Approximating limit from numeric table

WEBVTT
Kind: captions
Language: en

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- [Instructor] In this
video, we're going to try

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to get a sense of what the
limit as x approaches three

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of x to the third minus 3x squared

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over 5x minus 15 is.

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And when I say get a sense,
we're going to do that

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by seeing what values for
this expression we get

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as x gets closer and closer to three.

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Now, one thing that you
might want to try out,

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is, well, what happens to this expression

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when x is equal to three?

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Well, then, it's going to
be three to the third power

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minus three times three squared

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over five times three minus 15.

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So at x equals three, this
expression's going to be,

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and let's see, in the
numerator, you have 27 minus 27,

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zero, over 15 minus 15, over zero.

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So this expression is actually

00:00:47.030 --> 00:00:50.200
not defined as x equals three.

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We get this indeterminate form.

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We get zero over zero.

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But let's see, even though the function,

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even though the expression is not defined,

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let's see if we can get a sense
of what the limit might be.

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And to do that, I'm
going to set up a table.

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So, let me set up a table here.

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And actually I'm going
to set up two tables.

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So this is x and this is x to the third

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minus 3x squared over 5x minus 15.

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Actually, I'm going to do that again,

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and I'll tell you why in a second.

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So, this is gonna be x,
and this is x to the third

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minus 3x squared over 5x minus 15.

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The reason why I set up two tables,

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I didn't have to do two tables.

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I could have done it all in one table,

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but hopefully, this will make it

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a little bit more intuitive
what I'm trying to do,

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is on this left table,

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I'm gonna, let's try out x values

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that get closer and close to three

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from the left, from values
that are less than three.

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So for example, we could
go to 2.9 and figure out

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what the expression equals
when x is 2.9, but then

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we could try to get even a
little bit closer than that.

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We could go to 2.99,

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and then we could go
even closer than that.

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We could go to 2.999, and so,
one way to think about it here

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is as we try to figure out
what this expression equals

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as we get closer and closer to three,

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we're trying to approximate
the limit from the left.

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So, limit from the left.

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And why do I say the left?

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Well, if you would think about
this on a coordinate plane,

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these are the x values, so
they're to the left of three,

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but we're getting closer
and closer and closer.

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We're moving to the right,

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but these are the x values
that are on the left side

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of three, that are less than three.

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But we also, in order
for the limit to exist,

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we have to be approaching the
same thing from both sides,

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from both the left and the right.

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So we could also try to approximate

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the limit from the right,

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and so what values would those be?

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Well, those would be x
values larger than three,

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so we could say 3.1,

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but then we might wanna
get a little bit closer.

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We could go 3.01, but
then we might wanna get

00:03:06.261 --> 00:03:07.236
even closer to three.

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3.001, and every time we get
closer and closer to three,

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we're gonna get a better approximation,

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or we're gonna get a better sense

00:03:16.695 --> 00:03:19.887
of what we are actually approaching.

00:03:19.887 --> 00:03:21.401
So let's get a calculator out and do this,

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and you could keep going, 2.999999,

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3.000001.

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Now, one key idea here to point out

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before I even calculate
what these are going to be,

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sometimes when people say
the limit from both sides

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or the limit from the left
or the limit from the right,

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they imagine that the limit
from the left is negative values

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and the limit from the
right are positive values.

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But as you can see here,
the limit from the left

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are to the left of the x value

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that you're trying to find the limit at.

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So these aren't negative values.

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These are just approaching the three

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right over here from
values less than three.

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This is approaching the three
from values larger than three.

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So let's see.

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Let's calculate the
value of this expression

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when x is equal to 2.9.

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So we have 2.9

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to the third power

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minus three times 2.9 squared,

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which is, close the parentheses,

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so that's our numerator,
and then divided by,

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open parentheses, five times 2.9 minus 15,

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that's our denominator,
is equal to, let's see,

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I got about 1.682.

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1.682.

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So now let's keep going.

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So, let's see.

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If we do 2.99 to the third power

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minus 3 times 2.99 squared

00:05:03.555 --> 00:05:06.555
is equal to, and then we divide that

00:05:08.491 --> 00:05:11.971
by our denominator, open parentheses,

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five times 2.99 minus 15

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is equal to 1.78802.

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Well, I'll just found that to 1.788,

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approximately 1.788,

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and now let's get even closer to three.

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Remember, we're now approaching
three from the left,

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so 2.999 is even that
much closer to three.

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So let's get the calculator out again,

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and we get...

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Now we get 2.999 to the third power

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minus three times, open
parentheses, 2.999 squared,

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close parentheses.

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So that's our numerator.

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Divided by, open parenthesis,

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fives times 2.999 minus 15

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is equal to 1.7988 or
approximately 1.7988.

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Approximately 1.7988.

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So it seems to be getting
closer and closer to 1.8,

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and you could even verify.

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Try 2.9999, see if it's
getting closer to 1.8.

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So this is our limit from the left.

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Now we could try the limit from the right.

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So what happens when we try 3.1.

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So 3.1 to the third power

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minus three times, open
parentheses, 3.1 squared,

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close parentheses, that's our numerator,

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divided by, open
parentheses, five times 3.1

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minus 15, that's our
denominator, is equal to 1.922.

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1.922.

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All right, let's keep going.

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This is something
strangely fun about this.

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All right, so let's see when x is 3.01.

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So now we're approaching
x equals 3 from the right,

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from values that are larger.

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We're getting closer
and closer and closer.

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We're approaching from the right,

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but we're moving towards the left,

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towards x equals three, without
equaling x equals three.

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So now let's do 3.01 to the third power

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minus three times, open parentheses,

00:07:45.121 --> 00:07:47.871
3.01 squared, closed parentheses.

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That's our numerator.

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Divided by, open
parentheses, five times 3.01

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minus 15, close parentheses,
is equal to 1.81,

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I'll say roughly 1.812,
approximately 1.812.

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It does look like we're
approaching 1.8 from above, now,

00:08:13.628 --> 00:08:17.028
but let's just get even closer, 3.001.

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So we can finish at least
what I put on our tables here.

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If you're unsure, you can
just keep getting closer

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and closer and closer,
3.0000001 if you're interested.

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But let's just do this one.

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3.001 to the third power

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minus three times 3.001 squared,

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close parentheses, that's our numerator,

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divided by, open parentheses,
five times 3.001 minus 15,

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that's our denominator,
is equal to roughly 1.801.

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So approximately 1.801.

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So based on what we're
seeing here, I would make

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the estimate that this looks
like it's approaching 1.8,

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so is this equal to 1.8.

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As I said, in the future,
we're going to be able to find

00:09:10.250 --> 00:09:12.722
this out exactly, but if
you're not sure about this,

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you could try even closer
and closer and closer values.

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Actually, just for kicks,

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let's just do a super close value.

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Let's do 2.99999.

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Let's see what that gets us.

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So we're getting really close.

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We're approaching from the left,

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but we're getting really
close to three now.

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So let me get the calculator out here.

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Okay, so how many is that?

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That is five nines.

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So two point one, two, three, four, five,

00:09:42.354 --> 00:09:43.937
to the third power,

00:09:44.890 --> 00:09:46.390
minus three times,

00:09:48.797 --> 00:09:51.503
and we're gonna have two point

00:09:51.503 --> 00:09:54.586
one, two, three, four, five, squared,

00:09:55.469 --> 00:09:57.148
close parentheses.

00:09:57.148 --> 00:09:58.581
That's our numerator.

00:09:58.581 --> 00:10:01.164
Divided by five times two point

00:10:03.392 --> 00:10:05.887
one, two, three, four, five,

00:10:05.887 --> 00:10:09.387
2.99999, minus 15, that's our denominator,

00:10:11.584 --> 00:10:14.717
is equal to, notice, we got even closer.

00:10:14.717 --> 00:10:16.249
In fact, if we round this at all,

00:10:16.249 --> 00:10:18.998
this is going to be 1.8.

00:10:18.998 --> 00:10:22.144
So yeah, if we round it
to the nearest thousandth,

00:10:22.144 --> 00:10:23.381
this is going to...

00:10:23.381 --> 00:10:25.606
Yeah, if we round this to
the nearest thousandth,

00:10:25.606 --> 00:10:27.549
this is going to be 1.8.

00:10:27.549 --> 00:10:31.037
So we're getting very close here to 1.8,

00:10:31.037 --> 00:10:32.750
and if you're not feeling good about that,

00:10:32.750 --> 00:10:35.685
you could go even closer
and closer and closer

00:10:35.685 --> 00:10:39.026
to three to feel better and
better and better about it.

00:10:39.026 --> 00:10:40.071
Once again, we're approximating it.

00:10:40.071 --> 00:10:41.855
We're seeing, what is the
behavior as we get closer

00:10:41.855 --> 00:10:44.687
and closer to three from
the left and from the right.

00:10:44.687 --> 00:10:47.003
And not only is this valuable
for calculating limits,

00:10:47.003 --> 00:10:48.994
and sometimes you would have
to calculate them numerically

00:10:48.994 --> 00:10:51.649
like this, but it also
gives you an intuition

00:10:51.649 --> 00:10:54.149
for what a limit is all about.


Example approximating limit graphically

WEBVTT
Kind: captions
Language: en

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- [Instructor] The function h is defined

00:00:01.675 --> 00:00:06.018
for all real numbers in
the graph y is equal to

00:00:06.018 --> 00:00:08.118
h of x, right over here.

00:00:08.118 --> 00:00:09.890
That's what they're showing us.

00:00:09.890 --> 00:00:11.851
And they ask us, what
is a reasonable estimate

00:00:11.851 --> 00:00:15.585
for the limit as x approaches
negative seven of h of x,

00:00:15.585 --> 00:00:17.774
and they give us some choices
for those reasonable estimates

00:00:17.774 --> 00:00:20.699
including the possibility
that the limit does not exist.

00:00:20.699 --> 00:00:22.092
So encourage you to pause the video

00:00:22.092 --> 00:00:24.697
and see if you can work
through this yourself.

00:00:24.697 --> 00:00:27.293
All right, now let's think
about what's going on.

00:00:27.293 --> 00:00:31.539
So something is interesting
at x equals negative seven.

00:00:31.539 --> 00:00:35.095
At x equals negative seven,
we have the function.

00:00:35.095 --> 00:00:38.833
It is defined, but it is not continuous

00:00:38.833 --> 00:00:40.754
with the rest of the function.

00:00:40.754 --> 00:00:42.959
H of negative seven,

00:00:42.959 --> 00:00:43.792
h of

00:00:44.908 --> 00:00:46.408
negative seven is,

00:00:47.603 --> 00:00:49.690
looks like a little bit,

00:00:49.690 --> 00:00:52.403
it looks like it's like
negative four point something.

00:00:52.403 --> 00:00:56.084
So it's approximately negative four point,

00:00:56.084 --> 00:00:57.837
I don't know, looks like negative

00:00:57.837 --> 00:01:01.250
4.1 something, something, something.

00:01:01.250 --> 00:01:04.129
So that's what h of negative seven is.

00:01:04.129 --> 00:01:05.519
And sometimes there's this temptation

00:01:05.519 --> 00:01:07.137
to say, oh, whatever the function equals,

00:01:07.137 --> 00:01:08.935
that must be what the limit is.

00:01:08.935 --> 00:01:10.886
But this is actually a really good example

00:01:10.886 --> 00:01:12.586
for showing the difference,

00:01:12.586 --> 00:01:16.605
that many times, the limit
exists and is approaching

00:01:16.605 --> 00:01:19.104
or it is a value different
than what the value

00:01:19.104 --> 00:01:21.969
that the function takes on at that point.

00:01:21.969 --> 00:01:24.071
And so let's think about the limit.

00:01:24.071 --> 00:01:27.041
As x approaches negative seven

00:01:27.041 --> 00:01:29.303
from smaller values of x,

00:01:29.303 --> 00:01:32.649
so as x gets closer and
closer to negative seven,

00:01:32.649 --> 00:01:35.129
it looks like the value of
our function is approaching

00:01:35.129 --> 00:01:37.695
where we have this gap.

