the graph of sin(x) over the y-axis

00:08:26.983 --> 00:08:38.940
So we have reflected it over the y-axis

00:08:38.940 --> 00:08:43.050
This is the y-axis so hopefully you see that reflection

00:08:43.050 --> 00:08:47.218
that's what that -x has done

00:08:47.218 --> 00:08:51.117
So now let's think about kind of the combo

00:08:51.117 --> 00:08:54.185
Having the 2 out the front and the -x right over there

00:08:54.185 --> 00:08:58.502
so let me put the graph on the axis there one more time

00:08:58.502 --> 00:09:02.981
And now let's try to do what was asked of us

00:09:02.981 --> 00:09:04.533
So I'll do it in a new colour

00:09:04.533 --> 00:09:05.285
I'll do it in blue

00:09:05.285 --> 00:09:17.502
now let's graph y = 2sin(-x)

00:09:17.502 --> 00:09:19.789
so based on everything we've done

00:09:19.789 --> 00:09:21.010
how will this look

00:09:21.010 --> 00:09:24.215
what are the transformations we will do?

00:09:24.215 --> 00:09:32.648
If we are going from the original sin(x) to y=2sin(-x)

00:09:32.648 --> 00:09:35.062
Well there're 2 ways of thinking about it

00:09:35.062 --> 00:09:37.333
You could either take 2sin(x)

00:09:37.333 --> 00:09:41.421
so here we multiply it by 2 to get double the amplitude

00:09:41.421 --> 00:09:42.752
And you could say

00:09:42.752 --> 00:09:47.018
well I'm going to flip it over to get the negative side of x

00:09:47.018 --> 00:09:50.013
And so if you did that you'd get

00:09:50.013 --> 00:09:51.728
so let me make it clear what I'm flipping

00:09:51.728 --> 00:09:56.318
so if I took between 0 and -2π

00:09:56.318 --> 00:09:58.054
and I flip it over

00:09:58.054 --> 00:10:02.340
what used to be here you reflect it over the y-axis

00:10:02.340 --> 00:10:03.906
and you now have

00:10:03.906 --> 00:10:07.435
so we go negative first

00:10:07.435 --> 00:10:09.556
then we go back to 0 and then it'll go positive

00:10:09.556 --> 00:10:13.974
and then you get right over there

00:10:13.974 --> 00:10:17.268
So all I did to get from 2sin(x) to 2sin(-x)

00:10:17.268 --> 00:10:20.701
is I just reflected over the y-axis

00:10:20.701 --> 00:10:24.186
and then of course what is between 0 and -2pi

00:10:24.186 --> 00:10:26.610
you just have to look between 0 and 2pi

00:10:26.610 --> 00:10:29.102
so it's going to go up

00:10:29.102 --> 00:10:30.192
up and down

00:10:30.192 --> 00:10:32.000
and make it even a little bit better

00:10:32.000 --> 00:10:33.167
draw a little bit neater

00:10:33.167 --> 00:10:37.691
and then down and then down and up

00:10:37.691 --> 00:10:41.093
so it's a reflection of what was between 0 and 2π

00:10:41.093 --> 00:10:43.320
so you see that right over here

00:10:43.320 --> 00:10:47.140
or if you start with sin(-x) and you go to 2sin(-x)

00:10:47.140 --> 00:10:52.163
notice what happens between sin(-x) and 2sin(-x)

00:10:52.163 --> 00:10:55.570
What's the difference between this graph and this graph?

00:10:55.570 --> 00:10:58.460
Well we just have twice the amplitude

00:10:58.460 --> 00:11:01.955
we're multiplying this one by 2

00:11:01.955 --> 00:11:05.057
and so you get twice the amplitude

00:11:05.057 --> 00:11:07.937
And so the last thought or question I have for you

00:11:07.937 --> 00:11:13.791
is how does the period of 2sin(-x)

00:11:13.791 --> 00:11:17.866
how does that relate to the period of sin(x)

00:11:17.866 --> 00:11:19.340
Well there's 2 ways of thinking about it

00:11:19.340 --> 00:11:23.451
we could actually I'll you think about it for a second

00:11:23.451 --> 00:11:25.331
Well there's 2 ways to thinking about it

00:11:25.331 --> 00:11:28.150
you could refer to the graphs over here

00:11:28.150 --> 00:11:31.753
or you can think about the formula which is hopefully a little intuitive right now

00:11:31.753 --> 00:11:35.274
if you wanted to refer to the kind of classical formula

00:11:35.274 --> 00:11:37.742
the period is going to be 2π

00:11:37.742 --> 00:11:40.464
and you divide by the absolute value of the coefficient

00:11:40.464 --> 00:11:44.269
to figure out how much faster you're going to get to 2π

00:11:44.269 --> 00:11:52.020
So the absolute value of -1 is just 1

00:11:52.020 --> 00:11:53.435
so you get 2π

00:11:53.435 --> 00:11:58.973
So you get the exact same period as the period of sin(x)

00:11:58.973 --> 00:12:00.236
And you see that

00:12:00.236 --> 00:12:08.083
you complete one cycle every 2π ...

00:12:08.083 --> 00:12:09.975
Now what is the difference?

