Finding the slope of a line from its graph

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:04.740
Find the slope of the
line in the graph.

00:00:04.740 --> 00:00:08.020
And just as a bit of a review,
slope is just telling us how

00:00:08.020 --> 00:00:10.490
steep a line is.

00:00:10.490 --> 00:00:19.850
And the best way to view it,
slope is equal to change in y

00:00:19.850 --> 00:00:21.510
over change in x.

00:00:25.510 --> 00:00:31.080
And for a line, this will
always be constant.

00:00:31.080 --> 00:00:33.940
And sometimes you might see it
written like this: you might

00:00:33.940 --> 00:00:36.590
see this triangle, that's a
capital delta, that means

00:00:36.590 --> 00:00:40.730
change in, change in
y over change in x.

00:00:40.730 --> 00:00:42.560
That's just a fancy way
of saying change in y

00:00:42.560 --> 00:00:43.740
over change in x.

00:00:43.740 --> 00:00:47.760
So let's see what this change
in y is for any change in x.

00:00:47.760 --> 00:00:51.010
So let's start at some point
that seems pretty reasonable

00:00:51.010 --> 00:00:54.810
to read from this table right
here, from this graph.

00:00:54.810 --> 00:00:57.560
So let's see, we're starting
here-- let me do it in a more

00:00:57.560 --> 00:00:59.160
vibrant color-- so let's
say we start at

00:00:59.160 --> 00:01:00.520
that point right there.

00:01:00.520 --> 00:01:02.350
And we want to go to another
point that's pretty

00:01:02.350 --> 00:01:04.379
straightforward to read,
so we can move to

00:01:04.379 --> 00:01:05.450
that point right there.

00:01:05.450 --> 00:01:08.340
We could literally pick any
two points on this line.

00:01:08.340 --> 00:01:11.490
I'm just picking ones that are
nice integer coordinates, so

00:01:11.490 --> 00:01:12.610
it's easy to read.

00:01:12.610 --> 00:01:15.510
So what is the change in y and
what is the change in x?

00:01:15.510 --> 00:01:17.540
So first let's look at
the change in x.

00:01:17.540 --> 00:01:21.750
So if we go from there
to there, what is

00:01:21.750 --> 00:01:23.510
the change in x?

00:01:23.510 --> 00:01:26.660
My change in x is
equal to what?

00:01:26.660 --> 00:01:28.340
Well, I can just count it out.

00:01:28.340 --> 00:01:31.480
I went 1 steps, 2
steps, 3 steps.

00:01:31.480 --> 00:01:33.620
My change in x is 3.

00:01:33.620 --> 00:01:35.400
And you could even see
it from the x values.

00:01:35.400 --> 00:01:40.060
If I go from negative 3
to 0, I went up by 3.

00:01:40.060 --> 00:01:42.105
So my change in x is 3.

00:01:42.105 --> 00:01:48.200
So let me write this, change in
x, delta x is equal to 3.

00:01:48.200 --> 00:01:50.860
And what's my change in y?

00:01:50.860 --> 00:01:53.200
Well, my change in y, I'm going
from negative 3 up to

00:01:53.200 --> 00:01:57.580
negative 1, or you could
just say 1, 2.

00:01:57.580 --> 00:02:05.260
So my change in y, is
equal to positive 2.

00:02:05.260 --> 00:02:06.360
So let me write that down.

00:02:06.360 --> 00:02:09.710
Change in y is equal to 2.

00:02:09.710 --> 00:02:13.450
So what is my change in
y for a change in x?

00:02:13.450 --> 00:02:19.560
Well, when my change in x was
3, my change in y is 2.

00:02:19.560 --> 00:02:21.870
So this is my slope.

00:02:21.870 --> 00:02:23.490
And one thing I want to do, I
want to show you that I could

00:02:23.490 --> 00:02:25.910
have really picked any
two points here.

00:02:25.910 --> 00:02:31.670
Let's say I didn't pick-- let me
clear this out-- let's say

00:02:31.670 --> 00:02:33.970
I didn't pick those two points,
let me pick some other

00:02:33.970 --> 00:02:35.700
points, and I'll even go in
a different direction.

00:02:35.700 --> 00:02:38.510
I want to show you that you're
going to get the same answer.

00:02:38.510 --> 00:02:42.440
Let's say I've used this as my
starting point, and I want to

00:02:42.440 --> 00:02:46.300
go all the way over there.

00:02:46.300 --> 00:02:48.640
Well, let's think about the
change in y first. So the

00:02:48.640 --> 00:02:55.960
change in y, I'm going down
by how many units?

00:02:55.960 --> 00:03:01.490
1, 2, 3, 4 units, so my change
in y, in this example, is

00:03:01.490 --> 00:03:03.000
negative 4.

00:03:03.000 --> 00:03:06.710
I went from 1 to negative
3, that's negative 4.

00:03:06.710 --> 00:03:08.400
That's my change in y.

00:03:08.400 --> 00:03:10.950
Change in y is equal
to negative 4.

00:03:10.950 --> 00:03:12.820
Now what is my change in x?

00:03:12.820 --> 00:03:16.320
Well I'm going from this point,
or from this x value,

00:03:16.320 --> 00:03:20.020
all the way-- let me do that in
a different color-- all the

00:03:20.020 --> 00:03:23.080
way back like this.

00:03:23.080 --> 00:03:24.880
So I'm going to the left, so
it's going to be a negative

00:03:24.880 --> 00:03:31.060
change in x, and I went 1,
2, 3, 4, 5, 6 units back.

00:03:31.060 --> 00:03:34.845
So my change in x is equal
to negative 6.

00:03:38.380 --> 00:03:41.470
And you can even see I started
it at x is equal to 3, and I

00:03:41.470 --> 00:03:43.910
went all the way to x is
equal to negative 3.

00:03:43.910 --> 00:03:46.430
That's a change of negative 6.

00:03:46.430 --> 00:03:49.050
I went 6 to the left, or
a change of negative 6.

00:03:49.050 --> 00:03:52.270
So what is my change in
y over change in x?

00:03:52.270 --> 00:03:57.670
My change in y over change in x
is equal to negative 4 over

00:03:57.670 --> 00:03:58.790
negative 6.

00:03:58.790 --> 00:04:01.370
The negatives cancel out
and what's 4 over 6?

00:04:01.370 --> 00:04:03.380
Well, that's just 2 over 3.

00:04:03.380 --> 00:04:05.610
So it's the same value, you just
have to be consistent.

00:04:05.610 --> 00:04:08.200
If this is my start point,
I went down 4, and

00:04:08.200 --> 00:04:09.930
then I went back 6.

00:04:09.930 --> 00:04:11.330
Negative 4 over negative 6.

00:04:11.330 --> 00:04:13.800
If I viewed this as my starting
point, I could say

00:04:13.800 --> 00:04:21.040
that I went up 4, so it would
be a change in y would be 4,

00:04:21.040 --> 00:04:26.480
and then my change
in x would be 6.

00:04:26.480 --> 00:04:29.640
And either way, once again,
change in y over change in x

00:04:29.640 --> 00:04:32.370
is going to be 4 over 6, 2/3.

00:04:32.370 --> 00:04:34.440
So no matter which point you
choose, as long as you kind of

00:04:34.440 --> 00:04:36.570
think about it in a consistent
way, you're going to get the

00:04:36.570 --> 00:04:38.870
same value for slope.


Exploring nonlinear relationships

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:04.410
The graph below shows the
depreciation of a motorcycle's

00:00:04.410 --> 00:00:06.000
value over time.

00:00:06.000 --> 00:00:08.900
And they shows us the graph
and they say during which

00:00:08.900 --> 00:00:12.480
two-year span does the most
rapid-- let me underline

00:00:12.480 --> 00:00:15.220
that-- the most rapid
depreciation occur.

00:00:15.220 --> 00:00:18.290
So depreciation just means
loss in value.

00:00:18.290 --> 00:00:20.970
And this shows the motorcycle's
value.

00:00:20.970 --> 00:00:22.250
You can just eyeball
this here.

00:00:22.250 --> 00:00:24.360
Let's just look at what
this chart's saying.

00:00:24.360 --> 00:00:27.690
This is saying right when the
motorcycle is brand new, when

00:00:27.690 --> 00:00:30.840
it's zero days old, zero
years old, I should

00:00:30.840 --> 00:00:33.750
say, it is worth $10,000.

00:00:33.750 --> 00:00:36.560
And then when it's one year old,
it's looks like it's a

00:00:36.560 --> 00:00:38.630
little over $6,000.

00:00:38.630 --> 00:00:40.100
And when it's two years
old, it looks

00:00:40.100 --> 00:00:43.280
like it's about $5,000.

00:00:43.280 --> 00:00:46.450
Then at three years,
it's about $4,000.

00:00:46.450 --> 00:00:47.900
And so on and so forth.

00:00:47.900 --> 00:00:50.560
So this curve shows how
it depreciates, how

00:00:50.560 --> 00:00:52.180
it loses its value.

00:00:52.180 --> 00:00:53.760
And they're saying when
is the most rapid

00:00:53.760 --> 00:00:55.190
depreciation occurring?

00:00:55.190 --> 00:00:57.120
And they don't want to know
what year, they want a

00:00:57.120 --> 00:01:02.290
two-year span when we have the
most rapid depreciation.

00:01:02.290 --> 00:01:04.200
Now, one thing you can look at
here, you can just kind of

00:01:04.200 --> 00:01:04.760
eyeball it.

00:01:04.760 --> 00:01:07.130
You could say, look, the
beginning of this curve is

00:01:07.130 --> 00:01:08.500
this steepest part.

00:01:08.500 --> 00:01:11.350
It's steepest over these two
years than over any other

00:01:11.350 --> 00:01:12.180
two-year period.

00:01:12.180 --> 00:01:15.680
Out here, it's kind of a very
shallow-- it's almost flat in

00:01:15.680 --> 00:01:18.580
terms of how quickly it's losing
value, so it's not

00:01:18.580 --> 00:01:20.520
losing value too quickly.

00:01:20.520 --> 00:01:23.900
So if you just eyeballed it, you
would just say, well, the

00:01:23.900 --> 00:01:27.090
most rapid depreciation is
occurring over the first two

00:01:27.090 --> 00:01:30.490
years, from zero to two years
up to this point because you

00:01:30.490 --> 00:01:33.590
can see the curve is getting
less and less steep the entire

00:01:33.590 --> 00:01:37.290
time, but it's most steep over
those first two years.

00:01:37.290 --> 00:01:39.310
Now, if you didn't want to just
eyeball it and really

00:01:39.310 --> 00:01:43.390
verify that it loses the most
value over the first two years

00:01:43.390 --> 00:01:46.590
than any other two-year
period, let's actually

00:01:46.590 --> 00:01:48.010
calculate it.

00:01:48.010 --> 00:01:56.070
So after from zero-- so let me
write two-year period and

00:01:56.070 --> 00:02:00.750
let's write lost value, or we
could call it depreciation,

00:02:00.750 --> 00:02:01.300
either way.

00:02:01.300 --> 00:02:05.300
So from zero to two years, we
went from $10,000 to what

00:02:05.300 --> 00:02:07.720
looks like about
$5,000, right?

00:02:07.720 --> 00:02:09.919
That looks right at
about $5,000.

00:02:09.919 --> 00:02:14.340
So we lost $5,000
of lost value.

00:02:14.340 --> 00:02:16.680
Then if we go from one to three
years, what happened?

00:02:16.680 --> 00:02:19.230
One to three years, that's the
other two-year period.

00:02:19.230 --> 00:02:22.500
You go from one to three years,
we started at it looks

00:02:22.500 --> 00:02:26.560
like about $6,100, maybe
$6,200 or so,

00:02:26.560 --> 00:02:28.680
and we went to $4,000.

00:02:28.680 --> 00:02:35.550
So we lost about $2,200, so
definitely less than what we

00:02:35.550 --> 00:02:37.770
lost in the first two years.

00:02:37.770 --> 00:02:41.740
Then if we look at the second
through the fourth year, we

00:02:41.740 --> 00:02:47.850
went from about $5,000 to about
$4,000, so we lost about

00:02:47.850 --> 00:02:54.370
$1,000 in value, maybe
a little less.

00:02:54.370 --> 00:02:58.690
Yeah, that's about what we went
from years two to four,

00:02:58.690 --> 00:03:03.390
about $5,000, or maybe a little
under $4,000, so I'll

00:03:03.390 --> 00:03:06.340
give it a little bit more value,
so maybe it's like

00:03:06.340 --> 00:03:11.160
$3,800 right there, so maybe
on the order of $1,200.

00:03:11.160 --> 00:03:18.350
And then if you go from year
three to year five, we're

00:03:18.350 --> 00:03:23.834
going from about $4,000 to looks
like about $3,600 if I

00:03:23.834 --> 00:03:27.770
had to eyeball it, so it looks
like we lost about $400.

00:03:27.770 --> 00:03:30.350
So if anything, as we go further
and further in time,

00:03:30.350 --> 00:03:33.560
each two-year period we're
losing less value.

00:03:33.560 --> 00:03:36.380
So then finally if we go from
four to six years, it looks

00:03:36.380 --> 00:03:42.000
like we're at about $3,800 and
we end up at about-- I don't

00:03:42.000 --> 00:03:43.360
know what this value
looks like.

00:03:43.360 --> 00:03:48.730
It's about-- I would say it's
maybe $3,500 or maybe we're

00:03:48.730 --> 00:03:50.640
going from $3,700.

00:03:50.640 --> 00:03:52.740
We're losing about $300.

00:03:52.740 --> 00:03:55.930
It's shallower than
it was before.

00:03:55.930 --> 00:03:59.670
So the most rapid depreciation
occurs in the first two years.

00:03:59.670 --> 00:04:01.630
And you didn't have to go
through this exercise.

00:04:01.630 --> 00:04:03.210
You could have just looked at
the chart and said, hey, look

00:04:03.210 --> 00:04:06.460
how steep it is the first two
years compared to all of the

00:04:06.460 --> 00:04:08.420
years after that.


