Graphing a parabola in vertex form

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.080
We're asked to graph the
equation y is equal to negative

00:00:03.080 --> 00:00:06.470
2 times x minus
2 squared plus 5.

00:00:06.470 --> 00:00:09.400
So let me get by scratch pad out
so we could think about this.

00:00:09.400 --> 00:00:16.730
So y is equal to negative 2
times x minus 2 squared plus 5.

00:00:16.730 --> 00:00:21.420
So one thing, when you see a
quadratic or a parabola graph

00:00:21.420 --> 00:00:24.200
expressed in this way, the
thing that might jump out at you

00:00:24.200 --> 00:00:27.680
is that this term
right over here

00:00:27.680 --> 00:00:29.495
is always going to be
positive because it's

00:00:29.495 --> 00:00:30.690
some quantity squared.

00:00:30.690 --> 00:00:32.940
Or I should say, it's always
going to be non-negative.

00:00:32.940 --> 00:00:34.550
It could be equal to 0.

00:00:34.550 --> 00:00:36.677
So it's always going to
be some quantity squared.

00:00:36.677 --> 00:00:38.510
And then we're multiplying
it by a negative.

00:00:38.510 --> 00:00:41.380
So this whole quantity
right over here

00:00:41.380 --> 00:00:43.160
is going to be non positive.

00:00:43.160 --> 00:00:47.550
It's always going to be
less than or equal to 0.

00:00:47.550 --> 00:00:51.080
So this thing is always
less than or equal to 0,

00:00:51.080 --> 00:00:55.750
the maximum value
that y will take on

00:00:55.750 --> 00:00:58.430
is when this thing
actually does equal 0.

00:00:58.430 --> 00:01:03.590
So the maximum
value for y is at 5.

00:01:03.590 --> 00:01:06.430
The maximum value for y is 5.

00:01:06.430 --> 00:01:07.750
And when does that happen?

00:01:07.750 --> 00:01:10.485
Well, y hits 5 when
this whole thing is 0.

00:01:10.485 --> 00:01:11.860
And when does this
thing equal 0?

00:01:11.860 --> 00:01:15.570
Well, this whole thing equals
0 when x minus 2 is equal to 0.

00:01:15.570 --> 00:01:18.910
And x minus 2 is equal to
0 when x is equal to 2.

00:01:18.910 --> 00:01:21.870
So the point 2 comma
5 is the maximum point

00:01:21.870 --> 00:01:22.990
for this parabola.

00:01:22.990 --> 00:01:25.400
And it is actually
going to be the vertex.

00:01:25.400 --> 00:01:30.290
So if we were to graph this,
so the point 2 comma 5.

00:01:30.290 --> 00:01:32.690
So that's my y-axis.

00:01:32.690 --> 00:01:34.800
This is my x-axis.

00:01:34.800 --> 00:01:39.420
So this is 1, 2, 1, 2, 3, 4, 5.

00:01:39.420 --> 00:01:42.660
So this right here is
the point 2 comma 5.

00:01:42.660 --> 00:01:45.670
This is a maximum point,
it's a maximum point

00:01:45.670 --> 00:01:47.152
for this parabola.

00:01:47.152 --> 00:01:49.610
And now I want to find two more
points so that I can really

00:01:49.610 --> 00:01:50.568
determine the parabola.

00:01:50.568 --> 00:01:54.360
Three points completely
determine a parabola.

00:01:54.360 --> 00:01:57.021
So that's 1, the vertex,
that's interesting.

00:01:57.021 --> 00:01:58.520
Now, what I'd like
to do is just get

00:01:58.520 --> 00:02:01.430
two points that are
equidistant from the vertex.

00:02:01.430 --> 00:02:03.960
And the easiest way to do
that is to maybe figure out

00:02:03.960 --> 00:02:08.940
what happens when x is equal
to 1 and when x is equal to 3.

00:02:08.940 --> 00:02:12.140
So I could make a table here
actually, let me do that.

00:02:12.140 --> 00:02:17.540
So I care about x being equal
to 1, 2, and 3, and what

00:02:17.540 --> 00:02:19.150
the corresponding y is.

00:02:19.150 --> 00:02:22.422
We already know that when x is
equal to 2, y is equal to 5.

00:02:22.422 --> 00:02:24.720
2 comma 5 is our vertex.

00:02:24.720 --> 00:02:28.510
When x is equal to 1, 1
minus 2 is negative 1,

00:02:28.510 --> 00:02:30.850
squared is just 1.

00:02:30.850 --> 00:02:32.920
So this thing is going
to be negative 2 plus 5,

00:02:32.920 --> 00:02:34.410
so it's going to be 3.

00:02:34.410 --> 00:02:37.120
And when x is equal to
3, this is 3 minus 2,

00:02:37.120 --> 00:02:40.880
which is 1 squared is 1 times
negative 2 is negative 2 plus 5

00:02:40.880 --> 00:02:42.281
is 3 as well.

00:02:42.281 --> 00:02:43.280
So we have three points.

00:02:43.280 --> 00:02:49.980
We have the point 1 comma
3, the point 2 comma 5,

00:02:49.980 --> 00:02:55.530
and the point 3 comma
3 for this parabola.

00:02:55.530 --> 00:02:56.930
So let me go back
to the exercise

00:02:56.930 --> 00:02:59.710
and actually put
those three points in.

00:02:59.710 --> 00:03:07.670
And so we have the point 1 comma
3, we have the point 2 comma 5,

00:03:07.670 --> 00:03:10.470
and we have the point 3 comma 3.

00:03:10.470 --> 00:03:14.430
And we have now fully
determined our parabola.


Graphing a parabola using roots and vertex

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:03.120
We're asked to graph
the following equation y

00:00:03.120 --> 00:00:07.610
equals 5x squared
minus 20x plus 15.

00:00:07.610 --> 00:00:10.400
So let me get my
little scratch pad out.

00:00:10.400 --> 00:00:17.770
So it's y is equal to 5x
squared minus 20x plus 15.

00:00:17.770 --> 00:00:19.620
Now there's many
ways to graph this.

00:00:19.620 --> 00:00:22.060
You can just take three
values for x and figure

00:00:22.060 --> 00:00:23.960
out what the
corresponding values for y

00:00:23.960 --> 00:00:26.000
are and just graph
those three points.

00:00:26.000 --> 00:00:28.945
And three points actually
will determine a parabola.

00:00:28.945 --> 00:00:31.320
But I want to do something a
little bit more interesting.

00:00:31.320 --> 00:00:33.330
I want to find the places.

00:00:33.330 --> 00:00:37.770
So if we imagine our axes.

00:00:37.770 --> 00:00:38.900
This is my x-axis.

00:00:38.900 --> 00:00:40.600
That's my y-axis.

00:00:40.600 --> 00:00:43.240
And this is our curve.

00:00:43.240 --> 00:00:46.050
So the parabola might
look something like this.

00:00:46.050 --> 00:00:49.420
I want to first figure out where
does this parabola intersect

00:00:49.420 --> 00:00:50.880
the x-axis.

00:00:50.880 --> 00:00:52.690
And as we have already
seen, intersecting

00:00:52.690 --> 00:00:57.430
the x-axis is the same thing
as saying when it does this

00:00:57.430 --> 00:00:59.970
when does y equal
0 for this problem?

00:00:59.970 --> 00:01:01.740
Or another way of
saying it, when

00:01:01.740 --> 00:01:05.390
does this 5x squared
minus 20x plus 15,

00:01:05.390 --> 00:01:07.560
when does this equal 0?

00:01:07.560 --> 00:01:09.280
So I want to figure
out those points.

00:01:09.280 --> 00:01:11.446
And then I also want to
figure out the point exactly

00:01:11.446 --> 00:01:13.241
in between, which is the vertex.

00:01:13.241 --> 00:01:14.990
And if I can graph
those three points then

00:01:14.990 --> 00:01:17.690
I should be all set with
graphing this parabola.

00:01:17.690 --> 00:01:19.540
So as I just said,
we're going to try

00:01:19.540 --> 00:01:25.360
to solve the equation 5x
squared minus 20x plus 15 is

00:01:25.360 --> 00:01:26.892
equal to 0.

00:01:26.892 --> 00:01:28.600
Now the first thing
I like to do whenever

00:01:28.600 --> 00:01:31.330
I see a coefficient out here on
the x squared term that's not

00:01:31.330 --> 00:01:34.289
a 1, is to see if I can
divide everything by that term

00:01:34.289 --> 00:01:35.830
to try to simplify
this a little bit.

00:01:35.830 --> 00:01:38.880
And maybe this will get us
into a factor-able form.

00:01:38.880 --> 00:01:42.350
And it does look like every
term here is divisible by 5.

00:01:42.350 --> 00:01:44.740
So I will divide by 5.

00:01:44.740 --> 00:01:49.210
So I'll divide both sides
of this equation by 5.

00:01:49.210 --> 00:01:52.200
And so that will give
me-- these cancel out

00:01:52.200 --> 00:01:57.820
and I'm left with x squared
minus 20 over 5 is 4x.

00:01:57.820 --> 00:02:05.420
Plus 15 over 5 is 3 is
equal to 0 over 5 is just 0.

00:02:05.420 --> 00:02:08.389
And now we can attempt to
factor this left-hand side.

00:02:08.389 --> 00:02:11.356
We say are there two numbers
whose product is positive 3?

00:02:11.356 --> 00:02:12.980
The fact that their
product is positive

00:02:12.980 --> 00:02:15.070
tells you they both
must be positive.

00:02:15.070 --> 00:02:17.480
And whose sum is negative
4, which tells you well

00:02:17.480 --> 00:02:18.840
they both must be negative.

00:02:18.840 --> 00:02:21.082
If we're getting a
negative sum here.

00:02:21.082 --> 00:02:22.540
And the one that
probably jumps out

00:02:22.540 --> 00:02:23.915
of your mind--
and you might want

00:02:23.915 --> 00:02:26.590
to review the videos
on factoring quadratics

00:02:26.590 --> 00:02:30.230
if this is not so fresh-- is
a negative 3 and negative 1

00:02:30.230 --> 00:02:30.790
seem to work.

00:02:30.790 --> 00:02:33.050
Negative 3 times negative 1.

00:02:33.050 --> 00:02:35.400
Negative 3 times
negative 1 is 3.

00:02:35.400 --> 00:02:39.470
Negative 3 plus negative
1 is negative 4.

00:02:39.470 --> 00:02:44.730
So this will factor out as
x minus 3 times x minus 1.

00:02:44.730 --> 00:02:46.230
And on the right-hand
side, we still

00:02:46.230 --> 00:02:48.260
have that being equal to 0.

00:02:48.260 --> 00:02:51.550
And now we can think about what
x's will make this expression

00:02:51.550 --> 00:02:53.300
0, and if they make
this expression 0,

00:02:53.300 --> 00:02:55.190
well they're going to
make this expression 0.

00:02:55.190 --> 00:02:58.600
Which is going to make
this expression equal to 0.

00:02:58.600 --> 00:03:01.700
And so this will be true if
either one of these is 0.

00:03:01.700 --> 00:03:04.080
So x minus 3 is equal to 0.

00:03:04.080 --> 00:03:06.860
Or x minus 1 is equal to 0.

00:03:06.860 --> 00:03:09.460
This is true, and you can
add 3 to both sides of this.

00:03:09.460 --> 00:03:11.270
This is true when
x is equal to 3.

00:03:11.270 --> 00:03:13.640
This is true when
x is equal to 1.

00:03:13.640 --> 00:03:15.810
So we were able to figure
out these two points

00:03:15.810 --> 00:03:16.476
right over here.

00:03:16.476 --> 00:03:18.040
This is x is equal to 1.

00:03:18.040 --> 00:03:21.000
This is x is equal to 3.

00:03:21.000 --> 00:03:23.260
So this is the point 1 comma 0.

00:03:23.260 --> 00:03:26.072
This is the point 3 comma 0.

00:03:26.072 --> 00:03:27.780
And so the last one
I want to figure out,

00:03:27.780 --> 00:03:30.150
is this point right
over here, the vertex.

00:03:30.150 --> 00:03:33.160
Now the vertex always
sits exactly smack dab

00:03:33.160 --> 00:03:35.240
between the roots,
when you do have roots.

00:03:35.240 --> 00:03:37.740
Sometimes you might not
intersect the x-axis.

00:03:37.740 --> 00:03:40.650
So we already know what its
x-coordinate is going to be.

00:03:40.650 --> 00:03:41.940
It's going to be 2.

00:03:41.940 --> 00:03:43.690
And now we just have
to substitute back in

00:03:43.690 --> 00:03:45.980
to figure out its y-coordinate.

00:03:45.980 --> 00:03:54.230
When x equals 2, y is going to
be equal to 5 times 2 squared

00:03:54.230 --> 00:03:59.730
minus 20 times 2 plus 15,
which is equal to-- let's see,

00:03:59.730 --> 00:04:02.260
this is equal to 2 squared is 4.

00:04:02.260 --> 00:04:07.090
This is 20 minus 40 plus 15.

00:04:07.090 --> 00:04:08.970
So this is going
to be negative 20

00:04:08.970 --> 00:04:11.810
plus 15, which is
equal to negative 5.

00:04:11.810 --> 00:04:14.920
So this is the point
2 comma negative 5.

00:04:14.920 --> 00:04:16.690
And so now we can go
back to the exercise

00:04:16.690 --> 00:04:18.231
and actually plot
these three points.

00:04:18.231 --> 00:04:21.880
1 comma 0, 2 comma
negative 5, 3 comma 0.

00:04:21.880 --> 00:04:23.280
So let's do that.

00:04:23.280 --> 00:04:27.130
So first I'll do the vertex
at 2 comma negative 5,

00:04:27.130 --> 00:04:28.760
which is right there.

00:04:28.760 --> 00:04:30.740
And now we also know
one of the times

00:04:30.740 --> 00:04:33.590
it intersects the
x-axis is at 1 comma 0.

