Constructing exponential models: percent change

WEBVTT
Kind: captions
Language: en

00:00:00.330 --> 00:00:02.965
- [Voiceover] Chepi is an
ecologist who studies the change

00:00:02.965 --> 00:00:06.784
in the narwhal population of
the Arctic ocean over time.

00:00:06.784 --> 00:00:10.720
She observed that the
population population loses 5.6%

00:00:10.720 --> 00:00:14.144
of its size every 2.8 months.

00:00:14.144 --> 00:00:16.129
The population of narwhals can be modeled

00:00:16.129 --> 00:00:18.602
by a function, N, which
depends on the amount

00:00:18.602 --> 00:00:20.533
of time, t in months.

00:00:20.533 --> 00:00:22.889
When Chepi began the study, she observed

00:00:22.889 --> 00:00:27.765
that there were 89,000
narwhals in the Arctic ocean.

00:00:27.765 --> 00:00:30.771
Write a function that models
the population of narwhals

00:00:30.771 --> 00:00:33.898
t months since the
beginning of Chepi's study.

00:00:33.898 --> 00:00:35.175
Like always, pause the video

00:00:35.175 --> 00:00:36.312
and see if you can do it on your own

00:00:36.312 --> 00:00:38.530
before we work through it together.

00:00:38.530 --> 00:00:39.930
So let's now work through it together.

00:00:39.930 --> 00:00:42.032
To get my essence of what
this function is to do,

00:00:42.032 --> 00:00:45.711
it's always valuable to
see, to create a table

00:00:45.711 --> 00:00:48.033
for some interesting
inputs for the function

00:00:48.033 --> 00:00:52.131
and seeing how the function should behave.

00:00:52.131 --> 00:00:53.914
So first of all, if t is in months

00:00:53.914 --> 00:00:57.815
and N of t is, in N is the,

00:00:57.815 --> 00:01:01.878
that models, N is the number
of narwhals, the narwhals.

00:01:01.878 --> 00:01:04.200
So what, when T is equal to zero,

00:01:04.200 --> 00:01:05.593
what is N of zero?

00:01:05.593 --> 00:01:07.357
Well, we know a T equals zero.

00:01:07.357 --> 00:01:10.503
There are 89,000 narwhals in the ocean.

00:01:10.503 --> 00:01:13.882
So 89,000.

00:01:13.882 --> 00:01:16.819
And now, what's another interesting one?

00:01:16.819 --> 00:01:17.674
Well, T is in months

00:01:17.674 --> 00:01:19.822
and we know that the population decreases

00:01:19.822 --> 00:01:22.840
5.6% every 2.8 months.

00:01:22.840 --> 00:01:26.044
So let's think about when T is 2.8,

00:01:26.044 --> 00:01:27.425
2.8 months.

00:01:27.425 --> 00:01:28.900
Well then the population,

00:01:28.900 --> 00:01:32.847
it should have gone down 5.6%.

00:01:32.847 --> 00:01:36.179
So going down 5.6% is the same thing

00:01:36.179 --> 00:01:37.839
as retaining.

00:01:37.839 --> 00:01:40.799
What's one minus 5.6%?

00:01:40.799 --> 00:01:45.106
Retaining 94.4%.

00:01:45.106 --> 00:01:45.744
I'll be clear.

00:01:45.744 --> 00:01:47.908
100%,

00:01:47.908 --> 00:01:51.625
if you lose 5.6%,

00:01:51.625 --> 00:01:56.625
you are going to be left with 94.4%.

00:01:57.354 --> 00:01:59.606
The 0.6 plus 0.4 gets you a 95

00:01:59.606 --> 00:02:01.069
plus another five is 100.

00:02:01.069 --> 00:02:02.929
So another way of saying, this sentence,

00:02:02.929 --> 00:02:06.748
that the population loses 5.6%
of its size every 2.8 months

00:02:06.748 --> 00:02:09.465
is to say that the population is 94%,

00:02:09.465 --> 00:02:12.785
94.4% of its size every 2.8 months

00:02:12.785 --> 00:02:16.604
or shrinks to 94.4% of its original size

00:02:16.604 --> 00:02:19.669
every, or let me phrase this clearly.

00:02:19.669 --> 00:02:22.687
After every 2.8 months, the population,

00:02:22.687 --> 00:02:25.102
you can either say it shrinks 5.6%

00:02:25.102 --> 00:02:27.865
or you could say it has, it's gone from,

00:02:27.865 --> 00:02:31.348
it's 94.4% of the
population at the beginning

00:02:31.348 --> 00:02:32.902
of those 2.8 months.

00:02:32.902 --> 00:02:37.902
So after 2.8 months, the
population should be 89,000 times,

00:02:38.172 --> 00:02:40.413
I could write times 94.4%

00:02:40.413 --> 00:02:44.789
or I could write times 0.944.

00:02:44.789 --> 00:02:48.442
Now, if we go another 2.8 months,

00:02:48.442 --> 00:02:51.622
so two times 2.8.

00:02:51.622 --> 00:02:53.329
I have to say you could
just write that as,

00:02:53.329 --> 00:02:55.129
I could write that as 5.6 months

00:02:55.129 --> 00:02:56.951
but let me just write this is 2.8 months.

00:02:56.951 --> 00:02:57.706
Where are we gonna be?

00:02:57.706 --> 00:03:00.132
We're gonna be 89,000

00:03:00.132 --> 00:03:03.309
times 0.944.

00:03:03.309 --> 00:03:04.470
This is where we were before

00:03:04.470 --> 00:03:05.666
at the beginning of this period.

00:03:05.666 --> 00:03:08.046
Now, we're gonna be 94.4% of that.

00:03:08.046 --> 00:03:10.696
So we're gonna multiply by 94.4% again

00:03:10.696 --> 00:03:13.154
or 0.944 again

00:03:13.154 --> 00:03:16.868
or we can just say times 0.944 squared.

00:03:16.868 --> 00:03:20.998
And after three of these periods,

00:03:20.998 --> 00:03:24.235
well, we're gonna be times 0.944 again.

00:03:24.235 --> 00:03:26.812
So it's gonna be 89,000

00:03:26.812 --> 00:03:31.812
times 0.944 squared

00:03:32.133 --> 00:03:33.909
times 0.944

00:03:33.909 --> 00:03:37.171
which is gonna be 0.944
to the third power.

00:03:37.171 --> 00:03:39.331
And I think you might
see what's going on here.

00:03:39.331 --> 00:03:41.721
We have an exponential function.

00:03:41.721 --> 00:03:43.370
Between every 2.8 months,

00:03:43.370 --> 00:03:45.961
we are multiplying by this common ratio

00:03:45.961 --> 00:03:48.399
of 0.944.

00:03:48.399 --> 00:03:52.404
And so we could write our function N of t.

00:03:52.404 --> 00:03:57.311
Our initial value is 89,000

00:03:57.311 --> 00:04:02.261
times 0.944

00:04:02.261 --> 00:04:04.943
to the power of, however, many of these

00:04:04.943 --> 00:04:07.585
2.8 month period we've gone so far.

00:04:07.585 --> 00:04:09.396
So if we take the number of months

00:04:09.396 --> 00:04:11.567
and we divide by 2.8, that's how many

00:04:11.567 --> 00:04:14.446
2.8 month period we have gotten gone.

00:04:14.446 --> 00:04:16.815
And so notice, when t equals zero,

00:04:16.815 --> 00:04:18.115
all of this turns into one,

00:04:18.115 --> 00:04:19.752
erasing the zero part, that becomes one.

00:04:19.752 --> 00:04:21.029
You have 89,000.

00:04:21.029 --> 00:04:24.326
When t is equal to 2.8,
this exponent is one.

00:04:24.326 --> 00:04:27.448
Now we're gonna multiply by 0.944 once.

00:04:27.448 --> 00:04:30.908
When t is 5.6, the exponent
is going to be two.

00:04:30.908 --> 00:04:34.066
Now we're gonna multiply by 0.944 twice.

00:04:34.066 --> 00:04:36.285
And I'm just doing the values that make

00:04:36.285 --> 00:04:38.219
the exponent integers
but it's going to work

00:04:38.219 --> 00:04:39.346
for the ones in between.

00:04:39.346 --> 00:04:40.702
I encourage you to graph it

00:04:40.702 --> 00:04:43.581
or to try those values in
a calculator if you like.

00:04:43.581 --> 00:04:44.958
But there you have it, we're done.

00:04:44.958 --> 00:04:47.540
We have modeled our narwhals.

00:04:47.540 --> 00:04:50.553
So let me just underline
that and we're done.


Constructing exponential models

WEBVTT
Kind: captions
Language: en

00:00:00.260 --> 00:00:02.628
- [Voiceover] Derek sent a
chain letter to his friends,

00:00:02.628 --> 00:00:05.321
asking them to forward the
letter to more friends.

00:00:05.321 --> 00:00:07.504
The group of people who
receive the email gains

00:00:07.504 --> 00:00:10.662
9/10 of its size every three weeks,

00:00:10.662 --> 00:00:12.867
and can be modeled by a function, P,

00:00:12.867 --> 00:00:16.280
which depends on the
amount of time, t in weeks.

00:00:16.280 --> 00:00:19.241
Derek initially sent the
chain letter to 40 friends.

00:00:19.241 --> 00:00:21.539
Write a function that
models the group of people

00:00:21.539 --> 00:00:24.047
who receive the email t weeks

00:00:24.047 --> 00:00:26.763
since Derek initially
sent the chain letter.

00:00:26.763 --> 00:00:29.758
So pause the video if you
wanna have a go at this.

00:00:29.758 --> 00:00:31.906
All right, now the way I'd
like to think about this,

00:00:31.906 --> 00:00:33.508
let's just gonna table

00:00:33.508 --> 00:00:37.181
with values for t

00:00:37.181 --> 00:00:39.422
and our function P

00:00:39.422 --> 00:00:41.988
which is a function of t.

00:00:41.988 --> 00:00:43.729
For some values that we can just pull out

00:00:43.729 --> 00:00:45.343
of the description here.

00:00:45.343 --> 00:00:48.442
So when t is zero, when
it's been zero weeks

00:00:48.442 --> 00:00:50.659
since Derek initially
sent the chain letter,

00:00:50.659 --> 00:00:52.470
how many people have gotten it?

00:00:52.470 --> 00:00:53.086
Well, they tell us.

00:00:53.086 --> 00:00:55.268
He initially, Derek initially sent

00:00:55.268 --> 00:00:57.404
the chain letter to 40 friends.

00:00:57.404 --> 00:00:58.925
So t equals zero.

00:00:58.925 --> 00:01:01.653
P of t or P of zero is 40.

00:01:01.653 --> 00:01:04.486
Now, what's an interesting time period?

00:01:04.486 --> 00:01:06.007
It says that the email,
the number of people

00:01:06.007 --> 00:01:08.491
who receive the email gains 9/10

00:01:08.491 --> 00:01:11.950
or increases by 9/10 every three weeks,

00:01:11.950 --> 00:01:13.564
every three weeks.

00:01:13.564 --> 00:01:15.770
So after three weeks,

00:01:15.770 --> 00:01:17.453
so three weeks have gone by.

00:01:17.453 --> 00:01:20.146
So I'm just adding three to t.

00:01:20.146 --> 00:01:22.004
What is P of t going to be?

00:01:22.004 --> 00:01:25.321
Well, they tell us it's going
to gain 9/10 of its size.

00:01:25.321 --> 00:01:27.981
So it's going to be 40

00:01:27.981 --> 00:01:32.981
plus 9/10 times 40

00:01:33.183 --> 00:01:35.505
which is going to be equal to, what?

00:01:35.505 --> 00:01:37.741
Well, that's equal to 40.

00:01:37.741 --> 00:01:39.006
We factor out our 40.

00:01:39.006 --> 00:01:43.162
40 times one plus 9/10

00:01:43.162 --> 00:01:46.598
or you could say this
is equal to 40 times,

00:01:46.598 --> 00:01:51.184
whoops, 40 times 1.9.

00:01:51.184 --> 00:01:52.206
Another way of thinking about it,

00:01:52.206 --> 00:01:54.678
after three weeks, we've grown 90%.

00:01:54.678 --> 00:01:56.478
That's another way of saying
that the number of people

00:01:56.478 --> 00:01:59.568
who receive the email
gains 9/10 of its size.

00:01:59.568 --> 00:02:02.308
You could say the group of
people who receive the email

00:02:02.308 --> 00:02:06.303
grows 90% every three weeks.

00:02:06.303 --> 00:02:08.451
And so if we go another three weeks,

00:02:08.451 --> 00:02:09.856
so plus another three weeks,

00:02:09.856 --> 00:02:10.838
I could say, well, let me just write this,

00:02:10.838 --> 00:02:12.231
this is six weeks.

00:02:12.231 --> 00:02:14.739
Well, how many people would
have received the email?

00:02:14.739 --> 00:02:16.097
It was going to be this number.

00:02:16.097 --> 00:02:18.651
It's going to be grown another 90%.

00:02:18.651 --> 00:02:21.240
So we're gonna multiply
it times 1.9 again.

00:02:21.240 --> 00:02:22.691
So it's going to be 40

00:02:22.691 --> 00:02:26.116
times 1.9 times 1.9.

00:02:26.116 --> 00:02:28.252
We're gonna grow by another 9/10.

