Domain and range of a function given a formula

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:02.310
Determine the domain and range

00:00:02.310 --> 00:00:08.070
of the function f of x is equal to
3x squared plus 6x minus 2.

00:00:08.070 --> 00:00:12.845
So, the domain of the function is:
what is a set of all of the valid inputs,

00:00:12.845 --> 00:00:16.209
or all of the valid x values
for this function?

00:00:16.209 --> 00:00:19.760
And, I can take any real number,
square it, multiply it by 3,

00:00:19.760 --> 00:00:23.150
then add 6 times that real number
and then subtract 2 from it.

00:00:23.150 --> 00:00:27.690
So essentially any number if we're talking
about reals when we talk about any number.

00:00:27.690 --> 00:00:30.674
So, the domain, the set
of valid inputs, the set of

00:00:30.674 --> 00:00:35.280
inputs over which this function
is defined, is all real numbers.

00:00:35.280 --> 00:00:39.610
So, the domain here is
all real numbers.

00:00:44.260 --> 00:00:48.270
And, for those of you who might say, well,
you know, aren't all numbers real?

00:00:48.270 --> 00:00:50.960
You may or may not know that
there is a class of numbers,

00:00:50.960 --> 00:00:53.750
that are a little bit bizarre
when you first learn them,

00:00:53.750 --> 00:00:56.020
called imaginary numbers
and complex numbers.

00:00:56.020 --> 00:00:57.790
But, I won't go into that right now.

00:00:57.790 --> 00:01:00.397
But, most of the traditional
numbers that you know of,

00:01:00.397 --> 00:01:02.520
they are part of
the set of real numbers.

00:01:02.520 --> 00:01:05.340
It's pretty much
everything but complex numbers.

00:01:05.340 --> 00:01:07.868
So, you take any real number
and you put it here,

00:01:07.868 --> 00:01:11.440
you can square it, multiply it by 3,
then add 6 times it and subtract 2.

00:01:11.440 --> 00:01:14.672
Now, the range, at least the way we've
been thinking about it

00:01:14.672 --> 00:01:16.520
in this series of videos--

00:01:16.520 --> 00:01:21.790
The range is set of possible,
outputs of this function.

00:01:21.790 --> 00:01:26.770
Or if we said y equals f of x
on a graph, it's a set

00:01:26.770 --> 00:01:29.030
of all the possible y values.

00:01:29.030 --> 00:01:31.914
And, to get a flavor for this,
I'm going to try to graph

00:01:31.914 --> 00:01:33.450
this function right over here.

00:01:33.450 --> 00:01:35.438
And, if you're familiar with quadratics--

00:01:35.438 --> 00:01:38.770
and that's what this function is
right over here, it is a quadratic--

00:01:38.770 --> 00:01:41.300
you might already know
that it has a parabolic shape.

00:01:41.300 --> 00:01:43.610
And, so its shape might look
something like this.

00:01:43.610 --> 00:01:46.530
And, actually this one will
look like this, it's upward opening.

00:01:46.530 --> 00:01:48.540
But other parabolas
have shapes like that.

00:01:48.540 --> 00:01:51.201
And, you see when a parabola
has a shape like this,

00:01:51.201 --> 00:01:55.127
it won't take on any values
below its vertex when it's upward opening,

00:01:55.127 --> 00:01:59.960
and it won't take on any values above
its vertex when it is downward opening.

00:01:59.960 --> 00:02:04.450
So, let's see if we can graph
this and maybe get a sense of its vertex.

00:02:04.450 --> 00:02:07.420
There are ways
to calculate the vertex exactly,

00:02:07.420 --> 00:02:10.050
but let's see how we can
think about this problem.

00:02:10.050 --> 00:02:12.840
So, I'm gonna try some x and y values.

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There's other ways to directly compute the
vertex.

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Negative b over 2a is the formula for it.

00:02:18.850 --> 00:02:19.990
It comes straight out of the quadratic

00:02:19.990 --> 00:02:22.300
formula, which you get from completing the
square.

00:02:22.300 --> 00:02:27.140
Lets try some x values and lets see what f
of x is equal to.

00:02:27.140 --> 00:02:30.680
So, let's try, well this the values we've
been trying the last two videos.

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What happens when x is equal to negative
two?

00:02:33.610 --> 00:02:39.004
Then f of x is 3 times negative 2 squared,
which is 4, plus 6 times

00:02:39.004 --> 00:02:44.870
negative 2, which is 6 times negative 2,
so it's minus 12 minus 2.

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So, this is 12 minus 12 minus 2.

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So, it's equal to negative 2.

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Now, what happens when x is equal to
negative 1?

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So, this is going to be 3 times negative 1
squared, which is just 1, minus, or I

00:02:59.400 --> 00:03:02.434
should say plus 6 times negative 1 which
is

00:03:02.434 --> 00:03:05.690
minus 6 and then minus 2, and then minus
2.

00:03:07.450 --> 00:03:10.402
So, this is 3 minus 6 is negative 3 minus

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2 is equal negative 5, and that actually
is the vertex.

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And, you know the formula for the vertex,
once again, is negative b over 2 a.

00:03:19.130 --> 00:03:20.550
So, negative b.

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That's the coefficient on this term right
over here.

00:03:22.780 --> 00:03:26.860
It's negative 6 over 2 times this one
right over here, 2 times 3.

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2 times 3, this is equal to negative 1.

00:03:30.280 --> 00:03:34.070
So, that is the vertex, but let's just
keep on going right over here.

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So, what happens when x is equal to 0?

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These first two terms are 0, you're just
left with a negative 2.

00:03:40.170 --> 00:03:42.740
When x is equal to positive 1.

00:03:42.740 --> 00:03:44.555
And, this is where you can see that this

00:03:44.555 --> 00:03:46.920
is the vertex, and you start seeing the
symmetry.

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If you go one above the vertex, f of x is
equal to negative 2.

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If you go one x value below the vertex, or
below the x

00:03:53.790 --> 00:03:56.910
value of the vertex, f of x is equal to
negative 2 again.

00:03:56.910 --> 00:03:58.780
But, let's just keep going.

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We could try, let's do one more point over
here.

00:04:01.170 --> 00:04:04.250
So, we have, we could try, x is equal to
1.

00:04:04.250 --> 00:04:09.840
When x is equal to 1, you have 3 times one
squared which is 1.

00:04:09.840 --> 00:04:14.070
So, 3 times 1 plus 6 times 1, which is
just 6, minus 2.

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So, this is 9 minus 2 it's equal to 7.

00:04:16.800 --> 00:04:19.491
And, that I think is enough points to give

00:04:19.491 --> 00:04:22.730
us a scaffold of what this graph will look
like.

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What the graph of the function would look
like.

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So, it would look something like this.

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I do my best to draw it.

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So, this is a x equals negative 2.

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We draw the whole axis.

00:04:35.270 --> 00:04:40.495
This is x is equal to negative 1, this is
x is equal to, this is x is

00:04:40.495 --> 00:04:45.720
equal to 0 and then this is x is equal to
1 right over there and then

00:04:45.720 --> 00:04:50.980
when x is equal to, we go from negative 2
all the way to positive.

00:04:50.980 --> 00:04:54.530
Or, we should go from negative 5 all the
way to positive 7.

00:04:54.530 --> 00:05:00.210
So, let's say this is negative 1,2,3,4,5.

00:05:00.210 --> 00:05:03.450
That's negative five over there on the y
axis,

00:05:03.450 --> 00:05:06.760
y axis and then it will go to positive 7.

00:05:06.760 --> 00:05:12.330
One, two, three, four, five, six, seven.

00:05:12.330 --> 00:05:15.210
I could keep going, this is in the y, and
we're going

00:05:15.210 --> 00:05:18.540
to set y equal to whatever our output of
the function is.

00:05:18.540 --> 00:05:20.310
Y is equal to f of x.

00:05:20.310 --> 00:05:22.110
And this is one right here.

00:05:22.110 --> 00:05:23.210
So, lets plot the points.

00:05:23.210 --> 00:05:26.480
You have the point negative 2, negative 2.

00:05:26.480 --> 00:05:28.710
When x is negative 2, this is the x axis.

00:05:28.710 --> 00:05:32.400
When x is negative 2, y is negative 2.

00:05:32.400 --> 00:05:35.590
Y is negative 2 so that is that right over
3.

00:05:36.600 --> 00:05:41.500
So, that is the point, that is the point
negative 2, negative 2.

00:05:41.500 --> 00:05:42.258
Fair enough?

00:05:42.258 --> 00:05:47.130
Then, we have this point that we have this
pink or purplish color.

00:05:47.130 --> 00:05:49.900
Negative, when x is negative 1, f of x is
negative 5.

00:05:49.900 --> 00:05:53.010
When x is negative 1, f of x is negative
5.

00:05:53.010 --> 00:05:54.590
And, we already said that this is the
vertex.

00:05:54.590 --> 00:05:56.960
And, you'll see the symmetry around it in
a second.

00:05:56.960 --> 00:05:59.914
So, this is the point negative 1, negative
5.

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And then, with the point 0, negative 2.

00:06:03.550 --> 00:06:09.234
0, negative when x is a 0, y is negative
2, for f' of x is negative

00:06:09.234 --> 00:06:14.624
2 or f of 0 is negative 2, so this is the
point 0, negative 2,

00:06:14.624 --> 00:06:20.050
and then finally when x is equal to 1 and
f of 1 is 7, f of 1 is 7.

00:06:20.050 --> 00:06:24.895
So, that's right there it's a point 1, 7
and it gives us a

00:06:24.895 --> 00:06:30.740
scaffold for what this parabola, what this
curve will look like.

00:06:30.740 --> 00:06:34.560
So, I'll try my best to draw it
respectably.

00:06:34.560 --> 00:06:37.563
So, it would look something, something
like

00:06:37.563 --> 00:06:40.830
that, and keep on going in that direction.

00:06:40.830 --> 00:06:42.320
Keep on going in that direction.

00:06:42.320 --> 00:06:45.000
But, I think you see the symmetry around
the vertex.

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That if you were to.

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If you were to put a line right over here,
the

00:06:47.870 --> 00:06:50.190
two sides are kind of the mirror images of
each other.

00:06:50.190 --> 00:06:53.143
There, you can flip them over, and that's
how we know it's the vertex.

00:06:53.143 --> 00:06:56.839
And, that's how we also know, because this
is an upward opening parabola, I

00:06:56.839 --> 00:07:01.008
mean, there is formulas for vertex, and
there are multiple ways of calculating it.

00:07:01.008 --> 00:07:03.483
But, since it's an upward opening
parabola, where

00:07:03.483 --> 00:07:05.652
the vertex is going to be, the minimum
point.

00:07:05.652 --> 00:07:08.960
This is the minimum value that the
parabola will take on.

00:07:08.960 --> 00:07:12.210
So, going back to the original question,
this is all for trying to figure out

00:07:12.210 --> 00:07:16.150
the range, the set of y values, the set of
outputs that this function can generate.

00:07:16.150 --> 00:07:19.920
You see that the function, it can get as
low as negative 5.

00:07:19.920 --> 00:07:22.560
It got all the way down to negative 5
right at the vertex.

00:07:22.560 --> 00:07:25.699
But, as you go to the right, as x values
increase to

00:07:25.699 --> 00:07:30.160
the right or decrease to the left, then
the parabola goes upwards.

00:07:30.160 --> 00:07:32.440
So, the parabola can never give you
values--

00:07:32.440 --> 00:07:36.040
f of x is never going to be less than
negative 5.

00:07:36.040 --> 00:07:37.920
So, our domain,
but it can take on all the vaues.

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It can keep on increasing forever as x
gets

00:07:39.810 --> 00:07:42.750
larger, x gets smaller farther away from
the vertex.

00:07:42.750 --> 00:07:46.432
So, our range, so we already said our
domain is all real numbers.

00:07:46.432 --> 00:07:52.463
Our range, the possible y values
is all real

00:07:52.463 --> 00:07:58.866
numbers greater than or equal to negative
5.

00:07:58.866 --> 00:08:00.234
It can take on the value of any

00:08:00.234 --> 00:08:02.646
real number greater than or equal to
negative 5.

00:08:02.646 --> 00:08:05.007
Nothing less than negative 5.


Graphing Radical Functions

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:03.080
We're asked to graph the
function f of x is equal

00:00:03.080 --> 00:00:06.956
to the principal square
root of 0.5x minus 1.

00:00:06.956 --> 00:00:08.330
And to graph it,
we're just going

00:00:08.330 --> 00:00:10.500
to sample some points,
some x values that

00:00:10.500 --> 00:00:13.540
are within the domain of
this function definition.

00:00:13.540 --> 00:00:16.740
And see what the function is
equal to for those x values.

00:00:16.740 --> 00:00:18.230
And then we could
connect the dots

00:00:18.230 --> 00:00:20.270
and see what the
curve might look like.

00:00:20.270 --> 00:00:22.670
So let me do a
little table here.

00:00:22.670 --> 00:00:25.400
And so I'm going to
pick some x values.

