Asymptotes of rational functions

WEBVTT
Kind: captions
Language: en

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In this video, we're going
to see if we can

00:00:02.210 --> 00:00:04.245
graph a rational function.

00:00:04.245 --> 00:00:08.760
A rational function is just
a function that has an

00:00:08.760 --> 00:00:10.420
expression on the numerator
and the denominator.

00:00:10.420 --> 00:00:12.660
It has a polynomial in the
numerator-- Let's see, we have

00:00:12.660 --> 00:00:16.600
x squared over-- and another
polynomial in the denominator

00:00:16.600 --> 00:00:19.460
--x squared minus 16.

00:00:19.460 --> 00:00:22.090
We could obviously graph this by
just trying out a bunch of

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points and then connecting
the dots.

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That's what a calculator would
do for us, a graphing

00:00:25.890 --> 00:00:26.740
calculator.

00:00:26.740 --> 00:00:29.580
But what we want to do is,
before we try out some points

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to kind of fill in the gaps, I
want to understand the basic

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structure of this graph first.

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And to understand that, I want
to see what happens as x gets

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really big, so x gets really
big, or x gets really, really

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small, as x goes in the
negative direction.

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Or another way we could think
about it, I want to understand

00:00:47.240 --> 00:00:52.210
what happens as the magnitude of
x, or the absolute value of

00:00:52.210 --> 00:00:58.050
x, becomes really big as it
approaches really, really

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bigness, or as it approaches
infinity.

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So when the size of x
approaches infinity.

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Which essentially is saying,
as x goes really far in the

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positive direction, or x goes
really far in the negative

00:01:12.480 --> 00:01:14.430
direction, what is going
to happen to the

00:01:14.430 --> 00:01:17.020
value of this function?

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So let's get out a calculator.

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Won't use the graphing part of
it just yet, but let's just

00:01:21.470 --> 00:01:22.720
try out some values.

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What happens when x
is equal to 10?

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It's going to be the same thing
as when x is equal to

00:01:26.880 --> 00:01:29.220
negative 10, because when you
put a negative 10 here, you

00:01:29.220 --> 00:01:31.230
square it, you get 100,
just like 10.

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Same here, negative 10, you
square it, you get the same

00:01:33.700 --> 00:01:35.320
thing as a positive 10.

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So whether you go in the super
high positive direction or the

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super low negative direction,
as you approach positive or

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negative infinity, you're going
to approach the same

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thing because you're squaring
the values.

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But let's try out some values.

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If I get 10 squared divided
by 10 squared

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minus 16, I get 1.19.

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Now what happens if x gets
a little bit bigger?

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This is x is equal to 10.

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What happens when x
is equal to 100?

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We have 100 squared divided
by 100 squared minus 16.

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I'm getting even closer to 1.

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When x was 10, around here,
we're getting y is 1.19.

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When x is 100, 100 squared
over 100 squared

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minus 16, y is 1.0016.

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Just for fun, let's try 1,000.

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So it's 1,000 squared divided
by 1,000 squared minus 16.

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And we're even closer to 1.

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So as the size of x gets larger
and larger and larger,

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our y gets closer
and closer to 1.

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And that would also be true
if this was a negative 10,

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because negative 10 squared over
negative 10 squared minus

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16 is going to be the
exact same thing.

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Because the negative, when
you square it, is

00:02:59.490 --> 00:03:00.700
going to be a positive.

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It's going to be the same thing
as 10 squared, same

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thing over here.

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So whether x gets really big or
x gets really small, we're

00:03:07.350 --> 00:03:10.190
going to be approaching
y is equal to 1.

00:03:10.190 --> 00:03:11.916
You could try it with a million
if you want, and

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you're going to get a number
even closer to 1.

00:03:14.640 --> 00:03:19.870
So as the size of x approaches
infinity, the absolute value

00:03:19.870 --> 00:03:25.400
of x, or the distance from the
origin, approaches infinity, y

00:03:25.400 --> 00:03:29.690
is approaching 1.

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Or another way to think about
it is, the graph of this

00:03:32.940 --> 00:03:37.700
function is going to approach
the line y is equal to 1.

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So let me graph the line
y is equal to 1.

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So I'll do it in a dotted line
because this isn't the graph

00:03:43.520 --> 00:03:47.230
of our function, but this is
a line that our function is

00:03:47.230 --> 00:03:48.960
approaching.

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So that is the graph
of y is equal to 1.

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Now, this idea of a function,
or the graph of a function,

00:03:57.110 --> 00:03:59.810
approaching a line but never
quite touching it.

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So this is going to get closer
and closer and closer to this

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line, y equal to one, in that
direction, but never quite

00:04:04.280 --> 00:04:07.430
getting close enough to it.

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It'll approach 0, its distance
from this y equals 1, but

00:04:11.220 --> 00:04:12.900
it'll never quite get there.

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This line that the graph
is approaching

00:04:15.190 --> 00:04:16.440
is called an asymptote.

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And it'll be even more
clear once I

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actually graph the function.

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We're going to work up there.

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And since it's a horizontal
line, we call this a

00:04:26.060 --> 00:04:27.310
horizontal asymptote.

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This is what our graph
approaches as we go in the

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positive direction, or really
far in the negative direction.

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Let's think about some of the
other interesting things about

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this, about this function
right here.

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Well one thing that might pop
out at you is this is a

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difference of squares.

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This is x squared
minus 4 squared.

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So we can rewrite this as
x squared over x plus 4

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times x minus 4.

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So what's going to happen here,
as x approaches either

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positive 4, or x approaches
negative 4?

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Well, first of all, try
those values out.

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If x is equal to 4, what
is going to happen?

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This expression right here, this
term right here, is going

00:05:11.740 --> 00:05:12.900
to be equal to 0.

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And we're going to
be dividing by 0.

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We cannot do that.

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Similarly, if x is equal
to negative 4, we'd

00:05:18.810 --> 00:05:20.060
be dividing by 0.

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This expression, right here,
is going to be equal to 0.

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We can't do that.

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We could say that this function
is undefined at x is

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equal to plus or minus 4.

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It can't equal those values
because we'd be dividing by 0

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in either one of those
circumstances.

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Now, what happens as we
approach those values?

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What happens as x approaches
negative 4?

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Let's just do that
one for fun.

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What happens as x approaches
negative 4?

00:06:07.150 --> 00:06:11.155
Let's say we're approaching it
from the negative direction.

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So let's try it out
in our calculator.

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So let's say we want to go from
the negative direction.

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So let's start with
negative 4.1.

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So if we have negative 4.1
squared divided by negative

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4.1 squared minus 16,
what do we get?

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We get 20.75.

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So we get this number,
whatever.

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Let's see if we get even
closer to negative 4.

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So let me just get
that entry there.

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So let's get a little bit
closer to negative 4.

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So instead of negative 4.1,
let's do negative 4.01.

00:07:02.950 --> 00:07:09.752
So let me insert a
negative 4.01.

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And then over here, this
is negative 4.01,

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and see what it is.

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Now we went to 200, so
we're getting to

00:07:19.320 --> 00:07:20.490
larger and larger values.

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Let's try negative 4.001.

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Let's try that out.

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Whoops, that's not what
I wanted to do.

00:07:30.620 --> 00:07:32.630
I wanted to do that.

00:07:32.630 --> 00:07:34.670
So let's try.

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No that's not what
I want to do.

00:07:36.985 --> 00:07:37.840
Let's see.

00:07:37.840 --> 00:07:44.000
So we want to go to, instead of
4.01, I want to do 4.001,

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and over there, negative
4.001.

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And what do we get?

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We get 2,000.

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So as we get closer and closer
to negative 4 from the

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negative direction, we're
approaching larger and larger,

00:08:00.470 --> 00:08:01.440
super larger numbers.

00:08:01.440 --> 00:08:04.360
And you can try it, if it's
4.0000001, it's going to get

00:08:04.360 --> 00:08:07.130
to smaller and smaller numbers--
or sorry, larger and

00:08:07.130 --> 00:08:08.170
larger numbers here.

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If you do 4.001, it's probably
going to be 20,000.

00:08:11.490 --> 00:08:12.540
And then if you add
another 0 here.

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So as we get closer and closer,
it's getting to larger

00:08:14.860 --> 00:08:16.250
and larger numbers.

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So we could say, as x approaches
negative 4, we

00:08:19.850 --> 00:08:22.950
could say y is approaching
infinity.

00:08:22.950 --> 00:08:25.970
It's getting to a larger and
larger and larger value.

00:08:25.970 --> 00:08:28.650
But we can't ever quite get
to x is equal to 4.

00:08:28.650 --> 00:08:29.680
It's undefined there.

00:08:29.680 --> 00:08:32.440
That will make the denominator
here equal to 0.

00:08:32.440 --> 00:08:35.570
So what we want to do here is,
we can never quite equal x

00:08:35.570 --> 00:08:36.700
equal negative 4.

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So let me see, x is equal
to 1, 2, 3, 4.

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We can never quite get to x
is equal to negative 4.

00:08:44.220 --> 00:08:47.520
Let me draw x is equal to
negative 4 as a dotted line,

00:08:47.520 --> 00:08:49.710
right there.

00:08:49.710 --> 00:08:52.090
That is x is equal
to negative 4.

00:08:55.050 --> 00:08:58.290
We can never quite get there,
but as we approached it from

00:08:58.290 --> 00:09:01.430
the negative side, as we had
4.1, then 4.01, we went to

00:09:01.430 --> 00:09:03.230
larger and larger values.

00:09:03.230 --> 00:09:05.330
And we also know that as we went
on the left-hand side, as

00:09:05.330 --> 00:09:09.290
we go to larger and larger x
values, that y will get closer

00:09:09.290 --> 00:09:10.530
and closer to 1.

00:09:10.530 --> 00:09:13.120
So you have a general sense of
what this part of the graph

00:09:13.120 --> 00:09:13.920
will look like.

00:09:13.920 --> 00:09:15.200
This part of the graph
is going to look

00:09:15.200 --> 00:09:19.230
something like that.

00:09:19.230 --> 00:09:21.700
As x gets to super negative
numbers, it gets closer and

00:09:21.700 --> 00:09:24.980
closer to 1, as x gets closer
and closer to negative 4 from

00:09:24.980 --> 00:09:28.700
the negative direction, it's
going to go closer and closer

00:09:28.700 --> 00:09:29.720
to infinity.

00:09:29.720 --> 00:09:32.245
You're going to get closer and
closer to a very-- It's going

00:09:32.245 --> 00:09:33.750
to get larger and larger,
I guess, is an

00:09:33.750 --> 00:09:35.280
easy way to say it.

00:09:35.280 --> 00:09:38.710
Now, just like x equal negative
4, x equals 4 will

00:09:38.710 --> 00:09:40.760
also be a point where the
graph is undefined.

00:09:40.760 --> 00:09:42.620
So let me graph that here.

00:09:42.620 --> 00:09:44.600
1, 2, 3, 4.

00:09:44.600 --> 00:09:45.850
Right here.

00:09:49.380 --> 00:09:50.936
Right over here.

00:09:50.936 --> 00:09:53.280
x is equal to 4.

00:09:53.280 --> 00:09:55.820
And, once again, what happens
as we approach x equals 4,

00:09:55.820 --> 00:09:57.890
let's say from the positive
direction?

00:09:57.890 --> 00:10:05.470
So as x approaches 4 from the
positive direction, what's

00:10:05.470 --> 00:10:06.220
going to happen?

00:10:06.220 --> 00:10:09.890
So this is like trying out x
is equal to 4.01, or x is

00:10:09.890 --> 00:10:14.040
equal to 4.001, or x
is equal to 4.0001.

00:10:14.040 --> 00:10:17.420
So we're just getting closer and
closer and closer to x is

00:10:17.420 --> 00:10:18.570
equal to 4.

00:10:18.570 --> 00:10:21.260
Well, these values are the exact
same values that we just

00:10:21.260 --> 00:10:22.980
tried on our calculator, except
they are the negative

00:10:22.980 --> 00:10:24.580
version of them, right?

