Function symmetry introduction

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:02.240
- [Instructor] You've
likely heard the concept

00:00:02.240 --> 00:00:05.720
of even and odd numbers,
and what we're going to do

00:00:05.720 --> 00:00:09.720
in this video is think about
even and odd functions.

00:00:09.720 --> 00:00:12.290
And as you can see or as you will see,

00:00:12.290 --> 00:00:14.500
there's a little bit of a
parallel between the two,

00:00:14.500 --> 00:00:16.860
but there's also some differences.

00:00:16.860 --> 00:00:20.640
So let's first think about
what an even function is.

00:00:20.640 --> 00:00:23.500
One way to think about
an even function is that

00:00:23.500 --> 00:00:26.740
if you were to flip it over the y-axis,

00:00:26.740 --> 00:00:28.990
that the function looks the same.

00:00:28.990 --> 00:00:33.030
So here's a classic example
of an even function.

00:00:33.030 --> 00:00:35.870
It would be this right over here,

00:00:35.870 --> 00:00:38.270
your classic parabola

00:00:38.270 --> 00:00:40.510
where your vertex

00:00:40.510 --> 00:00:42.650
is on the y-axis.

00:00:42.650 --> 00:00:44.730
This is an even function.

00:00:44.730 --> 00:00:46.150
So this one is maybe the graph

00:00:46.150 --> 00:00:48.930
of f of x is equal to x squared.

00:00:48.930 --> 00:00:53.280
And notice, if you were to
flip it over the y-axis,

00:00:53.280 --> 00:00:56.220
you're going to get the exact same graph.

00:00:56.220 --> 00:00:58.980
Now, a way that we can talk
about that mathematically,

00:00:58.980 --> 00:01:00.200
and we've talked about this

00:01:00.200 --> 00:01:02.640
when we introduced the idea of reflection,

00:01:02.640 --> 00:01:05.040
to say that a function is equal

00:01:05.040 --> 00:01:08.440
to its reflection over the y-axis,

00:01:08.440 --> 00:01:13.300
that's just saying that f of
x is equal to f of negative x.

00:01:13.300 --> 00:01:15.820
Because if you were to replace
your x's with a negative x,

00:01:15.820 --> 00:01:20.190
that flips your function over the y-axis.

00:01:20.190 --> 00:01:22.930
Now, what about odd functions?

00:01:22.930 --> 00:01:25.310
So odd functions, you
get the same function

00:01:25.310 --> 00:01:29.040
if you flip over the y- and the x-axes.

00:01:29.040 --> 00:01:33.930
So let me draw a classic
example of an odd function.

00:01:33.930 --> 00:01:36.560
Our classic example would be

00:01:36.560 --> 00:01:40.420
f of x is equal to x to the third,

00:01:40.420 --> 00:01:42.000
is equal to x to the third,

00:01:42.000 --> 00:01:43.773
and it looks something like this.

00:01:44.750 --> 00:01:49.260
So notice, if you were to
flip first over the y-axis,

00:01:49.260 --> 00:01:51.810
you would get something
that looks like this.

00:01:51.810 --> 00:01:53.360
So I'll do it as a dotted line.

00:01:53.360 --> 00:01:55.580
If you were to flip just over the y-axis,

00:01:55.580 --> 00:01:56.467
it would look like this.

00:01:56.467 --> 00:01:59.780
And then if you were to
flip that over the x-axis,

00:01:59.780 --> 00:02:04.330
well, then you're going to
get the same function again.

00:02:04.330 --> 00:02:07.830
Now, how would we write
this down mathematically?

00:02:07.830 --> 00:02:11.920
Well, that means that our
function is equivalent to

00:02:12.840 --> 00:02:15.260
not only flipping it over the y-axis,

00:02:15.260 --> 00:02:16.680
which would be f of negative x,

00:02:16.680 --> 00:02:19.120
but then flipping that over the x-axis,

00:02:19.120 --> 00:02:21.550
which is just taking the negative of that.

00:02:21.550 --> 00:02:24.210
So this is doing two flips.

00:02:24.210 --> 00:02:26.270
So some of you might be noticing a pattern

00:02:26.270 --> 00:02:28.530
or think you might be on the
verge of seeing a pattern

00:02:28.530 --> 00:02:32.000
that connects the words even
and odd with the notions

00:02:32.000 --> 00:02:34.970
that we know from earlier
in our mathematical lives.

00:02:34.970 --> 00:02:37.200
I've just shown you an even function

00:02:37.200 --> 00:02:39.470
where the exponent is an even number,

00:02:39.470 --> 00:02:41.200
and I've just showed you an odd function

00:02:41.200 --> 00:02:43.680
where the exponent is an odd number.

00:02:43.680 --> 00:02:46.920
Now, I encourage you to try
out many, many more polynomials

00:02:46.920 --> 00:02:48.290
and try out the exponents,

00:02:48.290 --> 00:02:53.290
but it turns out that if you
just have f of x is equal to,

00:02:53.320 --> 00:02:56.870
if you just have f of x
is equal to x to the n,

00:02:56.870 --> 00:03:00.410
then this is going to be an
even function if n is even,

00:03:00.410 --> 00:03:03.290
and it's going to an odd
function if n is odd.

00:03:03.290 --> 00:03:04.940
So that's one connection.

00:03:04.940 --> 00:03:06.317
Now, some of you are thinking,

00:03:06.317 --> 00:03:07.857
"Wait, but there seem
to be a lot of functions

00:03:07.857 --> 00:03:10.170
"that are neither even nor odd."

00:03:10.170 --> 00:03:12.520
And that is indeed the case.

00:03:12.520 --> 00:03:16.853
For example, if you just had
the graph x squared plus two,

00:03:17.710 --> 00:03:21.590
this right over here is
still going to be even.

00:03:21.590 --> 00:03:22.680
'Cause if you flip it over,

00:03:22.680 --> 00:03:25.020
you have the symmetry around the y-axis.

00:03:25.020 --> 00:03:26.950
You're going to get back to itself.

00:03:26.950 --> 00:03:30.200
But if you had x minus two squared,

00:03:30.200 --> 00:03:33.000
which looks like this,

00:03:33.000 --> 00:03:35.930
x minus two, that would
shift two to the right,

00:03:35.930 --> 00:03:37.260
it'll look like that.

00:03:37.260 --> 00:03:40.180
That is no longer even.

00:03:40.180 --> 00:03:43.810
Because notice, if you
flip it over the y-axis,

00:03:43.810 --> 00:03:47.340
you're no longer getting
the same function.

00:03:47.340 --> 00:03:49.720
So it's not just the exponent.

00:03:49.720 --> 00:03:52.340
It also matters on the structure
of the expression itself.

00:03:52.340 --> 00:03:55.690
If you have something very
simple, like just x to the n,

00:03:55.690 --> 00:03:58.720
well, then that could be or
that would be even or odd

00:03:58.720 --> 00:04:00.850
depending on what your n is.

00:04:00.850 --> 00:04:04.360
Similarly, if we were
to shift this f of x,

00:04:04.360 --> 00:04:07.550
if we were to even shift
it up, it's no longer,

00:04:07.550 --> 00:04:09.810
it is no longer, so if
this is x to the third,

00:04:09.810 --> 00:04:11.580
let's say, plus three,

00:04:11.580 --> 00:04:14.040
this is no longer odd.

00:04:14.040 --> 00:04:18.190
Because you flip it over once,
you get right over there.

00:04:18.190 --> 00:04:22.230
But then you flip it again,
you're going to get this.

00:04:22.230 --> 00:04:25.330
You're going to get something like this.

00:04:25.330 --> 00:04:28.730
So you're no longer back
to your original function.

00:04:28.730 --> 00:04:30.510
Now, an interesting thing to think about,

00:04:30.510 --> 00:04:35.510
can you imagine a function
that is both even and odd?

00:04:36.130 --> 00:04:38.100
So I encourage you to pause that video,

00:04:38.100 --> 00:04:40.600
or pause the video and
try to think about it.

00:04:40.600 --> 00:04:44.190
Is there a function where f of
x is equal to f of negative x

00:04:44.190 --> 00:04:47.973
and f of x is equal to the
negative of f of negative x?

00:04:48.940 --> 00:04:50.730
Well, I'll give you a hint,

00:04:50.730 --> 00:04:52.590
or actually I'll just give you the answer.

00:04:52.590 --> 00:04:57.350
Imagine if f of x is just
equal to the constant zero.

00:04:57.350 --> 00:05:01.750
Notice, this thing is
just a horizontal line,

00:05:01.750 --> 00:05:04.830
just like that, at y is equal to zero.

00:05:04.830 --> 00:05:07.660
And if you flip it over the y-axis,

00:05:07.660 --> 00:05:09.210
you get back to where it was before.

00:05:09.210 --> 00:05:12.830
Then if you flip it over the x-axis,

00:05:12.830 --> 00:05:15.730
again, then you're still back
to where you were before.

00:05:15.730 --> 00:05:18.760
So this over here is both even and odd,

00:05:18.760 --> 00:05:20.953
a very interesting case.


Reflecting functions: examples

WEBVTT
Kind: captions
Language: en

00:00:00.200 --> 00:00:01.510
- [Instructor] What we're
going to do in this video

00:00:01.510 --> 00:00:04.840
is do some practice examples
of exercises on Khan Academy

00:00:04.840 --> 00:00:07.070
that deal with reflections of functions.

00:00:07.070 --> 00:00:09.870
So this first one says this
is the graph of function f.

00:00:09.870 --> 00:00:10.770
Fair enough.

00:00:10.770 --> 00:00:14.330
Function g is defined as g of
x is equal to f of negative x.

00:00:14.330 --> 00:00:15.500
Also fair enough.

00:00:15.500 --> 00:00:16.770
What is the graph of g?

00:00:16.770 --> 00:00:18.450
And on Khan Academy, it's multiple choice,

00:00:18.450 --> 00:00:20.210
but I thought for the sake of this video,

00:00:20.210 --> 00:00:22.630
it'd be fun to think about
what g would look like

00:00:22.630 --> 00:00:24.810
without having any choices,
just sketching it out.

00:00:24.810 --> 00:00:26.420
So pause this video and
try to think about it,

00:00:26.420 --> 00:00:27.920
at least in your head.

00:00:27.920 --> 00:00:29.850
All right, now let's work
through this together.

00:00:29.850 --> 00:00:32.340
So we've already gone
over that g of x is equal

00:00:32.340 --> 00:00:35.570
to f of negative x.

00:00:35.570 --> 00:00:38.780
So whatever the value of
f is at a certain value,

00:00:38.780 --> 00:00:41.750
we would expect g to take on that value

00:00:41.750 --> 00:00:43.530
at the negative of that.

00:00:43.530 --> 00:00:48.000
So for example, we can see
that f of four is equal to two,

00:00:48.000 --> 00:00:52.800
so we would expect g of
negative four to be equal to two

00:00:52.800 --> 00:00:55.160
because, once again, g of negative four,

00:00:55.160 --> 00:00:56.040
we could write it over here.

00:00:56.040 --> 00:00:59.250
G of negative four is going to be equal

00:00:59.250 --> 00:01:02.810
to f of the negative of negative four,

00:01:02.810 --> 00:01:05.650
which is equal to f of four.

00:01:05.650 --> 00:01:07.930
And so we could keep going with that.

00:01:07.930 --> 00:01:11.260
What would g of negative two be?

00:01:11.260 --> 00:01:12.950
Well, that would be the
same thing as f of two,

00:01:12.950 --> 00:01:15.690
which is zero, so it
would be right over there.

00:01:15.690 --> 00:01:18.580
What would g of zero be?

00:01:18.580 --> 00:01:20.300
Well, that would be the
same thing as f of zero

00:01:20.300 --> 00:01:21.590
'cause a negative zero is zero.

00:01:21.590 --> 00:01:24.320
And f of zero is right over there.

00:01:24.320 --> 00:01:25.590
Looks like negative two.

00:01:25.590 --> 00:01:28.240
And so you can already
see where this is going.

00:01:28.240 --> 00:01:30.810
And we've already talked
about it in previous videos

00:01:30.810 --> 00:01:33.810
that if you replace your
x with a negative x,

00:01:33.810 --> 00:01:37.210
you're essentially
reflecting over the y-axis.