00:04:33.590 --> 00:04:36.550
And the other time
is at 3 comma 0.

00:04:36.550 --> 00:04:38.610
And now we can check our answer.

00:04:38.610 --> 00:04:40.900
And we got it right.


Law of cosines

WEBVTT
Kind: captions
Language: en

00:00:00.512 --> 00:00:02.972
- [Voiceover] Let's say
that I've got a triangle,

00:00:03.482 --> 00:00:07.584
and this side has length b,

00:00:07.584 --> 00:00:10.045
which is equal to 12,

00:00:10.045 --> 00:00:13.667
12 units or whatever units
of measurement we're using.

00:00:14.177 --> 00:00:17.501
Let's say that this side right over here,

00:00:17.501 --> 00:00:20.181
this side right over here, has length c,

00:00:20.181 --> 00:00:22.292
and that happens to be equal to nine.

00:00:23.172 --> 00:00:27.511
And that we want to figure out the length

00:00:30.241 --> 00:00:33.019
of this side, and this side has length a,

00:00:33.019 --> 00:00:36.200
so we need to figure out what
a is going to be equal to.

00:00:36.200 --> 00:00:38.243
Now, we won't be able to figure this out

00:00:38.243 --> 00:00:40.147
unless we also know the angle here,

00:00:40.147 --> 00:00:44.071
because you could bring the blue side

00:00:44.071 --> 00:00:45.581
and the green side close together,

00:00:45.581 --> 00:00:46.718
and then a would be small,

00:00:46.718 --> 00:00:49.179
but if this angle was larger
than a would be larger.

00:00:49.179 --> 00:00:51.548
So we need to know what
this angle is as well.

00:00:51.548 --> 00:00:53.777
So let's say that we know that this angle,

00:00:53.777 --> 00:00:55.758
which we will call theta,

00:00:55.758 --> 00:00:59.548
is equal to 87 degrees.

00:00:59.548 --> 00:01:01.114
So how can we figure out a?

00:01:01.634 --> 00:01:04.318
I encourage you to pause this
and try this on your own.

00:01:04.318 --> 00:01:07.848
Well, lucky for us, we
have the Law of Cosines,

00:01:07.848 --> 00:01:10.378
which gives us a way for
determining a third side

00:01:10.378 --> 00:01:13.885
if we know two of the sides
and the angle between them.

00:01:13.885 --> 00:01:18.087
The Law of Cosines tells
us that a squared is going

00:01:18.087 --> 00:01:22.661
to be equal b squared plus c squared.

00:01:23.481 --> 00:01:25.355
Now, if we were dealing
with a pure right triangle,

00:01:25.355 --> 00:01:28.373
if this was 90 degrees, then
a would be the hypotenuse,

00:01:28.373 --> 00:01:30.579
and we would be done, this would
be the Pythagorean Theorem.

00:01:30.579 --> 00:01:32.344
But the Law of Cosines
gives us an adjustment

00:01:32.344 --> 00:01:34.618
to the Pythagorean Theorem,
so that we can do this

00:01:34.618 --> 00:01:36.477
for any arbitrary angle.

00:01:36.477 --> 00:01:38.868
So Law of Cosines tell
us a squared is going

00:01:38.868 --> 00:01:40.470
to be b squared plus c squared,

00:01:41.050 --> 00:01:44.116
minus two times bc,

00:01:48.276 --> 00:01:53.031
times the cosine of theta.

00:01:55.751 --> 00:01:59.882
And this theta is the angle that opens up

00:01:59.882 --> 00:02:02.273
to the side that we care about.

00:02:02.273 --> 00:02:05.199
So we can use theta because
we're looking for a.

00:02:05.199 --> 00:02:06.847
If they gave us another
angle right over here,

00:02:06.847 --> 00:02:08.450
that's not the angle that we would use.

00:02:08.450 --> 00:02:10.609
We care about the angle that opens up

00:02:10.609 --> 00:02:13.535
into the side that we
are going to solve for.

00:02:13.535 --> 00:02:14.951
So now let's solve for a,

00:02:14.951 --> 00:02:17.434
because we know what bc
and theta actually are.

00:02:18.224 --> 00:02:23.031
So a squared is going to
be equal to b squared...

00:02:24.411 --> 00:02:27.512
so it's going to be equal to 144,

00:02:27.512 --> 00:02:31.669
plus c squared which is 81, so plus 81,

00:02:32.219 --> 00:02:35.802
minus two times b times c.

00:02:35.802 --> 00:02:37.914
So, it's minus two,
I'll just write it out.

00:02:37.914 --> 00:02:42.675
Minus two times 12 times nine,

00:02:45.215 --> 00:02:49.268
times the cosine of 87 degrees.

00:02:52.458 --> 00:02:54.076
And this is going to be equal to,

00:02:56.216 --> 00:03:00.647
let's see, this is 225 minus,

00:03:00.647 --> 00:03:03.178
let's see, 12 times nine is 108.

00:03:03.178 --> 00:03:05.801
108 times two is 216.

00:03:05.801 --> 00:03:10.306
Minus 216 times the cosine of 87 degrees.

00:03:12.726 --> 00:03:13.951
Now, let's get our calculator out

00:03:13.951 --> 00:03:15.204
in order to approximate this.

00:03:15.204 --> 00:03:16.783
And remember, this is a squared.

00:03:16.783 --> 00:03:18.270
Actually, before I get my calculator out,

00:03:18.270 --> 00:03:19.408
let's just solve for a.

00:03:19.408 --> 00:03:21.473
So a is just going to be
the square root of this.

00:03:21.473 --> 00:03:24.818
So a is going to be equal
to the square root...

00:03:26.098 --> 00:03:28.533
of all of this business, which
I can just copy and paste.

00:03:29.233 --> 00:03:31.250
It's going to be equal to
the square root of that.

00:03:31.250 --> 00:03:34.012
So let me copy and paste it.

00:03:34.502 --> 00:03:36.171
So a is going to be equal
to the square root of that,

00:03:36.171 --> 00:03:39.074
which we can now use the
calculator to figure out.

00:03:39.074 --> 00:03:40.770
Let me increase this radical a little bit,

00:03:40.770 --> 00:03:42.047
so that we make sure we're
taking the square root

00:03:42.047 --> 00:03:43.208
of this whole thing.

00:03:43.208 --> 00:03:44.903
So let me get my calculator out.

00:03:45.483 --> 00:03:49.244
So I want to find that square root of 220.

00:03:49.244 --> 00:03:50.985
Actually, before I do that,
let me just make sure I'm

00:03:50.985 --> 00:03:53.609
in degree mode, and I am in degree mode.

00:03:53.609 --> 00:03:57.952
Because we're evaluating a
trig function in degrees here.

00:03:58.502 --> 00:03:59.878
So that's fine, so let me exit.

00:03:59.878 --> 00:04:04.842
So it's going to be 225 minus 216,

00:04:04.842 --> 00:04:08.465
times cosine

00:04:08.465 --> 00:04:10.015
of 87 degrees.

00:04:10.015 --> 00:04:12.511
Not 88 degrees, 87 degrees.

00:04:13.721 --> 00:04:16.040
And we deserve a drumroll now.

00:04:16.720 --> 00:04:20.624
This is going to be 14.61,

00:04:20.624 --> 00:04:22.564
or 14.618.

00:04:22.564 --> 00:04:24.492
If, say, we wanted to
round to the nearest tenth,

00:04:24.492 --> 00:04:25.769
just to get an approximation,

00:04:25.769 --> 00:04:28.277
it would be approximately 14.6.

00:04:28.737 --> 00:04:31.398
So a is approximately equal

00:04:31.398 --> 00:04:36.168
to 14.6, whatever units we're using long.


Complex number polar form intuition exercise

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:05.460
We're asked to adjust the angle
in radius of the orange plotted

00:00:05.460 --> 00:00:08.220
complex number-- so that's
this right over here--

00:00:08.220 --> 00:00:11.980
to match the blue plotted
complex number negative 3.5

00:00:11.980 --> 00:00:14.990
plus 6.06i.

00:00:14.990 --> 00:00:16.490
So this is this one
right over here.

00:00:16.490 --> 00:00:19.100
So just to get our bearings,
this vertical axis,

00:00:19.100 --> 00:00:21.650
this is the imaginary axis.

00:00:21.650 --> 00:00:25.240
This horizontal axis
is the real axis.

00:00:25.240 --> 00:00:28.320
The real part of this
number is negative 3.5,

00:00:28.320 --> 00:00:30.880
and so we've plotted that on
the real axis-- negative 1,

00:00:30.880 --> 00:00:33.630
negative 2, negative 3.5.

00:00:33.630 --> 00:00:37.440
And then the imaginary
part is 6.06.

00:00:37.440 --> 00:00:44.150
So we go up 1, 2, 3, 4, 5,
6, and then a very little bit

00:00:44.150 --> 00:00:45.910
to get to 6.06.

00:00:45.910 --> 00:00:47.970
So they've plotted it like that.

00:00:47.970 --> 00:00:49.670
And now they want
us to think about it

00:00:49.670 --> 00:00:55.370
in kind of the polar form is
one way of thinking about it.

00:00:55.370 --> 00:00:58.560
And so they tell us that we
can adjust the angle, which

00:00:58.560 --> 00:01:01.590
is often called the argument
when you're plotting up

00:01:01.590 --> 00:01:05.300
a complex number in this form,
and the radius, which is often

00:01:05.300 --> 00:01:06.470
called the modulus.

00:01:06.470 --> 00:01:08.030
So let's change that.

00:01:08.030 --> 00:01:10.507
So let's see, if we
increase the radius,

00:01:10.507 --> 00:01:12.590
so then the thing is going
to go out a little bit.

00:01:12.590 --> 00:01:16.590
If we increase the
angle, that's pi over 3.

00:01:16.590 --> 00:01:18.330
This is pi over 2.

00:01:18.330 --> 00:01:20.920
So we increase the
angle to 2 pi over 3.

00:01:20.920 --> 00:01:22.830
And this seems to be
the right direction,

00:01:22.830 --> 00:01:25.570
and we just have to increase
the radius a little bit.

00:01:25.570 --> 00:01:31.550
So there we have a radius
of 7, an angle of 2/3 pi.

00:01:31.550 --> 00:01:37.160
Now, let's verify for ourselves
that this complex number

00:01:37.160 --> 00:01:39.620
and this complex number
really are the same thing.

00:01:39.620 --> 00:01:43.090
We've already seen it right over
here in this actual diagram.

00:01:43.090 --> 00:01:44.590
We've plotted it
both in I guess you

00:01:44.590 --> 00:01:46.780
could say, rectangular
form and polar form,

00:01:46.780 --> 00:01:51.170
although this is a complex
axis, and this is a real axis.

00:01:51.170 --> 00:01:55.490
So let me copy and paste the
information that we have here.

00:01:55.490 --> 00:01:57.430
So let me just copy
and paste all of that.

00:01:57.430 --> 00:02:01.796
And then I can put it on
my little scratch pad.

00:02:01.796 --> 00:02:03.830
So let me throw it
right over there,

00:02:03.830 --> 00:02:05.600
and let's actually verify that.

00:02:08.910 --> 00:02:14.460
So here we have an argument, or
an angle I guess I could say.

00:02:14.460 --> 00:02:18.490
So this angle, often
referred to as phi

00:02:18.490 --> 00:02:20.490
when you're dealing
in the complex plane,

00:02:20.490 --> 00:02:22.210
this is equal to 2/3 pi.

00:02:22.210 --> 00:02:24.480
And this is, of
course, in radians.

00:02:24.480 --> 00:02:26.500
And the modulus, which
is really the distance

00:02:26.500 --> 00:02:31.650
from the origin to the complex
number, this right over here,

00:02:31.650 --> 00:02:36.050
by just manipulating that little
tool, we saw that to be 7.

00:02:36.050 --> 00:02:39.240
So let's verify that this is
really the same complex number

00:02:39.240 --> 00:02:44.310
as negative 3.5 plus 6.06i.

00:02:44.310 --> 00:02:48.170
Well, we can
represent this as 7--

00:02:48.170 --> 00:02:50.450
this is the modulus,
or the radius.

00:02:50.450 --> 00:02:59.600
7 times e to the 2/3 pi i, which
we know from Euler's identity

00:02:59.600 --> 00:03:00.852
is the exact same thing.

00:03:00.852 --> 00:03:03.060
This is equal to-- and this
is one of really the most

00:03:03.060 --> 00:03:05.230
mind-blowing things
in all of mathematics.

00:03:05.230 --> 00:03:11.580
This is the same thing as
7 times cosine of 2/3 pi

00:03:11.580 --> 00:03:16.990
plus i sine of 2/3 pi.

00:03:16.990 --> 00:03:20.200
And they already wrote that
down for us right over there.

00:03:20.200 --> 00:03:22.920
Now let's verify what
cosine of 2/3 pi is

00:03:22.920 --> 00:03:25.270
and what sine of 2/3 pi is.

00:03:25.270 --> 00:03:28.090
So if this is 2/3 as
an angle-- and now

00:03:28.090 --> 00:03:32.040
you can almost even
think of the unit circle.

00:03:32.040 --> 00:03:34.110
So this is 2/3 pi.

00:03:34.110 --> 00:03:37.356
So actually, just let me just
draw a unit circle like this

00:03:37.356 --> 00:03:38.980
because I don't want
us to confuse just

00:03:38.980 --> 00:03:40.730
our traditional
rectangular coordinates

00:03:40.730 --> 00:03:42.920
with what we're doing
on the polar axis.

00:03:42.920 --> 00:03:47.000
So this isn't a
polar-- we're not

00:03:47.000 --> 00:03:49.300
graphing imaginary
numbers right over here.

00:03:49.300 --> 00:03:51.360
This is just the
traditional unit circle.

00:03:51.360 --> 00:03:54.100
This is our x-axis
this is our y-axis.