00:01:37.695 --> 00:01:40.641
So it looks like our
function is getting closer,

00:01:40.641 --> 00:01:44.705
the value of our function
is approaching this value

00:01:44.705 --> 00:01:46.038
right over here.

00:01:47.249 --> 00:01:51.379
Similarly, if we were to
approach from the left,

00:01:51.379 --> 00:01:54.230
so for approach from more negative values,

00:01:54.230 --> 00:01:56.944
it looks like the value of our function

00:01:56.944 --> 00:01:58.915
is approaching that same thing.

00:01:58.915 --> 00:02:01.618
And since it looks like it's
approaching the same value

00:02:01.618 --> 00:02:03.588
whether we're coming
from more negative values

00:02:03.588 --> 00:02:05.907
or less negative values,

00:02:05.907 --> 00:02:08.277
we know, or we can assume,

00:02:08.277 --> 00:02:10.800
or we have a good sense
that this limit exists.

00:02:10.800 --> 00:02:12.707
We seem to be approaching the same value.

00:02:12.707 --> 00:02:14.012
Now, it's a value different

00:02:14.012 --> 00:02:15.759
than the actual value
of the function there.

00:02:15.759 --> 00:02:17.939
The value of the function
there is at negative 4.1

00:02:17.939 --> 00:02:19.432
or something like that.

00:02:19.432 --> 00:02:24.319
Now, this looks like the
function itself is approaching,

00:02:24.319 --> 00:02:27.397
this looks like, I don't, one point,

00:02:27.397 --> 00:02:28.230
1.2,

00:02:29.182 --> 00:02:31.728
1.3, something like that.

00:02:31.728 --> 00:02:33.398
So now, let's look at our choices.

00:02:33.398 --> 00:02:37.334
What's a reasonable
estimate for this limit?

00:02:37.334 --> 00:02:39.087
Well, it's definitely not negative seven.

00:02:39.087 --> 00:02:41.078
The function is definitely not approaching

00:02:41.078 --> 00:02:42.932
y equals negative seven.

00:02:42.932 --> 00:02:44.516
This is kind of a distractor choice

00:02:44.516 --> 00:02:46.956
because this is what x is approaching.

00:02:46.956 --> 00:02:48.459
X is approaching negative seven,

00:02:48.459 --> 00:02:50.821
but we want to know what is
the function approaching?

00:02:50.821 --> 00:02:52.627
What is the value of h approaching

00:02:52.627 --> 00:02:54.356
as x approaches negative seven?

00:02:54.356 --> 00:02:56.547
So let's rule that one out.

00:02:56.547 --> 00:02:57.630
Negative 4.1,

00:02:59.088 --> 00:03:02.227
so negative 4.1 is an interesting choice

00:03:02.227 --> 00:03:03.816
because that's a good approximation

00:03:03.816 --> 00:03:07.296
for what the actual value
of the function is there.

00:03:07.296 --> 00:03:09.257
The function is defined there,

00:03:09.257 --> 00:03:12.328
although it kind of jumps and
is defined right over here,

00:03:12.328 --> 00:03:14.396
and it's discontinuous.

00:03:14.396 --> 00:03:16.854
So this is more of what h,

00:03:16.854 --> 00:03:20.304
this could be a reasonable
estimate for h of negative seven,

00:03:20.304 --> 00:03:22.742
what the function
actually is defined to be

00:03:22.742 --> 00:03:24.408
at x equals negative seven.

00:03:24.408 --> 00:03:26.541
But as we approach x equals,

00:03:26.541 --> 00:03:30.093
as we approach negative
seven, it is not approaching,

00:03:30.093 --> 00:03:31.725
the value of the function does not seem

00:03:31.725 --> 00:03:33.726
to be approaching negative 4.1.

00:03:33.726 --> 00:03:35.707
So I'm gonna rule that one out.

00:03:35.707 --> 00:03:37.922
Now, 1.3, that is what,

00:03:37.922 --> 00:03:39.791
that is a pretty reasonable estimate

00:03:39.791 --> 00:03:42.151
for what the function
seems to be approaching

00:03:42.151 --> 00:03:44.830
as x gets closer and
closer to negative seven

00:03:44.830 --> 00:03:46.575
'cause, you know, 6.9, 6.99,

00:03:46.575 --> 00:03:50.509
or negative 6.9, negative
6.99, negative 6.9999,

00:03:50.509 --> 00:03:52.219
from the right or from the left,

00:03:52.219 --> 00:03:54.851
negative 7.1, negative 7.01,

00:03:54.851 --> 00:03:57.564
negative 7.001, so on and so forth.

00:03:57.564 --> 00:04:01.049
It looks like the value of
that function is approaching

00:04:01.049 --> 00:04:03.046
approximately 1.3.

00:04:03.046 --> 00:04:04.541
So I like

00:04:04.541 --> 00:04:07.336
that choice right over there.

00:04:07.336 --> 00:04:09.785
1.8, well, that's just another choice

00:04:09.785 --> 00:04:12.269
but this, looking,
approximating where this is,

00:04:12.269 --> 00:04:14.901
it definitely looks closer
to one than it does to two.

00:04:14.901 --> 00:04:16.326
So I'd rule that one out.

00:04:16.326 --> 00:04:17.962
And I would say that this limit does exist

00:04:17.962 --> 00:04:19.964
because it looks like we're
approaching the same value

00:04:19.964 --> 00:04:24.131
whether we're coming from below
or from above, so to speak.


Approximating limits

WEBVTT
Kind: captions
Language: en

00:00:00.320 --> 00:00:01.423
- [Instructor] What we're
gonna do in this video

00:00:01.423 --> 00:00:04.107
is see how we can approximate limits

00:00:04.107 --> 00:00:06.934
graphically and using tables.

00:00:06.934 --> 00:00:08.772
In the future, we're also going to be able

00:00:08.772 --> 00:00:10.528
to learn techniques where
we're going to be able

00:00:10.528 --> 00:00:13.783
to directly figure out
exactly what this limit is.

00:00:13.783 --> 00:00:16.111
But for now, let's think
about how to approximate it,

00:00:16.111 --> 00:00:17.741
and I'll build our intuition

00:00:17.741 --> 00:00:20.138
for even what a limit actually is.

00:00:20.138 --> 00:00:23.116
So we want to know the
limit as x approaches two

00:00:23.116 --> 00:00:27.283
of x to the third minus 2x
squared over 3x minus six.

00:00:28.626 --> 00:00:30.338
Now, the first thing you
might wanna check out

00:00:30.338 --> 00:00:32.955
is, well, what is this expression

00:00:32.955 --> 00:00:36.970
equal to when x is exactly equal to two?

00:00:36.970 --> 00:00:39.414
And we could do that by
substituting x with two,

00:00:39.414 --> 00:00:42.566
so it'll be two to the
third power minus two

00:00:42.566 --> 00:00:46.649
times two squared over
three times two minus six.

00:00:50.393 --> 00:00:51.420
Well, this numerator over here,

00:00:51.420 --> 00:00:54.108
this is eight minus two
times four minus eight,

00:00:54.108 --> 00:00:56.654
so that's going to be equal to zero,

00:00:56.654 --> 00:00:59.429
and then the denominator
over here is six minus six.

00:00:59.429 --> 00:01:00.758
Well, that's a zero,

00:01:00.758 --> 00:01:04.520
and we end up in the indeterminate
form right over here.

00:01:04.520 --> 00:01:08.687
So this expression is not
defined for x equals two,

00:01:09.705 --> 00:01:13.391
but we can think about what
does the expression approach

00:01:13.391 --> 00:01:15.308
as x approaches two.

00:01:16.732 --> 00:01:19.536
And first, let's think about it visually.

00:01:19.536 --> 00:01:24.163
So, if we were to graph it,
and I graph this on the site

00:01:24.163 --> 00:01:27.999
Desmos, which has a nice
graphing calculator,

00:01:27.999 --> 00:01:32.222
you see the curve, y equals
this expression right over here.

00:01:32.222 --> 00:01:36.037
So this is the curve of y is equal...

00:01:36.037 --> 00:01:38.451
Let me do that in a darker color.

00:01:38.451 --> 00:01:42.603
This is equal to y is
equal to x to the third

00:01:42.603 --> 00:01:45.520
minus 2x squared over 3x minus six.

00:01:47.933 --> 00:01:49.788
And you can see, at least everywhere

00:01:49.788 --> 00:01:51.169
where I've shown it here,
and it's actually true,

00:01:51.169 --> 00:01:54.908
it's defined everywhere except
for when x is equal to two,

00:01:54.908 --> 00:01:57.833
and that's why we have
this little gap over here,

00:01:57.833 --> 00:01:59.597
showing that it is not defined.

00:01:59.597 --> 00:02:01.946
And so what I wanna do
is I wanna approximate,

00:02:01.946 --> 00:02:04.579
well, as x gets closer and closer to two,

00:02:04.579 --> 00:02:06.749
either from lower values of two

00:02:06.749 --> 00:02:10.916
or from larger values of two,
what is the value of this

00:02:11.991 --> 00:02:15.824
expression or the value of
this function approaching?

00:02:15.824 --> 00:02:17.525
And at this level of zoom,

00:02:17.525 --> 00:02:21.692
it looks like it's approaching
this value right over here,

00:02:22.596 --> 00:02:23.937
so as x gets closer and closer to two,

00:02:23.937 --> 00:02:26.254
it looks like our function
is getting closer and closer

00:02:26.254 --> 00:02:27.684
to that value there,

00:02:27.684 --> 00:02:30.379
regardless of which direction
we are approaching from,

00:02:30.379 --> 00:02:33.351
and so just approximating
it visually right over here,

00:02:33.351 --> 00:02:34.184
let's see.

00:02:34.184 --> 00:02:36.313
This is zero, this is two.

00:02:36.313 --> 00:02:38.811
This is one, right over here.

00:02:38.811 --> 00:02:41.078
This would be 1.5.

00:02:41.078 --> 00:02:42.289
At this level of zoom,

00:02:42.289 --> 00:02:45.289
it looks like it's about 1.3 or 1.4,

00:02:46.683 --> 00:02:47.850
so 1.3 or 1.4.

00:02:50.749 --> 00:02:52.445
Let's zoom in a little bit more,

00:02:52.445 --> 00:02:54.333
and if you have access
to a graphing calculator

00:02:54.333 --> 00:02:57.088
or you go to a website like
Desmos or Wolfram Alpha,

00:02:57.088 --> 00:03:00.233
you can zoom in further
and further on this graph,

00:03:00.233 --> 00:03:02.070
so I encourage you to
try that out yourself

00:03:02.070 --> 00:03:04.262
or do it with other graphs.

00:03:04.262 --> 00:03:06.512
So let's zoom in even more.

00:03:07.839 --> 00:03:12.501
So once again, we're not
defined at x equals two,

00:03:12.501 --> 00:03:15.065
but here we get a slightly better read.

00:03:15.065 --> 00:03:17.663
Let's see, this is one, this is two.

00:03:17.663 --> 00:03:19.378
The value that we are approaching

00:03:19.378 --> 00:03:22.006
as x gets closer and closer to two,

00:03:22.006 --> 00:03:24.460
we're getting, our value is
getting closer and closer

00:03:24.460 --> 00:03:26.813
to that right over there,

00:03:26.813 --> 00:03:30.498
and if we look at what y value that is,

00:03:30.498 --> 00:03:32.324
let's see, if this is...

00:03:32.324 --> 00:03:34.757
This is split into one,
two, three, four, five.

00:03:34.757 --> 00:03:37.176
So this is 1.2 right over here.

00:03:37.176 --> 00:03:40.299
This is 1.4 right over here.

00:03:40.299 --> 00:03:42.087
So it looks like it's between...

00:03:42.087 --> 00:03:43.981
So this would be 1.3 right over there.

00:03:43.981 --> 00:03:47.232
So it's a little bit more than 1.3.

00:03:47.232 --> 00:03:49.399
So a little more than 1.3.

00:03:50.385 --> 00:03:52.468
So, approximately, 1.3...

00:03:54.522 --> 00:03:56.626
Now, I do that in a lighter color.