00:01:37.210 --> 00:01:41.360
So g is going to look something like this.

00:01:41.360 --> 00:01:42.970
It is going to look something like this.

00:01:42.970 --> 00:01:45.610
Once again, g of negative
six would be the same thing

00:01:45.610 --> 00:01:47.060
as f of six.

00:01:47.060 --> 00:01:49.760
And so that would be the graph of g.

00:01:49.760 --> 00:01:50.810
And if you're doing this on Khan Academy,

00:01:50.810 --> 00:01:52.470
you'd pick the choice
that looks like this,

00:01:52.470 --> 00:01:55.980
that would give a
reflection over the y-axis.

00:01:55.980 --> 00:01:57.973
Let's do another example.

00:01:59.010 --> 00:02:02.800
So here once again, this is
the graph of the function f.

00:02:02.800 --> 00:02:05.600
And then they say what is the graph of g?

00:02:05.600 --> 00:02:07.330
And so pause this video and at least try

00:02:07.330 --> 00:02:10.830
to sketch it out in your
mind what g should look like.

00:02:10.830 --> 00:02:13.120
All right, so in this situation,

00:02:13.120 --> 00:02:16.840
they didn't replace the x
with negative x in f of x.

00:02:16.840 --> 00:02:20.300
Instead, g of x is equal to
the negative of all of f of x.

00:02:20.300 --> 00:02:22.280
In fact, we could
rewrite g of x like this.

00:02:22.280 --> 00:02:24.700
We could say that g of
x is equal to, notice,

00:02:24.700 --> 00:02:25.750
all of this right over here,

00:02:25.750 --> 00:02:28.050
that was our definition of f of x.

00:02:28.050 --> 00:02:31.610
So g of x is equal to
the negative of f of x.

00:02:31.610 --> 00:02:33.460
So instead of it being f of negative x,

00:02:33.460 --> 00:02:35.820
it's equal to the negative of f of x.

00:02:35.820 --> 00:02:38.150
So one way to think about it is

00:02:38.150 --> 00:02:41.060
we can see that f of zero is two,

00:02:41.060 --> 00:02:43.250
but g of zero is going to
be the negative of that.

00:02:43.250 --> 00:02:46.150
So it's going to be equal to negative two.

00:02:46.150 --> 00:02:48.680
And so you could keep going with that.

00:02:48.680 --> 00:02:52.970
You could see that whatever
f of a certain value is,

00:02:52.970 --> 00:02:55.740
g of that value would
be the negative of that.

00:02:55.740 --> 00:02:57.430
So it would be down here.

00:02:57.430 --> 00:03:01.120
And so g of x would be
a reflection of f of x

00:03:01.120 --> 00:03:03.860
about the x-axis.

00:03:03.860 --> 00:03:05.710
So g of x is going to look

00:03:05.710 --> 00:03:07.477
something like that,

00:03:12.070 --> 00:03:15.140
a reflection about the x-axis.

00:03:15.140 --> 00:03:16.530
Once again, on Khan Academy,
you'd pick the choice

00:03:16.530 --> 00:03:18.620
that would actually look like that.

00:03:18.620 --> 00:03:19.453
Let's do another example.

00:03:19.453 --> 00:03:22.107
His is strangely fun.

00:03:22.107 --> 00:03:25.760
(laughs) All right, so here
we're told functions f,

00:03:25.760 --> 00:03:27.480
so that's in solid in this blue color,

00:03:27.480 --> 00:03:31.190
and g dashed, so that's right
over there, are graphed.

00:03:31.190 --> 00:03:34.970
What is the equation of g in terms of f?

00:03:34.970 --> 00:03:37.760
So pause this video and
try to think about it.

00:03:37.760 --> 00:03:42.060
So the key is to realize
how do we transform f of x,

00:03:42.060 --> 00:03:43.290
actually, they've labeled it over here,

00:03:43.290 --> 00:03:46.810
this is f of x right over
here, in order to get g?

00:03:46.810 --> 00:03:50.120
So f of negative x would be a reflection

00:03:50.120 --> 00:03:52.600
of f about the y-axis.

00:03:52.600 --> 00:03:54.720
And so it would intersect there.

00:03:54.720 --> 00:03:59.300
It would have this
straight portion like this.

00:04:00.360 --> 00:04:01.730
And I'm just experimenting right now.

00:04:01.730 --> 00:04:03.370
It'd have the straight portion like this.

00:04:03.370 --> 00:04:05.930
And then it would go up.

00:04:05.930 --> 00:04:07.183
And let's see.

00:04:08.240 --> 00:04:09.570
If f of negative x.

00:04:09.570 --> 00:04:11.460
So when you input six into it,

00:04:11.460 --> 00:04:13.770
that would be f of
negative six, which is six.

00:04:13.770 --> 00:04:15.290
So it would go up there.

00:04:15.290 --> 00:04:20.290
So f of negative x would
look something like this.

00:04:20.670 --> 00:04:23.110
Something like that.

00:04:23.110 --> 00:04:25.750
So the purple is f of negative x.

00:04:25.750 --> 00:04:28.910
Now, that doesn't quite get us to g,

00:04:28.910 --> 00:04:30.030
but it gets us a little bit closer

00:04:30.030 --> 00:04:31.670
'cause it looks like if I
were to take the reflection

00:04:31.670 --> 00:04:36.060
of f of negative x, f of
negative x about the x-axis,

00:04:36.060 --> 00:04:38.510
it looks like I'm going to get to g.

00:04:38.510 --> 00:04:40.980
And so how do you reflect
something about the x-axis?

00:04:40.980 --> 00:04:43.130
Well, we saw it in the example just now.

00:04:43.130 --> 00:04:45.590
You multiply the entire
function by a negative.

00:04:45.590 --> 00:04:47.390
So we could say that g is equal

00:04:47.390 --> 00:04:51.200
to the negative of f of negative x.

00:04:51.200 --> 00:04:52.530
It's equal to the negative of this.

00:04:52.530 --> 00:04:54.270
So we're doing both reflections.

00:04:54.270 --> 00:04:55.820
We're flipping over the y-axis,

00:04:55.820 --> 00:04:59.240
and we're flipping over
the x-axis to get to g.

00:04:59.240 --> 00:05:01.183
Let's do one more example.

00:05:02.810 --> 00:05:05.910
So once again, they've
graphed f, they've graphed g,

00:05:05.910 --> 00:05:08.880
and they've said f is defined
as this right over here.

00:05:08.880 --> 00:05:10.780
What is the equation of g?

00:05:10.780 --> 00:05:12.930
So they're not just
asking it in terms of f.

00:05:12.930 --> 00:05:14.940
They just wanna know what
is the equation of g?

00:05:14.940 --> 00:05:17.460
Pause this video and
try to think about it.

00:05:17.460 --> 00:05:19.540
Well, you can see pretty clearly

00:05:19.540 --> 00:05:23.260
that this is a reflection
across the y-axis.

00:05:23.260 --> 00:05:26.050
And a reflection across the y-axis,

00:05:26.050 --> 00:05:28.090
you can see pretty clearly,

00:05:28.090 --> 00:05:33.090
that g of x is equal to f of negative x.

00:05:33.100 --> 00:05:34.590
F of negative x.

00:05:34.590 --> 00:05:36.070
How do we know that?

00:05:36.070 --> 00:05:40.830
Well, for whenever we take f
of x and we get that value,

00:05:40.830 --> 00:05:43.570
g at the negative of that value

00:05:43.570 --> 00:05:46.500
takes on the same function
value, I guess I could say.

00:05:46.500 --> 00:05:47.860
Or another way to think about it is

00:05:47.860 --> 00:05:49.380
we could just pick this
point, negative eight.

00:05:49.380 --> 00:05:52.780
F of negative eight is
equal to a little over four,

00:05:52.780 --> 00:05:57.210
but g of eight is equal
to a little over four,

00:05:57.210 --> 00:05:59.180
is equal to that same value.

00:05:59.180 --> 00:06:01.140
And so what is the equation of g?

00:06:01.140 --> 00:06:03.340
Well, we just have to rewrite this

00:06:03.340 --> 00:06:05.800
so that we can write
it out as an equation.

00:06:05.800 --> 00:06:09.080
And so we could write
out g of x is equal to.

00:06:09.080 --> 00:06:10.930
If I were to replace all of the x's here

00:06:10.930 --> 00:06:12.340
with a negative x, what would I get?

00:06:12.340 --> 00:06:17.090
I would get four times the
square root of two minus.

00:06:17.090 --> 00:06:19.720
Instead of an x, I will have a negative x.

00:06:19.720 --> 00:06:23.030
And then the minus eight
is outside of the radical.

00:06:23.030 --> 00:06:25.240
And so we would have g of x is equal

00:06:25.240 --> 00:06:30.240
to four times the square root
of two plus x minus eight.

00:06:30.600 --> 00:06:31.433
And we're done.


Reflecting functions introduction

WEBVTT
Kind: captions
Language: en

00:00:00.160 --> 00:00:02.170
- [Instructor] So you see
here, this is a screenshot

00:00:02.170 --> 00:00:04.510
of the Desmos online graphing calculator.

00:00:04.510 --> 00:00:06.620
You can use it at desmos.com,

00:00:06.620 --> 00:00:08.540
and I encourage you to
use this after this video,

00:00:08.540 --> 00:00:10.370
or even while I'm doing this video,

00:00:10.370 --> 00:00:14.580
but the goal here is to think
about reflection of functions.

00:00:14.580 --> 00:00:16.660
So let's just start with some examples.

00:00:16.660 --> 00:00:19.260
Let's saying that I
had a function, f of x,

00:00:19.260 --> 00:00:24.200
and it is equal to the square root of x.

00:00:24.200 --> 00:00:26.010
So that's what it looks like.

00:00:26.010 --> 00:00:28.030
Fairly reasonable.

00:00:28.030 --> 00:00:30.790
Now, let's make another function, g of x,

00:00:30.790 --> 00:00:32.840
and I'll start off by also making

00:00:32.840 --> 00:00:34.740
that the square root of x.

00:00:34.740 --> 00:00:37.500
So no surprise there, g of x was graphed

00:00:37.500 --> 00:00:39.510
right on top of f of x.

00:00:39.510 --> 00:00:42.170
But what would happen if instead of it

00:00:42.170 --> 00:00:43.390
just being the square root of x,

00:00:43.390 --> 00:00:45.610
what would happen if we
put a negative out front

00:00:45.610 --> 00:00:46.640
right over there?

00:00:46.640 --> 00:00:49.723
What do you think is going
to happen when I do that?

00:00:50.910 --> 00:00:52.440
Well, let's just try it out.

00:00:52.440 --> 00:00:53.740
When I put the negative,

00:00:53.740 --> 00:00:57.570
it looks like it flipped
it over the x-axis.

00:00:57.570 --> 00:01:02.040
It looks like it reflected
it over the x-axis.

00:01:02.040 --> 00:01:04.400
Now instead of doing that way,

00:01:04.400 --> 00:01:06.980
what if we had another function, h of x,

00:01:06.980 --> 00:01:10.810
and I'll start off by making
it identical to f of x.

00:01:10.810 --> 00:01:12.490
So once again, it's right over there.

00:01:12.490 --> 00:01:14.620
It traces out f of x.

00:01:14.620 --> 00:01:16.290
Instead of putting the negative out

00:01:16.290 --> 00:01:18.080
in front of the radical sign,

00:01:18.080 --> 00:01:19.750
what if we put it under the radical sign?

00:01:19.750 --> 00:01:23.050
What if we replaced x with a negative x?

00:01:23.050 --> 00:01:25.610
What do you think is
going to happen there?

00:01:25.610 --> 00:01:27.630
Well, let's try it out.

00:01:27.630 --> 00:01:28.990
If we replace it,

00:01:28.990 --> 00:01:32.780
that shifted it over the y-axis.

00:01:32.780 --> 00:01:33.980
And then, pause this video,

00:01:33.980 --> 00:01:36.403
and think about how you
shifted over both axes.