00:03:54.100 --> 00:03:58.670
2/3 pi is the same
thing as 120 degrees.

00:03:58.670 --> 00:04:02.980
So it looks just like what
we did over here, so 2/3 pi.

00:04:02.980 --> 00:04:06.090
So let me draw it as neatly as
I can, which isn't that neatly.

00:04:06.090 --> 00:04:09.130
So that angle is 2/3 pi.

00:04:09.130 --> 00:04:14.530
Our cosine of 2/3 pi is going
to be our x-coordinate here.

00:04:14.530 --> 00:04:18.470
And sine of 2/3 pi is going to
be our y-coordinates over here.

00:04:18.470 --> 00:04:19.920
So what's our x-coordinate?

00:04:19.920 --> 00:04:23.370
Well, if this right over
here is 120 degrees,

00:04:23.370 --> 00:04:25.740
then this is 60 degrees.

00:04:25.740 --> 00:04:28.540
This is a 30-60-90 triangle.

00:04:28.540 --> 00:04:30.130
And we know this
is a unit circle.

00:04:30.130 --> 00:04:31.740
The radius is 1.

00:04:31.740 --> 00:04:34.780
So the 30-degree side
will have length 1/2.

00:04:34.780 --> 00:04:36.370
So this coordinate
right over here

00:04:36.370 --> 00:04:37.660
is going to be negative 1/2.

00:04:40.890 --> 00:04:43.500
And then the side opposite
the 60-degree side

00:04:43.500 --> 00:04:46.030
is square root of
3 times the 1/2.

00:04:46.030 --> 00:04:48.740
So this right over here is going
to be square root of 3 over 2.

00:04:48.740 --> 00:04:52.270
And you could verify that on
a calculator, if you like.

00:04:52.270 --> 00:04:55.750
So cosine of 2/3
pi is negative 1/2.

00:04:55.750 --> 00:04:59.780
So this is going to be equal
to 7 times negative 1/2.

00:04:59.780 --> 00:05:04.390
That's cosine of 2/3 pi
plus i times the square root

00:05:04.390 --> 00:05:06.700
of 3 over 2.

00:05:06.700 --> 00:05:08.200
So let's verify
that that's actually

00:05:08.200 --> 00:05:10.330
the same thing as
that over there.

00:05:10.330 --> 00:05:16.410
Well, 7 times negative 1/2
is negative 3.5 plus i.

00:05:16.410 --> 00:05:20.960
I'll just write it lik e7
square root of 3 over 2i.

00:05:20.960 --> 00:05:23.720
So we already see that the real
parts are definitely the same.

00:05:23.720 --> 00:05:25.364
It's negative 3.5.

00:05:25.364 --> 00:05:26.780
And I'll just get
a calculator out

00:05:26.780 --> 00:05:31.270
to evaluate this and see
if it is roughly 6.06.

00:05:31.270 --> 00:05:34.120
So let me get my
trusty TI-85 out.

00:05:34.120 --> 00:05:40.180
And then I want to evaluate
7 times the square root of 3,

00:05:40.180 --> 00:05:45.020
and then that divided
by 2 is indeed 6.06,

00:05:45.020 --> 00:05:47.420
if we round to the
nearest hundredths.

00:05:47.420 --> 00:05:54.180
So this is approximately equal
to negative 3.5 plus 6.06i,

00:05:54.180 --> 00:06:00.170
which is exactly the complex
number that we started with.


Using the focus and directrix to find the equation of a parabola

WEBVTT
Kind: captions
Language: en

00:00:00.890 --> 00:00:03.860
We're asked to first find
the focus and directrix

00:00:03.860 --> 00:00:06.310
of the parabola by
moving the orange point

00:00:06.310 --> 00:00:08.530
and line to their
correct positions,

00:00:08.530 --> 00:00:10.650
then use that
information to find

00:00:10.650 --> 00:00:12.510
the equation of the parabola.

00:00:12.510 --> 00:00:14.960
So just as a little
bit of review,

00:00:14.960 --> 00:00:17.070
a parabola is a
conic section that

00:00:17.070 --> 00:00:19.080
is formed from all
of the points that

00:00:19.080 --> 00:00:22.420
are equidistant to
some point that we call

00:00:22.420 --> 00:00:26.182
the focus and some line that
we called the directrix.

00:00:26.182 --> 00:00:28.140
And what we have right
over here-- what it does

00:00:28.140 --> 00:00:29.830
is, as we move
along the parabola,

00:00:29.830 --> 00:00:32.420
as we move our cursor
along the parabola

00:00:32.420 --> 00:00:34.920
right over here, it looks
at the distance to where

00:00:34.920 --> 00:00:38.690
the current dot is
and then it tries

00:00:38.690 --> 00:00:42.400
to plot that same distance to
some theoretical directrix.

00:00:42.400 --> 00:00:44.170
And that's where
that line drops down.

00:00:44.170 --> 00:00:45.930
And you can clearly
see right now

00:00:45.930 --> 00:00:49.030
that this isn't working out,
that this is not equidistant.

00:00:49.030 --> 00:00:51.890
So the first thing that I like
to do when I try to figure out

00:00:51.890 --> 00:00:54.930
where the focus is
is that the parabola

00:00:54.930 --> 00:00:57.770
is going to be symmetric
around the focus,

00:00:57.770 --> 00:01:00.410
or at least around the vertical
line that the focus is on,

00:01:00.410 --> 00:01:01.340
I should say.

00:01:01.340 --> 00:01:04.349
And so the focus is going to
sit right above the vertex.

00:01:04.349 --> 00:01:06.200
So that's the first thing to do.

00:01:06.200 --> 00:01:09.860
And so let me keep
manipulating it.

00:01:09.860 --> 00:01:12.275
So if I look below
down there-- and it's

00:01:12.275 --> 00:01:14.400
hard to do that because
when I move the cursor then

00:01:14.400 --> 00:01:16.275
that green thing
disappears-- but if you look

00:01:16.275 --> 00:01:19.229
at that vertical green line,
that shouldn't be moving--

00:01:19.229 --> 00:01:21.270
the bottom of it shouldn't
be moving up and down.

00:01:21.270 --> 00:01:23.410
It should be equidistant
to a flat line.

00:01:23.410 --> 00:01:26.260
So let me keep playing with it.

00:01:26.260 --> 00:01:28.450
It looks like that
might be an improvement.

00:01:28.450 --> 00:01:29.160
Let me see.

00:01:29.160 --> 00:01:31.960
So this actually
looks pretty good.

00:01:31.960 --> 00:01:34.660
Notice the bottom--
oops-- notice

00:01:34.660 --> 00:01:38.470
the bottom of the
vertical green line

00:01:38.470 --> 00:01:41.245
is not moving in the
vertical direction.

00:01:41.245 --> 00:01:43.120
It's just moving in the
horizontal direction.

00:01:43.120 --> 00:01:46.420
So if I move my directrix up
there, that should work out.

00:01:46.420 --> 00:01:49.530
So let me move it up there.

00:01:49.530 --> 00:01:50.940
And there you have it.

00:01:50.940 --> 00:01:53.230
Notice now my parabola--
these green lines

00:01:53.230 --> 00:01:55.410
show the distance to the
focus and the distance

00:01:55.410 --> 00:01:58.990
to the directrix, and notice
that they are always the same.

00:01:58.990 --> 00:02:03.393
So it looks like I got the right
focus and the right directrix

00:02:03.393 --> 00:02:04.550
now.

00:02:04.550 --> 00:02:07.400
So I got the right focus
and the right directrix.

00:02:07.400 --> 00:02:11.900
And the focus is 1/4
comma negative 3/8.

00:02:11.900 --> 00:02:15.880
And the directrix is
y equals negative 5/8.

00:02:15.880 --> 00:02:18.760
So now let's try to
use that information

00:02:18.760 --> 00:02:22.044
to actually find the equation
of the parabola in this form

00:02:22.044 --> 00:02:22.710
right over here.

00:02:22.710 --> 00:02:25.840
And this is going to involve a
little bit of algebra for us.

00:02:25.840 --> 00:02:29.890
So actually let me
copy and paste this.

00:02:29.890 --> 00:02:33.219
So I'm going to
copy and paste this

00:02:33.219 --> 00:02:34.760
so that we know what
we need, and let

00:02:34.760 --> 00:02:38.200
me open my little
scratch pad up.

00:02:38.200 --> 00:02:38.730
All right.

00:02:38.730 --> 00:02:40.210
So this is our
information, and we

00:02:40.210 --> 00:02:44.210
need to figure out the
equation of the parabola.

00:02:44.210 --> 00:02:48.329
So just as a little bit of
review, we have the point.

00:02:48.329 --> 00:02:49.620
So let's say this is our point.

00:02:49.620 --> 00:02:55.380
This is our focus-- is the
point 1/4 comma negative 3/8.

00:02:55.380 --> 00:02:59.130
And then we have the directrix
at y equals negative 5/8.

00:02:59.130 --> 00:03:03.440
So the directrix looks
something like this.

00:03:03.440 --> 00:03:07.860
That is y is equal
to negative 5/8.

00:03:07.860 --> 00:03:11.360
And then we have our parabola.

00:03:11.360 --> 00:03:12.800
And it looks
something like this.

00:03:12.800 --> 00:03:16.230
It's equidistant to
the 2, so our parabola

00:03:16.230 --> 00:03:19.120
looks something like that.

00:03:19.120 --> 00:03:22.542
And the parabola, the x's and
y's that satisfy this equation

00:03:22.542 --> 00:03:24.000
are all of the x's
and y's that are

00:03:24.000 --> 00:03:25.610
equidistant to these two things.

00:03:25.610 --> 00:03:28.450
So let's call that x comma y.

00:03:28.450 --> 00:03:30.880
So let's find an equation for
all of the x's and y's that

00:03:30.880 --> 00:03:33.350
are equidistant to this
point and to this line

00:03:33.350 --> 00:03:34.860
right over here.

00:03:34.860 --> 00:03:38.250
So, first, let's think about
the distance between this focus

00:03:38.250 --> 00:03:41.742
and this x, y right over here.

00:03:41.742 --> 00:03:44.200
Well, that distance just comes
straight out of the distance

00:03:44.200 --> 00:03:47.190
formula, which comes straight
out of the Pythagorean theorem.

00:03:47.190 --> 00:03:54.160
It's going to be x
minus 1/4 squared

00:03:54.160 --> 00:03:58.570
plus y minus
negative 3/8 squared.

00:03:58.570 --> 00:04:03.200
So that's going to be
y plus 3/8 squared.

00:04:03.200 --> 00:04:04.670
So this is our distance squared.

00:04:04.670 --> 00:04:07.660
We could take the square root if
we-- Why don't we just do that?

00:04:07.660 --> 00:04:09.160
So let me just take
the square root.

00:04:09.160 --> 00:04:13.660
So this is the distance between
any point on the parabola x, y

00:04:13.660 --> 00:04:15.120
and the focus.

00:04:15.120 --> 00:04:17.887
And the distance between that
same point and the directrix--

00:04:17.887 --> 00:04:20.220
well, that's just going to
be a straight, vertical line.

00:04:20.220 --> 00:04:24.880
So that's just going
to be our change in y.

00:04:24.880 --> 00:04:31.230
So this is going to be equal
to y minus negative 5/8,

00:04:31.230 --> 00:04:33.746
which is the same
thing as y plus 5/8.

00:04:33.746 --> 00:04:35.870
And just to make sure that
this is always positive,

00:04:35.870 --> 00:04:39.920
let's square it and then
take the square root.

00:04:39.920 --> 00:04:41.620
Now, to simplify
this, we could square

00:04:41.620 --> 00:04:43.100
both sides of this equation.

00:04:43.100 --> 00:04:49.730
And we get x minus
1/4 squared plus y--

00:04:49.730 --> 00:04:51.230
and actually let
me expand this out.

00:04:51.230 --> 00:04:52.980
This will be useful
for us in a second,

00:04:52.980 --> 00:04:54.130
so let me expand this out.

00:04:54.130 --> 00:04:57.100
So this is going to be y
squared plus-- let's see,

00:04:57.100 --> 00:05:06.550
2 times 3/8 is 3/4-- so
plus 3/4y plus 9/64 is

00:05:06.550 --> 00:05:10.486
equal to this thing squared,
which is just y-- well, let me

00:05:10.486 --> 00:05:11.860
actually expand
that out as well.

00:05:11.860 --> 00:05:19.880
So that's y squared plus 2
times 5/8 is going to be 5/4.

00:05:19.880 --> 00:05:26.664
5/4y plus 25 over 64.

00:05:26.664 --> 00:05:28.205
Now, let me see if
I can simplify it.

00:05:28.205 --> 00:05:30.040
I have a y squared
on both sides,

00:05:30.040 --> 00:05:31.600
so I can cancel those terms out.

00:05:31.600 --> 00:05:34.260
So y squared, y squared.

00:05:34.260 --> 00:05:36.520
Let me subtract 3/4y
from both sides.

00:05:42.070 --> 00:05:45.060
And let me subtract
9/64 from both sides.

00:05:45.060 --> 00:05:47.360
Minus 9/64.

00:05:47.360 --> 00:05:49.239
Minus 9/64.

00:05:49.239 --> 00:05:50.780
Actually, we had a
little more space.

00:05:50.780 --> 00:05:53.370
So that's going to be gone
and that's going to be gone.

00:05:53.370 --> 00:05:55.610
Our left hand side is
just going to simplify

00:05:55.610 --> 00:05:59.500
to x minus 1/4 squared.

00:05:59.500 --> 00:06:02.660
And that's going to be
equal to-- let's see.

00:06:02.660 --> 00:06:09.800
5/4 minus 3/4 is 2/4, which
is the same thing as 1/2y.

00:06:09.800 --> 00:06:16.850
And then 25/64
minus 9/64 is 16/64,

00:06:16.850 --> 00:06:19.510
which is just the
same thing as 1/4.