00:00:25.400 --> 00:00:28.780
And I'm going to figure out
what the f of x values are,

00:00:28.780 --> 00:00:29.792
given those x values.

00:00:29.792 --> 00:00:31.250
Actually, I'm going
to do something

00:00:31.250 --> 00:00:33.710
in between because I want
to have nice, clean numbers.

00:00:33.710 --> 00:00:36.700
I want to pick x values that
are nice, clean integers.

00:00:36.700 --> 00:00:39.284
And I want to get f of x values
that are nice, clean integers.

00:00:39.284 --> 00:00:41.449
So I'm going to be a little
bit more careful in what

00:00:41.449 --> 00:00:42.280
x values I pick.

00:00:42.280 --> 00:00:46.980
I want to pick x values so that
this part of the expression,

00:00:46.980 --> 00:00:48.700
this part of the
function definition,

00:00:48.700 --> 00:00:51.900
ends up being an integer.

00:00:51.900 --> 00:00:53.720
And if this needs
to be an integer,

00:00:53.720 --> 00:00:56.730
then what's underneath
the radical, 0.5x,

00:00:56.730 --> 00:00:58.949
that needs to be
a perfect square.

00:00:58.949 --> 00:01:01.240
So actually, let me make
another column right over here

00:01:01.240 --> 00:01:06.240
just to make our math
a little bit cleaner.

00:01:06.240 --> 00:01:08.120
You could pick just
arbitrary x points.

00:01:08.120 --> 00:01:11.090
You could pick x to be 1,
2, 3, just go like that.

00:01:11.090 --> 00:01:13.399
But on some of those, you
might get strange radicals

00:01:13.399 --> 00:01:14.940
over here, which
would be hard for us

00:01:14.940 --> 00:01:17.320
to graph without a calculator.

00:01:17.320 --> 00:01:18.320
So let's try.

00:01:18.320 --> 00:01:23.080
So let me write over
here what 0.5x is,

00:01:23.080 --> 00:01:25.230
and I'm doing this just
so I can pick this.

00:01:25.230 --> 00:01:28.640
We want this thing right over
here to be perfect squares

00:01:28.640 --> 00:01:31.590
because if 0.5x is
a perfect square,

00:01:31.590 --> 00:01:36.070
then the square root of that
is going to be an integer.

00:01:36.070 --> 00:01:38.160
So let's say that 0.5--
let's think about all

00:01:38.160 --> 00:01:40.420
of the perfect squares.

00:01:40.420 --> 00:01:42.460
Well, we could start with 0.

00:01:42.460 --> 00:01:43.930
Then, we could go to 1.

00:01:43.930 --> 00:01:47.060
Then, we could go to-- Let's
see, 2 is in a perfect square,

00:01:47.060 --> 00:01:48.390
3 is in a perfect square.

00:01:48.390 --> 00:01:49.848
I'm just thinking
of all the values

00:01:49.848 --> 00:01:51.990
that if I took the square
root, I get an integer.

00:01:51.990 --> 00:01:53.470
And then I go to 4.

00:01:53.470 --> 00:01:55.550
Then, I go to 9.

00:01:55.550 --> 00:01:57.870
Then, I go to 16.

00:01:57.870 --> 00:02:00.154
Then, I go to 25.

00:02:00.154 --> 00:02:02.320
These are definitely the
first five perfect squares.

00:02:02.320 --> 00:02:04.861
And, obviously, 0 you take the
principal square root of that,

00:02:04.861 --> 00:02:05.470
you get 0.

00:02:05.470 --> 00:02:08.330
So these would be
good values for 0.5x.

00:02:08.330 --> 00:02:11.440
But if those are what 0.5x
is equal to, what is x going

00:02:11.440 --> 00:02:12.350
to be?

00:02:12.350 --> 00:02:15.769
So if 0.5x is 0, then x
is also going to be 0.

00:02:15.769 --> 00:02:18.310
You could solve it separately
in an equation right over here.

00:02:18.310 --> 00:02:22.090
You could say 0.5x
is equal to 0.

00:02:22.090 --> 00:02:25.590
x is equal to 0 divided by 0.5.

00:02:25.590 --> 00:02:29.550
Just divide both sides by 0.5,
and you get that equal to 0.

00:02:29.550 --> 00:02:31.560
So in general, to get
what the x value is,

00:02:31.560 --> 00:02:34.966
we just have to take this
number and divide it by 0.5.

00:02:34.966 --> 00:02:37.090
Or, another way to think
about it, take this number

00:02:37.090 --> 00:02:39.080
and multiply it by 2.

00:02:39.080 --> 00:02:43.680
So when 0.5x is 1,
that means x is 2.

00:02:43.680 --> 00:02:45.826
0.5 time 2 is 1.

00:02:45.826 --> 00:02:47.735
0.5 times 8 is 4.

00:02:47.735 --> 00:02:50.313
0.5 times 18 is 9.

00:02:50.313 --> 00:02:54.601
0.5 times 32 is 16.

00:02:54.601 --> 00:02:57.942
0.5 times 50 is 25.

00:02:57.942 --> 00:02:59.650
So these are the x's
we're going to pick.

00:02:59.650 --> 00:03:01.233
And, once again, I
picked them just so

00:03:01.233 --> 00:03:03.220
that this part of the
expression, this part

00:03:03.220 --> 00:03:06.040
of the function definition,
comes out cleanly.

00:03:06.040 --> 00:03:09.130
And now let's figure out what
f of x is at those values.

00:03:09.130 --> 00:03:16.950
So f of 0 is going to be equal
to the square root of 0.5

00:03:16.950 --> 00:03:18.360
times 0.

00:03:18.360 --> 00:03:21.160
Let me make my radical
a little bit bigger.

00:03:21.160 --> 00:03:24.660
0.5 times 0 minus 1.

00:03:24.660 --> 00:03:27.795
Well, this part right
over here is just 0.

00:03:27.795 --> 00:03:32.210
0 minus 1 is just going
to be equal to negative 1.

00:03:32.210 --> 00:03:34.970
Now let's try it
when x is equal to 2.

00:03:34.970 --> 00:03:40.960
f of 2 is going to be equal to
the square root of 0.5 times 2

00:03:40.960 --> 00:03:42.060
minus 1.

00:03:42.060 --> 00:03:44.990
We already know that this part
right over here is equal to 1.

00:03:44.990 --> 00:03:49.000
So you have 1 minus
1 is equal to 0.

00:03:49.000 --> 00:03:50.590
Now let's do the next one.

00:03:50.590 --> 00:03:54.050
f of 4-- I've already
used that blue color.

00:03:54.050 --> 00:03:55.710
I'll use green.

00:03:55.710 --> 00:04:04.520
f of 4 is equal to
0.5 times 4 minus 1.

00:04:04.520 --> 00:04:07.180
So this is we already
know-- Oh, sorry.

00:04:07.180 --> 00:04:09.440
0.5 times 8.

00:04:09.440 --> 00:04:10.440
We're not taking f of 4.

00:04:10.440 --> 00:04:12.080
We're taking f of 8.

00:04:12.080 --> 00:04:13.240
Let me be careful.

00:04:13.240 --> 00:04:14.710
This is what the x values are.

00:04:14.710 --> 00:04:23.780
So f of 8 is equal to
0.5 times 8 minus 1.

00:04:23.780 --> 00:04:26.820
This part right over
here we'll evaluate too.

00:04:26.820 --> 00:04:29.380
So 0.5 times 8 is 4.

00:04:29.380 --> 00:04:31.840
You take the square root,
the principal root of 4.

00:04:31.840 --> 00:04:32.410
You get 2.

00:04:32.410 --> 00:04:37.480
2 minus 1 is equal to 1.

00:04:37.480 --> 00:04:40.810
Then over here, we'd
want to do this in pink.

00:04:40.810 --> 00:04:51.150
f of 18 is square root
of 0.5 times 18 minus 1.

00:04:51.150 --> 00:04:52.746
0.5 times 18 is 9.

00:04:52.746 --> 00:04:54.120
You take the square
root of that.

00:04:54.120 --> 00:04:54.940
You get 3.

00:04:54.940 --> 00:04:57.740
3 minus 1 is equal to 2.

00:04:57.740 --> 00:04:59.810
We'll do a couple more.

00:04:59.810 --> 00:05:10.370
f of 32 is equal to
0.5 times 32 minus 1.

00:05:10.370 --> 00:05:12.840
So it's the square
root of 16 minus 1.

00:05:12.840 --> 00:05:14.450
So that's 4 minus 1.

00:05:14.450 --> 00:05:17.470
The principal root of
16, which is 4 minus 1,

00:05:17.470 --> 00:05:20.170
which is equal to 3.

00:05:20.170 --> 00:05:23.740
And then, finally,
do one more of these.

00:05:23.740 --> 00:05:30.320
f of 50 is equal to the
square root of 0.5 times 50.

00:05:30.320 --> 00:05:32.040
0.5 times 50 is
this 25, and we're

00:05:32.040 --> 00:05:33.290
going to take the square root.

00:05:33.290 --> 00:05:35.850
So this whole thing is
going to evaluate to 5.

00:05:35.850 --> 00:05:41.250
5 minus 1 is equal to 4.

00:05:41.250 --> 00:05:43.960
So we've taken some
x's is in our domain.

00:05:43.960 --> 00:05:45.690
And it's very important
to realize here

00:05:45.690 --> 00:05:47.570
that our domain is restricted.

00:05:47.570 --> 00:05:49.282
It is not all real numbers.

00:05:49.282 --> 00:05:50.740
And the reason why
it is restricted

00:05:50.740 --> 00:05:53.384
is if we're going to
take the principal root,

00:05:53.384 --> 00:05:54.800
right-- for the
sake of this if we

00:05:54.800 --> 00:05:56.810
don't go into things
like imaginary numbers.

00:05:56.810 --> 00:05:59.320
And don't worry about that if
you don't know what those are.

00:05:59.320 --> 00:06:02.210
We have to assume that what's
underneath the radical sign

00:06:02.210 --> 00:06:02.900
is positive.

00:06:02.900 --> 00:06:06.200
And in order for that to be
positive x has to be positive.

00:06:06.200 --> 00:06:09.720
So in order for 0.5x to be
greater than or equal to 0,

00:06:09.720 --> 00:06:12.640
which is what the principal
root is defined over,

00:06:12.640 --> 00:06:15.920
for our purposes in this video,
then if you divide both sides

00:06:15.920 --> 00:06:18.930
of this inequality by 0.5,
you say that x has got to be

00:06:18.930 --> 00:06:20.450
greater than or equal to 0.

00:06:20.450 --> 00:06:23.060
So we sampled some
points in the domain.

00:06:23.060 --> 00:06:25.730
The domain is all
real numbers, all

00:06:25.730 --> 00:06:28.460
x's or all real numbers
greater than or equal to 0.

00:06:28.460 --> 00:06:29.800
We sampled some points there.

00:06:29.800 --> 00:06:31.820
We've strategically
picked them so that we'd

00:06:31.820 --> 00:06:33.770
get nice, clean
answers for f of x.

00:06:33.770 --> 00:06:35.550
But now that we
figured out what f of x

00:06:35.550 --> 00:06:38.870
is at each of these points,
we're ready to plot.

00:06:38.870 --> 00:06:41.630
So let me draw some axes here.

00:06:44.350 --> 00:06:46.660
So that's our x-axis.

00:06:46.660 --> 00:06:48.726
And let's make this our y-axis.

00:06:52.710 --> 00:06:55.820
And I'll say y is
equal to f of x.

00:06:55.820 --> 00:06:59.404
So whatever f of x
we get for a given x,

00:06:59.404 --> 00:07:01.320
we're going to set that
equal to y and plot it

00:07:01.320 --> 00:07:02.740
on the vertical axis.

00:07:02.740 --> 00:07:05.530
This is our x-axis
right over there.

00:07:05.530 --> 00:07:06.760
And let's go point by point.

00:07:06.760 --> 00:07:08.593
Well, actually, first
let me draw the scale.

00:07:08.593 --> 00:07:10.780
So the x-axis goes
all the way to 50.

00:07:10.780 --> 00:07:11.820
So let me divide this.

00:07:11.820 --> 00:07:13.640
So let's say that this is 10.

00:07:13.640 --> 00:07:15.250
This is 20.

00:07:15.250 --> 00:07:16.780
This is 30.

00:07:16.780 --> 00:07:18.500
This is 40.

00:07:18.500 --> 00:07:20.690
And this is 50.

00:07:20.690 --> 00:07:22.950
And then our y-axis,
we're going to have

00:07:22.950 --> 00:07:24.590
to go between negative 1 and 4.

00:07:24.590 --> 00:07:26.960
So let's make this negative 1.

00:07:26.960 --> 00:07:33.360
Let's make that 1, 2, 3, and 4.

00:07:33.360 --> 00:07:35.380
And now we're ready to
plot all the points.

00:07:35.380 --> 00:07:43.360
So when x is equal to 0, y,
or f of x, is negative 1.

