Worked example: Rewriting expressions by completing the square

WEBVTT
Kind: captions
Language: en

00:00:00.141 --> 00:00:01.473
- [Voiceover] Let's
see if we can take this

00:00:01.473 --> 00:00:05.351
quadratic expression here, X
squared plus 16 X plus nine

00:00:05.351 --> 00:00:07.091
and write it in this form.

00:00:07.091 --> 00:00:08.021
You might be saying, hey Sal,

00:00:08.021 --> 00:00:10.485
why do I even need to worry about this?

00:00:10.485 --> 00:00:12.900
And one, it is just
good algebraic practice

00:00:12.900 --> 00:00:14.386
to be able to manipulate things,

00:00:14.386 --> 00:00:16.661
but as we'll see in the
future, what we're about to do

00:00:16.661 --> 00:00:18.798
is called completing the square.

00:00:18.798 --> 00:00:21.525
It's a really valuable
technique for solving quadratics

00:00:21.525 --> 00:00:23.475
and it's actually the basis for the proof

00:00:23.475 --> 00:00:26.725
of the quadratic formula which
you'll learn in the future.

00:00:26.725 --> 00:00:30.059
So it's actually a pretty
interesting technique.

00:00:30.059 --> 00:00:33.181
So how do we write this in this form?

00:00:33.181 --> 00:00:36.003
Well one way to think about
is if we expanded this

00:00:36.003 --> 00:00:39.436
X plus A squared, we know if
we square X plus A it would be

00:00:39.436 --> 00:00:42.779
X squared plus two A

00:00:42.779 --> 00:00:45.775
X plus A squared,

00:00:45.775 --> 00:00:49.722
and then you still have that
plus B, right over there.

00:00:49.722 --> 00:00:51.390
So one way to think about it is,

00:00:51.390 --> 00:00:53.724
let's take this expression,

00:00:53.724 --> 00:00:56.160
this X squared plus 16 X plus nine,

00:00:56.160 --> 00:00:58.181
and I'm just gonna write
it with a few spaces in it.

00:00:58.181 --> 00:01:01.866
X squared plus 16 X

00:01:01.866 --> 00:01:05.581
and then plus nine, just like that.

00:01:05.581 --> 00:01:08.460
And so, if we say alright,
we have an X squared here.

00:01:08.460 --> 00:01:10.340
We have an X squared here.

00:01:10.340 --> 00:01:14.740
If we say that two A X is
the same thing as that,

00:01:14.740 --> 00:01:16.088
then what's A going to be?

00:01:16.088 --> 00:01:19.917
So this is two A times X.

00:01:19.917 --> 00:01:22.438
Well, that means two A is 16

00:01:22.438 --> 00:01:25.837
or that A is equal to 8.

00:01:25.837 --> 00:01:28.775
And so if I want to have
an A squared over here,

00:01:28.775 --> 00:01:33.466
well if A is eight, I
would add an eight squared

00:01:33.466 --> 00:01:35.693
which would be a 64.

00:01:35.693 --> 00:01:38.376
Well I can't just add numbers
willy nilly to an expression

00:01:38.376 --> 00:01:40.083
without changing the
value of an expression

00:01:40.083 --> 00:01:42.662
so if don't want to change
the value of the expression,

00:01:42.662 --> 00:01:45.448
I still need to subtract 64.

00:01:45.448 --> 00:01:47.642
So notice, all that I have done now,

00:01:47.642 --> 00:01:50.208
is I just took our original expression

00:01:50.208 --> 00:01:53.447
and I added 64 and I subtracted 64,

00:01:53.447 --> 00:01:56.720
so I have not changed the
value of that expression.

00:01:56.720 --> 00:01:59.318
But what was valuable about me doing that,

00:01:59.318 --> 00:02:02.067
is now this first part of the expression,

00:02:02.067 --> 00:02:04.329
this part right over here,

00:02:04.329 --> 00:02:07.790
it fits the pattern of a perfect square

00:02:07.790 --> 00:02:10.064
quadratic right over here.

00:02:10.064 --> 00:02:13.188
We have X squared plus
two A X, where A is 8,

00:02:13.188 --> 00:02:15.300
plus A squared, 64.

00:02:15.300 --> 00:02:16.949
Once again, how did I get 64?

00:02:16.949 --> 00:02:21.407
I took half of the 16 and I
squared it to get to the 64.

00:02:21.407 --> 00:02:24.645
And so the stuff that I just squared off,

00:02:24.645 --> 00:02:28.569
this is going to be X plus eight squared.

00:02:28.569 --> 00:02:32.180
X plus eight, squared.

00:02:32.180 --> 00:02:36.463
Once again I know that because
A is eight, A is eight,

00:02:36.463 --> 00:02:38.879
so this is X plus eight squared,

00:02:38.879 --> 00:02:41.768
and then all of this business
on the right hand side.

00:02:41.768 --> 00:02:44.787
What is nine minus 64?

00:02:44.787 --> 00:02:47.922
Well 64 minus nine is 55,

00:02:47.922 --> 00:02:50.859
so this is going to be negative 55.

00:02:50.859 --> 00:02:53.644
So minus 55, and we're done.

00:02:53.644 --> 00:02:56.302
We've written this
expression in this form,

00:02:56.302 --> 00:02:57.603
and what's also called

00:02:57.603 --> 00:03:00.535
completing the square.


Solving quadratics by factoring: leading coefficient Ã¢Â  Â  1

WEBVTT
Kind: captions
Language: en

00:00:00.352 --> 00:00:03.231
- [Voiceover] We have six
x squared, minus 120 x,

00:00:03.231 --> 00:00:05.925
plus 600, equals zero.

00:00:05.925 --> 00:00:07.782
Like always, pause this video,

00:00:07.782 --> 00:00:09.626
and see if you can solve for x,

00:00:09.626 --> 00:00:13.378
if you could find the x values
that satisfy this equation.

00:00:13.378 --> 00:00:15.777
Alright, let's work through this together.

00:00:16.967 --> 00:00:20.227
The numbers here don't seem
like outlandish numbers.

00:00:20.227 --> 00:00:22.944
They seem like something that
I might be able to deal with,

00:00:22.944 --> 00:00:26.334
and I might be able to factor,
so let's try to do that.

00:00:26.334 --> 00:00:28.261
The first thing I like
to do is see if I can get

00:00:28.261 --> 00:00:31.210
a coefficient of one, on
the second degree term,

00:00:31.210 --> 00:00:32.789
on the x-squared term.

00:00:32.789 --> 00:00:34.878
It looks like actually all of these terms

00:00:34.878 --> 00:00:36.208
are divisible by six.

00:00:36.208 --> 00:00:38.988
So if we divide both sides
of this equation by six,

00:00:38.988 --> 00:00:42.355
I'm still going to have
nice integer coefficients.

00:00:42.355 --> 00:00:43.237
Let's do that.

00:00:43.237 --> 00:00:45.304
Let's divide both sides by six.

00:00:45.304 --> 00:00:48.160
If we divide the left side
by six, divide by six,

00:00:48.160 --> 00:00:50.667
divide by six, divide by six.

00:00:50.667 --> 00:00:53.407
And I divide the right side by six.

00:00:53.407 --> 00:00:55.938
If I do that, and clearly
if I do the same thing

00:00:55.938 --> 00:00:57.092
to both sides of the equation,

00:00:57.092 --> 00:00:58.864
then the equality still holds.

00:00:58.864 --> 00:01:03.755
On the left-hand side, I am
going to be left with x squared,

00:01:04.970 --> 00:01:08.499
and then negative 120, divided by six.

00:01:08.499 --> 00:01:09.411
That is, let's see.

00:01:09.411 --> 00:01:12.071
120 divided by six is 20.

00:01:12.071 --> 00:01:15.813
So that's minus 20 x.

00:01:15.813 --> 00:01:18.625
Then 600 divided by six, is 100.

00:01:19.505 --> 00:01:23.580
So plus 100, is equal
to zero divided by six.

00:01:23.580 --> 00:01:25.728
Is equal to zero.

00:01:25.728 --> 00:01:27.443
So let's see if we can factor,

00:01:27.443 --> 00:01:30.276
if we can express this quadratic

00:01:30.276 --> 00:01:32.992
as a product of two expressions.

00:01:32.992 --> 00:01:34.385
The way we think about this,

00:01:34.385 --> 00:01:36.104
and we've done it multiple times,

00:01:36.104 --> 00:01:40.631
if we have something, if we have x plus a,

00:01:40.631 --> 00:01:44.254
times x plus b, and this is
hopefully review for you.

00:01:44.254 --> 00:01:47.365
If you multiply that out,
that is going to be equal to,

00:01:47.365 --> 00:01:52.365
that equals to x squared, plus a, plus b,

00:01:52.729 --> 00:01:55.614
x, plus a b.

00:01:55.614 --> 00:01:58.772
What we want to do is
see if we can factor this

00:01:58.772 --> 00:02:01.396
into an x plus a, and an x plus b.

00:02:02.416 --> 00:02:06.007
A plus b, needs to be
equal to negative 20.

00:02:06.007 --> 00:02:07.928
That needs to be a plus b.

00:02:07.928 --> 00:02:10.969
And then a times b, right over here,

00:02:10.969 --> 00:02:12.780
that needs to be equal
to the constant term.

00:02:12.780 --> 00:02:15.659
That needs to be a times
b, right over there.

00:02:15.659 --> 00:02:19.467
Can we think of two numbers,
that if we take their product,

00:02:19.467 --> 00:02:22.927
we get positive 100, and
if we take their sum,

00:02:22.927 --> 00:02:24.924
we get negative 20?

00:02:24.924 --> 00:02:26.642
Well since their product is positive,

00:02:26.642 --> 00:02:29.255
we know that they have the same sign.

00:02:29.255 --> 00:02:32.445
They're both going to have the same sign.

00:02:32.445 --> 00:02:34.511
They're either both going to be positive,

00:02:34.511 --> 00:02:36.090
or their both going to be negative,

00:02:36.090 --> 00:02:39.016
since we know that we
have a positive product.

00:02:39.016 --> 00:02:41.036
Since their sum is negative,

00:02:41.036 --> 00:02:43.567
well they both must both be negative.

00:02:43.567 --> 00:02:46.516
You can't add up two positive
numbers, and get a negative.

00:02:46.516 --> 00:02:49.859
So they both must be negative.

00:02:49.859 --> 00:02:51.345
Let's think about it a little bit.

00:02:51.345 --> 00:02:55.566
What negative numbers,
when I add them together

00:02:55.566 --> 00:02:59.115
I get negative 20, when
I multiply it, I get 100?

00:02:59.115 --> 00:03:00.753
Well you could try to factor 100.

00:03:00.753 --> 00:03:03.271
You could say, well negative
two times negative 50,

00:03:03.271 --> 00:03:06.475
or negative four times negative 25.

00:03:06.475 --> 00:03:07.892
But the one that might jump out at you

00:03:07.892 --> 00:03:12.118
is this is negative 10,
times, I'll write it this way,

00:03:12.118 --> 00:03:15.229
negative 10, times negative 10,

00:03:15.229 --> 00:03:19.339
and this is negative 10, plus negative 10.

00:03:19.339 --> 00:03:22.539
So in that case, both our a and
our b, would be negative 10.

00:03:22.539 --> 00:03:26.381
And so we can rewrite the
left side of this equation as,

00:03:26.381 --> 00:03:29.028
we can rewrite it as,

00:03:29.028 --> 00:03:31.234
x, and I'll write it this way at first,

00:03:31.234 --> 00:03:33.853
x plus negative 10,

00:03:35.181 --> 00:03:36.636
times,

00:03:39.206 --> 00:03:42.204
x plus negative 10 again.

00:03:42.204 --> 00:03:44.412
X plus negative 10,

00:03:45.527 --> 00:03:48.470
and that is going to be equal to zero.

00:03:48.470 --> 00:03:50.722
So all I've done is I've
factored this quadratic.

00:03:50.722 --> 00:03:54.205
Another way, these are both
the same thing as x minus 10.

00:03:54.205 --> 00:03:59.205
I could rewrite this
as x minus 10, squared,

00:03:59.662 --> 00:04:02.010
is equal to zero.

00:04:03.066 --> 00:04:04.840
The only way that the left-hand side

00:04:04.840 --> 00:04:06.186
is going to be equal to zero,

00:04:06.186 --> 00:04:08.230
is if x minus 10 is equal to zero.

00:04:08.230 --> 00:04:09.274
You could think of this as taking

00:04:09.274 --> 00:04:11.759
the square root of both sides.

00:04:11.759 --> 00:04:12.943
It doesn't matter if
you're taking the positive

00:04:12.943 --> 00:04:14.738
or negative square root, or both of them.

00:04:14.738 --> 00:04:16.588
The square root of zero, is zero.

00:04:16.588 --> 00:04:21.588
So, we would say, that x minus
10 needs to be equal to zero.

00:04:21.650 --> 00:04:24.785
So x , adding 10 to both sides of this,

00:04:24.785 --> 00:04:28.796
you have x is equal to 10, is the solution

00:04:28.796 --> 00:04:32.117
to this quadratic equation, up here.


Solving quadratics by taking square roots examples

WEBVTT
Kind: captions
Language: en

00:00:00.336 --> 00:00:01.661
- [Voiceover] So pause
the video and see if you

00:00:01.661 --> 00:00:03.030
can solve for x here.

00:00:03.030 --> 00:00:07.605
Figure out which x-values
will satisfy this equation.

00:00:07.605 --> 00:00:08.914
All right, let's work through this.

00:00:08.914 --> 00:00:11.452
So, the way I'm gonna do
this is I'm gonna isolate

00:00:11.452 --> 00:00:13.668
the x plus three squared on one side

00:00:13.668 --> 00:00:17.790
and the best way to do that
is to add four to both sides.

00:00:17.790 --> 00:00:20.359
So, adding four to both sides

00:00:20.359 --> 00:00:23.430
will get rid of this
four, subtracting four,

00:00:23.430 --> 00:00:25.417
this negative four on the left-hand side.

00:00:25.417 --> 00:00:28.913
And so we're just left
with x plus three squared.

00:00:28.913 --> 00:00:31.616
X plus three squared.