00:02:28.252 --> 00:02:30.817
Growing by 9/10 is the
same thing as multiplying

00:02:30.817 --> 00:02:33.011
by one and 9/10.

00:02:33.011 --> 00:02:34.439
The one is what you already are.

00:02:34.439 --> 00:02:36.517
Then you're growing by another 9/10.

00:02:36.517 --> 00:02:38.433
So this is the same thing as 40

00:02:38.433 --> 00:02:41.811
times 1.9 squared.

00:02:41.811 --> 00:02:44.690
We go another three weeks,

00:02:44.690 --> 00:02:45.816
nine weeks.

00:02:45.816 --> 00:02:47.492
We're gonna grow another 90%.

00:02:47.492 --> 00:02:48.955
So you're gonna multiply it by,

00:02:48.955 --> 00:02:50.092
you're gonna take this number

00:02:50.092 --> 00:02:51.938
and multiply by 1.9 again

00:02:51.938 --> 00:02:55.851
which is going to be
1.9 to the third power.

00:02:55.851 --> 00:02:58.056
And so what's going on over here?

00:02:58.056 --> 00:02:59.751
Well, we can see some
exponential function.

00:02:59.751 --> 00:03:01.318
We have our initial value.

00:03:01.318 --> 00:03:03.849
And every three weeks,
we're multiplying by 1.9.

00:03:03.849 --> 00:03:06.113
So 1.9 would be our common ratio.

00:03:06.113 --> 00:03:11.113
So we could say that P of t is
equal to our initial value 40

00:03:11.295 --> 00:03:14.309
times our common ratio 1.9.

00:03:14.309 --> 00:03:18.438
And we multiply by 1.9 every three weeks.

00:03:18.438 --> 00:03:19.402
So we could just say,

00:03:19.402 --> 00:03:22.199
how many three week
periods have passed by?

00:03:22.199 --> 00:03:25.357
Well, we will take t
and divide it by three.

00:03:25.357 --> 00:03:27.342
T divided by three is a
number of three week periods

00:03:27.342 --> 00:03:28.886
that have gone by.

00:03:28.886 --> 00:03:29.664
And there you have it.

00:03:29.664 --> 00:03:31.487
And notice, t equals zero,

00:03:31.487 --> 00:03:34.366
1.9 to the zeroth power is one

00:03:34.366 --> 00:03:35.759
so 40 times one.

00:03:35.759 --> 00:03:39.485
T equals three, that's gonna
be 1.9 to the first power,

00:03:39.485 --> 00:03:40.600
three over three.

00:03:40.600 --> 00:03:43.943
And so we're gonna grow by
90% and so on and so forth.

00:03:43.943 --> 00:03:46.376
So feeling pretty good about this.


Constructing exponential models: half life

WEBVTT
Kind: captions
Language: en

00:00:00.167 --> 00:00:02.210
- [Voiceover] We're told
carbon-14 is an element

00:00:02.210 --> 00:00:07.065
which loses exactly half of
its mass every 5730 years.

00:00:07.065 --> 00:00:09.188
The mass of a sample of carbon-14

00:00:09.188 --> 00:00:11.405
can be modeled by a function, M,

00:00:11.405 --> 00:00:14.620
which depends on its age, t, in years.

00:00:14.620 --> 00:00:17.149
We measure that the initial mass

00:00:17.149 --> 00:00:20.587
of a sample of carbon-14 is 741 grams.

00:00:20.587 --> 00:00:22.497
Write a function that models the mass

00:00:22.497 --> 00:00:25.210
of the carbon-14 sample remaining t years

00:00:25.210 --> 00:00:27.066
since the initial measurement.

00:00:27.066 --> 00:00:29.143
Alright, so, like always, pause the video

00:00:29.143 --> 00:00:30.978
and see if you can come
up with this function, M,

00:00:30.978 --> 00:00:33.067
that is going to be a function of t,

00:00:33.067 --> 00:00:36.457
the years since the initial measurement.

00:00:36.457 --> 00:00:37.908
Alright, let's work through it together.

00:00:37.908 --> 00:00:39.545
What I like to do is, I
always like to start off

00:00:39.545 --> 00:00:42.785
with a little bit of a table
to get a sense of things.

00:00:42.785 --> 00:00:46.266
So let's think about t, how much time,

00:00:46.266 --> 00:00:48.812
how many years have passed
since the initial measurement,

00:00:48.812 --> 00:00:53.625
and what the amount of
mass we're going to have.

00:00:53.625 --> 00:00:55.527
Well, we know that the initial,

00:00:55.527 --> 00:00:56.859
we know that the initial mass

00:00:56.859 --> 00:01:00.885
of a sample of carbon-14 is 741 grams,

00:01:00.885 --> 00:01:05.885
so at t equals zero, our mass is 741.

00:01:06.062 --> 00:01:10.010
Now, what's another interesting
t that we could think about?

00:01:10.010 --> 00:01:13.446
Well, we know at every 5730 years,

00:01:13.446 --> 00:01:17.150
we lose exactly half of
our mass of carbon-14.

00:01:17.150 --> 00:01:19.982
Every 5730 years.

00:01:19.982 --> 00:01:24.521
So let's think about what
happens when t is 5730.

00:01:24.521 --> 00:01:26.761
Well, we're going to
lose half of our mass,

00:01:26.761 --> 00:01:30.441
so we're going to multiply this times 1/2.

00:01:30.441 --> 00:01:34.702
So this is going to be 741 times 1/2.

00:01:34.702 --> 00:01:37.244
I'm not even gonna calculate
what that is right now.

00:01:37.244 --> 00:01:41.123
And then let's say we have
another 5730 years take place,

00:01:41.123 --> 00:01:42.445
so that's going to be,

00:01:42.445 --> 00:01:45.986
and I'm just gonna write two times 5730.

00:01:45.986 --> 00:01:47.589
I could calculate what it's going to be.

00:01:47.589 --> 00:01:48.319
10,000,

00:01:48.319 --> 00:01:51.303
11,460 or something like that.

00:01:51.303 --> 00:01:54.147
Alright, but let's just
go with two times 5730.

00:01:54.147 --> 00:01:55.145
Is it 10? Yeah.

00:01:55.145 --> 00:01:59.104
10,000 plus 1400 so 11,400 plus 60.

00:01:59.104 --> 00:02:00.764
Yeah. So 11,460.

00:02:00.764 --> 00:02:02.285
But let's just leave it like this.

00:02:02.285 --> 00:02:04.665
Well, then, it's gonna be this times 1/2.

00:02:04.665 --> 00:02:09.665
So it's gonna be 741 times 1/2 times 1/2.

00:02:10.318 --> 00:02:13.662
So we're gonna multiply by 1/2 again.

00:02:13.662 --> 00:02:18.662
And so this is the same thing
as 741 times 1/2 squared.

00:02:20.338 --> 00:02:21.765
And then, let's just think about

00:02:21.765 --> 00:02:25.062
if we wait another 5730 years,

00:02:25.062 --> 00:02:28.695
so three times 5730.

00:02:28.695 --> 00:02:31.760
Well, then it's going
to be 1/2 times this.

00:02:31.760 --> 00:02:34.183
So it's going to be 741,

00:02:34.183 --> 00:02:39.058
this times 1/2 is gonna
be 1/2 to the third power.

00:02:39.058 --> 00:02:41.903
So you might notice a little
bit of a pattern here.

00:02:41.903 --> 00:02:45.141
However many half-lifes we
have, we're gonna multiply,

00:02:45.141 --> 00:02:47.358
we're gonna raise 1/2 to that power

00:02:47.358 --> 00:02:51.237
and then we multiply it
times our initial mass.

00:02:51.237 --> 00:02:53.105
This is one half-life has gone by,

00:02:53.105 --> 00:02:55.543
two half-lifes, we have
an exponent of two,

00:02:55.543 --> 00:02:57.737
three half-lifes, we multiply by three.

00:02:57.737 --> 00:03:00.697
Sorry, we multiply by 1/2 three times.

00:03:00.697 --> 00:03:03.634
So what's going to be a
general way to express M of t?

00:03:03.634 --> 00:03:08.634
Well, M of t is going to be
our initial value, 741, times,

00:03:09.381 --> 00:03:11.518
and you might already be identifying this

00:03:11.518 --> 00:03:13.677
as an exponential function,

00:03:13.677 --> 00:03:16.063
we're going to multiply times this number,

00:03:16.063 --> 00:03:17.878
which we could call our common ratio,

00:03:17.878 --> 00:03:20.235
as many half-lifes has passed by.

00:03:20.235 --> 00:03:22.719
So how do we know how many
half-lives have passed by?

00:03:22.719 --> 00:03:24.298
Well, we could take t,

00:03:24.298 --> 00:03:26.501
and we could divide it by the half-life.

00:03:27.781 --> 00:03:29.446
And try to test this out.

00:03:29.446 --> 00:03:31.804
When t equals zero, it's gonna
be 1/2 to the zeroth power,

00:03:31.804 --> 00:03:34.984
which is just one, and
we're just gonna have 741.

00:03:34.984 --> 00:03:37.317
When t is equal to 5730,

00:03:37.317 --> 00:03:40.545
this exponent is going to be
one, which we want it to be.

00:03:40.545 --> 00:03:43.482
We're just gonna multiply our
initial value by 1/2 once.

00:03:43.482 --> 00:03:45.943
When this exponent is two times 5730,

00:03:45.943 --> 00:03:48.464
when t is two times 5730,

00:03:48.464 --> 00:03:50.099
well then the exponent is going to be two,

00:03:50.099 --> 00:03:52.177
and we're gonna multiply by 1/2 twice.

00:03:52.177 --> 00:03:54.047
It's going to be 1/2 to the second power.

00:03:54.047 --> 00:03:56.658
And it's going to work
for everything in between.

00:03:56.658 --> 00:03:58.701
When we are a fraction
of a half-life along,

00:03:58.701 --> 00:04:01.696
we're gonna get a non-integer exponent,

00:04:01.696 --> 00:04:03.763
and that, too, will work out.

00:04:03.763 --> 00:04:05.423
And so this is our function.

00:04:05.423 --> 00:04:07.107
We are,

00:04:07.107 --> 00:04:09.220
we are done.

00:04:09.220 --> 00:04:11.435
We have written our function, M,

00:04:11.435 --> 00:04:13.455
that models the mass
of carbon-14 remaining

00:04:13.455 --> 00:04:16.307
t years since the initial measurement.


Transforming exponential graphs

WEBVTT
Kind: captions
Language: en

00:00:00.258 --> 00:00:01.384
- [Voiceover] We're told the graph of y

00:00:01.384 --> 00:00:04.008
equals two to the x is
shown below, alright.

00:00:04.008 --> 00:00:05.377
Which of the following is the graph

00:00:05.377 --> 00:00:09.358
of y equals two to the
negative x minus five?

00:00:09.358 --> 00:00:10.577
So there's two changes here.

00:00:10.577 --> 00:00:13.271
Instead of two to the x, we
have two to the negative x

00:00:13.271 --> 00:00:14.431
and then, we're not leaving that alone,

00:00:14.431 --> 00:00:15.906
we, then, subtract five.

00:00:15.906 --> 00:00:17.636
So let's take 'em step by step.

00:00:17.636 --> 00:00:19.911
So let's first think about what y

00:00:19.911 --> 00:00:22.929
equals two to the negative
x would look like.

00:00:22.929 --> 00:00:25.670
Well, any input we now put into an x,

00:00:25.670 --> 00:00:27.306
we're now going to take the negative of.

00:00:27.306 --> 00:00:31.091
So if I input a two, it's
like taking the opposite

00:00:31.091 --> 00:00:35.572
of the two and then, inputting
that into two to the x.

00:00:35.572 --> 00:00:37.339
And so, what we're essentially going to do

00:00:37.339 --> 00:00:41.506
is flip this graph over the y-axis.

00:00:41.506 --> 00:00:44.885
So, here we have the point two comma four.

00:00:44.885 --> 00:00:46.544
Over here, we're going to have the point

00:00:46.544 --> 00:00:49.620
negative two comma four.

00:00:49.620 --> 00:00:52.303
When x is zero, they're going
to give us the same value.

00:00:52.303 --> 00:00:54.555
So they're both going to
have the same y-intercept.

00:00:54.555 --> 00:00:57.263
And so are graph is going to look like,

00:00:57.263 --> 00:01:01.327
our graph is going to
look something like, this.

00:01:01.327 --> 00:01:02.406
They're going to be mirror images

00:01:02.406 --> 00:01:04.820
flipped around the y-axis.

00:01:04.820 --> 00:01:09.820
So, it's going to look like that.

00:01:10.440 --> 00:01:15.440
That is the graph of y is
equal to two to the negative x.

00:01:15.687 --> 00:01:16.886
And then we have to worry about

00:01:16.886 --> 00:01:19.127
the subtracting five from it.

00:01:19.127 --> 00:01:20.868
Well, that's, you're subtracting five

00:01:20.868 --> 00:01:23.097
from your final y-value
so that's going to work.

00:01:23.097 --> 00:01:25.779
You're subtracting five
to get your y-value now,

00:01:25.779 --> 00:01:27.578
where your y-value is
going to be five lower,

00:01:27.578 --> 00:01:28.867
is I guess the best way to say it,

00:01:28.867 --> 00:01:31.699
so this is going to shift
the graph down by five.

00:01:31.699 --> 00:01:33.267
So instead of having
the y-intercept there,

00:01:33.267 --> 00:01:34.561
it's going to be five lower.