00:03:56.626 --> 00:04:00.793
So it's approximately 1.3
something, it looks like.

00:04:02.331 --> 00:04:04.636
Let's zoom in even more

00:04:04.636 --> 00:04:08.059
to see if we can get an
even better approximation.

00:04:08.059 --> 00:04:11.049
So now, once again, we're
approaching that same value.

00:04:11.049 --> 00:04:12.994
We're not defined at x equals two,

00:04:12.994 --> 00:04:17.161
but as we're approaching
x equals two, let's see.

00:04:18.033 --> 00:04:20.650
Let me get a darker color.

00:04:20.650 --> 00:04:23.567
So, it would be right around there,

00:04:24.640 --> 00:04:26.435
and this is, let's see, this is one, 1.1,

00:04:26.435 --> 00:04:28.752
1.2, 1.3, 1.4, 1.5.

00:04:28.752 --> 00:04:31.626
So once again, it looks
like it's about one point...

00:04:31.626 --> 00:04:34.260
If I were to just base it off of this,

00:04:34.260 --> 00:04:37.510
it looks like it's about 1.333 or 1.34,

00:04:40.329 --> 00:04:42.829
so I'd say approximately 1.33.

00:04:44.687 --> 00:04:46.604
So, approximately 1.33.

00:04:47.845 --> 00:04:49.691
If I were, it looks like it might be

00:04:49.691 --> 00:04:52.199
approaching 1 1/3, but
we don't know for sure.

00:04:52.199 --> 00:04:54.630
Remember, when you're
trying to figure out a limit

00:04:54.630 --> 00:04:57.201
from a graph the best
you can really do here

00:04:57.201 --> 00:04:59.116
is just approximate, try to eyeball, well,

00:04:59.116 --> 00:05:01.637
the closer x gets to two,
it looks like this function

00:05:01.637 --> 00:05:05.295
is approaching this value right over here.

00:05:05.295 --> 00:05:08.928
Now, another technique which
tends to be a little bit more

00:05:08.928 --> 00:05:13.095
precise is to try to approximate
this limit numerically.

00:05:14.023 --> 00:05:15.770
So let's do that.

00:05:15.770 --> 00:05:19.020
So let me get rid of these graphs here.

00:05:19.889 --> 00:05:21.972
We already got a sense of

00:05:23.365 --> 00:05:24.977
what the graphs can do for us.

00:05:24.977 --> 00:05:27.274
They got us to about 1.33,

00:05:27.274 --> 00:05:30.544
but now, let's try to do it numerically.

00:05:30.544 --> 00:05:33.235
So, I'm going to set up a table here,

00:05:33.235 --> 00:05:36.714
and I encourage you to do the same.

00:05:36.714 --> 00:05:38.316
So, on this column, I'll have my x,

00:05:38.316 --> 00:05:39.481
and on this column, I'm going to say,

00:05:39.481 --> 00:05:43.528
well, what is the expression
x minus 2x squared

00:05:43.528 --> 00:05:45.695
over 3x minute six equals?

00:05:46.559 --> 00:05:49.111
And we know that when x
is exactly equal to two,

00:05:49.111 --> 00:05:51.346
that this thing right
over here isn't defined,

00:05:51.346 --> 00:05:53.679
but let's see what happens
as we approach two.

00:05:53.679 --> 00:05:56.571
So let's see what happens
when we're at 1.9,

00:05:56.571 --> 00:05:57.404
or 1.99...

00:06:00.722 --> 00:06:01.972
I'll do 1.99...

00:06:02.810 --> 00:06:03.977
I'll do 1.999.

00:06:04.882 --> 00:06:06.631
And we could also see
from the other direction.

00:06:06.631 --> 00:06:10.798
We could say, well, what happens
when we approach at 2.001,

00:06:13.335 --> 00:06:15.743
and see if we are getting...

00:06:15.743 --> 00:06:18.073
If this, if both of
these values seem to be

00:06:18.073 --> 00:06:21.010
approaching something, and
so this is approaching it

00:06:21.010 --> 00:06:25.260
from lower values of x
and then we could say

00:06:25.260 --> 00:06:27.906
this was approaching it
from higher values of x.

00:06:27.906 --> 00:06:30.297
We could say 2.1, right over here.

00:06:30.297 --> 00:06:32.554
So let me get a calculator
out and evaluate these,

00:06:32.554 --> 00:06:35.391
and I encourage you to do the
same, get a calculator out,

00:06:35.391 --> 00:06:38.277
and see if you can evaluate these things.

00:06:38.277 --> 00:06:39.498
All right, so let's see.

00:06:39.498 --> 00:06:42.120
If we can evaluate it when x equals 1.9,

00:06:42.120 --> 00:06:45.370
it's going to be 1.9 to the third power

00:06:47.666 --> 00:06:49.166
minus two times...

00:06:52.580 --> 00:06:54.413
Two times 1.9 squared.

00:06:57.068 --> 00:07:00.901
1.9 squared is equal to,
so we get that value,

00:07:04.677 --> 00:07:08.427
and then we're going to
divide by, divide by,

00:07:09.677 --> 00:07:13.844
we get, we're gonna divide
by three times 1.9 minus six,

00:07:18.227 --> 00:07:20.136
and that's going to be equal to,

00:07:20.136 --> 00:07:23.173
looks like it's about
1.203, so I'll just write

00:07:23.173 --> 00:07:24.840
approximately 1.203.

00:07:28.838 --> 00:07:33.302
Now, let's try it with a
much closer value of x.

00:07:33.302 --> 00:07:34.253
So that was just 1.9.

00:07:34.253 --> 00:07:36.086
Now let's go to 1.999.

00:07:37.671 --> 00:07:40.754
So once again, we're gonna have 1.999

00:07:42.681 --> 00:07:46.431
to the third power, and
then minus two times,

00:07:48.822 --> 00:07:51.489
open parentheses, 1.999 squared,

00:07:54.735 --> 00:07:56.782
close the parentheses,

00:07:56.782 --> 00:07:59.424
is equal to, so that's my numerator,

00:07:59.424 --> 00:08:03.412
and then my denominator, so divided by,

00:08:03.412 --> 00:08:07.118
divided by, and now, open parentheses,

00:08:07.118 --> 00:08:09.451
three times 1.999 minus six,

00:08:14.029 --> 00:08:17.362
close my parentheses, is equal to 1.332.

00:08:18.457 --> 00:08:19.545
So this is interesting.

00:08:19.545 --> 00:08:21.462
So approximately 1.332.

00:08:23.488 --> 00:08:26.031
So numerically, I seem to be approaching

00:08:26.031 --> 00:08:28.135
that same value or
close to that same value

00:08:28.135 --> 00:08:30.087
that I was approaching graphically,

00:08:30.087 --> 00:08:34.254
and we can also do it from
values of x greater than two.

00:08:35.207 --> 00:08:38.019
Let me get my calculator
back, and I'll do some of it,

00:08:38.019 --> 00:08:40.885
and I encourage you to
finish this up on your own,

00:08:40.885 --> 00:08:43.642
and you could even try 1.999999

00:08:43.642 --> 00:08:46.045
to see what it actually is approaching.

00:08:46.045 --> 00:08:49.500
So, for example, if I wanted to try 2.1,

00:08:49.500 --> 00:08:52.833
so that would be 2.1 to the third power,

00:08:54.712 --> 00:08:57.545
minus two times, open parentheses,

00:08:59.329 --> 00:09:03.496
2.1 squared, and then close
parentheses is equal to,

00:09:06.172 --> 00:09:08.988
that's my numerator, divided by,

00:09:08.988 --> 00:09:11.738
open parentheses, three times 2.1

00:09:13.795 --> 00:09:16.618
minus 6, close parentheses.

00:09:16.618 --> 00:09:20.320
That's my denominator, and I get 1.47,

00:09:20.320 --> 00:09:22.153
so approximately 1.47.

00:09:23.516 --> 00:09:25.522
So now you can think I'm
getting closer and closer to two

00:09:25.522 --> 00:09:27.033
from values larger than two.

00:09:27.033 --> 00:09:30.001
Let's see if we seem to be
approaching the same value.

00:09:30.001 --> 00:09:33.104
So now let's get even closer to two, 2.01.

00:09:33.104 --> 00:09:35.367
And I'll do this one, and
then I'll leave it up to you

00:09:35.367 --> 00:09:38.378
to see if you can get even more precise.

00:09:38.378 --> 00:09:39.622
So, two point...

00:09:39.622 --> 00:09:41.031
Actually, let's get super precise.

00:09:41.031 --> 00:09:42.031
Let's, well,

00:09:42.892 --> 00:09:45.142
let's do three zeroes here.

00:09:46.452 --> 00:09:48.535
2.0001 to the third power

00:09:51.773 --> 00:09:53.106
minus two times,

00:09:55.473 --> 00:09:57.556
open parentheses, 2.0001.

00:09:59.618 --> 00:10:01.227
So we're getting really close.

00:10:01.227 --> 00:10:03.477
Squared, close parentheses,

00:10:06.085 --> 00:10:09.636
is equal to, so that's my numerator,

00:10:09.636 --> 00:10:12.053
divided by, open parentheses,

00:10:13.138 --> 00:10:14.721
three times 2.0001,

00:10:18.909 --> 00:10:22.136
minus six, this is my denominator.

00:10:22.136 --> 00:10:24.694
Is equal to, and notice, we're
getting closer and closer

00:10:24.694 --> 00:10:28.041
and closer to 1 1/3, 1.333...

00:10:28.041 --> 00:10:29.596
Now, five.

00:10:29.596 --> 00:10:32.429
So, we'll say approximately 1.333,

00:10:34.020 --> 00:10:35.078
and this is now for,

00:10:35.078 --> 00:10:37.411
this is actually for 2.0001.

00:10:40.101 --> 00:10:44.268
So it does indeed look like
we are approaching 1.333333,

00:10:46.100 --> 00:10:50.631
or close to, it looks like
we are approaching 1 1/3.

00:10:50.631 --> 00:10:52.598
But once again, these
are just approximations,

00:10:52.598 --> 00:10:55.386
both through the table or graphically.

00:10:55.386 --> 00:10:57.840
If you wanna find the
exact value of the limit,

00:10:57.840 --> 00:10:58.962
there are other techniques.

00:10:58.962 --> 00:11:02.926
We're going to explore those
techniques in future videos.


Justinian and the Byzantine Empire

WEBVTT
Kind: captions
Language: en

00:00:00.514 --> 00:00:03.472
- [Instructor] In previous
videos, we talk about how

00:00:03.472 --> 00:00:07.094
as we exit the fourth
century, in the 390s,

00:00:07.094 --> 00:00:11.261
the emperor Theodosius, he
actually splits the Roman Empire.

00:00:12.234 --> 00:00:15.787
We already had the city of Constantinople

00:00:15.787 --> 00:00:18.947
being established as a
capital of the empire.

00:00:18.947 --> 00:00:22.474
That was done by Constantine in 324,

00:00:22.474 --> 00:00:25.476
the city previously known as Byzantium.

00:00:25.476 --> 00:00:28.552
But as we enter into the fifth century,

00:00:28.552 --> 00:00:30.964
there is an official split between

00:00:30.964 --> 00:00:34.778
the Western Roman Empire and
the Eastern Roman Empire.

00:00:34.778 --> 00:00:37.747
They're actually governed separately.

00:00:37.747 --> 00:00:41.404
Now, also in other videos, we talk about

00:00:41.404 --> 00:00:45.237
the eventual fall of the
Western Roman Empire.

00:00:47.282 --> 00:00:49.387
Now, one thing to keep in mind,

00:00:49.387 --> 00:00:51.781
sometimes when we talk
about the fall of the west,

00:00:51.781 --> 00:00:54.683
it somehow implies that
there's no governance left

00:00:54.683 --> 00:00:56.024
or there's no kingdoms left.

00:00:56.024 --> 00:00:58.445
And nothing could be
further from the truth.

00:00:58.445 --> 00:01:01.388
One of the reasons why the
Western Roman Empire fell

00:01:01.388 --> 00:01:04.224
is repeated attacks by Germanic tribes,

00:01:04.224 --> 00:01:07.097
and after the Western
Roman Empire was gone,

00:01:07.097 --> 00:01:10.096
the Germanic tribes, for
the most part, took over.