00:07:43.360 --> 00:07:45.600
So you have the
point 0, negative 1.

00:07:45.600 --> 00:07:49.470
0, which is right over
there, 0, negative 1.

00:07:49.470 --> 00:07:52.990
So let me actually-- this
right over here is 0.

00:07:52.990 --> 00:07:55.780
But we want to plot the
point, 0, negative 1

00:07:55.780 --> 00:07:57.780
is going to sit
right over there.

00:07:57.780 --> 00:08:01.950
Let me give its coordinates
that is 0, negative 1.

00:08:01.950 --> 00:08:05.530
Then, we plot 2 comma 0.

00:08:05.530 --> 00:08:11.710
So 2 is going to be right
around-- So that is 5.

00:08:11.710 --> 00:08:13.520
2 is going to be
right around there.

00:08:13.520 --> 00:08:15.930
2 comma 0 is going
to be right there.

00:08:15.930 --> 00:08:20.950
So that point right over
there is 2 comma 0 in magenta.

00:08:20.950 --> 00:08:24.700
Now we go to 8.

00:08:24.700 --> 00:08:26.810
Let me switch to that
same green color.

00:08:26.810 --> 00:08:29.720
We go to 8 comma 1.

00:08:29.720 --> 00:08:31.890
8 is going to be
right around here.

00:08:31.890 --> 00:08:33.409
We go 1 in the
vertical direction,

00:08:33.409 --> 00:08:35.299
gets us right about there.

00:08:35.299 --> 00:08:38.320
That is the point 8 comma 1.

00:08:38.320 --> 00:08:43.270
Then, we go, in this pink
color, to 18 comma 2.

00:08:43.270 --> 00:08:46.320
When x is 18, f of x is 2.

00:08:46.320 --> 00:08:49.630
When x is 18-- 18 gets
us right about there.

00:08:49.630 --> 00:08:53.020
When x is 18, f of
x is equal to 2.

00:08:53.020 --> 00:08:55.300
So that puts us
right about there.

00:08:55.300 --> 00:08:58.200
So that is 18 comma 2.

00:08:58.200 --> 00:08:59.650
Then, what's our next color?

00:08:59.650 --> 00:09:00.835
We'll use orange then.

00:09:00.835 --> 00:09:06.610
Then, we have 32 comma
3 where f of 32 is 3.

00:09:06.610 --> 00:09:09.310
So 32 is right about there.

00:09:09.310 --> 00:09:11.530
And f of 32 is 3.

00:09:11.530 --> 00:09:14.460
So it gets us
right, right there.

00:09:14.460 --> 00:09:17.740
So that is the
point, 32 comma 3.

00:09:17.740 --> 00:09:22.780
And then we have the last sample
of our domain, which is x is 50

00:09:22.780 --> 00:09:25.900
and f of 50 is 4. x is 50.

00:09:25.900 --> 00:09:27.620
f of 50 is 4.

00:09:27.620 --> 00:09:30.340
So I'll put it right over there.

00:09:30.340 --> 00:09:34.720
So this is the point 50 comma 4.

00:09:34.720 --> 00:09:36.420
And now, we can
connect the dots to get

00:09:36.420 --> 00:09:37.524
the graph of the function.

00:09:37.524 --> 00:09:38.440
I'll do that in white.

00:09:38.440 --> 00:09:41.800
So if I connect the
dots, I get a curve

00:09:41.800 --> 00:09:44.600
that looks like half
of a sideways parabola.

00:09:44.600 --> 00:09:46.110
And that's essentially
what it is.

00:09:46.110 --> 00:09:49.637
So my curve will look
something like this.

00:09:49.637 --> 00:09:51.970
It's important to realize
that the graph of the function

00:09:51.970 --> 00:09:52.910
isn't just these point.

00:09:52.910 --> 00:09:54.326
It's actually any
of these points.

00:09:54.326 --> 00:09:56.710
If I were to pick an
arbitrary x values in domain--

00:09:56.710 --> 00:09:58.640
remember the x
values, the domain

00:09:58.640 --> 00:10:01.640
here is x is greater than or
equal to 0, or any real number

00:10:01.640 --> 00:10:03.210
greater than or equal to 0.

00:10:03.210 --> 00:10:06.620
So if I pick anything
on the x-axis that

00:10:06.620 --> 00:10:09.640
is greater than or equal
to 0 that's in the domain,

00:10:09.640 --> 00:10:11.420
and then if I
evaluated its function,

00:10:11.420 --> 00:10:13.250
it should sit on that curve.

00:10:13.250 --> 00:10:16.770
It should sit on that
curve that I just drew.


Graphing a  Basic Function

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.070
We're asked to
graph the function

00:00:02.070 --> 00:00:05.710
f of x is equal to 5x minus 4.

00:00:05.710 --> 00:00:08.540
And we'll do it by really
just sampling some points

00:00:08.540 --> 00:00:11.825
from the domain and seeing
what value our function takes.

00:00:11.825 --> 00:00:13.450
And then we'll just
graph those points.

00:00:13.450 --> 00:00:15.620
And then we'll connect the
dots and see what forms.

00:00:15.620 --> 00:00:17.040
And there's other
ways to do this,

00:00:17.040 --> 00:00:20.391
but this is the most simple
or the most basic way.

00:00:20.391 --> 00:00:22.390
So if you look at this
function right over here,

00:00:22.390 --> 00:00:24.320
it's actually defined
for any real number

00:00:24.320 --> 00:00:25.760
that you stick there for x.

00:00:25.760 --> 00:00:28.080
So when we talk about
sampling the domain,

00:00:28.080 --> 00:00:30.631
we can actually pick
any real number for x.

00:00:30.631 --> 00:00:32.464
I'm actually going to
pick real numbers that

00:00:32.464 --> 00:00:35.960
are fairly small in magnitude
so that they're easy to graph.

00:00:35.960 --> 00:00:39.080
So I don't have to plot 8
billion on a number line

00:00:39.080 --> 00:00:39.630
over here.

00:00:39.630 --> 00:00:43.740
So let's draw ourselves
some simple graph paper.

00:00:43.740 --> 00:00:45.780
Let's call that the x-axis.

00:00:45.780 --> 00:00:49.050
Let's call that the
y-axis, or, actually, I

00:00:49.050 --> 00:00:52.820
should call it the y
is equal to f of x.

00:00:52.820 --> 00:00:56.590
Whatever my x is, I input it
into this function definition.

00:00:56.590 --> 00:00:58.650
It'll tell me what
the function value is.

00:00:58.650 --> 00:00:59.640
It'll tell me f of x.

00:00:59.640 --> 00:01:02.050
And we're going to say
y is equal to that.

00:01:02.050 --> 00:01:05.000
Now let's draw a little table
here to try out some values.

00:01:05.000 --> 00:01:09.910
So let's say that this is x
values and then this is f of x.

00:01:09.910 --> 00:01:12.610
And I can even say this
is my y values, which

00:01:12.610 --> 00:01:17.010
are going to be equal to f of x.

00:01:17.010 --> 00:01:21.210
So let's start with-- I'm
going to try negative 2,

00:01:21.210 --> 00:01:27.390
negative 1, 0, 1, and 2.

00:01:27.390 --> 00:01:30.300
So let's try negative 2 first.

00:01:30.300 --> 00:01:33.810
if x is equal to negative
2, what is f of x?

00:01:33.810 --> 00:01:37.330
Well, f of x is going to be
5 times negative 2 minus 4.

00:01:37.330 --> 00:01:41.230
So it's 5 times
negative 2 minus 4,

00:01:41.230 --> 00:01:47.220
which is equal to negative 10
minus 4, which is negative 14.

00:01:47.220 --> 00:01:48.880
So let me plot that over here.

00:01:48.880 --> 00:01:50.550
And let me-- actually,
maybe I'm going

00:01:50.550 --> 00:01:52.950
to have a slightly
more scrunched up scale

00:01:52.950 --> 00:01:53.860
on the y-axis.

00:01:53.860 --> 00:01:56.130
And, actually, let me
give myself some space

00:01:56.130 --> 00:01:57.560
so that I can go further down.

00:01:57.560 --> 00:01:59.450
Let me clear this.

00:01:59.450 --> 00:02:01.350
Give myself a
little bit of space.

00:02:01.350 --> 00:02:03.975
So I have a feeling that I'll be
using a lot of negative values

00:02:03.975 --> 00:02:06.270
since I just got all the
way down to negative 14.

00:02:06.270 --> 00:02:08.150
So let me go further down here.

00:02:08.150 --> 00:02:10.590
And then let me mark off
some increments over here.

00:02:10.590 --> 00:02:15.580
So let's say that
this is negative 5.

00:02:15.580 --> 00:02:17.780
This is negative 10.

00:02:17.780 --> 00:02:21.650
And this is negative 15.

00:02:21.650 --> 00:02:24.590
And then this would
be positive 5.

00:02:24.590 --> 00:02:27.240
And then this would
be positive 10.

00:02:27.240 --> 00:02:39.710
So we have when x is negative 2,
f of x is equal to negative 14,

00:02:39.710 --> 00:02:41.070
which is right about there.

00:02:41.070 --> 00:02:42.580
So that is negative 14.

00:02:42.580 --> 00:02:45.720
So that right over there
is the point negative 2,

00:02:45.720 --> 00:02:49.400
negative 14, which we
got right from that.

00:02:49.400 --> 00:02:50.990
Now, let's try another point.

00:02:50.990 --> 00:02:52.861
Let's try negative 1.

00:02:52.861 --> 00:02:58.900
When x is negative 1, f of x
is 5 times negative 1 minus 4.

00:02:58.900 --> 00:03:04.160
5 times negative 1 is negative
5, minus 4 is negative 9.

00:03:04.160 --> 00:03:09.200
So when x is negative1,
f of x is negative 9.

00:03:09.200 --> 00:03:12.640
So negative 9 is
right about here.

00:03:12.640 --> 00:03:14.610
So this is negative 9.

00:03:14.610 --> 00:03:17.950
So this is the point
negative 1 negative 9

00:03:17.950 --> 00:03:21.910
that is on the graph of f of x.

00:03:21.910 --> 00:03:23.900
And now let's try 0.

00:03:23.900 --> 00:03:29.780
When x is 0, f of x is going
to be 5 times 0 minus 4.

00:03:29.780 --> 00:03:33.950
Well, that's just 0 minus 4,
which is equal to negative 4.

00:03:33.950 --> 00:03:37.850
So when x is 0, f of x
is equal to negative 4.

00:03:37.850 --> 00:03:39.660
So that's negative
4 right over there.

00:03:39.660 --> 00:03:42.326
And actually, our point is going
to be sitting right over there.

00:03:42.326 --> 00:03:43.835
That is 0, negative 4.

00:03:43.835 --> 00:03:45.510
I wrote it to the
left of the point.

00:03:45.510 --> 00:03:47.070
Let's keep going.

00:03:47.070 --> 00:03:49.290
Let's what happens
when x is equal to 1.

00:03:49.290 --> 00:03:55.490
When x is equal to 1, f of x
is equal to 5 times 1 minus 4.

00:03:55.490 --> 00:04:00.050
5 times 1 is 5 ,
minus 4 is equal to 1.

00:04:00.050 --> 00:04:02.190
So we get to the point 1,1.

00:04:02.190 --> 00:04:05.440
So 1 and then 1 is
right about there.

00:04:05.440 --> 00:04:07.840
So this is the point 1,1.

00:04:07.840 --> 00:04:09.750
And then let's
just try one more.

00:04:09.750 --> 00:04:13.590
Let's see what happens
when x is equal to 2.

00:04:13.590 --> 00:04:20.370
Then, f of x is 5 times 2
minus 4, which is 10 minus 4,

00:04:20.370 --> 00:04:24.090
which is equal to 6.

00:04:24.090 --> 00:04:28.630
So we have when x is equal
to 2, f of x is equal to 6.

00:04:28.630 --> 00:04:30.590
So it's right about there.

00:04:30.590 --> 00:04:32.340
That's 6 right there.

00:04:32.340 --> 00:04:36.140
So this is the point 2, 6.

00:04:36.140 --> 00:04:38.680
So once again, these are just
sample points from the domain.

00:04:38.680 --> 00:04:40.057
This isn't the entire domain.

00:04:40.057 --> 00:04:41.890
But it gives us enough
points to kind of see

00:04:41.890 --> 00:04:43.490
the general structure
of the graph.

00:04:43.490 --> 00:04:46.700
If we connect the dots,
we see a line forming.

00:04:46.700 --> 00:04:49.560
So If we connect the
dots-- let me draw.

00:04:49.560 --> 00:04:53.130
So if we connect the dots,
it looks something like this.

00:04:53.130 --> 00:04:56.240
And my shaky hand makes it
look a little bit curvy,

00:04:56.240 --> 00:04:58.510
but it should be an
absolute straight line.

00:04:58.510 --> 00:05:00.760
If I had this on
really good graph paper

00:05:00.760 --> 00:05:02.260
and I have had a
ruler, you will see

00:05:02.260 --> 00:05:03.468
that this is a straight line.