00:01:34.561 --> 00:01:38.908
One, two, one, each hash mark is two

00:01:38.908 --> 00:01:41.961
so this is one, two, three, four, five,

00:01:41.961 --> 00:01:43.319
is going to be right over there.

00:01:43.319 --> 00:01:47.417
So shift down by five, two, four, five.

00:01:47.417 --> 00:01:48.381
It's going to look like that.

00:01:48.381 --> 00:01:50.343
And the asymptote,
instead of the asymptote,

00:01:50.343 --> 00:01:54.243
going towards y equals zero, the asymptote

00:01:54.243 --> 00:01:58.341
is going to be at y is
equal to negative five.

00:01:58.341 --> 00:02:02.497
So the asymptote is going to
be y equals negative five.

00:02:02.497 --> 00:02:07.497
So it should look something like,

00:02:08.362 --> 00:02:13.214
something like what I
am drawing right now.

00:02:13.214 --> 00:02:17.324
Something like, something like that.

00:02:17.324 --> 00:02:18.984
So now we can look at which choices.

00:02:18.984 --> 00:02:21.457
So this should be the graph of y

00:02:21.457 --> 00:02:24.579
equals two to the negative x minus five.

00:02:24.579 --> 00:02:27.122
So let's see which of
these choices depict that.

00:02:28.267 --> 00:02:31.588
So this first choice
actually seems to be spot-on.

00:02:31.588 --> 00:02:33.364
It's exactly what we drew.

00:02:33.364 --> 00:02:35.697
Well, we can look at the
other ones, just in case.

00:02:35.697 --> 00:02:39.006
Well, this is, this looks like,
what did they do over here?

00:02:39.006 --> 00:02:40.863
Looks like they, instead of flipping over

00:02:40.863 --> 00:02:44.717
the y-axis, they took the,
they flipped over the x-axis

00:02:44.717 --> 00:02:47.004
and then they shifted
down, so that's not right.

00:02:47.004 --> 00:02:48.316
Here it looks like they got what we got

00:02:48.316 --> 00:02:50.835
but they flipped it over the x-axis.

00:02:50.835 --> 00:02:53.296
And this looks like they
flipped it over the y-axis

00:02:53.296 --> 00:02:54.922
but then they shifted,
instead of shifting down

00:02:54.922 --> 00:02:57.301
by five, it looks like they
shifted to the left by five.

00:02:57.301 --> 00:02:58.962
So we should feel pretty good especially

00:02:58.962 --> 00:03:00.208
'cause we essentially drew this

00:03:00.208 --> 00:03:02.877
before even looking at the choice.


Graphs of rational functions: zeros

WEBVTT
Kind: captions
Language: en

00:00:00.643 --> 00:00:02.322
- [Voiceover] So we're told let f of x

00:00:02.322 --> 00:00:06.787
equal two x squared minus 18 over g of x

00:00:06.787 --> 00:00:08.851
where g of x is a polynomial.

00:00:08.851 --> 00:00:10.395
And then they tell us,
which of the following is

00:00:10.395 --> 00:00:14.249
a possible graph of y is equal to f of x?

00:00:14.249 --> 00:00:16.954
And they gave us four choices here.

00:00:16.954 --> 00:00:19.613
And like always, I encourage
you to pause the video

00:00:19.613 --> 00:00:20.890
and see if you could give a go at it.

00:00:20.890 --> 00:00:22.213
Look at our f of x

00:00:22.213 --> 00:00:24.368
and think about which of these graphs

00:00:24.368 --> 00:00:27.085
actually match up or could match up

00:00:27.085 --> 00:00:29.081
to that f of x?

00:00:29.081 --> 00:00:31.287
All right, now let's work
through this together.

00:00:31.287 --> 00:00:32.959
And they don't give us
a lot of information.

00:00:32.959 --> 00:00:35.002
They don't tell us anything
about the denominator

00:00:35.002 --> 00:00:37.289
of this rational expression.

00:00:37.289 --> 00:00:38.868
They do tell us it's numerator.

00:00:38.868 --> 00:00:41.445
And like we've seen before,
it's useful to factor

00:00:41.445 --> 00:00:42.664
the numerator and see,

00:00:42.664 --> 00:00:44.928
and what x values do
interesting things happen.

00:00:44.928 --> 00:00:46.541
And particular, at what x values

00:00:46.541 --> 00:00:48.898
does the numerator equal zero?

00:00:48.898 --> 00:00:51.243
So if we factor the numerator up here,

00:00:51.243 --> 00:00:53.367
we could rewrite f of x

00:00:53.367 --> 00:00:56.444
as being equal to, let's see
we could factor out a two

00:00:56.444 --> 00:00:57.825
out of the numerator.

00:00:57.825 --> 00:01:01.540
So it's two times x squared minus nine.

00:01:01.540 --> 00:01:04.524
And that's all going to be over g of x.

00:01:04.524 --> 00:01:05.963
We don't know what the denominator is.

00:01:05.963 --> 00:01:08.285
We just know that it's a polynomial.

00:01:08.285 --> 00:01:09.334
Now see, here are the numerator

00:01:09.334 --> 00:01:11.133
x squared minus nine.

00:01:11.133 --> 00:01:12.189
You might recognize that.

00:01:12.189 --> 00:01:13.803
It's a difference of squares

00:01:13.803 --> 00:01:15.486
so we can factor that further.

00:01:15.486 --> 00:01:17.257
So we still have that original two

00:01:17.257 --> 00:01:19.915
and it's going to be x plus three

00:01:19.915 --> 00:01:22.423
times x minus three.

00:01:22.423 --> 00:01:23.920
We've seen that multiple times.

00:01:23.920 --> 00:01:25.778
If that looks unfamiliar to you,

00:01:25.778 --> 00:01:27.124
I encourage you to watch the video

00:01:27.124 --> 00:01:30.004
on differences of squares
or factoring polynomials.

00:01:30.004 --> 00:01:31.536
X squared minus nine is the same thing

00:01:31.536 --> 00:01:33.417
as x squared minus three squared.

00:01:33.417 --> 00:01:36.165
So it's x plus three times x minus three.

00:01:36.165 --> 00:01:40.835
And then all of that is still
going to be over g of x.

00:01:40.835 --> 00:01:42.318
So the first thing that we might realize,

00:01:42.318 --> 00:01:44.756
okay, when is our numerator equal zero?

00:01:44.756 --> 00:01:47.833
Well, when x is equal to negative three

00:01:47.833 --> 00:01:49.632
or when x is equal to positive three.

00:01:49.632 --> 00:01:51.191
If x is equal to negative three,

00:01:51.191 --> 00:01:53.625
this expression is going to be zero.

00:01:53.625 --> 00:01:55.420
If x is equal to positive three,

00:01:55.420 --> 00:01:57.382
this expression is equal to zero.

00:01:57.382 --> 00:01:59.785
So you might just say,
"Oh, maybe we have zeros

00:01:59.785 --> 00:02:01.619
"at plus or minus three."

00:02:01.619 --> 00:02:05.938
So maybe at x is maybe,

00:02:05.938 --> 00:02:08.759
maybe f of negative three

00:02:08.759 --> 00:02:10.001
is equal to zero

00:02:10.001 --> 00:02:13.251
and f of positive three

00:02:13.251 --> 00:02:14.250
is equal to zero.

00:02:14.250 --> 00:02:15.743
Those values sure look they made

00:02:15.743 --> 00:02:17.380
the numerator equal to zero.

00:02:17.380 --> 00:02:19.760
And then we look at our choices.

00:02:19.760 --> 00:02:21.444
And when we look at our choices,

00:02:21.444 --> 00:02:25.278
this choice A does seem to have a zero

00:02:25.278 --> 00:02:26.845
at positive three

00:02:26.845 --> 00:02:28.865
but it doesn't have one at negative three.

00:02:28.865 --> 00:02:31.164
It has a vertical asymptote
at negative three.

00:02:31.164 --> 00:02:34.289
So that seems a little bit confusing.

00:02:34.289 --> 00:02:36.622
This choice B does have a zero

00:02:36.622 --> 00:02:38.724
at positive three

00:02:38.724 --> 00:02:41.289
but it has nothing going on here at,

00:02:41.289 --> 00:02:43.541
or nothing interesting
on at negative three.

00:02:43.541 --> 00:02:44.842
It defines negative three.

00:02:44.842 --> 00:02:46.629
It doesn't even have a
vertical asymptote there.

00:02:46.629 --> 00:02:49.821
So once again, this looks
a little bit perplexing.

00:02:49.821 --> 00:02:54.821
Choice C has a removable discontinuity

00:02:54.941 --> 00:02:56.625
at positive three

00:02:56.625 --> 00:02:59.451
and then it has a vertical asymptote

00:02:59.451 --> 00:03:01.490
at negative two.

00:03:01.490 --> 00:03:03.070
So once again, this doesn't have anything

00:03:03.070 --> 00:03:05.996
interesting going on at x
is equal to negative three.

00:03:05.996 --> 00:03:07.795
So still a little perplexing.

00:03:07.795 --> 00:03:08.933
And this one has a zero

00:03:08.933 --> 00:03:11.115
at positive six and negative six.

00:03:11.115 --> 00:03:13.507
So none of the choices had zeros

00:03:13.507 --> 00:03:16.513
at both x equals positive three

00:03:16.513 --> 00:03:17.906
and x equals negative three.

00:03:17.906 --> 00:03:19.427
So, what's going on?

00:03:19.427 --> 00:03:20.669
Well, what we need to realize is

00:03:20.669 --> 00:03:22.608
just because something makes the numerator

00:03:22.608 --> 00:03:24.373
equal to zero doesn't mean

00:03:24.373 --> 00:03:27.507
that it's definitely going to
be a zero for that function.

00:03:27.507 --> 00:03:29.132
And you might say,
"Well, how can that be?"

00:03:29.132 --> 00:03:31.152
Well, think about situations
in which those values

00:03:31.152 --> 00:03:33.753
would also make the
denominator equal to zero.

00:03:33.753 --> 00:03:36.771
So let me write out some
potential f of x is here.

00:03:36.771 --> 00:03:39.035
So we just know that g
of x is a polynomial.

00:03:39.035 --> 00:03:41.426
So f of x could be, we know the numerator,

00:03:41.426 --> 00:03:43.307
two times x plus three

00:03:43.307 --> 00:03:47.261
times x minus three, over.

00:03:47.261 --> 00:03:48.837
Well, let's just say g of x is,

00:03:48.837 --> 00:03:50.079
I'm just gonna make up something,

00:03:50.079 --> 00:03:52.656
g of x is equal to x plus one.

00:03:52.656 --> 00:03:54.514
Well, in this situation,
none of the values

00:03:54.514 --> 00:03:56.476
that make the numerator equal zero

00:03:56.476 --> 00:03:58.054
make the denominator equal zero.

00:03:58.054 --> 00:04:00.852
So this is a situation where
you would have two zeros

00:04:00.852 --> 00:04:02.535
at x equals positive three

00:04:02.535 --> 00:04:03.812
and x equals negative three.

00:04:03.812 --> 00:04:07.318
So this would be the two zeros.

00:04:07.318 --> 00:04:10.975
Two zeros.

00:04:10.975 --> 00:04:13.738
And so let's look at
another similar situation.

00:04:13.738 --> 00:04:16.080
Let's look at a situation where f of x

00:04:16.080 --> 00:04:18.518
is equal to, you know, the numerator,

00:04:18.518 --> 00:04:20.464
x plus three

00:04:20.464 --> 00:04:22.514
times x minus three.

00:04:22.514 --> 00:04:25.022
And let's say that we do have,

00:04:25.022 --> 00:04:26.821
let's say that one of those x values,

00:04:26.821 --> 00:04:28.040
positive or negative three,

00:04:28.040 --> 00:04:31.116
do make the denominator equal to zero.

00:04:31.116 --> 00:04:34.112
So let's say, x plus three

00:04:34.112 --> 00:04:36.898
and then, say times x plus one.

00:04:36.898 --> 00:04:40.380
Well, you see here, now since x plus three

00:04:40.380 --> 00:04:42.714
is both the numerator and the denominator,

00:04:42.714 --> 00:04:43.956
you could divide x plus three

00:04:43.956 --> 00:04:45.361
divided by x plus three.

00:04:45.361 --> 00:04:46.580
They cancel out.

00:04:46.580 --> 00:04:48.576
And here x equals negative three

00:04:48.576 --> 00:04:50.770
would be a removable discontinuity.

00:04:50.770 --> 00:04:55.449
So this would have zero

00:04:55.449 --> 00:04:57.713
at x equals three

00:04:57.713 --> 00:05:02.496
and a removable discontinuity

00:05:02.496 --> 00:05:06.152
at x equals negative three.

00:05:06.152 --> 00:05:09.465
And so those values that make
the numerator equal to zero

00:05:09.465 --> 00:05:11.207
we now see it could be a zero

00:05:11.207 --> 00:05:14.492
or it could represent a
removable discontinuity.

00:05:14.492 --> 00:05:16.547
And here I just pick a
removable discontinuity

00:05:16.547 --> 00:05:17.476
be at negative three

00:05:17.476 --> 00:05:19.089
or it could be the other way around

00:05:19.089 --> 00:05:20.424
or it could be at both values.