Recognizing linear functions

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:03.690
Deirdre is working with a
function that contains the

00:00:03.690 --> 00:00:05.060
following points.

00:00:05.060 --> 00:00:07.350
These are the x values,
these are y values.

00:00:07.350 --> 00:00:12.460
They ask us, is this function
linear or non-linear?

00:00:12.460 --> 00:00:16.379
So linear functions, the way to
tell them is for any given

00:00:16.379 --> 00:00:19.610
change in x, is the change
in y always going

00:00:19.610 --> 00:00:20.760
to be the same value.

00:00:20.760 --> 00:00:26.180
For example, for any one-step
change in x, is the change in

00:00:26.180 --> 00:00:27.460
y always going to be 3?

00:00:27.460 --> 00:00:28.940
Is it always going to be 5?

00:00:28.940 --> 00:00:31.450
If it's always going to be the
same value, you're dealing

00:00:31.450 --> 00:00:32.670
with a linear function.

00:00:32.670 --> 00:00:36.570
If for each change in x--so over
here x is always changing

00:00:36.570 --> 00:00:40.380
by 1, so since x is always
changing by 1, the change in

00:00:40.380 --> 00:00:42.800
y's have to always
be the same.

00:00:42.800 --> 00:00:43.920
If they're not, then
we're dealing with

00:00:43.920 --> 00:00:45.200
a non-linear function.

00:00:45.200 --> 00:00:47.280
We can actually show
that plotting out.

00:00:47.280 --> 00:00:50.150
If the changes in x-- we're
going by different values, if

00:00:50.150 --> 00:00:53.190
this went from 1 to 2 and then
2 to 4-- what you'd want to

00:00:53.190 --> 00:00:57.310
do, then, is divide the change
in y by the change in x, and

00:00:57.310 --> 00:00:59.030
that should always
be a constant.

00:00:59.030 --> 00:01:00.280
In fact, let me write
that down.

00:01:03.580 --> 00:01:11.660
If something is linear, then
the change in y over the

00:01:11.660 --> 00:01:17.240
change in x always constant.

00:01:21.760 --> 00:01:23.960
Now, in this example,
the change in x's

00:01:23.960 --> 00:01:25.480
are always 1, right?

00:01:25.480 --> 00:01:28.890
We go from 1 to 2, 2 to
3, 3 to 4, 4 to 5.

00:01:28.890 --> 00:01:34.060
So in this example, the change
in x is always going to be 1.

00:01:34.060 --> 00:01:38.030
So in order for this function to
be linear, our change in y

00:01:38.030 --> 00:01:40.180
needs to be constant because
we're just going to take that

00:01:40.180 --> 00:01:41.310
and divide it by 1.

00:01:41.310 --> 00:01:43.910
So let's see if our change
in y is constant.

00:01:43.910 --> 00:01:48.480
When we go from 11 to
14, we go up by 3.

00:01:48.480 --> 00:01:51.500
When we go from 14 to 19, we go
up by 5, so I already see

00:01:51.500 --> 00:01:52.980
that it is not constant.

00:01:52.980 --> 00:01:55.190
We didn't go up by 3 this
time, we went up by 5.

00:01:55.190 --> 00:01:58.100
And here, we go up by 7.

00:01:58.100 --> 00:02:01.230
And here, we're going up by 9.

00:02:01.230 --> 00:02:04.780
So we're actually going up by
increasing amounts, so we're

00:02:04.780 --> 00:02:08.380
definitely dealing with
a non-linear function.

00:02:08.380 --> 00:02:10.780
And we can see that if
we graph it out.

00:02:10.780 --> 00:02:13.390
So let me draw-- I'll do
a rough graph here.

00:02:13.390 --> 00:02:17.410
So let me make that my vertical
axis, my y-axis.

00:02:17.410 --> 00:02:19.510
And we go all the
way up to 35.

00:02:19.510 --> 00:02:26.040
So I'll just do 10, 20, 30.

00:02:26.040 --> 00:02:29.470
Actually, I can it do a little
bit more granularly than that.

00:02:29.470 --> 00:02:44.650
I could do 5, 10, 15, 20,
25, 30, and then 35.

00:02:44.650 --> 00:02:46.690
And then our values
go 1 through 5.

00:02:46.690 --> 00:02:48.970
I'll do it on this
axis right here.

00:02:48.970 --> 00:02:52.010
They're not obviously the exact
same scale, so I'll do

00:02:52.010 --> 00:02:58.030
1, 2, 3, 4, and 5.

00:02:58.030 --> 00:02:59.610
So let's plot these points.

00:02:59.610 --> 00:03:03.800
So the first point is 1, 11,
when x is 1, y is 11.

00:03:03.800 --> 00:03:04.760
This is our x-axis.

00:03:04.760 --> 00:03:08.600
When x is 1, y is 11, that's
right about there.

00:03:08.600 --> 00:03:15.160
When x is 2, y is 14, that's
right about there.

00:03:15.160 --> 00:03:23.570
When x is 3, y is 19,
right about there.

00:03:23.570 --> 00:03:32.200
When x is 4, y is 26,
right about there.

00:03:32.200 --> 00:03:40.080
And then finally, when x is 5,
y is 35, right up there.

00:03:40.080 --> 00:03:41.810
So you can immediately
see that this is not

00:03:41.810 --> 00:03:44.480
tracing out a line.

00:03:44.480 --> 00:03:49.110
If this was a linear function,
then all the points would be

00:03:49.110 --> 00:03:51.250
on a line that looks something
like that.

00:03:51.250 --> 00:03:53.530
That's why it's called
a linear function.

00:03:53.530 --> 00:03:56.150
In this case, it's not,
it's non-linear.

00:03:56.150 --> 00:03:59.320
The rate of increase as
x changes is going up.


Modeling with linear equations example 1

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:03.630
On Monday morning, there
were 12 inches

00:00:03.630 --> 00:00:05.290
of snow on the ground.

00:00:05.290 --> 00:00:08.130
The weather warmed up, and
by Tuesday morning,

00:00:08.130 --> 00:00:09.520
2 inches had melted.

00:00:09.520 --> 00:00:11.440
All right, so we'll
have 10 left.

00:00:11.440 --> 00:00:14.350
2 more inches melted by
Wednesday morning.

00:00:14.350 --> 00:00:16.980
This pattern continued
throughout the week until no

00:00:16.980 --> 00:00:18.300
more snow was left.

00:00:18.300 --> 00:00:21.280
So they're essentially saying
that we had 12 inches of snow

00:00:21.280 --> 00:00:24.790
on the ground on Monday and that
every day after that, two

00:00:24.790 --> 00:00:25.640
inches melted.

00:00:25.640 --> 00:00:28.220
So after Tuesday, you'd have
10 inches, and after

00:00:28.220 --> 00:00:29.780
Wednesday, you'd have
eight inches, and

00:00:29.780 --> 00:00:31.220
that pattern continued.

00:00:31.220 --> 00:00:34.640
And what they say is create an
equation and a graph to show

00:00:34.640 --> 00:00:37.790
the relationship between the day
and the amount of snow on

00:00:37.790 --> 00:00:38.330
the ground.

00:00:38.330 --> 00:00:41.670
So let's define a variable that
tells us how far away we

00:00:41.670 --> 00:00:42.670
are from Monday.

00:00:42.670 --> 00:00:49.010
So let's let x equal
days after Monday.

00:00:57.040 --> 00:01:03.050
And then let y be equal to
inches of snow on the ground.

00:01:11.560 --> 00:01:17.110
So, one way to think about it
is, OK, when x is 0, when

00:01:17.110 --> 00:01:19.905
we're on Monday, when we're 0
days after Monday, we're going

00:01:19.905 --> 00:01:23.880
to have 12 inches of snow on
the ground, and every day

00:01:23.880 --> 00:01:27.280
after that, we're going
to lose two inches.

00:01:27.280 --> 00:01:30.550
So if we're on Tuesday, we're
going to have 2 inches times

00:01:30.550 --> 00:01:33.975
1, because Tuesday is one day,
so if x is 1, that means we're

00:01:33.975 --> 00:01:34.890
on Tuesday.

00:01:34.890 --> 00:01:40.130
If x is 2, that means we're 2
times 2, we've lost 4 inches,

00:01:40.130 --> 00:01:42.770
which is what the case
is on Wednesday.

00:01:42.770 --> 00:01:47.280
So this is our equation for the
relationship between the

00:01:47.280 --> 00:01:49.370
day and the amount of
snow on the ground.

00:01:49.370 --> 00:01:54.885
x is the day, how many days
after Monday, and then y is

00:01:54.885 --> 00:01:57.440
the inches of the snow
left on the ground.

00:01:57.440 --> 00:01:59.670
We start with 12, and
then every day we

00:01:59.670 --> 00:02:02.060
lose exactly two inches.

00:02:02.060 --> 00:02:03.310
Now let's graph this.

00:02:06.040 --> 00:02:11.009
So I'll make my vertical axis
the y-axis, that's inches on

00:02:11.009 --> 00:02:11.780
the ground.

00:02:11.780 --> 00:02:17.420
y is equal to inches
left on the ground.

00:02:17.420 --> 00:02:22.400
And then the horizontal axis,
that is our x-axis-- let me

00:02:22.400 --> 00:02:31.940
scroll down a little bit-- this
is days after Monday.

00:02:35.200 --> 00:02:40.790
And so we have 0 days after
Monday, we have 1, 2,

00:02:40.790 --> 00:02:46.000
3, 4, 5, and 6.

00:02:46.000 --> 00:02:51.630
And then on Monday, which is
exactly 0 days after Monday,

00:02:51.630 --> 00:02:54.380
that is Monday, we have 12
inches on the ground.

00:02:54.380 --> 00:02:59.970
So I'll do it up here, so we
have 12 inches on the ground

00:02:59.970 --> 00:03:00.210
right there.

00:03:00.210 --> 00:03:02.740
And actually, I could do
a table if you like.

00:03:02.740 --> 00:03:04.420
Let me draw this.

00:03:04.420 --> 00:03:11.610
So if we do x and y, this is
the days after Monday, so

00:03:11.610 --> 00:03:16.050
there's 0, 1, 2, 3, 4, 5, 6.

00:03:16.050 --> 00:03:19.730
And then on the first day, we
have 12 inches, on Monday, 0

00:03:19.730 --> 00:03:20.770
days after Monday.

00:03:20.770 --> 00:03:22.610
Then we lose two inches
each day.

00:03:22.610 --> 00:03:28.600
On day 1 we have 10, day
2, 8, 6, 4, 2, 0.

00:03:28.600 --> 00:03:29.690
So let's plot these points.

00:03:29.690 --> 00:03:33.240
We already plotted 0, 12
in that blue color.

00:03:33.240 --> 00:03:35.820
Now let's plot 1, 10.

00:03:35.820 --> 00:03:38.505
1, 10 is right about there.

00:03:38.505 --> 00:03:41.690
It'll be right over there.

00:03:41.690 --> 00:03:45.500
Then we can plot 2, 8.

00:03:45.500 --> 00:03:47.730
So 2 days after Monday,
we have 8

00:03:47.730 --> 00:03:50.450
inches left on the ground.

00:03:50.450 --> 00:03:53.580
So this is on Wednesday,
so that's 8 inches.

00:03:53.580 --> 00:04:00.650
And then on 3 days after Monday,
we have 6 inches on

00:04:00.650 --> 00:04:01.900
the ground.

00:04:05.950 --> 00:04:08.340
You can see that a line
is forming here.

00:04:08.340 --> 00:04:12.810
And then if we go to 4 days
after Monday, we have 4 inches

00:04:12.810 --> 00:04:14.060
on the ground.

00:04:16.809 --> 00:04:18.890
So that is 4.

00:04:18.890 --> 00:04:22.850
And then 5 days after
Monday, we have 2

00:04:22.850 --> 00:04:24.100
inches on the ground.

00:04:27.340 --> 00:04:31.350
And then finally, on the sixth
day, 6 days after Monday-- so

00:04:31.350 --> 00:04:34.650
what are we at, Sunday now-- we
are going to have no inches

00:04:34.650 --> 00:04:36.550
on the ground.

00:04:36.550 --> 00:04:38.450
So that's that right there.

00:04:38.450 --> 00:04:41.580
And you can see that there's
this line that formed, because

00:04:41.580 --> 00:04:43.720
this is a linear relationship.

00:04:43.720 --> 00:04:45.230
It looks a little curvy because
I didn't draw it

00:04:45.230 --> 00:04:47.690
perfectly, but that is a line.

00:04:47.690 --> 00:04:48.660
So we've done everything.

00:04:48.660 --> 00:04:50.010
We've created the equation.

00:04:50.010 --> 00:04:52.030
We start with 12 inches,
every day after

00:04:52.030 --> 00:04:53.500
that we lose two inches.

00:04:53.500 --> 00:04:57.050
And we showed a graph that
depicts the relationship.


Basic Linear Function

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:04.530
A photo finishing store charges
customers a rate of

00:00:04.530 --> 00:00:08.970
$0.29 per photo to
print pictures.

00:00:08.970 --> 00:00:11.140
For new customers, the
store offers a

00:00:11.140 --> 00:00:15.740
one-time discount of $3.

00:00:15.740 --> 00:00:18.850
Write a function representing
the amount that a new customer

00:00:18.850 --> 00:00:23.020
would have to pay to have x
number of photos printed.

00:00:23.020 --> 00:00:25.590
So x is the number of
photos printed.

00:00:25.590 --> 00:00:29.190
So just a regular customer would
have to pay the number

00:00:29.190 --> 00:00:32.040
of photos times $0.29 a photo.

00:00:32.040 --> 00:00:35.290
So if we just said, let y be
the amount that a customer

00:00:35.290 --> 00:00:39.580
would have to pay, a regular
customer would have to pay x

00:00:39.580 --> 00:00:41.640
times $0.29 per photo.