00:00:31.616 --> 00:00:33.417
And on the right-hand
side I'm just gonna have

00:00:33.417 --> 00:00:34.937
zero plus four.

00:00:34.937 --> 00:00:38.483
So, x plus three squared is equal to four.

00:00:38.483 --> 00:00:42.425
And so now, I could take the square root

00:00:42.425 --> 00:00:45.859
of both sides and, or, another
way of thinking about it,

00:00:45.859 --> 00:00:48.875
if I have something-squared equaling four,

00:00:48.875 --> 00:00:50.979
I could say that that something needs

00:00:50.979 --> 00:00:53.790
to either be positive or negative two.

00:00:53.790 --> 00:00:55.286
So, one way of thinking about it is,

00:00:55.286 --> 00:00:59.061
I'm saying that x plus
three is going to be equal

00:00:59.061 --> 00:01:03.024
to the plus or minus
square root of that four.

00:01:04.294 --> 00:01:06.216
And hopefully this makes
intuitive sense for you.

00:01:06.216 --> 00:01:08.255
If something-squared is equal to four,

00:01:08.255 --> 00:01:10.299
that means that the something,

00:01:10.299 --> 00:01:12.469
that means that this
something right over here,

00:01:12.469 --> 00:01:15.368
is going to be equal to the
positive square root of four

00:01:15.368 --> 00:01:16.850
or the negative square root of four.

00:01:16.850 --> 00:01:19.379
Or it's gonna be equal to
positive or negative two.

00:01:19.379 --> 00:01:20.890
And so we could write

00:01:20.890 --> 00:01:24.866
that x plus three could
be equal to positive two

00:01:24.866 --> 00:01:28.847
or x plus three could be
equal to negative two.

00:01:28.847 --> 00:01:31.410
Notice, if x plus three was positive two,

00:01:31.410 --> 00:01:33.196
two-squared is equal to four.

00:01:33.196 --> 00:01:35.086
If x plus three was negative two,

00:01:35.086 --> 00:01:37.158
negative two-squared is equal to four.

00:01:37.158 --> 00:01:40.268
So, either of these would
satisfy our equation.

00:01:40.268 --> 00:01:42.564
So, if x plus three is
equal to two, we could just

00:01:42.564 --> 00:01:46.586
subtract three from both
sides to solve for x

00:01:46.586 --> 00:01:50.797
and we're left with x is
equal to negative one.

00:01:50.797 --> 00:01:53.401
Or, over here we could subtract three

00:01:53.401 --> 00:01:56.670
from both sides to solve for x.

00:01:56.670 --> 00:02:01.102
So, or, x is equal to
negative two minus three

00:02:01.102 --> 00:02:02.735
is negative five.

00:02:02.735 --> 00:02:04.259
So, those are the two possible solutions

00:02:04.259 --> 00:02:05.364
and you can verify that.

00:02:05.364 --> 00:02:07.791
Take these x-values,
substitute it back in,

00:02:07.791 --> 00:02:09.264
and then you can see when
you substitute it back in

00:02:09.264 --> 00:02:11.198
if you substitute x equals negative one,

00:02:11.198 --> 00:02:12.863
then x plus three is equal to two,

00:02:12.863 --> 00:02:15.429
two-squared is four, minus four is zero.

00:02:15.429 --> 00:02:17.342
And when x is equal to negative five,

00:02:17.342 --> 00:02:20.047
negative five plus three is negative two,

00:02:20.047 --> 00:02:24.006
squared is positive four, minus
four is also equal to zero.

00:02:24.006 --> 00:02:25.988
So, these are the two possible x-values

00:02:25.988 --> 00:02:27.897
that satisfy the equation.

00:02:27.897 --> 00:02:30.386
Now let's do another one
that's presented to us

00:02:30.386 --> 00:02:33.016
in a slightly different way.

00:02:33.016 --> 00:02:37.942
So, we are told that f of x is
equal to x minus two squared

00:02:37.942 --> 00:02:39.151
minus nine.

00:02:39.151 --> 00:02:41.305
And then we're asked at what x-values

00:02:41.305 --> 00:02:45.687
does the graph of y equals
f of x intersect the x-axis.

00:02:46.577 --> 00:02:49.829
So, if I'm just generally
talking about some graph,

00:02:49.829 --> 00:02:52.477
so I'm not necessarily gonna
draw that y equals f of x.

00:02:53.557 --> 00:02:58.120
So if I'm just, so that's our
y-axis, this is our x-axis.

00:02:58.120 --> 00:03:01.106
And so if I just have the
graph of some function.

00:03:01.106 --> 00:03:02.792
If I have the graph of some function

00:03:02.792 --> 00:03:04.610
that looks something like that.

00:03:04.610 --> 00:03:07.567
Let's say that the y is
equal to some other function,

00:03:07.567 --> 00:03:09.275
not necessarily this f of x.

00:03:09.275 --> 00:03:11.568
Y is equal to g of x.

00:03:11.568 --> 00:03:14.576
The x-values where you intersect,

00:03:14.576 --> 00:03:18.333
where you intersect the x-axis.

00:03:18.333 --> 00:03:20.368
Well, in order to intersect the x-axis,

00:03:20.368 --> 00:03:22.989
y must be equal to zero.

00:03:22.989 --> 00:03:25.223
So, y is equal to zero there.

00:03:25.223 --> 00:03:27.804
Notice our y-coordinate
at either of those points

00:03:27.804 --> 00:03:29.762
are going to be equal to zero.

00:03:29.762 --> 00:03:32.681
And that means that our
function is equal to zero.

00:03:32.681 --> 00:03:35.598
So, figuring out the
x-values where the graph

00:03:35.598 --> 00:03:39.524
of y equals f of x intersects the x-axis,

00:03:39.524 --> 00:03:40.701
this is equivalent to saying,

00:03:40.701 --> 00:03:45.364
"For what x-values does
f of x equal zero?"

00:03:45.364 --> 00:03:47.020
So we could just say,

00:03:47.020 --> 00:03:50.963
"For what x-values does
this thing right over here

00:03:50.963 --> 00:03:52.326
"equal zero?"

00:03:52.326 --> 00:03:54.074
So, let me just write that down.

00:03:54.074 --> 00:03:56.629
So we could rewrite this as x,

00:03:56.629 --> 00:03:59.926
x minus two squared

00:03:59.926 --> 00:04:02.776
minus nine equals zero.

00:04:02.776 --> 00:04:05.180
We could add nine to both
sides and so we could get

00:04:05.180 --> 00:04:09.085
x minus two squared is equal to nine.

00:04:09.085 --> 00:04:11.014
And just like we saw before, that means

00:04:11.014 --> 00:04:13.504
that x minus two is equal to the positive

00:04:13.504 --> 00:04:16.064
or negative square root of nine.

00:04:16.064 --> 00:04:19.982
So, we could say x minus two
is equal to positive three

00:04:19.982 --> 00:04:23.998
or x minus two is equal to negative three.

00:04:23.998 --> 00:04:25.868
Well, you add two to both sides of this,

00:04:25.868 --> 00:04:28.659
you get x is equal to five,

00:04:28.659 --> 00:04:32.147
or x is equal to, if we
add two to both sides

00:04:32.147 --> 00:04:35.675
of this equation, you'll get
x is equal to negative one.

00:04:35.675 --> 00:04:36.857
And you can verify that.

00:04:36.857 --> 00:04:39.744
If x is equal to five,
five minus two is three,

00:04:39.744 --> 00:04:42.642
squared is nine, minus nine is zero.

00:04:42.642 --> 00:04:46.370
So, the point five comma zero

00:04:46.370 --> 00:04:48.630
is going to be on this graph.

00:04:48.630 --> 00:04:50.410
And also, if x is equal to negative one,

00:04:50.410 --> 00:04:53.356
negative one minus two, negative three.

00:04:53.356 --> 00:04:57.156
Squared is positive
nine, minus nine is zero.

00:04:57.156 --> 00:05:01.022
So, also the point negative one comma zero

00:05:01.022 --> 00:05:04.238
is on this graph.

00:05:04.238 --> 00:05:05.883
So those are the points where,

00:05:05.883 --> 00:05:08.059
those are the x-values where the function

00:05:08.059 --> 00:05:10.035
intersects the x-axis.


Extending geometric sequences

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.870
- [Voiceover] So we're told
that the first four terms

00:00:02.870 --> 00:00:05.148
of a geometric sequence are given.

00:00:05.148 --> 00:00:06.463
So they give us the first four terms.

00:00:06.463 --> 00:00:08.636
And they say what is the
fifth term in the sequence.

00:00:08.636 --> 00:00:09.956
And like always pause the video

00:00:09.956 --> 00:00:11.664
and see if you can come
up with the fifth term.

00:00:11.664 --> 00:00:13.960
Well what we have to remind ourselves

00:00:13.960 --> 00:00:18.960
is for a geometric sequence
each successive term

00:00:19.953 --> 00:00:23.326
is the previous term
multiplied by some number.

00:00:23.326 --> 00:00:25.073
And that number we call the common ratio.

00:00:25.073 --> 00:00:26.744
So let's think about it.

00:00:26.744 --> 00:00:29.115
To go from negative 1/32,
that's the first term

00:00:29.115 --> 00:00:33.261
to 1/8, what do we have to multiply by?

00:00:33.261 --> 00:00:35.211
What do we have to multiply by?

00:00:35.211 --> 00:00:35.842
Let's see.

00:00:35.842 --> 00:00:36.979
We're going to multiply,

00:00:36.979 --> 00:00:38.340
it's going to be multiplied by a negative

00:00:38.340 --> 00:00:39.833
since we went from a
negative to a positive.

00:00:39.833 --> 00:00:43.226
So we're going to multiply by negative

00:00:43.226 --> 00:00:46.274
and there's going to be a one over

00:00:46.274 --> 00:00:51.274
let's see to go from a 32 to an eight,

00:00:51.414 --> 00:00:53.039
actually it's not going to be a one over.

00:00:53.039 --> 00:00:56.900
It's going to be, this is
four times as large as that.

00:00:56.900 --> 00:00:58.647
It's going to be negative four.

00:00:58.647 --> 00:01:03.523
Negative 1/32 times negative
four is positive 1/8.

00:01:03.523 --> 00:01:05.433
Just to make that clear.

00:01:05.433 --> 00:01:10.206
Negative 1/32 times negative four

00:01:10.206 --> 00:01:13.417
that's the same thing
as times negative four

00:01:13.417 --> 00:01:14.474
over one.

00:01:14.474 --> 00:01:16.037
It's going to be positive.

00:01:16.037 --> 00:01:17.643
Negative times a negative is a positive.

00:01:17.643 --> 00:01:19.777
Positive four over 32.

00:01:19.777 --> 00:01:22.499
Which is equal to 1/8.

00:01:22.499 --> 00:01:23.881
Now let's see if that holds up.

00:01:23.881 --> 00:01:27.294
So to go from 1/8 to negative 1/2

00:01:27.294 --> 00:01:31.133
we once again would
multiply by negative four.

00:01:31.133 --> 00:01:33.714
Negative four times 1/8 is negative 4/8,

00:01:33.714 --> 00:01:35.319
which is negative 1/2

00:01:35.319 --> 00:01:39.484
and so then we multiply
by negative four again.

00:01:39.484 --> 00:01:42.307
So, let me make it clear.

00:01:42.307 --> 00:01:44.339
We're multiplying by
negative four each time.

00:01:44.339 --> 00:01:45.925
You multiply by negative four again,

00:01:45.925 --> 00:01:47.245
you get to positive two.

00:01:47.245 --> 00:01:51.207
Because negative four over negative two,

00:01:51.207 --> 00:01:53.015
you can do it that way, is positive two.

00:01:53.015 --> 00:01:55.738
And so to get the fifth
term in the sequence,

00:01:55.738 --> 00:01:59.029
we would multiply by negative four again.

00:01:59.029 --> 00:02:02.746
And so two times negative
four is negative eight.

00:02:02.746 --> 00:02:04.230
Negative four is the common ratio

00:02:04.230 --> 00:02:05.834
for this geometric sequence.

00:02:05.834 --> 00:02:06.993
But just to answer the question,

00:02:06.993 --> 00:02:08.090
What is the fifth term?

00:02:08.090 --> 00:02:11.198
It is going to be negative eight.


Using recursive formulas of geometric sequences

WEBVTT
Kind: captions
Language: en

00:00:00.352 --> 00:00:03.092
- [Voiceover] The
geometric sequence a sub i,

00:00:03.092 --> 00:00:06.041
is defined by the formula
where the first term,

00:00:06.041 --> 00:00:08.293
a sub one is equal to negative 1/8,

00:00:08.293 --> 00:00:11.613
and then every term after
that is defined as being,

00:00:11.613 --> 00:00:16.613
so a sub i is going to be two
times the term before that.

00:00:16.652 --> 00:00:20.646
A sub i is two times a sub i, minus one.

00:00:20.646 --> 00:00:24.813
What is a sub four, the
fourth term in the sequence?

00:00:24.813 --> 00:00:28.017
Pause the video, and see
if you can work this out.

00:00:28.017 --> 00:00:30.530
Well there's a couple of ways
that you could tackle this.

00:00:30.530 --> 00:00:34.047
One is to just directly
use these formulas.

00:00:34.047 --> 00:00:36.635
We could say that a sub four,

00:00:36.635 --> 00:00:38.579
well that's going to be
this case right over here.

00:00:38.579 --> 00:00:42.666
A sub four is going to be
equal to two times a sub three.

00:00:42.666 --> 00:00:45.800
Well, a sub three, if we
go and use this formula,

00:00:45.800 --> 00:00:49.590
is going to be equal
to two times a sub two.

00:00:49.590 --> 00:00:52.702
Each term is equal to two
times the term before it.

00:00:52.702 --> 00:00:54.138
Then we go back to this formula again,

00:00:54.138 --> 00:00:58.367
and say a sub two is going
to be two times a sub one.

00:00:58.367 --> 00:01:00.487
Two times a sub one.

00:01:00.487 --> 00:01:05.487
Lucky for us, we know that
a sub one is negative 1/8.

00:01:05.681 --> 00:01:09.842
It's going to be two times negative 1/8,

00:01:09.842 --> 00:01:13.008
which is equal to negative 1/4.

00:01:13.918 --> 00:01:15.538
Negative 1/4.