00:06:19.510 --> 00:06:24.170
So it's going to
be plus 1/4 there.

00:06:24.170 --> 00:06:26.630
Now, we're almost got
where we need to go.

00:06:26.630 --> 00:06:30.060
We have this x
minus 1/4 squared.

00:06:30.060 --> 00:06:32.460
We just need to get a
coefficient out here and write

00:06:32.460 --> 00:06:35.250
this in terms of
y minus something.

00:06:35.250 --> 00:06:36.670
So what we can do
right over here

00:06:36.670 --> 00:06:38.770
is we can factor out a 1/2.

00:06:38.770 --> 00:06:39.840
So let's do that.

00:06:39.840 --> 00:06:43.340
So this is going to be
x minus 1/4 squared is

00:06:43.340 --> 00:06:49.809
equal to-- factoring out a
1/2, it's 1/2 times y plus 1/2.

00:06:49.809 --> 00:06:52.350
And now, in order to put it in
the form that they care about,

00:06:52.350 --> 00:06:55.830
they just have y minus
something right over here

00:06:55.830 --> 00:06:59.420
and then something times
x minus something squared.

00:06:59.420 --> 00:07:01.990
So we need to get this 1/2
or get this coefficient

00:07:01.990 --> 00:07:03.089
on the other side.

00:07:03.089 --> 00:07:04.630
The easiest way I
know how to do that

00:07:04.630 --> 00:07:07.340
is to multiply both sides by 2.

00:07:07.340 --> 00:07:09.960
So you multiply both sides by 2.

00:07:09.960 --> 00:07:11.110
That cancels.

00:07:11.110 --> 00:07:13.450
And if we really want
to put it in the form

00:07:13.450 --> 00:07:17.140
that they care about,
we can swap the sides.

00:07:17.140 --> 00:07:18.970
Instead of writing
y plus 1/2, we

00:07:18.970 --> 00:07:23.730
can write that as y
minus negative 1/2.

00:07:23.730 --> 00:07:25.572
That's the same
thing as y plus 1/2--

00:07:25.572 --> 00:07:27.030
I'm going to do it
in the same form

00:07:27.030 --> 00:07:38.277
that they want-- is equal to
2 times x minus 1/4 squared.

00:07:38.277 --> 00:07:38.860
So let me see.

00:07:38.860 --> 00:07:39.750
Did I do that right?

00:07:39.750 --> 00:07:40.250
Yep.

00:07:40.250 --> 00:07:43.530
y minus negative 1/2 is the
same thing as y plus 1/2

00:07:43.530 --> 00:07:48.410
is equal to 2 times
x minus 1/4 squared.

00:07:48.410 --> 00:07:51.400
So let me see if we got
that actually right.

00:07:51.400 --> 00:07:52.840
So let's see.

00:07:52.840 --> 00:07:58.330
It's negative 1/2 here,
positive 2, and then 1/4.

00:07:58.330 --> 00:08:10.860
So we have it is negative
1/2, positive 2, and then 1/4.

00:08:10.860 --> 00:08:12.270
Let's see if we got it right.

00:08:12.270 --> 00:08:14.160
And we did.


Example 1: Factor a linear binomial by taking a common factor

WEBVTT
Kind: captions
Language: en

00:00:00.990 --> 00:00:05.280
What I want to do is start with
an expression like 4x plus 18

00:00:05.280 --> 00:00:08.020
and see if we can rewrite
this as the product of two

00:00:08.020 --> 00:00:08.560
expressions.

00:00:08.560 --> 00:00:11.110
Essentially, we're going
to try to factor this.

00:00:11.110 --> 00:00:14.030
And the key here
is to figure out

00:00:14.030 --> 00:00:17.420
are there any common
factors to both 4x and 18?

00:00:17.420 --> 00:00:20.100
And we can factor that
common factor out.

00:00:20.100 --> 00:00:21.890
We're essentially
going to be reversing

00:00:21.890 --> 00:00:23.690
the distributive property.

00:00:23.690 --> 00:00:26.159
So for example, what
is the largest number

00:00:26.159 --> 00:00:28.700
that is-- or I could really say
the largest expression-- that

00:00:28.700 --> 00:00:32.299
is divisible into
both 4x and 18?

00:00:32.299 --> 00:00:34.970
Well, 4x is divisible
by 2, because we

00:00:34.970 --> 00:00:37.000
know that 4 is divisible by 2.

00:00:37.000 --> 00:00:39.280
And 18 is also
divisible by 2, so we

00:00:39.280 --> 00:00:51.430
can rewrite 4x as
being 2 times 2x.

00:00:51.430 --> 00:00:54.070
If you multiply that side,
it's obviously going to be 4x.

00:00:54.070 --> 00:01:01.660
And then, we can write 18 as
the same thing as 2 times 9.

00:01:01.660 --> 00:01:03.450
And now it might
be clear that when

00:01:03.450 --> 00:01:04.957
you apply the
distributive property,

00:01:04.957 --> 00:01:06.540
you'll usually end
up with a step that

00:01:06.540 --> 00:01:07.290
looks something like this.

00:01:07.290 --> 00:01:09.400
Now we're just going to
undistribute the two right over

00:01:09.400 --> 00:01:09.900
here.

00:01:09.900 --> 00:01:11.849
We're going to
factor the two out.

00:01:11.849 --> 00:01:13.140
Let me actually just draw that.

00:01:13.140 --> 00:01:16.660
So we're going to
factor the two out,

00:01:16.660 --> 00:01:24.090
and so this is going to
be 2 times 2x plus 9.

00:01:26.690 --> 00:01:29.010
And if you were to-- wanted
to multiply this out,

00:01:29.010 --> 00:01:31.102
it would be 2 times
2x plus 2 times 9.

00:01:31.102 --> 00:01:32.560
It would be exactly
this, which you

00:01:32.560 --> 00:01:34.800
would simplify as
this, right up here.

00:01:34.800 --> 00:01:35.765
So there we have it.

00:01:35.765 --> 00:01:37.730
We have written
this as the product

00:01:37.730 --> 00:01:41.880
of two expressions,
2 times 2x plus 9.

00:01:41.880 --> 00:01:43.350
Let's do this again.

00:01:43.350 --> 00:01:53.930
So let's say that I
have 12 plus-- let

00:01:53.930 --> 00:01:59.140
me think of something
interesting-- 32x.

00:01:59.140 --> 00:02:03.260
Actually since we-- just to get
a little bit of variety here,

00:02:03.260 --> 00:02:07.340
let's put a y here, 12 plus 32y.

00:02:07.340 --> 00:02:09.180
Well, what's the
largest number that's

00:02:09.180 --> 00:02:10.949
divisible into both 12 and 32?

00:02:10.949 --> 00:02:15.040
2 is clearly divisible
into both, but so is 4.

00:02:15.040 --> 00:02:15.680
And let's see.

00:02:15.680 --> 00:02:18.000
It doesn't look like
anything larger than 4

00:02:18.000 --> 00:02:20.290
is divisible into
both 12 and 32.

00:02:20.290 --> 00:02:23.810
The greatest common
factor of 12 and 32 is 4,

00:02:23.810 --> 00:02:25.802
and y is only divisible
into the second term,

00:02:25.802 --> 00:02:27.510
not into this first
term right over here.

00:02:27.510 --> 00:02:30.110
So it looks like 4 is the
greatest common factor.

00:02:30.110 --> 00:02:33.379
So we could rewrite each
of these as a product of 4

00:02:33.379 --> 00:02:34.170
and something else.

00:02:34.170 --> 00:02:43.090
So for example, 12, we
can rewrite as 4 times 3.

00:02:43.090 --> 00:02:45.970
And 32, we can
rewrite-- since it's

00:02:45.970 --> 00:02:48.953
going to be plus-- 4 times.

00:02:51.580 --> 00:02:57.470
Well if you divide 32y by
4, it's going to be 8y.

00:02:57.470 --> 00:03:00.200
And now once again, we
can factor out the 4.

00:03:00.200 --> 00:03:06.465
So this is going to
be 4 times 3 plus 8y.

00:03:09.440 --> 00:03:11.710
And once you do more and
more examples of this,

00:03:11.710 --> 00:03:13.380
you're going to find
that you can just

00:03:13.380 --> 00:03:14.690
do this stuff all at once.

00:03:14.690 --> 00:03:17.200
You can say hey, what's
the largest number that's

00:03:17.200 --> 00:03:18.640
divisible into both of these?

00:03:18.640 --> 00:03:21.400
Well, it's 4, so let
me factor a 4 out.

00:03:21.400 --> 00:03:23.586
12 divided by 4 is 3.

00:03:23.586 --> 00:03:26.992
32y divided by 4 is 8y.


Example 2: Factoring a difference of squares with leading coefficient other than 1

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:02.820
Let's see if we can
factor the expression

00:00:02.820 --> 00:00:09.920
45x squared minus 125.

00:00:09.920 --> 00:00:11.560
So whenever I see
something like this--

00:00:11.560 --> 00:00:13.590
I have a second-degree
term here,

00:00:13.590 --> 00:00:17.274
I have a subtraction
sign-- my temptation

00:00:17.274 --> 00:00:19.190
is to look at this as a
difference of squares.

00:00:19.190 --> 00:00:20.815
We've already seen
this multiple times.

00:00:20.815 --> 00:00:24.850
We've already seen that if
we have something of the form

00:00:24.850 --> 00:00:30.290
a squared minus b squared, that
this can be factored as a plus

00:00:30.290 --> 00:00:34.170
b times a minus b.

00:00:34.170 --> 00:00:35.330
So let's look over here.

00:00:35.330 --> 00:00:37.250
Well, over here,
it's not obvious

00:00:37.250 --> 00:00:39.630
that this right over
here is a perfect square.

00:00:39.630 --> 00:00:42.110
Neither is it obvious
that this right over here

00:00:42.110 --> 00:00:42.950
is a perfect square.

00:00:42.950 --> 00:00:44.324
So it's not clear
to me that this

00:00:44.324 --> 00:00:46.080
is a difference of squares.

00:00:46.080 --> 00:00:50.240
But what is interesting
is that both 45 and 125

00:00:50.240 --> 00:00:52.210
have some factors in common.

00:00:52.210 --> 00:00:54.650
And the one that
jumps out at me is 5.

00:00:54.650 --> 00:00:57.220
So let's see if we can factor
out a 5, and by doing that,

00:00:57.220 --> 00:00:59.511
whether we can get something
that's a little bit closer

00:00:59.511 --> 00:01:01.150
to this pattern right over here.

00:01:01.150 --> 00:01:06.870
So if we factor out a 5,
this becomes 5 times-- well,

00:01:06.870 --> 00:01:12.320
45x squared divided by 5
is going to be 9x squared.

00:01:12.320 --> 00:01:17.570
And then 125 divided by 5 is 25.

00:01:17.570 --> 00:01:18.980
Now, this is interesting.

00:01:18.980 --> 00:01:21.580
9x squared-- that's
a perfect square.

00:01:21.580 --> 00:01:25.490
If we call this a
squared, then that

00:01:25.490 --> 00:01:29.100
tells us that a
would be equal to 3x.

00:01:29.100 --> 00:01:32.630
3x-- the whole thing
squared is 9x squared.

00:01:32.630 --> 00:01:37.660
Similarly-- I can never say
similarly correctly-- 25

00:01:37.660 --> 00:01:40.290
is clearly just 5 squared.

00:01:40.290 --> 00:01:43.380
So in this case, if we're
looking at this template,

00:01:43.380 --> 00:01:46.270
b would be equal to 5.

00:01:46.270 --> 00:01:48.160
So now this is a
difference of squares,

00:01:48.160 --> 00:01:50.340
and we can factor it completely.

00:01:50.340 --> 00:01:52.850
So we can't forget our 5 out
front that we factored out.

00:01:52.850 --> 00:01:56.535
So it's going to be
5 times a plus b.

00:01:56.535 --> 00:01:57.410
So let me write this.

00:01:57.410 --> 00:02:02.610
So it's going to be 5 times
a plus b times a minus b.

00:02:08.530 --> 00:02:13.180
So let me write the b's
down, plus b and minus b.

00:02:13.180 --> 00:02:13.920
And we're done.

00:02:13.920 --> 00:02:17.100
5 times 3x plus 5
times 3x minus 5

00:02:17.100 --> 00:02:22.200
is 45x squared minus
125 factored out.


Example 6: Factoring quadratics with two variables

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:03.170
Let's see if we can use our
existing factoring skills

00:00:03.170 --> 00:00:07.350
to factor 30x squared
plus 11xy plus y squared.

00:00:07.350 --> 00:00:09.170
And I encourage you
to pause the video

00:00:09.170 --> 00:00:12.219
and see if you can
handle it yourself.

00:00:12.219 --> 00:00:13.760
Now, the first hint
I will give you--

00:00:13.760 --> 00:00:16.970
and this might open up
what's going on here--

00:00:16.970 --> 00:00:18.650
is to maybe rearrange
this a little bit.

00:00:18.650 --> 00:00:21.340
We could rewrite
this as y squared

00:00:21.340 --> 00:00:28.430
plus 11xy plus 30x squared.

00:00:28.430 --> 00:00:29.850
And my whole
motivation for doing

00:00:29.850 --> 00:00:33.670
that-- there are ways to
factor a quadratic where

00:00:33.670 --> 00:00:37.350
your first coefficient, your
coefficient on this first term,

00:00:37.350 --> 00:00:38.940
is something other than 1.

00:00:38.940 --> 00:00:40.210
But we haven't seen that yet.

00:00:40.210 --> 00:00:42.844
And so rearranging it this way,
this got us a little bit more

00:00:42.844 --> 00:00:43.760
into our comfort zone.

00:00:43.760 --> 00:00:47.350
Now our coefficient is a
1 on the y squared term.

00:00:47.350 --> 00:00:51.080
So now we can start to think
of this in the same form

00:00:51.080 --> 00:00:53.590
that we've looked at some of
the other factoring problems.