00:01:37.310 --> 00:01:38.730
Well, we could do a,

00:01:38.730 --> 00:01:40.130
well, I'm running out of letters,

00:01:40.130 --> 00:01:43.870
maybe I will do a, I don't
know, k of x is equal to,

00:01:43.870 --> 00:01:46.600
so I'm gonna put the negative
outside the radical sign,

00:01:46.600 --> 00:01:48.810
and then, I'm gonna take the square root,

00:01:48.810 --> 00:01:50.980
and I'm gonna put a negative
inside the radical sign.

00:01:50.980 --> 00:01:53.200
And notice, it flipped it over both.

00:01:53.200 --> 00:01:56.750
It flipped it over both
the x-axis and the y-axis

00:01:56.750 --> 00:01:58.270
to go over here.

00:01:58.270 --> 00:02:00.020
Now, why does this happen?

00:02:00.020 --> 00:02:02.000
Well, let's just start with the g of x.

00:02:02.000 --> 00:02:04.200
So when you get put the
negative out in front,

00:02:04.200 --> 00:02:06.640
when you negate everything
that's in the expression

00:02:06.640 --> 00:02:08.150
that defines a function,

00:02:08.150 --> 00:02:11.630
whatever value you would've
gotten of the function before,

00:02:11.630 --> 00:02:14.600
you're now going to
get the opposite of it.

00:02:14.600 --> 00:02:17.660
So when x is zero, we get zero.

00:02:17.660 --> 00:02:19.670
When x is one, instead of one now,

00:02:19.670 --> 00:02:20.610
you're taking the negative of it

00:02:20.610 --> 00:02:22.300
so you're gonna get negative one.

00:02:22.300 --> 00:02:25.820
When x is four, instead
of getting positive two,

00:02:25.820 --> 00:02:27.970
you're now going to get negative two.

00:02:27.970 --> 00:02:30.600
When x is equal to nine, instead
of getting positive three,

00:02:30.600 --> 00:02:32.310
you now get negative three.

00:02:32.310 --> 00:02:33.350
So hopefully, that makes sense

00:02:33.350 --> 00:02:35.020
why putting a negative out front

00:02:35.020 --> 00:02:38.790
of an entire expression
is going to flip it over,

00:02:38.790 --> 00:02:42.040
flip its graph over the x-axis.

00:02:42.040 --> 00:02:45.060
Now what about replacing
an x with a negative x?

00:02:45.060 --> 00:02:46.610
Well, one way to think about it,

00:02:46.610 --> 00:02:50.300
now is, whenever you inputted one before,

00:02:50.300 --> 00:02:52.510
that would now be a negative one

00:02:52.510 --> 00:02:54.670
that you're trying to
evaluate the principle root of

00:02:54.670 --> 00:02:56.290
and we know that the
principle root function

00:02:56.290 --> 00:02:58.860
is not defined for negative one.

00:02:58.860 --> 00:03:01.630
But when x is equal to negative one,

00:03:01.630 --> 00:03:03.570
our original function wasn't defined there

00:03:03.570 --> 00:03:05.560
when x is equal to negative one,

00:03:05.560 --> 00:03:07.660
but if you take the negative of that,

00:03:07.660 --> 00:03:09.680
well now you're taking
this principle root of one.

00:03:09.680 --> 00:03:12.240
And so, that's why this is now defined.

00:03:12.240 --> 00:03:15.550
So, whatever value the
function would've taken on

00:03:15.550 --> 00:03:18.352
at a given value of x,
it now takes that value

00:03:18.352 --> 00:03:21.370
on the corresponding opposite value of x,

00:03:21.370 --> 00:03:23.640
and on the negative value of that x.

00:03:23.640 --> 00:03:27.410
And so that's why it
flips it over the y-axis.

00:03:27.410 --> 00:03:29.490
And this is true with
many types of functions.

00:03:29.490 --> 00:03:33.170
We don't have to do this just
with a square root function.

00:03:33.170 --> 00:03:35.300
Let's try another function.

00:03:35.300 --> 00:03:40.300
Let's say, we tried this
for e to the x power.

00:03:42.770 --> 00:03:43.640
So there you go.

00:03:43.640 --> 00:03:46.070
We have a very classic exponential there.

00:03:46.070 --> 00:03:49.610
Now let's say that g of x is
equal to negative e to the x.

00:03:49.610 --> 00:03:52.140
And if what we expect to happen happens,

00:03:52.140 --> 00:03:55.450
this will flip it over the x-axis.

00:03:55.450 --> 00:03:59.140
So negative e to the x power

00:03:59.140 --> 00:04:01.370
and indeed that is what happens.

00:04:01.370 --> 00:04:05.580
And then, how would we
flip it over the y-axis?

00:04:05.580 --> 00:04:08.270
Well, let's do an h of x.

00:04:08.270 --> 00:04:11.660
That's going to be equal to e to the,

00:04:11.660 --> 00:04:13.130
instead of putting an x there,

00:04:13.130 --> 00:04:15.660
we will put a negative x.

00:04:15.660 --> 00:04:16.720
Negative x.

00:04:16.720 --> 00:04:17.580
And there you have it.

00:04:17.580 --> 00:04:20.450
Notice, it flipped it over the y-axis.

00:04:20.450 --> 00:04:21.980
Now, both examples that I just did,

00:04:21.980 --> 00:04:23.360
these are very simple expressions.

00:04:23.360 --> 00:04:26.730
Let's imagine something that's
a little bit more complex.

00:04:26.730 --> 00:04:28.110
Let's say that f of x,

00:04:28.110 --> 00:04:30.840
let's give it a nice,
higher-degree polynomial,

00:04:30.840 --> 00:04:33.190
so let's say it's x to the third

00:04:33.190 --> 00:04:36.970
minus two x squared.

00:04:36.970 --> 00:04:38.200
That's a nice one

00:04:38.200 --> 00:04:40.340
and actually let's just
add another term here.

00:04:40.340 --> 00:04:43.080
So plus two x.

00:04:43.080 --> 00:04:43.913
And I wanna make it,

00:04:43.913 --> 00:04:45.340
make it minus two x.

00:04:45.340 --> 00:04:47.170
I wanna see it accentuates
some of those curves.

00:04:47.170 --> 00:04:49.750
All right, so that's a
pretty interesting graph.

00:04:49.750 --> 00:04:54.750
Now, how would I flip it over the x-axis?

00:04:55.030 --> 00:04:56.770
Well the way that I would do that

00:04:56.770 --> 00:04:59.440
is I could define a g of x.

00:04:59.440 --> 00:05:00.470
I could do it two ways.

00:05:00.470 --> 00:05:05.100
I could say g of x is equal
to the negative of f of x

00:05:05.100 --> 00:05:06.230
and we get that.

00:05:06.230 --> 00:05:08.410
So that's essentially just
taking this entire expression

00:05:08.410 --> 00:05:10.140
and multiplying it by negative one.

00:05:10.140 --> 00:05:11.760
And notice, it's multiplying,

00:05:11.760 --> 00:05:14.790
it's flipping it over the x-axis.

00:05:14.790 --> 00:05:16.820
Another way we could've
done it is instead of that,

00:05:16.820 --> 00:05:21.080
we could've said the
negative of x to the third

00:05:21.080 --> 00:05:26.020
minus two x squared,

00:05:26.020 --> 00:05:29.010
and then minus two x,

00:05:29.010 --> 00:05:31.110
and then we close those parentheses,

00:05:31.110 --> 00:05:33.330
and we get the same effect.

00:05:33.330 --> 00:05:36.110
Now, what if we wanted to
flip it over the y-axis?

00:05:36.110 --> 00:05:37.960
Well then instead of putting a negative

00:05:37.960 --> 00:05:39.660
on the entire expression,

00:05:39.660 --> 00:05:43.590
what we wanna do is replace
our x's with a negative x.

00:05:43.590 --> 00:05:45.260
So you could do it like this.

00:05:45.260 --> 00:05:49.020
You could say that that's
going to be f of negative x

00:05:49.020 --> 00:05:50.910
and that has the effect
of everywhere you saw

00:05:50.910 --> 00:05:53.230
an x before you replaced
it with a negative x.

00:05:53.230 --> 00:05:55.570
And notice, it did exactly what we expect.

00:05:55.570 --> 00:05:59.920
It flipped it over over the y-axis.

00:05:59.920 --> 00:06:01.690
Now, the other way we could've don't that

00:06:01.690 --> 00:06:02.760
just to make it clear,

00:06:02.760 --> 00:06:05.990
that's the same thing as
negative x to the third power

00:06:06.920 --> 00:06:11.140
minus two times negative x squared

00:06:12.400 --> 00:06:15.990
minus two times negative x.

00:06:15.990 --> 00:06:17.810
And of course, we could
simplify that expression,

00:06:17.810 --> 00:06:20.690
but notice, it has the exact same idea.

00:06:20.690 --> 00:06:22.130
And if we wanted to flip it over

00:06:22.130 --> 00:06:25.160
both the x and y-axis,

00:06:25.160 --> 00:06:27.190
well we've already flipped
it over the y-axis,

00:06:27.190 --> 00:06:28.780
to flip it over the x-axis,

00:06:28.780 --> 00:06:30.390
oh whoops, I just deleted it,

00:06:30.390 --> 00:06:33.620
to flip it over the,
I'm having issues here,

00:06:33.620 --> 00:06:37.360
to flip it over the x-axis as well,

00:06:37.360 --> 00:06:39.710
we would, oh and it gave
me a parentheses already,

00:06:39.710 --> 00:06:41.810
I would just put a negative out front.

00:06:41.810 --> 00:06:44.440
So I put a negative out
front and there you have it.

00:06:44.440 --> 00:06:47.740
This flipped it over
both the x and y-axis.

00:06:47.740 --> 00:06:49.210
You can do them in either order

00:06:49.210 --> 00:06:51.970
and you will get to this green curve.

00:06:51.970 --> 00:06:53.690
Now, an easier way of writing that

00:06:53.690 --> 00:06:58.007
would've been just the
negative of f of negative x

00:06:59.340 --> 00:07:02.860
and you would've gotten
to that same place.

00:07:02.860 --> 00:07:05.270
So go to Desmos, play around with it,

00:07:05.270 --> 00:07:07.120
really good to build this intuition,

00:07:07.120 --> 00:07:09.253
and really understand why it's happening.


Shifting functions introduction

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:03.450
- [Instructor] So I am here at desmos.com,

00:00:03.450 --> 00:00:05.630
which is an online graphing calculator,

00:00:05.630 --> 00:00:07.270
and the goal of this video is

00:00:07.270 --> 00:00:10.430
to explore how shifts in functions happen.

00:00:10.430 --> 00:00:12.620
How do things shift to the right or left

00:00:12.620 --> 00:00:15.120
or how do they shift up and down?

00:00:15.120 --> 00:00:16.920
And what we're going to start off doing

00:00:16.920 --> 00:00:19.140
is just graph a plain vanilla function,

00:00:19.140 --> 00:00:23.280
f of x is equal to x squared.

00:00:23.280 --> 00:00:25.930
That looks as we would expect it to look,

00:00:25.930 --> 00:00:30.050
but now let's think about how
we can shift it up or down.

00:00:30.050 --> 00:00:32.050
Well one thought is, well, to shift it up,

00:00:32.050 --> 00:00:34.690
we just have to make the
value of f of x higher

00:00:34.690 --> 00:00:36.510
so we can add a value,

00:00:36.510 --> 00:00:39.360
and that does look like
it shifted it up by one.

00:00:39.360 --> 00:00:40.970
Whatever f of x was before,

00:00:40.970 --> 00:00:42.400
we're now adding one to it

00:00:42.400 --> 00:00:45.070
so it shifts the graph up by
one, that's pretty intuitive.

00:00:45.070 --> 00:00:46.190
If we subtract one,

00:00:46.190 --> 00:00:47.790
or actually, let's subtract three.

00:00:47.790 --> 00:00:49.190
Notice, it shifted it down.

00:00:49.190 --> 00:00:51.710
The vertex was right
over here at zero, zero.

00:00:51.710 --> 00:00:54.530
Now it is at zero, negative three,

00:00:54.530 --> 00:00:55.920
so it shifted it down.