00:01:15.538 --> 00:01:17.740
So this is negative 1/4.

00:01:18.930 --> 00:01:23.318
Two times negative 1/4,
is equal to negative 2/4,

00:01:23.318 --> 00:01:25.792
or negative 1/2.

00:01:25.792 --> 00:01:27.800
A sub four is two times a sub three.

00:01:27.800 --> 00:01:29.776
A sub three is negative 1/2.

00:01:29.776 --> 00:01:33.024
So this is going to be
two times negative 1/2,

00:01:33.024 --> 00:01:35.973
which is going to be
equal to negative one.

00:01:35.973 --> 00:01:37.807
So that's one way to solve it.

00:01:37.807 --> 00:01:39.409
Another way to think about it is,

00:01:39.409 --> 00:01:42.797
look, we have our initial term.

00:01:42.797 --> 00:01:44.353
We also know our common ratio.

00:01:44.353 --> 00:01:46.211
We know each successive term

00:01:46.211 --> 00:01:48.904
is two times the term before it.

00:01:48.904 --> 00:01:51.551
So we could explicitly,

00:01:51.551 --> 00:01:54.964
this is a recursive definition
for our geometric series.

00:01:54.964 --> 00:01:57.797
We could explicitly write it as a sub i

00:01:57.797 --> 00:02:02.180
is going to be equal to our
initial term negative 1/8.

00:02:02.180 --> 00:02:04.827
Then we're going to multiply it by two.

00:02:06.063 --> 00:02:11.063
We're going to multiply it
by two, i minus one times.

00:02:11.937 --> 00:02:16.937
So we could say times two,
to the i minus oneth power.

00:02:17.841 --> 00:02:21.666
Let's make sure that makes sense.

00:02:21.666 --> 00:02:24.850
A sub one, based on this formula,

00:02:24.850 --> 00:02:27.311
a sub one would be negative 1/8,

00:02:27.311 --> 00:02:30.097
times two to the one minus one.

00:02:30.097 --> 00:02:31.839
Two to the zeroeth power.

00:02:31.839 --> 00:02:32.814
So that makes sense.

00:02:32.814 --> 00:02:34.346
That would be negative 1/8.

00:02:34.346 --> 00:02:35.670
Based on this formula,

00:02:35.670 --> 00:02:39.362
a sub two would be negative 1/8,

00:02:39.362 --> 00:02:41.800
times two to the two minus one.

00:02:41.800 --> 00:02:43.239
So two to the first power.

00:02:43.239 --> 00:02:44.725
We're going to take our initial term,

00:02:44.725 --> 00:02:46.815
and multiply it by two, once.

00:02:46.815 --> 00:02:47.976
Which is exactly right.

00:02:47.976 --> 00:02:50.860
A sub two is negative 1/4.

00:02:50.860 --> 00:02:53.014
So we want to find the
fourth term in the sequence,

00:02:53.014 --> 00:02:56.213
we could just say well,
using this explicit formula,

00:02:56.213 --> 00:03:01.006
we could say a sub four,
is equal to negative 1/8,

00:03:01.006 --> 00:03:04.979
times two to the four, minus one.

00:03:04.979 --> 00:03:08.043
Two to the four, minus one power.

00:03:08.043 --> 00:03:10.574
So this is equal to negative 1/8,

00:03:10.574 --> 00:03:13.131
times two to the third power.

00:03:14.321 --> 00:03:16.694
This is negative 1/8 times eight.

00:03:16.694 --> 00:03:21.541
Negative 1/8 times eight,
which is equal to negative one.

00:03:22.382 --> 00:03:25.099
You might be a little bit,
a toss up on which method

00:03:25.099 --> 00:03:28.280
you want to use, but for
sure this second method,

00:03:28.280 --> 00:03:30.811
right over here where we'd come
up with an explicit formula

00:03:30.811 --> 00:03:32.924
once we know the initial term,

00:03:32.924 --> 00:03:34.572
and we know the common ratio,

00:03:34.572 --> 00:03:37.080
this would be way easier, if
you were trying to find say,

00:03:37.080 --> 00:03:38.403
the 40th term.

00:03:38.403 --> 00:03:40.841
Because doing the 40th term
recurs to be like this,

00:03:40.841 --> 00:03:43.939
would take a lot of time,
and frankly, a lot of paper.


Using explicit formulas of geometric sequences

WEBVTT
Kind: captions
Language: en

00:00:00.440 --> 00:00:04.015
- [Voiceover] The geometric
sequence A sub I is defined

00:00:04.015 --> 00:00:06.924
by the formula and so they
tell us that the Ith term is

00:00:06.924 --> 00:00:09.100
going to be equal to three
times negative one fourth

00:00:09.100 --> 00:00:11.515
to the I minus one power.

00:00:11.515 --> 00:00:13.060
So given that, what is A sub five,

00:00:13.060 --> 00:00:14.885
the fifth term in the sequence?

00:00:14.885 --> 00:00:16.484
So pause the video and try to figure out

00:00:16.484 --> 00:00:18.951
what is A subscript five?

00:00:19.839 --> 00:00:21.889
Alright, well, we can
just use this formula.

00:00:21.889 --> 00:00:23.744
A...

00:00:23.744 --> 00:00:26.339
A sub five is going to be...

00:00:26.339 --> 00:00:28.835
is everywhere I see an
I or a place with a five

00:00:28.835 --> 00:00:32.843
is going to be equal to three
times negative one fourth

00:00:32.843 --> 00:00:36.233
to the five minus one power.

00:00:36.233 --> 00:00:40.751
Well that's equal to three
times negative one fourth

00:00:40.751 --> 00:00:43.052
to the fourth power.

00:00:43.992 --> 00:00:46.321
Well that's going to be equal to...

00:00:46.321 --> 00:00:48.559
lets see, we're raising
it to an even power

00:00:48.559 --> 00:00:50.254
so it's going to give us a positive value

00:00:50.254 --> 00:00:52.076
since we're gonna be
multiplying the negative

00:00:52.076 --> 00:00:54.655
an even number of times so
it's gonna be a positive value

00:00:54.655 --> 00:00:57.322
so it's gonna be three times...

00:00:57.322 --> 00:00:59.309
let's see, one to the one fourth is-

00:00:59.309 --> 00:01:01.665
oh, one to the fourth power is just one,

00:01:01.665 --> 00:01:03.841
and then four to the fourth power...

00:01:03.841 --> 00:01:05.898
let's see, four squared is 16,

00:01:05.898 --> 00:01:08.190
so four squared times four
squared is four to the fourth

00:01:08.190 --> 00:01:11.198
so it's 16 times 16 is 256.

00:01:11.198 --> 00:01:12.587
256.

00:01:12.587 --> 00:01:14.966
And once again I know
it's going to be positive

00:01:14.966 --> 00:01:18.490
because I'm multiplying a
negative times itself four times,

00:01:18.490 --> 00:01:20.495
or I'm multiplying four
negatives together,

00:01:20.495 --> 00:01:22.638
so that's going to give
me a positive value.

00:01:22.638 --> 00:01:24.622
So I get three

00:01:24.622 --> 00:01:27.262
over 256.

00:01:28.542 --> 00:01:29.897
And we're done.

00:01:29.897 --> 00:01:32.394
That's the fifth term in our sequence.

00:01:32.394 --> 00:01:35.257
Positive three over 256.


Recursive formulas for arithmetic sequences

WEBVTT
Kind: captions
Language: en

00:00:00.203 --> 00:00:01.381
- [Voiceover] g is a function

00:00:01.381 --> 00:00:03.873
that describes an arithmetic sequence.

00:00:03.873 --> 00:00:06.997
Here are the first few
terms of the sequence.

00:00:06.997 --> 00:00:08.589
So let's say the first term is four,

00:00:08.589 --> 00:00:10.924
second term is 3 4/5,

00:00:10.924 --> 00:00:12.479
third term is 3 3/5,

00:00:12.479 --> 00:00:14.587
fourth term is 3 2/5.

00:00:14.587 --> 00:00:17.674
Find the values of the
missing parameters A and B

00:00:17.674 --> 00:00:21.137
in the following recursive
definition of the sequence.

00:00:21.137 --> 00:00:22.469
So they say the nth term

00:00:22.469 --> 00:00:24.444
is going to be equal to A

00:00:24.444 --> 00:00:26.245
if n is equal to one

00:00:26.245 --> 00:00:30.804
and it's going to be
equal to g of n minus one

00:00:30.804 --> 00:00:33.990
plus B if n is greater than one.

00:00:33.990 --> 00:00:36.199
And so I encourage you to pause this video

00:00:36.199 --> 00:00:37.183
and see if you could figure out

00:00:37.183 --> 00:00:40.337
what A and B are going to be.

00:00:40.337 --> 00:00:41.509
Well, the first one to figure out,

00:00:41.509 --> 00:00:44.041
A is actually pretty straightforward.

00:00:44.041 --> 00:00:45.732
If n is equal to one,

00:00:45.732 --> 00:00:47.183
if n is equal to one,

00:00:47.183 --> 00:00:50.457
the first term when n equals one is four.

00:00:50.457 --> 00:00:53.242
So A is equal to four.

00:00:53.242 --> 00:00:56.122
So we could write this as g of n

00:00:56.122 --> 00:00:59.559
is equal to four

00:00:59.559 --> 00:01:03.589
if n is equal to one.

00:01:03.589 --> 00:01:05.532
And now let's think about the second line.

00:01:05.532 --> 00:01:06.946
The second line is interesting.

00:01:06.946 --> 00:01:09.833
It's saying it's going to be
equal to the previous term,

00:01:09.833 --> 00:01:11.511
g of n minus one.

00:01:11.511 --> 00:01:15.635
This means the n minus oneth term, plus B,

00:01:15.635 --> 00:01:17.937
will give you the nth term.

00:01:17.937 --> 00:01:18.979
Let's just think about what's happening

00:01:18.979 --> 00:01:20.270
with this arithmetic sequence.

00:01:20.270 --> 00:01:22.892
When I go from the first
term to the second term,

00:01:22.892 --> 00:01:25.023
what have I done?

00:01:25.023 --> 00:01:27.720
Looks like I have subtracted 1/5,

00:01:27.720 --> 00:01:29.472
so minus 1/5,

00:01:29.472 --> 00:01:30.984
and then it's an arithmetic sequence

00:01:30.984 --> 00:01:34.460
so I should subtract or add
the same amount to every time,

00:01:34.460 --> 00:01:37.127
and I am, I'm subtracting 1/5,

00:01:37.127 --> 00:01:40.964
and so I am subtracting 1/5.

00:01:40.964 --> 00:01:42.296
And so one way to think about it,

00:01:42.296 --> 00:01:43.937
if we were to go the other way,

00:01:43.937 --> 00:01:48.365
we could say, for example, that g of four

00:01:48.365 --> 00:01:52.999
is equal to g of three

00:01:52.999 --> 00:01:55.215
minus 1/5,

00:01:55.215 --> 00:01:57.246
minus 1/5.

00:01:57.246 --> 00:01:58.485
You see that right over here.

00:01:58.485 --> 00:02:00.593
g of three is this.

00:02:00.593 --> 00:02:04.512
You subtract 1/5, you get g of four.

00:02:04.512 --> 00:02:05.731
You see that right over there

00:02:05.731 --> 00:02:06.841
and of course I could have written this

00:02:06.841 --> 00:02:09.235
like g of four

00:02:09.235 --> 00:02:11.728
is equal to g of four

00:02:11.728 --> 00:02:14.998
minus one minus 1/5.

00:02:14.998 --> 00:02:16.016
So if you look at this way,

00:02:16.016 --> 00:02:18.651
you could see that if I'm
trying to find the nth term,

00:02:18.651 --> 00:02:21.153
it's gonna be the n minus oneth term

00:02:21.153 --> 00:02:23.534
plus negative 1/5,

00:02:23.534 --> 00:02:25.880
so B is negative 1/5.

00:02:25.880 --> 00:02:27.747
Once again, if I'm trying
to find the fourth term,

00:02:27.747 --> 00:02:29.636
if n is equal to four,

00:02:29.636 --> 00:02:30.884
I'm not gonna use this first case

00:02:30.884 --> 00:02:33.603
'cause this has to be for n equals one,

00:02:33.603 --> 00:02:36.079
so if n equals four, I
would use the second case,

00:02:36.079 --> 00:02:38.348
so then it would be g of four minus one,

00:02:38.348 --> 00:02:41.760
it would be g of three minus 1/5.

00:02:41.760 --> 00:02:43.373
And so we could say g of n

00:02:43.373 --> 00:02:46.873
is equal to g of n minus one,

00:02:46.873 --> 00:02:50.537
so the term right before that minus 1/5

00:02:50.537 --> 00:02:53.506
if n is greater than one.

00:02:53.506 --> 00:02:55.138
But for the sake of this problem,

00:02:55.138 --> 00:02:59.993
we see that A is equal to four

00:02:59.993 --> 00:03:04.132
and B is equal to negative 1/5.


Using arithmetic sequences formulas

WEBVTT
Kind: captions
Language: en

00:00:00.323 --> 00:00:00.857
- [Voiceover] All right,

00:00:00.857 --> 00:00:03.434
we're told that the
arithmetic sequence a-sub-i

00:00:03.434 --> 00:00:07.869
is defined by the formula where
the ith term in the sequence

00:00:07.869 --> 00:00:10.330
is going to be 4 plus 3

00:00:10.330 --> 00:00:12.698
times i minus 1.

00:00:12.698 --> 00:00:14.788
What is a-sub-20?

00:00:14.788 --> 00:00:17.226
And so a-sub-20 is the
20th term in the sequence

00:00:17.226 --> 00:00:18.642
and I encourage you to pause the video

00:00:18.642 --> 00:00:21.568
and figure out what is the 20th term?

00:00:21.568 --> 00:00:23.356
Well, we can just think
about it like this.

00:00:23.356 --> 00:00:27.930
A-sub-20, we just use this
definition of the ith term.