00:00:41.640 --> 00:00:46.750
So we'd write 0.29-- we could
put the dollar sign there if

00:00:46.750 --> 00:00:50.490
you want-- or 0.29-- we're
assuming everything is

00:00:50.490 --> 00:00:51.590
dollars-- x.

00:00:51.590 --> 00:00:54.080
That's how much a regular
customer would have to pay.

00:00:54.080 --> 00:00:56.525
But they're asking us a first
time, a new customer.

00:00:56.525 --> 00:00:58.830
And they say that the new
customer gets a one-time

00:00:58.830 --> 00:01:00.520
discount of $3.

00:01:00.520 --> 00:01:06.040
So a new customer will
pay this minus $3.

00:01:06.040 --> 00:01:11.260
So this right here is pretty
much a simplified version of a

00:01:11.260 --> 00:01:14.020
function that would describe
what a new customer would pay.

00:01:14.020 --> 00:01:16.400
If you wanted to get fancy,
this is a little bit

00:01:16.400 --> 00:01:20.850
inaccurate when the customer
ordered less than 10 photos

00:01:20.850 --> 00:01:23.540
because, let's say the customer
orders 10 photos.

00:01:23.540 --> 00:01:30.300
Then x is going to be 10 times
$0.29 is $2.90 minus 3 is

00:01:30.300 --> 00:01:31.790
negative $0.10.

00:01:31.790 --> 00:01:34.960
The customer is not going to
pay negative $0.10, which

00:01:34.960 --> 00:01:37.760
would be like the store paying
them $0.10 for them to print

00:01:37.760 --> 00:01:40.390
10 photos, the customer's not
going to pay anything.

00:01:40.390 --> 00:01:43.110
So what we could do if we wanted
to make this a little

00:01:43.110 --> 00:01:57.740
bit fancier, we could say that a
new customer would pay-- and

00:01:57.740 --> 00:02:10.110
then we can say that's y, so y
is equal to 0.29x minus 3 if x

00:02:10.110 --> 00:02:14.420
is greater than 10, if the
guy or the gal orders

00:02:14.420 --> 00:02:16.160
more than 10 photos.

00:02:16.160 --> 00:02:18.100
It's 0 otherwise.

00:02:21.810 --> 00:02:22.880
Then they don't pay anything.

00:02:22.880 --> 00:02:24.760
If it's less than 10 photos,
they pay nothing.

00:02:24.760 --> 00:02:27.090
It's not like the store's going
to pay them, because if

00:02:27.090 --> 00:02:30.640
you're less than 10, or 10 or
less, actually, then this is

00:02:30.640 --> 00:02:32.750
going to become negative and
we don't want a situation

00:02:32.750 --> 00:02:33.260
where it's negative.

00:02:33.260 --> 00:02:35.340
So they're either going to pay
nothing, but if they get more

00:02:35.340 --> 00:02:38.600
than 10 photos, then they're
going to pay based on this

00:02:38.600 --> 00:02:40.800
function right there.


Basic rate problem

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:04.720
We're told that as part of an
experiment about train speed,

00:00:04.720 --> 00:00:07.860
4 different train conductors
measured the distance that

00:00:07.860 --> 00:00:10.380
they covered over a certain
amount of time

00:00:10.380 --> 00:00:11.650
during a recent journey.

00:00:11.650 --> 00:00:14.300
So these are the 4 trains.

00:00:14.300 --> 00:00:18.260
This is how long it took them
to go this many miles.

00:00:18.260 --> 00:00:21.460
However, they all use different
time intervals.

00:00:21.460 --> 00:00:23.910
Yeah, you see this guy, he
did it over half an hour.

00:00:23.910 --> 00:00:25.740
This guy did it over 2 hours.

00:00:25.740 --> 00:00:28.890
And they're having a difficult
time comparing their data.

00:00:28.890 --> 00:00:33.080
According to the data provided,
were all 4 trains

00:00:33.080 --> 00:00:38.330
moving at the same speed
during the experiment?

00:00:38.330 --> 00:00:40.660
So we just have to figure out
what was the speed of all of

00:00:40.660 --> 00:00:43.790
these 4 trains based on
how far they went in

00:00:43.790 --> 00:00:45.300
this amount of time.

00:00:45.300 --> 00:00:48.170
And just to remind us, you might
already know this, but

00:00:48.170 --> 00:00:52.050
you know that distance is equal
to rate times time.

00:00:52.050 --> 00:00:55.300
Or if you were to divide both
sides of this equation by

00:00:55.300 --> 00:00:57.610
time, I could do that.

00:00:57.610 --> 00:01:01.770
If you divide both sides by
time, the times cancel out and

00:01:01.770 --> 00:01:04.295
you get rate is equal to
distance divided by time.

00:01:08.430 --> 00:01:11.850
Rate is the same thing as speed,
and it makes sense that

00:01:11.850 --> 00:01:13.720
it's distance divided by time.

00:01:13.720 --> 00:01:18.400
You've heard miles per hour
or meter per second or

00:01:18.400 --> 00:01:20.440
kilometers per hour.

00:01:20.440 --> 00:01:24.660
Rate speed is distance
divided by time.

00:01:24.660 --> 00:01:26.100
So we can just use this.

00:01:26.100 --> 00:01:29.270
We can just divide the distance
by the time for each

00:01:29.270 --> 00:01:32.250
of these trains and figure
out what their speed was.

00:01:32.250 --> 00:01:33.510
So let's do that.

00:01:33.510 --> 00:01:36.710
So let's write their speeds.

00:01:36.710 --> 00:01:41.160
I'll make another column for
their speed right here.

00:01:41.160 --> 00:01:42.260
So what's this first guy?

00:01:42.260 --> 00:01:46.650
He went 27.5 miles
in 0.5 hours.

00:01:46.650 --> 00:01:50.620
So once again, let's divide the
distance for this train 1.

00:01:50.620 --> 00:01:54.815
He went 27.5 miles in 0.5.

00:01:54.815 --> 00:01:58.470
So if you divide something by
0.5, this is the same thing as

00:01:58.470 --> 00:02:01.810
27.5 times 2.

00:02:01.810 --> 00:02:04.790
Let me do it this
way, times 2.

00:02:04.790 --> 00:02:07.840
What's 27.5 times 2?

00:02:07.840 --> 00:02:13.920
It is 55, and it is
miles per hour.

00:02:13.920 --> 00:02:19.120
So his speed is 55
miles per hour.

00:02:19.120 --> 00:02:21.610
Now, this guy, this is pretty
straightforward.

00:02:21.610 --> 00:02:25.620
He went 55 miles in
exactly 1 hour.

00:02:25.620 --> 00:02:28.380
The distance divided
by time is 55.

00:02:28.380 --> 00:02:30.020
So this guy's pretty
straightforward.

00:02:30.020 --> 00:02:31.630
He's also 55.

00:02:31.630 --> 00:02:32.990
And you can put it
in the formula.

00:02:32.990 --> 00:02:34.780
55 is the distance.

00:02:34.780 --> 00:02:36.320
The time is 1 hour.

00:02:36.320 --> 00:02:37.450
55 over 1.

00:02:37.450 --> 00:02:39.440
55 miles per hour.

00:02:39.440 --> 00:02:42.600
Now this guy went 82-1/2
miles in 1-1/2 hours.

00:02:42.600 --> 00:02:43.840
Let's see what that is.

00:02:43.840 --> 00:02:49.210
So he went 82-1/2 miles
in 1.5 hours.

00:02:49.210 --> 00:02:50.530
So let's figure out
what this is.

00:02:50.530 --> 00:02:53.120
And I'll have to do a little bit
of decimal long division

00:02:53.120 --> 00:02:54.020
to do this.

00:02:54.020 --> 00:03:01.950
So we have 1.5 goes into 82.5.

00:03:01.950 --> 00:03:05.020
Let's multiply both of these
numbers by 10, essentially

00:03:05.020 --> 00:03:07.810
shift their decimals one
over, so shift that

00:03:07.810 --> 00:03:08.730
decimal over here.

00:03:08.730 --> 00:03:14.360
So it becomes 15 goes into
825 how many times?

00:03:14.360 --> 00:03:18.000
So 15 doesn't go
into 8 at all.

00:03:18.000 --> 00:03:21.540
It goes into 82 five times.

00:03:21.540 --> 00:03:25.040
5 times 5 is 25, carry the 2.

00:03:25.040 --> 00:03:29.460
5 times 1 is 5, plus 2 is 70.

00:03:29.460 --> 00:03:30.590
Subtract 75.

00:03:30.590 --> 00:03:35.310
82 minus 75 is just
a 7, right?

00:03:35.310 --> 00:03:37.560
And then you bring
down this 5.

00:03:37.560 --> 00:03:40.570
15 goes into 75 five
times exactly.

00:03:40.570 --> 00:03:43.680
5 times 15 is 75.

00:03:43.680 --> 00:03:45.620
Subtract, no remainder.

00:03:45.620 --> 00:03:49.700
So 82.5 miles in an hour
and a half is also

00:03:49.700 --> 00:03:52.620
55 miles per hour.

00:03:52.620 --> 00:03:56.280
And then finally, if you do 110
miles 2 hours, 110 divided

00:03:56.280 --> 00:03:58.990
by 2 is 55.

00:03:58.990 --> 00:04:03.930
So they did indeed all go, or
they all did move at the same

00:04:03.930 --> 00:04:05.380
speed during the experiment.


Direct variation 1

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:04.370
We're told that the total cost
of filling up your car with

00:00:04.370 --> 00:00:10.540
gas varies directly with the
number of gallons of gasoline

00:00:10.540 --> 00:00:11.880
you are purchasing.

00:00:11.880 --> 00:00:15.800
So this first statement tells
us that if x is equal to the

00:00:15.800 --> 00:00:28.790
number of gallons purchased, and
y is equal to the cost of

00:00:28.790 --> 00:00:35.300
filling up the car, this first
statement tells us that y

00:00:35.300 --> 00:00:39.450
varies directly with the number
of gallons, with x.

00:00:39.450 --> 00:00:44.850
So that means that y is equal
to some constant, we'll just

00:00:44.850 --> 00:00:47.050
call that k, times x.

00:00:47.050 --> 00:00:49.310
This is what it means
to vary directly.

00:00:49.310 --> 00:00:51.670
If x goes up, y will go up.

00:00:51.670 --> 00:00:54.150
We don't know what the rate
is. k tells us the rate.

00:00:54.150 --> 00:00:57.100
If x goes down, y
will be down.

00:00:57.100 --> 00:00:58.850
Now, they give us more
information, and this will

00:00:58.850 --> 00:01:01.260
help us figure out what k is.

00:01:01.260 --> 00:01:07.570
If a gallon of gas costs $2.25,
how many gallons could

00:01:07.570 --> 00:01:09.700
you purchase for $18?

00:01:09.700 --> 00:01:13.350
So if x is equal to 1-- this
statement up here, a gallon of

00:01:13.350 --> 00:01:17.090
gas-- that tells us if we get 1
gallon, if x is equal to 1,

00:01:17.090 --> 00:01:20.060
then y is $2.25, right?

00:01:20.060 --> 00:01:21.390
y is what it costs.

00:01:21.390 --> 00:01:25.300
They tell us 1 gallon costs
$2.25, so you could write it

00:01:25.300 --> 00:01:33.190
right here, $2.25 is equal
to k times x, times 1.

00:01:33.190 --> 00:01:35.190
Well, I didn't even have to
write the times 1 there.

00:01:35.190 --> 00:01:37.330
It's essentially telling
us exactly what the

00:01:37.330 --> 00:01:38.900
rate is, what k is.

00:01:38.900 --> 00:01:40.260
We don't even have to
write that 1 there.

00:01:40.260 --> 00:01:42.880
k is equal to 2.25.

00:01:42.880 --> 00:01:44.450
That's what this told
us right there.

00:01:44.450 --> 00:01:51.220
So the equation, how y varies
with x, is y is equal to

00:01:51.220 --> 00:01:56.710
2.25x, where x is the number
of gallons we purchase.

00:01:56.710 --> 00:01:59.650
y is the cost of that
purchase, so

00:01:59.650 --> 00:02:01.390
it's $2.25 a gallon.

00:02:01.390 --> 00:02:06.320
And then they ask us, how
many gallons could you

00:02:06.320 --> 00:02:08.590
purchase for $18?

00:02:08.590 --> 00:02:12.690
So $18 is going to be our total
cost. It is y cost of

00:02:12.690 --> 00:02:13.990
filling the car.

00:02:13.990 --> 00:02:19.690
So 18 is going to be
equal to 2.25x.

00:02:19.690 --> 00:02:25.020
Now if we want to solve for x,
we can divide both sides by

00:02:25.020 --> 00:02:29.730
2.25, so let's do that.

00:02:29.730 --> 00:02:36.310
You divide 18 by 2.25,
divide 2.25x by 2.25,

00:02:36.310 --> 00:02:37.610
and what do we get?

00:02:37.610 --> 00:02:39.520
Let me scroll down
a little bit.

00:02:39.520 --> 00:02:43.650
The right-hand side, the 2.25's
cancel out, you get x.

00:02:43.650 --> 00:02:48.020
And then what is 18
divided by 2.25?

00:02:48.020 --> 00:02:49.090
So let me write this down.

00:02:49.090 --> 00:02:51.210
So first of all, I just like to
think of it as a fraction.

00:02:51.210 --> 00:02:57.930
2.25 is the same thing-- let me
write over here-- 2.25 is

00:02:57.930 --> 00:03:04.160
equal to 2 and 1/4, which is
the same thing as 9 over 4.

00:03:04.160 --> 00:03:12.480
So 18 divided by 2.25 is equal
to 18 divided by 9 over 4,

00:03:12.480 --> 00:03:18.960
which is equal to 18 times
4 over 9, or 18 over 1

00:03:18.960 --> 00:03:21.160
times 4 over 9.

00:03:21.160 --> 00:03:26.540
And let's see, 18 divided by 9
is 2, 9 divided by 9 is 1.