00:05:20.424 --> 00:05:22.478
If this was x plus three
times x minus three

00:05:22.478 --> 00:05:24.951
over x plus three times x minus three,

00:05:24.951 --> 00:05:26.600
then you would have a
removable discontinuity

00:05:26.600 --> 00:05:29.711
at both x is positive
three and negative three.

00:05:29.711 --> 00:05:31.092
And then you could go even further.

00:05:31.092 --> 00:05:33.159
F of x could look like this.

00:05:33.159 --> 00:05:36.409
It could be two times x plus three

00:05:36.409 --> 00:05:41.099
times x minus three over

00:05:41.099 --> 00:05:44.060
x plus three squared

00:05:44.060 --> 00:05:46.474
and I'll just make up some
other, another expression here,

00:05:46.474 --> 00:05:47.451
x plus one.

00:05:47.451 --> 00:05:48.751
So, what's going to happen here?

00:05:48.751 --> 00:05:50.318
Even if you divide the
numerator and the denominator

00:05:50.318 --> 00:05:51.978
by x plus three, you're
still going to have

00:05:51.978 --> 00:05:54.254
one x plus three left
over the denominator.

00:05:54.254 --> 00:05:56.877
That could cancel with
one of the x plus threes

00:05:56.877 --> 00:05:58.793
but you're still going
to have an x plus three.

00:05:58.793 --> 00:06:02.148
And so in this case, you would
have a vertical asymptote.

00:06:02.148 --> 00:06:06.234
So in this case, you would have a zero

00:06:06.234 --> 00:06:07.871
at x is equal to three.

00:06:07.871 --> 00:06:11.861
And you would have a vertical asymptote.

00:06:11.861 --> 00:06:14.138
I'll just shorten it shorthand.

00:06:14.138 --> 00:06:15.573
You would have a vertical,
I'll just write that out,

00:06:15.573 --> 00:06:17.917
vertical asymptote

00:06:17.917 --> 00:06:20.006
at x is equal to negative three.

00:06:20.006 --> 00:06:22.255
So these particular examples
that I just showed you

00:06:22.255 --> 00:06:24.055
showed you that any value that makes

00:06:24.055 --> 00:06:25.274
the numerator equals zero

00:06:25.274 --> 00:06:27.653
aren't necessarily zeros for the function.

00:06:27.653 --> 00:06:29.058
They could be zeros.

00:06:29.058 --> 00:06:31.380
They could be removable discontinuities

00:06:31.380 --> 00:06:33.377
or they could be vertical asymptotes.

00:06:33.377 --> 00:06:34.282
But they would all occur

00:06:34.282 --> 00:06:36.523
at x is equal to positive
or negative three.

00:06:36.523 --> 00:06:39.576
So if that lands, now let's
look at the choices again.

00:06:39.576 --> 00:06:43.418
So choice A has a zero at
x equals positive three

00:06:43.418 --> 00:06:47.122
and it has a vertical asymptote
at x equals negative three.

00:06:47.122 --> 00:06:51.020
So that's actually very
consistent with this

00:06:51.020 --> 00:06:52.915
situation that I just described.

00:06:52.915 --> 00:06:56.494
So choice A actually
is looking pretty good.

00:06:56.494 --> 00:07:00.217
Choice B has a zero at x equals three

00:07:00.217 --> 00:07:03.084
but its vertical asymptote
looks like it's an x equals two

00:07:03.084 --> 00:07:04.426
and nothing interesting is happening

00:07:04.426 --> 00:07:06.191
at x equals negative three.

00:07:06.191 --> 00:07:08.791
So we could rule that out.

00:07:08.791 --> 00:07:10.536
If I look at choice C,

00:07:10.536 --> 00:07:12.463
you have a removable discontinuity

00:07:12.463 --> 00:07:14.158
at x equals positive three

00:07:14.158 --> 00:07:16.225
which is completely possible.

00:07:16.225 --> 00:07:17.803
We've seen that situation where something

00:07:17.803 --> 00:07:19.580
that makes the numerator equals zero

00:07:19.580 --> 00:07:21.321
could be a removable discontinuity

00:07:21.321 --> 00:07:24.618
if you have that same
expression in the denominator.

00:07:24.618 --> 00:07:26.081
But then the vertical asymptote

00:07:26.081 --> 00:07:27.718
isn't that x equals negative three.

00:07:27.718 --> 00:07:29.343
It's an x equals negative two.

00:07:29.343 --> 00:07:30.318
So that rules it out.

00:07:30.318 --> 00:07:31.781
Once again, nothing interesting happening

00:07:31.781 --> 00:07:33.522
at x equals negative three.

00:07:33.522 --> 00:07:34.799
And here you have two zeros

00:07:34.799 --> 00:07:36.715
but they're not at x equals
positive or negative three.

00:07:36.715 --> 00:07:38.340
They're x equals positive or negative six.

00:07:38.340 --> 00:07:40.239
We can definitely rule that one out.

00:07:40.239 --> 00:07:42.758
So we should feel pretty good about,

00:07:42.758 --> 00:07:46.071
we should feel pretty good about choice A.


Graphs of rational functions: y-intercept

WEBVTT
Kind: captions
Language: en

00:00:00.446 --> 00:00:04.117
- [Voiceover] Let f of
x equal a times x to n

00:00:04.117 --> 00:00:06.474
plus b x plus 12

00:00:06.474 --> 00:00:09.620
over c times x to the m

00:00:09.620 --> 00:00:11.640
plus d x plus 12

00:00:11.640 --> 00:00:13.776
where m and n are integers

00:00:13.776 --> 00:00:17.177
and a, b, c and d are
unknown constants are.

00:00:17.177 --> 00:00:18.477
This is interesting.

00:00:18.477 --> 00:00:22.216
Which of the following is a possible graph

00:00:22.216 --> 00:00:25.199
of y is equal to f of x?

00:00:25.199 --> 00:00:27.892
And it tells us the dashed
lines indicate asymptotes.

00:00:27.892 --> 00:00:30.574
So they gave us four choices here.

00:00:30.574 --> 00:00:32.072
And so I encourage you like always.

00:00:32.072 --> 00:00:32.781
Pause this video.

00:00:32.781 --> 00:00:33.511
Give a go at it.

00:00:33.511 --> 00:00:35.741
See if you can figure
out which of these graphs

00:00:35.741 --> 00:00:38.913
could be the graph for y equals f of x

00:00:38.913 --> 00:00:40.852
where f of x, where they've given us

00:00:40.852 --> 00:00:43.707
this information about f of x.

00:00:43.707 --> 00:00:45.228
All right, now let's
work to this together.

00:00:45.228 --> 00:00:46.308
Now this is really interesting.

00:00:46.308 --> 00:00:48.572
It's, you know, they didn't give us a lot.

00:00:48.572 --> 00:00:50.875
They didn't even tell us
what the exponents on x are.

00:00:50.875 --> 00:00:52.627
They haven't even given
us the coefficients.

00:00:52.627 --> 00:00:55.389
All they've told us is this 12 here.

00:00:55.389 --> 00:00:58.374
So this 12 looks like a pretty big clue.

00:00:58.374 --> 00:00:59.923
So what can that tell us?

00:00:59.923 --> 00:01:00.816
The way they've written that

00:01:00.816 --> 00:01:04.845
we really can't deduce any
zeros for the function.

00:01:04.845 --> 00:01:06.589
We really can't deduce what x values

00:01:06.589 --> 00:01:08.296
make the numerator equals zero

00:01:08.296 --> 00:01:11.117
or what x values make the
denominator equal zero.

00:01:11.117 --> 00:01:13.021
So it's gonna be hard for us to deduce

00:01:13.021 --> 00:01:14.657
what are the zeros of the function

00:01:14.657 --> 00:01:16.573
or what are the removable discontinuities

00:01:16.573 --> 00:01:18.314
or what are the vertical asymptotes.

00:01:18.314 --> 00:01:21.437
But just these 12 sitting
here does tell us one thing.

00:01:21.437 --> 00:01:23.979
What happens when x equals zero?

00:01:23.979 --> 00:01:26.673
Because when x equal
zero, every other term

00:01:26.673 --> 00:01:28.414
in this rational expression is

00:01:28.414 --> 00:01:30.202
just going to be equal to zero.

00:01:30.202 --> 00:01:32.698
And so we can figure out f of zero.

00:01:32.698 --> 00:01:36.018
F of zero is going to be equal to,

00:01:36.018 --> 00:01:38.235
well a times zero to the n,

00:01:38.235 --> 00:01:40.418
well that's just going to be a zero

00:01:40.418 --> 00:01:42.484
plus b times zero.

00:01:42.484 --> 00:01:44.144
Well, that's just going to be zero.

00:01:44.144 --> 00:01:47.143
Plus 12 over

00:01:48.173 --> 00:01:50.835
c times zero to the m power.

00:01:50.835 --> 00:01:53.017
Well, that's just going to be zero.

00:01:53.017 --> 00:01:56.042
D times zero is going to be a zero.

00:01:56.042 --> 00:01:57.742
And then we have our 12 there.

00:01:57.742 --> 00:01:59.252
So we're actually able to figure out

00:01:59.252 --> 00:02:00.494
what f of zero is.

00:02:00.494 --> 00:02:02.769
It's 12 over 12

00:02:02.769 --> 00:02:04.367
or one.

00:02:04.367 --> 00:02:07.246
So we actually know the y
intercept for this function.

00:02:07.246 --> 00:02:08.848
Let's see if that's enough information

00:02:08.848 --> 00:02:10.821
for us to figure out
if any of these choices

00:02:10.821 --> 00:02:14.769
could be the graph of y equals f of x.

00:02:14.769 --> 00:02:16.301
So let's see here.

00:02:16.301 --> 00:02:20.538
So choice a, our y intercept is a two.

00:02:20.538 --> 00:02:23.406
When x equals zero, our graph goes to two.

00:02:23.406 --> 00:02:24.880
So we can rule that out.

00:02:24.880 --> 00:02:27.004
The y intercept needs to be one.

00:02:27.004 --> 00:02:29.767
So we can rule out, let's see, choice B

00:02:29.767 --> 00:02:31.625
does have a y intercept.

00:02:31.625 --> 00:02:34.840
It looks just eyeballing it at one,

00:02:34.840 --> 00:02:36.106
x equals zero, y is one.

00:02:36.106 --> 00:02:38.439
So this looks interesting.

00:02:38.439 --> 00:02:41.725
Choice C has its y intercept

00:02:41.725 --> 00:02:44.337
at y is equal to negative one.

00:02:44.337 --> 00:02:46.821
So once again, we can rule that out.

00:02:46.821 --> 00:02:50.164
And choice D has no y-intercept at all.

00:02:50.164 --> 00:02:51.999
So that was enough information.

00:02:51.999 --> 00:02:53.659
And lucky for us because
they really didn't give us

00:02:53.659 --> 00:02:55.214
a lot more information than just

00:02:55.214 --> 00:02:57.722
what be able to evaluate f of zero.

00:02:57.722 --> 00:02:59.463
We were able to figure
out any of the other,

00:02:59.463 --> 00:03:01.680
you know, the zeros or
the vertical asymptotes

00:03:01.680 --> 00:03:03.375
or their removable discontinuities.

00:03:03.375 --> 00:03:06.672
So we definitely feel
good about that choice.


Graphs of rational functions: horizontal asymptote

WEBVTT
Kind: captions
Language: en

00:00:00.306 --> 00:00:02.187
- [Voiceover] Let f of x
equal negative x squared

00:00:02.187 --> 00:00:03.406
plus a x plus b

00:00:03.406 --> 00:00:05.461
over x squared plus c x plus d

00:00:05.461 --> 00:00:09.582
where a, b, c, and d
are unknown constants.

00:00:09.582 --> 00:00:12.043
Which of the following is a possible graph

00:00:12.043 --> 00:00:14.400
of y is equal to f of x?

00:00:14.400 --> 00:00:17.105
And they told us dashed
lines indicate asymptotes.

00:00:17.105 --> 00:00:19.043
So this is really interesting here.

00:00:19.043 --> 00:00:20.301
And they gave us four choices.

00:00:20.301 --> 00:00:22.391
We see four of them,
three of them right now.

00:00:22.391 --> 00:00:24.190
Then if I scroll over bit over,

00:00:24.190 --> 00:00:25.955
you can see choice D.

00:00:25.955 --> 00:00:27.662
And so I encourage you to pause the video

00:00:27.662 --> 00:00:29.461
and think about how we can figure it out

00:00:29.461 --> 00:00:30.378
because it is interesting

00:00:30.378 --> 00:00:32.131
because they haven't
given us a lot of details.

00:00:32.131 --> 00:00:34.197
They haven't given us
what these coefficients

00:00:34.197 --> 00:00:36.833
or these constants are going to be.

00:00:36.833 --> 00:00:38.191
All right, now let's think about it.

00:00:38.191 --> 00:00:39.454
So one thing we could think about is

00:00:39.454 --> 00:00:40.830
horizontal asymptotes.

00:00:40.830 --> 00:00:42.064
So let's think about what happens

00:00:42.064 --> 00:00:45.007
as x approaches positive
or negative infinity.

00:00:45.007 --> 00:00:49.302
Well, as x, so as x approaches

00:00:49.302 --> 00:00:53.232
infinity or x approaches

00:00:53.232 --> 00:00:56.232
negative infinity, f of x.

00:00:56.232 --> 00:01:00.456
F of x is going to be
approximately equal to.