00:00:27.930 --> 00:00:30.368
Everywhere we see an i,
we would put a 20 in,

00:00:30.368 --> 00:00:35.368
so it's going to be 4 plus 3 times 20

00:00:35.755 --> 00:00:37.148
minus 1,

00:00:37.148 --> 00:00:40.933
so once again, just to be clear, a-sub-20,

00:00:40.933 --> 00:00:43.742
where instead of a-sub-i,
wherever we saw an i,

00:00:43.742 --> 00:00:45.460
we put a 20,

00:00:45.460 --> 00:00:47.620
and now we can just compute
what this is going to be

00:00:47.620 --> 00:00:48.641
equal to.

00:00:48.641 --> 00:00:50.453
This is going to be equal to

00:00:50.453 --> 00:00:54.377
4 plus 3 times 20 minus 1

00:00:54.377 --> 00:00:56.095
is 19.

00:00:56.095 --> 00:00:57.975
3 times 19,

00:00:57.975 --> 00:01:01.482
let's see, 3 times 19 is 57, right?

00:01:01.482 --> 00:01:03.107
It's 30 plus 27, yep.

00:01:03.107 --> 00:01:05.684
This is 57,

00:01:05.684 --> 00:01:10.467
and 4 plus 57 is equal to 61,

00:01:10.467 --> 00:01:13.718
so the 20th term in
this arithmetic sequence

00:01:13.718 --> 00:01:16.597
is going to be 61.

00:01:16.597 --> 00:01:18.826
Let's do another one of these.

00:01:18.826 --> 00:01:20.777
And here, they've told us
the arithmetic sequence

00:01:20.777 --> 00:01:23.168
a-sub-i is defined by the formula

00:01:23.168 --> 00:01:24.050
a-sub-1,

00:01:24.050 --> 00:01:25.560
they gave us the first term,

00:01:25.560 --> 00:01:27.231
and they say, every other term,

00:01:27.231 --> 00:01:30.691
so a-sub-i, they're defining
it in terms of the previous

00:01:30.691 --> 00:01:34.290
terms, so a-sub-i is going
to be a-sub-i minus 1

00:01:34.290 --> 00:01:36.055
minus 2,

00:01:36.055 --> 00:01:38.516
so this is actually a
recursive definition of

00:01:38.516 --> 00:01:39.886
our arithmetic sequence.

00:01:39.886 --> 00:01:41.558
Let's see what we can make of this,

00:01:41.558 --> 00:01:43.903
so a-sub-5,

00:01:43.903 --> 00:01:46.364
a-sub-5 is going to be equal to,

00:01:46.364 --> 00:01:48.941
we'll use this second line right here,

00:01:48.941 --> 00:01:51.542
a-sub-5 is going to be equal to

00:01:51.542 --> 00:01:54.746
a-sub-4 minus 2.

00:01:54.746 --> 00:01:56.998
Well, we don't know what
a-sub-4 is just yet,

00:01:56.998 --> 00:01:59.111
so let's try to figure that out.

00:01:59.111 --> 00:02:02.640
So we could say that a-sub-4 is equal to,

00:02:02.640 --> 00:02:04.614
well if we use the second line again,

00:02:04.614 --> 00:02:07.841
it's going to be a-sub-3, minus 2.

00:02:07.841 --> 00:02:10.233
We still don't know what a-sub-3 is.

00:02:10.233 --> 00:02:12.601
I'll keep switching colors
'cause it looks nice.

00:02:12.601 --> 00:02:15.829
A-sub-3 is going to be equal to

00:02:15.829 --> 00:02:17.802
a-sub-3 minus 1,

00:02:17.802 --> 00:02:20.774
so a-sub-2 minus 2.

00:02:20.774 --> 00:02:23.352
We still don't know what a-sub-2 is,

00:02:23.352 --> 00:02:25.883
and so, we could write,

00:02:25.883 --> 00:02:29.064
a-sub-2 is equal to

00:02:29.064 --> 00:02:30.643
a-sub-2 minus 1,

00:02:30.643 --> 00:02:33.684
so that's a-sub-1 minus 2.

00:02:33.684 --> 00:02:35.983
Now, luckily we know what a-sub-1 is.

00:02:35.983 --> 00:02:38.885
A-sub-1 is negative 7.

00:02:38.885 --> 00:02:41.207
A-sub-1 is negative 7,

00:02:41.207 --> 00:02:44.969
so if this is negative 7,

00:02:44.969 --> 00:02:48.103
then a-sub-2 is negative 7 minus 2,

00:02:48.103 --> 00:02:50.448
which is equal to negative 9.

00:02:50.448 --> 00:02:51.888
Well that starts helping us out

00:02:51.888 --> 00:02:54.163
because if a-sub-2 is negative 9,

00:02:54.163 --> 00:02:55.649
if this is negative 9,

00:02:55.649 --> 00:02:57.646
then a-sub-3 is negative 9

00:02:57.646 --> 00:02:58.877
minus 2,

00:02:58.877 --> 00:03:01.524
which is equal to negative 11.

00:03:01.524 --> 00:03:04.937
Well now that we know that
a-sub-3 is negative 11,

00:03:04.937 --> 00:03:06.864
so this is negative 11,

00:03:06.864 --> 00:03:09.047
we could figure out a-sub-4 is negative 11

00:03:09.047 --> 00:03:10.347
minus 2,

00:03:10.347 --> 00:03:13.644
which is equal to negative 13.

00:03:13.644 --> 00:03:14.875
And we're almost there.

00:03:14.875 --> 00:03:16.082
We know a-sub-4,

00:03:16.082 --> 00:03:17.893
the fourth term in this
arithmetic sequence

00:03:17.893 --> 00:03:19.518
is negative 13,

00:03:19.518 --> 00:03:20.633
so,

00:03:20.633 --> 00:03:21.376
we can now,

00:03:21.376 --> 00:03:23.257
so if this is negative 13,

00:03:23.257 --> 00:03:25.393
a-sub-5 is going to be a-sub-4,

00:03:25.393 --> 00:03:27.367
which is negative 13 minus 2,

00:03:27.367 --> 00:03:30.594
which is equal to negative 15,

00:03:30.594 --> 00:03:32.080
so the fifth term in the sequence

00:03:32.080 --> 00:03:35.145
is negative 15.

00:03:35.145 --> 00:03:36.579
And we're all done.


Generating inputs and outputs of a function

WEBVTT
Kind: captions
Language: en

00:00:00.606 --> 00:00:02.545
- [Voiceover] So we're asked
to pick any three pairs

00:00:02.545 --> 00:00:04.982
of corresponding input and output values

00:00:04.982 --> 00:00:09.175
of the following function, and
fill the table accordingly,

00:00:09.175 --> 00:00:11.804
and if necessary round our
answer to the nearest 0.1,

00:00:11.804 --> 00:00:14.683
and our function is
defined as if I input a t,

00:00:14.683 --> 00:00:17.248
what I'm going output
f of t is going to be

00:00:17.248 --> 00:00:19.187
negative two times my input,

00:00:19.187 --> 00:00:21.648
negative two times t plus three,

00:00:21.648 --> 00:00:25.247
and so we can just pick
three arbitrary t's

00:00:25.247 --> 00:00:26.803
and input them into our function

00:00:26.803 --> 00:00:29.042
and see what the function spits out.

00:00:29.042 --> 00:00:32.723
We could go picking some
really easy t's to compute.

00:00:32.723 --> 00:00:34.929
So for example, if t is equal to zero,

00:00:34.929 --> 00:00:36.704
then f of t, we could do this in our head,

00:00:36.704 --> 00:00:41.655
but it would be negative
two times zero plus three.

00:00:42.554 --> 00:00:45.642
Well, negative two times zero
is just going to be zero,

00:00:45.642 --> 00:00:49.160
plus three is just going to be three.

00:00:49.160 --> 00:00:54.140
If t is one, well then
f of t is going to be

00:00:54.140 --> 00:00:58.598
negative two times one, I'll
write it in parentheses,

00:00:58.598 --> 00:01:00.897
plus three.

00:01:00.897 --> 00:01:04.820
Negative two times one is negative two

00:01:04.820 --> 00:01:07.734
plus three is one.

00:01:07.734 --> 00:01:09.116
This is one.

00:01:09.116 --> 00:01:11.159
And just for kicks, we could
even put in negative one.

00:01:11.159 --> 00:01:14.224
Once again, I could
put in crazier numbers.

00:01:14.224 --> 00:01:18.415
I could have put in 2.30789,
but that would just be

00:01:18.415 --> 00:01:22.013
more computation, might as
well pick some easy numbers.

00:01:22.013 --> 00:01:23.755
We're just trying to get
a feel for this function.

00:01:23.755 --> 00:01:25.216
Depending on what I
input, what do I output,

00:01:25.216 --> 00:01:27.458
and I'm allowed to pick my inputs.

00:01:27.458 --> 00:01:30.115
So if it's negative one, it's going to be

00:01:30.115 --> 00:01:35.032
negative two, so t is now
negative one, plus three.

00:01:36.582 --> 00:01:38.891
So negative two times negative one is

00:01:38.891 --> 00:01:41.782
positive two plus three is five.

00:01:41.782 --> 00:01:45.032
So it's gonna be positive five.

00:01:45.032 --> 00:01:48.237
Let's do a few more of these.

00:01:48.237 --> 00:01:51.034
So this says choose any three
points on the following graph

00:01:51.034 --> 00:01:53.171
of y is equal to g of
x, and fill the table

00:01:53.171 --> 00:01:55.574
so each column represents a point.

00:01:55.574 --> 00:01:58.487
If necessary, round your
answer to the nearest 0.5.

00:01:58.487 --> 00:02:00.508
So once again we could see,

00:02:00.508 --> 00:02:02.156
there's a bunch of
interesting points here.

00:02:02.156 --> 00:02:05.059
If x is equal to zero, what is g of x?

00:02:05.059 --> 00:02:07.379
This is y is equal to g
of x and we see g of x

00:02:07.379 --> 00:02:10.132
is equal to negative one.

00:02:10.132 --> 00:02:11.931
Actually, let me make
this a little bit smaller

00:02:11.931 --> 00:02:15.286
so that we can see it all on one screen.

00:02:15.286 --> 00:02:16.981
So there we go, so now we should be

00:02:16.981 --> 00:02:19.326
able to see it a little bit easier.

00:02:19.326 --> 00:02:23.320
So if x is zero, and once again,

00:02:23.320 --> 00:02:25.827
I'm just picking zero
because it's easy to see

00:02:25.827 --> 00:02:29.948
that when x is zero, y
is equal to negative one.

00:02:29.948 --> 00:02:32.249
And y is equal to g of x,

00:02:32.249 --> 00:02:36.323
so g of zero is negative one.

00:02:36.323 --> 00:02:38.227
See that right from the graph.

00:02:38.227 --> 00:02:40.304
Let's see, some other points.

00:02:40.304 --> 00:02:41.675
So we have this point right over here.

00:02:41.675 --> 00:02:45.622
When x is negative one, y,
which is equal to g of x,

00:02:45.622 --> 00:02:49.518
is equal to two, the point
one comma two is on this line.

00:02:49.518 --> 00:02:53.860
When x is one, y is two,
and y is equal to g of x.

00:02:53.860 --> 00:02:57.505
So g of one is two.

00:02:57.505 --> 00:02:59.593
All right, let's pick another one here.

00:02:59.593 --> 00:03:04.215
Let's see, when x is
one, y is negative four.

00:03:04.215 --> 00:03:07.991
When x is, sorry, this was
negative one over here.

00:03:07.991 --> 00:03:11.090
When x is negative one, y is equal to two,

00:03:11.090 --> 00:03:14.701
and when x is positive
one, y would be equal

00:03:14.701 --> 00:03:16.871
to negative four, and actually, let me,

00:03:16.871 --> 00:03:19.171
these are the exact same
numbers that I used last time

00:03:19.171 --> 00:03:20.610
for the inputs.

00:03:20.610 --> 00:03:23.577
Let's just see that I
could use these numbers,

00:03:23.577 --> 00:03:25.120
but I could pick different numbers.

00:03:25.120 --> 00:03:27.639
I could say, let's see this
number, this point over here

00:03:27.639 --> 00:03:29.558
is interesting because it looks like

00:03:29.558 --> 00:03:32.093
both the x and y coordinates are integers,

00:03:32.093 --> 00:03:36.384
so if x is negative three, y is eight.

00:03:36.384 --> 00:03:39.928
So if x is negative three,

00:03:39.928 --> 00:03:43.632
y is eight.

00:03:43.632 --> 00:03:45.988
So once again, you should
try to pick some values

00:03:45.988 --> 00:03:50.214
where you can clearly see
what the corresponding

00:03:50.214 --> 00:03:53.093
x and y coordinates are of that line,

00:03:53.093 --> 00:03:55.149
but any of those would be valid pairs.

00:03:55.149 --> 00:03:58.938
For a given input, what's g
of x going to actually be?


Extending arithmetic sequences

WEBVTT
Kind: captions
Language: en

00:00:00.213 --> 00:00:01.931
- [Voiceover] We're told
the first four terms

00:00:01.931 --> 00:00:03.649
of an arithmetic sequence are given.

00:00:03.649 --> 00:00:06.454
So it goes from negative
eight to negative 14

00:00:06.454 --> 00:00:09.449
to negative 20 to negative 26.

00:00:09.449 --> 00:00:11.515
What is the fifth term in the sequence?

00:00:11.515 --> 00:00:13.578
So we just need to
figure out the next term.

00:00:13.578 --> 00:00:15.458
Well, in arithmetic sequence,

00:00:15.458 --> 00:00:19.220
each successive term is
separated by the same amount.

00:00:19.220 --> 00:00:22.029
So when we go from negative
eight to negative 14,

00:00:22.029 --> 00:00:25.466
we went down by six and
then we go down by six again

00:00:25.466 --> 00:00:28.391
to go to negative 20 and
then we go down by six again

00:00:28.391 --> 00:00:30.634
to go to negative 26, and
so we're gonna go down

00:00:30.634 --> 00:00:34.631
by six again to get to negative 32.

00:00:35.505 --> 00:00:38.677
Negative 32.

00:00:38.677 --> 00:00:40.595
Let's do a couple more of these.

00:00:40.595 --> 00:00:43.056
The first five terms of
the sequence are given.

00:00:43.056 --> 00:00:44.611
What is the sixth term?

00:00:44.611 --> 00:00:46.469
Let's see, to go from two to negative one,

00:00:46.469 --> 00:00:48.303
you subtracted three.