00:10:24.580 --> 00:10:27.440
And we already saw that, just
the way that this function is

00:10:27.440 --> 00:10:30.340
set up, the negative numbers,
they get squared, so whether

00:10:30.340 --> 00:10:32.410
you take the negative or the
positive x values, it's going

00:10:32.410 --> 00:10:33.620
to be the same thing.

00:10:33.620 --> 00:10:34.930
This graph is symmetric.

00:10:34.930 --> 00:10:36.900
When x is equal to negative
5, it's the same thing

00:10:36.900 --> 00:10:37.860
as x is equal 5.

00:10:37.860 --> 00:10:40.010
When x is equal to negative
10, it's the same

00:10:40.010 --> 00:10:41.420
thing as x equals 10.

00:10:41.420 --> 00:10:42.720
So the same thing is
going to happen.

00:10:42.720 --> 00:10:44.465
You could try it out with your
calculator, if you like.

00:10:44.465 --> 00:10:46.900
If you try out these values,
you're going to see, as we get

00:10:46.900 --> 00:10:50.330
closer and closer to 4, we're
going to approach larger and

00:10:50.330 --> 00:10:50.960
larger numbers.

00:10:50.960 --> 00:10:53.190
These same numbers over here.

00:10:53.190 --> 00:10:57.820
So the graph over here, as we
get closer and closer to 4,

00:10:57.820 --> 00:11:00.300
we're going to approach larger
and larger numbers.

00:11:00.300 --> 00:11:02.990
And then here, as x gets larger
and larger and larger,

00:11:02.990 --> 00:11:06.510
we saw over here, we had these
horizontal asymptotes, y gets

00:11:06.510 --> 00:11:08.650
closer and closer to 1.

00:11:08.650 --> 00:11:13.750
So just like we called this a
horizontal asymptote, these

00:11:13.750 --> 00:11:16.460
values-- or you can even view
these vertical lines: x is

00:11:16.460 --> 00:11:21.290
equal to negative 4 and x is
equal to 4 --we call these

00:11:21.290 --> 00:11:22.540
vertical asymptotes.

00:11:28.920 --> 00:11:31.730
These are lines, asymptotes,
once again, they are lines

00:11:31.730 --> 00:11:35.140
that the graph approaches,
but never quite touches.

00:11:35.140 --> 00:11:37.280
So that's what's
going on here.

00:11:37.280 --> 00:11:40.680
And then we can think about
what's happening to the graph

00:11:40.680 --> 00:11:41.600
inside of here.

00:11:41.600 --> 00:11:43.450
So you could think of it
in a couple of ways.

00:11:43.450 --> 00:11:50.750
You could say, well, what
happens as x approaches 4 from

00:11:50.750 --> 00:11:52.100
the negative direction?

00:11:52.100 --> 00:11:55.430
So let's try that out, from
the negative direction.

00:11:55.430 --> 00:12:03.990
So what happens if you do 3.9
squared divided by 3.9

00:12:03.990 --> 00:12:08.440
squared minus 16?

00:12:08.440 --> 00:12:12.010
You get negative 19.25.

00:12:12.010 --> 00:12:16.070
Now what happens
if we do 3.99?

00:12:16.070 --> 00:12:18.440
So let me put another 9 here.

00:12:18.440 --> 00:12:20.933
So we're going to get closer
and closer to 4, and we're

00:12:20.933 --> 00:12:23.890
going to do it from
the left-hand side

00:12:23.890 --> 00:12:25.470
as we approach 4.

00:12:25.470 --> 00:12:30.100
So insert another 9 here.

00:12:30.100 --> 00:12:31.255
So even more negative.

00:12:31.255 --> 00:12:34.220
So let's just do one more.

00:12:34.220 --> 00:12:36.400
So we're going to be
even more negative.

00:12:36.400 --> 00:12:40.610
So let me make it 3.999.

00:12:40.610 --> 00:12:42.245
Get even closer to 4.

00:12:45.980 --> 00:12:47.860
You're getting even
more negative.

00:12:47.860 --> 00:12:50.930
And this is also going to be
true if we did negative 3.9,

00:12:50.930 --> 00:12:56.380
or negative 3.99, or negative
3.999, because when we square

00:12:56.380 --> 00:12:58.540
it, the negatives and the
positives are the same thing.

00:12:58.540 --> 00:13:00.660
You square negative 1,
you get a positive 1.

00:13:00.660 --> 00:13:07.835
So as we approach 4 from-- you
go 3.9, 3.99, we get closer

00:13:07.835 --> 00:13:09.930
and closer to 4 --we're
getting more and

00:13:09.930 --> 00:13:10.780
more negative numbers.

00:13:10.780 --> 00:13:13.040
We approach negative infinity.

00:13:13.040 --> 00:13:16.380
So as we approach-- let
me just graph it here.

00:13:16.380 --> 00:13:18.390
As we approach from this
direction, we're going to get

00:13:18.390 --> 00:13:22.440
smaller-- want to not touch our
asymptote --it's going to

00:13:22.440 --> 00:13:25.250
look something like that.

00:13:25.250 --> 00:13:27.430
As we approach it from the
left-hand side, we're getting

00:13:27.430 --> 00:13:29.040
smaller and smaller numbers.

00:13:29.040 --> 00:13:31.840
And that's also going to be true
as we approach negative 4

00:13:31.840 --> 00:13:33.220
from the right-hand
side, right?

00:13:33.220 --> 00:13:37.210
As we get negative 3.9,
3.99, 3.999, we're

00:13:37.210 --> 00:13:37.970
going to drop down.

00:13:37.970 --> 00:13:40.800
It's going to look something
like that.

00:13:40.800 --> 00:13:43.950
And now that we have a general
sense of what the graph is,

00:13:43.950 --> 00:13:46.530
now is a good time where
we could maybe plot

00:13:46.530 --> 00:13:49.150
a few points here.

00:13:49.150 --> 00:13:52.190
And the easiest one is, what
happens when x is equal to 0?

00:13:52.190 --> 00:13:56.570
You have 0 squared over
0 squared minus 16.

00:13:56.570 --> 00:13:59.100
So the point when x is equal
to 0, we're going to have 0

00:13:59.100 --> 00:14:02.430
over, well, negative 16,
which is just 0.

00:14:02.430 --> 00:14:07.300
So the point 0, 0 is
on this curve.

00:14:07.300 --> 00:14:09.900
And then we could try some other
points if you like, but

00:14:09.900 --> 00:14:12.170
the general shape here
is going to look

00:14:12.170 --> 00:14:13.805
something like this.

00:14:18.010 --> 00:14:20.360
You could plot more points if
you really want to nail down

00:14:20.360 --> 00:14:23.430
exactly what the curve is doing
in between, but here is

00:14:23.430 --> 00:14:25.120
the general structure.

00:14:25.120 --> 00:14:28.390
And we tried out a lot of values
with the calculator.

00:14:28.390 --> 00:14:31.280
And I did that because I really
wanted to show you why

00:14:31.280 --> 00:14:32.500
it's dropping down like this.

00:14:32.500 --> 00:14:34.180
And if you think about it,
it makes complete sense.

00:14:34.180 --> 00:14:36.830
As you get closer and closer,
let's say you get closer and

00:14:36.830 --> 00:14:38.200
closer to 4.

00:14:38.200 --> 00:14:41.790
Either way, as you get closer
and closer to 4, this is going

00:14:41.790 --> 00:14:44.340
to become a smaller and smaller
and smaller number,

00:14:44.340 --> 00:14:47.140
because this is the difference
between x and 4.

00:14:47.140 --> 00:14:49.480
So this is becoming
a smaller and

00:14:49.480 --> 00:14:52.450
smaller and smaller number.

00:14:52.450 --> 00:14:56.100
Then, when you take 1
over that, right?

00:14:56.100 --> 00:15:00.015
You can essentially view this as
x squared over x plus 4, or

00:15:00.015 --> 00:15:02.270
times 1 over x minus 4.

00:15:02.270 --> 00:15:04.870
If this is becoming smaller and
smaller, this whole thing,

00:15:04.870 --> 00:15:08.730
1 over a super small number,
is a super large number.

00:15:08.730 --> 00:15:10.540
So as you can imagine, you're
going to get larger and

00:15:10.540 --> 00:15:13.010
larger, and depending on whether
you are approaching

00:15:13.010 --> 00:15:17.090
from the positive or negative,
so whether this is a super

00:15:17.090 --> 00:15:20.470
small negative number or super
small positive number, that's

00:15:20.470 --> 00:15:21.720
going to flip the sign.

00:15:21.720 --> 00:15:24.210
But either way, the magnitude--
So we're getting

00:15:24.210 --> 00:15:27.060
to a very large magnitude in the
negative direction because

00:15:27.060 --> 00:15:29.800
the difference between
x and 4 on this side

00:15:29.800 --> 00:15:31.560
is negative, right?

00:15:31.560 --> 00:15:35.010
3.9 minus 4 is 0.1.

00:15:35.010 --> 00:15:36.980
Take the inverse of
that, it's 10.

00:15:36.980 --> 00:15:38.640
So we're getting negative
numbers here.

00:15:38.640 --> 00:15:40.830
You take the inverse, you're
going to get super large

00:15:40.830 --> 00:15:41.460
negative numbers.

00:15:41.460 --> 00:15:43.590
So I really want to give
you that intuition.

00:15:43.590 --> 00:15:47.070
But the general way of being
able to graph these type of

00:15:47.070 --> 00:15:48.940
things, your first thing you
want to do is identify the

00:15:48.940 --> 00:15:50.480
horizontal asymptotes.

00:15:50.480 --> 00:15:54.450
What happens as we get very--
the magnitude of our x is very

00:15:54.450 --> 00:15:56.960
large, so super positive
values or

00:15:56.960 --> 00:15:58.910
super negative values.

00:15:58.910 --> 00:16:00.860
You could try it out on a
calculator, if you like.

00:16:00.860 --> 00:16:02.820
You literally, if you try out
the value of a million or a

00:16:02.820 --> 00:16:05.410
billion, it will kind of
give you the answer.

00:16:05.410 --> 00:16:11.170
But the way you could also think
about it is, as x gets

00:16:11.170 --> 00:16:15.680
really large, you could view
that this thing, these terms

00:16:15.680 --> 00:16:18.790
right here grow so much faster--
I mean this is just a

00:16:18.790 --> 00:16:19.720
constant term.

00:16:19.720 --> 00:16:21.190
This term doesn't
matter anymore.

00:16:21.190 --> 00:16:23.140
If this is a million
and a million, who

00:16:23.140 --> 00:16:24.330
cares about the 16?

00:16:24.330 --> 00:16:28.970
So as x gets really large,
you could say that y is

00:16:28.970 --> 00:16:32.000
approximately x squared
over x squared.

00:16:32.000 --> 00:16:33.690
These two terms dominate.

00:16:33.690 --> 00:16:35.880
You don't need to worry
about the 16 anymore.

00:16:35.880 --> 00:16:39.510
And of course, this is equal to
1, which is exactly what we

00:16:39.510 --> 00:16:42.360
got when we plugged in
really large numbers.

00:16:42.360 --> 00:16:44.320
So, in a problem like this,
where you have the same

00:16:44.320 --> 00:16:47.840
coefficient, or where you have
the same degree on the

00:16:47.840 --> 00:16:52.040
numerator and the denominator,
you look at the coefficient of

00:16:52.040 --> 00:16:55.470
those terms. So in this case,
the coefficient is 1 and 1.

00:16:55.470 --> 00:16:59.080
So our horizontal asymptote is
going to be 1 divided by 1, or

00:16:59.080 --> 00:17:00.380
y is equal to 1.

00:17:00.380 --> 00:17:03.510
If this was 2x squared over
x squared minus 16, our

00:17:03.510 --> 00:17:05.859
horizontal asymptote would
be y is equal to 2.

00:17:05.859 --> 00:17:07.930
We would approach that
line, up there.

00:17:07.930 --> 00:17:11.030
If it was a negative 2, our
horizontal asymptote would be

00:17:11.030 --> 00:17:13.599
y is equal to negative 2.