00:05:03.468 --> 00:05:05.704
And it would keep
going on and on and on.

00:05:05.704 --> 00:05:07.370
And I want to make
it clear that we just

00:05:07.370 --> 00:05:09.012
sampled points from the domain.

00:05:09.012 --> 00:05:11.470
But we could take any point in
the domain, which we already

00:05:11.470 --> 00:05:12.630
said is all real numbers.

00:05:12.630 --> 00:05:13.880
So if we took right over here.

00:05:13.880 --> 00:05:16.060
If we said 1.5 you could see.

00:05:16.060 --> 00:05:18.780
And if you graph
1.5, f of 1.5 it

00:05:18.780 --> 00:05:20.610
should sit on that
line right over there.

00:05:20.610 --> 00:05:25.230
If you graphed f of negative
0.5-- so this is negative 0.5.

00:05:25.230 --> 00:05:27.180
If you graphed f
of negative 0.5,

00:05:27.180 --> 00:05:29.785
it should sit on the
line right over there.

00:05:29.785 --> 00:05:30.410
But we're done.

00:05:30.410 --> 00:05:35.520
We have graphed the function
f of x is equal to 5x minus 4.


Evaluating Functions

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:05.830
Given g of x is equal to
8x plus 2, find g of zero,

00:00:05.830 --> 00:00:09.840
g of negative 5,
and g 2x plus 3.

00:00:09.840 --> 00:00:13.280
So all this is, when they say
g of x is equal to 8x plus 2,

00:00:13.280 --> 00:00:15.600
this is a function definition.

00:00:15.600 --> 00:00:17.970
They're saying
you give me any x,

00:00:17.970 --> 00:00:21.510
and you input it
into my function g,

00:00:21.510 --> 00:00:25.800
and my function g will output,
will associate with that number

00:00:25.800 --> 00:00:29.820
x, another number that
happens to be 8x plus 2.

00:00:29.820 --> 00:00:31.820
This is one way to
create an association.

00:00:31.820 --> 00:00:35.070
It's going to associate
if you were to put 1 here.

00:00:35.070 --> 00:00:38.070
If this x is 1, you're going
to do 8 times 1 plus 2.

00:00:38.070 --> 00:00:41.210
So you would
associate 1 through g.

00:00:41.210 --> 00:00:46.300
You'd associate 1 with 8
times 1 plus 2, which is 10.

00:00:46.300 --> 00:00:48.410
So it's just another way
of creating association

00:00:48.410 --> 00:00:50.310
that you could do with
sets of ordered pairs

00:00:50.310 --> 00:00:51.810
or you could do it graphically.

00:00:51.810 --> 00:00:53.680
But this is one of
the most typical ways

00:00:53.680 --> 00:00:54.960
you'll see a function defined.

00:00:54.960 --> 00:00:57.840
It says, look,
input an x over here

00:00:57.840 --> 00:01:00.584
and I'm going to
output 8x plus 2.

00:01:00.584 --> 00:01:02.250
So with that out of
the way, let's think

00:01:02.250 --> 00:01:04.550
about what the g of 0 is.

00:01:04.550 --> 00:01:06.540
g of 0 literally
just means we're

00:01:06.540 --> 00:01:08.990
inputting 0 into this function.

00:01:08.990 --> 00:01:11.040
And this function says,
whatever your inputting,

00:01:11.040 --> 00:01:13.950
I'm going to output
8 times that plus 2.

00:01:13.950 --> 00:01:16.940
So g of 0 is going to be--
let me write it this way

00:01:16.940 --> 00:01:19.770
so it becomes clear
what I'm doing. g of 0

00:01:19.770 --> 00:01:27.190
is going to be 8
times is 0 plus 2.

00:01:30.090 --> 00:01:32.420
So wherever we saw the
x, we replaced it what it

00:01:32.420 --> 00:01:33.860
with whatever we're inputting.

00:01:33.860 --> 00:01:37.020
That is our x in this
situation right over here.

00:01:37.020 --> 00:01:40.850
And so this is equal
to 8 times the 0 plus 2

00:01:40.850 --> 00:01:42.420
is just equal to 2.

00:01:42.420 --> 00:01:43.970
So g of 0 is equal to 2.

00:01:43.970 --> 00:01:47.700
Let me just write that
here. g of 0 is equal to 2.

00:01:47.700 --> 00:01:50.140
Now let's try g of negative 5.

00:01:50.140 --> 00:01:53.260
So let me write it this way.

00:01:53.260 --> 00:01:56.949
I'll write it, g of
negative 5 is equal

00:01:56.949 --> 00:01:58.990
to-- I'll try to do the
color-coding a little bit

00:01:58.990 --> 00:02:00.090
better this time.

00:02:00.090 --> 00:02:04.530
It's going to be equal
to 8 times our x, which

00:02:04.530 --> 00:02:06.630
in this case, is negative 5.

00:02:06.630 --> 00:02:09.190
8 times negative 5 plus 2.

00:02:12.170 --> 00:02:15.110
We're inputting x-- We're
inputting a negative 5.

00:02:15.110 --> 00:02:17.900
And we're going to output
8 times negative 5 plus 2.

00:02:17.900 --> 00:02:21.110
Wherever we saw the x, we just
replace it with a negative 5.

00:02:21.110 --> 00:02:23.950
And this is just
equal to 8 times

00:02:23.950 --> 00:02:31.430
negative 5 is negative 40 plus
2 is equal to negative 38.

00:02:31.430 --> 00:02:33.770
So I'll write this
over here on the right.

00:02:33.770 --> 00:02:40.920
We have g of negative 5
is equal to negative 38.

00:02:40.920 --> 00:02:43.440
Now, this last one is a
little bit interesting.

00:02:43.440 --> 00:02:44.440
So we just did this one.

00:02:44.440 --> 00:02:48.467
Now, let's do g of 2x plus 3.

00:02:48.467 --> 00:02:50.050
So now they're not
inputting a number.

00:02:50.050 --> 00:02:51.870
They're inputting an expression.

00:02:51.870 --> 00:02:54.400
Now, what we have to do is
everywhere where we see x here,

00:02:54.400 --> 00:02:56.960
we're going to replace
it with that expression.

00:02:56.960 --> 00:02:59.870
If I did g of a circle--
let me write this down.

00:02:59.870 --> 00:03:06.600
If I did g of an orange
circle right over here.

00:03:06.600 --> 00:03:10.220
By this function
definition, if I'm

00:03:10.220 --> 00:03:11.970
in putting an orange
circle, I should

00:03:11.970 --> 00:03:17.794
output an orange circle 8
times the orange circle,

00:03:17.794 --> 00:03:19.210
which doesn't make
a lot of sense.

00:03:19.210 --> 00:03:22.170
But hopefully you get the idea
that whatever you're inputting,

00:03:22.170 --> 00:03:24.690
you're just going to multiply
it times 8 and add two.

00:03:24.690 --> 00:03:28.710
So if we input 2x plus 3, we're
going to output this times 8

00:03:28.710 --> 00:03:29.300
plus 2.

00:03:29.300 --> 00:03:32.710
Wherever we see an x, we replace
it with this right over here.

00:03:32.710 --> 00:03:34.050
So let's try that out.

00:03:34.050 --> 00:03:46.560
So now we have g of 2x plus 3
is going to be equal to 8 times

00:03:46.560 --> 00:03:48.590
whatever we're
inputting, eight times

00:03:48.590 --> 00:03:51.052
whatever inputting plus 2.

00:03:51.052 --> 00:03:53.010
If this was an orange,
we'd put an orange here.

00:03:53.010 --> 00:03:55.093
If this is a negative 5,
we put a negative 5 here.

00:03:55.093 --> 00:03:56.860
If this is a 0, we
put a zero here.

00:03:56.860 --> 00:04:01.320
In this case, we're inputting
2x plus 3 into our function.

00:04:01.320 --> 00:04:04.660
So we put 2x plus 3 where
x is right over there.

00:04:04.660 --> 00:04:06.370
So 2x plus 3.

00:04:06.370 --> 00:04:09.740
Whatever we input, this function
says multiply by 8 and add 2.

00:04:09.740 --> 00:04:12.420
We're inputting 2x plus
3, multiply it by 8,

00:04:12.420 --> 00:04:13.690
and then add 2.

00:04:13.690 --> 00:04:18.950
And so this is going to be equal
to-- we just distribute the 8.

00:04:18.950 --> 00:04:22.740
And you get 16x plus 24.

00:04:22.740 --> 00:04:27.540
And then you still have
this plus 2 out over here

00:04:27.540 --> 00:04:33.140
So this is going to be
equal to 16x plus these two.

00:04:33.140 --> 00:04:35.182
You add them up, you get 26.

00:04:35.182 --> 00:04:36.640
So we can write
over here, and I'll

00:04:36.640 --> 00:04:38.580
do that same color so
it can be consistent.

00:04:38.580 --> 00:04:46.195
We have g of 2x plus 3
is equal to 16x plus 26.

00:04:46.195 --> 00:04:46.820
And we're done.

00:04:46.820 --> 00:04:48.270
I know it's going
to be confusing

00:04:48.270 --> 00:04:50.130
because you're inputting--
you're replacing

00:04:50.130 --> 00:04:52.661
x with something
that has x in it,

00:04:52.661 --> 00:04:54.160
but you really
should just view this

00:04:54.160 --> 00:04:57.390
as if this was an orange ball
or a star or even if this had

00:04:57.390 --> 00:04:57.890
y's is it.

00:04:57.890 --> 00:04:59.960
Wherever you see an
x, you replace it

00:04:59.960 --> 00:05:02.590
with whatever you
are inputting it.

00:05:02.590 --> 00:05:07.500
In every instance, we
replace the x with 2x plus 3.


Relations and functions

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:04.070
Is the relation given by the
set of ordered pairs shown

00:00:04.070 --> 00:00:05.930
below a function?

00:00:05.930 --> 00:00:08.400
So before we even attempt
to do this problem,

00:00:08.400 --> 00:00:11.210
right here, let's just remind
ourselves what a relation is

00:00:11.210 --> 00:00:14.770
and what type of relations
can be functions.

00:00:14.770 --> 00:00:17.770
So in a relation, you
have a set of numbers

00:00:17.770 --> 00:00:20.720
that you can kind of view as
the input into the relation.

00:00:20.720 --> 00:00:22.350
We call that the domain.

00:00:22.350 --> 00:00:24.240
You can view them as
the set of numbers

00:00:24.240 --> 00:00:26.490
over which that
relation is defined.

00:00:26.490 --> 00:00:27.920
And then you have
a set of numbers

00:00:27.920 --> 00:00:30.620
that you can view as the
output of the relation,

00:00:30.620 --> 00:00:33.330
or what the numbers that can
be associated with anything

00:00:33.330 --> 00:00:36.230
in domain, and we
call that the range.

00:00:36.230 --> 00:00:38.130
And it's a fairly
straightforward idea.

00:00:38.130 --> 00:00:42.360
So for example, let's say that
the number 1 is in the domain,

00:00:42.360 --> 00:00:47.240
and that we associate the
number 1 with the number 2

00:00:47.240 --> 00:00:48.847
in the range.

00:00:48.847 --> 00:00:50.430
So in this type of
notation, you would

00:00:50.430 --> 00:00:57.340
say that the relation
has 1 comma 2

00:00:57.340 --> 00:00:59.240
in its set of ordered pairs.

00:00:59.240 --> 00:01:02.400
These are two ways of
saying the same thing.

00:01:02.400 --> 00:01:05.730
Now the relation can also
say, hey, maybe if I have 2,

00:01:05.730 --> 00:01:08.850
maybe that is associated
with 2 as well.

00:01:08.850 --> 00:01:13.616
So 2 is also associated
with the number 2.

00:01:13.616 --> 00:01:15.990
And so notice, I'm just building
a bunch of associations.

00:01:15.990 --> 00:01:17.770
I've visually drawn
them over here.

00:01:17.770 --> 00:01:20.560
Here I'm just doing
them as ordered pairs.

00:01:20.560 --> 00:01:22.540
We could say that we
have the number 3.

00:01:22.540 --> 00:01:23.720
3 is in our domain.

00:01:23.720 --> 00:01:26.330
Our relation is
defined for number 3,

00:01:26.330 --> 00:01:31.510
and 3 is associated with,
let's say, negative 7.

00:01:31.510 --> 00:01:35.020
So this is 3 and negative 7.

00:01:35.020 --> 00:01:37.720
Now this type of
relation right over here,

00:01:37.720 --> 00:01:40.330
where if you give me any
member of the domain,

00:01:40.330 --> 00:01:43.650
and I'm able to tell you exactly
which member of the range

00:01:43.650 --> 00:01:48.200
is associated with it, this is
also referred to as a function.

00:01:48.200 --> 00:01:50.640
And in a few seconds,
I'll show you

00:01:50.640 --> 00:01:52.740
a relation that
is not a function.