00:00:48.303 --> 00:00:50.416
Two minus three is negative one.

00:00:50.416 --> 00:00:52.320
Negative one minus three is negative four.

00:00:52.320 --> 00:00:54.661
Negative four minus
three is negative seven.

00:00:54.661 --> 00:00:57.145
Negative seven minus three is negative 10.

00:00:57.145 --> 00:01:01.170
And so negative 10 minus
three is negative 13.

00:01:02.313 --> 00:01:05.721
So hopefully that gives
you the hang of things.


Interpreting units in formulas: novel units

WEBVTT
Kind: captions
Language: en

00:00:00.283 --> 00:00:02.048
- [Voiceover] So, we're
told, consider the formula

00:00:02.048 --> 00:00:04.648
Y is equal to two-C over P,

00:00:04.648 --> 00:00:08.084
where Y represents the carrot yield.

00:00:08.084 --> 00:00:11.515
So, Y represents the carrot yield.

00:00:13.045 --> 00:00:16.530
C represents the number

00:00:16.530 --> 00:00:18.980
of expected carrots,

00:00:18.980 --> 00:00:22.387
and P represents the number of plants.

00:00:22.387 --> 00:00:25.463
So, P represents the number of plants.

00:00:25.463 --> 00:00:28.098
Select an appropriate measurement unit

00:00:28.098 --> 00:00:30.583
for carrot yield.

00:00:30.583 --> 00:00:34.563
So, the key here is to realize
that we can treat units

00:00:34.563 --> 00:00:37.254
like we would treat algebraic variables.

00:00:37.254 --> 00:00:40.168
And so, for example, over here,

00:00:40.168 --> 00:00:42.327
if we're saying our yield is going to be

00:00:42.327 --> 00:00:46.390
two times the number of expected carrots

00:00:46.390 --> 00:00:48.886
divided by the number of plants,

00:00:48.886 --> 00:00:52.090
well, our units are going to be...

00:00:52.090 --> 00:00:54.911
Well actually we could put
some numbers in here actually

00:00:54.911 --> 00:00:56.896
just to make something more interesting.

00:00:56.896 --> 00:00:59.475
Let's just say that C is equal

00:00:59.475 --> 00:01:01.905
to 10 carrots,

00:01:01.905 --> 00:01:03.572
and I'm just making these numbers up.

00:01:03.572 --> 00:01:06.009
Just, hopefully, it makes
sense what I'm about to do.

00:01:06.009 --> 00:01:10.595
And let's say that P is equal to,

00:01:10.595 --> 00:01:13.045
I don't know, 30 plants.

00:01:13.045 --> 00:01:15.308
P is equal to 30 plants.

00:01:15.308 --> 00:01:18.628
And so using this formula, Y,

00:01:18.628 --> 00:01:21.944
Y would be equal to

00:01:21.944 --> 00:01:26.435
two times 10 carrots,

00:01:26.435 --> 00:01:28.345
two times

00:01:29.435 --> 00:01:33.107
10 carrots,

00:01:33.107 --> 00:01:37.577
divided by P,

00:01:37.577 --> 00:01:40.677
which for this case is 30 plants.

00:01:40.677 --> 00:01:43.677
30 plants.

00:01:43.677 --> 00:01:45.387
And so two times 10 carrots,

00:01:45.387 --> 00:01:47.663
that's going to be 20 carrots.

00:01:47.663 --> 00:01:50.611
So we can write the numerator
as, this is 20 carrots,

00:01:50.611 --> 00:01:52.585
divided by 30 plants,

00:01:52.585 --> 00:01:54.849
and so that is going to be equal to...

00:01:54.849 --> 00:01:56.532
You could divide the numbers.

00:01:56.532 --> 00:01:59.991
So, it would be 20 divided
by 30, which would be 2/3,

00:01:59.991 --> 00:02:04.991
2/3, and then the units
would be carrots per plant.

00:02:05.227 --> 00:02:09.000
2/3 carrots per plant.

00:02:09.000 --> 00:02:13.559
Carrots per plant.

00:02:14.909 --> 00:02:17.579
Carrots per plant.

00:02:17.579 --> 00:02:18.786
And once again, the whole point

00:02:18.786 --> 00:02:21.143
of what I just did is just to
see what the units would be

00:02:21.143 --> 00:02:22.907
for our carrot yield.

00:02:22.907 --> 00:02:26.578
And we can see the units
would be carrots per plant.

00:02:26.578 --> 00:02:28.039
And I put the numbers
there just so hopefully

00:02:28.039 --> 00:02:31.649
it makes a little bit intuitive
sense of what we just did.

00:02:32.569 --> 00:02:34.854
As we algebraically manipulate

00:02:34.854 --> 00:02:37.082
or mathematically manipulate the numbers,

00:02:37.082 --> 00:02:39.601
we do the same thing with the units.

00:02:39.601 --> 00:02:40.812
And so, when we try

00:02:40.812 --> 00:02:43.364
to find the appropriate
measure unit for carrot yield,

00:02:43.364 --> 00:02:46.074
which I had never heard
of before this video,

00:02:46.074 --> 00:02:49.676
we see it's carrots per plant is

00:02:49.676 --> 00:02:51.825
the appropriate measurement unit,

00:02:51.825 --> 00:02:54.078
not carrots per plant-squared.

00:02:54.078 --> 00:02:55.564
That would have been the
case if we were squaring

00:02:55.564 --> 00:02:57.038
this P, over here,

00:02:57.038 --> 00:03:00.056
or plants times the
square root of carrots.

00:03:00.056 --> 00:03:01.879
Well that would be more if we
were taking the square root

00:03:01.879 --> 00:03:03.446
up here and we weren't dividing by P,

00:03:03.446 --> 00:03:05.280
but we were multiplying by P.

00:03:05.280 --> 00:03:08.217
But the general theme that you
could see is this constant,

00:03:08.217 --> 00:03:11.154
this two, didn't affect
what happens to the units.

00:03:11.154 --> 00:03:13.883
But what did matter is
how these variables relate

00:03:13.883 --> 00:03:14.463
to each other.

00:03:14.463 --> 00:03:17.040
We're taking the variable,
C, dividing it by P.

00:03:17.040 --> 00:03:19.200
So, whatever the units were for C,

00:03:19.200 --> 00:03:21.545
we divide those units by the units for P

00:03:21.545 --> 00:03:23.599
and we get the units for yield,

00:03:23.599 --> 00:03:25.651
carrots per plant.


Interpreting units in formulas

WEBVTT
Kind: captions
Language: en

00:00:00.001 --> 00:00:02.621
- [Voiceover] Consider
the formula P is equal to

00:00:02.621 --> 00:00:06.547
W divided by T where P represents power,

00:00:06.547 --> 00:00:11.412
W represents work and has units of Joules

00:00:11.412 --> 00:00:13.741
and Joules can be expressed as kilograms

00:00:13.741 --> 00:00:17.583
times meters squared
per seconds squared and

00:00:17.583 --> 00:00:20.731
T represents time and
has units of seconds.

00:00:20.731 --> 00:00:22.814
And when you get to physics
class you'll get very familiar

00:00:22.814 --> 00:00:25.390
with things like Joules
which can be represented

00:00:25.390 --> 00:00:27.148
as kilograms times meters squared

00:00:27.148 --> 00:00:29.561
per seconds squared and things like power.

00:00:29.561 --> 00:00:31.892
But here we're going
to learn to manipulate

00:00:31.892 --> 00:00:34.101
these units so that they make sense.

00:00:34.101 --> 00:00:38.192
So it says select an appropriate
measurement unit for power.

00:00:38.192 --> 00:00:40.603
And what we've seen multiple times in our

00:00:40.603 --> 00:00:44.316
mathematical careers is that
on a certain level you can

00:00:44.316 --> 00:00:46.853
manipulate units in a lot of the same ways

00:00:46.853 --> 00:00:49.274
that you would manipulate
variables or numbers.

00:00:49.274 --> 00:00:53.402
So if power is equal
to work divided by time

00:00:53.402 --> 00:00:56.226
we could also say that the units for power

00:00:56.226 --> 00:00:59.334
are going to be the units for work

00:00:59.334 --> 00:01:01.909
divided by the units for time.

00:01:01.909 --> 00:01:05.224
So the units for work...the units

00:01:05.224 --> 00:01:09.680
for work right over here is Joules.

00:01:09.680 --> 00:01:11.602
And so we could write it's going to be

00:01:11.602 --> 00:01:16.510
Joules per and then the
unit for time is seconds.

00:01:17.566 --> 00:01:21.491
So you might want to say
it's Joules per second.

00:01:21.491 --> 00:01:25.206
But we don't see Joules per
second as a choice here,

00:01:25.206 --> 00:01:27.054
so we probably want to expand out Joules

00:01:27.054 --> 00:01:28.854
as being kilogram meters squared

00:01:28.854 --> 00:01:31.183
per seconds squared so lets do that.

00:01:31.183 --> 00:01:33.065
So this is going to be equal to Joules

00:01:33.065 --> 00:01:37.112
we can re-write as kilogram

00:01:39.848 --> 00:01:43.648
times meters squared over

00:01:43.648 --> 00:01:45.740
seconds squared and we are going

00:01:45.740 --> 00:01:48.251
to divide all of that by seconds.

00:01:52.245 --> 00:01:53.749
And so what's that going to be?

00:01:53.749 --> 00:01:57.475
Well we could re-write this,
this is going to be kilograms

00:01:57.475 --> 00:02:01.483
(and I'm intentionally trying
not to skip any steps),

00:02:01.483 --> 00:02:06.483
kilograms times meters
squared per seconds squared

00:02:06.512 --> 00:02:09.012
and dividing by seconds
is the same thing as

00:02:09.012 --> 00:02:10.976
multiplying by 1 over seconds.

00:02:10.976 --> 00:02:15.430
So times 1 over seconds and
so if we treat these units

00:02:15.430 --> 00:02:17.515
the way that we might
treat things like variables

00:02:17.515 --> 00:02:20.176
this would be equal to, in the numerator,

00:02:20.176 --> 00:02:25.176
we would have kilogram
times meters squared

00:02:25.521 --> 00:02:27.331
or kilogram times square meters

00:02:27.331 --> 00:02:29.531
over the denominator
you have seconds squared

00:02:29.531 --> 00:02:33.130
times seconds, you have
seconds to the third power.

00:02:33.130 --> 00:02:36.658
So a unit for power, one way
to express the units for power,

00:02:36.658 --> 00:02:39.925
could be kilogram meters
squared per second cubed.

00:02:39.925 --> 00:02:42.705
And we see that this is this first choice

00:02:42.705 --> 00:02:46.561
kilograms meters squared
per seconds cubed.


Linear equations with unknown coefficients

WEBVTT
Kind: captions
Language: en

00:00:00.193 --> 00:00:01.342
- [Voiceover] So we have an equation.

00:00:01.342 --> 00:00:05.786
It says, a-x plus three-x
is equal to b-x plus five.

00:00:05.786 --> 00:00:09.418
And what I want to do
together is to solve for x,

00:00:09.418 --> 00:00:11.047
and if we solve for x it's going to be

00:00:11.047 --> 00:00:14.391
in terms of a, b, and other numbers.

00:00:14.391 --> 00:00:17.038
So pause the video and
see if you can do that.

00:00:17.038 --> 00:00:18.640
All right now, let's do this together,

00:00:18.640 --> 00:00:19.260
and what I'm going to do,

00:00:19.260 --> 00:00:21.349
is I'm gonna try to
group all of the x-terms,

00:00:21.349 --> 00:00:23.353
let's group all the x-terms
on the left-hand side.

00:00:23.353 --> 00:00:25.953
So, I already have a-x and
three-x on the left-hand side.

00:00:25.953 --> 00:00:29.309
Let's get b-x onto the left-hand
side as well, and I can do

00:00:29.309 --> 00:00:34.057
that by subtracting b-x from both sides.

00:00:34.057 --> 00:00:37.551
And if I subtract b-x from both sides,

00:00:37.551 --> 00:00:41.499
I'm going to get on the right-hand side,

00:00:41.499 --> 00:00:44.192
I'm going to have a, or
on the left-hand side,

00:00:44.192 --> 00:00:47.984
a-x plus three-x

00:00:47.984 --> 00:00:52.701
minus b-x, so I can do
that in that color for fun,

00:00:52.701 --> 00:00:56.996
minus b-x, and that's
going to be equal to...

00:00:56.996 --> 00:00:59.155
Well, b-x minus b-x is just zero,

00:00:59.155 --> 00:01:00.211
and I have five.

00:01:00.211 --> 00:01:02.255
It is equal to five.

00:01:02.255 --> 00:01:04.251
And now what I can do is,

00:01:04.251 --> 00:01:07.015
I can factor an x out
of this left-hand side

00:01:07.015 --> 00:01:09.116
of this equation, out of all of the terms.

00:01:09.116 --> 00:01:12.796
So, I can rewrite this as x times...

00:01:12.796 --> 00:01:15.792
Well, a-x divided by x is a.

00:01:15.792 --> 00:01:18.589
Three-x divided by x is three,

00:01:18.589 --> 00:01:20.585
and then negative b-x divided by x

00:01:20.585 --> 00:01:23.847
is just going to be negative b.

00:01:23.847 --> 00:01:26.545
I could keep writing
it in that pink color.

00:01:26.545 --> 00:01:28.862
And that's all going to be equal to five.

00:01:28.862 --> 00:01:30.709
And now, to solve for x I can just divide

00:01:30.709 --> 00:01:35.166
both sides by, the thing that
x is being multiplied by,

00:01:35.166 --> 00:01:37.329
by a plus three minus b.

00:01:37.329 --> 00:01:40.013
So, I can divide both sides by

00:01:40.013 --> 00:01:44.237
a plus three minus b.

00:01:44.237 --> 00:01:47.175
A plus three minus b.

00:01:47.175 --> 00:01:49.415
On this side, they cancel out.

00:01:49.415 --> 00:01:52.515
And, I have x is equal to five

00:01:52.515 --> 00:01:56.598
over a plus three

00:01:56.598 --> 00:01:59.747
minus b, and we are done.

00:01:59.747 --> 00:02:02.394
Let's do one more of these.

00:02:02.394 --> 00:02:03.729
So, another equation here.