00:01:10.096 --> 00:01:12.586
On the Italian peninsula,

00:01:12.586 --> 00:01:15.761
you at first have the kingdom of Odoacer,

00:01:15.761 --> 00:01:17.955
but eventually you have rule,

00:01:17.955 --> 00:01:19.996
especially as we get into
the late fifth century,

00:01:19.996 --> 00:01:23.747
you have the Ostrogothic
Kingdom, Ostrogoths,

00:01:23.747 --> 00:01:27.300
the Eastern Goths, the
Eastern Germanic tribes.

00:01:27.300 --> 00:01:31.629
Here in North Africa,
you have the Vandals,

00:01:31.629 --> 00:01:33.622
another Germanic tribe.

00:01:33.622 --> 00:01:36.627
Here in the Iberian peninsula,
you have the Visigoths,

00:01:36.627 --> 00:01:39.977
the Western Goths, yet
another Germanic tribe.

00:01:39.977 --> 00:01:42.635
Here you have the Burgundians,
you have the Franks.

00:01:42.635 --> 00:01:46.631
The word France comes from the
notion of land of the Franks.

00:01:46.631 --> 00:01:49.506
Once again, all of these
are Germanic kingdoms

00:01:49.506 --> 00:01:51.905
that took over, where before you had

00:01:51.905 --> 00:01:53.824
the Western Roman Empire.

00:01:53.824 --> 00:01:56.506
Now, as you can see from this map,

00:01:56.506 --> 00:01:58.300
and we've talked about
it in previous videos,

00:01:58.300 --> 00:02:01.309
it was only the Western
Roman Empire that fell

00:02:01.309 --> 00:02:02.906
and fragmented and, for the most part,

00:02:02.906 --> 00:02:05.970
came into the hands of
these Germanic tribes.

00:02:05.970 --> 00:02:08.637
The Eastern Empire continues on,

00:02:09.536 --> 00:02:12.331
and it's often known
as the Byzantine Empire

00:02:12.331 --> 00:02:15.718
with its capital at Constantinople.

00:02:15.718 --> 00:02:18.313
Now, what we'll see is
that the Byzantine Empire

00:02:18.313 --> 00:02:21.396
lasts for almost another 1,000 years,

00:02:22.771 --> 00:02:26.051
but its importance in the region

00:02:26.051 --> 00:02:29.098
diminishes for most of that period.

00:02:29.098 --> 00:02:32.436
It does have one last hurrah, so to speak,

00:02:32.436 --> 00:02:35.436
with the reign of Emperor Justinian.

00:02:37.234 --> 00:02:40.734
Justinian is able to come to power in 527.

00:02:43.685 --> 00:02:46.568
He comes into power with
the help of his uncle,

00:02:46.568 --> 00:02:49.625
who's the previous Emperor Justin,

00:02:49.625 --> 00:02:53.625
and Justinian is able to
regain much of the land

00:02:54.802 --> 00:02:58.071
that was lost to the Germanic tribes.

00:02:58.071 --> 00:03:01.119
Over the course of Justinian's reign,

00:03:01.119 --> 00:03:05.286
as you can see, they're able
to retake the Italian peninsula

00:03:06.230 --> 00:03:08.534
and much of North Africa.

00:03:08.534 --> 00:03:11.176
Now, things weren't
always easy for Justinian,

00:03:11.176 --> 00:03:13.482
and they are very difficult
at the beginning of his rule

00:03:13.482 --> 00:03:17.121
and they get hard near the
end of his rule as well.

00:03:17.121 --> 00:03:20.545
In 532, there's significant
rioting in Constantinople,

00:03:20.545 --> 00:03:22.183
which is his capital.

00:03:22.183 --> 00:03:24.595
It all starts around chariot racing

00:03:24.595 --> 00:03:26.075
and the different groups or gangs

00:03:26.075 --> 00:03:29.070
that support the different
chariot racing teams,

00:03:29.070 --> 00:03:31.143
but it eventually
becomes an all-out revolt

00:03:31.143 --> 00:03:32.753
against Justinian.

00:03:32.753 --> 00:03:34.338
And many of Justinian's advisors,

00:03:34.338 --> 00:03:36.078
including Justinian himself,

00:03:36.078 --> 00:03:38.602
they're afraid that they're
gonna be overthrown and killed,

00:03:38.602 --> 00:03:41.940
and they think about
fleeing Constantinople.

00:03:41.940 --> 00:03:44.099
And this is only five
years into his reign.

00:03:44.099 --> 00:03:45.349
This is in 532.

00:03:46.410 --> 00:03:50.077
And at that point, his
wife Empress Theodora

00:03:50.998 --> 00:03:55.231
stands up and becomes a very
important actor in history.

00:03:55.231 --> 00:03:57.732
She has her famous speech
at the point in time

00:03:57.732 --> 00:04:01.544
when Justinian and his advisors
were thinking about fleeing,

00:04:01.544 --> 00:04:02.572
fear of death.

00:04:02.572 --> 00:04:04.697
They thought they were
going to be overthrown.

00:04:04.697 --> 00:04:08.864
And here is that famous
quote from Empress Theodora.

00:04:10.746 --> 00:04:13.130
The present occasion is too serious

00:04:13.130 --> 00:04:15.283
to allow me to follow the convention

00:04:15.283 --> 00:04:18.472
that a woman should not
speak in a man's council.

00:04:18.472 --> 00:04:22.521
Those whose interests are
threatened by extreme danger

00:04:22.521 --> 00:04:25.499
should think only of the
wisest course of action,

00:04:25.499 --> 00:04:27.840
not of conventions.

00:04:27.840 --> 00:04:31.661
In my opinion, flight
is not the right course,

00:04:31.661 --> 00:04:34.292
even if it should bring us to safety.

00:04:34.292 --> 00:04:36.609
It is impossible for a person,

00:04:36.609 --> 00:04:40.674
having been born into
this world, not to die,

00:04:40.674 --> 00:04:42.644
but for one who has reigned,

00:04:42.644 --> 00:04:45.404
it is intolerable to be a fugitive.

00:04:45.404 --> 00:04:48.625
May I never be deprived
of this purple robe,

00:04:48.625 --> 00:04:51.584
the purple robe signifying
royalty, being an empress,

00:04:51.584 --> 00:04:54.525
and may I never see the
day when those who meet me

00:04:54.525 --> 00:04:56.442
do not call me empress.

00:04:57.368 --> 00:04:59.386
If you wish to save yourself,

00:04:59.386 --> 00:05:02.331
so she's talking to Justinian
and Justinian's advisors,

00:05:02.331 --> 00:05:04.531
if you wish to save yourself, my lord,

00:05:04.531 --> 00:05:06.098
there is no difficulty.

00:05:06.098 --> 00:05:07.408
We are rich.

00:05:07.408 --> 00:05:10.434
Over there is the sea,
and there are the ships.

00:05:10.434 --> 00:05:13.233
Yet reflect for a moment whether,

00:05:13.233 --> 00:05:16.733
when you have once escaped
to a place of security,

00:05:16.733 --> 00:05:21.277
you would not gladly exchange
such safety for death.

00:05:21.277 --> 00:05:24.027
As for me, I agree with the adage

00:05:24.872 --> 00:05:28.191
that the royal purple
is the noblest shroud.

00:05:28.191 --> 00:05:30.655
So, she's saying I would
rather die as an empress

00:05:30.655 --> 00:05:32.607
than try to be a fugitive

00:05:32.607 --> 00:05:34.697
and just try to live my
life, the rest of my life,

00:05:34.697 --> 00:05:36.341
somehow comfortably.

00:05:36.341 --> 00:05:38.814
And she's able to convince Justinian,

00:05:38.814 --> 00:05:42.081
and they're able to put down
the uprising quite brutally,

00:05:42.081 --> 00:05:46.248
killing 30,000 of the people
who are rising up against them.

00:05:48.687 --> 00:05:51.048
Now, during that uprising,
much of Constantinople

00:05:51.048 --> 00:05:55.730
is burned, including the main
church, the Hagia Sophia,

00:05:55.730 --> 00:05:57.613
and it was under Justinian's rule

00:05:57.613 --> 00:06:01.316
that the Hagia Sophia gets rebuilt.

00:06:01.316 --> 00:06:05.338
And it still exists in
what is now Istanbul today.

00:06:05.338 --> 00:06:07.921
Constantinople eventually
gets renamed Istanbul

00:06:07.921 --> 00:06:10.319
once it's conquered by the Turks.

00:06:10.319 --> 00:06:11.956
And it eventually gets
turned into a mosque,

00:06:11.956 --> 00:06:13.844
and today, it is a museum.

00:06:13.844 --> 00:06:16.737
The Hagia Sophia literally
means holy wisdom.

00:06:16.737 --> 00:06:19.889
You might recognize the
word Sophia from philosophy,

00:06:19.889 --> 00:06:24.056
which literally means friend
of wisdom, philosophia.

00:06:25.150 --> 00:06:29.144
Now, perhaps what Justinian
is most famous for

00:06:29.144 --> 00:06:32.977
is his attempt to better
codify a set of laws.

00:06:34.430 --> 00:06:37.181
The Roman Empire, of
course, always had laws,

00:06:37.181 --> 00:06:39.163
but Justinian got legal experts

00:06:39.163 --> 00:06:41.805
to go and collect the laws of the land,

00:06:41.805 --> 00:06:43.458
to make them consistent,

00:06:43.458 --> 00:06:45.462
to throw out the laws
that are inconsistent,

00:06:45.462 --> 00:06:47.411
to add new laws as necessary.

00:06:47.411 --> 00:06:49.498
And he did this over his entire reign,

00:06:49.498 --> 00:06:50.545
and he spread them.

00:06:50.545 --> 00:06:54.712
And they are often referred
to as the Code of Justinian.

00:06:55.658 --> 00:06:58.967
And they would have an influence on the

00:06:58.967 --> 00:07:02.485
Byzantine Empire for the next 900 years

00:07:02.485 --> 00:07:04.626
and even some influence on some

00:07:04.626 --> 00:07:07.376
relatively modern schools of law.

00:07:09.972 --> 00:07:11.606
Now, as I've mentioned before,

00:07:11.606 --> 00:07:14.783
the reign of Justinian is
considered really the high point

00:07:14.783 --> 00:07:17.314
of the Byzantine Empire.

00:07:17.314 --> 00:07:19.560
And even as we get into roughly

00:07:19.560 --> 00:07:21.398
the second half of his reign,

00:07:21.398 --> 00:07:23.689
things start to slide downward,

00:07:23.689 --> 00:07:27.973
and over the next 900 years,
are just in a state of decline.

00:07:27.973 --> 00:07:30.497
Starting in the 540s,
there's a significant plague

00:07:30.497 --> 00:07:32.320
in the Byzantine Empire.

00:07:32.320 --> 00:07:34.388
Some accounts I've seen show that

00:07:34.388 --> 00:07:37.721
roughly 40% of Constantinople dies,

00:07:38.673 --> 00:07:40.287
significant chunk of the population

00:07:40.287 --> 00:07:42.359
of the Byzantine Empire dies.

00:07:42.359 --> 00:07:45.006
And that makes it more
susceptible to attack

00:07:45.006 --> 00:07:47.044
from the various Germanic tribes

00:07:47.044 --> 00:07:49.230
throughout Justinian's reign.

00:07:49.230 --> 00:07:51.006
And well after Justinian's reign,

00:07:51.006 --> 00:07:53.845
there's constant tension
and often conflict

00:07:53.845 --> 00:07:57.772
between the Sassanid Persians in the east.

00:07:57.772 --> 00:07:59.766
And then, as we go further on,

00:07:59.766 --> 00:08:03.125
especially as we get into the
seventh and eighth century,

00:08:03.125 --> 00:08:05.375
you have the rise of Islam.