00:01:52.740 --> 00:01:56.490
Because over here, you pick
any member of the domain,

00:01:56.490 --> 00:01:58.820
and the function really
is just a relation.

00:01:58.820 --> 00:02:01.090
It's really just an
association, sometimes called

00:02:01.090 --> 00:02:03.730
a mapping between
members of the domain

00:02:03.730 --> 00:02:05.360
and particular
members of the range.

00:02:05.360 --> 00:02:07.026
So you give me any
member of the domain,

00:02:07.026 --> 00:02:10.150
I'll tell you exactly which
member of the range it maps to.

00:02:10.150 --> 00:02:12.590
You give me 1, I say, hey,
it definitely maps it to 2.

00:02:12.590 --> 00:02:14.980
You give me 2, it definitely
maps to 2 as well.

00:02:14.980 --> 00:02:18.460
You give me 3, it's definitely
associated with negative 7

00:02:18.460 --> 00:02:19.200
as well.

00:02:19.200 --> 00:02:22.330
So this relation is both a--
it's obviously a relation--

00:02:22.330 --> 00:02:24.007
but it is also a function.

00:02:24.007 --> 00:02:26.090
Now to show you a relation
that is not a function,

00:02:26.090 --> 00:02:27.700
imagine something like this.

00:02:27.700 --> 00:02:30.760
So once again, I'll
draw a domain over here,

00:02:30.760 --> 00:02:32.790
and I do this big, fuzzy
cloud-looking thing

00:02:32.790 --> 00:02:34.710
to show you that I'm not
showing you all of the things

00:02:34.710 --> 00:02:35.293
in the domain.

00:02:35.293 --> 00:02:37.190
I'm just picking
specific examples.

00:02:37.190 --> 00:02:39.840
And let's say that this big,
fuzzy cloud-looking thing

00:02:39.840 --> 00:02:41.760
is the range.

00:02:41.760 --> 00:02:44.060
And let's say in this
relation-- and I'll

00:02:44.060 --> 00:02:46.940
build it the same way that
we built it over here-- let's

00:02:46.940 --> 00:02:51.440
say in this relation,
1 is associated with 2.

00:02:51.440 --> 00:02:53.490
So let's build the
set of ordered pairs.

00:02:53.490 --> 00:02:55.890
So 1 is associated with 2.

00:02:55.890 --> 00:03:00.420
Let's say that 2
is associated with,

00:03:00.420 --> 00:03:03.250
let's say that 2 is
associated with negative 3.

00:03:03.250 --> 00:03:06.270
So you'd have 2,
negative 3 over there.

00:03:06.270 --> 00:03:09.970
And let's say on top of
that, we also associate,

00:03:09.970 --> 00:03:14.390
we also associate 1
with the number 4.

00:03:14.390 --> 00:03:16.490
So we also created
an association with 1

00:03:16.490 --> 00:03:17.960
with the number 4.

00:03:17.960 --> 00:03:21.860
So we have the ordered
pair 1 comma 4.

00:03:21.860 --> 00:03:23.400
Now this is a relationship.

00:03:23.400 --> 00:03:25.576
We have, it's defined
for a certain--

00:03:25.576 --> 00:03:26.950
if this was a
whole relationship,

00:03:26.950 --> 00:03:29.940
then the entire domain is
just the numbers 1, 2--

00:03:29.940 --> 00:03:32.100
actually just the
numbers 1 and 2.

00:03:32.100 --> 00:03:34.797
It's definitely a relation, but
this is no longer a function.

00:03:34.797 --> 00:03:36.630
And the reason why it's
no longer a function

00:03:36.630 --> 00:03:39.160
is, if you tell me,
OK I'm giving you 1

00:03:39.160 --> 00:03:43.027
in the domain, what member of
the range is 1 associated with?

00:03:43.027 --> 00:03:45.610
Over here, you say, well I don't
know, is 1 associated with 2,

00:03:45.610 --> 00:03:46.870
or is it associated with 4?

00:03:46.870 --> 00:03:48.570
It could be either one.

00:03:48.570 --> 00:03:50.430
So you don't have a
clear association.

00:03:50.430 --> 00:03:52.430
If I give you 1 here,
you're like, I don't know,

00:03:52.430 --> 00:03:53.470
do I hand you a 2 or 4?

00:03:53.470 --> 00:03:54.914
That's not what a function does.

00:03:54.914 --> 00:03:56.580
A function says, oh,
if you give me a 1,

00:03:56.580 --> 00:03:58.180
I know I'm giving you a 2.

00:03:58.180 --> 00:03:59.930
If you give me 2, I
know I'm giving you 2.

00:03:59.930 --> 00:04:01.763
Now with that out of
the way, let's actually

00:04:01.763 --> 00:04:03.941
try to tackle the
problem right over here.

00:04:03.941 --> 00:04:05.940
So let's think about its
domain, and let's think

00:04:05.940 --> 00:04:07.240
about its range.

00:04:07.240 --> 00:04:10.920
So the domain here,
the possible, you

00:04:10.920 --> 00:04:13.190
can view them as x
values or inputs,

00:04:13.190 --> 00:04:15.960
into this thing that could be
a function, that's definitely

00:04:15.960 --> 00:04:19.279
a relation, you could
have a negative 3.

00:04:19.279 --> 00:04:21.250
You could have a negative 2.

00:04:21.250 --> 00:04:23.210
You could have a 0.

00:04:23.210 --> 00:04:25.650
You could have a, well, we
already listed a negative 2,

00:04:25.650 --> 00:04:27.000
so that's right over there.

00:04:27.000 --> 00:04:31.510
Or you could have a positive 3.

00:04:31.510 --> 00:04:34.910
Those are the possible values
that this relation is defined

00:04:34.910 --> 00:04:36.960
for, that you could
input into this relation

00:04:36.960 --> 00:04:38.920
and figure out what it outputs.

00:04:38.920 --> 00:04:42.150
Now the range here, these
are the possible outputs

00:04:42.150 --> 00:04:43.650
or the numbers
that are associated

00:04:43.650 --> 00:04:45.260
with the numbers in the domain.

00:04:45.260 --> 00:04:56.230
The range includes 2, 4,
5, 2, 4, 5, 6, 6, and 8.

00:04:56.230 --> 00:04:57.300
2, 4, 5, 6, and 8.

00:04:57.300 --> 00:05:00.277
I could have drawn this
with a big cloud like this,

00:05:00.277 --> 00:05:02.360
and I could have done this
with a cloud like this,

00:05:02.360 --> 00:05:04.060
but here we're showing
the exact numbers

00:05:04.060 --> 00:05:05.490
in the domain and the range.

00:05:05.490 --> 00:05:07.880
And now let's draw the
actual associations.

00:05:07.880 --> 00:05:12.120
So negative 3 is associated
with 2, or it's mapped to 2.

00:05:12.120 --> 00:05:15.630
So negative 3 maps
to 2 based on this

00:05:15.630 --> 00:05:17.360
ordered pair right over there.

00:05:17.360 --> 00:05:21.030
Then we have negative
2 is associated with 4.

00:05:21.030 --> 00:05:22.990
So negative 2 is
associated with 4

00:05:22.990 --> 00:05:25.015
based on this ordered
pair right over there.

00:05:25.015 --> 00:05:26.390
Actually that
first ordered pair,

00:05:26.390 --> 00:05:28.670
let me-- that
first ordered pair,

00:05:28.670 --> 00:05:30.450
I don't want to
get you confused.

00:05:30.450 --> 00:05:32.770
It should just be this
ordered pair right over here.

00:05:32.770 --> 00:05:35.260
Negative 3 is associated with 2.

00:05:35.260 --> 00:05:38.870
Then we have negative 2-- we'll
do that in a different color--

00:05:38.870 --> 00:05:41.990
we have negative 2
is associated with 4.

00:05:41.990 --> 00:05:44.090
Negative 2 is associated with 4.

00:05:44.090 --> 00:05:47.060
We have 0 is associated with 5.

00:05:47.060 --> 00:05:48.260
0 is associated with 5.

00:05:48.260 --> 00:05:51.600
Or sometimes people
say, it's mapped to 5.

00:05:51.600 --> 00:05:55.050
We have negative
2 is mapped to 6.

00:05:55.050 --> 00:05:56.050
Now this is interesting.

00:05:56.050 --> 00:05:57.830
Negative 2 is already
mapped to something.

00:05:57.830 --> 00:06:03.070
Now this ordered pair is
saying it's also mapped to 6.

00:06:03.070 --> 00:06:05.521
And then finally--
I'll do this in a color

00:06:05.521 --> 00:06:07.770
that I haven't used yet,
although I've used almost all

00:06:07.770 --> 00:06:11.830
of them-- we have
3 is mapped to 8.

00:06:11.830 --> 00:06:14.000
3 is mapped to 8.

00:06:14.000 --> 00:06:16.170
So the question here,
is this a function?

00:06:16.170 --> 00:06:18.850
And for it to be a function
for any member of the domain,

00:06:18.850 --> 00:06:21.180
you have to know what
it's going to map to.

00:06:21.180 --> 00:06:24.210
It can only map to one
member of the range.

00:06:24.210 --> 00:06:27.580
So negative 3, if you put
negative 3 as the input

00:06:27.580 --> 00:06:30.160
into the function, you know
it's going to output 2.

00:06:30.160 --> 00:06:32.800
If you put negative 2 into
the input of the function,

00:06:32.800 --> 00:06:34.340
all of a sudden
you get confused.

00:06:34.340 --> 00:06:37.680
Do I output 4, or do I output 6?

00:06:37.680 --> 00:06:40.740
So you don't know if you
output 4 or you output 6.

00:06:40.740 --> 00:06:42.780
And because there's
this confusion,

00:06:42.780 --> 00:06:44.380
this is not a function.

00:06:44.380 --> 00:06:46.430
You have a member
of the domain that

00:06:46.430 --> 00:06:49.180
maps to multiple
members of the range.

00:06:49.180 --> 00:06:56.700
So this right over here is not
a function, not a function.


Graphical relations and functions

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:03.250
Determine whether the
points on this graph

00:00:03.250 --> 00:00:05.390
represent a function.

00:00:05.390 --> 00:00:07.820
Now, just as a
refresher, a function

00:00:07.820 --> 00:00:11.860
is really just an association
between members of a set

00:00:11.860 --> 00:00:14.360
that we call the domain
and members of the set

00:00:14.360 --> 00:00:15.760
that we call a range.

00:00:15.760 --> 00:00:19.580
So if I take any member of
the domain, let's call that x,

00:00:19.580 --> 00:00:22.590
and I give it to the
function, the function

00:00:22.590 --> 00:00:26.830
should tell me what member of
my range is associated with it.

00:00:26.830 --> 00:00:29.430
So it should point
to some other value.

00:00:29.430 --> 00:00:30.730
This is a function.

00:00:30.730 --> 00:00:32.600
It would not be a
function if it says,

00:00:32.600 --> 00:00:34.020
well, it could point to y.

00:00:34.020 --> 00:00:36.940
Or it could point to z.

00:00:36.940 --> 00:00:40.470
Or maybe it could point
to e or whatever else.

00:00:40.470 --> 00:00:42.650
This would not be a function.

00:00:42.650 --> 00:00:46.040
So this right over
here not a function,

00:00:46.040 --> 00:00:49.700
because it's not clear if
you input x what member

00:00:49.700 --> 00:00:51.200
of the range you're
going to get.

00:00:51.200 --> 00:00:52.770
In order for it
to be a function,

00:00:52.770 --> 00:00:54.800
it has to be very clear.

00:00:54.800 --> 00:00:56.527
For any input into
the function, you

00:00:56.527 --> 00:00:58.110
have to be very clear
that you're only

00:00:58.110 --> 00:00:59.650
going to get one output.

00:00:59.650 --> 00:01:01.400
Now, with that out of
the way, let's think

00:01:01.400 --> 00:01:04.410
about this function that
is defined graphically.

00:01:04.410 --> 00:01:08.190
So the domains,
the valid inputs,

00:01:08.190 --> 00:01:10.990
are the x values where
this function is defined.

00:01:10.990 --> 00:01:14.320
So for example, it tells us
if x is equal to negative 1--

00:01:14.320 --> 00:01:16.270
if we assume that this
over here is the x-axis

00:01:16.270 --> 00:01:18.480
and this is the
y-axis-- it tells us,

00:01:18.480 --> 00:01:22.090
when x is equal to negative
1, we should output.

00:01:22.090 --> 00:01:25.280
Or y is going to be equal to 3.

00:01:25.280 --> 00:01:27.160
So one way to write
that mapping is you

00:01:27.160 --> 00:01:34.550
could say, if you
take negative 1

00:01:34.550 --> 00:01:37.450
and you input it into
our function-- I'll

00:01:37.450 --> 00:01:42.430
put a little f box right over
there-- you will get the number

00:01:42.430 --> 00:01:43.090
3.

00:01:43.090 --> 00:01:44.570
This is our x.

00:01:44.570 --> 00:01:46.860
And this is our y.