00:00:53.590 --> 00:00:59.760
Can we think of two numbers
whose product is 30x squared

00:00:59.760 --> 00:01:04.019
and whose sum is 11x?

00:01:04.019 --> 00:01:06.420
Notice, 11x is the
coefficient on y.

00:01:06.420 --> 00:01:09.390
We have y squared,
some coefficient on y.

00:01:09.390 --> 00:01:11.610
And then in terms
of y, this isn't

00:01:11.610 --> 00:01:12.920
in any way dependent on y.

00:01:12.920 --> 00:01:16.930
So one way to think about this,
if you knew what x was, then

00:01:16.930 --> 00:01:19.220
this would be a
quadratic in terms of y.

00:01:19.220 --> 00:01:21.440
And that's how we're really
thinking about it here.

00:01:21.440 --> 00:01:23.570
So can we find two
numbers whose product

00:01:23.570 --> 00:01:26.590
is 30x squared and two
numbers whose sum is

00:01:26.590 --> 00:01:32.120
the coefficient on this y term
right here, whose sum is 11x?

00:01:32.120 --> 00:01:35.160
So let's just think about all
of the different possibilities.

00:01:35.160 --> 00:01:37.290
If we were just thinking
about two numbers

00:01:37.290 --> 00:01:40.680
whose product was 30
and whose sum was 11,

00:01:40.680 --> 00:01:43.660
we would be thinking of 5 and 6.

00:01:43.660 --> 00:01:45.180
5 times 6 is 30.

00:01:45.180 --> 00:01:46.157
5 plus 6 is 11.

00:01:46.157 --> 00:01:47.240
It's some trial and error.

00:01:47.240 --> 00:01:49.590
You could have tried 3 and 10.

00:01:49.590 --> 00:01:52.510
Well, that would have been--
13 would be their sum.

00:01:52.510 --> 00:01:53.780
You could have tried 2 and 15.

00:01:53.780 --> 00:01:56.560
That wouldn't have worked.

00:01:56.560 --> 00:01:59.360
But 5 and 6 does work
here, so we've already

00:01:59.360 --> 00:02:00.950
seen that multiple times.

00:02:00.950 --> 00:02:06.810
So 5 and 6 would work for
30, but we have 30x squared.

00:02:06.810 --> 00:02:10.400
So what if we have 5x and 6x?

00:02:10.400 --> 00:02:17.260
Well, 5x times 6x is 30x
squared, and 5x plus 6x is 11x.

00:02:17.260 --> 00:02:19.130
So this actually works.

00:02:19.130 --> 00:02:23.120
So then our factoring
or our factorization

00:02:23.120 --> 00:02:25.760
of this expression
is just going to be

00:02:25.760 --> 00:02:35.260
y plus 5x times y plus 6x.

00:02:38.560 --> 00:02:41.460
And I'll leave it up to you to
verify that this does indeed,

00:02:41.460 --> 00:02:45.010
when you multiply it
out, equal this up here.


Example 5: Factoring quadratics with two variables (leading coefficient is 1)

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:02.470
We already have the
tools in our tool kit

00:00:02.470 --> 00:00:09.107
to factor something like
x squared plus 4x minus 5.

00:00:09.107 --> 00:00:11.690
And the way that we've already
thought about it is we've said,

00:00:11.690 --> 00:00:13.500
hey, let's think of
two numbers that if we

00:00:13.500 --> 00:00:16.309
were to take their product,
we'd get negative 5.

00:00:16.309 --> 00:00:19.530
And if we were to add the two
numbers, we'd get positive 4.

00:00:19.530 --> 00:00:21.322
And the fact that their
product is negative

00:00:21.322 --> 00:00:23.863
tells you one of them is going
to be positive and one of them

00:00:23.863 --> 00:00:24.960
is going to be negative.

00:00:24.960 --> 00:00:26.710
And so there's a couple of
ways you could think about it.

00:00:26.710 --> 00:00:28.835
Well, you could say, well,
maybe one of the numbers

00:00:28.835 --> 00:00:32.535
is negative 1 and then the
other one is positive 5.

00:00:32.535 --> 00:00:33.910
Actually, this
one seems to work.

00:00:33.910 --> 00:00:36.770
Negative 1 times
5 is negative 5.

00:00:36.770 --> 00:00:39.472
Negative 1 plus 5 is positive 4.

00:00:39.472 --> 00:00:40.930
So this one actually
seems to work.

00:00:40.930 --> 00:00:43.090
The other option would have
been-- since we're just

00:00:43.090 --> 00:00:46.510
going to deal with the factors
of 5, and 5's a prime number,

00:00:46.510 --> 00:00:48.260
the other option would
have been something

00:00:48.260 --> 00:00:51.520
like 1 and negative 5.

00:00:51.520 --> 00:00:53.210
There's only two factors for 5.

00:00:53.210 --> 00:00:55.250
So 1 and negative
5-- their product

00:00:55.250 --> 00:00:56.390
would have been negative 5.

00:00:56.390 --> 00:00:57.820
But if you were to
add these two numbers,

00:00:57.820 --> 00:01:00.130
you would have gotten a
negative 4 right over here.

00:01:00.130 --> 00:01:02.640
So we're going to go with
this right over here.

00:01:02.640 --> 00:01:04.379
And so this tells
us that if we want

00:01:04.379 --> 00:01:07.500
to factor this using the tools
that we already know about,

00:01:07.500 --> 00:01:09.540
we will get-- And let
me write these numbers

00:01:09.540 --> 00:01:12.010
in a different color so
we can keep track of them.

00:01:12.010 --> 00:01:14.070
So negative 1 and 5.

00:01:14.070 --> 00:01:15.710
We know that this
would factor out

00:01:15.710 --> 00:01:25.880
to be x minus 1 times x plus 5.

00:01:28.670 --> 00:01:30.620
And you can verify
this for yourself

00:01:30.620 --> 00:01:32.360
that if you were to
multiply this out,

00:01:32.360 --> 00:01:35.300
you will get x squared
plus 4x minus 5.

00:01:35.300 --> 00:01:38.470
You can even see this here.
x times x is x squared.

00:01:38.470 --> 00:01:42.300
Negative x plus 5x
is going to be 4x.

00:01:42.300 --> 00:01:45.250
And then negative 1
times 5 is negative 5.

00:01:45.250 --> 00:01:45.780
Fair enough.

00:01:45.780 --> 00:01:48.400
This is all review
for us at this point.

00:01:48.400 --> 00:01:52.660
Now I want to tackle something
a little bit more interesting.

00:01:52.660 --> 00:01:57.640
Let's say we wanted
to factor x squared

00:01:57.640 --> 00:02:04.700
plus 4xy minus 5y squared.

00:02:04.700 --> 00:02:06.430
And at first, this
looks really daunting.

00:02:06.430 --> 00:02:09.509
All of a sudden, I've introduced
a y and a y squared here.

00:02:09.509 --> 00:02:10.560
I have two variables.

00:02:10.560 --> 00:02:11.990
How would I tackle it?

00:02:11.990 --> 00:02:14.870
But the important thing is
to just take a deep breath

00:02:14.870 --> 00:02:16.690
and realize that we're
not fundamentally

00:02:16.690 --> 00:02:18.049
doing something different.

00:02:18.049 --> 00:02:19.590
Now, the one little
tricky thing I've

00:02:19.590 --> 00:02:20.940
done when I've written
it this way-- and I

00:02:20.940 --> 00:02:23.106
encourage you to pause this
and try this on your own

00:02:23.106 --> 00:02:24.580
before I explain any further.

00:02:24.580 --> 00:02:26.770
But the one tricky thing
I did right over here

00:02:26.770 --> 00:02:29.151
is I wrote the x before the y.

00:02:29.151 --> 00:02:30.650
And that tends to
be the convention.

00:02:30.650 --> 00:02:32.801
You just write them kind
of in alphabetical order.

00:02:32.801 --> 00:02:34.300
But if we wanted
it in a form that's

00:02:34.300 --> 00:02:36.830
a little bit closer to
this and something that

00:02:36.830 --> 00:02:38.910
would fit this mold
a little bit more

00:02:38.910 --> 00:02:40.922
is if we swapped these two.

00:02:40.922 --> 00:02:42.380
Because then we
could write it as x

00:02:42.380 --> 00:02:49.870
squared plus 4yx
minus 5y squared.

00:02:49.870 --> 00:02:53.270
And now it becomes pretty
clear that this 4y term

00:02:53.270 --> 00:02:55.180
right over here--
this right over here

00:02:55.180 --> 00:02:58.350
is the coefficient on the
x term, the same way that 4

00:02:58.350 --> 00:03:00.780
was the coefficient
on x right here.

00:03:00.780 --> 00:03:04.600
And this negative 5y
squared corresponds

00:03:04.600 --> 00:03:06.750
to the negative 5
right over here.

00:03:06.750 --> 00:03:09.410
So we can do the exact
same thought process.

00:03:09.410 --> 00:03:11.740
Let's think of two--
now not just numbers.

00:03:11.740 --> 00:03:13.406
They're going to have
variables in them.

00:03:13.406 --> 00:03:15.460
Let's think of two
terms or two expressions

00:03:15.460 --> 00:03:19.270
that if I multiply them,
I get negative 5y squared.

00:03:19.270 --> 00:03:22.310
And then when I take
the sum, I get 4y.

00:03:22.310 --> 00:03:24.800
So let's think about
how we could do this.

00:03:24.800 --> 00:03:29.310
So one option would be positive.

00:03:29.310 --> 00:03:37.430
Let's say positive
y and negative 5y.

00:03:37.430 --> 00:03:38.720
So where would this take us?

00:03:38.720 --> 00:03:42.490
Positive y times
negative 5y would indeed

00:03:42.490 --> 00:03:46.000
be equal to negative 5y squared.

00:03:46.000 --> 00:03:49.100
But then if I add
y to negative 5y,

00:03:49.100 --> 00:03:50.650
I'm going to get negative 4y.

00:03:50.650 --> 00:03:51.680
So this doesn't work.

00:03:51.680 --> 00:03:54.390
But let's see if we
swap the two signs.

00:03:54.390 --> 00:03:58.580
So what about negative
y and positive 5y?

00:03:58.580 --> 00:04:02.330
Well, here, if I take the
product of negative y and 5y,

00:04:02.330 --> 00:04:05.090
it will be negative 5y squared.

00:04:05.090 --> 00:04:11.020
And if I take the sum negative y
and 5y, it will be positive 4y.

00:04:11.020 --> 00:04:12.879
So we know how to factor it now.

00:04:12.879 --> 00:04:14.920
So once again, let me put
this in the same color.

00:04:14.920 --> 00:04:19.050
So this I'm going to put in this
mauve color, this light purple.

00:04:19.050 --> 00:04:21.290
And this I'll put
in a darker purple.

00:04:21.290 --> 00:04:23.810
So now we know how
to factor this.

00:04:23.810 --> 00:04:25.740
And this is the same
exact mold that we did up

00:04:25.740 --> 00:04:27.570
here, same exact idea.

00:04:27.570 --> 00:04:29.370
This is going to be x.

00:04:29.370 --> 00:04:32.267
Instead of just a minus 1
here, now we've factored-- here

00:04:32.267 --> 00:04:33.850
we factored into a
negative 1 and a 5.

00:04:33.850 --> 00:04:36.167
Here we factored into
a negative y and 5y.

00:04:36.167 --> 00:04:37.750
So instead of a
negative 1, it's going

00:04:37.750 --> 00:04:45.080
to be a negative y, x
minus y times x plus 5y.

00:04:50.480 --> 00:04:53.110
And we can verify that
when you multiply this out,

00:04:53.110 --> 00:04:57.970
it indeed does equal x
squared plus 4xy minus 5y.

00:04:57.970 --> 00:05:00.390
Let me do that here just
so we know for sure.

00:05:00.390 --> 00:05:06.020
So x times x is going
to be x squared.

00:05:06.020 --> 00:05:07.770
Let me do everything
in a different color.

00:05:07.770 --> 00:05:14.290
x times 5y is going
to be plus 5xy.

00:05:14.290 --> 00:05:20.650
Then negative y times
x is negative yx.

00:05:20.650 --> 00:05:25.800
And then finally-- and I'm
running out of colors--

00:05:25.800 --> 00:05:31.586
if we take negative y times
5y, that's negative 5y squared.

00:05:31.586 --> 00:05:32.960
And now we just
have to simplify.

00:05:32.960 --> 00:05:36.640
We have to combine these middle
two terms right over here.

00:05:36.640 --> 00:05:38.770
And at first, it looks a
little bit-- this is xy.

00:05:38.770 --> 00:05:39.340
This is yx.

00:05:39.340 --> 00:05:40.289
It's not so obvious.

00:05:40.289 --> 00:05:41.580
But we just have to rewrite it.

00:05:41.580 --> 00:05:47.310
This is the same
thing as 5yx minus yx.

00:05:47.310 --> 00:05:50.084
And so here, you're
saying, look, I have 5yx's.

00:05:50.084 --> 00:05:51.375
And I'm going to subtract yx's.

00:05:51.375 --> 00:05:53.760
So I'm going to have 4yx's.

00:05:53.760 --> 00:05:57.220
So this is just
going to be 4yx's.

00:05:57.220 --> 00:05:58.640
I have 5yx's.

00:05:58.640 --> 00:05:59.660
Take out another yx.

00:05:59.660 --> 00:06:01.690
I'm going to have 4yx's.

00:06:01.690 --> 00:06:07.720
So this is going to be x squared
plus 4yx minus 5y squared.

00:06:07.720 --> 00:06:10.420
And it all works out.


Introduction to point-slope form

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:03.279
So what I've drawn here
in yellow is a line.

00:00:03.279 --> 00:00:05.320
And let's say we know two
things about this line.

00:00:05.320 --> 00:00:08.170
We know that it
has a slope of m,

00:00:08.170 --> 00:00:11.525
and we know that the point
a, b is on this line.