00:01:00.456 --> 00:01:02.172
Well, we're going to look
at the highest degree terms

00:01:02.172 --> 00:01:03.832
because these are going to dominate

00:01:03.832 --> 00:01:05.759
as the magnitude of x,

00:01:05.759 --> 00:01:08.534
the absolute value of
x becomes very large.

00:01:08.534 --> 00:01:12.353
So f of x is going to be
approximately negative x squared

00:01:12.353 --> 00:01:16.997
over x squared which is equal to negative

00:01:16.997 --> 00:01:18.332
or we could another way to think about it.

00:01:18.332 --> 00:01:20.398
This is the same thing as negative one.

00:01:20.398 --> 00:01:24.578
So f of x is going to approach,

00:01:24.578 --> 00:01:28.162
f of x is going to approach negative one.

00:01:28.162 --> 00:01:30.542
In either direction, as
x approaches infinity

00:01:30.542 --> 00:01:32.422
or x approaches negative infinity.

00:01:32.422 --> 00:01:34.477
So we have a horizontal asymptote

00:01:34.477 --> 00:01:37.267
at y equals negative one.

00:01:37.267 --> 00:01:39.103
Let's see, choice A here,

00:01:39.103 --> 00:01:41.779
it does look like they
have a horizontal asymptote

00:01:41.779 --> 00:01:45.252
at y is equal to negative one

00:01:45.252 --> 00:01:46.425
right over there.

00:01:46.425 --> 00:01:47.992
And we can verify that

00:01:47.992 --> 00:01:51.451
because each hash mark is two.

00:01:51.451 --> 00:01:52.786
We go from two to zero

00:01:52.786 --> 00:01:54.992
to negative two to negative four.

00:01:54.992 --> 00:01:57.302
So this does look like
it's a negative one.

00:01:57.302 --> 00:02:00.007
So just based only on
the horizontal asymptote,

00:02:00.007 --> 00:02:02.004
choice A looks good.

00:02:02.004 --> 00:02:05.382
Choice B, we have a horizontal asymptote

00:02:05.382 --> 00:02:08.110
at y is equal to positive two.

00:02:08.110 --> 00:02:11.262
So we can rule that out.

00:02:11.262 --> 00:02:12.905
We know that a horizontal asymptote

00:02:12.905 --> 00:02:15.761
as x approaches positive
or negative infinity

00:02:15.761 --> 00:02:17.445
is at negative one,

00:02:17.445 --> 00:02:19.082
y equals negative one.

00:02:19.082 --> 00:02:21.241
Here, our horizontal asymptote is

00:02:21.241 --> 00:02:23.121
at y is equal to zero.

00:02:23.121 --> 00:02:24.805
The graph approaches,

00:02:24.805 --> 00:02:27.277
it approaches the x axis

00:02:27.277 --> 00:02:29.263
from either above or below.

00:02:29.263 --> 00:02:30.702
So it's not the horizontal asymptote.

00:02:30.702 --> 00:02:32.420
It's not y equals negative one.

00:02:32.420 --> 00:02:34.661
So we can rule that one out.

00:02:34.661 --> 00:02:37.424
And then similarly, over here,

00:02:37.424 --> 00:02:39.920
our horizontal asymptote is
not y equals negative one.

00:02:39.920 --> 00:02:42.300
Our horizontal asymptote
is y is equal to zero

00:02:42.300 --> 00:02:44.343
so we can rule that one out.

00:02:44.343 --> 00:02:45.341
And that makes sense

00:02:45.341 --> 00:02:47.837
because really they only
gave us enough information

00:02:47.837 --> 00:02:50.124
to figure out the horizontal asymptote.

00:02:50.124 --> 00:02:51.900
They didn't give us enough information

00:02:51.900 --> 00:02:53.142
to figure out how many roots

00:02:53.142 --> 00:02:54.535
or what happens in the interval

00:02:54.535 --> 00:02:55.673
and all of those types of things.

00:02:55.673 --> 00:02:57.298
How many zeros and all that

00:02:57.298 --> 00:03:00.917
because we don't know what the actual

00:03:00.917 --> 00:03:03.161
coefficients or constants
of the quadratic are.

00:03:03.161 --> 00:03:07.530
All we know is what happens as
the x squared terms dominate.

00:03:07.530 --> 00:03:10.441
This thing is going to
approach negative one.

00:03:10.441 --> 00:03:14.430
And so we picked choice A.


Graphs of rational functions: vertical asymptotes

WEBVTT
Kind: captions
Language: en

00:00:00.478 --> 00:00:01.999
- [Voiceover] We're told, let f of x

00:00:01.999 --> 00:00:05.540
equal g of x over x
squared minus x minus six,

00:00:05.540 --> 00:00:07.468
where g of x is a polynomial.

00:00:07.468 --> 00:00:09.825
Which of the following is a possible graph

00:00:09.825 --> 00:00:11.600
of y equals f of x?

00:00:11.600 --> 00:00:12.865
And they give us four choices.

00:00:12.865 --> 00:00:15.663
The fourth choice is off right over here.

00:00:15.663 --> 00:00:17.428
And like always, pause the video,

00:00:17.428 --> 00:00:19.870
and see if you can figure
it out or if you were

00:00:19.870 --> 00:00:22.052
having trouble with it as
I start to think about it

00:00:22.052 --> 00:00:25.066
with you, pause it anytime,
if you feel inspired.

00:00:25.066 --> 00:00:27.110
It's really important to
try to engage in the problem

00:00:27.110 --> 00:00:28.920
as opposed to just watch me do it.

00:00:28.920 --> 00:00:31.462
But let's start tackling
this now together.

00:00:31.462 --> 00:00:32.403
So, this is interesting.

00:00:32.403 --> 00:00:34.667
They don't give us a lot of
information about f of x.

00:00:34.667 --> 00:00:36.268
If fact, we don't know
what the numerator is.

00:00:36.268 --> 00:00:38.485
We just know that it is a polynomial.

00:00:38.485 --> 00:00:39.995
Well, that's somewhat valuable.

00:00:39.995 --> 00:00:41.968
But they do give us the denominator

00:00:41.968 --> 00:00:43.861
and so, we can think about what are

00:00:43.861 --> 00:00:45.069
the interesting numbers, what are

00:00:45.069 --> 00:00:46.757
the interesting x-values
for the denominator.

00:00:46.757 --> 00:00:48.858
In particular, what x-values will make

00:00:48.858 --> 00:00:50.926
the denominator equal to zero?

00:00:50.926 --> 00:00:53.432
And to do that, we can
factor out the denominator.

00:00:53.432 --> 00:00:55.731
So let's see, the coefficient
on the first degree term

00:00:55.731 --> 00:00:57.530
is essentially negative one.

00:00:57.530 --> 00:00:59.289
We can write negative
one there if we want.

00:00:59.289 --> 00:01:01.559
And the constant is negative six.

00:01:01.559 --> 00:01:02.988
So if we want to factor that, we can say,

00:01:02.988 --> 00:01:04.786
well, what two number,s they're product

00:01:04.786 --> 00:01:07.967
is negative six and they
add up to negative one?

00:01:07.967 --> 00:01:12.967
Well, negative three times
positive two is negative six.

00:01:14.120 --> 00:01:18.287
And negative three plus two
is equal to negative one.

00:01:18.287 --> 00:01:20.226
So I can rewrite f of x.

00:01:20.226 --> 00:01:25.226
I can say that f of x
is equal to g of x over

00:01:27.106 --> 00:01:32.106
x minus three times x plus two.

00:01:32.492 --> 00:01:34.452
So the denominator equals zero

00:01:34.452 --> 00:01:39.452
for x equals three or
x equals negative two.

00:01:39.688 --> 00:01:42.451
That's when the denominator is zero.

00:01:42.451 --> 00:01:47.451
So zero denominator. I'll
just write it like that.

00:01:47.525 --> 00:01:50.478
And so something makes the
denominator equal to zero.

00:01:50.478 --> 00:01:51.814
That tell us that we're either going

00:01:51.814 --> 00:01:53.990
to have a vertical asymptote at that point

00:01:53.990 --> 00:01:54.988
or we're going to have a

00:01:54.988 --> 00:01:57.148
removable discontinuity at that point.

00:01:57.148 --> 00:01:58.053
And the way that that would be

00:01:58.053 --> 00:01:59.691
a removable discontinuity, let's say,

00:01:59.691 --> 00:02:01.350
if we had a removable discontinuity

00:02:01.350 --> 00:02:04.195
at x equals three, well
that means that g of x

00:02:04.195 --> 00:02:07.237
could be factored into x minus three

00:02:07.237 --> 00:02:09.186
times a bunch of other stuff.

00:02:09.186 --> 00:02:11.113
If that was the case, the x equals three

00:02:11.113 --> 00:02:13.494
would a removable discontinuity.

00:02:13.494 --> 00:02:17.847
If x equals three does not
make g of x equal zero.

00:02:17.847 --> 00:02:21.213
So, for example, if g of
three does not equal zero,

00:02:21.213 --> 00:02:23.790
or g of negative two does not equal zero,

00:02:23.790 --> 00:02:26.588
then these would both
be vertical asymptotes.

00:02:26.588 --> 00:02:28.933
So let's look at the choices here.

00:02:28.933 --> 00:02:32.613
So choice A, we have
one vertical asymptote.

00:02:32.613 --> 00:02:36.827
That vertical asymptote is
at x equals negative two.

00:02:36.827 --> 00:02:41.204
So it seems, this line,
let me draw this line here.

00:02:41.204 --> 00:02:43.862
This vertical asymptote, right over there,

00:02:43.862 --> 00:02:46.706
that is a line, x is
equal to negative two.

00:02:46.706 --> 00:02:48.421
So at least to be, it
seems to be consistent

00:02:48.421 --> 00:02:51.358
with that over there but
what about x equals three?

00:02:51.358 --> 00:02:53.262
This one seems completely cool.

00:02:53.262 --> 00:02:55.915
This graph is defined at x equals three.

00:02:55.915 --> 00:02:57.887
X equals three is right over there

00:02:57.887 --> 00:02:59.733
and it seems to be defined there.

00:02:59.733 --> 00:03:02.892
And this, f of x, is
clearly not defined at f,

00:03:02.892 --> 00:03:06.932
at x is equal to three
because when x equals three,

00:03:06.932 --> 00:03:08.951
the denominator is zero and

00:03:08.951 --> 00:03:11.052
dividing by zero is not defined.

00:03:11.052 --> 00:03:13.385
So even though this has
one vertical asymptote

00:03:13.385 --> 00:03:15.383
at an interesting place,
we're going to rule it out

00:03:15.383 --> 00:03:18.412
because this graph is
defined at x is equals three,

00:03:18.412 --> 00:03:21.012
even though f of x is not.

00:03:21.012 --> 00:03:25.569
We would need to see either
a vertical asymptote there

00:03:25.569 --> 00:03:28.239
or a removable discontinuity.

00:03:28.239 --> 00:03:30.497
Alright, here we have a vertical asymptote

00:03:30.497 --> 00:03:33.863
at x is equal to negative two and we have

00:03:33.863 --> 00:03:38.863
another vertical asymptote at
x is equal to positive four.

00:03:39.122 --> 00:03:40.980
So that doesn't make sense either.

00:03:40.980 --> 00:03:42.257
This one, just like the last one,

00:03:42.257 --> 00:03:44.381
is actually defined at x equals three.

00:03:44.381 --> 00:03:48.781
We that x is equals three,
the function is equal to zero.

00:03:48.781 --> 00:03:51.497
But this function? F of
three is not equal to zero.

00:03:51.497 --> 00:03:54.784
F of three is undefined.
We're dividing by zero.

00:03:54.784 --> 00:03:56.723
So we could rule this out.

00:03:56.723 --> 00:03:58.579
Once again, at x equals
three, we need to see

00:03:58.579 --> 00:04:02.375
the removable discontinuity
or a vertical asymptote,

00:04:02.375 --> 00:04:04.674
because we're not defined there.

00:04:04.674 --> 00:04:06.322
Alright, let's see choice C.

00:04:06.322 --> 00:04:11.322
We see a vertical asymptote
at x is equal to negative two.

00:04:11.558 --> 00:04:13.218
So that looks pretty good.

00:04:13.218 --> 00:04:17.420
And we see a removable
discontinuity at x equals three.

00:04:17.420 --> 00:04:20.915
So this function is,
this graph is not defined

00:04:20.915 --> 00:04:24.617
for x equals three or for
x equals negative two,

00:04:24.617 --> 00:04:27.276
which is good because f is not defined

00:04:27.276 --> 00:04:29.156
at either of those
points because at either

00:04:29.156 --> 00:04:32.629
of those x-values, our f's
denominator is equal to zero.

00:04:32.629 --> 00:04:36.169
So this one looks quite interesting.

00:04:36.169 --> 00:04:37.388
And this would be consistent.

00:04:37.388 --> 00:04:41.416
This one would be consistent
with the, with f of x

00:04:41.416 --> 00:04:44.562
being something of the sort of,

00:04:44.562 --> 00:04:46.512
so the denominator, we already know.

00:04:46.512 --> 00:04:50.413
X minus three times x plus two.

00:04:50.413 --> 00:04:52.258
And in the numerator, we would have,

00:04:52.258 --> 00:04:54.522
since x minus three is not a vertical as-,

00:04:54.522 --> 00:04:57.181
since x equals three isn't
a vertical asymptote,

00:04:57.181 --> 00:04:59.456
it's a removable discontinuity, we must

00:04:59.456 --> 00:05:02.521
be able to factor, for this one, g of x

00:05:02.521 --> 00:05:06.364
into x minus three times something else.