00:03:26.540 --> 00:03:29.330
That simplifies pretty
nicely into 8.

00:03:29.330 --> 00:03:37.915
So 18 divided by 2.25 is 8, so
we can buy 8 gallons for $18.


Domain and range 2

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:03.980
We are asked to represent the
following function on the

00:00:03.980 --> 00:00:04.820
coordinate plane.

00:00:04.820 --> 00:00:08.560
And they give us domain, and
then they give us a range, a

00:00:08.560 --> 00:00:11.250
set of values that our
function can take on.

00:00:11.250 --> 00:00:13.970
And the way the question has
been phrased, I'm assuming

00:00:13.970 --> 00:00:19.820
they assume that the negative 8
maps to the 4, the negative

00:00:19.820 --> 00:00:24.260
6 maps to the 3, the negative 4
maps to the 2, the negative

00:00:24.260 --> 00:00:27.310
2 maps to 1, and the
0 maps to 0.

00:00:27.310 --> 00:00:30.120
Because you could have another
function with this exact same

00:00:30.120 --> 00:00:33.090
domain, and this exact same
range, where the mapping is

00:00:33.090 --> 00:00:35.480
different, where negative
8 maps to 3, and

00:00:35.480 --> 00:00:36.890
negative 6 maps to 4.

00:00:36.890 --> 00:00:39.650
So this domain and this
range could be for

00:00:39.650 --> 00:00:40.880
many different functions.

00:00:40.880 --> 00:00:44.070
But when they say represent the
following function, I'm

00:00:44.070 --> 00:00:47.470
assuming that they want us to
assume that they're kind of in

00:00:47.470 --> 00:00:48.690
order, the pairing.

00:00:48.690 --> 00:00:51.600
So let's just graph this
function, or represent it

00:00:51.600 --> 00:00:53.450
using their terminology.

00:00:53.450 --> 00:00:56.130
So let me draw some axes here.

00:00:56.130 --> 00:01:00.200
So this is the domain, the
independent variable, just

00:01:00.200 --> 00:01:04.720
like that, often known as the
x-axis or the horizontal axis.

00:01:04.720 --> 00:01:07.250
And all of our domain, all of
the values, these are all

00:01:07.250 --> 00:01:10.790
non-positive numbers, so let
me give a lot more space to

00:01:10.790 --> 00:01:11.320
quadrant II.

00:01:11.320 --> 00:01:12.980
All of our range are
positive numbers.

00:01:12.980 --> 00:01:15.940
They're going to be hanging out
in the top left quadrant,

00:01:15.940 --> 00:01:19.670
or quadrant II, just
like that.

00:01:19.670 --> 00:01:23.100
And the values that my domain
can take on are 0 through

00:01:23.100 --> 00:01:26.780
negative 8, and every even
negative number.

00:01:26.780 --> 00:01:31.260
So it's 0, let me do it like
this, so 0 is there.

00:01:31.260 --> 00:01:38.340
And then we have negative 2,
negative 4, negative 6, and

00:01:38.340 --> 00:01:42.160
then we have negative
8 just like that.

00:01:42.160 --> 00:01:44.920
And then our range goes
from 0 through 4.

00:01:44.920 --> 00:01:49.960
So this is 0, then you have 1,
then you have 2, then you have

00:01:49.960 --> 00:01:52.560
3, and then you have 4.

00:01:52.560 --> 00:01:55.870
So if we assume the mapping,
that if you give this function

00:01:55.870 --> 00:01:58.610
a negative 8, it's going
to map to 4, then

00:01:58.610 --> 00:01:59.590
we can plot it here.

00:01:59.590 --> 00:02:03.170
So if we take this first pair
right there, you give me a

00:02:03.170 --> 00:02:05.490
negative 8, this function
is going to give me a 4.

00:02:05.490 --> 00:02:09.539
So negative 8, 4 is that
point right there.

00:02:09.539 --> 00:02:12.390
So if you imagine that this is
the x-axis, this would be the

00:02:12.390 --> 00:02:16.700
f of x axis, if we're assuming
that f is this function.

00:02:16.700 --> 00:02:22.320
You give me a negative
6, I'll give you a 3.

00:02:22.320 --> 00:02:25.540
Negative 6 and 3.

00:02:25.540 --> 00:02:31.740
You give me a negative 4, this
function will pop out a 2.

00:02:31.740 --> 00:02:34.640
Negative 4 comma 2.

00:02:34.640 --> 00:02:38.040
And then you give a negative
2, this function

00:02:38.040 --> 00:02:39.060
will pop out a 1.

00:02:39.060 --> 00:02:40.910
F of negative 2 is 1.

00:02:44.090 --> 00:02:48.030
And then finally, f of 0 is 0.

00:02:48.030 --> 00:02:50.890
So we'll just plot
it right there.

00:02:50.890 --> 00:02:51.800
So that's the function.

00:02:51.800 --> 00:02:52.930
I've represented it.

00:02:52.930 --> 00:02:56.710
And remember, I just assumed
that negative 8 maps to 4,

00:02:56.710 --> 00:02:59.880
negative 6 maps to 3, but you
could have multiple functions

00:02:59.880 --> 00:03:01.610
that have this domain
and range.

00:03:01.610 --> 00:03:03.950
But just the way that they said
the following function,

00:03:03.950 --> 00:03:07.440
I'm assuming this is what
they want us to do.


Domain of modeling functions

WEBVTT
Kind: captions
Language: en

00:00:01.000 --> 00:00:04.810
Eric is selling raffle tickets
for a school fundraiser.

00:00:04.810 --> 00:00:09.160
Each ticket costs $3, and he
knows the amount of money he

00:00:09.160 --> 00:00:12.670
collects is a function of how
many tickets he sells.

00:00:12.670 --> 00:00:15.540
What are the domain and range
for that function?

00:00:15.540 --> 00:00:17.350
So let's write the
function for how

00:00:17.350 --> 00:00:19.040
much money he collects.

00:00:19.040 --> 00:00:23.850
So I'll call that m, the
function will be m, and it's a

00:00:23.850 --> 00:00:26.690
function of how many
tickets he sells.

00:00:26.690 --> 00:00:29.290
So it's a function of
t, for tickets.

00:00:29.290 --> 00:00:32.030
So m, the amount of money
collected, is a function of

00:00:32.030 --> 00:00:33.820
the number of tickets
he sells.

00:00:33.820 --> 00:00:35.940
That's a pretty straightforward
function.

00:00:35.940 --> 00:00:37.690
Every ticket costs $3.

00:00:37.690 --> 00:00:39.920
He gets $3 for every ticket.

00:00:39.920 --> 00:00:45.630
So it's going to be 3 times
t dollars or 3t dollars.

00:00:45.630 --> 00:00:47.420
That's how much money
he collects.

00:00:47.420 --> 00:00:50.490
Now they ask us, what
are the domain and

00:00:50.490 --> 00:00:52.850
range for that function?

00:00:52.850 --> 00:00:55.250
It sounds all fancy and
difficult, but just remember

00:00:55.250 --> 00:01:00.070
the domain, this just
means, what can I

00:01:00.070 --> 00:01:01.320
input into the function?

00:01:08.860 --> 00:01:12.140
So another way to think about
is what are the possible t's

00:01:12.140 --> 00:01:14.520
that can be input into
this function?

00:01:14.520 --> 00:01:17.270
The range is, what are the
possible values that the

00:01:17.270 --> 00:01:19.190
function can take on?

00:01:19.190 --> 00:01:20.710
So think about it.

00:01:20.710 --> 00:01:22.980
You might at first say I could
put any t there, but think

00:01:22.980 --> 00:01:25.820
about the actual reality
of what he's doing.

00:01:25.820 --> 00:01:28.540
He's selling tickets,
and so he can't

00:01:28.540 --> 00:01:29.670
sell negative tickets.

00:01:29.670 --> 00:01:31.880
He might sell 0 tickets,
and he might

00:01:31.880 --> 00:01:33.280
sell a gazillion tickets.

00:01:33.280 --> 00:01:35.290
I guess he could sell an
infinite amount of tickets.

00:01:35.290 --> 00:01:37.670
At some point that becomes
unrealistic.

00:01:37.670 --> 00:01:40.590
But he definitely can't
sell negative tickets.

00:01:40.590 --> 00:01:42.970
He also is not going to
sell half of a ticket.

00:01:42.970 --> 00:01:46.520
Every ticket he sells is a whole
number, it's an integer.

00:01:46.520 --> 00:01:58.590
So the domain for this function,
we could say t has

00:01:58.590 --> 00:02:09.389
to be a non-negative integer.

00:02:09.389 --> 00:02:12.160
I think that covers what
I just talked about.

00:02:12.160 --> 00:02:13.150
Non-negative.

00:02:13.150 --> 00:02:14.730
Instead of saying positive,
because it could be 0.

00:02:14.730 --> 00:02:17.620
He might literally
sell no tickets.

00:02:17.620 --> 00:02:21.550
He can't sell a negative 1
ticket or negative 2, so it's

00:02:21.550 --> 00:02:22.730
anything that's non-negative.

00:02:22.730 --> 00:02:25.450
It has to be an integer, he
can't sell half of a ticket,

00:02:25.450 --> 00:02:28.730
so, and he definitely-- it
has to be an integer.

00:02:28.730 --> 00:02:30.200
So that's our domain,.

00:02:30.200 --> 00:02:32.050
And let's think about
what our range is.

00:02:32.050 --> 00:02:36.960
A range is the possible values
that we can take on.

00:02:36.960 --> 00:02:40.850
If t is always going to be a
non-negative integer, then

00:02:40.850 --> 00:02:42.910
what's 3t always going to be?

00:02:42.910 --> 00:02:46.850
Well it's going to be a
non-negative multiple of 3.

00:02:46.850 --> 00:02:58.130
So non-negative multiples
of 3.

00:02:58.130 --> 00:02:59.450
Think about it.

00:02:59.450 --> 00:03:02.530
He'll never be able to collect
$2, because he could either

00:03:02.530 --> 00:03:04.920
sell 0 tickets and
get nothing.

00:03:04.920 --> 00:03:06.790
Let me write this down.

00:03:06.790 --> 00:03:11.480
He might sell 0 tickets, so m
of 0, he's going to get $0.

00:03:11.480 --> 00:03:16.050
m of, if he sells one ticket,
he's going to get $3.

00:03:16.050 --> 00:03:19.800
If he sells two tickets,
he's going to get $6.

00:03:19.800 --> 00:03:22.710
So he's never going to be
able to get 2 or 4.

00:03:22.710 --> 00:03:25.440
Every possible value for the
amount of money he collects

00:03:25.440 --> 00:03:28.370
for our function has to
be a multiple of 3.

00:03:28.370 --> 00:03:30.660
It's going to be a non-negative
multiple of 3

00:03:30.660 --> 00:03:33.610
because the domain is
non-negative integers.


Testing if a relationship is a function

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:02.900
We're asked: Do the points
on the graph

00:00:02.900 --> 00:00:05.390
below represent a function?

00:00:05.390 --> 00:00:09.140
So in order for the points to
represent a function, for

00:00:09.140 --> 00:00:12.610
every input into our function,
we can only get one value.

00:00:12.610 --> 00:00:14.990
So if we look here, they've
graphed the point--it looks

00:00:14.990 --> 00:00:19.340
like negative 1, 3, so that's
the point negative 1, 3.

00:00:19.340 --> 00:00:23.740
So if we assume that this is our
x-axis and that is our f

00:00:23.740 --> 00:00:26.540
of x axis, and I'm just assuming
it's a function, I

00:00:26.540 --> 00:00:29.320
don't know whether it really
is just now, this point is

00:00:29.320 --> 00:00:32.820
telling us that if you put
negative 1 into our function,

00:00:32.820 --> 00:00:34.760
or that thing that might be
a function, or maybe our

00:00:34.760 --> 00:00:38.520
relation, you'll get a 3 So
it's telling us that f of

00:00:38.520 --> 00:00:41.580
negative 1 is equal to 3.

00:00:41.580 --> 00:00:43.220
So far it could be a reasonable
function.

00:00:43.220 --> 00:00:46.960
You give me negative 1 and
I will map it to 3.

00:00:46.960 --> 00:00:51.720
Then they have if x is 2, then
our value is negative 2.

00:00:51.720 --> 00:00:54.500
This is the point 2, negative
2, so that still seems

00:00:54.500 --> 00:00:56.510
consistent with being
a function.

00:00:56.510 --> 00:01:01.170
If you pass me 2, I will map
you or I will point you to

00:01:01.170 --> 00:01:02.560
negative 2.

00:01:02.560 --> 00:01:03.930
Seems fair enough.

00:01:03.930 --> 00:01:05.230
Let's see this next
value here.

00:01:05.230 --> 00:01:09.250
This is the point 3 comma
2 right there.

00:01:09.250 --> 00:01:13.190
So once again, that says that,
look, if you give me 3 into my

00:01:13.190 --> 00:01:17.800
function, into my black box,
I will output a 2.

00:01:17.800 --> 00:01:18.780
Pretty reasonable.

00:01:18.780 --> 00:01:21.200
No reason why these points
can't represent

00:01:21.200 --> 00:01:22.970
a function so far.

00:01:22.970 --> 00:01:28.480
Now, what about when we input
4 into the function?

00:01:28.480 --> 00:01:29.720
Let me do this in magenta.

00:01:29.720 --> 00:01:32.440
So what happens if I input
4 into my function?

00:01:32.440 --> 00:01:34.390
So this is 4 right here.

00:01:34.390 --> 00:01:38.120
Well, according to these points,
there's two points

00:01:38.120 --> 00:01:41.440
that relate to 4 that
4 can be mapped to.

00:01:41.440 --> 00:01:46.900
I could map it to the
point 4 comma 5.

00:01:46.900 --> 00:01:49.720
So that says if you give me
a 4, I'll give you a 5.