00:00:55.920 --> 00:00:58.440
And we can set up a slider here

00:00:58.440 --> 00:01:00.700
to make that a little bit clearer,

00:01:00.700 --> 00:01:03.150
so if I just replace this with,

00:01:03.150 --> 00:01:07.460
if I just replace this
with the variable k,

00:01:07.460 --> 00:01:10.010
then let me delete this little thing here,

00:01:10.010 --> 00:01:12.110
that little subscript thing that happened.

00:01:14.600 --> 00:01:17.600
Then we can add a slider k here,

00:01:17.600 --> 00:01:21.320
and this is just allowing us
to set what k is equal to,

00:01:21.320 --> 00:01:23.550
so here, k is equal to one,

00:01:23.550 --> 00:01:25.430
so this is x squared plus one,

00:01:25.430 --> 00:01:26.740
and notice, we have shifted up,

00:01:26.740 --> 00:01:29.250
and if we increase the value of k,

00:01:29.250 --> 00:01:31.490
notice how it shifts the graph up,

00:01:31.490 --> 00:01:33.620
and as we decrease the value of k,

00:01:33.620 --> 00:01:36.240
if k is zero, we're back where our vertex

00:01:36.240 --> 00:01:37.850
is right at the origin,

00:01:37.850 --> 00:01:39.770
and as we decrease the value of k,

00:01:39.770 --> 00:01:42.690
it shifts our graph down.

00:01:42.690 --> 00:01:44.120
And that's pretty intuitive,

00:01:44.120 --> 00:01:47.210
'cause we're adding or subtracting
that amount to x squared

00:01:47.210 --> 00:01:49.730
so it changes, we could say the y value,

00:01:49.730 --> 00:01:51.360
it shifts it up or down.

00:01:51.360 --> 00:01:55.310
But how do we shift to
the left or to the right?

00:01:55.310 --> 00:01:58.020
So what's interesting here
is to shift to the left

00:01:58.020 --> 00:02:01.050
or the right, we can replace our x

00:02:01.050 --> 00:02:03.650
with an x minus something,

00:02:03.650 --> 00:02:05.410
so let's see how that might work.

00:02:05.410 --> 00:02:07.680
So I'm gonna replace our x

00:02:07.680 --> 00:02:11.610
with an x minus, let's replace
it with an x minus one.

00:02:11.610 --> 00:02:12.540
What do you think is going to happen?

00:02:12.540 --> 00:02:14.810
Do you think that's going
to shift it one to the right

00:02:14.810 --> 00:02:17.120
or one to the left?

00:02:17.120 --> 00:02:19.460
So let's just put the one in.

00:02:19.460 --> 00:02:20.330
Well, that's interesting.

00:02:20.330 --> 00:02:22.800
Before, our vertex was at zero, zero.

00:02:22.800 --> 00:02:25.320
Now our vertex is at one, zero.

00:02:25.320 --> 00:02:29.230
So by replacing our x with an x minus one,

00:02:29.230 --> 00:02:32.910
we actually shifted one to the right.

00:02:32.910 --> 00:02:35.440
Now why does that make sense?

00:02:35.440 --> 00:02:36.610
Well, one way to think about it,

00:02:36.610 --> 00:02:38.160
before we put this x,

00:02:38.160 --> 00:02:41.180
before we replaced our
x with an x minus one,

00:02:41.180 --> 00:02:44.730
the vertex was when we were squaring zero.

00:02:44.730 --> 00:02:46.740
Now, in order to square zero,

00:02:46.740 --> 00:02:50.810
squaring zero happens
when x is equal to one.

00:02:50.810 --> 00:02:53.880
When x is equal to one,
you do one minus one,

00:02:53.880 --> 00:02:57.570
you get zero, and then that's
when you are squaring zero.

00:02:57.570 --> 00:02:59.840
So it makes sense that you
have a similar behavior

00:02:59.840 --> 00:03:02.750
of the graph at the vertex
now when x equals one

00:03:02.750 --> 00:03:05.190
as before you had when x equals zero.

00:03:05.190 --> 00:03:08.280
And to see how this can be generalized,

00:03:08.280 --> 00:03:10.300
let's put another variable here

00:03:10.300 --> 00:03:12.560
and let's add a slider for h.

00:03:12.560 --> 00:03:16.320
And then we can see that
when h is zero and k is zero,

00:03:16.320 --> 00:03:18.880
our function is really
then just x squared,

00:03:18.880 --> 00:03:22.540
and then if h increases,

00:03:22.540 --> 00:03:25.860
we are replacing our x with
x minus a larger value.

00:03:25.860 --> 00:03:28.090
That's shifting to the right

00:03:28.090 --> 00:03:31.130
and that is, as h decreases,

00:03:31.130 --> 00:03:32.460
as it becomes negative,

00:03:32.460 --> 00:03:33.930
that shifts to the left.

00:03:33.930 --> 00:03:36.450
Now right here, h is
equal to negative five.

00:03:36.450 --> 00:03:39.260
You typically won't see
x minus negative five.

00:03:39.260 --> 00:03:42.530
You would see that written as x plus five,

00:03:42.530 --> 00:03:45.540
so if you replace your
x's with an x plus five,

00:03:45.540 --> 00:03:49.450
that actually shifts everything
five units to the left.

00:03:49.450 --> 00:03:51.040
And of course, we can shift both

00:03:51.040 --> 00:03:53.820
of them together, like this.

00:03:53.820 --> 00:03:55.890
So here, we're shifting it up,

00:03:55.890 --> 00:03:57.410
and then we are,

00:03:57.410 --> 00:04:00.330
we could get back to our
neutral horizontal shift

00:04:00.330 --> 00:04:03.640
and then we can shift it
to the right like that.

00:04:03.640 --> 00:04:05.330
And everything we did just now

00:04:05.330 --> 00:04:09.800
is with the x squared
function as our core function,

00:04:09.800 --> 00:04:11.670
but you could do it with
all sorts of functions.

00:04:11.670 --> 00:04:13.970
You could do it with an
absolute value function.

00:04:13.970 --> 00:04:16.550
Let's do absolute value,
that's always a fun one.

00:04:16.550 --> 00:04:18.960
So instead of squaring all this business,

00:04:18.960 --> 00:04:22.160
let's have an absolute value here.

00:04:22.160 --> 00:04:25.430
So I'm gonna put an absolute, whoops.

00:04:25.430 --> 00:04:27.810
Absolute value,

00:04:27.810 --> 00:04:28.680
and there you have it.

00:04:28.680 --> 00:04:32.130
You can start at, let me
make both of these variables

00:04:32.130 --> 00:04:36.160
equal to zero, so that
would just be the graph of

00:04:36.160 --> 00:04:39.140
f of x is equal to the
absolute value of x.

00:04:39.140 --> 00:04:41.280
But let's say you wanted to shift it

00:04:41.280 --> 00:04:44.500
so that this point right over
here that's at the origin

00:04:44.500 --> 00:04:46.670
is at the point negative
five, negative five,

00:04:46.670 --> 00:04:48.330
which is right over there.

00:04:48.330 --> 00:04:51.020
So what you would do is
you would replace your x

00:04:51.020 --> 00:04:53.890
with x plus five,

00:04:53.890 --> 00:04:56.730
or you would make this h variable

00:04:56.730 --> 00:04:59.260
to negative five right
over here, 'cause notice,

00:04:59.260 --> 00:05:01.690
if you replace your h
with a negative five,

00:05:01.690 --> 00:05:05.090
inside the absolute value,
you would have an x plus five,

00:05:05.090 --> 00:05:06.700
and then if you want to shift it down,

00:05:06.700 --> 00:05:08.410
you just reduce the value of k,

00:05:08.410 --> 00:05:10.090
and if you want to shift it down by five,

00:05:10.090 --> 00:05:12.150
you reduce it by five,

00:05:12.150 --> 00:05:14.010
and you could get something like that.

00:05:14.010 --> 00:05:15.950
So I encourage you, go to desmos.com.

00:05:15.950 --> 00:05:17.230
Try this out for yourself,

00:05:17.230 --> 00:05:19.370
and really play around
with these functions

00:05:19.370 --> 00:05:22.990
to give yourself an
intuition of how things

00:05:22.990 --> 00:05:24.810
and why things shift up or down

00:05:24.810 --> 00:05:25.890
when you add a constant,

00:05:25.890 --> 00:05:28.070
and why things shift to
the left or the right

00:05:28.070 --> 00:05:32.410
when you replace your x's
with an x minus, in this case,

00:05:32.410 --> 00:05:33.490
an x minus h,

00:05:33.490 --> 00:05:37.123
but it really could be x
minus some type of a constant.


Solving equations by graphing: word problems

WEBVTT
Kind: captions
Language: en

00:00:00.400 --> 00:00:03.240
- [Instructor] We're told to
study the growth of bacteria,

00:00:03.240 --> 00:00:06.970
a scientist measure the
area in square millimeters

00:00:06.970 --> 00:00:10.750
occupied by a sample population.

00:00:10.750 --> 00:00:13.070
The growth of the
population can be modeled

00:00:13.070 --> 00:00:18.070
by f of t is equal to 24 times
e to the 0.4 times t power

00:00:20.220 --> 00:00:23.560
where t is the number of hours
since the experiment began.

00:00:23.560 --> 00:00:25.170
Here is the graph of f.

00:00:25.170 --> 00:00:27.320
So I guess f is going to be,

00:00:27.320 --> 00:00:29.190
the output of this function
is going to be the number

00:00:29.190 --> 00:00:33.060
of square millimeters
after t hours, all right?

00:00:33.060 --> 00:00:35.220
So here, we have the graph.

00:00:35.220 --> 00:00:38.340
We see how as time goes on,

00:00:38.340 --> 00:00:41.660
the square millimeters of our
little bacterial population

00:00:41.660 --> 00:00:43.420
keeps going, and it clearly is growing,

00:00:43.420 --> 00:00:44.780
or it looks like it's
growing exponentially.

00:00:44.780 --> 00:00:46.540
In fact, we know it's growing exponential

00:00:46.540 --> 00:00:48.970
'cause it's an exponential
function right over here,

00:00:48.970 --> 00:00:51.240
and they say

00:00:51.240 --> 00:00:55.050
when does the population
first occupy an area

00:00:55.050 --> 00:00:57.730
of 400 square millimeters?

00:00:57.730 --> 00:01:00.080
So pause this video and
try to figure that out.

00:01:01.060 --> 00:01:02.700
All right, and this is a screenshot

00:01:02.700 --> 00:01:04.290
from the Khan Academy exercise,

00:01:04.290 --> 00:01:07.050
so we want to say when does
the population first occupy

00:01:07.050 --> 00:01:09.150
an area of 400 square millimeters?

00:01:09.150 --> 00:01:09.983
Well, let's see.

00:01:09.983 --> 00:01:12.950
400 square millimeters
is right over there,

00:01:12.950 --> 00:01:17.720
and so it looks like after seven hours,

00:01:17.720 --> 00:01:21.970
that we are going to be 400
square millimeters or larger.

00:01:21.970 --> 00:01:24.130
So it first hits it after seven hours,

00:01:24.130 --> 00:01:26.640
so seven hours just like that.

00:01:26.640 --> 00:01:31.280
Now let's do the next few
examples that build on this.

00:01:31.280 --> 00:01:34.380
So if I go back up to the top,

00:01:34.380 --> 00:01:37.800
and now, we're told the same thing

00:01:37.800 --> 00:01:40.660
where you're using square
millimeters, square millimeters

00:01:40.660 --> 00:01:42.080
to study the growth.

00:01:42.080 --> 00:01:43.700
This is the function,

00:01:43.700 --> 00:01:45.330
but then they add this next line.

00:01:45.330 --> 00:01:47.430
Here is the graph,

00:01:47.430 --> 00:01:49.090
here is the graph of f

00:01:49.090 --> 00:01:52.170
and the graph of the line y equals 600.

00:01:52.170 --> 00:01:54.730
So they added that graph there,

00:01:54.730 --> 00:01:58.420
and then they say which
statement represents the meaning

00:01:58.420 --> 00:02:01.990
of the intersection point of the graphs?

00:02:01.990 --> 00:02:04.403
All right, so let's look
at the choices here.

00:02:05.600 --> 00:02:08.090
So and it says choose all that apply,

00:02:08.090 --> 00:02:11.320
so pause this video and
see if you can answer that.