00:01:46.860 --> 00:01:48.400
So that seems reasonable.

00:01:48.400 --> 00:01:51.030
Negative 1 very clear
that you get to 3.

00:01:51.030 --> 00:01:53.220
Let's see what happens
when we go over here.

00:01:53.220 --> 00:01:57.020
If you put 2 into the function,
when x is 2, y is negative 2.

00:01:57.020 --> 00:01:59.940
Once again, when x
is 2 the function

00:01:59.940 --> 00:02:04.960
associates 2 for x, which
is a member of the domain.

00:02:04.960 --> 00:02:06.570
It's defined for 2.

00:02:06.570 --> 00:02:07.720
It's not defined for 1.

00:02:07.720 --> 00:02:09.389
We don't know what our
function is equal to at 1.

00:02:09.389 --> 00:02:10.472
So it's not defined there.

00:02:10.472 --> 00:02:12.150
So 1 isn't part of the domain.

00:02:12.150 --> 00:02:12.700
2 is.

00:02:12.700 --> 00:02:15.580
It tells us when
x is 2, then y is

00:02:15.580 --> 00:02:17.280
going to be equal to negative 2.

00:02:17.280 --> 00:02:20.470
So it maps it or associates
it with negative 2.

00:02:20.470 --> 00:02:23.380
That doesn't seem too
troublesome just yet.

00:02:23.380 --> 00:02:24.670
Now, let's look over here.

00:02:24.670 --> 00:02:28.980
Our function is also
defined at x is equal to 3.

00:02:28.980 --> 00:02:36.520
Our function associates or maps
3 to the value y is equal to 2.

00:02:36.520 --> 00:02:38.430
That seems pretty
straightforward.

00:02:38.430 --> 00:02:41.140
And then we get to x
is equal to 4, where

00:02:41.140 --> 00:02:43.820
it seems like this thing
that could be a function

00:02:43.820 --> 00:02:46.280
is somewhat defined.

00:02:46.280 --> 00:02:48.820
It does try to
associate 4 with things.

00:02:48.820 --> 00:02:51.220
But what's interesting here
is it tries to associate 4

00:02:51.220 --> 00:02:52.500
with two different things.

00:02:52.500 --> 00:02:54.844
All of a sudden in this
thing that we think

00:02:54.844 --> 00:02:56.260
might have been a
function, but it

00:02:56.260 --> 00:02:58.770
looks like it might
not be, we don't know.

00:02:58.770 --> 00:03:02.590
Do we associate 4 with 5?

00:03:02.590 --> 00:03:05.810
Or do we associate
it with negative 1?

00:03:05.810 --> 00:03:08.590
So this thing right over
here is actually a relation.

00:03:08.590 --> 00:03:13.050
You can have one member of
the domain being related

00:03:13.050 --> 00:03:15.140
to multiple members
of the range.

00:03:15.140 --> 00:03:16.640
But if you do have
that, then you're

00:03:16.640 --> 00:03:17.890
not dealing with a function.

00:03:17.890 --> 00:03:22.800
So once again, because of
this, this is not a function.

00:03:22.800 --> 00:03:26.280
It's not clear that when
you input 4 into it,

00:03:26.280 --> 00:03:27.560
should you output 5?

00:03:27.560 --> 00:03:29.257
Or should you output negative 1?

00:03:29.257 --> 00:03:31.590
And sometimes there's something
called the vertical line

00:03:31.590 --> 00:03:33.790
test that tells you whether
something is a function.

00:03:33.790 --> 00:03:36.610
When it's graphically defined
like this, you literally

00:03:36.610 --> 00:03:40.280
say, OK, when x is 4, if
I draw a vertical line,

00:03:40.280 --> 00:03:43.230
do I intersect the function
at two places or more?

00:03:43.230 --> 00:03:44.860
It could be two or more places.

00:03:44.860 --> 00:03:48.200
And if you do, that means that
there's two or more values that

00:03:48.200 --> 00:03:51.610
are related to that
value in the domain.

00:03:51.610 --> 00:03:54.450
There's two or more
outputs for the input 4.

00:03:54.450 --> 00:03:56.840
And if there are two or more
outputs for that one input,

00:03:56.840 --> 00:03:58.506
then you're not dealing
with a function.

00:03:58.506 --> 00:04:00.140
You're just dealing
with a relation.

00:04:00.140 --> 00:04:02.760
A function is a special
case of a relation.

00:04:02.760 --> 00:04:05.862
Or you could view it as
a well-behaved relation.


Domain and range of a relation

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:05.750
Determine the domain and range
for the relation described

00:00:05.750 --> 00:00:07.380
by the table.

00:00:07.380 --> 00:00:09.220
And so what they
want us to say, what

00:00:09.220 --> 00:00:11.520
they want us to figure out,
when they say the domain,

00:00:11.520 --> 00:00:15.020
they're saying, what are all
of the possible inputs that we

00:00:15.020 --> 00:00:17.050
could put into-- in
this case, a relation,

00:00:17.050 --> 00:00:19.342
and later we'll see into
functions-- and so over here--

00:00:19.342 --> 00:00:20.924
or I guess one way
to think about it--

00:00:20.924 --> 00:00:23.500
what are all of the inputs
that this relationship is

00:00:23.500 --> 00:00:24.960
defined for?

00:00:24.960 --> 00:00:27.640
And so you can view
the x as the input.

00:00:27.640 --> 00:00:30.300
So when x is negative 1, y is 3.

00:00:30.300 --> 00:00:32.780
When x is 3, y is negative 2.

00:00:32.780 --> 00:00:35.210
When x is 3 again, now y is 2.

00:00:35.210 --> 00:00:37.610
And that's why we can't
describe this as a function

00:00:37.610 --> 00:00:40.460
here, because we have two
different y values for a given

00:00:40.460 --> 00:00:42.670
x value, but it
can be a relation.

00:00:42.670 --> 00:00:44.680
When x is 4, y is 8.

00:00:44.680 --> 00:00:47.170
When x is 6, y is negative 1.

00:00:47.170 --> 00:00:48.630
So to answer the
first part, when

00:00:48.630 --> 00:00:53.660
they ask us what is the domain
of this relation, they're

00:00:53.660 --> 00:00:55.590
really just saying,
what are all the inputs?

00:00:55.590 --> 00:00:59.500
What are all the x values for
which this relation is defined?

00:00:59.500 --> 00:01:01.640
And they list the x
values right over here.

00:01:01.640 --> 00:01:04.310
So it is the set-- and that's
what these curly brackets mean

00:01:04.310 --> 00:01:06.570
that I'm about to
describe a set-- it

00:01:06.570 --> 00:01:14.690
is a set of the numbers,
negative 1, 3, 4, and 6.

00:01:14.690 --> 00:01:16.260
So all we're saying
here is if we

00:01:16.260 --> 00:01:19.340
say the domain of this
relation is these four numbers,

00:01:19.340 --> 00:01:23.180
it says that this
relation is defined

00:01:23.180 --> 00:01:24.760
for any of these four numbers.

00:01:24.760 --> 00:01:27.290
If you give any of these
four numbers as an x value,

00:01:27.290 --> 00:01:31.340
there is a y, at least one
y value, associated with it.

00:01:31.340 --> 00:01:37.030
Now, when they talk about
the range of this relation--

00:01:37.030 --> 00:01:39.380
and the idea also
applies to functions,

00:01:39.380 --> 00:01:42.900
which are a more specific
class of relations.

00:01:42.900 --> 00:01:45.670
You can view them as a
well-behaved relation.

00:01:45.670 --> 00:01:48.940
The range is, what are
all the possible outputs

00:01:48.940 --> 00:01:50.910
that this relation can give you?

00:01:50.910 --> 00:01:55.810
Or given the inputs, what are
all of the possible values

00:01:55.810 --> 00:01:57.500
that this relation can take on?

00:01:57.500 --> 00:02:00.360
So here you'll take a look at
all of the possible y values

00:02:00.360 --> 00:02:01.790
that it can take on.

00:02:01.790 --> 00:02:04.365
So we can get-- and I'll write--
we could write them in order

00:02:04.365 --> 00:02:05.630
or we don't have to
write them in order,

00:02:05.630 --> 00:02:07.270
but I'll write them in order
just for the sake of it.

00:02:07.270 --> 00:02:08.580
Actually, let's just
go straight this way.

00:02:08.580 --> 00:02:09.871
It doesn't have to be in order.

00:02:09.871 --> 00:02:11.760
A set does not imply
some type of order.

00:02:11.760 --> 00:02:14.320
It just means a
collection of things.

00:02:14.320 --> 00:02:17.730
So the range here, well, our y
value can take on the value 3.

00:02:17.730 --> 00:02:19.760
It can take on the
value negative 2.

00:02:19.760 --> 00:02:21.400
It can take on the value 2.

00:02:21.400 --> 00:02:23.260
It can take on the value 8.

00:02:23.260 --> 00:02:26.400
And it can take on
the value negative 1.

00:02:26.400 --> 00:02:27.780
And we're done.

00:02:27.780 --> 00:02:31.210
These are the x values for
which this relation is defined

00:02:31.210 --> 00:02:34.330
that you can actually find an
association or relationship.

00:02:34.330 --> 00:02:36.160
And these are all the y values.

00:02:36.160 --> 00:02:39.460
These are all of the outputs of
the relation that it can take

00:02:39.460 --> 00:02:43.100
on, and we just looked right
over here to find them.


Area of a regular hexagon

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:05.060
We're told that ABCDEF
is a regular hexagon.

00:00:05.060 --> 00:00:07.129
And this regular part--
hexagon obviously

00:00:07.129 --> 00:00:08.920
tells us that we're
dealing with six sides.

00:00:08.920 --> 00:00:10.169
And you could just count that.

00:00:10.169 --> 00:00:12.136
You didn't have to be
told it's a hexagon.

00:00:12.136 --> 00:00:13.510
But the regular
part lets us know

00:00:13.510 --> 00:00:16.920
that all of the sides, all six
sides, have the same length

00:00:16.920 --> 00:00:20.760
and all of the interior
angles have the same measure.

00:00:20.760 --> 00:00:21.262
Fair enough.

00:00:21.262 --> 00:00:23.470
And then they give us the
length of one of the sides.

00:00:23.470 --> 00:00:24.735
And since this is
a regular hexagon,

00:00:24.735 --> 00:00:27.026
they're actually giving us
the length of all the sides.

00:00:27.026 --> 00:00:28.640
They say it's 2
square roots of 3.

00:00:28.640 --> 00:00:31.079
So this side right over
here is 2 square roots of 3.

00:00:31.079 --> 00:00:32.870
This side over here is
2 square roots of 3.

00:00:32.870 --> 00:00:34.495
And I could just go
around the hexagon.

00:00:34.495 --> 00:00:36.570
Every one of their sides
is 2 square roots of 3.

00:00:36.570 --> 00:00:39.020
They want us to find the
area of this hexagon.

00:00:39.020 --> 00:00:42.460
Find the area of ABCDEF.

00:00:42.460 --> 00:00:43.890
And the best way
to find the area,

00:00:43.890 --> 00:00:45.306
especially of
regular polygons, is

00:00:45.306 --> 00:00:46.760
try to split it
up into triangles.

00:00:46.760 --> 00:00:48.710
And hexagons are a
bit of a special case.

00:00:48.710 --> 00:00:50.168
Maybe in future
videos, we'll think

00:00:50.168 --> 00:00:52.354
about the more general
case of any polygon.

00:00:52.354 --> 00:00:54.270
But with a hexagon, what
you could think about

00:00:54.270 --> 00:00:57.660
is if we take this
point right over here.

00:00:57.660 --> 00:00:59.730
And let's call this
point G. And let's say

00:00:59.730 --> 00:01:01.335
it's the center of the hexagon.

00:01:01.335 --> 00:01:03.640
And when I'm talking about
a center of a hexagon,

00:01:03.640 --> 00:01:04.879
I'm talking about a point.

00:01:04.879 --> 00:01:07.000
It can't be equidistant
from everything

00:01:07.000 --> 00:01:08.746
over here, because
this isn't a circle.

00:01:08.746 --> 00:01:10.120
But we could say
it's equidistant

00:01:10.120 --> 00:01:12.910
from all of the vertices,
so that GD is the same thing

00:01:12.910 --> 00:01:16.120
as GC is the same thing as GB,
which is the same thing as GA,

00:01:16.120 --> 00:01:19.310
which is the same thing as GF,
which is the same thing as GE.

00:01:19.310 --> 00:01:22.220
So let me draw some of those
that I just talked about.

00:01:22.220 --> 00:01:24.100
So that is GE.

00:01:24.100 --> 00:01:25.690
There's GD.

00:01:25.690 --> 00:01:26.750
There's GC.

00:01:26.750 --> 00:01:29.410
All of these lengths are
going to be the same.

00:01:29.410 --> 00:01:31.040
So there's a point
G which we can

00:01:31.040 --> 00:01:34.562
call the center of this polygon.