00:12:09.975 --> 00:12:11.051
Well the period's the same.

00:12:11.051 --> 00:12:14.310
But remember this negative is not completely ignored.

00:12:14.310 --> 00:12:18.203
It doesn't change the period but it does change how the graph looks

00:12:18.203 --> 00:12:20.314
When you start getting increased x's

00:12:20.314 --> 00:12:22.763
instead of sin being positive as it would be

00:12:22.763 --> 00:12:25.805
in the case of a traditional sine function

00:12:25.805 --> 00:12:27.616
here as x grows

00:12:27.616 --> 00:12:29.492
you're taking the sine of -x

00:12:29.492 --> 00:12:32.056
You're taking the sine of a negative angle

00:12:32.056 --> 00:12:35.439
And that's why you start off having negative values of sine

00:12:35.439 --> 00:12:36.696
and that's also another way

00:12:36.696 --> 00:12:37.673
if you want to think about it

00:12:37.673 --> 00:12:39.861
that it's just a reflection over the y-axis

00:12:39.861 --> 00:12:41.158
of just sin(x)

00:12:41.158 --> 00:12:43.804
These two are reflections and these two are reflections

00:12:43.804 --> 00:12:47.296
this one is two times the amplitude of this one

00:12:47.296 --> 00:12:51.296
and that one is two times the amplitude of that one


3 Letter symmetry

WEBVTT
Kind: captions
Language: en

00:00:00.569 --> 00:00:02.657
- [Voiceover] If this pages
was folded along the dotted

00:00:02.657 --> 00:00:04.811
line in the figure above,
the left half of the letter

00:00:04.811 --> 00:00:08.041
W would exactly coincide
with the right half of W.

00:00:08.041 --> 00:00:10.011
You see that right over there.

00:00:10.011 --> 00:00:11.950
If you fold it, this point
would go to that point

00:00:11.950 --> 00:00:13.410
right over here, this
point would go to that

00:00:13.410 --> 00:00:15.434
point right over there,
that point would go to that

00:00:15.434 --> 00:00:16.680
point right over there.

00:00:16.680 --> 00:00:17.795
And I could keep going.

00:00:17.795 --> 00:00:19.322
You have this axis of symmetry right

00:00:19.322 --> 00:00:21.448
over here right down the center.

00:00:21.448 --> 00:00:24.082
Which of the following letters, as shown,

00:00:24.082 --> 00:00:27.240
cannot be folded along
a vertical line so that

00:00:27.240 --> 00:00:30.328
it's left half would
coincide with its right half?

00:00:30.328 --> 00:00:31.640
I encourage you to pause the video

00:00:31.640 --> 00:00:34.182
and think about this on your own.

00:00:34.182 --> 00:00:35.947
Well, let's just think
about each of these.

00:00:35.947 --> 00:00:38.977
If I were to drop a vertical
line own the center of A,

00:00:38.977 --> 00:00:40.904
it looks like A is symmetric about that.

00:00:40.904 --> 00:00:42.575
There are mirror images on the left

00:00:42.575 --> 00:00:44.061
and the right hand side.

00:00:44.061 --> 00:00:48.541
Same thing for the letter I,
same thing for the letter O,

00:00:48.541 --> 00:00:50.596
mirror images on the two sides,

00:00:50.596 --> 00:00:53.533
same thing for the letter U.

00:00:53.533 --> 00:00:56.319
Now, E gets interesting.

00:00:56.319 --> 00:00:58.963
We don't have a mirror image.

00:00:58.963 --> 00:01:00.403
If I were to take the
mirror image of this point

00:01:00.403 --> 00:01:01.331
right over here, we would get

00:01:01.331 --> 00:01:03.026
to this point right over there.

00:01:03.026 --> 00:01:04.826
If I were to take the
mirror image of that point

00:01:04.826 --> 00:01:06.601
right over there, well
that does get you to there,

00:01:06.601 --> 00:01:09.364
but this point over here,
there's no corresponding

00:01:09.364 --> 00:01:11.223
point on the other side here.

00:01:11.223 --> 00:01:12.837
If I were to go here, there's no

00:01:12.837 --> 00:01:14.860
corresponding point right over there.

00:01:14.860 --> 00:01:17.240
In order for E to have been symmetric,

00:01:17.240 --> 00:01:19.928
it should've either looked like this,

00:01:21.820 --> 00:01:24.786
where if I were to drop
a vertical line there,

00:01:24.786 --> 00:01:28.805
this would be symmetric,
or if it looked like this.

00:01:30.718 --> 00:01:35.329
But obviously this is not
what the letter E looks like.

00:01:35.329 --> 00:01:36.861
It does not look like
either of these symbols.

00:01:36.861 --> 00:01:38.763
It looks like this and this
one does not have a line

00:01:38.763 --> 00:01:41.702
of symmetry down its center axis.

00:01:41.702 --> 00:01:44.465
A vertical line cannot be drawn down

00:01:44.465 --> 00:01:47.459
its center to have a
line of symmetry here.

00:01:47.459 --> 00:01:50.443
This is the one that cannot be
folded along a vertical line

00:01:50.443 --> 00:01:54.079
so its left half would
coincide with its right half.