00:00:11.525 --> 00:00:13.900
And so the question that we're
going to try to answer is,

00:00:13.900 --> 00:00:16.530
can we easily come
up with an equation

00:00:16.530 --> 00:00:20.220
for this line using
this information?

00:00:20.220 --> 00:00:21.270
Well, let's try it out.

00:00:21.270 --> 00:00:26.050
So any point on this line,
or any x, y on this line,

00:00:26.050 --> 00:00:28.110
would have to
satisfy the condition

00:00:28.110 --> 00:00:30.170
that the slope
between that point--

00:00:30.170 --> 00:00:32.020
so let's say that this
is some point x, y.

00:00:32.020 --> 00:00:34.120
It's an arbitrary
point on the line--

00:00:34.120 --> 00:00:36.190
the fact that it's
on the line tells us

00:00:36.190 --> 00:00:42.540
that the slope between a, b
and x, y must be equal to m.

00:00:42.540 --> 00:00:45.950
So let's use that knowledge to
actually construct an equation.

00:00:45.950 --> 00:00:49.890
So what is the slope
between a, b and x, y?

00:00:49.890 --> 00:00:52.090
Well, our change in y--
remember slope is just

00:00:52.090 --> 00:00:53.400
change in y over change in x.

00:00:53.400 --> 00:00:54.150
Let me write that.

00:00:54.150 --> 00:00:58.900
Slope is equal to change
in y over change in x.

00:00:58.900 --> 00:01:01.930
This little triangle character,
that's the Greek letter Delta,

00:01:01.930 --> 00:01:03.900
shorthand for change in.

00:01:03.900 --> 00:01:05.510
Our change in y--
well let's see.

00:01:05.510 --> 00:01:11.890
If we start at y is equal
to b, and if we end up

00:01:11.890 --> 00:01:15.290
at y equals this arbitrary
y right over here,

00:01:15.290 --> 00:01:18.497
this change in y right over
here is going to be y minus b.

00:01:21.060 --> 00:01:23.130
Let me write it in
those same colors.

00:01:23.130 --> 00:01:28.170
So this is going to be y
minus my little orange b.

00:01:28.170 --> 00:01:30.860
And that's going to be
over our change in x.

00:01:30.860 --> 00:01:33.360
And the exact same logic--
we start at x equals a.

00:01:33.360 --> 00:01:35.410
We finish at x
equals this arbitrary

00:01:35.410 --> 00:01:37.460
x, whatever x we
happen to be at.

00:01:37.460 --> 00:01:40.370
So that change in x is
going to be that ending

00:01:40.370 --> 00:01:44.000
point minus our starting
point-- minus a.

00:01:44.000 --> 00:01:47.320
And we know this is the slope
between these two points.

00:01:47.320 --> 00:01:50.310
That's the slope between
any two points on this line.

00:01:50.310 --> 00:01:52.150
And that's going
to be equal to m.

00:01:52.150 --> 00:01:54.950
So this is going
to be equal to m.

00:01:54.950 --> 00:01:57.960
And so what we've
already done here

00:01:57.960 --> 00:02:00.959
is actually create an equation
that describes this line.

00:02:00.959 --> 00:02:03.250
It might not be in any form
that you're used to seeing,

00:02:03.250 --> 00:02:05.160
but this is an
equation that describes

00:02:05.160 --> 00:02:08.130
any x, y that satisfies this
equation right over here

00:02:08.130 --> 00:02:11.670
will be on the line because
any x, y that satisfies this,

00:02:11.670 --> 00:02:16.460
the slope between that x, y
and this point right over here,

00:02:16.460 --> 00:02:21.306
between the point a, b,
is going to be equal to m.

00:02:21.306 --> 00:02:24.450
So let's actually now
convert this into forms

00:02:24.450 --> 00:02:27.050
that we might
recognize more easily.

00:02:27.050 --> 00:02:29.620
So let me paste that.

00:02:29.620 --> 00:02:33.070
So to simplify this expression
a little bit, or at least

00:02:33.070 --> 00:02:35.660
to get rid of the x minus
a in the denominator, let's

00:02:35.660 --> 00:02:39.590
multiply both
sides by x minus a.

00:02:39.590 --> 00:02:44.535
So if we multiply both sides
by x minus a-- so x minus a

00:02:44.535 --> 00:02:49.622
on the left-hand side and
x minus a on the right.

00:02:53.290 --> 00:02:55.660
Let me put some
parentheses around it.

00:02:55.660 --> 00:02:58.360
So we're going to multiply
both sides by x minus a.

00:02:58.360 --> 00:03:01.160
The whole point of that is you
have x minus a divided by x

00:03:01.160 --> 00:03:04.478
minus a, which is just
going to be equal to 1.

00:03:04.478 --> 00:03:06.186
And then on the
right-hand side, you just

00:03:06.186 --> 00:03:07.490
have m times x minus a.

00:03:07.490 --> 00:03:08.970
So this whole thing
has simplified

00:03:08.970 --> 00:03:21.080
to y minus b is equal
to m times x minus a.

00:03:21.080 --> 00:03:24.180
And right here, this is
a form that people, that

00:03:24.180 --> 00:03:28.050
mathematicians, have
categorized as point-slope form.

00:03:28.050 --> 00:03:32.380
So this right over here
is the point-slope form

00:03:32.380 --> 00:03:34.670
of the equation that
describes this line.

00:03:34.670 --> 00:03:36.720
Now, why is it called
point-slope form?

00:03:36.720 --> 00:03:39.090
Well, it's very easy to
inspect this and say, OK.

00:03:39.090 --> 00:03:41.860
Well look, this is the
slope of the line in green.

00:03:41.860 --> 00:03:43.340
That's the slope of the line.

00:03:43.340 --> 00:03:45.850
And I can put the two points in.

00:03:45.850 --> 00:03:49.710
If the point a, b
is on this line,

00:03:49.710 --> 00:03:54.770
I'll have the slope times x
minus a is equal to y minus b.

00:03:54.770 --> 00:03:56.700
Now, let's see
why this is useful

00:03:56.700 --> 00:03:58.970
or why people like to
use this type of thing.

00:03:58.970 --> 00:04:01.440
Let's not use just a, b
and a slope of m anymore.

00:04:01.440 --> 00:04:03.850
Let's make this a little
bit more concrete.

00:04:03.850 --> 00:04:08.010
Let's say that someone tells you
that I'm dealing with some line

00:04:08.010 --> 00:04:15.634
where the slope is equal
to 2, and let's say

00:04:15.634 --> 00:04:17.300
it goes through the
point negative 7, 5.

00:04:21.990 --> 00:04:24.785
So very quickly, you
could use this information

00:04:24.785 --> 00:04:28.500
and your knowledge
of point-slope form

00:04:28.500 --> 00:04:30.090
to write this in this form.

00:04:30.090 --> 00:04:31.900
You would just say,
well, an equation that

00:04:31.900 --> 00:04:39.440
contains this point and has this
slope would be y minus b, which

00:04:39.440 --> 00:04:42.720
is 5-- y minus the
y-coordinate of the point

00:04:42.720 --> 00:04:53.760
that this line contains--
is equal to my slope times x

00:04:53.760 --> 00:04:57.730
minus the x-coordinate
that this line contains.

00:04:57.730 --> 00:05:00.230
So x minus negative 7.

00:05:00.230 --> 00:05:03.180
And just like that, we have
written an equation that

00:05:03.180 --> 00:05:05.300
has a slope of 2
and that contains

00:05:05.300 --> 00:05:07.140
this point right over here.

00:05:07.140 --> 00:05:09.800
And if we don't like the x minus
negative 7 right over here,

00:05:09.800 --> 00:05:12.000
we could obviously
rewrite that as x plus 7.

00:05:12.000 --> 00:05:14.084
But this is kind of the
purest point-slope form.

00:05:14.084 --> 00:05:15.750
If you want to simplify
it a little bit,

00:05:15.750 --> 00:05:22.370
you could write it as y minus
5 is equal to 2 times x plus 7.

00:05:22.370 --> 00:05:25.090
And if you want to see that this
is just one way of expressing

00:05:25.090 --> 00:05:27.010
the equation of this line--
there are many others,

00:05:27.010 --> 00:05:28.718
and the one that we're
most familiar with

00:05:28.718 --> 00:05:31.130
is y-intercept form--
this can easily

00:05:31.130 --> 00:05:32.840
be converted to
y-intercept form.

00:05:32.840 --> 00:05:34.760
To do that, we just have
to distribute this 2.

00:05:34.760 --> 00:05:40.250
So we get y minus 5 is equal
to 2 times x plus 2 times 7, so

00:05:40.250 --> 00:05:41.430
that's equal to 14.

00:05:41.430 --> 00:05:43.680
And then we can get rid of
this negative 5 on the left

00:05:43.680 --> 00:05:47.290
by adding 5 to both
sides of this equation.

00:05:47.290 --> 00:05:50.750
And then we are left with,
on the left-hand side, y

00:05:50.750 --> 00:05:54.680
and, on the right-hand
side, 2x plus 19.

00:05:54.680 --> 00:05:57.520
So this right over here
is slope-intercept form.

00:05:57.520 --> 00:06:00.150
You have your slope
and your y-intercept.

00:06:00.150 --> 00:06:03.050
So this is slope-intercept form.

00:06:03.050 --> 00:06:06.920
And this right up here
is point-slope form.


Converting from point slope to slope intercept form

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:02.760
We're asked to convert
the following equation

00:00:02.760 --> 00:00:05.957
from point-slope to
slope-intercept form.

00:00:05.957 --> 00:00:07.790
And so this is the
equation right over here.

00:00:07.790 --> 00:00:09.460
Let me get my little
scratch pad out here.

00:00:09.460 --> 00:00:10.876
Let me see if I
can remember this.

00:00:10.876 --> 00:00:15.930
It's y plus 13 is equal
to 4 times x plus 2.

00:00:15.930 --> 00:00:17.060
So let's write that down.

00:00:17.060 --> 00:00:24.680
We have y plus 13 is
equal to 4 times x plus 2.

00:00:24.680 --> 00:00:26.640
So this is in point-slope form.

00:00:26.640 --> 00:00:28.990
And just inspecting
this, I can tell you

00:00:28.990 --> 00:00:33.480
that the slope of the line
described by this equation

00:00:33.480 --> 00:00:34.640
is 4.

00:00:34.640 --> 00:00:38.915
I could tell you that the
point x equals negative 2,

00:00:38.915 --> 00:00:42.960
y equals negative
13 is on this line.

00:00:42.960 --> 00:00:45.510
So we know the point
x equals negative 2, y

00:00:45.510 --> 00:00:47.895
equals negative 13
is on this line.

00:00:47.895 --> 00:00:50.020
Because when you put
something in point-slope form,

00:00:50.020 --> 00:00:51.728
it's of the form where
this is the slope,

00:00:51.728 --> 00:00:54.040
and this would be x
minus one of the points.

00:00:54.040 --> 00:00:57.070
So x minus negative
2 is positive 2,

00:00:57.070 --> 00:01:01.440
and y minus the
y-coordinate of that point,

00:01:01.440 --> 00:01:04.610
so y minus negative
13 is positive 13.

00:01:04.610 --> 00:01:07.890
But let's actually convert
this into slope-intercept form.

00:01:07.890 --> 00:01:10.730
And that of course,
is a form y is

00:01:10.730 --> 00:01:14.830
equal to mx plus b,
where m is the slope

00:01:14.830 --> 00:01:15.921
and b is the y-intercept.

00:01:15.921 --> 00:01:17.420
And we're going to
do this just with

00:01:17.420 --> 00:01:18.586
some algebraic manipulation.

00:01:18.586 --> 00:01:20.240
And you can imagine,
once we do this,

00:01:20.240 --> 00:01:23.010
the m should still be this 4.

00:01:23.010 --> 00:01:24.190
So let's do that.

00:01:24.190 --> 00:01:25.950
So the first thing
I'm going to do

00:01:25.950 --> 00:01:29.970
is distribute this
4 over the x plus 2.

00:01:29.970 --> 00:01:34.706
So this is going to be equal
to 4x plus 4 times 2 is 8.

00:01:34.706 --> 00:01:38.050
4 times x plus 4 times
2, so it's 4x plus 8.

00:01:38.050 --> 00:01:41.730
And we still, of course,
have our y plus 13 here.

00:01:41.730 --> 00:01:44.420
And now we're pretty
close to getting it

00:01:44.420 --> 00:01:45.940
into slope-intercept form.

00:01:45.940 --> 00:01:48.027
We just have to subtract
13 from both sides.

00:01:48.027 --> 00:01:49.610
And I'm going to
have it of the form y

00:01:49.610 --> 00:01:52.370
is equal to something
times x plus something.

00:01:52.370 --> 00:01:53.200
So let's do that.

00:01:53.200 --> 00:01:56.730
Let's subtract 13
from both sides.

00:01:56.730 --> 00:02:00.070
And we will get on the
left-hand side just y.

00:02:00.070 --> 00:02:03.660
And on the right-hand side,
we are going to have 4x.

00:02:03.660 --> 00:02:08.130
And then 8 minus
13 is negative 5.

00:02:08.130 --> 00:02:10.930
So we get y equals 4x minus 5.

00:02:10.930 --> 00:02:15.090
And let's just type that in and
see if we got the right result.

00:02:15.090 --> 00:02:17.140
So y is equal to 4x minus 5.

00:02:17.140 --> 00:02:22.000
So y is equal to 4 times x, and
they just have a positive here.

00:02:22.000 --> 00:02:24.550
So I would write
plus negative 5,

00:02:24.550 --> 00:02:26.790
the same thing as 4x minus 5.

00:02:26.790 --> 00:02:29.820
So let's check our answer,
and we got it right.


How to find the domain and the range of a function given its graph (example)

WEBVTT
Kind: captions
Language: en

00:00:00.880 --> 00:00:03.210
The function f of x is graphed.