00:17:13.599 --> 00:17:16.050
So that's how you identify the
horizontal asymptotes where

00:17:16.050 --> 00:17:18.190
you have the same degree in
the numerator and the

00:17:18.190 --> 00:17:19.115
denominator.

00:17:19.115 --> 00:17:22.220
If the denominator has a
larger degree, then the

00:17:22.220 --> 00:17:24.769
denominator is going to get
larger much faster than the

00:17:24.769 --> 00:17:26.329
numerator, and your asymptote
is going to be 0.

00:17:26.329 --> 00:17:28.359
I'll show an example of
that in the future.

00:17:28.359 --> 00:17:30.870
And obviously, if your numerator
has a higher degree

00:17:30.870 --> 00:17:34.400
than your denominator, it's
going to grow way faster than

00:17:34.400 --> 00:17:36.230
your denominator, and you won't
have any asymptote.

00:17:36.230 --> 00:17:38.340
You'll just keep growing,
or keep going in

00:17:38.340 --> 00:17:39.020
the negative direction.

00:17:39.020 --> 00:17:40.970
And that's actually the case
with all of the polynomials

00:17:40.970 --> 00:17:41.300
we've seen.

00:17:41.300 --> 00:17:43.620
You can do them all
as being over 1.

00:17:43.620 --> 00:17:48.070
In which case, there was no
horizontal asymptote.

00:17:48.070 --> 00:17:50.520
Now the vertical asymptotes you
identify by essentially

00:17:50.520 --> 00:17:53.110
just factoring the denominator
and figuring out where

00:17:53.110 --> 00:17:54.150
does it equal 0.

00:17:54.150 --> 00:17:57.870
Those are the points where the
function is not defined.

00:17:57.870 --> 00:17:59.660
And I'll show you in the future,
there are some special

00:17:59.660 --> 00:18:03.030
cases where they won't be
vertical asymptotes, and I

00:18:03.030 --> 00:18:06.870
guess that special case is,
for example, if you had--

00:18:06.870 --> 00:18:08.950
Well, I won't show you the
special case right now.

00:18:08.950 --> 00:18:10.430
I'll show you that in
a future video.

00:18:10.430 --> 00:18:14.460
But in general, if you factor
the bottom terms and they

00:18:14.460 --> 00:18:17.430
don't cancel out with anything
on the numerator, then you're

00:18:17.430 --> 00:18:20.030
going to be dealing with
a vertical asymptote.

00:18:20.030 --> 00:18:23.800
If I had another x minus 4 up
here, if my numerator was x

00:18:23.800 --> 00:18:29.500
squared times x minus 4, and
then these canceled out, and

00:18:29.500 --> 00:18:33.500
my expression simplified to
this, the equation would still

00:18:33.500 --> 00:18:36.500
be undefined at x is equal to 4,
because you would give the

00:18:36.500 --> 00:18:37.700
0 in the denominator.

00:18:37.700 --> 00:18:40.825
But since that x minus 4 cancels
out with the x minus 4

00:18:40.825 --> 00:18:42.560
in the numerator, it
would not have

00:18:42.560 --> 00:18:44.920
been a vertical asymptote.

00:18:44.920 --> 00:18:46.890
We'll look at that
in the future.

00:18:46.890 --> 00:18:49.070
But this equation wasn't that.

00:18:49.070 --> 00:18:51.310
So the general rule of thumb for
identifying the vertical

00:18:51.310 --> 00:18:55.140
asymptotes, factor the
denominator, figure out where

00:18:55.140 --> 00:18:58.440
the denominator equals 0, and
if those terms don't cancel

00:18:58.440 --> 00:19:00.800
out with any terms of the
numerator, then those are

00:19:00.800 --> 00:19:02.020
vertical asymptotes.

00:19:02.020 --> 00:19:05.050
And then to figure out the
behavior, I guess, within the

00:19:05.050 --> 00:19:07.370
asymptotes, you can
plot some points.

00:19:07.370 --> 00:19:08.860
You can try out some points.

00:19:08.860 --> 00:19:11.180
You can actually substitute
values for x and

00:19:11.180 --> 00:19:12.600
figure out what y is.

00:19:12.600 --> 00:19:15.100
Now just to validate that we
hopefully got the right

00:19:15.100 --> 00:19:21.240
answer, let's actually graph
our rational function.

00:19:21.240 --> 00:19:22.830
So let me turn it on.

00:19:22.830 --> 00:19:24.525
Let me graph it.

00:19:24.525 --> 00:19:32.450
And we say y is equal to
x squared divided by x

00:19:32.450 --> 00:19:35.980
squared minus 16.

00:19:35.980 --> 00:19:38.890
And let's see what we get.

00:19:38.890 --> 00:19:42.370
Nope, I just want to graph it.

00:19:42.370 --> 00:19:45.490
My range is off.

00:19:45.490 --> 00:19:47.150
Let me do my range.

00:19:47.150 --> 00:19:51.120
Let me see, x minimum value I
want for x, let's say it's

00:19:51.120 --> 00:19:53.140
negative 10.

00:19:53.140 --> 00:19:55.850
My maximum value I
want for x is 10.

00:19:55.850 --> 00:20:01.750
x scale is 1. y minimum value,
I want negative 10.

00:20:01.750 --> 00:20:04.900
y maximum value, I want 10.

00:20:04.900 --> 00:20:06.630
Then y scale, I want 1.

00:20:06.630 --> 00:20:09.150
Now let me graph it.

00:20:09.150 --> 00:20:09.990
There we go.

00:20:09.990 --> 00:20:10.930
Look at that.

00:20:10.930 --> 00:20:12.160
Just like what we drew.

00:20:12.160 --> 00:20:15.830
We have an asymptote, as x gets
really large, or x gets

00:20:15.830 --> 00:20:19.530
really, really small, that
asymptote is y is equal to 1.

00:20:19.530 --> 00:20:20.910
We have our vertical
asymptote.

00:20:20.910 --> 00:20:24.110
It graphed it because it tried
to connect the dots, but it

00:20:24.110 --> 00:20:26.170
essentially graphed our
asymptotes for us, but that

00:20:26.170 --> 00:20:27.860
wouldn't actually be
part of the graph.

00:20:27.860 --> 00:20:31.910
But as we approach 4 from
0, I guess we can

00:20:31.910 --> 00:20:33.560
say, we go super negative.

00:20:33.560 --> 00:20:36.230
As we approach negative 4 from
0, we get super negative.

00:20:36.230 --> 00:20:39.220
Because in either of those
situations, as you approach 4

00:20:39.220 --> 00:20:41.910
from this side, this term
is going to be negative.

00:20:41.910 --> 00:20:45.150
As you approach negative 4 from
thid side, this term,

00:20:45.150 --> 00:20:50.330
right here, is going to be--
Well, this term right here is

00:20:50.330 --> 00:20:53.440
going to be positive, but then
this term right here is going

00:20:53.440 --> 00:20:53.980
to be negative.

00:20:53.980 --> 00:20:55.350
Negative times a positive,
you could play

00:20:55.350 --> 00:20:56.090
with it as you like.

00:20:56.090 --> 00:20:58.550
But we approach negative
infinity in either case.

00:20:58.550 --> 00:21:01.270
And then as x approaches
infinity, this thing

00:21:01.270 --> 00:21:02.220
asymptotes away.

00:21:02.220 --> 00:21:04.960
So hopefully you
found that fun.


Proportionality

WEBVTT
Kind: captions
Language: en

00:00:00.440 --> 00:00:02.080
Let's talk about
proportionality.

00:00:08.670 --> 00:00:12.190
All this means, if we say
that two variables are

00:00:12.190 --> 00:00:15.590
proportional, it just means
that they are constant

00:00:15.590 --> 00:00:17.040
multiples of each other.

00:00:17.040 --> 00:00:23.920
So if we say that y is equal to
3 times x, we can say that

00:00:23.920 --> 00:00:27.330
y is-- and we can be a little
bit more specific-- we could

00:00:27.330 --> 00:00:29.380
say y is directly
proportional.

00:00:29.380 --> 00:00:41.670
y is directly proportional
to x.

00:00:41.670 --> 00:00:45.110
And of course, we could divide
both sides of this equation by

00:00:45.110 --> 00:00:50.550
3, and we would get 1/3
y is equal to x.

00:00:50.550 --> 00:00:53.050
So once again, x is a constant
multiple of y.

00:00:53.050 --> 00:00:56.600
We could write this as
x is equal to 1/3 y.

00:00:56.600 --> 00:01:10.090
So we could also write that x is
directly proportional to y.

00:01:10.090 --> 00:01:13.540
If y is directly proportional to
x, then x is also directly

00:01:13.540 --> 00:01:15.920
proportional to y,
by a different

00:01:15.920 --> 00:01:18.570
proportionality constant.

00:01:18.570 --> 00:01:21.220
So this term right here,
this is the constant of

00:01:21.220 --> 00:01:24.100
proportionality between y and
x, or between x and y.

00:01:24.100 --> 00:01:26.990
This is the constant
of proportionality.

00:01:26.990 --> 00:01:35.140
So in general, if I say that a
is directly proportional--

00:01:35.140 --> 00:01:37.500
sometimes you'll just see it
written as a is proportional--

00:01:37.500 --> 00:01:44.500
a is directly proportional
to b.

00:01:44.500 --> 00:01:48.630
In general, from this statement,
we could write that

00:01:48.630 --> 00:01:53.840
a is equal to some constant
times b, where this constant

00:01:53.840 --> 00:01:56.465
is equivalent to that constant
or that constant.

00:01:56.465 --> 00:01:59.040
It's equal to some number.

00:01:59.040 --> 00:02:01.710
And you could also say that b is
directly proportional to a,

00:02:01.710 --> 00:02:05.280
where if you divide both sides
of this equation by k, you

00:02:05.280 --> 00:02:09.900
would get b is equal to 1/k a.

00:02:09.900 --> 00:02:12.690
This is direct proportionality.

00:02:12.690 --> 00:02:16.180
Now another thing you might
see-- and we'll see it later

00:02:16.180 --> 00:02:21.170
in this video-- is being
inversely proportional.

00:02:21.170 --> 00:02:22.960
So let me draw a little
line here.

00:02:26.620 --> 00:02:29.520
So let me give you a few
examples of inverse

00:02:29.520 --> 00:02:30.910
proportionality.

00:02:30.910 --> 00:02:39.060
Let's say that y is equal
to 5 times 1/x.

00:02:39.060 --> 00:02:40.270
Notice what's going on here.

00:02:40.270 --> 00:02:43.980
In this situation, y is
proportional to the

00:02:43.980 --> 00:02:46.210
inverse of x, right?

00:02:46.210 --> 00:02:55.630
Or you could say that y is
inversely proportional-- I'll

00:02:55.630 --> 00:02:58.890
abbreviate it-- proportional
to x.

00:02:58.890 --> 00:03:02.510
It is directly proportional
to the inverse of x.

00:03:02.510 --> 00:03:06.820
So here is our proportionality
constant, but it is directly

00:03:06.820 --> 00:03:10.530
proportional to the
inverse of x.

00:03:10.530 --> 00:03:13.020
Here's another example,
and I'm just playing

00:03:13.020 --> 00:03:14.260
with letters here.

00:03:14.260 --> 00:03:19.310
I could write a is
equal to 10/b.

00:03:19.310 --> 00:03:23.270
Once again, a is inversely
proportional to b.

00:03:23.270 --> 00:03:28.710
You could rewrite this as a
is equal to 10 times 1/b.

00:03:28.710 --> 00:03:32.130
a is proportional to the
inverse of b, or it's

00:03:32.130 --> 00:03:35.280
inversely proportional to b.

00:03:35.280 --> 00:03:37.660
So now that we have that out
of the way, let's do some

00:03:37.660 --> 00:03:43.240
examples using our newly-found
knowledge of direct

00:03:43.240 --> 00:03:47.210
proportionality and inverse
proportionality.

00:03:47.210 --> 00:03:49.940
So here I have a problem.