00:01:34.562 --> 00:01:37.020
And we know that this length
is equal to that length, which

00:01:37.020 --> 00:01:38.940
is equal to that length, which
is equal to that length, which

00:01:38.940 --> 00:01:41.260
is equal to that length,
which is equal to that length.

00:01:41.260 --> 00:01:44.400
We also know that
if we go all the way

00:01:44.400 --> 00:01:50.410
around the circle like that,
we've gone 360 degrees.

00:01:50.410 --> 00:01:54.500
And we know that these
triangles are all

00:01:54.500 --> 00:01:56.341
going to be congruent
to each other.

00:01:56.341 --> 00:01:58.340
And there's multiple ways
that we could show it.

00:01:58.340 --> 00:02:00.484
But the easiest way is,
look, they have two sides.

00:02:00.484 --> 00:02:02.150
All of them have this
side and this side

00:02:02.150 --> 00:02:04.570
be congruent to each other
because G is in the center.

00:02:04.570 --> 00:02:07.220
And they all have
this third common side

00:02:07.220 --> 00:02:08.400
of 2 square roots of 3.

00:02:08.400 --> 00:02:10.530
So all of them,
by side-side-side,

00:02:10.530 --> 00:02:13.910
they are all congruent.

00:02:13.910 --> 00:02:16.520
What that tells us is,
if they're all congruent,

00:02:16.520 --> 00:02:19.270
then this angle, this interior
angle right over here,

00:02:19.270 --> 00:02:24.970
is going to be the
same for all six

00:02:24.970 --> 00:02:26.450
of these triangles over here.

00:02:26.450 --> 00:02:27.950
And let me call that x.

00:02:27.950 --> 00:02:28.730
That's angle x.

00:02:28.730 --> 00:02:29.750
That's x.

00:02:29.750 --> 00:02:30.600
That's x.

00:02:30.600 --> 00:02:31.440
That's x.

00:02:31.440 --> 00:02:32.340
That's x.

00:02:32.340 --> 00:02:35.030
And if you add them all up,
we've gone around the circle.

00:02:35.030 --> 00:02:36.720
We've gone 360 degrees.

00:02:36.720 --> 00:02:38.520
And we have six of these x's.

00:02:38.520 --> 00:02:41.890
So you get 6x is
equal to 360 degrees.

00:02:41.890 --> 00:02:43.350
You divide both sides by 6.

00:02:43.350 --> 00:02:48.830
You get x is equal
to 60 degrees.

00:02:48.830 --> 00:02:51.024
All of these are
equal to 60 degrees.

00:02:51.024 --> 00:02:52.440
Now there's something
interesting.

00:02:52.440 --> 00:02:55.934
We know that these triangles--
for example, triangle GBC--

00:02:55.934 --> 00:02:58.100
and we could do that for
any of these six triangles.

00:02:58.100 --> 00:03:00.204
It looks kind of like a
Trivial Pursuit piece.

00:03:00.204 --> 00:03:02.370
We know that they're
definitely isosceles triangles,

00:03:02.370 --> 00:03:04.900
that this distance is
equal to this distance.

00:03:04.900 --> 00:03:08.360
So we can use that
information to figure out

00:03:08.360 --> 00:03:09.560
what the other angles are.

00:03:09.560 --> 00:03:12.200
Because these two base angles--
it's an isosceles triangle.

00:03:12.200 --> 00:03:13.350
The two legs are the same.

00:03:13.350 --> 00:03:15.205
So our two base
angles, this angle

00:03:15.205 --> 00:03:16.830
is going to be
congruent to that angle.

00:03:16.830 --> 00:03:19.570
If we could call that
y right over there.

00:03:19.570 --> 00:03:22.600
So you have y plus
y, which is 2y,

00:03:22.600 --> 00:03:28.090
plus 60 degrees is going
to be equal to 180.

00:03:28.090 --> 00:03:30.040
Because the interior
angles of any triangle--

00:03:30.040 --> 00:03:32.040
they add up to 180.

00:03:32.040 --> 00:03:34.090
And so subtract 60
from both sides.

00:03:34.090 --> 00:03:36.610
You get 2y is equal to 120.

00:03:36.610 --> 00:03:37.830
Divide both sides by 2.

00:03:37.830 --> 00:03:40.819
You get y is equal
to 60 degrees.

00:03:40.819 --> 00:03:41.860
Now, this is interesting.

00:03:41.860 --> 00:03:43.985
I could have done this with
any of these triangles.

00:03:43.985 --> 00:03:46.690
All of these triangles
are 60-60-60 triangles,

00:03:46.690 --> 00:03:48.710
which tells us-- and
we've proven this earlier

00:03:48.710 --> 00:03:50.110
on when we first
started studying

00:03:50.110 --> 00:03:52.400
equilateral triangles--
we know that all

00:03:52.400 --> 00:03:55.270
of the angles of a
triangle are 60 degrees,

00:03:55.270 --> 00:03:57.520
then we're dealing with an
equilateral triangle, which

00:03:57.520 --> 00:03:59.980
means that all the sides
have the same length.

00:03:59.980 --> 00:04:02.795
So if this is 2 square
roots of 3, then so is this.

00:04:02.795 --> 00:04:04.430
This is also 2
square roots of 3.

00:04:04.430 --> 00:04:06.350
And this is also 2
square roots of 3.

00:04:06.350 --> 00:04:08.070
So pretty much all
of these green lines

00:04:08.070 --> 00:04:09.069
are 2 square roots of 3.

00:04:09.069 --> 00:04:11.300
And we already knew, because
it's a regular hexagon,

00:04:11.300 --> 00:04:14.020
that each side of
the hexagon itself

00:04:14.020 --> 00:04:15.930
is also 2 square roots of 3.

00:04:15.930 --> 00:04:20.579
So now we can essentially
use that information

00:04:20.579 --> 00:04:22.560
to figure out--
actually, we don't even

00:04:22.560 --> 00:04:23.770
have to figure this part out.

00:04:23.770 --> 00:04:25.210
I'll show you in a
second-- to figure out

00:04:25.210 --> 00:04:27.030
the area of any one
of these triangles.

00:04:27.030 --> 00:04:29.100
And then we can
just multiply by 6.

00:04:29.100 --> 00:04:32.350
So let's focus on this
triangle right over here

00:04:32.350 --> 00:04:34.080
and think about how
we can find its area.

00:04:34.080 --> 00:04:37.110
We know that length of DC
is 2 square roots of 3.

00:04:37.110 --> 00:04:39.820
We can drop an
altitude over here.

00:04:39.820 --> 00:04:42.910
We can drop an altitude
just like that.

00:04:42.910 --> 00:04:44.730
And then if we drop
an altitude, we

00:04:44.730 --> 00:04:49.591
know that this is an
equilateral triangle.

00:04:49.591 --> 00:04:51.340
And we can show very
easily that these two

00:04:51.340 --> 00:04:52.370
triangles are symmetric.

00:04:52.370 --> 00:04:54.250
These are both 90-degree angles.

00:04:54.250 --> 00:04:57.050
We know that these two are
60-degree angles already.

00:04:57.050 --> 00:04:59.324
And then if you look
at each of these two

00:04:59.324 --> 00:05:01.490
independent triangles, you'd
have to just say, well,

00:05:01.490 --> 00:05:02.615
they have to add up to 180.

00:05:02.615 --> 00:05:04.490
So this has to be 30 degrees.

00:05:04.490 --> 00:05:05.920
This has to be 30 degrees.

00:05:05.920 --> 00:05:07.500
All the angles are the same.

00:05:07.500 --> 00:05:09.360
They also share
a side in common.

00:05:09.360 --> 00:05:11.600
So these two are
congruent triangles.

00:05:11.600 --> 00:05:16.090
So if we want to find the area
of this little slice of the pie

00:05:16.090 --> 00:05:17.600
right over here,
we can just find

00:05:17.600 --> 00:05:20.760
the area of this slice,
or this sub-slice,

00:05:20.760 --> 00:05:22.070
and then multiply by 2.

00:05:22.070 --> 00:05:24.430
Or we could just find this
area and multiply by 12

00:05:24.430 --> 00:05:26.050
for the entire hexagon.

00:05:26.050 --> 00:05:28.250
So how do we figure out
the area of this thing?

00:05:28.250 --> 00:05:31.660
Well, this is going to be
half of this base length,

00:05:31.660 --> 00:05:33.380
so this length right over here.

00:05:33.380 --> 00:05:35.820
Let me call this
point H. DH is going

00:05:35.820 --> 00:05:37.890
to be the square root of 3.

00:05:37.890 --> 00:05:39.835
And hopefully we've
already recognized

00:05:39.835 --> 00:05:42.070
that this is a
30-60-90 triangle.

00:05:42.070 --> 00:05:43.690
Let me draw it over here.

00:05:43.690 --> 00:05:49.000
So this is a 30-60-90 triangle.

00:05:49.000 --> 00:05:52.849
We know that this length over
here is square root of 3.

00:05:52.849 --> 00:05:54.390
And we already
actually did calculate

00:05:54.390 --> 00:05:56.010
that this is 2
square roots of 3.

00:05:56.010 --> 00:05:57.734
Although we don't
really need it.

00:05:57.734 --> 00:06:00.150
What we really need to figure
out is this altitude height.

00:06:00.150 --> 00:06:03.300
And from 30-60-90
triangles, we know

00:06:03.300 --> 00:06:07.380
that the side opposite
the 60-degree side

00:06:07.380 --> 00:06:09.300
is the square root
of 3 times the side

00:06:09.300 --> 00:06:11.180
opposite the 30-degree side.

00:06:11.180 --> 00:06:13.110
So this is going
to be square root

00:06:13.110 --> 00:06:17.830
of 3 times the square root of 3.

00:06:17.830 --> 00:06:19.740
Square root of 3 times
the square root of 3

00:06:19.740 --> 00:06:21.990
is obviously just 3.

00:06:21.990 --> 00:06:26.800
So this altitude right over
here is just going to be 3.

00:06:26.800 --> 00:06:28.980
So if we want the area
of this triangle right

00:06:28.980 --> 00:06:32.290
over here, which is this
triangle right over here,

00:06:32.290 --> 00:06:34.560
it's just 1/2 base times height.

00:06:34.560 --> 00:06:36.370
So the area of this
little sub-slice

00:06:36.370 --> 00:06:40.200
is just 1/2 times our base,
just the base over here.

00:06:40.200 --> 00:06:41.740
Actually, let's
take a step back.

00:06:41.740 --> 00:06:43.615
We don't even have to
worry about this thing.

00:06:43.615 --> 00:06:46.230
Let's just go straight to
the larger triangle, GDC.

00:06:46.230 --> 00:06:48.545
So let me rewind
this a little bit.

00:06:48.545 --> 00:06:51.170
Because now we have the base and
the height of the whole thing.

00:06:51.170 --> 00:06:57.670
If we care about the area
of triangle GDC-- so now

00:06:57.670 --> 00:07:02.490
I'm looking at this entire
triangle right over here.

00:07:02.490 --> 00:07:06.230
This is equal to 1/2
times base times height,

00:07:06.230 --> 00:07:08.589
which is equal to
1/2-- what's our base?

00:07:08.589 --> 00:07:09.630
Our base we already know.

00:07:09.630 --> 00:07:11.369
It's one of the
sides of our hexagon.

00:07:11.369 --> 00:07:12.410
It's 2 square roots of 3.

00:07:12.410 --> 00:07:14.510
It's this whole thing
right over here.

00:07:14.510 --> 00:07:16.965
So times 2 square roots of 3.

00:07:16.965 --> 00:07:19.090
And then we want to multiply
that times our height.

00:07:19.090 --> 00:07:21.910
And that's what we just figured
out using 30-60-90 triangles.

00:07:21.910 --> 00:07:23.710
Our height is 3.

00:07:23.710 --> 00:07:25.560
So times 3.

00:07:25.560 --> 00:07:27.420
1/2 and 2 cancel out.

00:07:27.420 --> 00:07:29.860
We're left with 3
square roots of 3.

00:07:29.860 --> 00:07:32.440
That's just the area of
one of these little wedges

00:07:32.440 --> 00:07:33.270
right over here.

00:07:33.270 --> 00:07:36.000
If we want to find the
area of the entire hexagon,

00:07:36.000 --> 00:07:38.460
we just have to
multiply that by 6,

00:07:38.460 --> 00:07:40.690
because there are six of
these triangles there.

00:07:40.690 --> 00:07:45.840
So this is going to be equal
to 6 times 3 square roots of 3,

00:07:45.840 --> 00:07:48.810
which is 18 square roots of 3.

00:07:48.810 --> 00:07:50.650
And we're done.


Problem involving angle derived from square and circle

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:05.330
So we're told that quadrilateral
ABCD is a square, which

00:00:05.330 --> 00:00:08.220
tells us that the four
sides have equal length

00:00:08.220 --> 00:00:11.830
and that all the interior
angles are 90 degrees.

00:00:11.830 --> 00:00:18.182
We also know that FG is a
perpendicular bisector of BC.