00:01:49.720 --> 00:01:54.130
But it also says if you give me
a 4, I could also give you

00:01:54.130 --> 00:01:58.330
a negative 1 because that's
the point 4, negative 1 So

00:01:58.330 --> 00:02:00.030
this is not a function.

00:02:00.030 --> 00:02:04.250
It cannot be a function if for
some input into the function

00:02:04.250 --> 00:02:06.460
you could give me two
different values.

00:02:06.460 --> 00:02:07.560
And you can see that
right here.

00:02:07.560 --> 00:02:10.800
And an easy test is to just
see, look, for one value I

00:02:10.800 --> 00:02:12.900
have two points for
this relationship.

00:02:12.900 --> 00:02:14.940
So this cannot be a function.

00:02:14.940 --> 00:02:20.060
So this is not a function!

00:02:20.060 --> 00:02:22.320
I'll put an exclamation mark.


Recognize functions from tables

WEBVTT
Kind: captions
Language: en

00:00:00.830 --> 00:00:03.160
We're asked to look at
the table below.

00:00:03.160 --> 00:00:06.530
From the information given,
is there a functional

00:00:06.530 --> 00:00:14.250
relationship between each person
and his or her height?

00:00:14.250 --> 00:00:16.050
So a good place to start is
just think about what a

00:00:16.050 --> 00:00:17.890
functional relationship means.

00:00:17.890 --> 00:00:19.430
Now, there's definitely
a relationship.

00:00:19.430 --> 00:00:21.740
They say, hey, if you're
Joelle, you're 5-6.

00:00:21.740 --> 00:00:23.450
If you're Nathan, you're 4-11.

00:00:23.450 --> 00:00:25.000
If you're Stewart,
you're 5-11.

00:00:25.000 --> 00:00:26.990
That is a relationship.

00:00:26.990 --> 00:00:31.050
Now, in order for it to be a
functional relationship, for

00:00:31.050 --> 00:00:34.820
every instance or every example
of the independent

00:00:34.820 --> 00:00:38.000
variable, you can only have one
example of the value of

00:00:38.000 --> 00:00:39.390
the function for it.

00:00:39.390 --> 00:00:48.730
So if you say if this is a
height function, in order for

00:00:48.730 --> 00:00:52.610
this to be a functional
relationship, no matter whose

00:00:52.610 --> 00:00:55.610
name you put inside of the
height function, you need to

00:00:55.610 --> 00:00:58.050
only be able to get one value.

00:00:58.050 --> 00:01:01.020
If there were two values
associated with one person's

00:01:01.020 --> 00:01:04.040
name, it would not be a
functional relationship.

00:01:04.040 --> 00:01:08.915
So if I were to ask you what
is the height of Nathan?

00:01:12.680 --> 00:01:14.870
Well, you'd look at the table
and say, well, Nathan's height

00:01:14.870 --> 00:01:19.910
is 4 foot 11.

00:01:19.910 --> 00:01:22.350
There are not two heights
for Nathan.

00:01:22.350 --> 00:01:23.810
There is only one height.

00:01:23.810 --> 00:01:26.790
And for any one of these people
that we can input into

00:01:26.790 --> 00:01:29.760
the function, there's only one
height associated with them,

00:01:29.760 --> 00:01:31.660
so it is a functional
relationship.

00:01:31.660 --> 00:01:33.880
We can even see that
on a graph.

00:01:33.880 --> 00:01:35.290
Let me graph that out for you.

00:01:38.860 --> 00:01:41.340
Let's see, the highest height
here is 6 foot 1.

00:01:41.340 --> 00:01:49.370
So if we start off with one
foot, two feet, three feet,

00:01:49.370 --> 00:01:57.330
four feet, five feet,
and six feet.

00:01:57.330 --> 00:01:59.680
And then if I were to plot the
different names, the different

00:01:59.680 --> 00:02:04.600
people that I could put into our
height function, we have--

00:02:04.600 --> 00:02:06.580
I'll just put the first letters
of their names.

00:02:06.580 --> 00:02:12.990
We have Joelle, we have Nathan,
we have Stewart, we

00:02:12.990 --> 00:02:17.540
have LJ, and then we have
Tariq right there.

00:02:17.540 --> 00:02:19.090
So lets plot them.

00:02:19.090 --> 00:02:24.540
So you have Joelle, Joelle's
height is 5-6, so 5-6 is right

00:02:24.540 --> 00:02:26.430
about there.

00:02:26.430 --> 00:02:29.800
Then you have Nathan.

00:02:29.800 --> 00:02:31.860
Let me do it in a
different color.

00:02:31.860 --> 00:02:37.390
Nathan's height is 4-11.

00:02:37.390 --> 00:02:41.220
We will plot to him
right over there.

00:02:41.220 --> 00:02:42.730
Then you have Stewart.

00:02:42.730 --> 00:02:47.010
Stewart's height is 5-11.

00:02:47.010 --> 00:02:48.910
He is pretty close
to six feet.

00:02:48.910 --> 00:02:51.430
So Stewart's height-- I made him
like six feet; let me make

00:02:51.430 --> 00:02:53.630
it a little lower-- is 5-11.

00:02:53.630 --> 00:02:57.060
Then you have LJ.

00:02:57.060 --> 00:02:59.560
LJ's height is 5-6.

00:02:59.560 --> 00:03:02.340
So you have two people with a
height of 5-6, but that's OK,

00:03:02.340 --> 00:03:05.090
as long as for each person
you only have one height.

00:03:05.090 --> 00:03:07.950
And then finally, Tariq
is 6 foot 1.

00:03:07.950 --> 00:03:09.375
He's the tallest guy here.

00:03:09.375 --> 00:03:13.040
Tariq is right up here
at 6 foot 1.

00:03:13.040 --> 00:03:16.600
So notice, for any one of the
inputs into our function, we

00:03:16.600 --> 00:03:20.810
only have one value, so this is
a functional relationship.

00:03:20.810 --> 00:03:23.695
Now, you might say OK, well,
isn't everything a functional

00:03:23.695 --> 00:03:24.180
relationship?

00:03:24.180 --> 00:03:24.620
No!

00:03:24.620 --> 00:03:27.530
If I gave you the situation, if
I also wrote here-- let's

00:03:27.530 --> 00:03:36.030
say the table was like this and
I also wrote that Stewart

00:03:36.030 --> 00:03:37.760
is 5 foot 3 inches.

00:03:37.760 --> 00:03:40.560
If this was our table, then
we would no longer have a

00:03:40.560 --> 00:03:43.950
functional relationship because
for the input of

00:03:43.950 --> 00:03:46.130
Stewart, we would have
two different values.

00:03:46.130 --> 00:03:49.980
If we were to graph this, we
have Stewart here at 5-11, and

00:03:49.980 --> 00:03:52.980
then all of a sudden, we would
also have Stewart at 5-3.

00:03:52.980 --> 00:03:55.300
Now, this doesn't make a lot of
sense, so we would plot it

00:03:55.300 --> 00:03:56.510
right over here.

00:03:56.510 --> 00:03:59.050
So for Stewart, you would have
two values, and so this

00:03:59.050 --> 00:04:02.880
wouldn't be a valid functional
relationship because you

00:04:02.880 --> 00:04:06.210
wouldn't know what value to give
if you were to take the

00:04:06.210 --> 00:04:10.120
height of Stewart.

00:04:10.120 --> 00:04:11.600
In order for this to be a
function, there can only be

00:04:11.600 --> 00:04:12.890
one value for this.

00:04:12.890 --> 00:04:14.960
You don't know in this situation
when I add this,

00:04:14.960 --> 00:04:17.240
whether it's 5-3 or 5-11.

00:04:17.240 --> 00:04:20.620
Now, this wasn't the case, so
that isn't there and so we

00:04:20.620 --> 00:04:24.810
know that the height of Stewart
is 5-11 and this is a

00:04:24.810 --> 00:04:26.300
functional relationship.

00:04:26.300 --> 00:04:29.220
I think to some level, it might
be confusing, because

00:04:29.220 --> 00:04:30.800
it's such a simple idea.

00:04:30.800 --> 00:04:33.520
Each of these values can
only have one height

00:04:33.520 --> 00:04:34.530
associated with it.

00:04:34.530 --> 00:04:35.600
That's what makes
it a function.

00:04:35.600 --> 00:04:37.430
If you had more than one height
associated with it, it

00:04:37.430 --> 00:04:39.450
would not be a function.


Finding the 100th term in a sequence

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:05.170
We are asked, what is the value
of the 100th term in

00:00:05.170 --> 00:00:06.000
this sequence?

00:00:06.000 --> 00:00:09.640
And the first term is 15, then
9, then 3, then negative 3.

00:00:09.640 --> 00:00:11.820
So let's write it like
this, in a table.

00:00:11.820 --> 00:00:15.890
So if we have the term, just
so we have things straight,

00:00:15.890 --> 00:00:18.070
and then we have the value.

00:00:18.070 --> 00:00:20.810
and then we have the
value of the term.

00:00:20.810 --> 00:00:23.260
I'll do a nice little
table here.

00:00:23.260 --> 00:00:26.770
So our first term
we saw is 15.

00:00:26.770 --> 00:00:30.730
Our second term is 9.

00:00:30.730 --> 00:00:33.330
Our third term is 3.

00:00:33.330 --> 00:00:35.940
I'm just really copying this
down, but I'm making sure we

00:00:35.940 --> 00:00:37.720
associate it with
the right term.

00:00:37.720 --> 00:00:42.510
And then our fourth term
is negative 3.

00:00:42.510 --> 00:00:46.550
And they want us to figure out
what the 100th term of this

00:00:46.550 --> 00:00:47.700
sequence is going to be.

00:00:47.700 --> 00:00:50.670
So let's see what's happening
here, if we can discern some

00:00:50.670 --> 00:00:51.900
type of pattern.

00:00:51.900 --> 00:00:53.300
So when we went from the
first term to the

00:00:53.300 --> 00:00:54.770
second term, what happened?

00:00:54.770 --> 00:00:56.270
15 to 9.

00:00:56.270 --> 00:00:57.670
Looks like we went down by 6.

00:00:57.670 --> 00:00:59.920
It's always good to think
about just how much the

00:00:59.920 --> 00:01:00.910
numbers changed by.

00:01:00.910 --> 00:01:02.860
That's always the simplest
type of pattern.

00:01:02.860 --> 00:01:06.070
So we went down by 6,
we subtracted 6.

00:01:06.070 --> 00:01:09.470
Then to go from 9 to 3, well,
we subtracted 6 again.

00:01:12.760 --> 00:01:16.380
And then to go from 3 to
negative 3, well, we

00:01:16.380 --> 00:01:17.900
subtracted 6 again.

00:01:21.750 --> 00:01:26.610
So it looks like, every
term, you subtract 6.

00:01:26.610 --> 00:01:29.900
So the second term is going to
be 6 less than the first term.

00:01:29.900 --> 00:01:35.330
The third term is going to be
12 from the first term, or

00:01:35.330 --> 00:01:37.690
negative 6 subtracted twice.

00:01:37.690 --> 00:01:41.010
So in the third term, you
subtract negative 6 twice.

00:01:41.010 --> 00:01:43.270
In the fourth term,
you subtract

00:01:43.270 --> 00:01:45.710
negative 6 three times.

00:01:45.710 --> 00:01:48.770
So whatever term you're looking
at, you subtract

00:01:48.770 --> 00:01:51.610
negative 6 one less than
that many times.

00:01:51.610 --> 00:01:55.210
Let me write this down just so--
Notice when your first

00:01:55.210 --> 00:01:57.070
term, you have 15, and
you don't subtract

00:01:57.070 --> 00:01:58.220
negative 6 at all.

00:01:58.220 --> 00:02:01.430
Or you could say you subtract
negative 6 0 times.

00:02:01.430 --> 00:02:07.850
So you can say this is 15 minus
negative 6 times-- or

00:02:07.850 --> 00:02:11.290
let me write it better
this way --minus 0

00:02:11.290 --> 00:02:12.860
times negative 6.

00:02:12.860 --> 00:02:14.890
That's what that first
term is right there.

00:02:14.890 --> 00:02:16.530
What's the second term?

00:02:16.530 --> 00:02:18.680
This is 15.

00:02:18.680 --> 00:02:21.140
We just subtracted negative
6 once, or you could say,

00:02:21.140 --> 00:02:24.980
minus 1 times 6.

00:02:24.980 --> 00:02:26.890
Or you could say plus
1 times negative 6.

00:02:26.890 --> 00:02:30.700
Either way, we're subtracting
the 6 once.

00:02:30.700 --> 00:02:32.330
Now what's happening here?

00:02:32.330 --> 00:02:38.970
This is 15 minus 2 times
negative 6-- or, sorry

00:02:38.970 --> 00:02:40.695
--minus 2 times 6.

00:02:43.850 --> 00:02:46.300
We're subtracting a 6 twice.

00:02:46.300 --> 00:02:47.960
What's the fourth term?

00:02:47.960 --> 00:02:52.740
This is 15 minus-- We're
subtracting the 6 three times

00:02:52.740 --> 00:02:56.870
from the 15, so minus
3 times 6.

00:02:56.870 --> 00:03:01.370
So, if you see the pattern here,
when we have our fourth

00:03:01.370 --> 00:03:04.940
term, we have the term
minus 1 right there.

00:03:04.940 --> 00:03:06.530
The fourth term, we have a 3.

00:03:06.530 --> 00:03:09.080
The third term, we have a 2.

00:03:09.080 --> 00:03:11.800
The second term, we have a 1.

00:03:11.800 --> 00:03:16.250
So if we had the nth term, if we
just had the nth term here,

00:03:16.250 --> 00:03:17.390
what's this going to be?

00:03:17.390 --> 00:03:21.890
It's going to be 15 minus--
You see it's going to be n

00:03:21.890 --> 00:03:23.320
minus 1 right here.

00:03:23.320 --> 00:03:23.630
Right?

00:03:23.630 --> 00:03:25.960
When n is 4, n minus 1 is 3.