00:00:03.210 --> 00:00:05.030
What is its domain?

00:00:05.030 --> 00:00:06.810
So the way it's graphed
right over here,

00:00:06.810 --> 00:00:08.920
we could assume that this
is the entire function

00:00:08.920 --> 00:00:11.040
definition for f of x.

00:00:11.040 --> 00:00:12.670
So for example, if
we say, well, what

00:00:12.670 --> 00:00:14.770
does f of x equal when x
is equal to negative 9?

00:00:14.770 --> 00:00:15.689
Well, we go up here.

00:00:15.689 --> 00:00:16.980
We don't see it's graphed here.

00:00:16.980 --> 00:00:20.140
It's not defined for x
equals negative 9 or x equals

00:00:20.140 --> 00:00:22.410
negative 8 and 1/2 or
x equals negative 8.

00:00:22.410 --> 00:00:24.990
It's not defined for
any of these values.

00:00:24.990 --> 00:00:29.410
It only starts getting defined
at x equals negative 6.

00:00:29.410 --> 00:00:32.950
At x equals negative 6,
f of x is equal to 5.

00:00:32.950 --> 00:00:34.880
And then it keeps
getting defined. f of x

00:00:34.880 --> 00:00:37.470
is defined for x all
the way from x equals

00:00:37.470 --> 00:00:40.550
negative 6 all the
way to x equals 7.

00:00:40.550 --> 00:00:43.190
When x equals 7, f
of x is equal to 5.

00:00:43.190 --> 00:00:45.650
You can take any x value
between negative 6,

00:00:45.650 --> 00:00:48.280
including negative
6, and positive 7,

00:00:48.280 --> 00:00:50.300
including positive
7, and you just

00:00:50.300 --> 00:00:52.200
have to see-- you
just have to move up

00:00:52.200 --> 00:00:54.060
above that number,
wherever you are,

00:00:54.060 --> 00:00:59.590
to find out what the value of
the function is at that point.

00:00:59.590 --> 00:01:03.440
So the domain of this
function definition?

00:01:03.440 --> 00:01:06.445
Well, f of x is
defined for any x that

00:01:06.445 --> 00:01:08.650
is greater than or
equal to negative 6.

00:01:08.650 --> 00:01:11.830
Or we could say negative 6
is less than or equal to x,

00:01:11.830 --> 00:01:13.760
which is less than
or equal to 7.

00:01:13.760 --> 00:01:16.810
If x satisfies this
condition right over here,

00:01:16.810 --> 00:01:19.400
the function is defined.

00:01:19.400 --> 00:01:20.460
So that's its domain.

00:01:20.460 --> 00:01:21.680
So let's check our answer.

00:01:21.680 --> 00:01:24.550
Let's do a few more of these.

00:01:24.550 --> 00:01:26.180
The function f of x is graphed.

00:01:26.180 --> 00:01:27.440
What is its domain?

00:01:27.440 --> 00:01:29.340
Well, exact similar argument.

00:01:29.340 --> 00:01:33.240
This function is not defined
for x is negative 9, negative 8,

00:01:33.240 --> 00:01:36.470
all the way down or all the way
up I should say to negative 1.

00:01:36.470 --> 00:01:39.030
At negative 1, it
starts getting defined.

00:01:39.030 --> 00:01:42.300
f of negative 1 is negative 5.

00:01:42.300 --> 00:01:45.410
So it's defined for negative
1 is less than or equal to x.

00:01:45.410 --> 00:01:48.390
And it's defined all the
way up to x equals 7,

00:01:48.390 --> 00:01:50.190
including x equals 7.

00:01:50.190 --> 00:01:51.850
So this right over
here, negative 1

00:01:51.850 --> 00:01:54.230
is less than or equal to x
is less than or equal to 7,

00:01:54.230 --> 00:01:56.240
the function is
defined for any x that

00:01:56.240 --> 00:02:00.350
satisfies this double
inequality right over here.

00:02:00.350 --> 00:02:02.750
Let's do a few more.

00:02:02.750 --> 00:02:04.370
The function f of x is graphed.

00:02:04.370 --> 00:02:06.350
What is its range?

00:02:06.350 --> 00:02:08.330
So now, we're not
thinking about the x's

00:02:08.330 --> 00:02:10.490
for which this
function is defined.

00:02:10.490 --> 00:02:14.400
We're thinking about
the set of y values.

00:02:14.400 --> 00:02:17.894
Where do all of the
y values fall into?

00:02:17.894 --> 00:02:18.560
Well, let's see.

00:02:18.560 --> 00:02:22.540
The lowest possible y value
or the lowest possible value

00:02:22.540 --> 00:02:25.770
of f of x that we get
here looks like it's 0.

00:02:25.770 --> 00:02:29.230
The function never goes below 0.

00:02:29.230 --> 00:02:32.300
So f of x-- so 0 is less
than or equal to f of x.

00:02:32.300 --> 00:02:36.070
It does equal 0 right over
here. f of negative 4 is 0.

00:02:36.070 --> 00:02:38.770
And then the highest y
value or the highest value

00:02:38.770 --> 00:02:42.780
that f of x obtains in this
function definition is 8.

00:02:42.780 --> 00:02:44.430
f of 7 is 8.

00:02:44.430 --> 00:02:47.150
It never gets above 8, but it
does equal 8 right over here

00:02:47.150 --> 00:02:48.820
when x is equal to 7.

00:02:48.820 --> 00:02:55.010
So 0 is less than f of x, which
is less than or equal to 8.

00:02:55.010 --> 00:02:57.227
So that's its range.

00:02:57.227 --> 00:02:58.060
Let's do a few more.

00:02:58.060 --> 00:02:59.740
This is kind of fun.

00:02:59.740 --> 00:03:01.310
The function f of x is graphed.

00:03:01.310 --> 00:03:02.400
What is its domain?

00:03:02.400 --> 00:03:06.020
So once again, this function
is defined for negative 2.

00:03:06.020 --> 00:03:09.660
Negative 2 is less than or
equal to x, which is less than

00:03:09.660 --> 00:03:11.190
or equal to 5.

00:03:11.190 --> 00:03:14.340
If you give me an x anywhere
in between negative 2 and 5,

00:03:14.340 --> 00:03:17.090
I can look at this graph to see
where the function is defined.

00:03:17.090 --> 00:03:19.130
f of negative 2 is negative 4.

00:03:19.130 --> 00:03:22.074
f of negative 1 is negative 3.

00:03:22.074 --> 00:03:23.490
So on and so forth,
and I can even

00:03:23.490 --> 00:03:26.010
pick the values in
between these integers.

00:03:26.010 --> 00:03:29.380
So negative 2 is less than or
equal to x, which is less than

00:03:29.380 --> 00:03:32.120
or equal to 5.


Testing solutions to linear equations

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:03.140
Some ordered pairs
for a linear function

00:00:03.140 --> 00:00:06.340
of x are given in
the table below.

00:00:06.340 --> 00:00:08.650
Which equation was used
to generate this table?

00:00:08.650 --> 00:00:10.275
So they give us a
bunch of data points.

00:00:10.275 --> 00:00:11.960
When x is 4, y is negative 8.

00:00:11.960 --> 00:00:15.310
When x is 7, y is negative
20, so on and so forth.

00:00:15.310 --> 00:00:18.530
And these points
have been generated

00:00:18.530 --> 00:00:21.170
by one of these equations
right over here,

00:00:21.170 --> 00:00:23.120
so let's just see which
of these equations

00:00:23.120 --> 00:00:25.531
actually could have generated
all of these points.

00:00:25.531 --> 00:00:27.530
So let's go to this first
point right over here,

00:00:27.530 --> 00:00:30.850
4 comma negative 8, and let's
go equation by equation.

00:00:30.850 --> 00:00:36.177
If x is 4 here, negative 2
times 4 would be negative 8.

00:00:36.177 --> 00:00:37.260
Let me just write it down.

00:00:40.020 --> 00:00:42.310
We'd multiply negative 2
times 4 to get negative 8,

00:00:42.310 --> 00:00:45.380
and then I would add
1 to get negative 7.

00:00:45.380 --> 00:00:46.570
I would get negative 7.

00:00:46.570 --> 00:00:49.940
So for this equation,
when x is 4,

00:00:49.940 --> 00:00:53.620
y would be negative
7, not negative 8.

00:00:53.620 --> 00:00:55.980
So this equation
definitely did not

00:00:55.980 --> 00:00:58.312
generate this top
point right over here.

00:00:58.312 --> 00:00:59.520
So we could just rule it out.

00:00:59.520 --> 00:01:01.150
Whatever equation
should have been

00:01:01.150 --> 00:01:02.670
able to generate
for any given x,

00:01:02.670 --> 00:01:05.140
for any of these x's, should
generate this y right over

00:01:05.140 --> 00:01:06.080
here.

00:01:06.080 --> 00:01:08.030
Now let's go to
this next equation.

00:01:08.030 --> 00:01:11.070
y is equal to
negative 2x plus 0.

00:01:11.070 --> 00:01:14.430
So when x is 4, y would
be negative 2 times

00:01:14.430 --> 00:01:15.760
4, which is negative 8.

00:01:15.760 --> 00:01:18.200
So this second
equation seems to be

00:01:18.200 --> 00:01:20.620
capable of generating
this first set of points.

00:01:20.620 --> 00:01:22.700
When x is 4, y is negative 8.

00:01:22.700 --> 00:01:24.640
But let's see if it
works for this second one

00:01:24.640 --> 00:01:25.810
right over here.

00:01:25.810 --> 00:01:28.760
When x is 7, you
would have negative 2

00:01:28.760 --> 00:01:34.100
times 7, which is negative
14, but here, when x is 7,

00:01:34.100 --> 00:01:35.830
y is negative 20.

00:01:35.830 --> 00:01:39.720
So this one, we can rule
out, because when x is 7,

00:01:39.720 --> 00:01:42.361
y does not equal negative
20 like this point right

00:01:42.361 --> 00:01:42.860
over here.

00:01:42.860 --> 00:01:44.870
So we'll rule that
second one out.

00:01:44.870 --> 00:01:46.260
Now let's look at
this third one.

00:01:46.260 --> 00:01:48.600
y is equal to
negative 4x plus 8.

00:01:48.600 --> 00:01:50.830
Well, we could go to
this first point again.

00:01:50.830 --> 00:01:53.800
When x is 4, let's think about
what happens when x is 4.

00:01:53.800 --> 00:01:56.390
So when x is 4,
you have negative 4

00:01:56.390 --> 00:02:03.830
times 4, which is negative 16,
plus 8, which is negative 8.

00:02:03.830 --> 00:02:07.460
So it seems to be able to
generate this first point.

00:02:07.460 --> 00:02:11.490
When x is 4, y is negative 8 for
this equation right over here.

00:02:11.490 --> 00:02:15.590
Now let's see what
happens when x is 7.

00:02:15.590 --> 00:02:20.560
So negative 4 times
7 is negative 28.

00:02:20.560 --> 00:02:24.010
Negative 28 plus
8 is negative 20,

00:02:24.010 --> 00:02:26.870
so this candidate is
starting to look pretty good.

00:02:26.870 --> 00:02:27.650
It's pretty good.

00:02:27.650 --> 00:02:30.780
It satisfies these two
points, and frankly,

00:02:30.780 --> 00:02:33.720
for any of these linear
functions, if it satisfies

00:02:33.720 --> 00:02:36.710
any of these two points, if some
linear function generated all

00:02:36.710 --> 00:02:41.130
of these, if some function
can generate any two of them,

00:02:41.130 --> 00:02:44.521
it will satisfy all of these,
because two points define

00:02:44.521 --> 00:02:45.020
a line.

00:02:45.020 --> 00:02:47.200
But we can verify
it right over here.

00:02:47.200 --> 00:02:49.650
So let's see what
happens when x equals 8.

00:02:49.650 --> 00:02:57.700
Negative 4 times 8 is negative
32, plus 8 is negative 24.

00:02:57.700 --> 00:03:01.970
So that satisfies so that when x
is 8, y does equal negative 24.

00:03:01.970 --> 00:03:05.340
And then finally, let's
look at that last point.

00:03:05.340 --> 00:03:10.070
When x is 9, negative 4
times 9 is negative 36,

00:03:10.070 --> 00:03:12.950
plus 8 is equal to negative 28.

00:03:12.950 --> 00:03:15.690
And that's exactly what
we see here in this table.

00:03:15.690 --> 00:03:18.170
So we don't even have
to look any further.

00:03:18.170 --> 00:03:22.120
Choice 3 satisfies
the conditions

00:03:22.120 --> 00:03:24.640
that when x is any
one of these things,

00:03:24.640 --> 00:03:28.270
the corresponding y that's
defined by this equation

00:03:28.270 --> 00:03:31.360
is going to be when x
is 4, y is negative 8.

00:03:31.360 --> 00:03:35.760
When x is 7, y is negative
20, so on and so forth.


Area of shaded region made from equilateral triangles

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:02.780
Let's say I have an
equilateral triangle where

00:00:02.780 --> 00:00:06.472
the length of each side is 14.

00:00:06.472 --> 00:00:07.930
So this is an
equilateral triangle.

00:00:07.930 --> 00:00:10.120
All of the sides have length 14.

00:00:10.120 --> 00:00:14.290
And inside that I have another
equilateral triangle-- right

00:00:14.290 --> 00:00:17.980
over here-- where the length
of each of the sides is 4.

00:00:17.980 --> 00:00:20.720
Now what I'm curious
about, is the area

00:00:20.720 --> 00:00:23.730
of the region-- let me color
this in a different color--

00:00:23.730 --> 00:00:26.980
is the area of the region that
I'm shading in right here.

00:00:26.980 --> 00:00:30.640
So it's the area inside the
larger equilateral triangle,

00:00:30.640 --> 00:00:34.850
but outside of the smaller
equilateral triangle.