00:08:08.871 --> 00:08:11.650
Early Islamic empires are able
to take over the Sassanids

00:08:11.650 --> 00:08:15.296
and significantly eat
into the Byzantine Empire

00:08:15.296 --> 00:08:17.259
and then even some of
the Germanic kingdoms

00:08:17.259 --> 00:08:18.983
in the Iberian peninsula.

00:08:18.983 --> 00:08:20.121
And so, this is taking us now,

00:08:20.121 --> 00:08:22.341
we're in the middle of the eighth century.

00:08:22.341 --> 00:08:25.888
And if we go further on,
we see this once again

00:08:25.888 --> 00:08:28.364
gradual decline continue on,

00:08:28.364 --> 00:08:32.857
where you have further invasions
into the Byzantine Empire

00:08:32.857 --> 00:08:37.024
from Turkish Muslim kingdoms
or Turkish Muslim conquerors,

00:08:38.960 --> 00:08:41.710
all the way until we get to 1453,

00:08:42.870 --> 00:08:46.620
when Constantinople
itself falls to the Turks

00:08:48.652 --> 00:08:51.735
and eventually gets renamed Istanbul.


Reflecting and scaling absolute value function

WEBVTT
Kind: captions
Language: en

00:00:00.031 --> 00:00:01.088
- [Instructor] The graph of y

00:00:01.088 --> 00:00:03.003
is equal to absolute value of x

00:00:03.003 --> 00:00:04.812
is reflected across the x-axis

00:00:04.812 --> 00:00:07.726
and then scaled vertically
by a factor of seven.

00:00:07.726 --> 00:00:10.603
What is the equation of the new graph?

00:00:10.603 --> 00:00:13.068
So pause the video and see
if you can figure that out.

00:00:13.068 --> 00:00:15.254
Alright, let's work
through it together now.

00:00:15.254 --> 00:00:17.508
Now, you might not need
to draw it visually

00:00:17.508 --> 00:00:19.883
but I will just so that
we can all together

00:00:19.883 --> 00:00:22.378
visualize what is going on.

00:00:22.378 --> 00:00:24.879
So let's say that's my x-axis

00:00:24.879 --> 00:00:26.712
and that is my y-axis.

00:00:29.837 --> 00:00:32.253
y equals the absolute value of x.

00:00:32.253 --> 00:00:35.032
So for non-negative values of x,

00:00:35.032 --> 00:00:36.547
y is going to be equal to x.

00:00:36.547 --> 00:00:37.858
Absolute value of zero is zero.

00:00:37.858 --> 00:00:39.017
Absolute value of one is one.

00:00:39.017 --> 00:00:40.730
Absolute value of two is two.

00:00:40.730 --> 00:00:42.462
So it's gonna look like this.

00:00:42.462 --> 00:00:44.923
It's gonna have a slope of one

00:00:44.923 --> 00:00:46.328
and then for negative values,

00:00:46.328 --> 00:00:47.684
when you take the absolute value,

00:00:47.684 --> 00:00:48.583
you're gonna take the opposite.

00:00:48.583 --> 00:00:50.062
You're gonna get the positive.

00:00:50.062 --> 00:00:52.175
So it's gonna look like this.

00:00:52.175 --> 00:00:55.012
Let me see if I can draw
that a little bit cleaner.

00:00:55.012 --> 00:00:58.032
This is a hand drawn
sketch so bear with me

00:00:58.032 --> 00:00:59.078
but hopefully this is familiar.

00:00:59.078 --> 00:00:59.911
You've seen the graph

00:00:59.911 --> 00:01:04.296
of y is equal to absolute
value of x before.

00:01:04.296 --> 00:01:06.708
Now, let's think about the
different transformations.

00:01:06.708 --> 00:01:10.875
So first, they say is
reflected across the x-axis.

00:01:12.349 --> 00:01:13.599
So for example,

00:01:14.558 --> 00:01:18.315
if I have some x value right over here,

00:01:18.315 --> 00:01:20.242
before, I would take
the absolute value of x

00:01:20.242 --> 00:01:21.878
and I would end up there

00:01:21.878 --> 00:01:25.131
but now we wanna reflect across the x-axis

00:01:25.131 --> 00:01:28.748
so we wanna essentially get
the negative of that value

00:01:28.748 --> 00:01:32.192
associated with that corresponding x

00:01:32.192 --> 00:01:33.705
and so for example, this x,

00:01:33.705 --> 00:01:35.921
before, we would get
the absolute value of x

00:01:35.921 --> 00:01:37.969
but now we wanna flip across the x-axis

00:01:37.969 --> 00:01:40.512
and we wanna get the negative of it.

00:01:40.512 --> 00:01:42.838
So in general, what we are doing

00:01:42.838 --> 00:01:47.005
is we are getting the negative
of the absolute value of x.

00:01:48.843 --> 00:01:51.743
In general, if you're
flipping over the x-axis,

00:01:51.743 --> 00:01:53.093
you're getting the negative.

00:01:53.093 --> 00:01:57.791
You're scaling the expression
or the function by a negative.

00:01:57.791 --> 00:02:00.240
So this is going to be y
is equal to the negative

00:02:00.240 --> 00:02:02.200
of the absolute value of x.

00:02:02.200 --> 00:02:03.919
Once again, whatever absolute value of x

00:02:03.919 --> 00:02:05.881
was giving you before for given x,

00:02:05.881 --> 00:02:08.062
we now wanna get the negative of it.

00:02:08.062 --> 00:02:11.334
We now wanna get the negative of it.

00:02:11.334 --> 00:02:16.212
So that's what reflecting
across the x-axis does for us

00:02:16.212 --> 00:02:20.379
but then they say scaled
vertically by a factor of seven

00:02:22.915 --> 00:02:26.028
and the way I view that is if
you're scaling it vertically

00:02:26.028 --> 00:02:27.417
by a factor of seven,

00:02:27.417 --> 00:02:30.167
whatever y value you got for given x,

00:02:30.167 --> 00:02:33.358
you now wanna get seven times the y value,

00:02:33.358 --> 00:02:35.275
seven times the y value

00:02:39.450 --> 00:02:40.617
for a given x.

00:02:43.655 --> 00:02:45.781
And so if you think
about that algebraically,

00:02:45.781 --> 00:02:48.407
well, if I want seven times the y value,

00:02:48.407 --> 00:02:51.514
I'd have to multiply this thing by seven.

00:02:51.514 --> 00:02:55.516
So I would get y is
equal to negative seven

00:02:55.516 --> 00:02:58.292
times the absolute value of x

00:02:58.292 --> 00:02:59.984
and that's essentially
what they're asking,

00:02:59.984 --> 00:03:01.742
what is the equation of the new graph,

00:03:01.742 --> 00:03:03.175
and so that's what it would be.

00:03:03.175 --> 00:03:05.795
The negative flips us over the x-axis

00:03:05.795 --> 00:03:09.027
and then the seven scales
vertically by a factor of seven

00:03:09.027 --> 00:03:11.516
but just to understand
what this would look like,

00:03:11.516 --> 00:03:13.695
well, you multiply zero times seven,

00:03:13.695 --> 00:03:15.263
it doesn't change anything

00:03:15.263 --> 00:03:17.843
but whatever x this is,

00:03:17.843 --> 00:03:20.049
this was equal to negative x

00:03:20.049 --> 00:03:23.079
but now we're gonna get
to negative seven x.

00:03:23.079 --> 00:03:27.760
So let's see, two, three,
four, five, six, seven

00:03:27.760 --> 00:03:29.972
so it'd put it something around that.

00:03:29.972 --> 00:03:32.545
So our graph is now going to look,

00:03:32.545 --> 00:03:34.623
is now going to look like this.

00:03:34.623 --> 00:03:39.334
It's going to be stretched
along the vertical axis.

00:03:39.334 --> 00:03:41.701
If we were scaling vertically

00:03:41.701 --> 00:03:44.654
by something that had an
absolute value less than one

00:03:44.654 --> 00:03:46.697
then it would make the graph less tall.

00:03:46.697 --> 00:03:47.798
It would make it look,

00:03:47.798 --> 00:03:49.552
it would make it look wider.

00:03:49.552 --> 00:03:52.245
Let me make it at least look
a little bit more symmetric.

00:03:52.245 --> 00:03:54.670
So it's gonna look something,

00:03:54.670 --> 00:03:57.047
something like that but the key issue

00:03:57.047 --> 00:03:58.876
and the reason why I'm
drawing is so you can see

00:03:58.876 --> 00:04:00.707
that it looks like it's
being scaled vertically.

00:04:00.707 --> 00:04:02.668
It's being stretched in
the vertical direction

00:04:02.668 --> 00:04:06.232
by a factor of seven and the
way we do that algebraically

00:04:06.232 --> 00:04:07.418
is we multiply by seven

00:04:07.418 --> 00:04:11.585
and the negative here is what
flipped us over the x-axis.


Sums and products of irrational numbers

WEBVTT
Kind: captions
Language: en

00:00:00.145 --> 00:00:02.737
- [Instructor] Let's say
that we have some number a

00:00:02.737 --> 00:00:06.487
and to that we are going
to add some number b

00:00:09.283 --> 00:00:12.533
and that sum is going to be equal to c.

00:00:15.492 --> 00:00:17.743
Let's say that we're also told

00:00:17.743 --> 00:00:20.493
that both a and b are irrational.

00:00:22.323 --> 00:00:23.240
Irrational.

00:00:24.827 --> 00:00:27.020
So based on the information
that I've given you,

00:00:27.020 --> 00:00:29.235
a and b are both irrational.

00:00:29.235 --> 00:00:34.168
Is their sum, c, is that going
to be rational or irrational?

00:00:34.168 --> 00:00:36.014
I encourage you to pause the video

00:00:36.014 --> 00:00:38.931
and try to answer that on your own.

00:00:40.494 --> 00:00:42.019
I'm guessing that you might have struggled

00:00:42.019 --> 00:00:44.377
with this a little bit
because the answer is

00:00:44.377 --> 00:00:46.508
that we actually don't know.

00:00:46.508 --> 00:00:50.675
It depends on what irrational
numbers a and b actually are.

00:00:52.194 --> 00:00:54.240
What do I mean by that?

00:00:54.240 --> 00:00:57.260
Well, I can pick two irrational numbers

00:00:57.260 --> 00:01:01.343
where their sum actually
is going to be rational.

00:01:02.603 --> 00:01:03.704
What do I mean?

00:01:03.704 --> 00:01:06.121
Well what if a is equal to pi

00:01:08.288 --> 00:01:10.871
and b is equal to one minus pi?

00:01:15.155 --> 00:01:18.324
Now both of these are irrational numbers.

00:01:18.324 --> 00:01:22.335
Pi is irrational and one minus
pi, whatever this value is,

00:01:22.335 --> 00:01:24.368
this is irrational as well.

00:01:24.368 --> 00:01:26.792
But if we add these two things together,

00:01:26.792 --> 00:01:29.375
if we add pi plus one minus pi,

00:01:31.645 --> 00:01:34.149
one minus pi, well these are gonna add up

00:01:34.149 --> 00:01:35.732
to be equal to one,

00:01:37.096 --> 00:01:40.697
which is clearly going
to be a rational number.

00:01:40.697 --> 00:01:42.614
So we were able to find one scenario

00:01:42.614 --> 00:01:44.781
in which we added two irrationals

00:01:44.781 --> 00:01:47.448
and the sum gives us a rational.

00:01:48.291 --> 00:01:49.840
In general you could do this trick

00:01:49.840 --> 00:01:51.910
with any irrational number.

00:01:51.910 --> 00:01:54.741
Instead of pi you could've had

00:01:54.741 --> 00:01:58.908
square root of two plus one
minus the square root of two.

00:02:00.508 --> 00:02:01.647
Both of these,

00:02:01.647 --> 00:02:03.721
what we have in this
orange color is irrational,

00:02:03.721 --> 00:02:05.821
what we have in this
blue color is irrational,

00:02:05.821 --> 00:02:08.126
but the sum is going to be rational.

00:02:08.126 --> 00:02:09.885
And you could do this,
instead of having one minus,

00:02:09.885 --> 00:02:12.113
you could have this as 1/2 minus.

00:02:12.113 --> 00:02:14.311
You could have done it a bunch
of different combinations

00:02:14.311 --> 00:02:17.460
so that you could end up
with a sum that is rational.

