WEBVTT
Kind: captions
Language: en
00:00:00.000 --> 00:00:03.720
Half of sinx and you get 3sinx.
00:00:04.460 --> 00:00:06.180
What does that do to the graph?
00:00:06.640 --> 00:00:10.540
Because you can transform any of the graphs, with the rules of f(x)
00:00:10.800 --> 00:00:13.200
Now let's do it on the graph we know..
00:00:13.200 --> 00:00:14.660
You can do the x squared
00:00:15.040 --> 00:00:16.480
You can do the square root of x
00:00:16.580 --> 00:00:17.140
So sin(x)
00:00:17.860 --> 00:00:18.900
Well regular sin(x)
00:00:24.560 --> 00:00:25.780
Kinda looks like that
00:00:30.720 --> 00:00:31.880
So 3 sin(x),
00:00:32.020 --> 00:00:33.060
Oh and this is pi
00:00:34.080 --> 00:00:35.120
and this is 2pi
00:00:36.240 --> 00:00:37.440
and that's minus 1
00:00:37.500 --> 00:00:38.100
And that's 1.
00:00:38.100 --> 00:00:39.740
That's one of my better sine graphs
00:00:39.800 --> 00:00:41.400
Didn't it come out okay?
00:00:43.960 --> 00:00:46.040
You can come up and draw it for us
00:00:53.280 --> 00:00:55.040
So what's 3 sin(x) look like?
00:00:55.220 --> 00:00:55.720
well
00:00:55.720 --> 00:01:00.420
It's going to look just like sin(x), because we stretched it vertically by a factor of 3
00:01:02.860 --> 00:01:05.600
so now... it'd be like that
00:01:06.380 --> 00:01:07.420
That's still pi
00:01:08.540 --> 00:01:10.420
still 2pi, now we go up to 3.
00:01:11.940 --> 00:01:12.920
and down to negative 3
00:01:38.420 --> 00:01:40.340
okay now we can do other stuff with a graph
00:01:45.440 --> 00:01:48.220
I'll uncover that in a second
00:02:03.400 --> 00:02:04.660
I can do something like that
00:02:06.040 --> 00:02:07.640
So remember that's sin(x)
00:02:08.740 --> 00:02:10.380
All 3sin(x) graph is
00:02:12.860 --> 00:02:16.060
You took the sin(x) graph and you just stretched it
00:02:16.220 --> 00:02:18.020
So it's now 3 up and 3 down
00:02:18.980 --> 00:02:21.540
Well what is I do sin of x minus pi, over 4
00:02:24.220 --> 00:02:28.120
Now I am going to take this sin(x) graph and I am going to move it pi/4 to the right
00:02:28.120 --> 00:02:31.440
We're going to transform it with a horizontal shift
00:02:33.720 --> 00:02:35.980
Horizontal means you can push it 4 tothe right
00:02:36.300 --> 00:02:38.540
Essentially all of these numbers
00:02:38.540 --> 00:02:41.780
are going to say pi/4 now. I'm going to cover that for a minute
00:02:45.240 --> 00:02:47.760
So now, the sine graph kind of looks like that
00:02:50.440 --> 00:02:51.880
Where that's pi/4
00:02:52.980 --> 00:02:55.060
That's pi plus pi/4
00:02:57.560 --> 00:02:59.360
This is five pi/4
00:03:01.680 --> 00:03:05.120
Thats 2pi plus pi/4
00:03:06.280 --> 00:03:07.720
So it will be 9pi/4
00:03:10.320 --> 00:03:11.840
But it still goes up to 1
00:03:13.060 --> 00:03:14.500
and down to negative 1
00:03:16.100 --> 00:03:17.380
Cause I took sin of x
00:03:17.720 --> 00:03:19.440
I could mutiply by 3
00:03:19.760 --> 00:03:21.640
So that doesn't affect the x values
00:03:21.640 --> 00:03:23.920
It just makes it taller in both directions
00:03:24.380 --> 00:03:26.220
I could shift it to the right
00:03:26.560 --> 00:03:28.800
Which doesn't effect the y values
00:03:29.360 --> 00:03:31.260
It only effects where it shows up on the x axis
00:03:31.260 --> 00:03:33.240
And of course, what am I going to do next?
00:03:35.600 --> 00:03:37.360
I am going to do both, okay?