00:03:49.940 --> 00:03:55.090
It says that x is proportional
to y.

00:03:55.090 --> 00:04:00.620
So that means that x is equal
to some constant times y.

00:04:00.620 --> 00:04:03.150
And just so you're familiar with
other notation that you

00:04:03.150 --> 00:04:06.150
might see when you're talking
about proportionality.

00:04:06.150 --> 00:04:08.570
Other ways to write this
statement here. x is

00:04:08.570 --> 00:04:09.510
proportional to y.

00:04:09.510 --> 00:04:11.730
You could write it like this.
x is equal to some

00:04:11.730 --> 00:04:13.490
constant times y.

00:04:13.490 --> 00:04:19.100
Or you could write it as
x is proportional to y.

00:04:19.100 --> 00:04:22.680
That little, I don't know, left
facing fish looking thing

00:04:22.680 --> 00:04:23.820
means proportional.

00:04:23.820 --> 00:04:25.550
This and this mean
the same thing.

00:04:25.550 --> 00:04:28.660
This means that x is equal
to some constant times y.

00:04:28.660 --> 00:04:33.400
Or sometimes you'll even see
it x squiggly line y. x is

00:04:33.400 --> 00:04:35.790
equal to some constant
times y.

00:04:35.790 --> 00:04:39.880
This, this, and this, and that
are all equivalent statements.

00:04:39.880 --> 00:04:41.590
But this is the most useful
because you say, oh, there's

00:04:41.590 --> 00:04:46.360
some constant, maybe we can
solve for that constant.

00:04:46.360 --> 00:04:49.320
Then the next statement here
says, x is proportional to y

00:04:49.320 --> 00:04:52.540
and inversely proportional
to z.

00:04:52.540 --> 00:04:57.180
So that second statement says
that x is equal to k.

00:04:57.180 --> 00:04:58.930
It's going to be a different
constant, not necessarily the

00:04:58.930 --> 00:04:59.760
same constant.

00:04:59.760 --> 00:05:02.520
So let me put a different--
well, I'll call it k2, I'll

00:05:02.520 --> 00:05:06.300
call this k1-- is equal to
some other constant-- not

00:05:06.300 --> 00:05:08.220
necessarily some other, but I'm
guessing it's going to be

00:05:08.220 --> 00:05:13.940
some other constant-- times the
inverse of z, times 1/z.

00:05:13.940 --> 00:05:16.530
That's what inversely
proportional to z means.

00:05:16.530 --> 00:05:20.530
And other ways we could write
this is, x is proportional to

00:05:20.530 --> 00:05:22.380
the inverse of z.

00:05:22.380 --> 00:05:26.460
x is inversely proportional
to z, or x is inversely

00:05:26.460 --> 00:05:27.800
proportional to z.

00:05:27.800 --> 00:05:30.150
These things in orange
are all equivalent.

00:05:30.150 --> 00:05:31.950
I just wanted to make you
familiar with it in case you

00:05:31.950 --> 00:05:33.430
ever see it.

00:05:33.430 --> 00:05:37.710
And then they tell us, OK, if
x is proportional to y,

00:05:37.710 --> 00:05:40.830
inversely proportional to z-- we
wrote that already-- and x

00:05:40.830 --> 00:05:44.250
is equal to 2 when
y is equal to 10.

00:05:44.250 --> 00:05:45.992
So let's do that.

00:05:45.992 --> 00:05:49.490
x is equal to 2 when
y is equal to 10.

00:05:49.490 --> 00:05:57.010
So x is equal to 2 when
k1 is multiplied by y.

00:05:57.010 --> 00:06:00.920
That's what that statement tells
us, and y is 10 when k1

00:06:00.920 --> 00:06:02.710
is multiplied by 10.

00:06:02.710 --> 00:06:04.770
x is 2 when y is 10.

00:06:04.770 --> 00:06:06.310
I just put those numbers
in there.

00:06:06.310 --> 00:06:08.850
And then we can actually use
this now to solve for k1.

00:06:08.850 --> 00:06:11.130
But before that, let's
see what this next

00:06:11.130 --> 00:06:11.810
statement tells us.

00:06:11.810 --> 00:06:16.190
It says x is 2 when y
is 10 and z is 25.

00:06:16.190 --> 00:06:34.710
So x is 2 when k2 is multiplied
by 1/z or 1/25.

00:06:34.710 --> 00:06:36.760
So that's what that second
statement tells us.

00:06:36.760 --> 00:06:40.020
So they say find x when
y is equal to 8 and

00:06:40.020 --> 00:06:42.380
z is equal to 35.

00:06:42.380 --> 00:06:44.460
So essentially what they're
asking us, hey, why don't you

00:06:44.460 --> 00:06:47.280
use this information that we
gave you in the green and the

00:06:47.280 --> 00:06:50.680
red, solve for the different
proportionality constants, and

00:06:50.680 --> 00:06:54.490
then use that to write, I guess,
the specific equations

00:06:54.490 --> 00:06:57.820
for the proportionality or the
inverse proportionality, and

00:06:57.820 --> 00:06:59.940
then solve for y or z.

00:06:59.940 --> 00:07:01.390
Let's just do it.

00:07:01.390 --> 00:07:04.170
So here, up here in green, let
me rewrite it over here.

00:07:04.170 --> 00:07:10.320
We know that x is 2 when k1 is
multiplied by y where y is 10.

00:07:10.320 --> 00:07:16.850
Divide both sides of this
equation by 10.

00:07:16.850 --> 00:07:18.740
You get 2 over 10 is 1/5.

00:07:18.740 --> 00:07:21.880
You get k1 is equal to 1/5.

00:07:21.880 --> 00:07:24.460
So that tells us that this first
equation right here--

00:07:24.460 --> 00:07:27.400
I'll write it in that original
color-- is x is

00:07:27.400 --> 00:07:31.760
equal to 1/5 y.

00:07:31.760 --> 00:07:38.930
Now, that second statement in
red, we know that 2 is equal

00:07:38.930 --> 00:07:43.040
to k2 times 1 over 25.

00:07:43.040 --> 00:07:47.460
Let's multiply both sides
of this equation by 25.

00:07:47.460 --> 00:07:48.750
This cancels out.

00:07:48.750 --> 00:07:53.730
And we're left with k2--
k2 is equal to 50.

00:07:53.730 --> 00:07:57.650
k2 is equal to 50, so we can
write this equation right here

00:07:57.650 --> 00:08:05.420
that x is equal to 50 times
1/z or is equal to 50/z.

00:08:05.420 --> 00:08:08.660
So we've now solved for
the two constants of

00:08:08.660 --> 00:08:10.930
proportionality for these two
equations, so now we can

00:08:10.930 --> 00:08:13.880
answer the second part
of this question.

00:08:13.880 --> 00:08:18.190
Find x when y is equal to
8 and z is equal to 35.

00:08:18.190 --> 00:08:21.220
So the situation when y is equal
to 8-- we just go right

00:08:21.220 --> 00:08:27.980
here-- x is equal to 1/5 times
8, which is equal to 8/5.

00:08:27.980 --> 00:08:31.460
So that's what x is equal
to when y is equal to 8.

00:08:31.460 --> 00:08:39.289
When z is equal to 35, once
again, x is equal to 50/z, is

00:08:39.289 --> 00:08:43.046
equal to 50/35.

00:08:43.046 --> 00:08:45.170
Now let's see, we can divide
the numerator and the

00:08:45.170 --> 00:08:51.070
denominator here by
5, so we get 10/7.

00:08:51.070 --> 00:08:54.270
So when z is equal to
35, we get 10/7.

00:08:54.270 --> 00:08:58.720
When y is equal to 8,
x is equal to 8/5.

00:08:58.720 --> 00:09:02.140
So those are our two answers,
right there.

00:09:02.140 --> 00:09:05.010
Let's do another one.

00:09:05.010 --> 00:09:06.510
Ohm's Law.

00:09:06.510 --> 00:09:11.180
Ohm's Law states that current
flowing in a wire is inversely

00:09:11.180 --> 00:09:14.170
proportional to resistance
of the wire.

00:09:14.170 --> 00:09:18.160
So current flowing in a wire is
inversely proportional to

00:09:18.160 --> 00:09:20.100
resistance of a wire.

00:09:20.100 --> 00:09:23.690
So let's use I for current.

00:09:23.690 --> 00:09:25.960
I is equal to current.

00:09:25.960 --> 00:09:29.550
And we'll use R for
resistance.

00:09:29.550 --> 00:09:32.732
R is equal to resistance.

00:09:32.732 --> 00:09:36.700
And you might be wondering why
I picked I, but later on when

00:09:36.700 --> 00:09:39.060
you start studying electricity
and maybe you become an

00:09:39.060 --> 00:09:42.920
electrical engineer, you'll see
that I is the conventional

00:09:42.920 --> 00:09:44.330
letter used for current.

00:09:44.330 --> 00:09:47.620
And I won't go into the details
of why, just yet.

00:09:47.620 --> 00:09:51.310
But it tells us that current,
that I, is inversely

00:09:51.310 --> 00:09:52.680
proportional to resistance.

00:09:52.680 --> 00:09:55.140
So that first statement I
underlined in yellow says that

00:09:55.140 --> 00:09:57.855
current is inversely
proportional.

00:09:57.855 --> 00:10:02.970
It's equal to some constant
times the inverse of

00:10:02.970 --> 00:10:03.900
resistance.

00:10:03.900 --> 00:10:08.440
It's inversely proportional
to resistance of the wire.

00:10:08.440 --> 00:10:13.530
If the current is 2.5 amperes
when the resistance is 20

00:10:13.530 --> 00:10:21.510
ohms. So the current is 2.5 when
k is multiplied by 1 over

00:10:21.510 --> 00:10:27.340
a resistance of 20 ohms, 1/20
ohms. And I'm assuming,

00:10:27.340 --> 00:10:29.650
because they're going to keep
the units in amperes and ohms

00:10:29.650 --> 00:10:33.680
later on, so we could write the
units here if we wanted

00:10:33.680 --> 00:10:36.240
to, but I'll keep it simple
and not write the units.

00:10:36.240 --> 00:10:39.680
They ask, find the resistance
when the current is 5 amperes.

00:10:39.680 --> 00:10:41.420
So this first statement,
right here.

00:10:41.420 --> 00:10:45.070
The current is 2.5 amperes
when the resistance is 20

00:10:45.070 --> 00:10:50.180
ohms. That's this equation,
right here.

00:10:50.180 --> 00:10:52.320
So we can use this
to solve for k.

00:10:52.320 --> 00:10:58.690
You multiply both sides
of this by 20.

00:10:58.690 --> 00:11:01.870
These cancel out, and you're
left with k is equal to--

00:11:01.870 --> 00:11:04.980
what's 20 times 2.5?

00:11:04.980 --> 00:11:07.500
20 times 2 is 40.

00:11:07.500 --> 00:11:09.950
20 times 0.5 is 10.

00:11:09.950 --> 00:11:12.420
So it's going to be
50, 40 plus 10.

00:11:12.420 --> 00:11:14.380
So k is equal to 50.

00:11:14.380 --> 00:11:21.200
So in this situation, this
equation is I is equal to 50

00:11:21.200 --> 00:11:25.190
times 1/R or 50/R.

00:11:25.190 --> 00:11:28.300
That's our relationship,
right there.

00:11:28.300 --> 00:11:30.390
And then they ask, find the
resistance when the

00:11:30.390 --> 00:11:31.940
current is 5 amperes.

00:11:38.140 --> 00:11:42.610
So when our current is 5
amperes, so 5 is equal to 50

00:11:42.610 --> 00:11:45.220
over the resistance.

00:11:45.220 --> 00:11:48.930
Let's multiply both sides
of this equation by the

00:11:48.930 --> 00:11:55.210
resistance, and you get
5R is equal to 50.

00:11:55.210 --> 00:11:59.900
I just multiplied both sides
of this equation by R.

00:11:59.900 --> 00:12:02.870
These canceled out, so I
got 5R is equal to 50.