00:02:17.460 --> 00:02:20.509
But you could also easily
add two irrational numbers

00:02:20.509 --> 00:02:23.799
and still end up with
an irrational number.

00:02:23.799 --> 00:02:25.716
For example, if a is pi

00:02:28.354 --> 00:02:29.354
and b is pi,

00:02:31.213 --> 00:02:36.106
well then their sum is
going to be equal to two pi,

00:02:36.106 --> 00:02:38.922
which is still irrational.

00:02:38.922 --> 00:02:40.422
Or if you added pi

00:02:42.154 --> 00:02:44.487
plus the square root of two,

00:02:45.532 --> 00:02:48.615
this is still going to be irrational.

00:02:49.751 --> 00:02:52.506
In fact, mathematically
I would just express this

00:02:52.506 --> 00:02:54.656
as pi plus the square root of two.

00:02:54.656 --> 00:02:56.986
This is some number right over here,

00:02:56.986 --> 00:02:59.448
but this is still going to be irrational.

00:02:59.448 --> 00:03:01.967
So the big takeaway is
if you're taking the sums

00:03:01.967 --> 00:03:03.938
of two irrational numbers

00:03:03.938 --> 00:03:05.505
and people don't tell you anything else,

00:03:05.505 --> 00:03:06.655
they don't tell you which

00:03:06.655 --> 00:03:08.396
specific irrational numbers they are,

00:03:08.396 --> 00:03:09.986
you don't know whether their sum

00:03:09.986 --> 00:03:13.153
is going to be rational or irrational.

00:03:14.410 --> 00:03:17.619
Now let's think about products.

00:03:17.619 --> 00:03:21.775
Similar exercise, let's
say we have a times b

00:03:21.775 --> 00:03:25.158
is equal to c, ab is equal to c,

00:03:25.158 --> 00:03:27.158
a times b is equal to c.

00:03:28.077 --> 00:03:30.501
And once again, let's
say someone tells you

00:03:30.501 --> 00:03:33.251
that both a and b are irrational.

00:03:34.898 --> 00:03:37.034
Pause this video and think about

00:03:37.034 --> 00:03:40.674
whether c must be rational, irrational,

00:03:40.674 --> 00:03:41.900
or whether we just don't know.

00:03:41.900 --> 00:03:43.346
Try to figure out some examples

00:03:43.346 --> 00:03:46.679
like we just did when we looked at sums.

00:03:47.852 --> 00:03:49.534
Alright, so let's think about,

00:03:49.534 --> 00:03:51.466
let's see if we can construct examples

00:03:51.466 --> 00:03:54.049
where c ends up being rational.

00:03:55.255 --> 00:03:57.966
Well one thing, as you
can tell I like to use pi,

00:03:57.966 --> 00:04:00.597
pi might be my favorite irrational number.

00:04:00.597 --> 00:04:02.264
If a was one over pi

00:04:04.146 --> 00:04:05.146
and b is pi,

00:04:07.236 --> 00:04:09.644
well, what's their product going to be?

00:04:09.644 --> 00:04:13.093
Well, their product is going
to be one over pi times pi,

00:04:13.093 --> 00:04:17.213
that's just going to be pi
over pi, which is equal to one.

00:04:17.213 --> 00:04:21.691
Here we got a situation where
the product of two irrationals

00:04:21.691 --> 00:04:23.691
became, or is, rational.

00:04:25.464 --> 00:04:28.047
But what if I were to multiply,

00:04:29.424 --> 00:04:30.613
and in general you could this

00:04:30.613 --> 00:04:31.679
with a lot of irrational numbers,

00:04:31.679 --> 00:04:34.802
one over square root of two
times the square of two,

00:04:34.802 --> 00:04:36.653
that would be one.

00:04:36.653 --> 00:04:39.486
What if instead I had pi times pi?

00:04:42.505 --> 00:04:46.672
Pi times pi, that you could
just write as pi squared,

00:04:47.652 --> 00:04:51.569
and pi squared is still
going to be irrational.

00:04:52.884 --> 00:04:55.467
This is irrational, irrational.

00:04:56.965 --> 00:04:58.287
It isn't even always the case

00:04:58.287 --> 00:05:01.904
that if you multiply the
same irrational number,

00:05:01.904 --> 00:05:03.344
if you square an irrational number

00:05:03.344 --> 00:05:04.736
that it's always going to be irrational.

00:05:04.736 --> 00:05:08.548
For example, if I have
square root of two times,

00:05:08.548 --> 00:05:10.240
I think you see where this is going,

00:05:10.240 --> 00:05:11.969
times the square root of two,

00:05:11.969 --> 00:05:14.509
I'm taking the product of
two irrational numbers.

00:05:14.509 --> 00:05:16.743
In fact, they're the
same irrational number,

00:05:16.743 --> 00:05:18.815
but the square root of two
times the square root of two,

00:05:18.815 --> 00:05:21.034
well, that's just going
to be equal to two,

00:05:21.034 --> 00:05:23.951
which is clearly a rational number.

00:05:25.568 --> 00:05:27.692
So once again, when
you're taking the product

00:05:27.692 --> 00:05:30.121
of two irrational numbers,

00:05:30.121 --> 00:05:32.442
you don't know whether the product

00:05:32.442 --> 00:05:35.844
is going to be rational or irrational

00:05:35.844 --> 00:05:39.729
unless someone tells you
the specific numbers.

00:05:39.729 --> 00:05:41.514
Whether you're taking the product

00:05:41.514 --> 00:05:43.546
or the sum of irrational numbers,

00:05:43.546 --> 00:05:47.182
in order to know whether
the resulting number

00:05:47.182 --> 00:05:50.376
is irrational or rational,
you need to know something

00:05:50.376 --> 00:05:54.543
about what you're taking
the sum or the product of.


Example scaling parabola

WEBVTT
Kind: captions
Language: en

00:00:00.194 --> 00:00:01.832
- [Instructor] Function
G can be thought of

00:00:01.832 --> 00:00:05.413
as a scaled version of F
of X is equal to X squared.

00:00:05.413 --> 00:00:07.463
Write the equation for G of X.

00:00:07.463 --> 00:00:08.995
So like always, pause this video and see

00:00:08.995 --> 00:00:10.880
if you can do it on your own.

00:00:10.880 --> 00:00:13.140
Alright now, let's work
through this together.

00:00:13.140 --> 00:00:15.344
So the first thing that
we might appreciate

00:00:15.344 --> 00:00:19.780
is that G seems not only to
be flipped over the x-axis,

00:00:19.780 --> 00:00:22.391
but then flipped over
and then stretched wider.

00:00:22.391 --> 00:00:24.300
So let's do these in steps.

00:00:24.300 --> 00:00:27.037
So first let's flip over,

00:00:27.037 --> 00:00:28.787
flip over the x-axis.

00:00:33.049 --> 00:00:37.637
So if we were to do this
visually it would look like this.

00:00:37.637 --> 00:00:40.105
Instead when X is equal to zero,

00:00:40.105 --> 00:00:42.003
Y is still gonna be equal to zero.

00:00:42.003 --> 00:00:43.513
But when X is equal to negative one,

00:00:43.513 --> 00:00:45.044
instead of Y being equal to one,

00:00:45.044 --> 00:00:46.775
it'd now be equal to negative one.

00:00:46.775 --> 00:00:49.312
When X is equal to one,
instead of squaring one

00:00:49.312 --> 00:00:52.081
and getting one, you then
take the negative of that

00:00:52.081 --> 00:00:53.998
to get to negative one.

00:00:55.152 --> 00:00:58.205
So when you flip it, it looks like this.

00:00:58.205 --> 00:00:59.980
Y when is X is equal to negative two

00:00:59.980 --> 00:01:01.838
instead of Y being equal to four,

00:01:01.838 --> 00:01:05.804
it would now be equal to negative four.

00:01:05.804 --> 00:01:08.346
So it would look like this.

00:01:08.346 --> 00:01:10.679
So as we just talk through
as we're trying to draw

00:01:10.679 --> 00:01:12.532
this flipped over version,

00:01:12.532 --> 00:01:15.963
whatever Y value we were
getting before for a given X,

00:01:15.963 --> 00:01:19.718
we would now get the opposite
of it, or the negative of it.

00:01:19.718 --> 00:01:21.988
So this green function right over here

00:01:21.988 --> 00:01:26.311
is going to be Y is equal
to the negative of F of X,

00:01:26.311 --> 00:01:29.871
or we could say Y is equal
to negative X squared.

00:01:29.871 --> 00:01:31.634
Whatever the X is, you square it,

00:01:31.634 --> 00:01:33.221
and then you take the negative of it.

00:01:33.221 --> 00:01:35.216
Whatever X is, you square it,

00:01:35.216 --> 00:01:37.289
and then you take the negative of it,

00:01:37.289 --> 00:01:40.825
and you see that that will
flip it over the x-axis.

00:01:40.825 --> 00:01:44.633
But that by itself does
not get us to G of X.

00:01:44.633 --> 00:01:47.634
G of X also seems to be stretched

00:01:47.634 --> 00:01:50.126
in the horizontal direction.

00:01:50.126 --> 00:01:52.105
And so let's think about,
can we multiply this

00:01:52.105 --> 00:01:56.275
times some scaling factor so
that it does that stretching

00:01:56.275 --> 00:01:58.809
so that we can match up to G of X?

00:01:58.809 --> 00:02:01.389
And the best way to do
this is to pick a point

00:02:01.389 --> 00:02:04.633
that we know sits on G of X,
and they in fact give us one.

00:02:04.633 --> 00:02:06.967
They show us right over
here that at the point

00:02:06.967 --> 00:02:10.195
two comma negative one, sits on G of X.

00:02:10.195 --> 00:02:12.563
When X is equal to two,

00:02:12.563 --> 00:02:15.768
Y is equal to negative one on G of X.

00:02:15.768 --> 00:02:18.966
So you could say G of two is negative one.

00:02:18.966 --> 00:02:22.908
Now on our green function,
when X is equal to two

00:02:22.908 --> 00:02:26.072
Y is equal to negative four.

00:02:26.072 --> 00:02:26.957
So let's see.

00:02:26.957 --> 00:02:31.124
Maybe we can just multiply
this by 1/4 to get our G.

00:02:32.428 --> 00:02:33.261
So let's see.

00:02:33.261 --> 00:02:36.996
If we were to, let's
see if we scale by 1/4,

00:02:36.996 --> 00:02:38.935
does that do the trick?

00:02:38.935 --> 00:02:41.563
Scale by 1/4.

00:02:41.563 --> 00:02:43.863
So in that case, we're gonna have Y

00:02:43.863 --> 00:02:45.767
is equal to not just negative X squared,

00:02:45.767 --> 00:02:48.934
but negative 1/4 X squared.

00:02:51.745 --> 00:02:54.184
And if you're saying hey,
so how did you get 1/4?

00:02:54.184 --> 00:02:56.982
Well I looked at when X is equal to two.

00:02:56.982 --> 00:02:59.046
On our green function,
when X is equal to two

00:02:59.046 --> 00:03:01.081
I get to negative four.

00:03:01.081 --> 00:03:03.246
Well we want that when X is equal to two

00:03:03.246 --> 00:03:05.103
to be equal to negative one.

00:03:05.103 --> 00:03:08.451
Well negative one is 1/4 of negative four,

00:03:08.451 --> 00:03:10.802
so that's why I said
okay, well let's up take

00:03:10.802 --> 00:03:12.123
to see if we could take
our green function,

00:03:12.123 --> 00:03:14.567
and if I multiply it by 1/4,

00:03:14.567 --> 00:03:18.713
that seems like it will
match up with G of X.

00:03:18.713 --> 00:03:20.606
And so let's verify that.

00:03:20.606 --> 00:03:23.493
When X is equal to
zero, well this is still

00:03:23.493 --> 00:03:25.818
all gonna be equal to
zero so that makes sense.