00:03:25.960 --> 00:03:29.000
When n is 3, n minus 1 is 2.

00:03:29.000 --> 00:03:32.080
When n is 2, n minus 1 is 1.

00:03:32.080 --> 00:03:35.960
When n is 1, n minus 1 is 0.

00:03:35.960 --> 00:03:39.420
So we're going to have this term
right here is n minus 1.

00:03:39.420 --> 00:03:44.330
So minus n minus 1 times 6.

00:03:44.330 --> 00:03:46.970
So if you want to figure out
the 100th term of this

00:03:46.970 --> 00:03:49.780
sequence, I didn't even have
to write it in this general

00:03:49.780 --> 00:03:51.450
term, you can just look
at this pattern.

00:03:51.450 --> 00:03:54.590
It's going to be-- and I'll do
it in pink --the 100th term in

00:03:54.590 --> 00:03:56.640
our sequence-- I'll continue
our table down

00:03:56.640 --> 00:03:57.600
--is going to be what?

00:03:57.600 --> 00:04:06.310
It's going to be 15 minus 100
minus 1, which is 99, times 6.

00:04:06.310 --> 00:04:06.620
right?

00:04:06.620 --> 00:04:07.900
I just follow the pattern.

00:04:07.900 --> 00:04:09.340
1, you had a 0 here.

00:04:09.340 --> 00:04:10.440
2, you had a 1 here.

00:04:10.440 --> 00:04:11.730
3, you had a 2 here.

00:04:11.730 --> 00:04:13.830
100, you're going to
have a 99 here.

00:04:13.830 --> 00:04:15.950
So let's just calculate
what this is.

00:04:15.950 --> 00:04:18.060
What's 99 times 6?

00:04:18.060 --> 00:04:22.680
So 99 times 6-- Actually you
can do this in your head.

00:04:22.680 --> 00:04:27.160
You could say that's going to
be 6 less than 100 times 6,

00:04:27.160 --> 00:04:31.360
which is 600, and
6 less is 594.

00:04:31.360 --> 00:04:33.120
But if you didn't want to do
it that way, you just do it

00:04:33.120 --> 00:04:34.310
the old-fashioned way.

00:04:34.310 --> 00:04:38.090
6 times 9 is 54.

00:04:38.090 --> 00:04:39.240
Carry the 5.

00:04:39.240 --> 00:04:42.490
9 times 6, or 6 times 9 is 54.

00:04:42.490 --> 00:04:46.600
54 plus 5 is 594.

00:04:46.600 --> 00:04:51.670
So this right here is 594.

00:04:51.670 --> 00:04:55.610
And then to figure out what 15--
So we want to figure out

00:04:55.610 --> 00:05:00.080
what 15 minus 594 is.

00:05:00.080 --> 00:05:03.160
And this can sometimes be
confusing, but the way I

00:05:03.160 --> 00:05:06.040
always process this in my head
is, I say that this is the

00:05:06.040 --> 00:05:13.970
exact same thing as the negative
of 594 minus 15.

00:05:13.970 --> 00:05:15.870
And if you don't believe
me, distribute out

00:05:15.870 --> 00:05:16.920
this negative sign.

00:05:16.920 --> 00:05:20.580
Negative 1 times 594
is negative 594.

00:05:20.580 --> 00:05:23.970
Negative 1 times negative
15 is positive 15.

00:05:23.970 --> 00:05:25.880
So these two statements
are equivalent.

00:05:25.880 --> 00:05:29.160
This is much easier for my
brain to understand.

00:05:29.160 --> 00:05:32.270
So what's 594 minus 15?

00:05:32.270 --> 00:05:33.120
We can do this in our heads.

00:05:33.120 --> 00:05:40.530
594 minus 14 would be 580,
and then 580 minus 1

00:05:40.530 --> 00:05:42.820
more would be 579.

00:05:42.820 --> 00:05:47.270
So that right there is 579, and
then we have this negative

00:05:47.270 --> 00:05:48.900
sign sitting out there.

00:05:48.900 --> 00:05:54.510
So the 100th term in our
sequence will be negative 579.


Equations of sequence patterns

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:05.130
Our question asks us, what
equation describes the growth

00:00:05.130 --> 00:00:08.550
pattern of this sequence
of a block?

00:00:08.550 --> 00:00:12.460
So we want to figure out, if I
know that x is equal to 10,

00:00:12.460 --> 00:00:14.860
how many blocks am
I going to have?

00:00:14.860 --> 00:00:17.240
So let's just look at
this pattern here.

00:00:17.240 --> 00:00:20.440
So our first term in our
sequence, or our first object,

00:00:20.440 --> 00:00:23.360
or our first pattern of blocks
right here, we just have 1

00:00:23.360 --> 00:00:24.350
block right there.

00:00:24.350 --> 00:00:28.470
So let me write, the term--
write it up here --so I have

00:00:28.470 --> 00:00:35.350
the term and, then I'll have
the number of blocks.

00:00:41.830 --> 00:00:45.520
So in our first term,
we had one block.

00:00:45.520 --> 00:00:47.780
And then our second term-- I'll
just write this down,

00:00:47.780 --> 00:00:50.150
just so we have it --what
happened here?

00:00:50.150 --> 00:00:52.940
So it looks just like our first
term, but we added a

00:00:52.940 --> 00:00:54.545
column here of four blocks.

00:00:54.545 --> 00:00:57.140
So it's like 1 plus
4 right there.

00:00:57.140 --> 00:00:59.500
So we're going to have five
blocks right there.

00:00:59.500 --> 00:01:01.330
We added 4 to it.

00:01:01.330 --> 00:01:03.850
Then in our third term
what happened?

00:01:03.850 --> 00:01:05.269
What happened in
our third term?

00:01:05.269 --> 00:01:07.650
Well it just looks just like the
second term, but we added

00:01:07.650 --> 00:01:09.980
another column of four
blocks here.

00:01:09.980 --> 00:01:10.170
Right?

00:01:10.170 --> 00:01:11.450
We added this column
right there.

00:01:11.450 --> 00:01:13.540
If you imagine they were being
added to the left-hand side of

00:01:13.540 --> 00:01:14.660
the pattern.

00:01:14.660 --> 00:01:16.030
So we added four more blocks.

00:01:16.030 --> 00:01:17.930
We have nine blocks now.

00:01:17.930 --> 00:01:20.020
We have nine blocks, so it looks
like each time we're

00:01:20.020 --> 00:01:20.810
adding four blocks.

00:01:20.810 --> 00:01:22.900
And on this fourth
term, same thing.

00:01:22.900 --> 00:01:25.520
The third term is just
this right here.

00:01:25.520 --> 00:01:28.250
This right here is what the
third term looked like, and

00:01:28.250 --> 00:01:30.650
then we added another column
of four blocks right here.

00:01:30.650 --> 00:01:33.570
So we added four more, so we're
going to have 13 blocks.

00:01:33.570 --> 00:01:36.800
So our fourth term is 13.

00:01:36.800 --> 00:01:38.870
So let's see if we can come up
with a formula, either looking

00:01:38.870 --> 00:01:42.020
at the graphics, or maybe
looking at the numbers

00:01:42.020 --> 00:01:43.530
themselves.

00:01:43.530 --> 00:01:47.675
So one way to think about it, so
we start off with-- So when

00:01:47.675 --> 00:01:52.470
x is equal to 1, let's say that
x is equal to the term,

00:01:52.470 --> 00:01:54.510
we add just this 1 there.

00:01:54.510 --> 00:01:59.440
Then when x is equal to 2, we
added one column of four.

00:01:59.440 --> 00:02:03.860
So this is when x is equal to 2,
we have one column of four.

00:02:03.860 --> 00:02:06.690
Then when x is equal
to 3, we have two

00:02:06.690 --> 00:02:09.520
columns of 4, right there.

00:02:09.520 --> 00:02:13.390
And you could even say when x
is equal to 1, you had zero

00:02:13.390 --> 00:02:14.270
columns, right?

00:02:14.270 --> 00:02:17.300
We had no, nothing, no extra
columns of four blocks.

00:02:17.300 --> 00:02:18.510
We didn't have any.

00:02:18.510 --> 00:02:22.670
And then when x is equal to
4, we had three columns.

00:02:22.670 --> 00:02:26.410
We had three columns there,
when x is equal to 4.

00:02:26.410 --> 00:02:27.750
So what's the pattern here?

00:02:27.750 --> 00:02:30.530
Or how can we express the number
of blocks we're going

00:02:30.530 --> 00:02:34.920
to have, given the term
that we have?

00:02:34.920 --> 00:02:36.800
Well, it looks like we're
always going to have one

00:02:36.800 --> 00:02:39.010
block, so let me write
it this way.

00:02:39.010 --> 00:02:41.550
If I write the number of
blocks-- let me write it this

00:02:41.550 --> 00:02:49.650
way --it looks like
we're always going

00:02:49.650 --> 00:02:51.050
to have one, right?

00:02:51.050 --> 00:02:54.500
We have this one right here,
that one right there, that one

00:02:54.500 --> 00:02:56.030
right there, that
one right there.

00:02:56.030 --> 00:03:00.420
Looks like we always have one
plus a certain number of

00:03:00.420 --> 00:03:02.710
columns of four, but how many
columns do we have?

00:03:02.710 --> 00:03:06.120
When x is equal to 1, we have
no columns of four blocks.

00:03:06.120 --> 00:03:08.740
When x is equal to 2,
we have one column.

00:03:08.740 --> 00:03:11.330
When x is equal to 3,
we have two columns.

00:03:11.330 --> 00:03:13.770
So when x is equal to anything,
it looks like we

00:03:13.770 --> 00:03:16.300
have one less number
of columns.

00:03:16.300 --> 00:03:22.740
So it's going to be
x minus 1, right?

00:03:22.740 --> 00:03:25.360
When x is 2, x minus 1 is 1.

00:03:25.360 --> 00:03:30.220
When x is 3, x minus 1, so this
right here is x minus 1.

00:03:30.220 --> 00:03:31.950
x is 2, this is x minus 1.

00:03:31.950 --> 00:03:33.670
This is x minus 1.

00:03:33.670 --> 00:03:36.650
This is x minus 1, and x minus
1 will tell us the number of

00:03:36.650 --> 00:03:37.990
columns we have, right?

00:03:37.990 --> 00:03:40.180
Here we have one, two,
three columns.

00:03:40.180 --> 00:03:42.150
Here we have one, two columns.

00:03:42.150 --> 00:03:43.330
Here we only have one column.

00:03:43.330 --> 00:03:44.780
Here we have zero columns.

00:03:44.780 --> 00:03:47.010
So it even works for
the first term.

00:03:47.010 --> 00:03:50.670
And in every one of these
columns, so this right here, x

00:03:50.670 --> 00:03:57.600
minus 1 is the number of
columns, and then in each

00:03:57.600 --> 00:03:59.430
column we have four blocks.

00:03:59.430 --> 00:04:03.530
So it's the number of columns
times 4, right?

00:04:03.530 --> 00:04:05.340
For each of these columns,
we have one column.

00:04:05.340 --> 00:04:07.710
We have one, two, three,
four blocks.

00:04:07.710 --> 00:04:11.590
So this is the equation that
describes the growth pattern.

00:04:11.590 --> 00:04:14.400
So let me write this, let me
simplify this a little bit.

00:04:14.400 --> 00:04:18.390
If I were to multiply 4 times x
minus 1, I get the number of

00:04:18.390 --> 00:04:21.940
blocks being equal to
1 plus 4 times x.

00:04:21.940 --> 00:04:22.960
I have to distribute it.

00:04:22.960 --> 00:04:28.050
4 times x is 4x, and then
4 times negative

00:04:28.050 --> 00:04:31.680
1 is negative 4.

00:04:31.680 --> 00:04:34.980
So that's equal to the
number of blocks.

00:04:38.230 --> 00:04:39.600
And we could simplify this.

00:04:39.600 --> 00:04:42.690
We have a 1 and we have a minus
4, or I guess we're

00:04:42.690 --> 00:04:47.420
subtracting 4 from it, so this
is going to be equal to 4x

00:04:47.420 --> 00:04:51.250
minus 3 is the number of blocks
given our x term.

00:04:51.250 --> 00:04:56.190
So if we're on term 50, it's
going to be 4 times 50, which

00:04:56.190 --> 00:05:00.120
is 200 minus 3, which
is 197 blocks.

00:05:00.120 --> 00:05:01.890
Now another way you could have
done it is you could have just

00:05:01.890 --> 00:05:05.100
said, look, every time we're
adding 4, this is a linear

00:05:05.100 --> 00:05:10.050
relationship, and you could
essentially find the slope of

00:05:10.050 --> 00:05:13.370
the line that connects this,
but assume that our line is

00:05:13.370 --> 00:05:15.560
only defined on integers.

00:05:15.560 --> 00:05:17.520
And that might be a little bit
more complicated, but the way

00:05:17.520 --> 00:05:23.330
that you think about it is,
every one, every time we added

00:05:23.330 --> 00:05:27.930
a block, we added-- or every
time we added a term we added

00:05:27.930 --> 00:05:28.660
four blocks.

00:05:28.660 --> 00:05:30.180
So we could write it this way.

00:05:30.180 --> 00:05:33.490
We could just write change--
so this the triangle right

00:05:33.490 --> 00:05:34.520
here means change.

00:05:34.520 --> 00:05:48.240
Delta means change in blocks
divided by change in x.

00:05:48.240 --> 00:05:49.490
Now you might recognize this.

00:05:49.490 --> 00:05:51.290
This is slope.

00:05:51.290 --> 00:05:54.160
And if you don't worry, if slope
is a completely foreign

00:05:54.160 --> 00:05:56.100
concept to you, you can just
do it the way we did it the

00:05:56.100 --> 00:05:57.190
first part of this video.

00:05:57.190 --> 00:05:59.180
And that's a completely
legitimate way, and hopefully

00:05:59.180 --> 00:06:01.550
it will make some connections
between what slope is.