00:03:48.860 --> 00:03:51.280
We're going to do 3 sine
00:03:53.120 --> 00:03:54.520
x minus pi/4
00:04:07.080 --> 00:04:08.440
I get something like that
00:04:11.000 --> 00:04:12.040
Now it goes up to 3
00:04:12.060 --> 00:04:13.180
And down minus 3
00:04:13.500 --> 00:04:15.140
and moved pi/4 to the right
00:04:16.440 --> 00:04:17.400
How are we doing so far
00:04:18.780 --> 00:04:19.660
Loving this?
00:04:20.020 --> 00:04:20.820
Not loving?
00:04:25.400 --> 00:04:29.400
Ah good question! How many points should we show on the test?
00:04:29.400 --> 00:04:32.580
Enough that it is clear to us that you know what you are doing
00:04:32.880 --> 00:04:34.400
So I would say at least 3
00:04:35.340 --> 00:04:37.020
Somewhere between 3 and 5
00:04:38.960 --> 00:04:41.440
Good the other two things you can show
00:04:42.220 --> 00:04:44.200
Where the maximum and minimum are
00:04:44.200 --> 00:04:48.940
Which you want to do is you want to demonstrate to us if you understand how to do the transformation
00:04:59.300 --> 00:05:01.760
You can lave the graph the same and change the numbers
00:05:02.260 --> 00:05:03.420
You can rearrange the graph
00:05:03.420 --> 00:05:05.220
We wont scale it, as far as I know we wont scale it
00:05:05.220 --> 00:05:07.200
I dont even know if there will be a graph
00:05:07.300 --> 00:05:10.580
We may give you a graph and say whats been done to it
00:05:11.220 --> 00:05:12.660
Which is the same idea
00:05:13.160 --> 00:05:13.660
or not
00:05:15.660 --> 00:05:16.700
You dont like that idea?
00:05:17.540 --> 00:05:18.040
no?
00:05:22.300 --> 00:05:25.100
I am pretty sure we will have some sort of graph on the test
00:05:27.660 --> 00:05:31.660
I think so, I am giving you a non definite answer abou the exam
00:05:37.700 --> 00:05:39.500
Why dont we try one and see what happens?
00:05:43.060 --> 00:05:46.260
if its sine or cosine theres more than one answer
00:06:01.600 --> 00:06:04.200
Okay thats a sign graph, what did I do to it?
00:06:05.140 --> 00:06:06.900
Thats a 5 and thats a minus 5
00:06:06.980 --> 00:06:09.860
That says pi over 6, 7pi over 6 and 13 pi over 6
00:06:17.440 --> 00:06:18.240
Doing okay?
00:06:18.880 --> 00:06:20.320
Well look what we did?
00:06:21.000 --> 00:06:24.840
Its up 5 and down 5, so thats a multiple of 5
00:06:24.940 --> 00:06:27.180
and I moved it pi over 6 to the right
00:06:27.940 --> 00:06:30.760
So its y equals pi sine
00:06:34.060 --> 00:06:37.040
x minus pi over 6. That's not so bad right?
00:06:37.560 --> 00:06:40.100
Seems scarier then it was. Let's do a couple more
00:07:02.820 --> 00:07:04.260
Alright do that graph
00:07:17.900 --> 00:07:18.620
Ahh okay so
00:07:22.520 --> 00:07:24.520
We will do it here so you can see
00:07:25.060 --> 00:07:27.300
So what you do, you want to take the normal graph
00:07:30.480 --> 00:07:32.320
and you want to transform it
00:07:32.620 --> 00:07:33.420
This is zero
00:07:34.300 --> 00:07:38.600
This is pi, 2pi. Positive 1 and negative 1
00:07:38.720 --> 00:07:41.600
This is the graph y equals sin(x)
00:07:41.600 --> 00:07:43.620
and now we are going to transform it
00:07:44.660 --> 00:07:45.540
to a new graph
00:07:46.400 --> 00:07:50.600
y equals one half, sin(x + pi/4)
00:07:51.260 --> 00:07:53.260
So what does the x + pi/4 do?
00:07:55.280 --> 00:07:56.900
Move it to the left. How much?
00:07:57.600 --> 00:07:58.460
Pi/4.
00:07:58.860 --> 00:08:03.520
The whole thing is shifted to the left by some amount right?