00:02:11.320 --> 00:02:15.470
All right, so choice A
says it describes the time

00:02:15.470 --> 00:02:19.610
when the population occupies
600 square millimeters.

00:02:19.610 --> 00:02:21.410
So which statement represents the meaning

00:02:21.410 --> 00:02:23.940
of the intersection point of the graph?

00:02:23.940 --> 00:02:26.080
So they're talking about,

00:02:26.080 --> 00:02:29.490
they're talking about this
point right over there.

00:02:29.490 --> 00:02:31.270
So does that describe the time

00:02:31.270 --> 00:02:34.583
when the population occupies
600 square millimeters?

00:02:35.590 --> 00:02:39.120
So that is the time when
the population has, indeed,

00:02:39.120 --> 00:02:40.890
reached 600 square millimeters,

00:02:40.890 --> 00:02:44.090
'cause that's the line y is
equal to 600 square millimeters.

00:02:44.090 --> 00:02:47.820
So I like that choice, I will select it.

00:02:47.820 --> 00:02:48.670
The next choice.

00:02:48.670 --> 00:02:51.630
It gives the solution to the equation 24

00:02:51.630 --> 00:02:55.760
times e to the 0.4 t is equal to 600.

00:02:55.760 --> 00:02:58.140
Well, if you think about it,

00:02:58.140 --> 00:03:00.960
this right over here in blue,
we've already talked about it,

00:03:00.960 --> 00:03:05.960
that is y is equal to 24
times e to the 0.4 t power.

00:03:09.520 --> 00:03:11.830
This is y is equal to 600,

00:03:11.830 --> 00:03:16.170
so the t value at which
these two graphs equal,

00:03:16.170 --> 00:03:19.530
that means that they're both
equal to the same y value.

00:03:19.530 --> 00:03:20.980
Or another way to think about it,

00:03:20.980 --> 00:03:24.670
it means that that is equal to that

00:03:24.670 --> 00:03:29.530
or that 24 times e to the
0.4 t power is, indeed,

00:03:29.530 --> 00:03:31.280
equal to 600.

00:03:31.280 --> 00:03:33.940
So I like this too.

00:03:33.940 --> 00:03:35.840
It gives a t value where this is true,

00:03:35.840 --> 00:03:38.520
so that's the solution to that equation.

00:03:38.520 --> 00:03:41.380
It describes the situation where the area

00:03:41.380 --> 00:03:46.380
of the population occupies is
equal to the number of hours.

00:03:46.720 --> 00:03:49.000
That's definitely not the case,

00:03:49.000 --> 00:03:51.790
'cause the area here is
600 square millimeters.

00:03:51.790 --> 00:03:54.520
The hours looks like it's
a little bit after eight,

00:03:54.520 --> 00:03:56.120
so they're definitely not equal.

00:03:57.210 --> 00:04:01.330
It describes the area
the population occupies

00:04:01.330 --> 00:04:03.640
after 600 hours.

00:04:03.640 --> 00:04:04.930
No, we don't even have to look up there.

00:04:04.930 --> 00:04:07.740
This t axis doesn't even go to 600 hours,

00:04:07.740 --> 00:04:10.500
so we wouldn't select that, as well.

00:04:10.500 --> 00:04:12.360
Now let's keep building

00:04:12.360 --> 00:04:15.230
and go to the next part of this,

00:04:15.230 --> 00:04:16.923
and it says,

00:04:18.954 --> 00:04:20.203
it says,

00:04:22.210 --> 00:04:25.360
so once again, we measure the
area in square millimeters

00:04:25.360 --> 00:04:27.200
to figure out the growth
of the population.

00:04:27.200 --> 00:04:29.240
The growth of, oh, it's here.

00:04:29.240 --> 00:04:30.660
We have two populations here.

00:04:30.660 --> 00:04:33.490
It says the growth of population A

00:04:33.490 --> 00:04:36.960
can be modeled by f of t is equal to that.

00:04:36.960 --> 00:04:38.290
We've seen that already,

00:04:38.290 --> 00:04:40.700
but now they are entering
another population.

00:04:40.700 --> 00:04:44.710
The growth of population
B can be modeled by g of t

00:04:44.710 --> 00:04:46.310
is equal to this,

00:04:46.310 --> 00:04:48.830
where t is the number of hours
since the experiment began.

00:04:48.830 --> 00:04:51.400
Here, the graphs of f and g.

00:04:51.400 --> 00:04:53.120
So now we have two populations.

00:04:53.120 --> 00:04:55.760
They're both growing exponentially,
but at different rates.

00:04:55.760 --> 00:04:56.810
And then it says

00:04:56.810 --> 00:05:01.810
when do the populations
occupy the same area?

00:05:02.010 --> 00:05:03.950
It says round your answer
to the nearest integer

00:05:03.950 --> 00:05:05.350
and you could pause this video

00:05:05.350 --> 00:05:07.990
and try to think about that, if you like.

00:05:07.990 --> 00:05:09.540
Well, you can see very clearly

00:05:09.540 --> 00:05:13.680
that it looks like they
intersect right around there.

00:05:13.680 --> 00:05:15.010
So that's the point at which they're going

00:05:15.010 --> 00:05:16.610
to occupy the same area.

00:05:16.610 --> 00:05:19.870
It looks like it's about
175 square millimeters,

00:05:19.870 --> 00:05:21.080
but they're not asking about the area.

00:05:21.080 --> 00:05:22.600
They're saying when does it happen?

00:05:22.600 --> 00:05:25.660
And it looks like it happens
after about five hours.

00:05:25.660 --> 00:05:29.860
So round to the nearest
integer and say five hours.

00:05:29.860 --> 00:05:33.130
Now let's do the last part.

00:05:33.130 --> 00:05:34.980
So it's the same setup,

00:05:34.980 --> 00:05:39.140
but now they are asking
us a different question.

00:05:39.140 --> 00:05:43.230
They are asking us which
statements represent the meaning

00:05:43.230 --> 00:05:45.770
of the intersection points of the graphs?

00:05:45.770 --> 00:05:47.830
All right, so choice A it says,

00:05:47.830 --> 00:05:48.840
and then pause the video again

00:05:48.840 --> 00:05:51.100
and try to answer these on your own.

00:05:51.100 --> 00:05:53.710
All right, choice A says it
means that the populations

00:05:53.710 --> 00:05:58.710
both occupied about 180 square
millimeters at the same time.

00:05:59.210 --> 00:06:00.930
So let's see this.

00:06:00.930 --> 00:06:01.763
That looks about right.

00:06:01.763 --> 00:06:03.350
I had estimated 175,

00:06:03.350 --> 00:06:04.590
but we could call that 180,

00:06:04.590 --> 00:06:06.430
and it looks like that does roughly happen

00:06:06.430 --> 00:06:08.180
around the fifth hour.

00:06:08.180 --> 00:06:10.560
So it looks like they're
occupying the same area

00:06:10.560 --> 00:06:11.680
at around the same time,

00:06:11.680 --> 00:06:14.023
so I like that choice.

00:06:15.100 --> 00:06:16.480
It means that at the beginning,

00:06:16.480 --> 00:06:19.683
population A was larger than population B.

00:06:20.520 --> 00:06:23.320
Well, the point of
intersection doesn't tell us

00:06:23.320 --> 00:06:26.070
what population was larger to begin with.

00:06:26.070 --> 00:06:28.140
We could try to answer
it by looking over here

00:06:28.140 --> 00:06:31.310
when time t equals zero.

00:06:31.310 --> 00:06:33.570
When time t equals zero,

00:06:33.570 --> 00:06:35.920
population A is the blue curve.

00:06:35.920 --> 00:06:36.980
It is f,

00:06:36.980 --> 00:06:39.500
and so it does look like
population A was larger

00:06:39.500 --> 00:06:42.530
than population B at time t
equals zero at the beginning,

00:06:42.530 --> 00:06:44.890
but that's not what the point
of intersection tells us.

00:06:44.890 --> 00:06:47.000
So they're not just asking
us for true statements.

00:06:47.000 --> 00:06:49.200
They're saying which statements
represent the meaning,

00:06:49.200 --> 00:06:52.260
the meaning of the intersection
point of the graphs?

00:06:52.260 --> 00:06:53.093
But that doesn't tell us

00:06:53.093 --> 00:06:56.060
about what the starting situation was.

00:06:56.060 --> 00:06:59.690
It gives the solution
to equation 24 times e

00:06:59.690 --> 00:07:03.163
to the 0.4 t is equal to
nine times e to the 0.6 t.

00:07:04.840 --> 00:07:07.750
Well, we already talked about
that in the last example

00:07:07.750 --> 00:07:09.120
where we only had one curve,

00:07:09.120 --> 00:07:11.060
and that actually is the case

00:07:11.060 --> 00:07:16.060
because y is equal to
24 times e to the 0.4 t

00:07:17.890 --> 00:07:20.950
is the curve for population A,

00:07:20.950 --> 00:07:25.950
and then y is equal to
nine times e to the 0.6 t

00:07:28.460 --> 00:07:30.380
is the curve for population B,

00:07:30.380 --> 00:07:34.820
and so the point at which
these two curves intersect,

00:07:34.820 --> 00:07:39.170
that's the point at which both this

00:07:39.170 --> 00:07:42.050
where the t value that gives the same,

00:07:42.050 --> 00:07:45.090
so that this expression will
give you the same y value

00:07:45.090 --> 00:07:46.300
as this expression.

00:07:46.300 --> 00:07:49.060
Or another way to say it
is we're at the t value

00:07:49.060 --> 00:07:52.240
where this is equal to this.

00:07:52.240 --> 00:07:55.840
So it does, indeed, give the
solution to the equation,

00:07:55.840 --> 00:07:58.490
and then the last choice
is it gives the solution

00:07:58.490 --> 00:08:03.490
to the equation 24 times e to
the 0.4 t is equal to zero?

00:08:05.700 --> 00:08:07.790
No, that would happen,

00:08:07.790 --> 00:08:10.260
if you want to know
when it's equal to zero,

00:08:10.260 --> 00:08:12.560
you would look at the curve y equals zero.

00:08:12.560 --> 00:08:13.730
I'll do that in a different color,

00:08:13.730 --> 00:08:16.140
which is right over here,

00:08:16.140 --> 00:08:19.930
and see where it
intersects the function f,

00:08:19.930 --> 00:08:22.223
which is equal to 24 times e to the 0.4 t,

00:08:23.280 --> 00:08:26.140
but that's not what this point
of intersection represents,

00:08:26.140 --> 00:08:28.943
so we definitely wouldn't
pick that one either.


Solving equations by graphing: graphing calculator

WEBVTT
Kind: captions
Language: en

00:00:00.470 --> 00:00:01.750
- [Instructor] We are
told we want to solve

00:00:01.750 --> 00:00:03.560
the following equation,

00:00:03.560 --> 00:00:05.674
the negative natural log of two x

00:00:05.674 --> 00:00:08.070
is equal to two times the absolute value

00:00:08.070 --> 00:00:11.440
of x minus four, all of that minus seven.

00:00:11.440 --> 00:00:14.930
One of the solutions is x is equal to 0.5.

00:00:14.930 --> 00:00:16.450
Find the other solution.

00:00:16.450 --> 00:00:18.310
They say hint, use a graphing calculator

00:00:18.310 --> 00:00:21.170
and round your answer
to the nearest tenth.

00:00:21.170 --> 00:00:24.730
So pause this video and have
a go at this if you like,

00:00:24.730 --> 00:00:26.070
and then we'll work on this together.

00:00:26.070 --> 00:00:27.795
And I encourage you to have a go on it,

00:00:27.795 --> 00:00:29.500
(laughing) a go at it.

00:00:29.500 --> 00:00:31.570
All right now let's work on this together.

00:00:31.570 --> 00:00:33.530
Now the key here is to
realize that we might be able

00:00:33.530 --> 00:00:34.560
to solve this by graphing,

00:00:34.560 --> 00:00:38.130
or at least approximate the
solutions to this by graphing.

00:00:38.130 --> 00:00:39.680
And the way we do that is,

00:00:39.680 --> 00:00:40.800
if we have an equation,

00:00:40.800 --> 00:00:43.195
especially a hairy equation
like this, in one variable,

00:00:43.195 --> 00:00:45.810
we can set y equal to the left

00:00:45.810 --> 00:00:47.300
and then set y equal to the right

00:00:47.300 --> 00:00:49.680
and then graph each of those functions

00:00:49.680 --> 00:00:51.940
and then think about where they intersect.