00:05:06.364 --> 00:05:08.558
So that's consistent
with this one over here.

00:05:08.558 --> 00:05:09.985
So I like this choice.

00:05:09.985 --> 00:05:12.092
Now let's look at this choice, choice D.

00:05:12.092 --> 00:05:13.949
Choice D has two vertical asymptotes.

00:05:13.949 --> 00:05:17.687
One at x is equal to negative one.

00:05:17.687 --> 00:05:20.334
This is negative two, so
this, x equals negative one.

00:05:20.334 --> 00:05:22.156
And this x is equal to six.

00:05:22.156 --> 00:05:24.153
Neither of them are, would coincide

00:05:24.153 --> 00:05:26.348
with what make our denominator equal zero,

00:05:26.348 --> 00:05:28.147
so we could rule this out as well.

00:05:28.147 --> 00:05:32.796
So we could feel really
good about choice C.


Vertices & direction of a hyperbola

WEBVTT
Kind: captions
Language: en

00:00:00.246 --> 00:00:01.407
- [ Voiceover] Which
of the following graphs

00:00:01.407 --> 00:00:03.764
can represent the hyperbola, y squared

00:00:03.764 --> 00:00:08.764
over nine minus x squared
over four is equal to one.

00:00:09.254 --> 00:00:10.623
And we have our four choices here.

00:00:10.623 --> 00:00:13.584
Choices A and C open up to the top

00:00:13.584 --> 00:00:16.567
and the bottom or up and down.

00:00:16.567 --> 00:00:19.307
Choices B and D, you could see D here,

00:00:19.307 --> 00:00:21.385
open to the left and the right.

00:00:21.385 --> 00:00:22.685
And you can see within the ones

00:00:22.685 --> 00:00:23.892
that open up to the left, to the right

00:00:23.892 --> 00:00:27.445
or the up, down ones, they
have different vertices.

00:00:27.445 --> 00:00:28.941
So I encourage you to pause the video

00:00:28.941 --> 00:00:30.532
and see if you can figure out,

00:00:30.532 --> 00:00:33.063
which of the following graphs represent

00:00:33.063 --> 00:00:35.210
the equation of a hyperbola, or the graphs

00:00:35.210 --> 00:00:37.904
of this equation right over here.

00:00:37.904 --> 00:00:40.052
Alright, so there's a bunch
of ways to think about it.

00:00:40.052 --> 00:00:40.981
One thing you might say, well,

00:00:40.981 --> 00:00:43.001
what's the center of this hyperbola?

00:00:43.001 --> 00:00:44.881
And since in our equation, we just have

00:00:44.881 --> 00:00:47.482
a simple y squared or a simple x squared

00:00:47.482 --> 00:00:50.326
we know that the center is
going to be at zero, zero.

00:00:50.326 --> 00:00:51.829
If the center was anywhere else,

00:00:51.829 --> 00:00:56.829
if the center was at the point h comma k,

00:00:57.726 --> 00:01:01.267
then this equation
would be y squared minus

00:01:01.267 --> 00:01:02.939
the y-coordinate of the center.

00:01:02.939 --> 00:01:06.584
Sorry, not y-squared,
it would be y minus k,

00:01:06.584 --> 00:01:11.181
so y minus the y-coordinate of
the center squared over nine

00:01:11.181 --> 00:01:14.002
minus x minus the x-coordinate

00:01:14.002 --> 00:01:17.822
of the center squared
over four is equal to one.

00:01:17.822 --> 00:01:19.958
And in this case, well,
this is just a case,

00:01:19.958 --> 00:01:23.022
where k and h, or h and
k, are both equal to zero.

00:01:23.022 --> 00:01:24.178
So we just get, you could view this

00:01:24.178 --> 00:01:27.927
as y minus zero squared
and x minus zero squared.

00:01:27.927 --> 00:01:31.596
So the center in this case
is going to be at zero, zero.

00:01:31.596 --> 00:01:34.567
And you see that for all of them.

00:01:34.567 --> 00:01:35.972
Now the next question you might ask,

00:01:35.972 --> 00:01:38.237
well, is this going to
be opening up and down,

00:01:38.237 --> 00:01:40.325
or is it going to be
opening left and right?

00:01:40.325 --> 00:01:42.010
And the key thing to realize is that

00:01:42.010 --> 00:01:43.913
you just need to look at whichever term,

00:01:43.913 --> 00:01:45.828
and when it's written in
standard form like this,

00:01:45.828 --> 00:01:49.671
when you have y minus k
squared over something squared

00:01:49.671 --> 00:01:52.294
minus x minus h squared over something

00:01:52.294 --> 00:01:53.449
squared is equal to one.

00:01:53.449 --> 00:01:54.657
Or it could be the other way around.

00:01:54.657 --> 00:01:57.257
The x-term might be positive

00:01:57.257 --> 00:01:58.929
and then the y-term would be negative,

00:01:58.929 --> 00:02:00.531
if we're dealing with the hyperbola.

00:02:00.531 --> 00:02:03.456
So the key is to just look at
whichever term is positive.

00:02:03.456 --> 00:02:06.393
That will tell you which
direction the hyperbola opens in.

00:02:06.393 --> 00:02:09.714
Since the y-term here is
the one that is positive,

00:02:09.714 --> 00:02:11.931
it tells us that this hyperbola is

00:02:11.931 --> 00:02:16.192
going to open up and down.

00:02:16.192 --> 00:02:18.131
Now you could just memorize that

00:02:18.131 --> 00:02:19.720
but I, that's never too satisfying.

00:02:19.720 --> 00:02:22.681
I always want to know why does that work.

00:02:22.681 --> 00:02:26.976
And the key thing to
realize is, if the y-term

00:02:26.976 --> 00:02:30.656
is positive than you can set
the other term equal to zero.

00:02:30.656 --> 00:02:31.956
And the way you would set the other term

00:02:31.956 --> 00:02:34.116
equal to zero in this case is by making

00:02:34.116 --> 00:02:38.062
your x equal to the
x-coordinate of your center

00:02:38.062 --> 00:02:41.558
and that's zero, so if
x is, if you, x is equal

00:02:41.558 --> 00:02:43.264
to the x squared of your center

00:02:43.264 --> 00:02:46.699
and this term becomes
zero, you can actually

00:02:46.699 --> 00:02:48.185
solve this equation.

00:02:48.185 --> 00:02:51.703
You can solve y squared
over nine equals one.

00:02:51.703 --> 00:02:54.803
So if x is equal to zero, so x is equal

00:02:54.803 --> 00:02:56.800
to the x-coordinate of its center,

00:02:56.800 --> 00:02:59.261
then this term goes away and you would get

00:02:59.261 --> 00:03:03.324
y squared over nine is equal to one.

00:03:03.324 --> 00:03:06.655
Or y squared is equal to nine.

00:03:06.655 --> 00:03:10.324
Or y is equal to plus or minus three.

00:03:10.324 --> 00:03:13.860
So you know that the coordinates,

00:03:13.860 --> 00:03:17.870
the x-coordinate of the
center, plus or minus three.

00:03:17.870 --> 00:03:20.156
That they're on the
hyperbola, that they're,

00:03:20.156 --> 00:03:21.479
and so you know that it's going

00:03:21.479 --> 00:03:23.257
to open upwards and downwards so you go

00:03:23.257 --> 00:03:25.160
to the center, the
x-coordinate of the center

00:03:25.160 --> 00:03:28.979
plus three and minus three
are on the hyperbola.

00:03:28.979 --> 00:03:29.768
Notice over here.

00:03:29.768 --> 00:03:33.448
Plus three and minus three
are not on this hyperbola.

00:03:33.448 --> 00:03:34.825
In fact, if plus three and minus three

00:03:34.825 --> 00:03:36.207
were on this hyperbola, you wouldn't

00:03:36.207 --> 00:03:39.260
be able to open up to
the right and the left.

00:03:39.260 --> 00:03:42.186
And so that's why,
whichever term is positive,

00:03:42.186 --> 00:03:46.202
that is the direction that
you open up or down with,

00:03:46.202 --> 00:03:48.686
or if, so if the x-term was
positive we would be opening

00:03:48.686 --> 00:03:51.321
to the left and the right,
for the exact same reason.

00:03:51.321 --> 00:03:53.914
And you can see, if we did
it the other way around,

00:03:53.914 --> 00:03:57.559
if we had the y equaling the
y-coordinate of the center,

00:03:57.559 --> 00:04:00.589
so this term, if the
y-term was zeroed out,

00:04:00.589 --> 00:04:02.249
you would end up with negative x

00:04:02.249 --> 00:04:04.652
squared over four is equal to one.

00:04:04.652 --> 00:04:06.870
Which is the same thing
as x squared over four

00:04:06.870 --> 00:04:10.560
is equal to negative
one, which is equal to

00:04:10.560 --> 00:04:13.021
x squared is equal to negative four.

00:04:13.021 --> 00:04:15.459
I just multiplied both
sides by negative one there.

00:04:15.459 --> 00:04:17.386
And then I multiplied both sides by four

00:04:17.386 --> 00:04:19.604
and this has no solution.

00:04:19.604 --> 00:04:22.099
And so that's why we
know that we're not going

00:04:22.099 --> 00:04:26.302
to intercept the, we're not
going to intercept the line.

00:04:26.302 --> 00:04:28.624
We're never going to
have a situation where

00:04:28.624 --> 00:04:33.279
y is equal to the
y-coordinate of the center.

00:04:33.279 --> 00:04:38.279
So that's, so y is never
going to be equal to zero.

00:04:38.515 --> 00:04:41.359
In this case, in cases B and D, y,

00:04:41.359 --> 00:04:43.263
there are points where y equals zero.

00:04:43.263 --> 00:04:45.585
So the thing to realize is whichever term

00:04:45.585 --> 00:04:48.359
is positive, that, and
whatever variable that is,

00:04:48.359 --> 00:04:50.797
so if it's the y-variable,
that's the direction

00:04:50.797 --> 00:04:52.759
we're going to open up in.

00:04:52.759 --> 00:04:54.059
And when I figured out what the

00:04:54.059 --> 00:04:56.799
actual vertices are, we
saw that the point zero,

00:04:56.799 --> 00:04:58.656
plus or minus three are on the graph,

00:04:58.656 --> 00:05:01.013
so A looks like a really good candidate.

00:05:01.013 --> 00:05:03.079
If we look at the other
choice, it opens up and down.

00:05:03.079 --> 00:05:05.958
It doesn't have zero, plus
or minus three on the graph.

00:05:05.958 --> 00:05:07.978
It has zero, plus or minus two.

00:05:07.978 --> 00:05:12.978
So we can feel pretty good
that, about, choice A.


Picking hyperbola equation

WEBVTT
Kind: captions
Language: en

00:00:00.581 --> 00:00:01.360
- [Voiceover] So we're asked to choose

00:00:01.360 --> 00:00:02.741
the equation that can represent

00:00:02.741 --> 00:00:04.483
the hyperbola graphed below.

00:00:04.483 --> 00:00:06.176
And so this is the hyperbola graphed

00:00:06.176 --> 00:00:08.116
in blue and I encourage
you to pause the video

00:00:08.116 --> 00:00:09.904
and figure out which of these equations

00:00:09.904 --> 00:00:13.340
are represented by the graph here.

00:00:13.340 --> 00:00:14.536
Alright, so let's think about it.

00:00:14.536 --> 00:00:16.461
This graph opens to
the left and the right.

00:00:16.461 --> 00:00:17.702
Well, I guess, the first
thing we could realize,

00:00:17.702 --> 00:00:19.920
it's centered at zero.

00:00:19.920 --> 00:00:21.638
So definitely, it's just going to have

00:00:21.638 --> 00:00:25.980
the form, x squared and y squared over

00:00:25.980 --> 00:00:29.393
two different things equaling one.

00:00:29.393 --> 00:00:32.896
And we know that it opens
to the left and the right.

00:00:32.896 --> 00:00:35.891
You can think of it opens
along the x-direction

00:00:35.891 --> 00:00:37.540
and so we know that the x-term is going

00:00:37.540 --> 00:00:39.618
to be positive here,
which tells us that the

00:00:39.618 --> 00:00:42.659
y-term is going to be negative.

00:00:42.659 --> 00:00:44.852
And we know that the vertices here are

00:00:44.852 --> 00:00:47.152
five to the right of the center

00:00:47.152 --> 00:00:49.416
and five to the left of the center

00:00:49.416 --> 00:00:51.331
and so since the distance
from the vertices

00:00:51.331 --> 00:00:55.197
to the center is five in
the horizontal direction,

00:00:55.197 --> 00:00:56.635
we know that this right over here

00:00:56.635 --> 00:01:01.635
is going to be five squared or 25.

00:01:01.837 --> 00:01:04.856
And this we don't quite know, just yet.

00:01:04.856 --> 00:01:09.856
I just call this a squared.
We don't know what a is.

00:01:11.670 --> 00:01:13.552
Now let's look at these choices here.

00:01:13.552 --> 00:01:16.151
X squared over 25 minus y squared

00:01:16.151 --> 00:01:18.496
over nine equals one.