00:12:02.870 --> 00:12:06.560
Divide both sides of
this equation by 5.

00:12:06.560 --> 00:12:12.010
We get R is equal to 50
divided by 5 is 10.

00:12:12.010 --> 00:12:15.090
So the answer is, when the
current is 5 amperes, the

00:12:15.090 --> 00:12:19.650
resistance will be equal to 10
ohms. And we could write it

00:12:19.650 --> 00:12:24.080
like this, 10 ohms,
just like that.

00:12:24.080 --> 00:12:25.330
Let's do one more.

00:12:28.420 --> 00:12:36.520
The intensity of light is
inversely proportional to

00:12:36.520 --> 00:12:41.050
the-- now let's be careful
here-- to the square of the

00:12:41.050 --> 00:12:47.680
distance between the light
source and the object being

00:12:47.680 --> 00:12:48.860
illuminated.

00:12:48.860 --> 00:12:54.580
So let's just say, well,
we could use I again.

00:12:54.580 --> 00:13:03.490
Let's say I is equal to
intensity of light.

00:13:03.490 --> 00:13:08.260
And let's say that D-- I'm going
to do it in a different

00:13:08.260 --> 00:13:17.160
color-- D is equal to the
distance between the light

00:13:17.160 --> 00:13:21.890
source and object being
illuminated.

00:13:25.430 --> 00:13:26.070
So that's what?

00:13:26.070 --> 00:13:28.250
D, the distance between the
light source and the object

00:13:28.250 --> 00:13:28.850
being illuminated.

00:13:28.850 --> 00:13:30.840
Now what does this first
sentence tell us?

00:13:30.840 --> 00:13:33.980
The intensity of the light is
inversely proportional.

00:13:33.980 --> 00:13:37.760
So the intensity of the light,
I, is inversely proportional.

00:13:37.760 --> 00:13:41.070
So it's going to be some
proportionality constant times

00:13:41.070 --> 00:13:43.730
the inverse.

00:13:43.730 --> 00:13:46.020
But notice, the intensity
of light is inversely

00:13:46.020 --> 00:13:48.800
proportional to the square
of the distance.

00:13:48.800 --> 00:13:50.350
Not just to the distance.

00:13:50.350 --> 00:13:52.330
So to the square of
the distance.

00:13:52.330 --> 00:13:55.010
So to distance squared.

00:13:55.010 --> 00:13:56.740
If it just said to the
distance, we would

00:13:56.740 --> 00:13:58.080
just have a D here.

00:13:58.080 --> 00:14:00.280
But it says the intensity
of light is inversely

00:14:00.280 --> 00:14:00.780
proportional.

00:14:00.780 --> 00:14:04.690
So 1 over the square of the
distance between the light

00:14:04.690 --> 00:14:07.050
source and the object
being illuminated.

00:14:07.050 --> 00:14:10.600
So that's what this first
statement is going to give us,

00:14:10.600 --> 00:14:12.530
this equation right here.

00:14:12.530 --> 00:14:18.160
Now, they tell us, a light meter
that is 10 meters-- so

00:14:18.160 --> 00:14:21.670
they're saying, when distance
is equal to 10 meters-- a

00:14:21.670 --> 00:14:24.700
light meter that is 10 meters
from a light source

00:14:24.700 --> 00:14:27.640
registers 35 lux.

00:14:27.640 --> 00:14:30.780
So when the distance is equal to
10 meters, they're telling

00:14:30.780 --> 00:14:33.280
us that the intensity-- the
lux is a measure of

00:14:33.280 --> 00:14:39.830
intensity-- the intensity
is equal to 35 lux.

00:14:39.830 --> 00:14:43.400
So what intensity would register
25 meters from the

00:14:43.400 --> 00:14:44.340
light source?

00:14:44.340 --> 00:14:47.220
So this first statement they
gave us, that I wrote down the

00:14:47.220 --> 00:14:50.140
information in this purple
color, we can use that to

00:14:50.140 --> 00:14:53.070
solve for the proportionality
constant.

00:14:53.070 --> 00:14:55.680
And then once we have that
constant, we can solve for I

00:14:55.680 --> 00:14:58.980
or D, given one or the other.

00:14:58.980 --> 00:15:04.020
So that statement told us, they
told us, that when D is

00:15:04.020 --> 00:15:06.400
equal to 10, I is equal to 35.

00:15:06.400 --> 00:15:13.690
So I is equal to 35, so 35 is
equal to K times 1 over D

00:15:13.690 --> 00:15:15.780
squared, 1 over 10 squared.

00:15:15.780 --> 00:15:23.790
Or we could say that 35 is
equal to k times 1/100,

00:15:23.790 --> 00:15:27.600
multiply-- 10 squared is just
100-- multiply both sides of

00:15:27.600 --> 00:15:30.970
this equation by 100.

00:15:30.970 --> 00:15:32.700
These cancel out.

00:15:32.700 --> 00:15:36.260
And we're left with k is equal
to-- what is this?

00:15:36.260 --> 00:15:40.160
3,500.

00:15:40.160 --> 00:15:43.230
So we've solved, we've used this
statement here where they

00:15:43.230 --> 00:15:45.740
gave us some values
to solve for our

00:15:45.740 --> 00:15:47.430
proportionality constant.

00:15:47.430 --> 00:15:50.790
So now we know that the equation
is-- the intensity is

00:15:50.790 --> 00:15:58.230
equal to 3,500-- that's k
times 1 over D squared.

00:15:58.230 --> 00:16:01.440
Or you can just write
over D squared.

00:16:01.440 --> 00:16:05.790
Now, they say, what intensity
would it register 25 meters

00:16:05.790 --> 00:16:07.000
from the light source?

00:16:07.000 --> 00:16:09.350
So they're saying D is 25.

00:16:09.350 --> 00:16:18.300
So intensity is equal to 3,500
divided by 25 squared.

00:16:18.300 --> 00:16:20.530
Our distance is 25 meters.

00:16:20.530 --> 00:16:31.870
So the intensity is going to be
equal to 3,500 divided by--

00:16:31.870 --> 00:16:34.270
well, actually, let's
just keep it simple.

00:16:34.270 --> 00:16:39.570
Let me see, well, this is
divided by 25 times 25.

00:16:39.570 --> 00:16:41.200
It's 625.

00:16:41.200 --> 00:16:43.410
Let's just get the
calculator out.

00:16:43.410 --> 00:16:48.920
So we get, it's going to be
3,500 divided by 25 times 25,

00:16:48.920 --> 00:16:54.010
which is 625, which
is equal to 5.6.

00:16:54.010 --> 00:16:58.930
It's equal to 5.6 lux,
which is the

00:16:58.930 --> 00:17:01.000
unit for light intensity.

00:17:01.000 --> 00:17:03.110
Hopefully you found
that useful.


Box-and-whisker Plot

WEBVTT
Kind: captions
Language: en

00:00:00.512 --> 00:00:01.812
- In this video, we're going to talk about

00:00:01.812 --> 00:00:06.564
box-and-whisker diagrams.

00:00:09.933 --> 00:00:12.725
And these are actually quite common.

00:00:12.725 --> 00:00:13.864
Box-and-whisker diagrams.

00:00:13.864 --> 00:00:17.418
Especially if you ever
do any stock trading.

00:00:17.418 --> 00:00:20.913
If you go to most stock chart providers,

00:00:23.205 --> 00:00:25.870
a box-and-whisker diagram is
usually one of the options.

00:00:25.870 --> 00:00:28.280
So let's understand how to generate one

00:00:28.280 --> 00:00:30.149
and essentially how to read one using

00:00:30.149 --> 00:00:32.323
just some data that we have up here.

00:00:32.323 --> 00:00:34.253
So the first thing we
want to do is order it

00:00:34.253 --> 00:00:36.793
because it we're going to
be finding a lot of medians

00:00:36.793 --> 00:00:39.190
when we want to generate
our box-and-whisker diagram.

00:00:39.190 --> 00:00:41.831
So let's just order the numbers up here.

00:00:41.831 --> 00:00:45.001
So let's see, it looks
like the lowest number here

00:00:45.001 --> 00:00:49.125
is 49...so I have that 49.

00:00:49.125 --> 00:00:52.274
Then I have another 49 right here,

00:00:52.274 --> 00:00:57.274
then I have another 49 right over there,

00:00:58.227 --> 00:01:00.828
and then do I have any 50s?

00:01:00.828 --> 00:01:02.637
Yes, I have two 50s.

00:01:02.637 --> 00:01:04.282
One, two 50s.

00:01:04.282 --> 00:01:07.918
I have two 50s...one, two...

00:01:07.918 --> 00:01:09.178
Do I have any 51s?

00:01:09.178 --> 00:01:12.408
I have one 51 as far as I can tell.

00:01:12.408 --> 00:01:14.359
So one 51.

00:01:14.359 --> 00:01:16.045
Do I have a 52?

00:01:16.045 --> 00:01:18.860
52? One 52.

00:01:19.352 --> 00:01:21.023
I have a 52...a 53?

00:01:21.023 --> 00:01:24.417
I have one 53...two 53s.

00:01:24.417 --> 00:01:29.417
53 and 53, 54...there's one.

00:01:30.125 --> 00:01:33.437
54...I don't have any other ones.

00:01:33.437 --> 00:01:36.870
55, I don't see one 55 here.

00:01:36.870 --> 00:01:40.641
56? I see a 56 right there.

00:01:42.025 --> 00:01:46.257
56, and then 57. I have
two--three of them.

00:01:46.257 --> 00:01:49.264
One, two, three.

00:01:49.264 --> 00:01:54.099
One, two, three 57s.

00:01:54.099 --> 00:01:56.192
And I've got a 58.

00:01:56.699 --> 00:02:00.658
A 58, a 59...

00:02:00.947 --> 00:02:03.325
a 59--let me scroll over a little bit.

00:02:03.325 --> 00:02:06.433
And then I have a 67 and I'm done,

00:02:06.785 --> 00:02:08.511
I've ordered all of these numbers

00:02:08.511 --> 00:02:11.843
and now it's a lot easier
to find the median.

00:02:11.843 --> 00:02:14.240
Now, the median as you
may or may not remember

00:02:14.240 --> 00:02:16.129
is the middle of these numbers.

00:02:16.129 --> 00:02:17.876
So how many total numbers do we have?

00:02:18.676 --> 00:02:21.635
We have one, two, three, four, five, six,

00:02:21.635 --> 00:02:24.766
seven, eight, nine, ten,
eleven, twelve, thirteen,

00:02:24.766 --> 00:02:28.931
fourteen, fifteen,
sixteen, seventeen numbers.

00:02:28.931 --> 00:02:31.654
So the middle number, if
you have seventeen numbers,

00:02:31.654 --> 00:02:33.056
there's two way to think about it:

00:02:33.056 --> 00:02:36.389
however many numbers you
have, we have seventeen,

00:02:36.389 --> 00:02:40.776
you add 1 to 17 and you divide by 2

00:02:40.776 --> 00:02:42.361
and you count that many spaces from

00:02:42.361 --> 00:02:44.922
either the left or the right-hand side

00:02:44.922 --> 00:02:46.993
and you will get the middle number.

00:02:46.993 --> 00:02:48.781
So this is eighteen divided by 2

00:02:48.781 --> 00:02:50.509
which is equaled to 9.

00:02:50.509 --> 00:02:53.049
So one, two, three, four, five, six,

00:02:53.049 --> 00:02:56.339
seven, eight, nine, 53.

00:02:56.339 --> 00:02:57.986
Should work from this side as well.

00:02:57.986 --> 00:03:00.727
One, two, three, four, five, six,

00:03:00.727 --> 00:03:04.384
seven, eight, nine, you got 53.

00:03:04.384 --> 00:03:06.071
And if you ever have a
situation where it's like

00:03:06.071 --> 00:03:08.448
let's say you only have sixteen numbers

00:03:08.448 --> 00:03:10.520
and it would be sixteen
plus one divided by 2

00:03:10.520 --> 00:03:11.861
you get eight and a half?