00:06:01.550 --> 00:06:05.670
So what is the change in blocks
for a change in x.

00:06:05.670 --> 00:06:11.050
So when we went from x going
from 1 to 2-- so our change in

00:06:11.050 --> 00:06:15.350
x here would be 2 minus 1, we
increased by 1 --what was our

00:06:15.350 --> 00:06:16.700
change in blocks?

00:06:16.700 --> 00:06:19.580
It would be 4, or 5 minus 1.

00:06:19.580 --> 00:06:21.740
It's 5 minus 1.

00:06:21.740 --> 00:06:22.560
And what is this equal to?

00:06:22.560 --> 00:06:28.330
This is equal to 4 over 1,
which is equal to 4.

00:06:28.330 --> 00:06:29.710
Let me scroll over
a little bit.

00:06:29.710 --> 00:06:34.380
So our change in blocks, or
change in x is 4, or our slope

00:06:34.380 --> 00:06:35.890
is equal to 4.

00:06:35.890 --> 00:06:39.920
So if you want to do this kind
of the setting up the equation

00:06:39.920 --> 00:06:42.900
of a line way, you would say
that our equation-- If, well

00:06:42.900 --> 00:06:43.460
let me write it.

00:06:43.460 --> 00:06:54.130
Number of blocks are going to
be equal to 4 times the term

00:06:54.130 --> 00:06:56.970
that we're dealing with,
the term in our

00:06:56.970 --> 00:07:00.490
pattern, plus some constant.

00:07:00.490 --> 00:07:02.260
This right here is the
equation of a line.

00:07:02.260 --> 00:07:05.135
If it's completely foreign to
you, just do it the way we did

00:07:05.135 --> 00:07:07.330
it earlier in the video.

00:07:07.330 --> 00:07:10.370
And so, how do we solve
for this constant?

00:07:10.370 --> 00:07:12.320
Well, we use one of
our terms here.

00:07:12.320 --> 00:07:14.960
We know that when we had one--
In our first term we

00:07:14.960 --> 00:07:16.700
only had one block.

00:07:16.700 --> 00:07:17.750
So let's put that here.

00:07:17.750 --> 00:07:22.760
So in our first term-- we're
going to have that b right

00:07:22.760 --> 00:07:25.390
there --we only had one block.

00:07:25.390 --> 00:07:28.830
So we have 1 is equal
to 4 plus b.

00:07:28.830 --> 00:07:33.260
If you subtract 4 from both
sides of this equation, so you

00:07:33.260 --> 00:07:36.240
subtract 4 from both sides,
what do you get?

00:07:36.240 --> 00:07:40.360
On the left-hand side, 1 minus
4 is negative 3, and that's

00:07:40.360 --> 00:07:43.260
equal to-- these 4's cancel
out --and and

00:07:43.260 --> 00:07:45.030
that's equal to b.

00:07:45.030 --> 00:07:46.970
So another way to get the
equation of a line, we have

00:07:46.970 --> 00:07:49.910
just solved that b is
equal to negative 3.

00:07:49.910 --> 00:07:53.960
We said how much do the number
of blocks change for a certain

00:07:53.960 --> 00:07:56.660
change in x, which is a change
in the number blocks for a

00:07:56.660 --> 00:07:58.680
change in x, we saw
it's always 4.

00:07:58.680 --> 00:08:00.230
4 per change in x.

00:08:00.230 --> 00:08:03.510
When x changes by 1,
we change by 4.

00:08:03.510 --> 00:08:05.630
That gave us our slope.

00:08:05.630 --> 00:08:08.535
And then to solve for-- If you
view this as a line, although

00:08:08.535 --> 00:08:12.120
this is only defined
on integers, I

00:08:12.120 --> 00:08:14.550
guess positive integers.

00:08:14.550 --> 00:08:17.230
In this situation, you could
view this as a y-intercept.

00:08:17.230 --> 00:08:20.520
To solve for this constant, we
just use one of our terms. You

00:08:20.520 --> 00:08:21.320
could have used any of them.

00:08:21.320 --> 00:08:22.540
We used 1 and 1.

00:08:22.540 --> 00:08:23.680
You could use 3 and 9.

00:08:23.680 --> 00:08:24.390
You could use anything.

00:08:24.390 --> 00:08:27.980
We solved b is equal to negative
3, and so if you put

00:08:27.980 --> 00:08:31.990
b back here, you get four x
minus 3, which is what we got

00:08:31.990 --> 00:08:33.789
earlier in the video,
right there.

00:08:33.789 --> 00:08:35.799
Hopefully you found that fun.


Patterns in sequences 2

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:04.830
Our question asks us, how many
toothpicks will be needed to

00:00:04.830 --> 00:00:09.780
form the 50th figure
in this sequence?

00:00:09.780 --> 00:00:12.010
So let's look at
this sequence.

00:00:12.010 --> 00:00:15.500
So the first figure in our
sequence-- they look like

00:00:15.500 --> 00:00:17.470
little houses-- how many
toothpicks are here?

00:00:17.470 --> 00:00:24.600
We have one, two, three, four,
five, six toothpicks.

00:00:24.600 --> 00:00:28.550
So our first object here, or
our first figure, has six

00:00:28.550 --> 00:00:29.660
toothpicks.

00:00:29.660 --> 00:00:31.640
How many do we have in
our second figure?

00:00:31.640 --> 00:00:33.350
Well we're going to have
these six that we had

00:00:33.350 --> 00:00:34.320
in the first, right?

00:00:34.320 --> 00:00:35.850
This is the first figure,
right here.

00:00:35.850 --> 00:00:37.580
Let me just trace it.

00:00:37.580 --> 00:00:40.820
That's the first figure,
right there.

00:00:40.820 --> 00:00:42.290
And then how many
new toothpicks

00:00:42.290 --> 00:00:43.380
are we going to have?

00:00:43.380 --> 00:00:48.410
We have one, two, three,
four, five.

00:00:48.410 --> 00:00:50.830
So we have six plus
another five.

00:00:50.830 --> 00:00:53.930
So we have the six in the
original one, plus another

00:00:53.930 --> 00:00:56.670
five, which is equal
to 11 toothpicks.

00:00:56.670 --> 00:01:00.440
Now what happens in this third
object, or this third little

00:01:00.440 --> 00:01:03.430
toothpick house looking
drawing thing?

00:01:03.430 --> 00:01:07.280
So we have our figure
from the two houses.

00:01:07.280 --> 00:01:09.690
So we have those right there.

00:01:09.690 --> 00:01:12.630
This is the exact same
thing that we had

00:01:12.630 --> 00:01:14.450
as our second object.

00:01:14.450 --> 00:01:15.660
So that's that right there.

00:01:15.660 --> 00:01:16.850
And how many more do we have?

00:01:16.850 --> 00:01:17.840
Well it's going to be
the same thing.

00:01:17.840 --> 00:01:20.340
We've just added on this other
extension to our house.

00:01:20.340 --> 00:01:21.630
You can view it that way.

00:01:21.630 --> 00:01:26.060
So we have one, two,
three, four, five.

00:01:26.060 --> 00:01:30.810
So it looks like every time we
go down this sequence, or we

00:01:30.810 --> 00:01:33.870
add a new term to the sequence,
or a new object to

00:01:33.870 --> 00:01:37.420
the sequence, we're adding
five toothpicks.

00:01:37.420 --> 00:01:39.500
So here, we're going
to have 11 plus 5.

00:01:39.500 --> 00:01:41.780
There's 11 toothpicks in this
part of it, and then we

00:01:41.780 --> 00:01:42.420
have another 5.

00:01:42.420 --> 00:01:45.680
11 plus 5, which is 16.

00:01:45.680 --> 00:01:46.970
Same thing over here.

00:01:46.970 --> 00:01:49.840
We have 16 in this part
of the drawing.

00:01:49.840 --> 00:01:53.320
That part of the drawing is
going to be 16 toothpicks, if

00:01:53.320 --> 00:01:54.920
you draw all of that.

00:01:54.920 --> 00:01:56.680
And then we're going to have
another five right here.

00:01:56.680 --> 00:02:00.960
So this is going to be
16 plus 5 or 21.

00:02:00.960 --> 00:02:04.340
So how can we figure out how
many toothpicks we're going to

00:02:04.340 --> 00:02:05.840
have in the 50th figure?

00:02:05.840 --> 00:02:07.840
I mean, we could draw 50 of
them, but it's going to take

00:02:07.840 --> 00:02:08.720
us forever.

00:02:08.720 --> 00:02:10.440
So all we have to do is
realize the pattern.

00:02:10.440 --> 00:02:12.780
Every one, we're just
going to add 5.

00:02:12.780 --> 00:02:17.180
Or, we could even come up with
a formula for the nth figure.

00:02:17.180 --> 00:02:19.940
How many toothpicks
in the nth figure?

00:02:19.940 --> 00:02:23.020
So here, in the first figure,
we have-- you could view it

00:02:23.020 --> 00:02:28.140
this way-- in this first figure,
we have 1 plus 5

00:02:28.140 --> 00:02:28.730
toothpicks.

00:02:28.730 --> 00:02:29.010
Right?

00:02:29.010 --> 00:02:30.170
We have this 1.

00:02:30.170 --> 00:02:31.980
You could imagine that was
always there, and then you

00:02:31.980 --> 00:02:33.140
added 5 right there.

00:02:33.140 --> 00:02:35.860
So that's our first term
in our sequence.

00:02:35.860 --> 00:02:38.600
Our second term in our sequence,
what do we have?

00:02:38.600 --> 00:02:41.090
We have 6, which is 1 plus 5.

00:02:41.090 --> 00:02:45.160
So we have that 1 plus
5, plus another 5.

00:02:45.160 --> 00:02:48.510
Plus this 5 right here,
plus another 5.

00:02:48.510 --> 00:02:51.810
And then our third term in our
sequence, what do we have?

00:02:51.810 --> 00:02:53.540
We have all of this business
that we had

00:02:53.540 --> 00:02:54.880
in the second term.

00:02:54.880 --> 00:02:59.355
So 1 plus 5 plus 5, and then
we have another 5.

00:02:59.355 --> 00:03:01.410
Let me do it in green.

00:03:01.410 --> 00:03:03.390
We have another 5.

00:03:03.390 --> 00:03:06.630
And then, finally, in this
fourth term, what do we have?

00:03:06.630 --> 00:03:09.270
We have-- so the fourth one-- we
have everything that we had

00:03:09.270 --> 00:03:14.840
in the third figure, so 1
plus 5 plus 5 plus 5.

00:03:14.840 --> 00:03:16.700
And then we add another 5.

00:03:16.700 --> 00:03:18.460
We add a fourth 5.

00:03:18.460 --> 00:03:20.930
We add a fourth 5 right there.

00:03:20.930 --> 00:03:22.420
So why did I do it this way?

00:03:22.420 --> 00:03:26.060
Because I wrote this first one
as 1 plus 5 instead of just a

00:03:26.060 --> 00:03:28.220
6 because you could say, well
look, I had a 1, and

00:03:28.220 --> 00:03:29.280
then I had one 5.

00:03:29.280 --> 00:03:32.780
Now I have a 1, and
I have two 5's.

00:03:32.780 --> 00:03:35.630
Now I have a 1 and
I have three 5's.

00:03:35.630 --> 00:03:38.850
Now I have a 1 and
I have four 5's.

00:03:38.850 --> 00:03:42.020
So you might see the pattern.

00:03:42.020 --> 00:03:46.790
If I had the nth term, if I had
the nth term right here, I

00:03:46.790 --> 00:03:50.400
will have a 1 plus
n 5's, right?

00:03:50.400 --> 00:03:51.620
The first term has one 5.

00:03:51.620 --> 00:03:52.780
The second term has two 5's.

00:03:52.780 --> 00:03:54.060
Third term has three 5's.

00:03:54.060 --> 00:03:55.900
Fourth term has four 5's.

00:03:55.900 --> 00:03:58.730
So the nth term, if this is, you
know, if this is 10, it's

00:03:58.730 --> 00:04:00.690
going to be 1 plus 10 5's.

00:04:00.690 --> 00:04:03.970
Or if it's n, if I'm kind of
abstracting a little bit, I'm

00:04:03.970 --> 00:04:06.730
going to have 1 plus
10-- sorry-- 1

00:04:06.730 --> 00:04:10.660
plus 5 times n, right?

00:04:10.660 --> 00:04:11.420
And try it out.

00:04:11.420 --> 00:04:14.010
If n is equal 1, it's going
to be 5 times 1.

00:04:14.010 --> 00:04:17.260
If n is equal to 2, it's
going to be 5 times 2.

00:04:17.260 --> 00:04:21.350
If n is equal to 3, it's
going to be 5 times 3.

00:04:21.350 --> 00:04:26.520
So that when the 50th term, if
we're talking about the 50th

00:04:26.520 --> 00:04:29.890
term, how many toothpicks
are we going to have?

00:04:29.890 --> 00:04:35.930
Well we're going to have 1 plus
5 times 50, right? n is

00:04:35.930 --> 00:04:37.690
50 here, which is
equal to what?

00:04:37.690 --> 00:04:41.240
5 times 50 is 250.

00:04:41.240 --> 00:04:47.390
That's 250 and then we add 1,
we're going to have 251

00:04:47.390 --> 00:04:50.390
toothpicks will be needed
to form the 50th

00:04:50.390 --> 00:04:53.300
figure in this sequence.


Patterns in sequences 1

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:04.350
The question we're given asks
us, how many students attended

00:00:04.350 --> 00:00:07.370
Einstein School in 2006?

00:00:07.370 --> 00:00:10.000
And in this little table, they
tell us how many attended in

00:00:10.000 --> 00:00:16.480
2001, 150 students, and in 2002,
225, 2003, we had 300

00:00:16.480 --> 00:00:19.170
students, but they don't
tell us 2006.