00:03:25.818 --> 00:03:27.735
When X is equal to one,

00:03:28.572 --> 00:03:31.572
let me do this in another color,

00:03:31.572 --> 00:03:33.489
when X is equal to one,

00:03:35.475 --> 00:03:37.542
then one squared times negative 1/4,

00:03:37.542 --> 00:03:40.746
well that does indeed look
like negative 1/4 right there.

00:03:40.746 --> 00:03:44.531
When X is equal to two,
two squared is four,

00:03:44.531 --> 00:03:48.698
times negative 1/4 is indeed
equal to negative one.

00:03:50.026 --> 00:03:51.272
Let's try this point
here 'cause it looks like

00:03:51.272 --> 00:03:54.732
this is sitting on our graph as well.

00:03:54.732 --> 00:03:58.399
When X is equal to four,
four squared is 16.

00:03:59.715 --> 00:04:04.022
16 times negative 1/4 is
indeed equal to negative four.

00:04:04.022 --> 00:04:08.341
And it does work also for the
negative values of X as well.

00:04:08.341 --> 00:04:09.354
So I'm feeling really good

00:04:09.354 --> 00:04:11.458
that this is the equation of G of X.

00:04:11.458 --> 00:04:16.281
G of X is equal to negative
1/4 times X squared.

00:04:16.281 --> 00:04:18.529
And so in general, that
when we were saying

00:04:18.529 --> 00:04:21.359
we were scaling it, we're
scaling it by negative value.

00:04:21.359 --> 00:04:24.025
This is what flips it over the x-axis,

00:04:24.025 --> 00:04:27.805
and then multiplying it by this fraction

00:04:27.805 --> 00:04:30.626
that has an absolute value less than one,

00:04:30.626 --> 00:04:32.931
this is actually stretching it wider.

00:04:32.931 --> 00:04:34.850
If this value right over here,

00:04:34.850 --> 00:04:37.464
its absolute value was greater than one,

00:04:37.464 --> 00:04:39.275
then it would stretch it vertically,

00:04:39.275 --> 00:04:43.442
or would make it thinner in
the horizontal direction.


Example translating parabola

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.481
- [Instructor] Function g can
be thought of as a translated

00:00:02.481 --> 00:00:05.619
or shifted version of f of
x is equal to x squared.

00:00:05.619 --> 00:00:07.588
Write the equation for g of x.

00:00:07.588 --> 00:00:08.463
Now, pause this video,

00:00:08.463 --> 00:00:10.981
and see if you can work
this out on your own.

00:00:10.981 --> 00:00:14.300
All right, so whenever I think
about shifting a function,

00:00:14.300 --> 00:00:16.270
and in this case, we're
shifting a parabola,

00:00:16.270 --> 00:00:18.441
I like to look for a distinctive point.

00:00:18.441 --> 00:00:20.744
And on a parabola, the vertex is going

00:00:20.744 --> 00:00:22.577
to be our most distinctive point.

00:00:22.577 --> 00:00:24.412
And if I focus on the vertex of f,

00:00:24.412 --> 00:00:28.196
it looks like if I shift that to the right

00:00:28.196 --> 00:00:29.029
by three,

00:00:30.338 --> 00:00:33.377
and then if I were to shift that down

00:00:33.377 --> 00:00:34.210
by four,

00:00:35.218 --> 00:00:37.702
at least our vertices would overlap.

00:00:37.702 --> 00:00:40.380
I would be able to shift the vertex

00:00:40.380 --> 00:00:42.713
to where the vertex of g is.

00:00:43.752 --> 00:00:45.093
And it does look,

00:00:45.093 --> 00:00:47.937
and we'll validate this, at
least visually, in a little bit,

00:00:47.937 --> 00:00:50.424
so I'm gonna go minus four
in the vertical direction,

00:00:50.424 --> 00:00:53.235
that not only would it
make the vertices overlap,

00:00:53.235 --> 00:00:55.599
but it would make the
entire curve overlap.

00:00:55.599 --> 00:01:00.191
So we're going to make,
we're gonna first shift to

00:01:00.191 --> 00:01:01.024
the right

00:01:02.158 --> 00:01:03.322
by three.

00:01:03.322 --> 00:01:06.031
And we're gonna think about how
would we change our equation

00:01:06.031 --> 00:01:08.581
so it shifts f to the right by three,

00:01:08.581 --> 00:01:11.434
and then we're gonna shift down by four.

00:01:11.434 --> 00:01:12.267
Shift down

00:01:14.146 --> 00:01:14.979
by four.

00:01:16.796 --> 00:01:19.264
Now, some of you might
already be familiar with this,

00:01:19.264 --> 00:01:20.550
and I go into the intuition

00:01:20.550 --> 00:01:22.617
in a lot more depth in other videos.

00:01:22.617 --> 00:01:23.670
But in general,

00:01:23.670 --> 00:01:27.614
when you shift to the right by some value,

00:01:27.614 --> 00:01:30.070
in this case, we're shifting
to the right by three,

00:01:30.070 --> 00:01:34.523
you would replace x with x minus three.

00:01:34.523 --> 00:01:37.767
So one way to think about this
would be y is equal to f of

00:01:37.767 --> 00:01:39.540
x minus three,

00:01:39.540 --> 00:01:42.582
or y is equal to, instead
of it being x squared,

00:01:42.582 --> 00:01:46.727
you would replace x with x minus three.

00:01:46.727 --> 00:01:49.213
So it'd be x minus three squared.

00:01:49.213 --> 00:01:52.066
Now, when I first learned this,
this was counterintuitive.

00:01:52.066 --> 00:01:54.010
I'm shifting to the right by three.

00:01:54.010 --> 00:01:57.477
The x-coordinate of my vertex
is increasing by three,

00:01:57.477 --> 00:02:00.654
but I'm replacing x with x minus three.

00:02:00.654 --> 00:02:02.910
Why does this make sense?

00:02:02.910 --> 00:02:05.695
Well, let's graph the shifted version,

00:02:05.695 --> 00:02:07.338
just to get a little
bit more intuition here.

00:02:07.338 --> 00:02:09.702
Once again, I go into much more
depth in other videos here.

00:02:09.702 --> 00:02:12.127
This is more of a worked example.

00:02:12.127 --> 00:02:14.707
So this is what the shifted
curve is gonna look like.

00:02:14.707 --> 00:02:16.363
Think about the behavior that we want,

00:02:16.363 --> 00:02:18.306
right over here, at x equals three.

00:02:18.306 --> 00:02:20.636
We want the same value
that we used to have

00:02:20.636 --> 00:02:22.252
when x equals zero.

00:02:22.252 --> 00:02:23.994
When x equals zero for the original f,

00:02:23.994 --> 00:02:25.146
zero squared was zero.

00:02:25.146 --> 00:02:26.035
Y equals zero.

00:02:26.035 --> 00:02:27.924
We still want y equals zero.

00:02:27.924 --> 00:02:29.116
Well, the way that we can do that is

00:02:29.116 --> 00:02:30.625
if we are squaring zero,

00:02:30.625 --> 00:02:32.869
and the way that we're gonna square zero

00:02:32.869 --> 00:02:35.440
is if we subtract three from x.

00:02:35.440 --> 00:02:37.512
And you can validate that at other points.

00:02:37.512 --> 00:02:40.214
Think about what happens
now, when x equals four.

00:02:40.214 --> 00:02:42.608
Four minus three is one squared.

00:02:42.608 --> 00:02:44.083
It does indeed equal one.

00:02:44.083 --> 00:02:46.404
The same behavior that you used to get

00:02:46.404 --> 00:02:48.404
at x is equal to one.

00:02:49.535 --> 00:02:52.428
So it does look like we have
indeed shifted to the right

00:02:52.428 --> 00:02:56.456
by three when we replace
x with x minus three.

00:02:56.456 --> 00:02:58.515
If you replaced x with x plus three,

00:02:58.515 --> 00:03:00.186
it would have had the opposite effect.

00:03:00.186 --> 00:03:02.558
You would have shifted
to the left by three,

00:03:02.558 --> 00:03:04.154
and I encourage to think about

00:03:04.154 --> 00:03:06.513
why that actually makes sense.

00:03:06.513 --> 00:03:08.441
So now that we've shifted
to the right by three,

00:03:08.441 --> 00:03:11.067
the next step is to shift down by four,

00:03:11.067 --> 00:03:13.359
and this one is little bit more intuitive.

00:03:13.359 --> 00:03:15.193
So let's start with our
shifted to the right.

00:03:15.193 --> 00:03:18.795
So that's y is equal to
x minus three squared.

00:03:18.795 --> 00:03:21.250
But now, whatever y value we were getting,

00:03:21.250 --> 00:03:23.335
we want to get four less than that.

00:03:23.335 --> 00:03:25.785
So when x equals three, instead
of getting y equals zero,

00:03:25.785 --> 00:03:29.198
we want to get y equals
four less, or negative four.

00:03:29.198 --> 00:03:30.863
When x equals four,
instead of getting one,

00:03:30.863 --> 00:03:33.374
we want to get y is
equal to negative three.

00:03:33.374 --> 00:03:35.970
So whatever y value we were getting,

00:03:35.970 --> 00:03:39.220
we want to now get four less than that.

00:03:40.082 --> 00:03:42.167
So the shifting in the vertical direction

00:03:42.167 --> 00:03:43.459
is a little bit more intuitive.

00:03:43.459 --> 00:03:46.195
If we shift down, we subtract that amount.

00:03:46.195 --> 00:03:49.436
If we shift up, we add that amount.

00:03:49.436 --> 00:03:52.850
So this, right over here,
is the equation for g of x.

00:03:52.850 --> 00:03:57.410
G of x is going to be equal
to x minus three squared

00:03:57.410 --> 00:03:58.965
minus four.

00:03:58.965 --> 00:04:00.798
And once again, just to review,

00:04:00.798 --> 00:04:04.858
replacing the x with x
minus three, on f of x,

00:04:04.858 --> 00:04:07.114
that's what shifted,

00:04:07.114 --> 00:04:07.947
shifted

00:04:08.942 --> 00:04:09.775
right

00:04:10.769 --> 00:04:11.602
by three,

00:04:12.599 --> 00:04:13.752
by three.

00:04:13.752 --> 00:04:16.999
And then, subtracting the four,

00:04:16.999 --> 00:04:19.296
that shifted us down by four,

00:04:19.296 --> 00:04:20.819
shifted

00:04:20.819 --> 00:04:21.652
down

00:04:23.229 --> 00:04:24.062
by four,

00:04:24.991 --> 00:04:26.794
to give us this next graph.

00:04:26.794 --> 00:04:30.098
And you can visualize, or
you can verify visually,

00:04:30.098 --> 00:04:33.285
that if you shift each of these
points exactly down by four,

00:04:33.285 --> 00:04:34.370
we are,

00:04:34.370 --> 00:04:36.044
we are indeed

00:04:36.044 --> 00:04:38.877
going to overlap on top of g of x.


Worked examples: Forms & features of quadratic functions

WEBVTT
Kind: captions
Language: en

00:00:00.503 --> 00:00:01.827
- [Instructor] The function m is given

00:00:01.827 --> 00:00:04.549
in three equivalent forms.

00:00:04.549 --> 00:00:08.775
Which form most quickly
reveals the y-intercept?

00:00:08.775 --> 00:00:11.010
So let's just remind ourselves.

00:00:11.010 --> 00:00:13.531
If I have a function,

00:00:13.531 --> 00:00:16.114
the graph y is equal to m of x.

00:00:17.499 --> 00:00:20.664
And these are all equivalent
forms, they tell us that.

00:00:20.664 --> 00:00:22.551
The function m is given
in three equivalent forms.

00:00:22.551 --> 00:00:23.889
I should be able to
algebraically manipulate

00:00:23.889 --> 00:00:26.815
anyone of these to get
any of the other ones.

00:00:26.815 --> 00:00:29.543
And so, if I wanted to
graph y is equal to m of x,

00:00:29.543 --> 00:00:32.591
and let's say it looks
something like this.

00:00:32.591 --> 00:00:34.724
I actually know it's a
downward opening parabola,

00:00:34.724 --> 00:00:36.273
because I can look at
this form right over here

00:00:36.273 --> 00:00:37.414
and say, "Hey look, the coefficient

00:00:37.414 --> 00:00:40.507
"on the second degree
term here is a negative."