00:03:11.861 --> 00:03:15.112
That means count eight
spaces and then go halfway

00:03:15.112 --> 00:03:17.693
between that number and the
number right after that.

00:03:17.693 --> 00:03:19.785
Which we saw that in
a few diagrams before.

00:03:19.785 --> 00:03:22.710
But anyway, 53 is the median.

00:03:22.710 --> 00:03:26.266
53 right here is the median.

00:03:27.696 --> 00:03:29.274
Now, the next thing we want to do

00:03:29.274 --> 00:03:32.219
when we want to generate
a box-and-whisker diagram

00:03:32.219 --> 00:03:35.571
is figure out the median of the two halves

00:03:35.571 --> 00:03:38.578
of the data, notice, if you pick 53,

00:03:38.578 --> 00:03:41.931
as kind of dividing line,
half the data is above it,

00:03:41.931 --> 00:03:43.799
half the data is below it.

00:03:43.799 --> 00:03:46.562
If someone, if, you know,
these are people's ages

00:03:46.562 --> 00:03:49.468
and they say they're 53,
they're the 50th percentile.

00:03:49.468 --> 00:03:54.100
50 percent of the people
are above their age

00:03:54.100 --> 00:03:57.066
50 percent of the people are below it.

00:03:57.066 --> 00:03:58.651
This is the middle number.

00:03:58.651 --> 00:04:01.658
What we want to do now is find
the middles of the middle.

00:04:01.658 --> 00:04:05.783
Or find essentially what is the
25th and the 75th percentile

00:04:05.783 --> 00:04:08.545
So what's the number where only one-fourth

00:04:08.545 --> 00:04:10.455
of the numbers are less than that?

00:04:10.455 --> 00:04:12.832
And there's actually a
little ambiguity here.

00:04:12.832 --> 00:04:15.149
So if I want to find the middle of this

00:04:15.149 --> 00:04:17.180
left-hand portion right here,

00:04:17.180 --> 00:04:20.512
it's debatable whether I want to include

00:04:20.512 --> 00:04:22.341
this median right here.

00:04:22.341 --> 00:04:24.941
And actually both of those are valid ways

00:04:24.941 --> 00:04:27.400
of generating box-and-whisker diagrams.

00:04:27.400 --> 00:04:29.858
The typical algorithm,
an algorithm is just

00:04:29.858 --> 00:04:32.804
a way, a process for doing things,

00:04:32.804 --> 00:04:36.441
is to include this
median in the bottom list

00:04:36.441 --> 00:04:40.240
and to include this
median in the top list.

00:04:40.240 --> 00:04:43.389
So let's find the median
of this list right here.

00:04:43.389 --> 00:04:46.538
So we have one, two, three, four,

00:04:46.538 --> 00:04:50.826
five, six, seven, eight, nine numbers.

00:04:51.979 --> 00:04:56.979
9 + 1 over 2 is equal to
10 over 2 is equal to 5.

00:04:57.155 --> 00:05:01.076
So we count one, two, three, four, five.

00:05:01.076 --> 00:05:04.936
Right there, that is the
median of this bottom list.

00:05:04.936 --> 00:05:07.638
And we actually call this, right here,

00:05:07.638 --> 00:05:11.231
we call this the first quartile,

00:05:11.231 --> 00:05:13.837
we call this the 1st.

00:05:13.837 --> 00:05:18.837
This number right here
is the 1st Quartile.

00:05:19.121 --> 00:05:21.406
And what it does is it separates its--

00:05:21.406 --> 00:05:23.155
there's a bunch of ways you could view it.

00:05:23.155 --> 00:05:27.034
You could say that all,
that twenty-five percent

00:05:27.034 --> 00:05:30.569
or one-fourth, that's where
the word quartile comes from,

00:05:30.569 --> 00:05:34.044
you have one-fourth of the
values are below this number.

00:05:34.044 --> 00:05:37.010
Or you could say
three-fourths of the values

00:05:37.010 --> 00:05:38.534
are above this number.

00:05:38.534 --> 00:05:41.642
Now the other way to do a
box-and-whisker diagram,

00:05:41.642 --> 00:05:43.674
is you don't include this median,

00:05:43.674 --> 00:05:45.523
but I think you would've
gotten the same answer.

00:05:45.523 --> 00:05:46.721
But if you did include this median,

00:05:46.721 --> 00:05:48.712
you'd have one, two, three, four, five,

00:05:48.712 --> 00:05:50.776
six, seven, eight values...

00:05:50.776 --> 00:05:54.311
8 + 1 divided by 2 is 4.5

00:05:54.311 --> 00:05:57.074
so you'd go one, two,
three, four and a half.

00:05:57.074 --> 00:05:59.431
So you go halfway between these two 50s,

00:05:59.431 --> 00:06:01.301
but you would still get 50.

00:06:01.301 --> 00:06:03.657
That's the number that's
between a 50 and a 50, is a 50.

00:06:03.657 --> 00:06:06.400
So either way our 1st Quartile
number wouldn't be different.

00:06:06.400 --> 00:06:09.061
Now, the second quartile is the median.

00:06:09.061 --> 00:06:13.343
This is the 2nd Quartile.

00:06:14.223 --> 00:06:15.684
I always imagined you have twenty-five,

00:06:15.684 --> 00:06:18.083
this is twenty-five
percent of the numbers,

00:06:18.083 --> 00:06:23.083
25% or 1/4 of the number,
is the 1st Quartile.

00:06:23.364 --> 00:06:26.392
Or it's the first section
fourth of the numbers

00:06:26.392 --> 00:06:29.073
and it's upper bound is the 1st Quartile.

00:06:29.073 --> 00:06:32.649
Then you have this
one-fourth of the numbers.

00:06:32.649 --> 00:06:35.595
This is also 25% of the total numbers

00:06:35.595 --> 00:06:39.553
but it's upper bound is the 2nd Quartile

00:06:39.553 --> 00:06:41.869
or it's the median, this is from the

00:06:41.869 --> 00:06:44.165
25th to the 50th percentile.

00:06:44.165 --> 00:06:46.847
And then if we want to go from 50th...

00:06:46.847 --> 00:06:50.905
So this is 25th percentile,

00:06:50.905 --> 00:06:52.855
this is 50th percentile,

00:06:52.855 --> 00:06:55.171
which means that 50% of
the values are below it.

00:06:55.171 --> 00:06:57.304
This means 25% of the values are below it.

00:06:57.304 --> 00:06:59.742
And then if we want the 75th percentile,

00:06:59.742 --> 00:07:02.465
we find the median of
these values over here.

00:07:02.465 --> 00:07:04.334
And we'll include this median right there.

00:07:04.334 --> 00:07:06.610
So once again we have
one, two, three, four,

00:07:06.610 --> 00:07:09.657
five, six, seven, eight, nine values.

00:07:09.657 --> 00:07:11.872
9 + 1 over 2 is 5.

00:07:11.872 --> 00:07:14.269
So you go one, two, three, four, five.

00:07:14.269 --> 00:07:19.226
So the median of this higher
half of our data is 57.

00:07:19.226 --> 00:07:22.823
So this right here, this
number right there, the 57...

00:07:25.361 --> 00:07:29.690
The 57, we call that our 3rd Quartile.

00:07:30.490 --> 00:07:33.542
Our 3rd Quartile and it
represents the upper bound

00:07:33.542 --> 00:07:36.173
on the section from our 50th percentile

00:07:36.173 --> 00:07:37.981
to 75th percentile.

00:07:37.981 --> 00:07:40.663
So that's 25% of our numbers there.

00:07:40.663 --> 00:07:42.573
You can imagine that's what's
called the 3rd Quartile.

00:07:42.573 --> 00:07:45.884
One quartile, two
quartiles, three quartiles,

00:07:45.884 --> 00:07:48.485
and of course, the
number 57, you would say

00:07:48.485 --> 00:07:50.679
it's in the 75th percentile.

00:07:50.679 --> 00:07:54.825
75th percentile,

00:07:54.825 --> 00:07:58.056
which means that 75 percent of numbers

00:07:58.056 --> 00:08:00.087
where it is the 75th percentile,

00:08:00.087 --> 00:08:03.562
75 percent of numbers are below it.

00:08:03.562 --> 00:08:04.781
And if you count these, you would see

00:08:04.781 --> 00:08:06.833
that's indeed the case.

00:08:06.833 --> 00:08:09.921
Anyway, so we've been able to classify

00:08:09.921 --> 00:08:11.485
some interesting numbers, here.

00:08:11.485 --> 00:08:13.639
Let's actually plot it, now,

00:08:13.639 --> 00:08:17.072
using a box-and-whisker diagram.

00:08:18.056 --> 00:08:19.429
So what I'm going to do here..

00:08:19.429 --> 00:08:22.517
let me draw my x-axis just like that.

00:08:22.517 --> 00:08:24.102
It doesn't have to be
horizontal like this,

00:08:24.102 --> 00:08:25.443
it could be in any form.

00:08:25.443 --> 00:08:28.612
So let's say that this right here is 0.

00:08:28.612 --> 00:08:32.696
That's 0, that right there is 100,

00:08:32.696 --> 00:08:35.378
halfway between 0 and 100,
I'm just making a scale,

00:08:35.378 --> 00:08:39.726
right here is 50, this
right here would be 75.

00:08:39.726 --> 00:08:41.432
Now, what I'm doing has nothing--

00:08:41.432 --> 00:08:43.058
well I want to include these numbers

00:08:43.058 --> 00:08:44.442
but I'm just trying to make a nice scale

00:08:44.442 --> 00:08:47.173
to measure by--this is 25.

00:08:47.173 --> 00:08:51.867
And so let's draw this data
using a box-and-whisker diagram.

00:08:51.867 --> 00:08:54.711
So the median is 53.

00:08:54.711 --> 00:08:58.429
So the median is 53, maybe 53
is sitting right about there.

00:08:58.429 --> 00:09:00.217
Right, that's right about 53.

00:09:00.217 --> 00:09:01.538
That's our median.

00:09:01.538 --> 00:09:05.763
Our 1st Quartile is 50,
so 50 is right above--

00:09:05.763 --> 00:09:07.795
right there, actually.

00:09:07.795 --> 00:09:11.870
Our 3rd Quartile is 57,

00:09:11.870 --> 00:09:14.978
so 57 might sit right around here.

00:09:14.978 --> 00:09:17.782
I might want to scale this
a little bit differently.

00:09:17.782 --> 00:09:19.692
And I'm going to draw a box here.

00:09:19.692 --> 00:09:21.805
I'm going to draw a box here.

00:09:21.805 --> 00:09:24.162
So this right there, that is 53,

00:09:24.162 --> 00:09:28.245
that right there is 57, that's 57,

00:09:28.245 --> 00:09:30.765
and that right there is 50.

00:09:30.765 --> 00:09:32.878
So what I'm essentially
doing is representing

00:09:32.878 --> 00:09:36.413
the middle 50 percent of
the data is in this box.

00:09:36.413 --> 00:09:39.704
And this line represents
the actual median.

00:09:39.704 --> 00:09:43.178
Now, it's called a box-and-whisker diagram

00:09:43.178 --> 00:09:44.702
and I've only drawn a box.

00:09:44.702 --> 00:09:47.851
Let me draw the whisker or the whiskers.

00:09:47.851 --> 00:09:49.477
The whiskers essentially show us

00:09:49.477 --> 00:09:52.382
the entire range of data,
so the lowest data point

00:09:52.382 --> 00:09:55.856
is 49, which is right over here.

00:09:55.856 --> 00:09:57.786
I should zoom in on this a little bit.

00:09:57.786 --> 00:10:02.195
Highest data point is 67, which
is maybe right over there.

00:10:02.195 --> 00:10:04.410
So we draw whiskers.

00:10:04.410 --> 00:10:07.478
So what a box-and-whisker
diagram is showing us,

00:10:07.478 --> 00:10:09.753
it says, okay, most of
the data is sitting inside

00:10:09.753 --> 00:10:12.780
of the box, 50 percent of
the data, this 25 percent

00:10:12.780 --> 00:10:15.848
and this 25 percent is within this box

00:10:15.848 --> 00:10:18.042
but just to get an idea
of the range of the data

00:10:18.042 --> 00:10:19.871
it shows whiskers to show the low point

00:10:19.871 --> 00:10:22.289
all the way to the high point.