00:02:03.729 --> 00:02:04.936
We have a...

00:02:04.936 --> 00:02:09.487
Here we have a times the
quantity, five minus x,

00:02:09.487 --> 00:02:11.496
is equal to b-x minus eight.

00:02:11.496 --> 00:02:12.796
So, once again, pause the video

00:02:12.796 --> 00:02:16.020
and see if you can solve for x.

00:02:16.020 --> 00:02:17.428
Well, the way I like to approach these

00:02:17.428 --> 00:02:20.028
is let's just expand everything all out.

00:02:20.028 --> 00:02:21.386
So, let me just distribute this a,

00:02:21.386 --> 00:02:23.454
and then I'm gonna collect
all the x-terms on one side,

00:02:23.454 --> 00:02:25.670
and all of the non-x-terms
on the other side,

00:02:25.670 --> 00:02:28.526
and essentially do what I
just did in the last example.

00:02:28.526 --> 00:02:31.068
So, let's first distribute this a.

00:02:31.068 --> 00:02:33.343
So the left-hand side becomes

00:02:33.343 --> 00:02:36.571
five-a, I could say a
times five or five-a,

00:02:36.571 --> 00:02:40.567
minus a-x, a-x,

00:02:40.567 --> 00:02:43.269
that is going to be equal to

00:02:43.269 --> 00:02:45.743
b-x minus eight.

00:02:45.743 --> 00:02:48.854
Now we can subtract b-x from both sides.

00:02:48.854 --> 00:02:51.303
So, we're gonna subtract
b-x from the left-hand side,

00:02:51.303 --> 00:02:52.812
b-x from the right-hand side.

00:02:52.812 --> 00:02:53.950
And, once again, the whole
reason I'm doing that,

00:02:53.950 --> 00:02:55.702
I want all the x-terms on the left,

00:02:55.702 --> 00:02:58.082
and all the non-x-terms on the right.

00:02:58.082 --> 00:03:01.530
And, actually, since I want all
the non-x-terms on the right,

00:03:01.530 --> 00:03:04.894
I can actually subtract five-a
from both sides, as well.

00:03:04.894 --> 00:03:06.634
So I'm kind of doing
two steps at once, here,

00:03:06.634 --> 00:03:07.621
but hopefully it makes sense.

00:03:07.621 --> 00:03:09.483
I'm trying to get rid of the b-x here,

00:03:09.483 --> 00:03:11.247
and I'm trying to get
rid of the five-a here.

00:03:11.247 --> 00:03:13.290
So, I subtract five-a there,

00:03:13.290 --> 00:03:15.809
and I'll subtract five-a there,

00:03:15.809 --> 00:03:19.164
and then let's see what this give us.

00:03:19.164 --> 00:03:21.207
So, the five-a's cancel out.

00:03:21.207 --> 00:03:22.264
And, on the left-hand side,

00:03:22.264 --> 00:03:24.539
I have negative a-x,

00:03:24.539 --> 00:03:26.769
negative a-x,

00:03:26.769 --> 00:03:28.659
minus b-x,

00:03:29.510 --> 00:03:33.269
minus, you know, in that same green color,

00:03:33.269 --> 00:03:35.104
minus b-x.

00:03:35.104 --> 00:03:37.148
And on the right-hand side,

00:03:37.148 --> 00:03:37.982
I have...

00:03:37.982 --> 00:03:38.888
This is going to be equal to,

00:03:38.888 --> 00:03:40.420
the b-x's cancel out,

00:03:40.420 --> 00:03:44.359
and I have negative eight minus five-a.

00:03:44.359 --> 00:03:47.304
Negative eight minus,

00:03:47.304 --> 00:03:49.707
in that same magenta color,

00:03:49.707 --> 00:03:52.586
minus five-a.

00:03:52.586 --> 00:03:55.977
And let's see, I have all my x's

00:03:55.977 --> 00:03:58.484
on one side, all my
non-x's on the other side.

00:03:58.484 --> 00:04:01.386
And here I can factor out an x,

00:04:01.386 --> 00:04:04.555
and if I factor out an x, what do I have?

00:04:04.555 --> 00:04:05.681
And, actually, one thing
that might be nice.

00:04:05.681 --> 00:04:08.317
Let me just multiply both
sides by negative one.

00:04:08.317 --> 00:04:10.139
If I multiple both sides by negative one,

00:04:10.139 --> 00:04:13.785
I get a-x plus b-x,

00:04:13.785 --> 00:04:17.941
plus b-x is equal to

00:04:17.941 --> 00:04:20.274
eight plus five-a.

00:04:20.274 --> 00:04:22.945
That just gets rid of all
of those negative signs.

00:04:22.945 --> 00:04:25.800
And now I can factor out an x here.

00:04:25.800 --> 00:04:28.795
So let me factor out an x,

00:04:28.795 --> 00:04:30.967
and I get x times

00:04:30.967 --> 00:04:34.847
a plus b.

00:04:34.847 --> 00:04:39.220
A plus b is going to

00:04:39.220 --> 00:04:43.563
be equal to eight plus five-a.

00:04:43.563 --> 00:04:45.118
Eight plus five-a.

00:04:45.118 --> 00:04:46.691
And we're in the home stretch now.

00:04:46.691 --> 00:04:50.902
We can just divide both sides by a plus b.

00:04:50.902 --> 00:04:54.962
So we could divide both sides by a plus b.

00:04:54.962 --> 00:04:56.958
A plus b.

00:04:56.958 --> 00:04:57.934
And we're going to be left with,

00:04:57.934 --> 00:05:00.241
x is equal to

00:05:01.161 --> 00:05:03.738
eight plus five-a,

00:05:03.738 --> 00:05:07.486
plus five-a, over...

00:05:09.276 --> 00:05:12.294
Actually, I'll write a and
b in our original colors.

00:05:12.294 --> 00:05:16.709
A plus, so this was that a, that a,

00:05:16.709 --> 00:05:18.459
plus b.

00:05:18.459 --> 00:05:22.556
So this is the b, that's a and b,

00:05:22.556 --> 00:05:24.375
a plus b, and we're done.

00:05:24.375 --> 00:05:25.738
We have now solved for x

00:05:25.738 --> 00:05:30.579
in terms of a's and b's and other things.

00:05:30.579 --> 00:05:33.196
And we are all done.


Energy equation

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.737
- [Voiceover] In recent
years the amount of C O two

00:00:02.737 --> 00:00:07.150
released by humans into
the planet has approached

00:00:07.150 --> 00:00:12.150
40 billion, 40 billion tons and if

00:00:12.407 --> 00:00:15.882
you wanted to break that
down based on people

00:00:15.882 --> 00:00:18.543
well we've recently crossed seven billion

00:00:18.543 --> 00:00:21.060
people on the planet so that's going to be

00:00:21.060 --> 00:00:26.052
approximately seven billion
people, seven billion people

00:00:26.772 --> 00:00:30.696
times the amount of tons per person

00:00:30.696 --> 00:00:32.180
and so if you work out the math,

00:00:32.180 --> 00:00:35.898
40 divided by seven is
going to be roughly 5.7

00:00:35.898 --> 00:00:40.898
5.7 tons per person, tons per person.

00:00:43.653 --> 00:00:45.548
And this might already seem like a lot,

00:00:45.548 --> 00:00:49.977
you can imagine 5.7 tons
of wood just being burned,

00:00:49.977 --> 00:00:52.781
actually it would have to be
more than 5.7 tons of wood

00:00:52.781 --> 00:00:55.361
because wood contains some water.

00:00:55.361 --> 00:00:58.146
But it's even more than that because,

00:00:58.146 --> 00:00:59.843
or at least for many of you all watching,

00:00:59.843 --> 00:01:01.754
it's going to be a lot more than that,

00:01:01.754 --> 00:01:03.887
because that's the
average of the developed

00:01:03.887 --> 00:01:06.204
and the developing world and it turns out

00:01:06.204 --> 00:01:09.028
that the average person
in the developed world,

00:01:09.028 --> 00:01:14.028
so developed, developed average,

00:01:14.208 --> 00:01:15.734
or if you're a wealthy person or a

00:01:15.734 --> 00:01:18.814
middle class person in
even the developing world,

00:01:18.814 --> 00:01:22.938
your average is going
to be closer to 20 tons,

00:01:22.938 --> 00:01:26.656
20 tons, per person, per year

00:01:26.656 --> 00:01:28.688
and in the developing world, or if you're

00:01:28.688 --> 00:01:30.537
not as wealthy of a
person or you're living

00:01:30.537 --> 00:01:32.223
in a rural place and
you're not using your car

00:01:32.223 --> 00:01:34.336
to get to where you need to get to

00:01:34.336 --> 00:01:36.124
or your house doesn't have electricity,

00:01:36.124 --> 00:01:41.049
so the poor, the poor, are using closer to

00:01:41.049 --> 00:01:46.049
one ton of C O two per person, per year.

00:01:47.014 --> 00:01:49.594
And so this already when
you even see this dynamic,

00:01:49.594 --> 00:01:51.624
it leads to something interesting.

00:01:51.624 --> 00:01:53.230
The number of people on the planet

00:01:53.230 --> 00:01:57.781
is only likely to increase
and we can see that.

00:01:57.781 --> 00:01:59.365
We have a little chart here from,

00:01:59.365 --> 00:02:02.170
based on data from the UN,
I know the numbers here

00:02:02.170 --> 00:02:03.896
are a little bit small, it starts in 1800

00:02:03.896 --> 00:02:06.781
and it goes all the way to 2100

00:02:06.781 --> 00:02:08.589
and the blue are the actuals, if you

00:02:08.589 --> 00:02:11.068
want to see where we are right now,

00:02:11.068 --> 00:02:14.786
so we are not quite at 2020 yet,

00:02:14.786 --> 00:02:18.423
so we're gonna be
roughly right over there.

00:02:18.423 --> 00:02:20.718
That's where we are right now,

00:02:20.718 --> 00:02:23.746
we've recently crossed
seven billion people

00:02:23.746 --> 00:02:25.090
and if you wanted to see where these

00:02:25.090 --> 00:02:27.486
UN predictions have us for 2050,

00:02:27.486 --> 00:02:30.920
2050 is right over here and so

00:02:30.920 --> 00:02:34.836
you can see the high UN prediction has us

00:02:34.836 --> 00:02:38.291
getting to a world population of about

00:02:38.291 --> 00:02:41.501
ten and a half billion,
the medium prediction

00:02:41.501 --> 00:02:44.529
has us crossing nine billion,
this is like a little bit,

00:02:44.529 --> 00:02:49.101
maybe 9.3, 9.4 billion and
then the low prediction

00:02:49.101 --> 00:02:51.132
still has a population increasing a bit,

00:02:51.132 --> 00:02:55.804
it's crossing eight
billion people worlwide.

00:02:55.804 --> 00:02:59.117
So this number is
clearly going to increase

00:02:59.117 --> 00:03:01.474
and so you can also imagine
we want it to happen

00:03:01.474 --> 00:03:03.789
that more people get out of poverty,

00:03:03.789 --> 00:03:07.039
that they are able to use, to
have more goods and services

00:03:07.039 --> 00:03:10.514
and so in that case well, they
might be using more energy.

00:03:10.514 --> 00:03:12.688
And if we don't change how
we produce that energy,

00:03:12.688 --> 00:03:16.202
well this has the potential
of really going up a lot,

00:03:16.202 --> 00:03:18.762
which could make this number go up a lot.

00:03:18.762 --> 00:03:21.424
And to help us think about
that a little bit more

00:03:21.424 --> 00:03:25.142
Bill Gates gave his famous
TED talk a couple of years ago

00:03:25.142 --> 00:03:28.230
where he talked about his energy equation,

00:03:28.230 --> 00:03:30.282
and it's a pretty
straight forward equation,

00:03:30.282 --> 00:03:34.285
it talks about the total
sum of C O two emissions.

00:03:34.285 --> 00:03:37.698
So this is the capital Greek
letter sigma right over here,

00:03:37.698 --> 00:03:40.812
it's shorthand math,
math shorthand for sum.

00:03:40.812 --> 00:03:43.888
So the annual sum of C O two emissions,

00:03:43.888 --> 00:03:45.621
and so that's that 40 billion number

00:03:45.621 --> 00:03:47.407
that I was talking about before,

00:03:47.407 --> 00:03:49.662
well you can derive this in any given year

00:03:49.662 --> 00:03:52.690
by taking the product of several things.

00:03:52.690 --> 00:03:56.429
First you can think about
the world's population, P,

00:03:56.429 --> 00:03:58.481
I'll even call that the average population

00:03:58.481 --> 00:04:00.735
because the population is changing,

00:04:00.735 --> 00:04:04.047
so the average population in a year,

00:04:04.047 --> 00:04:07.846
average population in a year and then

00:04:07.846 --> 00:04:10.813
you can multiply that times the amount of

00:04:10.813 --> 00:04:15.465
services on average each
person is using per year.

00:04:16.135 --> 00:04:21.135
So times the services,
let me write that down,

00:04:22.028 --> 00:04:27.028
the average services,
average, average services

00:04:28.086 --> 00:04:32.009
per person, per person and really

00:04:32.009 --> 00:04:34.569
we could say average services and goods.

00:04:34.569 --> 00:04:36.113
Because we are talking
about things that are,

00:04:36.113 --> 00:04:38.754
that are going to need energy

00:04:38.754 --> 00:04:41.091
and any good or service pretty much,

00:04:41.091 --> 00:04:42.594
that you look around is going to need

00:04:42.594 --> 00:04:44.565
energy for you to consume.

00:04:44.565 --> 00:04:45.908
There's some of the obvious things like

00:04:45.908 --> 00:04:49.118
transportation, obviously
your car might consume

00:04:49.118 --> 00:04:52.023
some fuel, it needs energy to do that.

00:04:52.023 --> 00:04:54.883
But even watching this
video, I'm consuming energy

00:04:54.883 --> 00:04:56.872
producing this video for you, I had to

00:04:56.872 --> 00:05:00.900
eat some food, some energy
which is, the reason

00:05:00.900 --> 00:05:03.399
I eat it is for energy
and some energy was used

00:05:03.399 --> 00:05:06.122
even to produce that food,
to get it to my table,

00:05:06.122 --> 00:05:09.088
I am using electricity
on my computer right now,

00:05:09.088 --> 00:05:12.339
you would have to use electricity
to consume this video,

00:05:12.339 --> 00:05:13.883
so there's a lot more energy being

00:05:13.883 --> 00:05:15.447
used in the goods and services you

00:05:15.447 --> 00:05:18.357
consume than you might realize.