00:00:40.507 --> 00:00:42.177
So it's going to be a
downward opening parabola.

00:00:42.177 --> 00:00:44.314
That's a messy drawing of it.

00:00:44.314 --> 00:00:46.529
And so if we're talking
about the y intercept,

00:00:46.529 --> 00:00:50.269
we're saying, "Hey where does
it intersect the y-axis?"

00:00:50.269 --> 00:00:55.225
So what is the y value
when x is equal to zero?

00:00:55.225 --> 00:00:56.812
So it boils down to how quickly

00:00:56.812 --> 00:00:58.998
can we evaluate m of zero?

00:00:58.998 --> 00:01:02.543
What is m of x, when x is equal to zero?

00:01:02.543 --> 00:01:06.030
So how quickly can you evaluate m of zero?

00:01:06.030 --> 00:01:09.287
Well in this top one, I
can substitute zero for x

00:01:09.287 --> 00:01:10.995
and so it'll be negative two

00:01:10.995 --> 00:01:13.696
times negative three times negative nine.

00:01:13.696 --> 00:01:15.481
So it's not too hard to figure out,

00:01:15.481 --> 00:01:18.244
but there's going to be
some calculation in my head.

00:01:18.244 --> 00:01:20.846
Similarly in the second choice,

00:01:20.846 --> 00:01:24.515
for x equals zero, I then
get negative six-squared,

00:01:24.515 --> 00:01:26.505
which is positive 36 times negative two,

00:01:26.505 --> 00:01:27.542
which is negative 72,

00:01:27.542 --> 00:01:30.032
and then I have to add
that to positive 18.

00:01:30.032 --> 00:01:31.129
I can do that,

00:01:31.129 --> 00:01:34.124
but it's a little bit of computation.

00:01:34.124 --> 00:01:36.051
But here for this last one,

00:01:36.051 --> 00:01:38.203
and this is known as standard form,

00:01:38.203 --> 00:01:40.399
if I say x equals zero
that term disappears,

00:01:40.399 --> 00:01:43.244
that term disappears and
I'm just left with m of zero

00:01:43.244 --> 00:01:45.244
is equal to negative 54.

00:01:46.127 --> 00:01:49.748
So standard form, this is
standard form right over here,

00:01:49.748 --> 00:01:52.351
was by far the easiest.

00:01:52.351 --> 00:01:56.949
So we know the y-intercept
is zero comma negative 54.

00:01:56.949 --> 00:01:58.573
Now one rule of caution.

00:01:58.573 --> 00:02:03.038
Sometimes you might look at
what is called, vertex form.

00:02:03.038 --> 00:02:04.500
And as we'll see this is the easiest one

00:02:04.500 --> 00:02:06.615
where it is to identify the vertex.

00:02:06.615 --> 00:02:08.525
But when you see this
little plus 18 hanging out,

00:02:08.525 --> 00:02:10.521
it looks a lot like this negative 54

00:02:10.521 --> 00:02:11.414
that was hanging out.

00:02:11.414 --> 00:02:12.674
And you say, "Hey, when x equals zero,

00:02:12.674 --> 00:02:14.381
"maybe I can just cross that out

00:02:14.381 --> 00:02:16.818
"the same way that I
cross these terms out."

00:02:16.818 --> 00:02:18.528
And be very, very careful there,

00:02:18.528 --> 00:02:19.629
because if x equals zero,

00:02:19.629 --> 00:02:21.581
this whole thing does not equal zero.

00:02:21.581 --> 00:02:23.818
When x is equal to zero, as I just said,

00:02:23.818 --> 00:02:24.956
you have negative six-squared,

00:02:24.956 --> 00:02:26.171
which is 36 times negative two.

00:02:26.171 --> 00:02:28.164
This is equal to negative 72.

00:02:28.164 --> 00:02:31.291
So m of zero is definitely not 18.

00:02:31.291 --> 00:02:33.087
So be very, very careful.

00:02:33.087 --> 00:02:35.488
But we can see that the best choice

00:02:35.488 --> 00:02:37.597
is this one, standard form,

00:02:37.597 --> 00:02:40.347
not vertex form or factored form.

00:02:42.066 --> 00:02:44.041
Factored form, as you can imagine,

00:02:44.041 --> 00:02:48.149
is very good for figuring out the zeros.

00:02:48.149 --> 00:02:50.752
Let's do another example.

00:02:50.752 --> 00:02:53.756
And actually this is the same m of x,

00:02:53.756 --> 00:02:55.302
but we're going to ask something else.

00:02:55.302 --> 00:02:58.500
So it's given in those same three forms.

00:02:58.500 --> 00:03:01.763
Which form most quickly
reveals the vertex?

00:03:01.763 --> 00:03:04.774
Well I just called this
vertex form before,

00:03:04.774 --> 00:03:07.294
but what's valuable about the vertex form

00:03:07.294 --> 00:03:08.229
is you can really say,

00:03:08.229 --> 00:03:10.099
"Okay, this is going to be,

00:03:10.099 --> 00:03:12.006
"this is going to achieve its vertex

00:03:12.006 --> 00:03:16.155
"when this thing over
here is equal to zero."

00:03:16.155 --> 00:03:17.128
How do I know that?

00:03:17.128 --> 00:03:18.750
Well, once you get used to vertex form,

00:03:18.750 --> 00:03:21.061
it'll just become a bit of second nature.

00:03:21.061 --> 00:03:23.991
But if this is a
downward-opening parabola,

00:03:23.991 --> 00:03:27.241
the vertex is when you
hit that maximum point.

00:03:27.241 --> 00:03:28.340
And as you can see here,

00:03:28.340 --> 00:03:31.462
x minus six-squared is always
going to be non-negative.

00:03:31.462 --> 00:03:33.009
You multiply that times the negative two,

00:03:33.009 --> 00:03:34.875
it's always going to be non-positive.

00:03:34.875 --> 00:03:36.823
It's either going to be
zero or a negative value,

00:03:36.823 --> 00:03:40.721
so this is always going
to take away from this 18.

00:03:40.721 --> 00:03:43.508
And so if you want to find the vertex,

00:03:43.508 --> 00:03:45.149
the maximum point here,

00:03:45.149 --> 00:03:48.274
it would be the x value that
makes this thing equal to zero.

00:03:48.274 --> 00:03:49.792
Because for any other x value,

00:03:49.792 --> 00:03:51.130
this thing is going to be negative,

00:03:51.130 --> 00:03:52.875
it's going to take away from that 18.

00:03:52.875 --> 00:03:54.420
And so you can see by inspection,

00:03:54.420 --> 00:03:56.975
well what x value will
make this equal to zero?

00:03:56.975 --> 00:03:58.627
Well if x is equal to six,

00:03:58.627 --> 00:04:00.265
six minus six is equal to zero,

00:04:00.265 --> 00:04:01.507
zero-squared is zero,

00:04:01.507 --> 00:04:02.999
times negative two is zero.

00:04:02.999 --> 00:04:06.040
And so m of six is equal to 18.

00:04:06.040 --> 00:04:08.263
So this lets us know very quickly

00:04:08.263 --> 00:04:11.559
that the vertex is going to
happen at x is equal to six

00:04:11.559 --> 00:04:14.141
and then the y value there or the m of six

00:04:14.141 --> 00:04:16.391
is going to be equal to 18.

00:04:17.252 --> 00:04:18.837
You can do it with these other ones.

00:04:18.837 --> 00:04:20.747
The hardest one would be standard form.

00:04:20.747 --> 00:04:23.087
Standard form you could
complete the square

00:04:23.087 --> 00:04:24.604
or do some other techniques

00:04:24.604 --> 00:04:26.762
or you could try to
get into factored form.

00:04:26.762 --> 00:04:29.367
Factored form you can find the zeros

00:04:29.367 --> 00:04:31.524
and then you'd know that the
x-coordinate of the vertex

00:04:31.524 --> 00:04:35.503
is halfway between the x-coordinate
of our two x-intercepts

00:04:35.503 --> 00:04:37.532
and then you could figure
out the y value there.

00:04:37.532 --> 00:04:39.997
But this one is definitely the easiest.

00:04:39.997 --> 00:04:42.044
The vertex form and what is the vertex?

00:04:42.044 --> 00:04:46.309
Well it's going to happen
at the point six comma 18.

00:04:46.309 --> 00:04:49.068
Let's do one last example.

00:04:49.068 --> 00:04:50.936
So this is a different function.

00:04:50.936 --> 00:04:53.421
The function f is given
in three equivalent forms,

00:04:53.421 --> 00:04:55.770
which form most quickly reveals the zeros

00:04:55.770 --> 00:04:58.371
or roots of the function?

00:04:58.371 --> 00:05:00.116
So once again,

00:05:00.116 --> 00:05:03.120
when we're talking about zeros or roots,

00:05:03.120 --> 00:05:06.029
if we have, let's say that's the x axis

00:05:06.029 --> 00:05:08.972
and if you have a parabola
that looks like that,

00:05:08.972 --> 00:05:12.265
the roots are, or the
zeros are the x values

00:05:12.265 --> 00:05:14.501
that make that function equal to zero.

00:05:14.501 --> 00:05:18.984
Or they're the x values
of the x-intercepts,

00:05:18.984 --> 00:05:20.528
you could say.

00:05:20.528 --> 00:05:22.521
And so what x values make

00:05:22.521 --> 00:05:24.520
or which one is easy to figure out

00:05:24.520 --> 00:05:26.745
when this function is equal to zero?

00:05:26.745 --> 00:05:28.600
Which of these forms, because
they're all equivalent.

00:05:28.600 --> 00:05:30.148
You can expand out these first two

00:05:30.148 --> 00:05:32.491
and you should get this
last one in standard form.

00:05:32.491 --> 00:05:36.039
Which one is easy to identify the zeros?

00:05:36.039 --> 00:05:37.519
Well in factored form, I could just say,

00:05:37.519 --> 00:05:40.062
"Well, what makes either this thing zero

00:05:40.062 --> 00:05:41.776
"or that thing zero?"

00:05:41.776 --> 00:05:44.239
Because an x that makes
this first one zero

00:05:44.239 --> 00:05:45.866
or the second one zero

00:05:45.866 --> 00:05:48.791
is going to make this
whole expression zero.

00:05:48.791 --> 00:05:49.685
So you can quickly say,

00:05:49.685 --> 00:05:51.394
"Well if x is equal to negative one,

00:05:51.394 --> 00:05:53.014
"this is going to be zero.

00:05:53.014 --> 00:05:55.776
"Or if x is equal to negative 11,

00:05:55.776 --> 00:05:56.990
"this is going to be zero."

00:05:56.990 --> 00:06:01.508
So this is a very fast
way to find out the zeros.

00:06:01.508 --> 00:06:03.777
This one here, a lot harder.

00:06:03.777 --> 00:06:08.004
You would have to solve three
times x plus six-squared

00:06:08.004 --> 00:06:10.347
minus 75 is equal to zero.

00:06:10.347 --> 00:06:11.848
Do some algebraic manipulation

00:06:11.848 --> 00:06:13.837
and you would eventually
get to these answers.

00:06:13.837 --> 00:06:17.008
So I would rule the vertex
form right over here.

00:06:17.008 --> 00:06:19.506
And this form, standard form,

00:06:19.506 --> 00:06:20.622
the first step I would do,

00:06:20.622 --> 00:06:23.031
is try to get it into factored form.

00:06:23.031 --> 00:06:26.571
And then from factored form,
I would find the zeros.

00:06:26.571 --> 00:06:28.741
So once again, this is
definitely more work

00:06:28.741 --> 00:06:30.553
than if you already have
it in factored form.

00:06:30.553 --> 00:06:32.640
So factored form is
definitely what you want

00:06:32.640 --> 00:06:34.484
when you're trying to find the zeros.

00:06:34.484 --> 00:06:36.010
And here it says write one of the zeros,

00:06:36.010 --> 00:06:37.589
I could write x equals negative one

00:06:37.589 --> 00:06:40.911
or I could have written
x equals negative 11.