00:00:19.170 --> 00:00:22.600
So what I think they want us
to do is to see the pattern

00:00:22.600 --> 00:00:26.520
and how many students the
population of students grew by

00:00:26.520 --> 00:00:29.650
every year, and just assume that
that pattern continues

00:00:29.650 --> 00:00:31.530
through 2006.

00:00:31.530 --> 00:00:34.570
So we need to figure out what
the pattern is from one year

00:00:34.570 --> 00:00:38.360
to the next, and then we're
going to figure out 2004,

00:00:38.360 --> 00:00:42.980
2005, and then 2006.

00:00:42.980 --> 00:00:45.340
So the assumption that we're
making in this problem, and I

00:00:45.340 --> 00:00:47.880
think that the problem wants us
to make, is that whatever

00:00:47.880 --> 00:00:51.000
the pattern of growth was in
these three years, that they

00:00:51.000 --> 00:00:54.090
continue all the way to 2006.

00:00:54.090 --> 00:00:55.940
So let's see what's
happening here.

00:00:55.940 --> 00:00:58.290
When we go the first year, what
happens to our student

00:00:58.290 --> 00:00:58.920
population?

00:00:58.920 --> 00:01:02.370
We go from 150 to
225 students.

00:01:02.370 --> 00:01:04.390
And the easiest pattern to think
about is, well just how

00:01:04.390 --> 00:01:06.500
much are we increasing
by every year?

00:01:06.500 --> 00:01:11.270
So how much do we have to
go to from 150 to 225?

00:01:11.270 --> 00:01:12.100
We increased.

00:01:12.100 --> 00:01:14.290
Well, if you increase by 50,
you get to 200, so we

00:01:14.290 --> 00:01:15.600
increased another 25.

00:01:15.600 --> 00:01:19.270
So we increased by
75 students.

00:01:19.270 --> 00:01:21.970
And then they go from
225 to 300, how many

00:01:21.970 --> 00:01:23.270
did we increase by?

00:01:23.270 --> 00:01:24.450
Looks like 75 again.

00:01:24.450 --> 00:01:26.700
75 plus 225 is 300.

00:01:26.700 --> 00:01:30.330
So it looks like we increased
by another 75.

00:01:30.330 --> 00:01:34.570
So the pattern here is that each
year, 75 more students

00:01:34.570 --> 00:01:37.720
attend the Einstein School.

00:01:37.720 --> 00:01:39.560
So let's just continue
that pattern.

00:01:39.560 --> 00:01:42.290
So let's add another 75.

00:01:42.290 --> 00:01:46.060
Let's add another 75
to the 300 in 2003,

00:01:46.060 --> 00:01:51.420
and we'll get 375.

00:01:51.420 --> 00:01:55.440
In 2004, if we assume the
pattern continues, let's add

00:01:55.440 --> 00:02:00.070
another 75.

00:02:00.070 --> 00:02:01.120
And then what will
that give us?

00:02:01.120 --> 00:02:06.450
That'll give us 450 students
in 2005, if we assume the

00:02:06.450 --> 00:02:07.760
pattern continues.

00:02:07.760 --> 00:02:09.550
And we're almost there.

00:02:09.550 --> 00:02:14.870
Let's add a final
75, 75 plus 450.

00:02:14.870 --> 00:02:16.440
If you add 50, you get
to 500, so it's

00:02:16.440 --> 00:02:20.140
another 25, so it's 525.

00:02:20.140 --> 00:02:23.300
So that would be the answer to
our question, if we assume the

00:02:23.300 --> 00:02:27.820
pattern continued, that 525
students attended Einstein

00:02:27.820 --> 00:02:30.500
School in 2006.


Absolute value equations 1

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:06.570
We're told, solve the absolute
value of 3x minus 9 is equal

00:00:06.570 --> 00:00:10.490
to 0, and graph the solution
on a number line.

00:00:10.490 --> 00:00:13.460
So let's just rewrite the
absolute value equation.

00:00:13.460 --> 00:00:16.880
They told us that the absolute
value of 3x minus

00:00:16.880 --> 00:00:20.950
9 is equal to 0.

00:00:20.950 --> 00:00:23.010
So we're told that the
absolute value of the

00:00:23.010 --> 00:00:26.730
something-- in this case the
something is 3x minus 9-- is

00:00:26.730 --> 00:00:27.840
equal to 0.

00:00:27.840 --> 00:00:29.660
If I told you that the absolute
value of something is

00:00:29.660 --> 00:00:32.580
equal to 0, I'm telling you
that something has to be

00:00:32.580 --> 00:00:36.900
exactly 0 away from 0, or 0 away
from the origin on the

00:00:36.900 --> 00:00:37.750
number line.

00:00:37.750 --> 00:00:40.850
So the only thing that that
something could be is 0.

00:00:40.850 --> 00:00:44.040
If I told you that the absolute
value of x is equal

00:00:44.040 --> 00:00:46.800
to 0, you know that x has
to be equal to 0.

00:00:46.800 --> 00:00:50.720
That's the only value whose
absolute value is 0.

00:00:50.720 --> 00:00:53.480
So if I told you that the
absolute value of 3x minus 9

00:00:53.480 --> 00:00:59.820
is 0, than we know that 3x minus
9 has to be equal to 0,

00:00:59.820 --> 00:01:02.810
and that's kind of unique about
the 0 is that, it's the

00:01:02.810 --> 00:01:06.060
only number the has a unique,
that's only the

00:01:06.060 --> 00:01:07.800
absolute value of 0.

00:01:07.800 --> 00:01:09.980
If you had, say, a 1 here, you
could say, oh well, then this

00:01:09.980 --> 00:01:12.420
thing could be a 1
or a negative 1.

00:01:12.420 --> 00:01:16.130
But here, if you have a 0,
this thing can only be 0.

00:01:16.130 --> 00:01:18.632
So solving this equation is
fairly straightforward.

00:01:18.632 --> 00:01:22.330
If we want to isolate the 3x,
get rid of the negative 9 on

00:01:22.330 --> 00:01:27.180
the left-hand side, we add 9 to
both sides of the equation.

00:01:27.180 --> 00:01:30.140
Add 9 to both sides
of the equation,

00:01:30.140 --> 00:01:31.460
these 9's cancel out.

00:01:31.460 --> 00:01:32.560
That's the whole point.

00:01:32.560 --> 00:01:36.460
On the left-hand side, you're
just left with 3x, and on the

00:01:36.460 --> 00:01:40.350
right-hand side, you are
just left with 9.

00:01:40.350 --> 00:01:44.230
Now we want to solve for x,
so we have 3 times x.

00:01:44.230 --> 00:01:47.390
Let's divide it by 3, because 3
times x divided by 3 is just

00:01:47.390 --> 00:01:48.450
going to be x.

00:01:48.450 --> 00:01:50.990
But if we divide the left side
by 3, we have to divide the

00:01:50.990 --> 00:01:52.600
right side by 3.

00:01:52.600 --> 00:01:56.232
So we are left with-- these
guys cancel out.

00:01:56.232 --> 00:02:00.355
x is equal to 9 over
3, which is 3.

00:02:00.355 --> 00:02:02.210
And that's our solution.

00:02:02.210 --> 00:02:02.920
Now let's try out.

00:02:02.920 --> 00:02:04.760
Let's make sure that this
actually works.

00:02:04.760 --> 00:02:07.870
Let's substitute it back into
our original equation.

00:02:07.870 --> 00:02:11.490
So we have the absolute
value of 3 times x.

00:02:11.490 --> 00:02:14.490
Instead of x, I'll just put in
our actual answer that we got,

00:02:14.490 --> 00:02:21.670
3 times 3 minus 9 has got
to be equal to 0.

00:02:21.670 --> 00:02:22.860
So what's this going
to be equal to?

00:02:22.860 --> 00:02:24.390
3 times 3 is 9.

00:02:24.390 --> 00:02:28.020
So it's the absolute value of
9 minus 9, which is the

00:02:28.020 --> 00:02:31.130
absolute value of 0, which
is, indeed, 0.

00:02:31.130 --> 00:02:35.730
So it does, indeed, equal
0, and we are done.


Absolute value smallest to biggest

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:04.790
We're told to list in order from
smallest to largest. So

00:00:04.790 --> 00:00:07.290
each of these quantities, it
looks like we have expressions

00:00:07.290 --> 00:00:09.420
inside of absolute
value signs.

00:00:09.420 --> 00:00:12.320
And just as a bit of review,
absolute value just means your

00:00:12.320 --> 00:00:13.880
distance from 0.

00:00:13.880 --> 00:00:17.040
Or another way to think about
it is, if it's a negative

00:00:17.040 --> 00:00:18.290
number inside of the
absolute value

00:00:18.290 --> 00:00:19.470
sign, it becomes positive.

00:00:19.470 --> 00:00:21.800
If it's already positive,
it stays positive.

00:00:21.800 --> 00:00:23.590
So let's think about
these numbers.

00:00:23.590 --> 00:00:29.020
So the first one is the
absolute value of 5.

00:00:29.020 --> 00:00:30.890
How far is 5 away from 0?

00:00:30.890 --> 00:00:33.440
Well, it's 5 away from 0.

00:00:33.440 --> 00:00:35.900
So it's equal to 5.

00:00:35.900 --> 00:00:37.160
So I were to actually
draw a number line,

00:00:37.160 --> 00:00:38.610
you would see that.

00:00:38.610 --> 00:00:42.240
0 is here, 5 is over
here, this distance

00:00:42.240 --> 00:00:43.440
right there is 5.

00:00:43.440 --> 00:00:46.230
So the absolute value
of 5 is 5.

00:00:46.230 --> 00:00:50.620
Now, the next quantity they want
us to figure out is the

00:00:50.620 --> 00:00:55.590
absolute value of 9 minus 7.

00:00:55.590 --> 00:00:58.540
Well, this is the same thing
as the absolute value of 2.

00:00:58.540 --> 00:01:00.190
9 minus 7 is 2.

00:01:00.190 --> 00:01:04.099
And once again, 2 is just
2 units away from 0,

00:01:04.099 --> 00:01:05.239
so it's just 2.

00:01:05.239 --> 00:01:07.720
If you have a positive value in
the absolute value sign, it

00:01:07.720 --> 00:01:08.980
just is itself.

00:01:08.980 --> 00:01:11.760
The absolute value of 2 is 2.

00:01:11.760 --> 00:01:21.500
Then we have the absolute
value of 5 minus 15.

00:01:21.500 --> 00:01:22.810
Well, that's going to
be the same thing as

00:01:22.810 --> 00:01:23.610
the absolute value.

00:01:23.610 --> 00:01:26.660
5 minus 15 is negative 10, so
it's the same thing as the

00:01:26.660 --> 00:01:28.930
absolute value of negative 10.

00:01:28.930 --> 00:01:31.130
Now, there's two ways you
can think about it.

00:01:31.130 --> 00:01:33.210
If it's a negative number inside
of the absolute value

00:01:33.210 --> 00:01:35.560
sign, it just becomes the
positive version of it, so it

00:01:35.560 --> 00:01:37.010
just becomes 10.

00:01:37.010 --> 00:01:39.480
Another way to think about it
is, if you had negative 10,

00:01:39.480 --> 00:01:41.310
you would plot it-- it would
be out here someplace,

00:01:41.310 --> 00:01:44.160
negative 10, we'd have to extend
the number line-- it is

00:01:44.160 --> 00:01:46.120
10 to the left of 0.

00:01:46.120 --> 00:01:48.840
That's what the absolute
value is telling us.

00:01:48.840 --> 00:01:52.270
Then we have the absolute
value of 0.

00:01:55.210 --> 00:01:57.330
Well, 0 is just 0 away
from the number line.

00:01:57.330 --> 00:02:00.220
Absolute value of 0 is just 0.

00:02:00.220 --> 00:02:01.270
That is just 0.

00:02:01.270 --> 00:02:02.800
That's just right there.

00:02:02.800 --> 00:02:05.490
It has no distance
from the origin.

00:02:05.490 --> 00:02:09.340
And then finally, we have the
absolute value of negative 3.

00:02:12.640 --> 00:02:15.870
That's 3 to the left of 0, or
you can just think of it as

00:02:15.870 --> 00:02:19.630
getting rid of this negative
sign, so it is equal to 3.

00:02:19.630 --> 00:02:22.820
So, now that we've expressed
them all as just simple

00:02:22.820 --> 00:02:26.120
integers, let's list them in
order from smallest to

00:02:26.120 --> 00:02:26.720
largest.

00:02:26.720 --> 00:02:29.670
So of all of these values,
which is the smallest?

00:02:29.670 --> 00:02:32.680
This one is the smallest,
the absolute value of 0.

00:02:32.680 --> 00:02:36.050
So let me write that down,
absolute value of 0.

00:02:36.050 --> 00:02:38.160
What's the next smallest?

00:02:38.160 --> 00:02:38.860
What's the next one?

00:02:38.860 --> 00:02:41.570
Since we have a 2 there,
this is the next

00:02:41.570 --> 00:02:43.300
smallest, right there.

00:02:43.300 --> 00:02:45.870
And that original expression
was the absolute value of 9

00:02:45.870 --> 00:02:50.810
minus 7, so the absolute
value of 9 minus 7.

00:02:50.810 --> 00:02:52.040
And then what's the
next smallest?

00:02:52.040 --> 00:02:54.360
We have this 3, and then
we have a 5 and a 10.

00:02:54.360 --> 00:02:58.230
So the next smallest is
this 3, right here.

00:02:58.230 --> 00:03:01.140
That 3, and that original
expression was the absolute

00:03:01.140 --> 00:03:02.390
value of negative 3.

00:03:05.190 --> 00:03:08.460
Then we have this 5 over here,
which is just the absolute

00:03:08.460 --> 00:03:10.630
value of 5.

00:03:10.630 --> 00:03:13.600
And then finally, we have this
10 over here, which was the

00:03:13.600 --> 00:03:19.200
absolute value of 5 minus 15.

00:03:19.200 --> 00:03:21.530
And we are done.