00:00:51.940 --> 00:00:54.210
Because they'll intersect at an x-value

00:00:54.210 --> 00:00:56.050
that gives us the same y-value,

00:00:56.050 --> 00:00:58.410
and that means that the
two sides are the same.

00:00:58.410 --> 00:01:00.030
So what do I mean by that?

00:01:00.030 --> 00:01:01.720
Well we could set y is equal

00:01:01.720 --> 00:01:04.000
to the negative natural log of two x.

00:01:04.000 --> 00:01:08.300
So we could have one equation
or one function like that.

00:01:08.300 --> 00:01:12.000
And then we could have
another equation or function

00:01:12.000 --> 00:01:14.710
that y is equal to two times

00:01:14.710 --> 00:01:18.520
the absolute value of x
minus four minus seven.

00:01:18.520 --> 00:01:19.930
And let's see where they intersect.

00:01:19.930 --> 00:01:21.640
And the x-values where they intersect,

00:01:21.640 --> 00:01:23.770
those are going to be solutions to that.

00:01:23.770 --> 00:01:27.910
So I'm going to use Desmos
as my graphing calculator.

00:01:27.910 --> 00:01:30.010
So let's type in the two sides.

00:01:30.010 --> 00:01:31.670
So first I'll do the left side.

00:01:31.670 --> 00:01:36.250
So if y is equal to the negative

00:01:36.250 --> 00:01:40.340
natural log of two x.

00:01:40.340 --> 00:01:43.880
And actually, let me make
my color to be the same

00:01:43.880 --> 00:01:47.740
or as close as I can, so maybe
closer to that bluer color.

00:01:47.740 --> 00:01:52.740
Okay and then the net
one, I want y is equal

00:01:52.810 --> 00:01:56.520
to two times,

00:01:56.520 --> 00:01:59.830
two times the absolute value,

00:01:59.830 --> 00:02:02.610
actually I don't know
whether Desmos prefers,

00:02:02.610 --> 00:02:04.460
I'll use that, actually that works, okay,

00:02:04.460 --> 00:02:09.430
x minus four, and then I
will close my absolute value,

00:02:09.430 --> 00:02:12.130
and then I have minus seven.

00:02:12.130 --> 00:02:14.720
And I will do this in the red color

00:02:14.720 --> 00:02:16.853
so that we can keep track of things.

00:02:18.220 --> 00:02:19.800
Okay.

00:02:19.800 --> 00:02:22.200
So those are my two graphs.

00:02:22.200 --> 00:02:24.890
And now, I just need to think
about where they intersect.

00:02:24.890 --> 00:02:27.300
And one of the solutions
is x equal to 0.5.

00:02:27.300 --> 00:02:28.580
That's not the one they want.

00:02:28.580 --> 00:02:31.510
They want the other solution so to speak.

00:02:31.510 --> 00:02:33.313
So let's see.

00:02:34.280 --> 00:02:36.240
So let's see, we have one solution,

00:02:36.240 --> 00:02:38.670
actually let me zoom in a little bit.

00:02:38.670 --> 00:02:42.570
So when x is equal to
0.5, that's this solution,

00:02:42.570 --> 00:02:44.650
that's this solution right over here.

00:02:44.650 --> 00:02:48.610
It looks like y is equal to zero there.

00:02:48.610 --> 00:02:50.990
But then the other point of intersection

00:02:50.990 --> 00:02:52.920
seems to be right over here,

00:02:52.920 --> 00:02:54.880
and actually Desmos has
a nice little feature

00:02:54.880 --> 00:02:57.920
where it'll tell us that
point right over there.

00:02:57.920 --> 00:02:59.300
But you could even approximate it.

00:02:59.300 --> 00:03:01.860
You can see that x is over six

00:03:01.860 --> 00:03:04.360
and that each of these,
let's see one, two, three,

00:03:04.360 --> 00:03:07.330
four, five, each of those is .2.

00:03:07.330 --> 00:03:10.550
So it's going to be 6.2 something

00:03:10.550 --> 00:03:12.286
is what I would do, and
they want us to round

00:03:12.286 --> 00:03:14.230
to the nearest tenth anyway.

00:03:14.230 --> 00:03:15.670
So you don't even need
to use that feature,

00:03:15.670 --> 00:03:18.010
but you can see very
clearly that when x is equal

00:03:18.010 --> 00:03:21.250
to approximately 6.238,

00:03:21.250 --> 00:03:25.470
that we get a y is equal to negative 2.54.

00:03:25.470 --> 00:03:28.140
Or another way to think about it is,

00:03:28.140 --> 00:03:32.100
when x is approximately equal to,

00:03:32.100 --> 00:03:35.790
when x is approximately equal to 6.2,

00:03:36.772 --> 00:03:38.635
that the two sides of this equation

00:03:38.635 --> 00:03:40.990
are going to be equal to each other,

00:03:40.990 --> 00:03:43.320
are approximately equal to each other.

00:03:43.320 --> 00:03:45.999
And we're done, we've just solved using a,

00:03:45.999 --> 00:03:49.030
or at least approximated
a solution using graphing

00:03:49.030 --> 00:03:50.280
in a graphing calculator.


Khan Academy now offers Official Praxis® Core Prep.

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:02.180
- Hello aspiring teachers,

00:00:02.180 --> 00:00:04.940
Sal here from the
not-for-profit Khan Academy.

00:00:04.940 --> 00:00:07.570
And all of us at Khan Academy and ETS

00:00:07.570 --> 00:00:10.263
makers of the Praxis Core Exam recognize

00:00:10.263 --> 00:00:14.050
that teachers are the most essential part

00:00:14.050 --> 00:00:15.700
of a learning experience

00:00:15.700 --> 00:00:18.810
and that's why today
we're excited to announce

00:00:18.810 --> 00:00:22.040
the launch of Official Praxis Core Prep

00:00:22.040 --> 00:00:26.540
from Khan Academy and ETS to
truly level the playing field

00:00:26.540 --> 00:00:31.000
and broaden the pipeline of
teachers throughout the country.

00:00:31.000 --> 00:00:32.640
Aspiring teachers are going to be able

00:00:32.640 --> 00:00:35.490
to come to Khan Academy, get diagnostics,

00:00:35.490 --> 00:00:38.250
and work on math, reading, and writing,

00:00:38.250 --> 00:00:39.787
and understand what your gaps are

00:00:39.787 --> 00:00:41.973
and the software will adapt to allow you

00:00:41.973 --> 00:00:45.460
to work on what will be
highest leverage for you

00:00:45.460 --> 00:00:48.700
to prepare for the Praxis Core and beyond.

00:00:48.700 --> 00:00:51.730
This is completely free,
it is not for profit

00:00:51.730 --> 00:00:54.690
and we're confident that if
we give you this little help

00:00:54.690 --> 00:00:56.230
that you'll be able to run with it

00:00:56.230 --> 00:00:59.603
and impact generations
of students to come.


Approximating solutions with graphing calculator

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:02.450
- [Instructor] We're
told this is the graph

00:00:02.450 --> 00:00:04.420
of y is equal to 3/2 to the x,

00:00:04.420 --> 00:00:06.060
and that's it right over there.

00:00:06.060 --> 00:00:09.150
Use the graph to find
an approximate solution

00:00:09.150 --> 00:00:12.170
to 3/2 to the x is equal to five.

00:00:12.170 --> 00:00:14.280
So pause this video and
try to do this on your own

00:00:14.280 --> 00:00:16.530
before we work on this together.

00:00:16.530 --> 00:00:19.040
All right, now let's work on this.

00:00:19.040 --> 00:00:21.780
So they already give us a
hint of how to solve it.

00:00:21.780 --> 00:00:24.540
They have the graph of y
is equal to 3/2 to the x,

00:00:24.540 --> 00:00:27.220
they graph it right over here,
and this gives us a hint,

00:00:27.220 --> 00:00:28.800
and especially because they want us

00:00:28.800 --> 00:00:30.670
to find an approximate solution,

00:00:30.670 --> 00:00:32.280
that maybe we can solve this equation,

00:00:32.280 --> 00:00:35.430
or approximate a solution to
this equation through graphing.

00:00:35.430 --> 00:00:37.970
And the way we could do that
is we could take each side

00:00:37.970 --> 00:00:41.340
of this equation and set
them up as a function.

00:00:41.340 --> 00:00:44.080
We could set y equals to each side of it.

00:00:44.080 --> 00:00:46.730
So if we set y equals
to the left hand side,

00:00:46.730 --> 00:00:51.730
we get y is equal to 3/2 to the x power,

00:00:51.930 --> 00:00:54.350
which is what they originally
give us, the graph of that.

00:00:54.350 --> 00:00:58.110
And if we set y equal
to the right hand side,

00:00:58.110 --> 00:01:02.750
we get y is equal to five,
and we can graph that.

00:01:02.750 --> 00:01:06.960
And what's interesting here
is if we can find the x-value

00:01:06.960 --> 00:01:11.370
that gives us the same y-value
on both of these equations,

00:01:11.370 --> 00:01:13.810
well that means that those
graphs are going to intersect.

00:01:13.810 --> 00:01:16.840
And if I'm getting the same
y-value for that x-value

00:01:16.840 --> 00:01:19.018
in both of these, well then that means

00:01:19.018 --> 00:01:24.018
that 3/2 to the x is
going to be equal to five.

00:01:25.020 --> 00:01:27.500
And so we can look at where they intersect

00:01:27.500 --> 00:01:31.560
and get an approximate sense
of what x-value that is.

00:01:31.560 --> 00:01:33.660
And we can see it, at least over here,

00:01:33.660 --> 00:01:38.020
it looks like x is roughly equal to four.

00:01:38.020 --> 00:01:42.800
So x is approximately equal to four.

00:01:42.800 --> 00:01:45.750
And if we wanted to, and
we'd be done at that point,

00:01:45.750 --> 00:01:48.380
if you wanted to, you
could try to test it out.

00:01:48.380 --> 00:01:50.390
You could say hey does
that actually work out.

00:01:50.390 --> 00:01:54.360
3/2 to the fourth power,
is that equal to five.

00:01:54.360 --> 00:01:56.970
Let's see, three to the fourth is 81,

00:01:56.970 --> 00:02:01.970
two to the fourth is 16, it
gets us pretty close to five.

00:02:03.310 --> 00:02:07.110
16 times five is 80, so it's not exact,

00:02:07.110 --> 00:02:08.790
but it gets up pretty close.

00:02:08.790 --> 00:02:10.260
And if you had a graphing calculator

00:02:10.260 --> 00:02:12.930
that could really zoom in
and zoom in and zoom in,

00:02:12.930 --> 00:02:14.880
you would get a value, you would see

00:02:14.880 --> 00:02:19.880
that x is slightly different
than x equals four.

00:02:19.990 --> 00:02:21.010
But let's do another example.

00:02:21.010 --> 00:02:23.330
The key here is that we
can approximate solutions

00:02:23.330 --> 00:02:25.203
to equations through graphing.

00:02:26.390 --> 00:02:31.330
So here we are told this is
the graph of y is equal to,

00:02:31.330 --> 00:02:34.630
so we have this third degree
polynomial right over here,

00:02:34.630 --> 00:02:37.810
use the graph to answer
the following questions.

00:02:37.810 --> 00:02:40.650
How many solutions does the equation

00:02:40.650 --> 00:02:42.820
x to the third minus two x squared minus x

00:02:42.820 --> 00:02:45.770
plus one equals negative one have?

00:02:45.770 --> 00:02:48.650
Pause the video and try
to think about that.

00:02:48.650 --> 00:02:50.110
When we think about solutions to this,

00:02:50.110 --> 00:02:52.960
we can say all right, well
let's imagine two functions.

00:02:52.960 --> 00:02:56.500
One is y is equal to x to
the third minus two x squared

00:02:56.500 --> 00:03:00.120
minus x plus one, which we
already have graphed here.

00:03:00.120 --> 00:03:02.800
And let's say that the other equation,

00:03:02.800 --> 00:03:05.660
or the other function, is
y is equal to negative one.