00:01:18.496 --> 00:01:20.667
Well, that seems to match the pattern

00:01:20.667 --> 00:01:23.128
that I was able to
generate really quickly,

00:01:23.128 --> 00:01:26.054
just looking at the
graph, so I like this one.

00:01:26.054 --> 00:01:28.410
This has the x-term being negative.

00:01:28.410 --> 00:01:30.128
So this graph over
here, this would open up

00:01:30.128 --> 00:01:33.054
up and down, not to
the left and the right,

00:01:33.054 --> 00:01:35.747
so we could rule this out.

00:01:35.747 --> 00:01:38.870
This one over here. This
has x squared over nine.

00:01:38.870 --> 00:01:41.412
That would imply that our x-intercepts

00:01:41.412 --> 00:01:45.232
are plus or minus three
to the right and left

00:01:45.232 --> 00:01:46.729
of the center, not five.

00:01:46.729 --> 00:01:49.148
Clearly, they aren't plus or minus three

00:01:49.148 --> 00:01:51.872
so we could rule this one out.

00:01:51.872 --> 00:01:54.082
This one has the y-term being positive

00:01:54.082 --> 00:01:55.000
and the x-term being negative,

00:01:55.000 --> 00:01:57.019
so once again, this
would open up and down.

00:01:57.019 --> 00:01:59.724
So we could rule that one out as well.

00:01:59.724 --> 00:02:01.639
And so our first choice
that we like that matched

00:02:01.639 --> 00:02:04.094
our pattern, we could
feel pretty good about it.

00:02:04.094 --> 00:02:05.718
Now if you wanted to verify the nine

00:02:05.718 --> 00:02:07.553
or if you want, you might want to try out

00:02:07.553 --> 00:02:09.817
some other points or solve some points,

00:02:09.817 --> 00:02:11.535
if it wasn't multiple
choice but in this case,

00:02:11.535 --> 00:02:13.230
we are able to pick out.

00:02:13.230 --> 00:02:15.540
This is the only one that even matches

00:02:15.540 --> 00:02:19.474
the general structure that
we were able to deduce.


Analyzing vertical asymptotes of rational functions

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.239
- [Voiceover] We're asked to describe

00:00:01.239 --> 00:00:03.134
the behavior of the function q around

00:00:03.134 --> 00:00:06.083
its vertical asymptote at x = -3,

00:00:06.083 --> 00:00:07.712
and like always, if
you're familiar with this,

00:00:07.712 --> 00:00:09.113
I encourage you to pause it and see if you

00:00:09.113 --> 00:00:10.725
can get some practice, and if you're not,

00:00:10.725 --> 00:00:12.499
well, I'm about to do it with you.

00:00:12.499 --> 00:00:14.572
All right, so this is q of x, it's defined

00:00:14.572 --> 00:00:16.105
by a rational expression,
and whenever I'm dealing

00:00:16.105 --> 00:00:18.621
with asymptotes, I like
to factor the numerators

00:00:18.621 --> 00:00:21.092
and the denominators so I can
make more sense of things.

00:00:21.092 --> 00:00:24.068
So the numerator here, what
two numbers if I were...

00:00:24.068 --> 00:00:26.151
Their product is two
and their sum is three,

00:00:26.151 --> 00:00:28.238
well that's two and one,
so I can factor this

00:00:28.238 --> 00:00:32.862
as x + 1 times x + 2, if
that's unfamiliar to you,

00:00:32.862 --> 00:00:34.606
I recommend you watch the videos

00:00:34.606 --> 00:00:37.355
on Khan Academy about
factoring quadratics,

00:00:37.355 --> 00:00:42.343
and over x + 3, and when

00:00:42.343 --> 00:00:46.257
x = -1, or x = -2, it would
make the numerator equal zero

00:00:46.257 --> 00:00:48.246
without making the denominator equal zero,

00:00:48.246 --> 00:00:49.518
so those are points where the function

00:00:49.518 --> 00:00:52.325
is equal to zero, but when x = -3,

00:00:52.325 --> 00:00:55.121
the denominator equals
zero, while the numerator

00:00:55.121 --> 00:00:57.480
is not equal to zero, so
we're dividing by zero,

00:00:57.480 --> 00:01:01.474
and so that's a pretty good
sign of a vertical asymptote,

00:01:01.474 --> 00:01:03.063
that our function as
we approach that value

00:01:03.063 --> 00:01:05.042
is either going to pop up like that,

00:01:05.042 --> 00:01:07.398
or it's going to pop down, or it's going

00:01:07.398 --> 00:01:10.572
to pop down like that, or maybe,

00:01:10.572 --> 00:01:13.585
or either way, or could pop up like that,

00:01:13.585 --> 00:01:15.466
or it could go down something like that,

00:01:15.466 --> 00:01:17.119
but we're going to have
a vertical asymptote

00:01:17.119 --> 00:01:19.745
that the function as you approach x = -3,

00:01:19.745 --> 00:01:21.495
that the function is going to approach

00:01:21.495 --> 00:01:23.962
either positive infinity
or negative infinity

00:01:23.962 --> 00:01:25.572
or it might do positive infinity

00:01:25.572 --> 00:01:26.943
from one direction or negative infinity

00:01:26.943 --> 00:01:29.225
from another direction and if this idea

00:01:29.225 --> 00:01:30.933
of directionality is a
little bit confusing,

00:01:30.933 --> 00:01:32.940
well that's where we're,
that's what we're about

00:01:32.940 --> 00:01:34.455
to address in this video.

00:01:34.455 --> 00:01:37.484
Let me just draw a number line here

00:01:37.484 --> 00:01:41.341
that focuses on these interesting values.

00:01:41.341 --> 00:01:45.416
So we care about x = -3, and then

00:01:45.416 --> 00:01:48.882
the other interesting values may be -2,

00:01:48.882 --> 00:01:52.876
- 1, it's all there, now what does it mean

00:01:52.876 --> 00:01:55.978
to be approaching x, what does it mean

00:01:55.978 --> 00:01:59.741
for x to be approaching -3
from the negative direction?

00:01:59.741 --> 00:02:00.920
And just to be clear, this little

00:02:00.920 --> 00:02:02.267
superscript right over here,

00:02:02.267 --> 00:02:06.257
that means we're approaching
from the negative direction.

00:02:06.257 --> 00:02:07.881
So that means we're
approaching from values

00:02:07.881 --> 00:02:10.171
more negative than -3, so those

00:02:10.171 --> 00:02:12.667
are these values right over here.

00:02:12.667 --> 00:02:16.511
We are approaching, we are
approaching from that direction.

00:02:16.511 --> 00:02:18.685
Another way to think
about it is we approach

00:02:18.685 --> 00:02:20.925
from the negative direction
where on the interval

00:02:20.925 --> 00:02:24.279
x is less than -3, so let's think about

00:02:24.279 --> 00:02:27.847
what the sign of q of x is going

00:02:27.847 --> 00:02:31.186
to be as we're approaching -3 from that

00:02:31.186 --> 00:02:34.977
negative direction, from the left.

00:02:34.977 --> 00:02:38.430
Well, if we have something less than -3,

00:02:38.430 --> 00:02:41.930
and you add one, this
is going to be negative.

00:02:41.930 --> 00:02:43.413
If you have something less than -3,

00:02:43.413 --> 00:02:45.779
and you add two, that's
going to be negative as well,

00:02:45.779 --> 00:02:47.318
and if you have something less than -3,

00:02:47.318 --> 00:02:50.016
and you add three, well that's
going to be negative as well.

00:02:50.016 --> 00:02:51.692
So a negative times a
negative is a positive,

00:02:51.692 --> 00:02:53.288
but then you divide by
a negative, it's going

00:02:53.288 --> 00:02:55.670
to be a negative, so q of x is going

00:02:55.670 --> 00:02:57.916
to be negative on that interval.

00:02:57.916 --> 00:03:00.136
So as we approach, we have
our vertical asymptote.

00:03:00.136 --> 00:03:03.274
As we approach -3 from the left hand side,

00:03:03.274 --> 00:03:04.691
well q of x is going to be negative

00:03:04.691 --> 00:03:07.506
and so it's going to
approach negative infinity.

00:03:07.506 --> 00:03:11.520
So as x approaches -3 from
the negative direction,

00:03:11.520 --> 00:03:14.267
q of x is going to
approach negative infinity.

00:03:14.267 --> 00:03:19.245
So at least this accurate,
this is accurate,

00:03:19.245 --> 00:03:23.401
and this is accurate right over here,

00:03:23.401 --> 00:03:25.730
and you can validate
that, try some values out.

00:03:25.730 --> 00:03:29.191
Try negative, let's see, if you did

00:03:29.191 --> 00:03:31.180
negative three point...

00:03:31.180 --> 00:03:36.180
q of negative three point,

00:03:37.275 --> 00:03:40.890
- 3.1, once again,
that's on the left side,

00:03:40.890 --> 00:03:42.898
it might be like right over there.

00:03:42.898 --> 00:03:44.534
Actually, it'd probably
be a little bit further.

00:03:44.534 --> 00:03:45.844
It might be something like this,

00:03:45.844 --> 00:03:48.351
like the scale that I've drawn it on.

00:03:48.351 --> 00:03:50.968
q of -3.1, if you wanted to verify it,

00:03:50.968 --> 00:03:55.889
it's going to be -3.1 + 1, which is -2.1,

00:03:55.889 --> 00:04:00.889
times -3.1 + 2, which is negative,

00:04:01.316 --> 00:04:05.028
- 1.1, 1.1, I could
put that in parentheses

00:04:05.028 --> 00:04:07.089
just to make it clear, and then all

00:04:07.089 --> 00:04:08.850
of that is going to be over,

00:04:08.850 --> 00:04:12.800
well -3.1 + 3 is going to be,

00:04:12.800 --> 00:04:17.416
is going to be -0.1.

00:04:17.416 --> 00:04:20.485
So notice whatever we get up
here that's a positive value,

00:04:20.485 --> 00:04:22.394
we're essentially going
to, if we're dividing it

00:04:22.394 --> 00:04:27.393
by -0.1, that's like
multiplying it by -10,

00:04:27.393 --> 00:04:30.136
so it's going to become
a very negative value,

00:04:30.136 --> 00:04:33.894
and if instead of it
being -3.1, imagine if it

00:04:33.894 --> 00:04:37.352
was -3.01, and this would be a 01 here.

00:04:37.352 --> 00:04:40.570
This would be a 01 here,
and this would be a 01 here.

00:04:40.570 --> 00:04:44.219
And so this denominator,
you're dividing by -0.01,

00:04:44.219 --> 00:04:46.887
it's going to be even
larger negative value,

00:04:46.887 --> 00:04:49.452
so you're going to
approach negative infinity.

00:04:49.452 --> 00:04:50.960
So it's going to be one
of these two choices.

00:04:50.960 --> 00:04:52.358
Now let's think about what happens

00:04:52.358 --> 00:04:54.942
as we approach x from
the positive direction,

00:04:54.942 --> 00:04:57.871
and that's what this
notation over there means,

00:04:57.871 --> 00:04:59.433
this superscript on the right hand

00:04:59.433 --> 00:05:01.154
side, the positive direction,

00:05:01.154 --> 00:05:04.400
so we're going to approach x
from the positive direction,

00:05:04.400 --> 00:05:06.554
and I'm going to pay
attention in particular

00:05:06.554 --> 00:05:10.138
to the interval between -2 and -3,

00:05:10.138 --> 00:05:11.642
because then we know
we don't have any weird

00:05:11.642 --> 00:05:13.579
sign changes going on in the numerator.

00:05:13.579 --> 00:05:18.579
So I care about the interval -3<x<-2.

00:05:19.476 --> 00:05:20.841
So I can draw an open circle here

00:05:20.841 --> 00:05:23.981
to say we're not considering
when we're at -2,

00:05:23.981 --> 00:05:25.709
and of course we're
not going to include -3

00:05:25.709 --> 00:05:28.390
because our function isn't defined there,

00:05:28.390 --> 00:05:31.932
but over this interval, so x + 1 is still

00:05:31.932 --> 00:05:33.338
going to be negative,

00:05:33.338 --> 00:05:36.552
x +1 is still going to be negative.

00:05:36.552 --> 00:05:41.068
If you took a -2.5 + 1,
it's going to be -1.5.

00:05:41.068 --> 00:05:43.066
x + 2 is still going to be negative,

00:05:43.066 --> 00:05:46.190
you're taking values that
are more negative than -2,

00:05:46.190 --> 00:05:49.057
that are less than -2, so you add two

00:05:49.057 --> 00:05:50.857
to that and you're still
going to be negative,

00:05:50.857 --> 00:05:53.475
and then let's see, if you add three,

00:05:53.475 --> 00:05:55.283
if you add three to
these values, remember,

00:05:55.283 --> 00:05:57.464
they're greater than -3, or you could

00:05:57.464 --> 00:06:00.785
say they're less negative than -3,

00:06:00.785 --> 00:06:03.514
well then this is going to
give you a positive value.

00:06:03.514 --> 00:06:06.813
This is going to give
you a positive value,

00:06:06.813 --> 00:06:09.155
and think about it, q of,

00:06:09.155 --> 00:06:13.971
q of, I don't know, -2.99,

00:06:13.971 --> 00:06:15.511
what is that going to be equal to?