00:05:18.357 --> 00:05:20.038
So then you want to take, and remember

00:05:20.038 --> 00:05:22.375
we want to get to the
total sum of C O two,

00:05:22.375 --> 00:05:24.630
you multiply that times the average energy

00:05:24.630 --> 00:05:29.364
per service, or you could say
per good or service consumed.

00:05:29.364 --> 00:05:33.264
So average, I'll just
services, average energy,

00:05:33.264 --> 00:05:38.264
average energy per service, per service.

00:05:39.648 --> 00:05:44.648
I'll just write the word
per, so per, per service

00:05:46.119 --> 00:05:47.918
right over there, as I
said almost any service

00:05:47.918 --> 00:05:51.860
you use, you consume is using some energy.

00:05:51.860 --> 00:05:54.135
When you go to school your teacher,

00:05:54.135 --> 00:05:57.426
he or she is consuming
energy, to just be alive,

00:05:57.426 --> 00:05:58.808
they had to eat some food, they had to

00:05:58.808 --> 00:06:01.977
use transportation to
get there, your classroom

00:06:01.977 --> 00:06:04.720
might be heated or it might
be cooled in some way.

00:06:04.720 --> 00:06:07.829
And then finally to get
us to the sum of C O two

00:06:07.829 --> 00:06:12.055
released in a year, we
have to multiply by,

00:06:12.055 --> 00:06:16.199
we have to multiply by, let me do that in

00:06:16.199 --> 00:06:19.125
the color I'm using for the C O two,

00:06:19.125 --> 00:06:23.940
we have to multiply
that times the C O two,

00:06:23.940 --> 00:06:28.940
C O two released, released per

00:06:29.666 --> 00:06:34.666
unit of energy, per unit of energy.

00:06:36.635 --> 00:06:39.275
And so, you can, we
can think about how all

00:06:39.275 --> 00:06:40.941
of these dynamics are going to change

00:06:40.941 --> 00:06:44.070
but if we ever want this thing
on the left to get to zero,

00:06:44.070 --> 00:06:45.919
and unless it gets to zero, things like

00:06:45.919 --> 00:06:48.174
global warming are only
going to get worse,

00:06:48.174 --> 00:06:49.616
in order for that to get
to zero and you're taking

00:06:49.616 --> 00:06:52.542
the product of four things, at least one

00:06:52.542 --> 00:06:54.391
of these things have to be zero.

00:06:54.391 --> 00:06:55.328
That's the only way that you're

00:06:55.328 --> 00:06:57.360
going to get a product to be zero.

00:06:57.360 --> 00:06:58.948
Now we could look at each of these,

00:06:58.948 --> 00:07:01.182
each of these elements
of the energy equation

00:07:01.182 --> 00:07:03.416
and think about what
they are likely to do.

00:07:03.416 --> 00:07:05.448
The average world population, well we see

00:07:05.448 --> 00:07:08.008
what's going on here,
even the medium prediction

00:07:08.008 --> 00:07:11.503
has the population growing by roughly 30%

00:07:11.503 --> 00:07:14.401
so we can say look, this
is going to increase,

00:07:14.401 --> 00:07:19.401
this is going to increase
by, by a factor of 1.3,

00:07:20.198 --> 00:07:22.433
it's going to increase by roughly 30%,

00:07:22.433 --> 00:07:24.384
once again these are all
predictions, this is all very rough

00:07:24.384 --> 00:07:27.248
but we can think directionally
where things are going.

00:07:27.248 --> 00:07:29.036
Now, average goods and services,

00:07:29.036 --> 00:07:31.616
or average services per
person, well we want the poor

00:07:31.616 --> 00:07:34.969
to be able to consume
more, now maybe some of us

00:07:34.969 --> 00:07:36.305
in the developed world, maybe we consume

00:07:36.305 --> 00:07:38.290
a little bit too much, you know,

00:07:38.290 --> 00:07:40.820
that's a philisophical
thing for folks to debate

00:07:40.820 --> 00:07:43.177
but on average we want the world, we want

00:07:43.177 --> 00:07:46.510
more people to have better transportation,

00:07:46.510 --> 00:07:48.276
better access to healthcare, better access

00:07:48.276 --> 00:07:51.527
to education, better access to leisure.

00:07:51.527 --> 00:07:53.416
And so this is likely to go up,

00:07:53.416 --> 00:07:55.631
in fact this will go up
as the world grows richer

00:07:55.631 --> 00:07:57.216
and some of the predictions,
and this is what was,

00:07:57.216 --> 00:08:00.365
I believe what Bill cited in his TED talk,

00:08:00.365 --> 00:08:03.209
was that this could go
up by a factor of two,

00:08:03.209 --> 00:08:04.428
this could go up by a factor of two,

00:08:04.428 --> 00:08:06.927
and this is good, we want
the world to become richer,

00:08:06.927 --> 00:08:10.523
we want more people to
consume goods and services.

00:08:10.523 --> 00:08:13.054
Now, the average energy per service.

00:08:13.054 --> 00:08:14.761
Well the good thing is technology is

00:08:14.761 --> 00:08:17.809
making things more
efficient, and so we might

00:08:17.809 --> 00:08:22.054
have better insulation,
better efficient engines,

00:08:22.054 --> 00:08:23.822
whether they're electric engines or

00:08:23.822 --> 00:08:26.463
some other type of fuel engines,

00:08:26.463 --> 00:08:28.150
and so the average energy per service

00:08:28.150 --> 00:08:30.466
is likely to go down, it depends on

00:08:30.466 --> 00:08:32.315
how optimistic you are,
it could go down by

00:08:32.315 --> 00:08:35.403
20% and depending on
your time frame, 50% and

00:08:35.403 --> 00:08:37.313
some even more optimistic predictions

00:08:37.313 --> 00:08:39.385
might be even more than 50%, so

00:08:39.385 --> 00:08:42.554
it's gonna go down, so let's
just say this is times,

00:08:42.554 --> 00:08:46.049
let's just use 0.5,
let's just be optimistic.

00:08:46.049 --> 00:08:47.635
But even if you were to multiply this,

00:08:47.635 --> 00:08:49.401
you know, just 1.3 times two, that's

00:08:49.401 --> 00:08:51.698
a factor of 2.6 and even if you were

00:08:51.698 --> 00:08:54.399
to take that by half you still have

00:08:54.399 --> 00:08:57.183
this entire product increasing.

00:08:57.183 --> 00:08:58.727
And even if we were to
get really efficient,

00:08:58.727 --> 00:09:03.460
if we were to, if we were
to decrease our energy used

00:09:03.460 --> 00:09:07.585
per service by 80% or even 90%

00:09:07.585 --> 00:09:10.836
you still have many
tens of billions of tons

00:09:10.836 --> 00:09:13.660
of C O two being released every year.

00:09:13.660 --> 00:09:15.935
So the only way we're going
to be able to get that,

00:09:15.935 --> 00:09:18.818
to get this thing down to zero,

00:09:18.818 --> 00:09:22.831
is if, is if C, right over here,

00:09:22.831 --> 00:09:25.900
the C O two released per unit of energy

00:09:25.900 --> 00:09:30.900
goes to zero, it needs, so
this needs to go to zero.

00:09:33.433 --> 00:09:36.987
Which is a fairly bold
thing to say because

00:09:36.987 --> 00:09:38.755
we're nowhere close to that right now

00:09:38.755 --> 00:09:40.766
but if we're serious about this,

00:09:40.766 --> 00:09:43.001
it's possible that humanity can get there.

00:09:43.001 --> 00:09:46.658
There are energy sources
that are, that don't

00:09:46.658 --> 00:09:50.640
release C O two or get pretty
close to being at zero.

00:09:50.640 --> 00:09:52.245
And just to be clear,
even if you're driving

00:09:52.245 --> 00:09:54.317
an electric car, you have to think about

00:09:54.317 --> 00:09:56.004
where does that electricity come from?

00:09:56.004 --> 00:09:58.584
If that electricity
was generated using say

00:09:58.584 --> 00:10:01.225
something like coal, well
that's going to release C O two

00:10:01.225 --> 00:10:04.821
or if it was generated using natural gas,

00:10:04.821 --> 00:10:08.438
that is going to release
carbon into the atmosphere.

00:10:08.438 --> 00:10:10.744
And so you need to think about,

00:10:10.744 --> 00:10:11.499
even if you are using something

00:10:11.499 --> 00:10:13.978
like an electric motor, how is that

00:10:13.978 --> 00:10:16.477
electricity actually produced?

00:10:16.477 --> 00:10:18.509
And you have energy sources, like nuclear,

00:10:18.509 --> 00:10:19.951
nuclear has it's own set of issues

00:10:19.951 --> 00:10:21.719
that we think about, like safety

00:10:21.719 --> 00:10:23.202
and what you do with the waste

00:10:23.202 --> 00:10:24.279
but people are working on things

00:10:24.279 --> 00:10:26.371
like clean burning nuclear
that are much safer

00:10:26.371 --> 00:10:30.536
and generate a very small
fraction of the waste.

00:10:30.536 --> 00:10:33.178
You have all your other
renewable souces of energy

00:10:33.178 --> 00:10:35.087
but this is a big challenge for humanity.

00:10:35.087 --> 00:10:37.668
If we want this thing
on the left to be zero,

00:10:37.668 --> 00:10:41.754
if we wanna slow, or if we wanna

00:10:41.754 --> 00:10:44.111
stop the heating of the planet

00:10:44.111 --> 00:10:46.020
we have to figure out how to do this.

00:10:46.020 --> 00:10:49.251
And the general sense
is, if as a civilization,

00:10:49.251 --> 00:10:54.011
we put our minds to it that we can do it.


Ellipse standard equation from graph

WEBVTT
Kind: captions
Language: en

00:00:00.224 --> 00:00:02.348
- [Voiceover] We have an
ellipse graphed right over here.

00:00:02.348 --> 00:00:03.486
What we're going to try to do is find

00:00:03.486 --> 00:00:05.378
the equation for this ellipse.

00:00:05.378 --> 00:00:07.062
Like always, pause this video and see

00:00:07.062 --> 00:00:09.601
if you can figure it out on your own.

00:00:09.601 --> 00:00:10.345
All right,

00:00:10.345 --> 00:00:12.143
let's just remind ourselves

00:00:12.143 --> 00:00:14.199
the form of an equation of an ellipse.

00:00:14.199 --> 00:00:17.321
Let's say our ellipse is centered

00:00:17.321 --> 00:00:18.679
at the point ...

00:00:18.679 --> 00:00:20.363
I'm going to speak in generalities first

00:00:20.363 --> 00:00:21.976
and then we'll think
about the specific numbers

00:00:21.976 --> 00:00:23.660
for this particular ellipse.

00:00:23.660 --> 00:00:27.538
Say the center is at the point H,K

00:00:27.538 --> 00:00:31.902
and let's say that you
have a horizontal radius.

00:00:31.902 --> 00:00:33.957
So the radius in the X direction,

00:00:33.957 --> 00:00:36.895
horizontal radius,

00:00:38.055 --> 00:00:41.817
radius is equal to A.

00:00:41.817 --> 00:00:45.587
Let's say your vertical radius,

00:00:45.937 --> 00:00:48.692
let's say your vertical radius,

00:00:49.732 --> 00:00:54.516
radius is equal to B.

00:00:54.516 --> 00:00:56.525
Then the equation of this ellipse

00:00:56.525 --> 00:00:58.221
is going to be,

00:00:58.221 --> 00:01:03.221
is going to be X - H,

00:01:04.082 --> 00:01:09.082
X - H squared

00:01:09.237 --> 00:01:12.116
over your horizontal radius squared,

00:01:12.116 --> 00:01:14.600
so your radius in the X direction squared,

00:01:14.600 --> 00:01:15.909
plus,

00:01:16.539 --> 00:01:18.361
plus, now we'll think
about what we're doing

00:01:18.361 --> 00:01:20.103
in the vertical direction.

00:01:20.103 --> 00:01:22.192
Y - ,

00:01:22.192 --> 00:01:26.476
Y - the Y coordinate of our center,

00:01:26.476 --> 00:01:31.116
so Y - K squared,

00:01:31.736 --> 00:01:33.592
over the vertical radius squared,

00:01:33.592 --> 00:01:34.777
B squared

00:01:34.777 --> 00:01:36.692
is equal to 1.

00:01:36.692 --> 00:01:39.014
Is equal to 1.

00:01:39.014 --> 00:01:42.335
What are H and K and A
and B in this situation?

00:01:42.335 --> 00:01:44.656
Well, H and K are pretty
easy to figure out.

00:01:44.656 --> 00:01:48.394
The center of this ellipse
is at the point ...

00:01:48.394 --> 00:01:51.163
See the X coordinate is -4

00:01:51.853 --> 00:01:55.069
and the Y coordinate is 3.

00:01:55.069 --> 00:01:57.971
So this right over here is -4

00:01:57.971 --> 00:02:01.617
and this right over here is positive 3.

00:02:01.617 --> 00:02:04.008
What is A going to be?

00:02:04.008 --> 00:02:05.552
A is your horizontal radius,

00:02:05.552 --> 00:02:07.096
your radius in the horizontal direction,

00:02:07.096 --> 00:02:10.648
so it's the length of
this line right over here.

00:02:10.648 --> 00:02:14.851
We can see it's 1, 2, 3, 4, 5 units long.

00:02:14.851 --> 00:02:17.672
A in this case is equal to 5.

00:02:17.672 --> 00:02:20.795
This is going to be 5 squared.

00:02:20.795 --> 00:02:23.708
And B is our radius in
the vertical direction.

00:02:23.708 --> 00:02:26.552
We can see it's 1, 2, 3, 4 units.

00:02:26.552 --> 00:02:29.327
So B is equal to 4.

00:02:29.327 --> 00:02:31.069
So that is 4.