00:00:18.182 --> 00:00:20.140
So we've already shown
that it's perpendicular,

00:00:20.140 --> 00:00:21.570
that this is a 90-degree angle.

00:00:21.570 --> 00:00:24.540
But it also bisects
BC, so this length

00:00:24.540 --> 00:00:27.270
is equal to that length
right over there.

00:00:27.270 --> 00:00:31.520
Then they tell us that arc AC--
that's a curve on top-- arc AC

00:00:31.520 --> 00:00:36.200
is part of circle B. So this
is a circle centered at B.

00:00:36.200 --> 00:00:38.220
So if this is the
center of the circle,

00:00:38.220 --> 00:00:41.220
this is part of that circle.

00:00:41.220 --> 00:00:44.400
It's really kind of the bottom
left quarter of that circle.

00:00:44.400 --> 00:00:46.020
And then given that
information, they

00:00:46.020 --> 00:00:51.260
want us to find what the
measure of angle BED is.

00:00:51.260 --> 00:00:54.252
So what is BED?

00:00:54.252 --> 00:00:56.190
So we need to figure
out the measure

00:00:56.190 --> 00:00:57.920
of this angle right over here.

00:00:57.920 --> 00:01:00.550
And I encourage you to
pause it and try it out.

00:01:00.550 --> 00:01:02.707
And you might
imagine-- well, you

00:01:02.707 --> 00:01:04.540
could pause it and try
it without any hints.

00:01:04.540 --> 00:01:07.410
And now I will give you a hint
if you tried it the first time

00:01:07.410 --> 00:01:08.659
and you weren't able to do it.

00:01:08.659 --> 00:01:11.330
And you should pause again
after this hint-- try

00:01:11.330 --> 00:01:13.660
to draw some triangles
that maybe split up

00:01:13.660 --> 00:01:15.740
this angle into a couple
of different angles.

00:01:15.740 --> 00:01:17.240
And it might be a
little bit easier.

00:01:17.240 --> 00:01:20.160
You might be able to use some
of what we know about triangles.

00:01:20.160 --> 00:01:22.920
Now, with that said, I
will try to solve it.

00:01:22.920 --> 00:01:24.430
And you should
pause at any point

00:01:24.430 --> 00:01:27.420
where you think that you know
exactly how to do this and try

00:01:27.420 --> 00:01:29.030
to do it yourself.

00:01:29.030 --> 00:01:31.730
So the trick here to realize
that this is a circle.

00:01:31.730 --> 00:01:35.590
And so any line that goes
between B and any point

00:01:35.590 --> 00:01:38.470
on this arc is going to be equal
to the radius of the circle.

00:01:38.470 --> 00:01:40.900
So AB is equal to the
radius of the circle.

00:01:40.900 --> 00:01:43.272
BE is equal to the
radius of the circle.

00:01:43.272 --> 00:01:44.980
And we can keep drawing
other things that

00:01:44.980 --> 00:01:46.563
are equal to the
radius of the circle.

00:01:46.563 --> 00:01:49.689
BC is equal to the
radius of the circle.

00:01:49.689 --> 00:01:51.230
So let's think about
it a little bit.

00:01:51.230 --> 00:01:53.650
If we were to draw it-- and
a lot of trickier geometry

00:01:53.650 --> 00:01:56.470
problems really all revolve
around drawing the right lines

00:01:56.470 --> 00:01:59.190
or visualizing the
right triangles.

00:01:59.190 --> 00:02:02.040
And I'll do one right here that
might open up a lot for you

00:02:02.040 --> 00:02:04.740
in terms of thinking about
how to do this problem.

00:02:04.740 --> 00:02:07.500
So let me draw segment EC.

00:02:07.500 --> 00:02:10.050
I'll draw that as
straight as possible.

00:02:10.050 --> 00:02:15.120
I can draw a better job
of that, so segment EC.

00:02:15.120 --> 00:02:17.600
Now something
becomes interesting.

00:02:17.600 --> 00:02:24.700
Because what is the relationship
between triangle EBG

00:02:24.700 --> 00:02:27.170
and triangle ECG?

00:02:27.170 --> 00:02:31.510
Well, they both definitely
share this side right over here.

00:02:31.510 --> 00:02:33.810
They both share side EG.

00:02:33.810 --> 00:02:36.814
And then BG is equal
to GC, and they both

00:02:36.814 --> 00:02:37.730
have 90-degree angles.

00:02:37.730 --> 00:02:39.320
You have a 90-degree angle here.

00:02:39.320 --> 00:02:41.060
You have a 90-degree
angle there.

00:02:41.060 --> 00:02:45.980
So you see by side-angle-side
that these two triangles

00:02:45.980 --> 00:02:47.620
are going to be congruent.

00:02:47.620 --> 00:02:59.380
So we know that triangle EBG is
congruent to triangle ECG-- I

00:02:59.380 --> 00:03:02.310
should emphasize
the C, not the E--

00:03:02.310 --> 00:03:05.045
by side-angle-side congruency.

00:03:09.520 --> 00:03:12.720
And that also tells us that
all of the corresponding angles

00:03:12.720 --> 00:03:14.610
and sides are going
to be the same.

00:03:14.610 --> 00:03:22.565
So that tells us right there
that EC is equal to EB.

00:03:25.730 --> 00:03:31.770
So we know that
EB is equal to EC.

00:03:31.770 --> 00:03:34.855
And what also is
equal to that length?

00:03:34.855 --> 00:03:36.980
Well, once again, this is
the radius of the circle.

00:03:36.980 --> 00:03:40.050
BE is one radius of the
circle going from the center

00:03:40.050 --> 00:03:41.130
to the arc.

00:03:41.130 --> 00:03:42.230
But so is BC.

00:03:42.230 --> 00:03:44.790
It is also a radius of the
circle going from the center

00:03:44.790 --> 00:03:45.860
to the arc.

00:03:45.860 --> 00:03:49.700
So this is also equal to BC.

00:03:49.700 --> 00:03:52.559
So I could draw the
three things right here.

00:03:52.559 --> 00:03:54.350
I'm referring to the
entire thing, not just

00:03:54.350 --> 00:03:56.970
one of the segments, all of BC.

00:03:56.970 --> 00:03:58.720
So what kind of a
triangle is this right

00:03:58.720 --> 00:04:01.630
over here, triangle BEC?

00:04:01.630 --> 00:04:08.450
Triangle BEC is equilateral.

00:04:08.450 --> 00:04:10.610
And we know that because
all three sides are equal,

00:04:10.610 --> 00:04:13.310
so that tells us that all
of its angles are equal.

00:04:13.310 --> 00:04:15.100
So that tells us that
the measure of angle

00:04:15.100 --> 00:04:21.680
BEC-- we're not done yet, but it
gets us close-- is 60 degrees.

00:04:21.680 --> 00:04:26.199
So the measure of angle BEC
right over there is 60 degrees.

00:04:26.199 --> 00:04:27.740
So that gives us
part of the problem.

00:04:27.740 --> 00:04:31.160
BEC is part of the angle BED.

00:04:31.160 --> 00:04:34.521
If we can just figure out
the measure of angle CED now,

00:04:34.521 --> 00:04:36.520
if we can figure out this
angle right over here,

00:04:36.520 --> 00:04:39.050
we just add that to 60
degrees, and we're done.

00:04:39.050 --> 00:04:43.150
We've figured out
the entire BED.

00:04:43.150 --> 00:04:45.850
Now, let's think about how we
can do this right over here.

00:04:45.850 --> 00:04:47.770
So there's a couple
of interesting things

00:04:47.770 --> 00:04:49.640
that we already do know.

00:04:49.640 --> 00:04:51.280
We know that this
right over here

00:04:51.280 --> 00:04:54.730
is equal to the
radius of the circle.

00:04:54.730 --> 00:04:57.290
And we also know that this
length down here-- this

00:04:57.290 --> 00:05:00.810
is a square-- we know that
this length down here is

00:05:00.810 --> 00:05:04.590
the same as this length
up here, that these

00:05:04.590 --> 00:05:06.610
are the exact same length.

00:05:06.610 --> 00:05:08.754
And this is equal to the
radius of the circle.

00:05:08.754 --> 00:05:10.420
We already put these
three slashes here.

00:05:10.420 --> 00:05:13.930
BC is the same as that length
is the same as that length.

00:05:13.930 --> 00:05:16.700
And so all four sides are
going to be that same length

00:05:16.700 --> 00:05:19.730
because this is a square.

00:05:19.730 --> 00:05:21.500
So let me write this down.

00:05:25.674 --> 00:05:26.840
I'll just write it this way.

00:05:26.840 --> 00:05:31.600
Because it's a square,
we know that CD

00:05:31.600 --> 00:05:34.666
is equal to BC,
which is equal to--

00:05:34.666 --> 00:05:36.040
and we already
established this--

00:05:36.040 --> 00:05:40.691
is equal to EC, which is equal
to EB, which is equal to EC.

00:05:40.691 --> 00:05:42.440
But the important thing
here is to realize

00:05:42.440 --> 00:05:44.950
that this and this
are the same length.

00:05:44.950 --> 00:05:46.940
And the reason why
that is interesting

00:05:46.940 --> 00:05:51.890
is it lets us know that this
is an isosceles triangle.

00:05:51.890 --> 00:05:55.120
So isosceles triangle, if
you have your two legs of it,

00:05:55.120 --> 00:05:57.610
the two base angles are
going to be congruent.

00:05:57.610 --> 00:05:59.220
So whatever angle
this green angle is,

00:05:59.220 --> 00:06:01.270
this angle is going
to be as well.

00:06:01.270 --> 00:06:05.550
So if somehow we can figure
out this angle right over here,

00:06:05.550 --> 00:06:08.400
we can subtract that from
180, and then divide by 2

00:06:08.400 --> 00:06:09.400
to figure out these two.

00:06:09.400 --> 00:06:10.983
Because we know that
they're the same.

00:06:10.983 --> 00:06:12.630
So how can we figure
out this angle?

00:06:12.630 --> 00:06:14.870
Well, we know all the
angles of this thing.

00:06:14.870 --> 00:06:18.480
We can figure out the angles of
all of this larger one up here.

00:06:18.480 --> 00:06:20.610
We know this is an
equilateral triangle.

00:06:20.610 --> 00:06:23.350
So this over here has to
be 60 degrees, as well.

00:06:23.350 --> 00:06:24.630
That's 60 degrees.

00:06:24.630 --> 00:06:26.430
And that is also 60 degrees.

00:06:26.430 --> 00:06:28.000
In fact, I could
write over here,

00:06:28.000 --> 00:06:31.035
which is equal to the
measure of angle BCE.

00:06:38.530 --> 00:06:40.464
So if this is 60
degrees, we know

00:06:40.464 --> 00:06:42.880
we're dealing with a square,
so this whole angle over here

00:06:42.880 --> 00:06:44.390
is a right angle.

00:06:44.390 --> 00:06:47.417
What is the measure
of angle ECD?

00:06:47.417 --> 00:06:48.875
What is this angle
right over here?

00:06:48.875 --> 00:06:50.720
Let me do this in a new color.

00:06:50.720 --> 00:06:54.510
This right over here is going
to have to be 30 degrees,

00:06:54.510 --> 00:06:56.790
so that is going
to be 30 degrees.

00:06:56.790 --> 00:07:01.660
And now, we're ready to solve
for these two base angles.

00:07:01.660 --> 00:07:05.260
If we call these x-- and we
know they have to be the same--

00:07:05.260 --> 00:07:07.730
we have x plus x
plus 30 degrees.

00:07:07.730 --> 00:07:12.260
x plus x plus 30
degrees is going

00:07:12.260 --> 00:07:14.720
to be equal to 180 degrees.

00:07:14.720 --> 00:07:17.640
That's the sum of all of the
interior angles of a triangle.

00:07:17.640 --> 00:07:25.580
So you get 2x plus 30
is equal to 180 degrees.

00:07:25.580 --> 00:07:27.790
Now, you can subtract
30 from both sides.

00:07:27.790 --> 00:07:33.470
And we are left with
2x is equal to 150.

00:07:33.470 --> 00:07:37.650
Divide both sides by 2,
you get x is equal to 75.

00:07:37.650 --> 00:07:40.190
So we've figured out
that x is equal to 75.

00:07:40.190 --> 00:07:41.720
And now we're at
the home stretch.

00:07:41.720 --> 00:07:43.453
We need to figure out angle BED.

00:07:46.160 --> 00:07:50.470
So x is equal to the
measure of angle CED.

00:07:50.470 --> 00:08:00.040
So BED is CED plus BEC, so the
60 degrees plus 75 degrees.

00:08:00.040 --> 00:08:03.270
So it's going to be-- ready for
the drum roll-- this is going

00:08:03.270 --> 00:08:08.250
to be equal to 75 degrees
plus 60 degrees, which

00:08:08.250 --> 00:08:11.660
is equal to 135 degrees.

00:08:11.660 --> 00:08:14.340
And we are done.