00:06:15.511 --> 00:06:20.391
If you add one to that, that is -1.99,

00:06:20.391 --> 00:06:22.893
and if you add two to that,

00:06:22.893 --> 00:06:25.991
times, times

00:06:25.991 --> 00:06:29.491
- 0.99

00:06:29.491 --> 00:06:33.726
all of that over -2.99 + 3,

00:06:33.726 --> 00:06:37.077
well that's going to be
0.01, so you're going

00:06:37.077 --> 00:06:38.785
to get a positive value on
top and then you're going

00:06:38.785 --> 00:06:41.458
to divide it by 0.01, that's the same

00:06:41.458 --> 00:06:43.493
thing as multiplying
by 100, so you're going

00:06:43.493 --> 00:06:45.500
to get larger and larger values.

00:06:45.500 --> 00:06:47.115
You're going to approach
infinity as you get

00:06:47.115 --> 00:06:50.479
closer and closer to it
from the right hand side.

00:06:50.479 --> 00:06:54.028
So q of x is going to
approach positive infinity.

00:06:54.028 --> 00:06:55.409
So this is the choice that is correct.

00:06:55.409 --> 00:06:56.837
This one is wrong, this says we're going

00:06:56.837 --> 00:06:58.999
to approach negative
infinity, so that's incorrect,

00:06:58.999 --> 00:07:03.122
and we will go with that choice.


Discontinuities of rational functions

WEBVTT
Kind: captions
Language: en

00:00:00.219 --> 00:00:01.438
- [Voiceover] So we have
this function, f of x,

00:00:01.438 --> 00:00:04.259
expressed as a rational expression here,

00:00:04.259 --> 00:00:06.243
or defined with a rational expression.

00:00:06.243 --> 00:00:07.602
And we are told at each
of the following values

00:00:07.602 --> 00:00:09.959
of x, select whether f has a zero,

00:00:09.959 --> 00:00:14.324
a vertical asymptote, or
a removable discontinuity.

00:00:14.324 --> 00:00:15.997
And before even looking at the choices,

00:00:15.997 --> 00:00:17.620
what I am going to do,
because you're not going

00:00:17.620 --> 00:00:19.200
to always going to have
these choices here.

00:00:19.200 --> 00:00:20.362
Sometimes you might just have to identify

00:00:20.362 --> 00:00:22.323
the vertical asymptotes, or the zeros,

00:00:22.323 --> 00:00:24.563
or the removable discontinuities,

00:00:24.563 --> 00:00:26.362
is I am going to factor this out.

00:00:26.362 --> 00:00:27.918
And hopefully, or I'm going
to factor the numerator

00:00:27.918 --> 00:00:29.520
and the denominator to hopefully

00:00:29.520 --> 00:00:31.459
makes things a little bit more clear.

00:00:31.459 --> 00:00:33.200
And we are to think
about what x values make

00:00:33.200 --> 00:00:36.646
the numerator and, or the
denominator equal to zero.

00:00:36.646 --> 00:00:39.212
So the numerator, can I factor that?

00:00:39.212 --> 00:00:39.908
Well, let's see.

00:00:39.908 --> 00:00:42.950
What two numbers, their
product is negative 24,

00:00:42.950 --> 00:00:45.028
and they add to negative two.

00:00:45.028 --> 00:00:49.973
See, that would be negative
six and positive four.

00:00:49.973 --> 00:00:54.630
So we could write x minus
six times x plus four.

00:00:54.630 --> 00:00:55.743
Did I do that right?

00:00:55.743 --> 00:00:58.654
Negative six times four is negative 24.

00:00:58.654 --> 00:01:01.764
Negative six x plus four
x is negative two x.

00:01:01.764 --> 00:01:03.214
Yeah, that's right.

00:01:03.214 --> 00:01:06.267
Alright, and now in the denominator,

00:01:06.267 --> 00:01:10.225
let's see, six times four
is 24, six plus four is 10.

00:01:10.225 --> 00:01:15.225
So we can say, six, x
plus six times x plus 4.

00:01:16.753 --> 00:01:19.076
So let's look at the numerator first.

00:01:19.076 --> 00:01:24.076
The numerator zero if x is equal to

00:01:26.958 --> 00:01:31.958
positive six, or x is
equal to negative four.

00:01:32.739 --> 00:01:37.739
The denominator is zero if x is equal

00:01:39.598 --> 00:01:44.598
to negative six, or x is
equal to negative four.

00:01:46.401 --> 00:01:47.890
So, let's think about it.

00:01:47.890 --> 00:01:52.755
Can we simplify this, what
we have up here a little bit?

00:01:52.755 --> 00:01:55.458
And I, in other words, keep in
mind what I just wrote down.

00:01:55.458 --> 00:01:56.955
Well it might be, just
jumping at it, you get

00:01:56.955 --> 00:01:59.363
an x plus four divided by an x plus four,

00:01:59.363 --> 00:02:01.801
why can't we simplify this
expression a little bit?

00:02:01.801 --> 00:02:04.100
And just write this whole thing, write

00:02:04.100 --> 00:02:06.862
our original expression as being

00:02:06.862 --> 00:02:11.862
x minus six over x plus
six, where x, if we want it

00:02:12.121 --> 00:02:15.491
to be algebraically equivalent
we have to constrain

00:02:15.491 --> 00:02:19.786
the domain, where x cannot
be equal to negative four.

00:02:19.786 --> 00:02:22.584
Well this is interesting
because before, when x equals

00:02:22.584 --> 00:02:25.626
negative four, we had all this
weird zero over zero stuff.

00:02:25.626 --> 00:02:28.087
Well we were able to
remove it, and we were able

00:02:28.087 --> 00:02:30.305
to make it this kind of,
this extra condition.

00:02:30.305 --> 00:02:32.151
And if you were to graph
it, it would show up

00:02:32.151 --> 00:02:35.193
as a little bit of a
removable discontinuity.

00:02:35.193 --> 00:02:36.714
A little bit of a point discontinuity,

00:02:36.714 --> 00:02:37.944
just one point, where the function

00:02:37.944 --> 00:02:41.752
is not actually, is actually not defined.

00:02:41.752 --> 00:02:44.294
So in this situation,
and this is typical where

00:02:44.294 --> 00:02:47.905
there's a thing that you can factor out,

00:02:47.905 --> 00:02:49.449
it was making the numerator
and the denominator

00:02:49.449 --> 00:02:51.190
equal zero but you can factor that out,

00:02:51.190 --> 00:02:52.351
so it no longer does it.

00:02:52.351 --> 00:02:54.847
That's going to be
removable discontinuity.

00:02:54.847 --> 00:02:59.792
So x equals negative four is
a removable discontinuity.

00:02:59.792 --> 00:03:01.464
And once you've factored
out all the things

00:03:01.464 --> 00:03:03.507
that would make it a
removable discontinuity,

00:03:03.507 --> 00:03:06.142
then you can think about
what's going to be a zero

00:03:06.142 --> 00:03:08.592
and what's going to be
a vertical asymptote.

00:03:08.592 --> 00:03:10.670
If it makes, once you've
factored out everything

00:03:10.670 --> 00:03:13.085
that's common to the
numerator and the denominator,

00:03:13.085 --> 00:03:15.766
if of what's left, something
makes the numerator

00:03:15.766 --> 00:03:17.647
equal to zero, well,
then it's going to make

00:03:17.647 --> 00:03:20.073
this whole expression equal to zero

00:03:20.073 --> 00:03:23.243
and so you're dealing with
a zero, so x equals six.

00:03:23.243 --> 00:03:25.262
Six would make the
numerator equal to zero.

00:03:25.262 --> 00:03:27.143
Six minus six is zero.

00:03:27.143 --> 00:03:30.405
So there you're dealing with a zero.

00:03:30.405 --> 00:03:32.497
And to make the denominator equal to zero,

00:03:32.497 --> 00:03:34.390
you would get x equals negative six.

00:03:34.390 --> 00:03:35.609
That would make the denominator equal zero

00:03:35.609 --> 00:03:39.195
so that's a vertical asymptote.

00:03:39.195 --> 00:03:40.391
It's called a vertical asymptote

00:03:40.391 --> 00:03:42.469
because as you approach this value,

00:03:42.469 --> 00:03:44.629
as approach negative
six, either from values

00:03:44.629 --> 00:03:47.206
smaller than negative six
or larger than negative six,

00:03:47.206 --> 00:03:51.629
the denominator is going to
become either very, very, very,

00:03:51.629 --> 00:03:54.845
it's either going to become a
very, very, very, very small

00:03:54.845 --> 00:03:57.213
positive number or a very, very,
very small negative number,

00:03:57.213 --> 00:03:59.953
so we're going to approach
zero either from above or below

00:03:59.953 --> 00:04:02.494
and so when you divide by
that, you're going to get

00:04:02.494 --> 00:04:05.606
very large positive values or
very large negative values.

00:04:05.606 --> 00:04:09.286
And so your graph does stuff like that.

00:04:09.286 --> 00:04:12.547
So you have a vertical asymptote.

00:04:12.547 --> 00:04:14.046
It might either do something like that

00:04:14.046 --> 00:04:18.144
or it might do something like this,

00:04:18.144 --> 00:04:20.210
where this is your vertical asymptote,

00:04:20.210 --> 00:04:21.232
and then when you approach from here,

00:04:21.232 --> 00:04:22.950
it goes, it bumps up like that.

00:04:22.950 --> 00:04:25.491
So either way, so that's why
that's a vertical asymptote.

00:04:25.491 --> 00:04:27.652
Let's do another one.

00:04:27.652 --> 00:04:32.586
Alright, so same drill and like before,

00:04:32.586 --> 00:04:34.546
let, we'll always pause the video

00:04:34.546 --> 00:04:36.602
and see if you can work
it out on your own.

00:04:36.602 --> 00:04:38.529
Let me factor it.

00:04:38.529 --> 00:04:40.851
So let's see, two numbers,
product is negative 32.

00:04:40.851 --> 00:04:43.881
They're going to have
different signs, eight, four.

00:04:43.881 --> 00:04:45.646
And then we want the
large one to be positive

00:04:45.646 --> 00:04:48.142
since they have to add
up to positive four x.

00:04:48.142 --> 00:04:53.142
So x plus eight times x minus four.

00:04:53.575 --> 00:04:54.956
Yep, I think that's right.

00:04:54.956 --> 00:04:59.391
And then divided by, this
is four times four is 16.

00:04:59.391 --> 00:05:01.993
Four, negative four plus
negative four is negative eight

00:05:01.993 --> 00:05:06.392
so this x minus four times x minus four.

00:05:06.392 --> 00:05:08.505
Now this is really, really interesting

00:05:08.505 --> 00:05:10.084
because you might say,
oh, I have an x minus four

00:05:10.084 --> 00:05:13.044
x minus four, maybe, x equals four

00:05:13.044 --> 00:05:15.389
is a removable discontinuity.

00:05:15.389 --> 00:05:17.305
And it would have been unless you had,

00:05:17.305 --> 00:05:18.826
if you didn't have this over here,

00:05:18.826 --> 00:05:21.322
because x equals four, even
after you've factored this.

00:05:21.322 --> 00:05:23.770
You still can't, the
expression, the function,

00:05:23.770 --> 00:05:26.871
is still not going to be
defined when x equals four.

00:05:26.871 --> 00:05:30.202
And so this function is
actually equivalent to

00:05:30.202 --> 00:05:32.710
x plus eight over x minus 4.

00:05:32.710 --> 00:05:35.810
I don't have to put that
extra little constraint

00:05:35.810 --> 00:05:37.307
like I did before.

00:05:37.307 --> 00:05:39.408
I don't have to put, I don't have to put

00:05:39.408 --> 00:05:41.011
this little constraint that describes

00:05:41.011 --> 00:05:43.392
this removable discontinuity
because the constraint

00:05:43.392 --> 00:05:45.830
is still there, after I've factored out,

00:05:45.830 --> 00:05:48.035
after I've cancelled out this x minus four

00:05:48.035 --> 00:05:49.417
with this x minus four.

00:05:49.417 --> 00:05:52.528
And so here, we've, this is
actually an algebraically

00:05:52.528 --> 00:05:56.696
equivalent expression to
this one right over here.

00:05:56.696 --> 00:05:59.412
And so now, we can think about

00:05:59.412 --> 00:06:02.071
what the zeros or the vertical asymptotes

00:06:02.071 --> 00:06:04.706
or the removable
discontinuities are going to be.

00:06:04.706 --> 00:06:06.772
Well, something that
makes the numerator zero

00:06:06.772 --> 00:06:10.371
without doing it to the
denominator is going to be a zero.

00:06:10.371 --> 00:06:12.565
And x equals negative
eight makes the numerator

00:06:12.565 --> 00:06:14.469
equal to zero without
doing anything to, without

00:06:14.469 --> 00:06:18.335
making the denominator equal to zero.

00:06:18.335 --> 00:06:20.065
It would make the
denominator to negative 12.

00:06:20.065 --> 00:06:23.071
So you could literally just
evaluate h of negative eight

00:06:23.071 --> 00:06:28.071
is equal to zero over negative
12, which is equal to zero.

00:06:29.410 --> 00:06:30.907
So that's why we call it a zero.

00:06:30.907 --> 00:06:32.544
What about x equals four?

00:06:32.544 --> 00:06:34.367
Well, x equals four is going
to make the denominator

00:06:34.367 --> 00:06:36.747
only equal to zero, so that's going

00:06:36.747 --> 00:06:40.006
to give us a vertical asymptote.

00:06:42.110 --> 00:06:44.490
And we're all done.