00:00:34.850 --> 00:00:36.600
So let's think about
how we would do this.

00:00:36.600 --> 00:00:39.960
And I encourage you to pause
this and try this on your own.

00:00:39.960 --> 00:00:44.870
Well the shaded
area is going to be

00:00:44.870 --> 00:00:47.980
equal to the large
equilateral triangle's

00:00:47.980 --> 00:00:58.025
area minus the area of the
small equilateral triangle.

00:01:04.294 --> 00:01:05.710
So we just have
to figure out what

00:01:05.710 --> 00:01:08.650
the area of each of these
equilateral triangles are.

00:01:08.650 --> 00:01:12.020
And so to do it, we remember
that the area of a triangle

00:01:12.020 --> 00:01:17.582
is equal to 1/2
base times height.

00:01:17.582 --> 00:01:19.040
But how do we figure
out the height

00:01:19.040 --> 00:01:20.360
of an equilateral triangle?

00:01:20.360 --> 00:01:23.310
So for example, if I have
an equilateral triangle

00:01:23.310 --> 00:01:25.490
like this-- let me
draw it big so I

00:01:25.490 --> 00:01:28.240
can dissect it little
bit-- so I have

00:01:28.240 --> 00:01:30.950
an equilateral
triangle like this.

00:01:30.950 --> 00:01:33.920
The length of each
of the sides are s.

00:01:33.920 --> 00:01:35.950
And I always have to
re-prove it for myself.

00:01:35.950 --> 00:01:38.220
Just because I always
forget the formula.

00:01:38.220 --> 00:01:42.560
We remember that the angles are
60 degrees, 60 degrees, and 60

00:01:42.560 --> 00:01:43.060
degrees.

00:01:43.060 --> 00:01:44.190
They're all equal.

00:01:44.190 --> 00:01:46.870
And what I like to do to
find out the area of this,

00:01:46.870 --> 00:01:49.910
in order to figure out the
height, is I drop an altitude.

00:01:49.910 --> 00:01:52.640
So I drop an altitude
just like here,

00:01:52.640 --> 00:01:54.220
and it would split
the side in two.

00:01:54.220 --> 00:01:55.886
I know it doesn't
look like it perfectly

00:01:55.886 --> 00:01:57.860
because I didn't
draw it to scale.

00:01:57.860 --> 00:01:59.312
But it would split it in two.

00:01:59.312 --> 00:02:01.597
It would form
these right angles.

00:02:01.597 --> 00:02:03.180
And what's neat about
this is I've now

00:02:03.180 --> 00:02:09.690
split my equilateral triangle
into two 30-60-90 triangles.

00:02:13.170 --> 00:02:14.920
And that's useful
because I know the ratio

00:02:14.920 --> 00:02:17.260
of the sides of a
30-60-90 triangle.

00:02:17.260 --> 00:02:20.120
If this is s and I've
just split this in two,

00:02:20.120 --> 00:02:25.760
this orange section right
over here is going to be s/2.

00:02:25.760 --> 00:02:28.380
This is also going to
be s/2 right over here.

00:02:28.380 --> 00:02:30.260
They obviously add up to s.

00:02:30.260 --> 00:02:32.920
And then we know from
30-60-90 triangles,

00:02:32.920 --> 00:02:35.550
that the side opposite
the 60-degree side

00:02:35.550 --> 00:02:38.530
is square root of 3
times the shortest side.

00:02:38.530 --> 00:02:42.430
So this altitude
right over here,

00:02:42.430 --> 00:02:44.420
is going to be
square root of 3s/2.

00:02:47.920 --> 00:02:50.210
And now we can figure
out a generalized formula

00:02:50.210 --> 00:02:52.360
for the area of an
equilateral triangle.

00:02:52.360 --> 00:02:55.350
It's going to be equal
to 1/2 times the base.

00:02:55.350 --> 00:02:57.560
Well the base is going to be s.

00:02:57.560 --> 00:03:00.220
So the base is s.

00:03:00.220 --> 00:03:02.700
And the height is square
root of 3s over 2.

00:03:08.670 --> 00:03:10.990
And so this will
simplify to, let's

00:03:10.990 --> 00:03:14.210
see we have in the numerator
we have the square root of 3s

00:03:14.210 --> 00:03:17.360
squared over four.

00:03:17.360 --> 00:03:19.090
And now we can apply
this to figure out

00:03:19.090 --> 00:03:21.361
the areas of each
of these triangles.

00:03:21.361 --> 00:03:23.610
So this is going to be equal
to the area of the larger

00:03:23.610 --> 00:03:30.200
triangle, is going to be square
root of 3/4 times 14 squared.

00:03:30.200 --> 00:03:31.700
And the area of the
smaller triangle

00:03:31.700 --> 00:03:36.810
is going to be square root
of 3/4 times 4 squared.

00:03:36.810 --> 00:03:39.450
And let's see, we could factor
out a square root of 3/4.

00:03:39.450 --> 00:03:45.160
So this is going to be equal
to square root of 3/4 times

00:03:45.160 --> 00:03:52.030
14 squared minus 4 squared.

00:03:52.030 --> 00:03:54.710
Which of course we
know is to be, is 16.

00:03:54.710 --> 00:03:56.740
But now let's actually
evaluate this,

00:03:56.740 --> 00:03:58.250
to actually get a number here.

00:03:58.250 --> 00:04:00.780
And I could try to
simplify it by hand.

00:04:00.780 --> 00:04:03.330
But instead let me actually
just get my-- Actually,

00:04:03.330 --> 00:04:06.150
let's just simplify
it by hand first.

00:04:06.150 --> 00:04:08.704
So in case you haven't
memorized your 14 times tables,

00:04:08.704 --> 00:04:09.870
we could just work that out.

00:04:09.870 --> 00:04:12.910
14 times 14, 4 times 14.

00:04:12.910 --> 00:04:14.820
4 times 4 is 16.

00:04:14.820 --> 00:04:16.470
And then carry the one.

00:04:16.470 --> 00:04:20.230
4 times 4 is 4
plus 1, so it's 56.

00:04:20.230 --> 00:04:23.180
You put a 0 right there, because
we're multiplying by 10 now.

00:04:23.180 --> 00:04:26.790
10 times 14 is 140.

00:04:26.790 --> 00:04:28.810
And so this is 196.

00:04:28.810 --> 00:04:41.060
So this is equal to square
root of 3/4 times 196 minus 16,

00:04:41.060 --> 00:04:43.610
which is equal to 180.

00:04:43.610 --> 00:04:46.960
So this right over
here is equal to 180.

00:04:46.960 --> 00:04:49.370
And 180 is divisible
by 4, so this

00:04:49.370 --> 00:04:52.480
is going to be equal to
the square root of 3 times,

00:04:52.480 --> 00:04:57.360
let's see, 180 divided
by 4 is going to be 45.

00:04:57.360 --> 00:05:01.510
So it's going to be
45 square roots of 3.

00:05:01.510 --> 00:05:03.320
Did I do that right?

00:05:03.320 --> 00:05:06.640
180 divided by 4 times
45 is 160 plus 20.

00:05:06.640 --> 00:05:08.110
Is exactly 180.

00:05:08.110 --> 00:05:09.730
So it's 45 square roots of 3.

00:05:09.730 --> 00:05:12.450
And if I wanted to get
an approximate answer

00:05:12.450 --> 00:05:13.990
as a decimal.

00:05:13.990 --> 00:05:15.950
So let me get my calculator out.

00:05:15.950 --> 00:05:23.420
So let's go 45 times the square
root of 3, would get us to 77.

00:05:23.420 --> 00:05:25.820
If I want to round to the
nearest hundredth, say,

00:05:25.820 --> 00:05:27.540
that's 77.94.

00:05:27.540 --> 00:05:33.450
So this is approximately
equal to 77.94 square units.

00:05:33.450 --> 00:05:36.220
The area of the shaded region.


Fractions cut and copy 1 exercise

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:04.660
The starting block below
is 16/9 units long.

00:00:04.660 --> 00:00:08.330
So this right here
is 16/9 units long.

00:00:08.330 --> 00:00:11.660
Use the tools to the right
to convert the starting block

00:00:11.660 --> 00:00:13.750
into the goal block
that is 1 unit long.

00:00:13.750 --> 00:00:16.129
So this right over
here is 1 unit long.

00:00:16.129 --> 00:00:18.340
You can see your progress
as the current block.

00:00:18.340 --> 00:00:21.140
So how do we convert
this into 1 unit?

00:00:21.140 --> 00:00:23.860
Well, one we can start
cutting it into some pieces.

00:00:23.860 --> 00:00:25.200
Let's just see what happens.

00:00:25.200 --> 00:00:27.270
So if we cut it into
2 pieces, how big

00:00:27.270 --> 00:00:29.710
is this red section now?

00:00:29.710 --> 00:00:33.280
So if we cut this in
half, a 1/2 times 16/9

00:00:33.280 --> 00:00:35.910
this is now going to be 8/9.

00:00:35.910 --> 00:00:44.280
If we cut it into 4 pieces,
then this is going to be 4/9.

00:00:44.280 --> 00:00:47.000
If we cut it into,
looks like we're

00:00:47.000 --> 00:00:49.240
going to have to go all
the way to 16 pieces

00:00:49.240 --> 00:00:51.190
to get it into 1/9.

00:00:51.190 --> 00:00:54.840
So let's cut this
into 16 pieces.

00:00:54.840 --> 00:00:57.600
So now if you take
16/9 and if you

00:00:57.600 --> 00:01:00.190
were to cut it into
16 pieces, this

00:01:00.190 --> 00:01:03.120
is going to be 1/9
right over here.

00:01:03.120 --> 00:01:05.519
So if I have 1/9 and I
want to make it a whole,

00:01:05.519 --> 00:01:06.420
how many do I need?

00:01:06.420 --> 00:01:08.100
Well, I need 9 of them.

00:01:08.100 --> 00:01:15.570
So let's do 2/9, 3/9, 4/9,
5/9, 6/9, 7/9, 8/9, and 9/9.

00:01:15.570 --> 00:01:17.810
And let's check our answer.

00:01:17.810 --> 00:01:20.450
Let's do a couple more of these.

00:01:20.450 --> 00:01:25.140
So the starting block right over
here is 1 and 9/12 units long.

00:01:25.140 --> 00:01:27.980
And we're going to use these
tools to convert the block

00:01:27.980 --> 00:01:33.870
into a goal block that
is 1 and 8/12 units long.

00:01:33.870 --> 00:01:36.590
You can see your progress
as the current block.

00:01:36.590 --> 00:01:38.810
So the most obvious
thing to me is

00:01:38.810 --> 00:01:41.870
if I were to just convert the
starting block into 12ths.

00:01:41.870 --> 00:01:44.620
And then I could replicate, I
could copy those 12ths enough

00:01:44.620 --> 00:01:46.681
to get to 1 and 8/12.

00:01:46.681 --> 00:01:48.680
So how many times do I
have to cut this in order

00:01:48.680 --> 00:01:50.830
to get it into just 1/12?

00:01:50.830 --> 00:01:53.570
So 1 and 9/12 is the
same thing as 12/12

00:01:53.570 --> 00:01:57.320
plus 9/12, which is 21/12.

00:01:57.320 --> 00:02:01.370
So if I cut this into
21 blocks, each of them

00:02:01.370 --> 00:02:03.170
are going to be 1/12.

00:02:03.170 --> 00:02:05.210
Because this is 21/12
right over here.

00:02:05.210 --> 00:02:06.440
So let me cut it into 21.

00:02:10.729 --> 00:02:13.230
So right now, I cut
this into 21 blocks.

00:02:13.230 --> 00:02:14.520
This was 21/12.

00:02:14.520 --> 00:02:19.120
If I cut it, this is 1/21 of
that, so it's going to be 1/12.

00:02:19.120 --> 00:02:20.620
So this is 1/12.

00:02:20.620 --> 00:02:22.580
And now I just have to
copy it enough times

00:02:22.580 --> 00:02:25.030
to get to 1 and 8/12.

00:02:25.030 --> 00:02:30.040
And 1 and 8/12 is the same thing
as 12/12 plus 8/12 is 20/12.

00:02:30.040 --> 00:02:31.131
So this is 1/12.

00:02:31.131 --> 00:02:33.630
If I get 20 of these, then it's
going to be the same length.

00:02:33.630 --> 00:02:34.380
So let me do that.

00:02:34.380 --> 00:02:41.950
So it's 1, 2, 3, 4, 5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15, 16,

00:02:41.950 --> 00:02:44.710
17, 18, 19, 20.

00:02:44.710 --> 00:02:46.670
And we are spot on.

00:02:46.670 --> 00:02:47.996
Let's check our answer.

00:02:47.996 --> 00:02:49.120
Let's do one more of these.

00:02:49.120 --> 00:02:50.170
This is fun.

00:02:50.170 --> 00:02:53.260
So this starting block
is 11/8 units long.

00:02:53.260 --> 00:02:56.350
We want to convert into a
block that is 1 unit long.

00:02:56.350 --> 00:02:59.440
So if I can convert the starting
block into eighths, and the way

00:02:59.440 --> 00:03:01.110
that I could make
this into 1/8 is

00:03:01.110 --> 00:03:03.350
by essentially
dividing it by 11.

00:03:03.350 --> 00:03:06.440
Now I have 11/8 and I divide by
11, I'm only going to have 1/8.

00:03:06.440 --> 00:03:10.030
So let me cut into 11 pieces.

00:03:10.030 --> 00:03:11.570
So now this is 1/8.

00:03:11.570 --> 00:03:15.430
And then if I take this
1/8 and if I replicate it

00:03:15.430 --> 00:03:18.560
so I have 8 of them, 8/8
is going to be equal to 1.

00:03:18.560 --> 00:03:19.880
So let me do that.

00:03:19.880 --> 00:03:23.990
2, 3, 4, 5, 6, 7, 8.

00:03:23.990 --> 00:03:26.180
And we got it right.