00:10:22.289 --> 00:10:24.178
Now let me draw this one a little bit--

00:10:24.178 --> 00:10:26.657
Let me zoom in a little
bit because I realized that

00:10:26.657 --> 00:10:29.135
this is a little bit all squenched up.

00:10:30.581 --> 00:10:33.585
So let me draw it a little bit nicer.

00:10:33.585 --> 00:10:36.775
So let me change my scale a little bit.

00:10:36.775 --> 00:10:39.131
Let's say that this right here is

00:10:39.131 --> 00:10:42.321
let's say that this is...

00:10:42.321 --> 00:10:45.836
45, 46, 47, 48, 49, 50,

00:10:45.836 --> 00:10:48.518
51, 52, 53, 54, 55, 56,

00:10:48.518 --> 00:10:51.301
57, 58, 59, 60, 61, 62,

00:10:51.301 --> 00:10:54.634
63, 64, 65, 66, 67, 68

00:10:54.634 --> 00:10:57.235
so this right here is 67.

00:10:57.235 --> 00:11:02.235
46, 47, 48, 49, 50 that's 50 right there.

00:11:02.355 --> 00:11:04.407
So if we use this as our scale,

00:11:04.407 --> 00:11:06.724
I think it'll be a little
bit clearer what's going on

00:11:06.724 --> 00:11:09.239
with the box-and-whisker diagram.

00:11:09.239 --> 00:11:11.819
We have our median, our 2nd Quartile,

00:11:11.819 --> 00:11:14.887
our 2nd Quartile is
53, one, two, three, 53

00:11:14.887 --> 00:11:16.533
It's right here.

00:11:16.533 --> 00:11:19.052
53 is right there.

00:11:19.052 --> 00:11:21.328
Our 1st Quartile, the number that is

00:11:21.328 --> 00:11:25.046
greater than 25 percent
of the values is 50.

00:11:25.430 --> 00:11:28.764
It's 50 right there, that's that.

00:11:28.764 --> 00:11:30.308
Let me color code it.

00:11:30.308 --> 00:11:32.197
This number, right there,

00:11:32.197 --> 00:11:35.332
is this number, right there.

00:11:35.332 --> 00:11:38.145
This number, our 3rd Quartile,

00:11:38.145 --> 00:11:40.481
the number that is larger than

00:11:40.481 --> 00:11:43.041
75 percent of the values is 57.

00:11:43.041 --> 00:11:47.389
One, two, three, four, five,
six, seven, it is this value.

00:11:47.389 --> 00:11:52.001
Right here, the median,
just to be clear was 53,

00:11:52.001 --> 00:11:54.296
that is this value right here.

00:11:54.296 --> 00:11:56.369
So now we can draw our box.

00:11:56.369 --> 00:11:59.274
We can draw our box.

00:11:59.274 --> 00:12:02.322
The box is telling us is
that 50 percent of the values

00:12:02.322 --> 00:12:06.791
are between here and
here, between 50 and 57.

00:12:06.791 --> 00:12:08.417
And then we draw our whiskers to show

00:12:08.417 --> 00:12:10.509
the entire range of the values.

00:12:10.509 --> 00:12:13.963
So the highest value here is 67.

00:12:13.963 --> 00:12:15.365
The highest value here--

00:12:15.365 --> 00:12:18.291
let me do that in another color--is 67.

00:12:18.291 --> 00:12:22.901
Is 67, so draw a whisker
that goes all the way to 67.

00:12:22.901 --> 00:12:25.794
And then the lowest value here is 49.

00:12:25.794 --> 00:12:30.331
So we draw a whisker that goes to 49.

00:12:30.331 --> 00:12:32.952
So what you see when you
look at a box-and-whiskers--

00:12:32.952 --> 00:12:34.354
so let's say you didn't
even need to see this data,

00:12:34.354 --> 00:12:36.649
you just looked at this
box-and-whisker diagram

00:12:36.649 --> 00:12:37.564
you'd immediately say,

00:12:37.564 --> 00:12:40.228
"Okay, the middle number looks like 53,

00:12:41.458 --> 00:12:43.760
"most of the numbers are
50 percent, I should say,

00:12:43.760 --> 00:12:47.214
"the numbers are between 50 and 57,

00:12:47.214 --> 00:12:49.023
"so most of them are
squenched up over here

00:12:49.023 --> 00:12:51.847
"but the entire range
goes all the way to 67

00:12:51.847 --> 00:12:54.772
"but it doesn't go too
much below the 50th,

00:12:54.772 --> 00:12:58.409
"the inner quartile range,
or the range between

00:12:58.409 --> 00:13:03.001
"the 25th and 75th percentile."

00:13:03.001 --> 00:13:04.829
So hopefully that gives
you a good overview

00:13:04.829 --> 00:13:06.536
and now when you see
these in stock charts--

00:13:06.536 --> 00:13:08.649
although in stock charts,
you'll see these verticle

00:13:08.649 --> 00:13:10.315
as opposed to horizontal like that--

00:13:10.315 --> 00:13:11.737
you'll have a good understanding

00:13:11.737 --> 00:13:13.750
of what they're telling you.


Histograms

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:03.000
In this video we're going to
talk about another way of

00:00:03.000 --> 00:00:09.860
visualizing data called the
histogram, which is a very

00:00:09.860 --> 00:00:14.290
fancy word for a not
so fancy thing.

00:00:14.290 --> 00:00:18.140
I think it's probably fair to
say that the histogram is the

00:00:18.140 --> 00:00:22.840
most used way of representing
statistical data.

00:00:22.840 --> 00:00:25.390
Let me just show you how to
figure out a histogram for

00:00:25.390 --> 00:00:26.800
some data, and I think you're
going to get the

00:00:26.800 --> 00:00:28.520
point pretty easily.

00:00:28.520 --> 00:00:30.740
So I have some data
here and I want to

00:00:30.740 --> 00:00:33.530
represent it with a histogram.

00:00:33.530 --> 00:00:35.880
What we're going to see
is how frequent are

00:00:35.880 --> 00:00:37.050
each of these numbers.

00:00:37.050 --> 00:00:39.570
And in order to figure that
out, let me just write the

00:00:39.570 --> 00:00:41.700
numbers down, let me just
categorize them in their

00:00:41.700 --> 00:00:43.370
respective buckets.

00:00:43.370 --> 00:00:45.820
So I have a 1 here.

00:00:48.440 --> 00:00:54.010
I have a 4, so I want to leave
space for the 2, the 3, and

00:00:54.010 --> 00:00:55.430
put a 4 there.

00:00:55.430 --> 00:00:57.820
I have a 2.

00:00:57.820 --> 00:00:59.220
I have a 1.

00:00:59.220 --> 00:01:02.280
Let me put that 1 on a stack
right above that 1.

00:01:02.280 --> 00:01:06.880
I have a 0-- let me put the
0 to the left of the 1.

00:01:06.880 --> 00:01:08.370
I want to put them in order.

00:01:08.370 --> 00:01:10.650
I have a 2, another 2.

00:01:10.650 --> 00:01:12.930
Let me stack that above
my first 2.

00:01:12.930 --> 00:01:14.100
I have another 1.

00:01:14.100 --> 00:01:16.750
Let me stack that above
my other two 1's.

00:01:16.750 --> 00:01:17.930
I have another 0.

00:01:17.930 --> 00:01:20.180
Let me stack it there.

00:01:20.180 --> 00:01:21.430
I have another 1.

00:01:24.120 --> 00:01:27.360
Then I have another 2.

00:01:27.360 --> 00:01:29.700
Another 1.

00:01:29.700 --> 00:01:32.020
I have two more 0's.

00:01:32.020 --> 00:01:33.910
0, 0.

00:01:33.910 --> 00:01:35.175
I have two more 2's.

00:01:37.770 --> 00:01:41.110
I have a 3.

00:01:41.110 --> 00:01:42.360
I have two more 1's.

00:01:44.840 --> 00:01:46.830
Another 3.

00:01:46.830 --> 00:01:49.510
And then I have a 6.

00:01:49.510 --> 00:01:52.050
So no 5's, and then
I have a 6.

00:01:52.050 --> 00:01:54.480
And that space right there
was unnecessary.

00:01:54.480 --> 00:01:56.670
But right here I've listed--
I've just rewritten those

00:01:56.670 --> 00:01:58.620
numbers and I've essentially
categorized them.

00:01:58.620 --> 00:02:01.950
Now what I want to do is
calculate how many of each of

00:02:01.950 --> 00:02:06.660
these numbers we have. So
let me go down here.

00:02:06.660 --> 00:02:08.539
So I want to look at
the frequency of

00:02:08.539 --> 00:02:09.789
each of these numbers.

00:02:12.780 --> 00:02:16.110
So I have one, two,
three, four 0's.

00:02:16.110 --> 00:02:20.660
I have one, two, three, four,
five, six, seven 1's.

00:02:20.660 --> 00:02:24.250
I have one, two, three,
four, five 2's.

00:02:24.250 --> 00:02:26.740
I have one, two 3's.

00:02:26.740 --> 00:02:31.370
I have one 4, and one 6.

00:02:31.370 --> 00:02:32.520
So we could write it this way.

00:02:32.520 --> 00:02:37.200
We could write the number,
and then we

00:02:37.200 --> 00:02:38.450
can have the frequency.

00:02:43.230 --> 00:02:49.060
So I have the numbers 0, 1, 2,
3, 4-- we could even throw 5

00:02:49.060 --> 00:02:52.420
in there, although 5 has
a frequency of 0.

00:02:52.420 --> 00:02:53.780
And we have a 6.

00:02:53.780 --> 00:02:57.950
So the 0 showed up four times
in this data set.

00:02:57.950 --> 00:03:01.570
1 showed up seven times
in this data set.

00:03:01.570 --> 00:03:08.890
2 showed up five times, 3 showed
up to two times, 4

00:03:08.890 --> 00:03:12.560
showed up one time, 5 didn't
show up, and 6

00:03:12.560 --> 00:03:13.820
showed up one time.

00:03:13.820 --> 00:03:16.740
All I did is I counted this
data set, and I did this

00:03:16.740 --> 00:03:18.760
first. But you could say how
many times do I see a 0?

00:03:18.760 --> 00:03:21.600
I see it one, two, three,
four times.

00:03:21.600 --> 00:03:22.940
How many times do I see a 1?

00:03:22.940 --> 00:03:27.290
One, two, three, four, five,
six, seven times.

00:03:27.290 --> 00:03:29.690
That's what we mean
by frequency.

00:03:29.690 --> 00:03:34.260
Now a histogram is really just
a plot, kind of a bar graph,

00:03:34.260 --> 00:03:36.980
plotting the frequency of
each of these numbers.

00:03:36.980 --> 00:03:38.780
It's going to look a lot
like this original

00:03:38.780 --> 00:03:40.400
thing that I drew.

00:03:40.400 --> 00:03:44.090
So let me draw some axes here.

00:03:44.090 --> 00:03:46.040
So the different buckets
here are the numbers.

00:03:46.040 --> 00:03:47.930
And that worked out because
we're dealing with very clean

00:03:47.930 --> 00:03:49.570
integers that tend to repeat.

00:03:49.570 --> 00:03:54.520
If you're dealing with things
that the exact number doesn't

00:03:54.520 --> 00:03:56.650
repeat, oftentimes people
will put the numbers

00:03:56.650 --> 00:03:58.450
into buckets or ranges.

00:03:58.450 --> 00:04:00.190
But here they fit into
nice little buckets.

00:04:00.190 --> 00:04:08.540
You have the numbers 0,
1, 2, 3, 4, 5, and 6.

00:04:08.540 --> 00:04:09.870
This is the actual numbers.

00:04:09.870 --> 00:04:11.505
And then on the vertical
axis we're

00:04:11.505 --> 00:04:15.190
going to plot the frequency.