00:03:05.660 --> 00:03:08.210
And then how many times
do these intersect?

00:03:08.210 --> 00:03:09.920
That would tell us how
many solutions we have.

00:03:09.920 --> 00:03:13.200
So that is y is equal to negative one.

00:03:13.200 --> 00:03:15.580
And so every time they intersect,

00:03:15.580 --> 00:03:18.030
that means we have a solution
to our original equation.

00:03:18.030 --> 00:03:21.770
And they intersect one,
two, and three times.

00:03:21.770 --> 00:03:24.340
So this has three solutions.

00:03:24.340 --> 00:03:26.300
What about the second situation?

00:03:26.300 --> 00:03:28.200
How many solutions does the equation

00:03:28.200 --> 00:03:30.363
all of this business equal two have?

00:03:31.290 --> 00:03:34.580
Well same drill, we could set
y equals to x to the third

00:03:34.580 --> 00:03:36.380
minus two x squared minus x plus one

00:03:36.380 --> 00:03:39.410
and then we could think
about another function,

00:03:39.410 --> 00:03:41.280
what if y is equal to two.

00:03:41.280 --> 00:03:45.640
Well y equals two would be up over there,

00:03:45.640 --> 00:03:47.510
y equals two, and we can see

00:03:47.510 --> 00:03:51.720
it only intersects y equals
all of this business once.

00:03:51.720 --> 00:03:55.380
So this is only going
to have one solution.

00:03:55.380 --> 00:03:57.440
So the key here, and
I'll just write it out,

00:03:57.440 --> 00:03:58.980
and these are screenshots
from the exercise

00:03:58.980 --> 00:04:00.750
on Khan Academy where
you'd have to type in one,

00:04:00.750 --> 00:04:03.440
or in the previous example
you would type in four,

00:04:03.440 --> 00:04:05.350
but these are examples where you can

00:04:05.350 --> 00:04:07.150
take an equation of one variable,

00:04:07.150 --> 00:04:10.740
set both sides of them
independently equal to y,

00:04:10.740 --> 00:04:13.550
graph them, and then think
about where they intersect

00:04:13.550 --> 00:04:15.550
because the x-values where they intersect

00:04:15.550 --> 00:04:18.440
will be solutions to
your original equation.

00:04:18.440 --> 00:04:20.600
And a graph is a useful way

00:04:20.600 --> 00:04:22.530
of approximating what a solution will be,

00:04:22.530 --> 00:04:24.300
especially if you have
a graphing calculator,

00:04:24.300 --> 00:04:26.083
or Desmos, or something like that.


Extraneous solutions

WEBVTT
Kind: captions
Language: en

00:00:00.350 --> 00:00:01.210
- [Instructor] In this video,

00:00:01.210 --> 00:00:03.940
we're going to talk about
extraneous solutions.

00:00:03.940 --> 00:00:06.710
If you've never heard the term before,

00:00:06.710 --> 00:00:08.750
I encourage you to review some videos

00:00:08.750 --> 00:00:11.130
on Khan Academy on extraneous solutions.

00:00:11.130 --> 00:00:13.320
But just as a bit of a refresher,
it's the idea that you do

00:00:13.320 --> 00:00:15.960
a bunch of legitimate
algebraic operations,

00:00:15.960 --> 00:00:18.670
you get a solution or
some solutions at the end,

00:00:18.670 --> 00:00:21.020
but then when you test it
in the original equation,

00:00:21.020 --> 00:00:24.020
it doesn't satisfy the original equation.

00:00:24.020 --> 00:00:25.760
And so the key of this video

00:00:25.760 --> 00:00:30.560
is why do extraneous solutions even occur?

00:00:30.560 --> 00:00:35.160
And it all is due to the
notion of reversibility.

00:00:35.160 --> 00:00:39.010
There's certain operations in algebra

00:00:39.010 --> 00:00:40.910
that you can do in one
direction, but you can't,

00:00:40.910 --> 00:00:42.460
and it'll always be true in one direction,

00:00:42.460 --> 00:00:46.990
but it isn't always true
in the other direction.

00:00:46.990 --> 00:00:49.080
And I'll show you those two operations.

00:00:49.080 --> 00:00:52.370
One is squaring and the other
is multiplying both sides

00:00:52.370 --> 00:00:54.310
by a variable expression.

00:00:54.310 --> 00:00:57.050
So let's just see the example of squaring,

00:00:57.050 --> 00:00:59.600
and then we're going to see
it in an actual scenario

00:00:59.600 --> 00:01:02.890
where you're dealing with
an extraneous solution.

00:01:02.890 --> 00:01:06.700
So we know, for example,
that if a is equal to b,

00:01:06.700 --> 00:01:09.980
I could square both sides

00:01:09.980 --> 00:01:13.790
and then a squared is going
to be equal to b squared.

00:01:13.790 --> 00:01:16.320
But the other way is not true.

00:01:16.320 --> 00:01:21.190
For example, if a squared
is equal to b squared,

00:01:21.190 --> 00:01:26.190
it is not always the case
that a is equal to b.

00:01:26.360 --> 00:01:28.050
What's a example that shows

00:01:28.050 --> 00:01:29.340
that this is not always the case?

00:01:29.340 --> 00:01:31.840
Actually, pause the video,
try to think about it.

00:01:31.840 --> 00:01:34.100
Well, negative two squared

00:01:34.100 --> 00:01:36.480
is indeed equal to two squared,

00:01:36.480 --> 00:01:40.070
but negative two is not equal to two.

00:01:40.070 --> 00:01:44.080
So this shows that you can
square both sides of an equation

00:01:45.090 --> 00:01:48.860
and deduce something that is true,

00:01:48.860 --> 00:01:50.130
but the other way around

00:01:50.130 --> 00:01:53.120
is not necessarily going to be true.

00:01:53.120 --> 00:01:56.660
Another non-reversible operation sometimes

00:01:56.660 --> 00:02:00.400
is multiplying both sides
by a variable expression.

00:02:00.400 --> 00:02:04.440
So multiplying both sides,

00:02:04.440 --> 00:02:06.260
actually, like I said,
that will make us confused,

00:02:06.260 --> 00:02:07.480
looks like an x.

00:02:07.480 --> 00:02:12.480
Multiply both sides

00:02:12.810 --> 00:02:14.390
by variable.

00:02:14.390 --> 00:02:15.270
I'll just write variable,

00:02:15.270 --> 00:02:18.900
but it could be a variable
expression as well.

00:02:18.900 --> 00:02:23.900
For example, we know
that if a is equal to b,

00:02:23.940 --> 00:02:25.800
that if we multiply both
sides by a variable,

00:02:25.800 --> 00:02:26.870
that's still going to be true.

00:02:26.870 --> 00:02:30.143
xa is going to be equal to xb.

00:02:32.150 --> 00:02:34.930
But the other, the reverse,
isn't always the case.

00:02:34.930 --> 00:02:37.890
If xa is equal to xb,

00:02:37.890 --> 00:02:41.940
is it always the case
that a is equal to b?

00:02:41.940 --> 00:02:43.230
Well, the simple answer is no,

00:02:43.230 --> 00:02:45.000
and I always encourage
you, pause this video

00:02:45.000 --> 00:02:46.790
and see if you can find an example

00:02:46.790 --> 00:02:48.380
where this doesn't work.

00:02:48.380 --> 00:02:52.330
Well, if a was two and b is three

00:02:52.330 --> 00:02:55.660
and the variable x just happened
to take on the value zero.

00:02:55.660 --> 00:02:57.200
So we know that zero times two

00:02:57.200 --> 00:02:59.820
is indeed equal to zero times three,

00:02:59.820 --> 00:03:04.380
but two is not equal to three.

00:03:04.380 --> 00:03:05.920
Now, how does all this connect

00:03:05.920 --> 00:03:08.740
to the extraneous solutions you've seen

00:03:08.740 --> 00:03:10.710
when you were solving radical equations

00:03:10.710 --> 00:03:12.510
or when you were solving some rational,

00:03:12.510 --> 00:03:15.880
or equations with rational
expressions on both sides?

00:03:15.880 --> 00:03:17.090
Well, let's look at an example.

00:03:17.090 --> 00:03:19.010
Let's solve a radical equation.

00:03:19.010 --> 00:03:22.530
If I wanted to solve the equation

00:03:22.530 --> 00:03:27.510
the square root of five x minus four

00:03:27.510 --> 00:03:29.990
is equal to x minus two,

00:03:29.990 --> 00:03:32.380
a typical first step is, hey,
let's get rid of this radical

00:03:32.380 --> 00:03:35.520
by squaring both sides.

00:03:35.520 --> 00:03:39.160
So I'm going to square both sides,

00:03:39.160 --> 00:03:41.730
and then I'm going to
get five x minus four

00:03:41.730 --> 00:03:45.030
is equal to x squared
minus four x plus four.

00:03:45.030 --> 00:03:47.060
Once again, if this looks
completely unfamiliar to you,

00:03:47.060 --> 00:03:48.810
we go into much more depth in other videos

00:03:48.810 --> 00:03:51.610
where we introduce the
idea of radical equations.

00:03:51.610 --> 00:03:55.860
And let's see, we can subtract
five x from both sides.

00:03:55.860 --> 00:03:57.450
We can add four to both sides.

00:03:57.450 --> 00:04:00.480
I'm just trying to get a
zero on the left hand side.

00:04:00.480 --> 00:04:01.740
And so I'm going to be left with

00:04:01.740 --> 00:04:06.210
zero is equal to x squared
minus nine x plus eight,

00:04:06.210 --> 00:04:11.210
or zero is equal to x minus
eight times x minus one,

00:04:11.640 --> 00:04:14.920
or we could say that x
minus one is equal to zero,

00:04:14.920 --> 00:04:18.230
or x minus eight is equal to zero.

00:04:18.230 --> 00:04:22.110
We get x equals one or x equals eight.

00:04:22.110 --> 00:04:23.520
So let's test these solutions.

00:04:23.520 --> 00:04:26.240
If x equals eight, we would get,

00:04:26.240 --> 00:04:28.000
and I'll color code it a little bit,

00:04:28.000 --> 00:04:30.770
for x equals eight, if I test
it in the original equation,

00:04:30.770 --> 00:04:35.410
I get the square root
of 36 is equal to six,

00:04:35.410 --> 00:04:38.370
which absolutely true, so that one works.

00:04:38.370 --> 00:04:40.210
But what about x equals one?

00:04:40.210 --> 00:04:45.020
I get the square root of five
times one minus four is one

00:04:45.020 --> 00:04:49.030
is equal to one minus two,
which is equal to negative one.

00:04:49.030 --> 00:04:50.760
That did not work.

00:04:50.760 --> 00:04:55.180
This right over here is
an extraneous solution.

00:04:55.180 --> 00:04:56.760
If someone said, what are all the x values

00:04:56.760 --> 00:04:59.330
that satisfy this equation,
you would not say x equals one,

00:04:59.330 --> 00:05:01.860
even though you got there with
legitimate algebraic steps.

00:05:01.860 --> 00:05:03.930
And the reason that is true is,

00:05:03.930 --> 00:05:06.220
actually, pause this video, look back.

00:05:06.220 --> 00:05:09.920
For which of these steps
does x equal one still work

00:05:09.920 --> 00:05:12.150
and what step does it not work?

00:05:12.150 --> 00:05:14.960
Well, you'll see that x equals one

00:05:14.960 --> 00:05:19.070
works for all of these equations
below this purple line.

00:05:19.070 --> 00:05:20.900
It just doesn't work for

00:05:20.900 --> 00:05:23.180
the square root of five minus four x

00:05:23.180 --> 00:05:25.410
is equal to x minus two.

00:05:25.410 --> 00:05:27.670
In fact, you could start with x minus one

00:05:27.670 --> 00:05:31.520
and then you could deduce all
the way up to this line here.

00:05:31.520 --> 00:05:32.890
But the issue here is that

00:05:32.890 --> 00:05:36.040
squaring is not a reversible operation.

00:05:36.040 --> 00:05:38.030
This is analogous to saying, hey,

00:05:38.030 --> 00:05:40.070
we know that a squared
is equal to b squared.

00:05:40.070 --> 00:05:42.860
We know that this is equal to this.