00:02:31.069 --> 00:02:33.635
We can rewrite this as,

00:02:33.635 --> 00:02:37.616
we can rewrite this as X - -4,

00:02:37.616 --> 00:02:40.332
and we can simplify that in a second.

00:02:40.332 --> 00:02:45.047
X - -4 squared over 5 squared

00:02:45.047 --> 00:02:47.588
over our horizontal radius squared,

00:02:47.588 --> 00:02:49.794
so it's going to be 25

00:02:49.794 --> 00:02:54.794
plus Y - 3 squared.

00:02:55.112 --> 00:02:59.592
Y minus the Y coordinate of our center.

00:02:59.592 --> 00:03:03.565
Y - 3 squared

00:03:04.875 --> 00:03:07.346
over our vertical radius squared,

00:03:07.346 --> 00:03:09.773
so B squared is going to be 16,

00:03:09.773 --> 00:03:13.593
and that is going to be equal to 1.

00:03:13.593 --> 00:03:15.902
Of course we could
simplify this a little bit.

00:03:15.902 --> 00:03:17.272
If I subtract a negative,

00:03:17.272 --> 00:03:20.105
that's the same thing
as adding a positive,

00:03:20.105 --> 00:03:22.705
so I can get rid of ...

00:03:22.705 --> 00:03:25.201
Instead of saying X - -4,

00:03:25.201 --> 00:03:28.150
I could just say X + 4.

00:03:28.150 --> 00:03:28.951
There you have it.

00:03:28.951 --> 00:03:32.480
We have the equation for this ellipse.


Ellipse graph from standard equation

WEBVTT
Kind: captions
Language: en

00:00:00.098 --> 00:00:01.619
- [Voiceover] We're asked
which ellipse is represented

00:00:01.619 --> 00:00:05.104
by the equation x minus
four squared over 16

00:00:05.104 --> 00:00:08.527
plus y minus one squared
over 49 is equal to one.

00:00:08.527 --> 00:00:10.486
And we're given a bunch of choices here.

00:00:10.486 --> 00:00:11.846
We're given four choices here.

00:00:11.846 --> 00:00:13.879
So let's just think about
what's going on here.

00:00:13.879 --> 00:00:18.879
So the center of the ellipse
is going to be four comma one.

00:00:18.965 --> 00:00:19.870
How do I know that?

00:00:19.870 --> 00:00:21.310
Well the equation of the
ellipse is going to be

00:00:21.310 --> 00:00:25.465
x minus the x-coordinate for
the center squared over here,

00:00:25.465 --> 00:00:29.903
over the horizontal axis is
horizontal radius squared

00:00:29.903 --> 00:00:33.941
plus y minus the y-coordinate
of the center squared

00:00:33.941 --> 00:00:36.379
over the vertical radius squared.

00:00:36.379 --> 00:00:39.015
So the center is going
to be four comma one.

00:00:39.015 --> 00:00:41.755
So the center here is not four comma one.

00:00:41.755 --> 00:00:46.318
The center over here
is not four comma one.

00:00:46.318 --> 00:00:48.547
Not four comma one.

00:00:48.547 --> 00:00:52.213
The only choice that has
a center at four comma one

00:00:52.213 --> 00:00:54.189
is this one over here.

00:00:54.189 --> 00:00:55.873
So we already know this is the choice

00:00:55.873 --> 00:00:58.137
without even looking at the horizontal

00:00:58.137 --> 00:00:59.820
and the vertical radius.

00:00:59.820 --> 00:01:01.503
But we can verify that this works out

00:01:01.503 --> 00:01:03.651
'cause a horizontal
radius right over here,

00:01:03.651 --> 00:01:05.950
notice it goes this orange line,

00:01:05.950 --> 00:01:08.864
which can represent the horizontal radius.

00:01:08.864 --> 00:01:13.864
It has a length of four and so
the horizontal radius is four

00:01:14.658 --> 00:01:18.524
and so we see indeed that 16

00:01:18.524 --> 00:01:21.856
is the horizontal radius
squared, this is four squared.

00:01:21.856 --> 00:01:25.559
And if we look at the
vertical radius here,

00:01:25.559 --> 00:01:26.674
we see it has a length of seven.

00:01:26.674 --> 00:01:29.100
We're going from y equals
one to y equals eight,

00:01:29.100 --> 00:01:30.494
has a length of seven.

00:01:30.494 --> 00:01:32.142
And we see in that equation

00:01:32.142 --> 00:01:35.230
that this indeed is seven squared.

00:01:35.230 --> 00:01:36.795
So that was pretty straight forward.


Graphing a circle from its standard equation

WEBVTT
Kind: captions
Language: en

00:00:00.225 --> 00:00:01.920
- [Voiceover] Whereas to graph the circle

00:00:01.920 --> 00:00:05.579
x plus five squared plus y
minus 5 squared equals four.

00:00:05.579 --> 00:00:06.612
I know what you're thinking.

00:00:06.612 --> 00:00:08.319
What's all of this silliness
on the right hand side?

00:00:08.319 --> 00:00:11.059
This is actually just the view we use

00:00:11.059 --> 00:00:13.717
when we're trying to debug
things on Khan Academy.

00:00:13.717 --> 00:00:15.458
But we can still do the exercise.

00:00:15.458 --> 00:00:17.850
So it says drag the center point
and perimeter of the circle

00:00:17.850 --> 00:00:18.895
to graph the equation.

00:00:18.895 --> 00:00:20.137
So the first thing we
want to think about is

00:00:20.137 --> 00:00:22.075
well what's the center of this equation?

00:00:22.075 --> 00:00:24.270
Well the standard form of a circle is

00:00:24.270 --> 00:00:28.611
x minus the x coordinate
of the center squared,

00:00:28.611 --> 00:00:32.918
plus y minus the y coordinate
of the center squared

00:00:32.918 --> 00:00:35.391
is equal to the radius squared.

00:00:35.391 --> 00:00:39.268
So x minus the x coordinate of the center.

00:00:39.268 --> 00:00:40.592
So the x coordinate of the center

00:00:40.592 --> 00:00:42.368
must be negative five.

00:00:42.368 --> 00:00:43.796
Cause the way we can get
a positive five here's

00:00:43.796 --> 00:00:46.083
by subtracting a negative five.

00:00:46.083 --> 00:00:50.018
So the x coordinate must be negative five

00:00:50.018 --> 00:00:52.816
and the y coordinate
must be positive five.

00:00:52.816 --> 00:00:56.902
Cause y minus the y
coordinate of the center.

00:00:56.902 --> 00:00:59.236
So y coordinate is positive five

00:00:59.236 --> 00:01:02.800
and then the radius squared
is going to be equal to four.

00:01:02.800 --> 00:01:04.796
So that means that the radius

00:01:04.796 --> 00:01:05.795
is equal to two.

00:01:05.795 --> 00:01:06.840
And the way it's drawn right now,

00:01:06.840 --> 00:01:08.883
we could drag this out like this,

00:01:08.883 --> 00:01:09.916
but this the way it's drawn,

00:01:09.916 --> 00:01:12.377
the radius is indeed equal to two.

00:01:12.377 --> 00:01:14.200
And so we're done.

00:01:14.200 --> 00:01:15.999
And I really want to hit the point home

00:01:15.999 --> 00:01:17.276
of what I just did.

00:01:17.276 --> 00:01:20.425
So let me get my little

00:01:20.425 --> 00:01:23.167
scratch pad out.

00:01:23.167 --> 00:01:26.107
Sorry for knocking the
microphone just now.

00:01:26.107 --> 00:01:27.558
That equation was

00:01:27.558 --> 00:01:30.717
x plus five squared

00:01:30.717 --> 00:01:34.593
plus y minus five squared

00:01:34.593 --> 00:01:36.567
is equal to four squared.

00:01:36.567 --> 00:01:37.786
So I want to rewrite this as,

00:01:37.786 --> 00:01:41.563
this is x minus negative five,

00:01:41.563 --> 00:01:46.078
x minus negative five squared,

00:01:46.078 --> 00:01:50.142
plus y minus positive five,

00:01:50.142 --> 00:01:54.495
positive five squared

00:01:54.495 --> 00:01:57.084
is equal to, instead of writing it as four

00:01:57.084 --> 00:02:00.404
I'll write it as two squared.

00:02:00.404 --> 00:02:02.064
So this right over here tells us

00:02:02.064 --> 00:02:03.562
that the center of the
circle is going to be

00:02:03.562 --> 00:02:05.142
x equals negative five

00:02:05.142 --> 00:02:06.721
y equals five

00:02:06.721 --> 00:02:09.344
and the radius is going
to be equal to two.

00:02:09.344 --> 00:02:11.341
And once again, this is no magic here.

00:02:11.341 --> 00:02:12.806
This is not just, I don't want you to just

00:02:12.806 --> 00:02:13.642
memorize this formula.

00:02:13.642 --> 00:02:15.359
I want you to appreciate that this formula

00:02:15.359 --> 00:02:18.830
comes straight out of
the Pythagorean Theorem,

00:02:18.830 --> 00:02:20.374
straight out of the distance formula,

00:02:20.374 --> 00:02:22.429
which comes out of the
Pythagorean Theorem.

00:02:22.429 --> 00:02:24.217
Remember, if you have some center,

00:02:24.217 --> 00:02:27.827
in this case is the point
negative five comma five,

00:02:27.827 --> 00:02:30.973
so negative five comma five,

00:02:30.973 --> 00:02:33.388
and you want to find
all of the x's and y's

00:02:33.388 --> 00:02:35.768
that are two away from it.

00:02:35.768 --> 00:02:38.658
So you want to find all the x's and y's

00:02:38.658 --> 00:02:40.179
that are two away from it.

00:02:40.179 --> 00:02:41.897
So that would be one of them,

00:02:41.897 --> 00:02:44.056
x comma y.

00:02:44.056 --> 00:02:45.856
This distance is two.

00:02:45.856 --> 00:02:47.319
And there's going to be a bunch of them.

00:02:47.319 --> 00:02:50.743
And when you plot all of them together,

00:02:50.743 --> 00:02:52.914
you're going to get a
circle with radius two

00:02:52.914 --> 00:02:54.284
around that center.

00:02:54.284 --> 00:02:57.639
Plus think about how we
got that actual formula.

00:02:57.639 --> 00:02:59.115
Well the distance between that coordinate,

00:02:59.115 --> 00:03:00.915
between any of these x's and y's,

00:03:00.915 --> 00:03:02.760
it could be an x and y here,
it could be an x and y here,

00:03:02.760 --> 00:03:04.490
and this is going to be two.

00:03:04.490 --> 00:03:06.278
So we can have our change in x.

00:03:06.278 --> 00:03:08.890
So we have x minus

00:03:08.890 --> 00:03:10.550
negative five.

00:03:10.550 --> 00:03:13.147
So that's our change
in x between any point

00:03:13.147 --> 00:03:16.305
x comma y, negative five comma five.

00:03:16.305 --> 00:03:21.299
So our change in x squared

00:03:22.028 --> 00:03:23.692
plus our change in y squared,

00:03:23.692 --> 00:03:25.424
so it's going to be y minus

00:03:25.424 --> 00:03:28.443
the y coordinate over here, squared

00:03:28.443 --> 00:03:30.242
is going to be equal to

00:03:30.242 --> 00:03:32.785
the radius squared.

00:03:32.785 --> 00:03:34.654
So the change in y

00:03:34.654 --> 00:03:36.310
is going to be from this y

00:03:36.310 --> 00:03:37.412
to that y,

00:03:37.412 --> 00:03:38.933
this is the end point.

00:03:38.933 --> 00:03:42.868
The end minus the beginning y minus five,

00:03:42.868 --> 00:03:45.759
y minus five squared.

00:03:45.759 --> 00:03:48.789
And so this shows for
any x y that is two away

00:03:48.789 --> 00:03:51.470
from the center, this equation will hold.

00:03:51.470 --> 00:03:53.258
And it becomes,

00:03:53.258 --> 00:03:55.579
I'll just write this in neutral color,

00:03:55.579 --> 00:03:58.063
x plus five squared

00:03:58.063 --> 00:04:01.047
plus y minus five squared

00:04:01.047 --> 00:04:03.137
is equal to the radius squared,

00:04:03.137 --> 00:04:05.006
is equal to two,

00:04:05.006 --> 00:04:07.929
or let me just write
that, is equal to four.

00:04:07.929 --> 00:04:10.682
And let me make it, I
really want to, you know,

00:04:10.682 --> 00:04:13.397
I dislike it when formulas
are just memorized.

00:04:13.397 --> 00:04:15.394
You don't see the
connection to other things.

00:04:15.394 --> 00:04:18.291
Notice we can construct

00:04:18.291 --> 00:04:21.120
a nice little right triangle here.

00:04:21.120 --> 00:04:23.941
So our change in x is
that right over there.

00:04:23.941 --> 00:04:26.820
So that is our change in x.

00:04:26.820 --> 00:04:28.260
Change in x.

00:04:28.260 --> 00:04:30.384
And our change in y

00:04:30.384 --> 00:04:32.903
our change in y, not
the change in y squared,

00:04:32.903 --> 00:04:34.401
but our change in y,

00:04:34.401 --> 00:04:36.386
is that right over there.

00:04:36.386 --> 00:04:37.303
Change in y,

00:04:37.303 --> 00:04:38.661
our change in y

00:04:38.661 --> 00:04:40.623
you could view that as

00:04:40.623 --> 00:04:45.221
y minus, so this is change in y

00:04:45.221 --> 00:04:47.345
is going to be y

00:04:47.345 --> 00:04:49.755
minus five.

00:04:49.755 --> 00:04:51.821
And our change in x

00:04:51.821 --> 00:04:55.223
is x minus negative five.

00:04:55.223 --> 00:04:57.533
x minus negative five.

00:04:57.533 --> 00:04:59.680
So this is just change in x squared

00:04:59.680 --> 00:05:01.213
plus change in y squared

00:05:01.213 --> 00:05:03.592
is equal to the hypotenuse squared,

00:05:03.592 --> 00:05:06.274
which is the length of,
which is this radius.

00:05:06.274 --> 00:05:08.178
So once again, comes straight out of

00:05:08.178 --> 00:05:09.757
the Pythagorean Theorem.

00:05:09.757 --> 00:05:13.025
Hopefully that makes sense.