00:04:15.190 --> 00:04:20.430
So one, two, three, four,
five, six, seven.

00:04:20.430 --> 00:04:25.690
So that's 7, 6, 5, 4, 3, 2, 1.

00:04:25.690 --> 00:04:28.370
So 0 shows up four times.

00:04:28.370 --> 00:04:29.860
So we'll draw a little
bar graph here.

00:04:29.860 --> 00:04:31.980
0 shows up four times.

00:04:31.980 --> 00:04:34.140
Draw it just like that.

00:04:34.140 --> 00:04:36.210
0 shows up four times.

00:04:36.210 --> 00:04:40.150
That is that information
right there.

00:04:40.150 --> 00:04:44.540
1 shows up seven times.

00:04:44.540 --> 00:04:45.680
So I'll do a little bar graph.

00:04:45.680 --> 00:04:48.520
1 shows up seven times.

00:04:48.520 --> 00:04:50.630
Just like that.

00:04:50.630 --> 00:04:53.740
I want to make it a little bit
straighter than that-- 1 shows

00:04:53.740 --> 00:04:55.190
up seven times.

00:04:58.890 --> 00:05:03.335
2-- I'll do it in a different
color-- 2 shows up five times.

00:05:06.930 --> 00:05:10.260
Do a bar graph, go all
the way up to five.

00:05:10.260 --> 00:05:12.910
2 shows up five times.

00:05:12.910 --> 00:05:14.650
3 shows up two times.

00:05:20.460 --> 00:05:24.210
We have one 3, two 3's.

00:05:24.210 --> 00:05:27.240
4 shows up one time here.

00:05:31.100 --> 00:05:33.680
5 doesn't show up at all.

00:05:33.680 --> 00:05:37.100
So it doesn't even get
any height there.

00:05:37.100 --> 00:05:44.140
And then finally, 6
shows up one time.

00:05:44.140 --> 00:05:47.140
So I'll do 6 showing
up one time.

00:05:47.140 --> 00:05:51.460
What I just plotted here,
this is a histogram.

00:05:51.460 --> 00:05:53.850
This right here is
a histogram.

00:05:53.850 --> 00:05:56.890
Very fancy word, but I think you
will agree it's a fairly

00:05:56.890 --> 00:05:58.100
simple idea.

00:05:58.100 --> 00:06:01.940
Figure out the frequency of each
of these numbers and then

00:06:01.940 --> 00:06:04.840
plot the frequency of each of
these numbers and you get

00:06:04.840 --> 00:06:06.090
yourself a histogram.


Stem and Leaf Plots

WEBVTT
Kind: captions
Language: en

00:00:00.460 --> 00:00:02.280
In this video, we're going
to talk about how

00:00:02.280 --> 00:00:03.590
to visualize data.

00:00:03.590 --> 00:00:08.000
In particular, we're going to
use something called a stem

00:00:08.000 --> 00:00:13.340
and leaf plot.

00:00:13.340 --> 00:00:16.520
And what's useful about this is
it essentially allows you

00:00:16.520 --> 00:00:20.770
to show all of your data and
show a lot of detail about all

00:00:20.770 --> 00:00:23.690
of the numbers in your data set,
while at the same time

00:00:23.690 --> 00:00:28.010
having a good sense about how
those numbers in your set are

00:00:28.010 --> 00:00:29.080
distributed.

00:00:29.080 --> 00:00:31.150
Right here, I have
a set of numbers.

00:00:31.150 --> 00:00:33.110
And when I just look at this,
it really doesn't make a lot

00:00:33.110 --> 00:00:33.730
of sense to me.

00:00:33.730 --> 00:00:35.330
It's just a bunch of
numbers to me.

00:00:35.330 --> 00:00:39.230
I have no sense are more of
them closer to 20 or 30?

00:00:39.230 --> 00:00:41.570
Are more of them closer to 80?

00:00:41.570 --> 00:00:46.170
Is their median or their
mode 30 or 40 or 84?

00:00:46.170 --> 00:00:48.640
I have no idea, just looking
at this list of numbers.

00:00:48.640 --> 00:00:52.030
But using a stem and leaf plot,
hopefully, some of this

00:00:52.030 --> 00:00:55.210
will become a little
bit more apparent.

00:00:55.210 --> 00:01:01.270
And the idea here is to list the
most significant digit in

00:01:01.270 --> 00:01:03.620
each of those numbers
in the stem.

00:01:03.620 --> 00:01:06.500
So when we talk about the most
significant digit, these

00:01:06.500 --> 00:01:08.180
numbers right here,
they range-- let's

00:01:08.180 --> 00:01:10.650
see, we have a 2 here.

00:01:10.650 --> 00:01:12.200
That looks like-- we have a 1.

00:01:12.200 --> 00:01:13.920
That looks like our
lowest number.

00:01:13.920 --> 00:01:18.490
And it goes all the
way up to 84.

00:01:18.490 --> 00:01:21.400
It has a bunch of numbers in
between in the teens, in the

00:01:21.400 --> 00:01:23.970
twenties, in the thirties,
forties, fifties, sixties.

00:01:23.970 --> 00:01:25.400
so on and so forth.

00:01:25.400 --> 00:01:28.280
So it looks like the most
significant digit here is the

00:01:28.280 --> 00:01:29.720
tens digit.

00:01:29.720 --> 00:01:31.110
So let's write it right here.

00:01:31.110 --> 00:01:32.380
Let's write our tens digit.

00:01:35.960 --> 00:01:41.430
And the tens digit could be a
0, 1, 2, 3, 4-- let me write

00:01:41.430 --> 00:01:42.950
this a little bit bigger.

00:01:42.950 --> 00:01:52.610
0, 1, 2, 3, 4, 5, 6, 7 or-- let
me scroll down a little

00:01:52.610 --> 00:01:57.080
bit-- an 8.

00:01:57.080 --> 00:01:59.830
These digits will represent
our stem.

00:01:59.830 --> 00:02:03.520
So I'm going to draw this
little line around here.

00:02:03.520 --> 00:02:10.120
This is going to be a our
stem right there in the

00:02:10.120 --> 00:02:14.090
stem and leaf plot.

00:02:14.090 --> 00:02:17.040
For a number like 1, the
tens digit is 0.

00:02:17.040 --> 00:02:18.890
You could imagine it as 01.

00:02:18.890 --> 00:02:23.540
For a number like 84, the tens
digit is obviously 8.

00:02:23.540 --> 00:02:26.140
So what I'm going to do is
rewrite all of these numbers

00:02:26.140 --> 00:02:29.200
here using the stem
and leaf plot.

00:02:29.200 --> 00:02:31.890
And the technique here, and I'm
just going to show you by

00:02:31.890 --> 00:02:34.320
example, let's go through
each of these numbers.

00:02:34.320 --> 00:02:36.290
Let's say I want to write 35.

00:02:36.290 --> 00:02:39.310
So 35-- let me cross it out.

00:02:39.310 --> 00:02:42.530
35, so it has a tens
digit of 3.

00:02:42.530 --> 00:02:44.490
I'll put a 5 right here.

00:02:44.490 --> 00:02:47.100
That tells me that 35 is
one of my numbers.

00:02:47.100 --> 00:02:48.960
Then I have 42.

00:02:48.960 --> 00:02:50.640
Cross it out, let me
know I used it.

00:02:50.640 --> 00:02:55.100
42, I'm writing the ones
digit as leaves.

00:02:55.100 --> 00:02:55.950
You're going to see what
I'm going to do.

00:02:55.950 --> 00:02:56.170
Let's see.

00:02:56.170 --> 00:02:57.880
38.

00:02:57.880 --> 00:02:59.080
Well, I have a 30.

00:02:59.080 --> 00:03:00.570
I'm going to write an 8 here.

00:03:00.570 --> 00:03:02.520
So I didn't have to
rewrite the 30.

00:03:02.520 --> 00:03:06.190
I'm just saying, look, I have
a 35 and I have a 38.

00:03:06.190 --> 00:03:10.270
I'm just writing the ones digits
within the three stem,

00:03:10.270 --> 00:03:13.060
within the tens place, within
the thirties place, however

00:03:13.060 --> 00:03:15.190
you want view it, the
thirties bucket.

00:03:15.190 --> 00:03:19.330
Then we have a 57.

00:03:19.330 --> 00:03:22.500
We have a 2, which you could
just view as 02.

00:03:22.500 --> 00:03:24.390
We have a 2.

00:03:24.390 --> 00:03:35.580
We have 24, 27, 36.

00:03:35.580 --> 00:03:43.680
We have a 35, a 38,
and a 36, 45, 60.

00:03:43.680 --> 00:03:45.460
I think you get the idea,
but just for the sake of

00:03:45.460 --> 00:03:47.140
completion, let's finish
all of these.

00:03:47.140 --> 00:04:05.230
38, 40, another 40, a 44, a 1,
another 44, a 48-- there's a

00:04:05.230 --> 00:04:16.240
lot of things in this forties
bucket-- 48, an 84, a 38, a

00:04:16.240 --> 00:04:25.070
20, a 4-- put that right
there, 04-- a 2.

00:04:25.070 --> 00:04:26.370
It's getting a little
monotonous, but

00:04:26.370 --> 00:04:27.390
let's just finish it.

00:04:27.390 --> 00:04:36.760
48, another 48 right
there, a 58, a 3.

00:04:36.760 --> 00:04:37.535
Stick it up there.

00:04:37.535 --> 00:04:39.180
It's 03.

00:04:39.180 --> 00:04:57.140
Then we have a 20, a 6,
a 40, 22, 26, 17.

00:04:57.140 --> 00:04:58.775
That's our first number
in the teens bucket.

00:04:58.775 --> 00:05:14.590
An 18, a 40-- a lot of things in
the forties-- a 51, 62, 31,

00:05:14.590 --> 00:05:37.750
27, 48, 35-- almost done--
27, 37, 58 and 21.

00:05:37.750 --> 00:05:41.100
And we have our 21.

00:05:41.100 --> 00:05:43.690
And just like that, it was a
little monotonous, but we

00:05:43.690 --> 00:05:46.860
essentially just rewrote this
list of numbers down here.

00:05:46.860 --> 00:05:48.350
I have every number
represented.

00:05:48.350 --> 00:05:50.950
We haven't lost any
information here.

00:05:50.950 --> 00:05:51.700
I know every number.

00:05:51.700 --> 00:05:54.620
I could go start with this,
and I could tell you every

00:05:54.620 --> 00:05:57.580
number that I had in my original
list. I have a 6, I

00:05:57.580 --> 00:06:00.430
have a 27, I have a 38, a 35.

00:06:00.430 --> 00:06:02.510
So we haven't lost
any information.

00:06:02.510 --> 00:06:05.760
But what's neater about this
representation is when the

00:06:05.760 --> 00:06:09.240
numbers are written this way, we
actually save ink, because

00:06:09.240 --> 00:06:11.790
you don't have to rewrite
the 40 to 45.

00:06:11.790 --> 00:06:13.110
You just write the ones digit.

00:06:13.110 --> 00:06:15.490
But more importantly, we kind of
understand how the numbers

00:06:15.490 --> 00:06:18.360
are distributed, that we have
a bunch of numbers in the

00:06:18.360 --> 00:06:20.250
forties and thirties
and twenties.

00:06:20.250 --> 00:06:21.520
That's where most of
the numbers are.

00:06:21.520 --> 00:06:24.530
We have very few numbers
up in the eighties.

00:06:24.530 --> 00:06:26.090
We have no numbers
in the seventies.

00:06:26.090 --> 00:06:26.910
We have very few.

00:06:26.910 --> 00:06:29.730
We have a lot in the ones, but
still, most of them are

00:06:29.730 --> 00:06:32.800
focused in this range
over here.

00:06:32.800 --> 00:06:36.780
So this is just one of many,
many, many ways to visualize

00:06:36.780 --> 00:06:40.290
data that you will run into
in your mathematical or

00:06:40.290 --> 00:06:41.270
statistical futures.

00:06:41.270 --> 00:06:43.980
But I just wanted to
expose you to this.