00:05:42.860 --> 00:05:46.970
But then that doesn't mean that
a is necessarily equal to b

00:05:46.970 --> 00:05:49.220
for x equals one.

00:05:49.220 --> 00:05:52.760
And we could do the same
thing with a rational,

00:05:52.760 --> 00:05:55.900
or an equation that deals
with rational expressions.

00:05:55.900 --> 00:05:58.650
So, for example, we
might have to deal with,

00:05:58.650 --> 00:06:01.150
and let me make sure I
have some space here.

00:06:01.150 --> 00:06:05.870
If I had to solve x
squared over x minus one

00:06:05.870 --> 00:06:08.770
is equal to one over x minus one,

00:06:08.770 --> 00:06:10.270
the first thing I might wanna do

00:06:10.270 --> 00:06:13.720
is multiply both sides by x minus one.

00:06:13.720 --> 00:06:18.700
So multiply by x minus one.

00:06:18.700 --> 00:06:20.340
Now, notice I'm multiplying both sides

00:06:20.340 --> 00:06:21.940
by a variable expression, so we have to be

00:06:21.940 --> 00:06:23.500
a little bit contentious now,

00:06:23.500 --> 00:06:26.070
but if I'd multiply both
sides by x minus one,

00:06:26.070 --> 00:06:28.440
I'm going to get x
squared is equal to one,

00:06:28.440 --> 00:06:30.550
or I could say that x equals one

00:06:30.550 --> 00:06:32.560
or x is equal to negative one.

00:06:32.560 --> 00:06:34.800
Well, we could test these.

00:06:34.800 --> 00:06:37.210
For x equals one, if I go up here,

00:06:37.210 --> 00:06:39.340
I'm dividing by zero on both sides.

00:06:39.340 --> 00:06:41.480
So this is an extraneous solution.

00:06:41.480 --> 00:06:44.480
They key here is that
we multiplied both sides

00:06:44.480 --> 00:06:45.670
by a variable expression.

00:06:45.670 --> 00:06:49.330
In this case, we multiplied
both sides by x minus one.

00:06:49.330 --> 00:06:50.530
You can do that.

00:06:50.530 --> 00:06:53.390
You can multiply both sides
by a variable expression,

00:06:53.390 --> 00:06:55.610
and it is a legitimate
algebraic operation.

00:06:55.610 --> 00:07:00.530
It's completely analogous to
what we saw right over here.

00:07:00.530 --> 00:07:03.670
Just because zero times two
is equal to zero times three

00:07:03.670 --> 00:07:06.370
does not mean that two is equal to three.

00:07:06.370 --> 00:07:07.620
It's completely analogous

00:07:07.620 --> 00:07:09.540
because we multiplied
by a variable expression

00:07:09.540 --> 00:07:12.370
that actually takes on the value zero

00:07:12.370 --> 00:07:14.790
when x is equal to one.

00:07:14.790 --> 00:07:18.420
So the big takeaway here
is, hopefully you understand

00:07:18.420 --> 00:07:21.870
why extraneous solutions
happen a little bit more.

00:07:21.870 --> 00:07:24.360
When you square, when
you multiply both sides

00:07:24.360 --> 00:07:26.760
by a variable expression,
completely legitimate

00:07:26.760 --> 00:07:28.500
as long as you do it properly,

00:07:28.500 --> 00:07:30.410
but it's not always the case

00:07:30.410 --> 00:07:32.980
that the reverse is true.

00:07:32.980 --> 00:07:34.970
You could add or subtract anything

00:07:34.970 --> 00:07:36.060
from both sides of an equation,

00:07:36.060 --> 00:07:38.100
and that's always going to be reversible.

00:07:38.100 --> 00:07:41.020
And so that's not going to
lead to extraneous solutions.

00:07:41.020 --> 00:07:42.390
You could multiply or divide

00:07:42.390 --> 00:07:45.360
by a non-zero constant value.

00:07:45.360 --> 00:07:48.200
That's also not going to
lead to anything shady,

00:07:48.200 --> 00:07:49.750
but if you're squaring both sides

00:07:49.750 --> 00:07:52.200
or multiplying both sides
by a variable expression,

00:07:52.200 --> 00:07:53.950
you should be a little bit careful.


Solving equations by graphing: intro

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:02.450
- [Instructor] We're
told, this is the graph

00:00:02.450 --> 00:00:04.420
of y is equal to 3/2 to the x.

00:00:04.420 --> 00:00:06.060
And that's it right over there.

00:00:06.060 --> 00:00:09.150
Use the graph to find
an approximate solution

00:00:09.150 --> 00:00:12.170
to 3/2 to the x is equal to five.

00:00:12.170 --> 00:00:14.280
So pause this video and
try to do this on your own

00:00:14.280 --> 00:00:16.530
before we work on this together.

00:00:16.530 --> 00:00:19.040
All right, now let's work on this.

00:00:19.040 --> 00:00:21.780
So they already give us a
hint of how to solve it.

00:00:21.780 --> 00:00:24.540
They have the graph of y
is equal to 3/2 to the x.

00:00:24.540 --> 00:00:25.950
They graph it right over here.

00:00:25.950 --> 00:00:28.140
And this gives us a hint,
and especially because it's,

00:00:28.140 --> 00:00:30.670
they want us to find an
approximate solution,

00:00:30.670 --> 00:00:32.280
that maybe we can solve this equation

00:00:32.280 --> 00:00:35.430
or approximate a solution to
this equation through graphing.

00:00:35.430 --> 00:00:37.350
And the way we could do
that is we could take

00:00:37.350 --> 00:00:41.340
each side of this equation
and set them up as a function.

00:00:41.340 --> 00:00:44.080
We could set y equals to each side of it.

00:00:44.080 --> 00:00:46.730
So if we set y equals
to the left-hand side,

00:00:46.730 --> 00:00:51.730
we get y is equal to 3/2 to the x power,

00:00:51.930 --> 00:00:53.150
which is what they originally give us,

00:00:53.150 --> 00:00:54.350
the graph of that.

00:00:54.350 --> 00:00:57.810
And if we set y equal
to the right-hand side,

00:00:57.810 --> 00:01:00.290
we get y is equal to five.

00:01:00.290 --> 00:01:02.750
And we can graph that.

00:01:02.750 --> 00:01:05.630
And what's interesting
here is if we can find

00:01:05.630 --> 00:01:09.110
the x-value that gives us the same y-value

00:01:09.110 --> 00:01:11.370
on both of these equations,

00:01:11.370 --> 00:01:13.810
well that means that those
graphs are going to intersect.

00:01:13.810 --> 00:01:16.840
And if I'm getting the same
y-value for that x-value

00:01:16.840 --> 00:01:19.080
in both of these, well then that means

00:01:19.080 --> 00:01:22.040
that 3/2 to the x

00:01:22.040 --> 00:01:25.020
is going to be equal to five.

00:01:25.020 --> 00:01:27.500
And so we could look
at where they intersect

00:01:27.500 --> 00:01:31.560
and get an approximate sense
of what x-value that is.

00:01:31.560 --> 00:01:33.660
And we can see it, at least over here,

00:01:33.660 --> 00:01:38.020
it looks like x is roughly equal to four.

00:01:38.020 --> 00:01:42.800
So x is approximately equal to four.

00:01:42.800 --> 00:01:45.750
And if we wanted to, and
we'd be done at that point.

00:01:45.750 --> 00:01:48.380
If you wanted to, you
could try to test it out.

00:01:48.380 --> 00:01:50.307
You could say, "Hey does
that actually work out?

00:01:50.307 --> 00:01:54.360
"3/2 to the fourth power,
is that equal to five?"

00:01:54.360 --> 00:01:56.970
Let's see, three to the fourth is 81.

00:01:56.970 --> 00:01:59.210
Two to the fourth is 16.

00:01:59.210 --> 00:02:03.310
It gets us, it gets us
pretty close to five.

00:02:03.310 --> 00:02:05.970
16 times five is 80.

00:02:05.970 --> 00:02:08.790
So it's not exact, but
it gets us pretty close.

00:02:08.790 --> 00:02:10.260
And if you had a graphing calculator

00:02:10.260 --> 00:02:12.930
that could really zoom in
and zoom in and zoom in,

00:02:12.930 --> 00:02:16.060
you would get a value,
you would see that x is

00:02:16.060 --> 00:02:19.990
slightly different than x equals four.

00:02:19.990 --> 00:02:21.010
But let's do another example.

00:02:21.010 --> 00:02:23.330
The key here is that we
can approximate solutions

00:02:23.330 --> 00:02:25.203
to equations through graphing.

00:02:26.390 --> 00:02:29.650
So here we are told, this is the graph

00:02:29.650 --> 00:02:33.320
of y is equal to, so we have
this third-degree polynomial

00:02:33.320 --> 00:02:34.630
right over here.

00:02:34.630 --> 00:02:37.810
Use the graph to answer
the following questions.

00:02:37.810 --> 00:02:40.338
How many solutions does the equation

00:02:40.338 --> 00:02:43.410
x to the third minus two
x squared minus x plus one

00:02:43.410 --> 00:02:45.770
equals negative one have?

00:02:45.770 --> 00:02:47.676
Pause this video and
try to think about that.

00:02:47.676 --> 00:02:50.110
Well when we think
about solutions to this,

00:02:50.110 --> 00:02:52.960
we could say, all right, well
let's imagine two functions,

00:02:52.960 --> 00:02:56.500
one is y is equal to x to
the third minus two x squared

00:02:56.500 --> 00:03:00.120
minus x plus one, which we
already have graphed here.

00:03:00.120 --> 00:03:02.800
And let's say that the other equation

00:03:02.800 --> 00:03:05.660
or the other function is y
is equal to negative one.

00:03:05.660 --> 00:03:08.210
And then how many times
do these intersect?

00:03:08.210 --> 00:03:09.920
That would tell us how
many solutions we have.

00:03:09.920 --> 00:03:13.200
So that is y is equal to negative one.

00:03:13.200 --> 00:03:15.580
And so every time they intersect,

00:03:15.580 --> 00:03:18.030
that means we have a solution
to our original equation.

00:03:18.030 --> 00:03:21.770
And they intersect one,
two, and three times.

00:03:21.770 --> 00:03:24.340
So this has three solutions.

00:03:24.340 --> 00:03:26.300
What about the second situation?

00:03:26.300 --> 00:03:29.040
How many solutions does the
equation all of this business

00:03:29.040 --> 00:03:30.494
equal two have?

00:03:30.494 --> 00:03:33.953
Well same drill, we could set y equals to

00:03:33.953 --> 00:03:36.380
x to the third minus two x
squared minus x plus one.

00:03:36.380 --> 00:03:39.410
And then we could think
about another function,

00:03:39.410 --> 00:03:41.280
what if y is equal to two?

00:03:41.280 --> 00:03:45.640
Well y equals two would be up over there,

00:03:45.640 --> 00:03:46.860
y equals two.

00:03:46.860 --> 00:03:49.830
And we could see it
only intersects y equals

00:03:49.830 --> 00:03:51.720
all of this business once.

00:03:51.720 --> 00:03:55.380
So this is only going
to have one solution.

00:03:55.380 --> 00:03:57.440
So the key here, and
I'll just write it out,

00:03:57.440 --> 00:03:58.980
and these are screenshots
from the exercise

00:03:58.980 --> 00:04:00.750
on Khan Academy where
you'd have to type in one,

00:04:00.750 --> 00:04:03.440
or in the previous example,
you would type in four.

00:04:03.440 --> 00:04:06.130
But these are examples where
you can take an equation

00:04:06.130 --> 00:04:08.830
of one variable, set both sides of them

00:04:08.830 --> 00:04:12.090
independently equal to y, graph them,

00:04:12.090 --> 00:04:13.550
and then think about where they intersect.

00:04:13.550 --> 00:04:15.550
Because the x-values where they intersect

00:04:15.550 --> 00:04:18.440
will be solutions to
your original equation.

00:04:18.440 --> 00:04:21.690
And a graph is a useful
way of approximating

00:04:21.690 --> 00:04:23.290
what a solution will be,
especially if you have

00:04:23.290 --> 00:04:26.083
a graphing calculator or
Desmos or something like that.