00:08:05.120 --> 00:08:06.060
By the way it keeps going
00:08:06.540 --> 00:08:08.300
We are just doing one curve
00:08:09.700 --> 00:08:14.380
So what happened? So this point zero moved over here
00:08:15.220 --> 00:08:17.180
So its now a negative pi/4
00:08:19.560 --> 00:08:22.800
this point pi moved to the right
00:08:23.520 --> 00:08:27.840
how much? To the left, sorry. So you took pi and pi/4 to the left
00:08:27.840 --> 00:08:29.520
So you subtracted pi/4.
00:08:31.220 --> 00:08:31.940
You took pi
00:08:32.960 --> 00:08:34.960
and you took away pi/4
00:08:38.040 --> 00:08:40.580
The other way you can think of it, you can say
00:08:41.300 --> 00:08:42.020
Same thing
00:08:42.660 --> 00:08:48.140
You can say x + pi/4 becomes pi and you can subtract
00:08:52.040 --> 00:08:54.960
Pi - pi/4. How do you find pi?
00:08:54.960 --> 00:08:56.260
Remember its just fractions
00:08:56.820 --> 00:09:01.120
One minus a quarter, 1 minus a quarter is 3/4
00:09:02.140 --> 00:09:04.700
So thats 3pi/4. So thats this point
00:09:10.640 --> 00:09:14.560
And then 2pi also moves to the left
00:09:15.360 --> 00:09:19.000
So that one is 2pi minus pi/4
00:09:19.820 --> 00:09:23.420
Which is 8/4 minus 1/4, which is 7pi/4
00:09:24.620 --> 00:09:26.100
very good. Okay?
00:09:27.220 --> 00:09:28.420
This is fractions
00:09:31.180 --> 00:09:32.140
This is why you learn fractions
00:09:46.740 --> 00:09:48.340
Uhh no that wouldnt work
00:09:49.040 --> 00:09:50.720
Okay I wont say it out loud
00:09:51.020 --> 00:09:58.380
essentially what you want to is you want to think of every point has this number being taken away from it
00:09:58.720 --> 00:10:03.360
Because you are taking the old access and moving it pi/4 to the left
00:10:04.880 --> 00:10:11.880
Now what happens to the amplitude, well instead of going up one and down one, it now moves up a half and down a half
00:10:14.180 --> 00:10:18.140
So we took this and we squashed it by a factor of a half
00:10:40.020 --> 00:10:41.460
Lets do a cosine graph
00:10:41.460 --> 00:10:45.880
By the way sine and cosine graphs are really the same thing of course
00:10:54.700 --> 00:10:56.220
What if I had that graph
00:10:59.720 --> 00:11:01.400
Y equals 2 cosine x minus 1
00:11:07.220 --> 00:11:10.180
Well lets figure out what cosine x looks like
00:11:12.500 --> 00:11:13.060
Cosine x
00:11:26.480 --> 00:11:34.880
Okay, cosine is just like a sine graph but it is shifted and it starts at the top and goes down and up and keeps going
00:11:36.920 --> 00:11:40.080
and theres and pi and this is a pi over 2, ther
00:11:41.660 --> 00:11:42.920
Thats a 3pi over 2
00:11:45.840 --> 00:11:47.200
So what does the 2 do?
00:11:47.200 --> 00:11:49.920
It changes it and stretches up 2 and down 2
00:11:50.940 --> 00:11:52.360
And now what about that take away 1?
00:11:53.620 --> 00:11:54.220
What does that do?
00:11:57.280 --> 00:11:58.840
The whole thing moves down one, perfect
00:12:01.560 --> 00:12:03.620
So one thing you can do to make your life easier
00:12:03.920 --> 00:12:06.120
If its shifted down one
00:12:07.700 --> 00:12:09.940
Draw a dotted line at negative one
00:12:12.180 --> 00:12:14.100
The whole graph just drops 1.
00:12:14.740 --> 00:12:17.380
And then it goes up 2 and down 2
00:12:17.980 --> 00:12:21.260
It goes up to positive 1 and then down to negative 3
00:12:23.160 --> 00:12:28.120
Okay because before it was going up 2 and down 2 and drop everything by 1.
00:12:30.620 --> 00:12:31.600
Everybody see that?
00:12:32.460 --> 00:12:36.020
So you go to the middle and go up 2 and dow 2
00:12:38.600 --> 00:12:40.040
and it looks like that
00:12:43.300 --> 00:12:50.180
What did we do to this graph? well we dropped it and then we stretched it. or we stretched it and dropped it
00:12:54.380 --> 00:12:55.300
You can do it either way
00:12:56.280 --> 00:12:58.560
As long as nothing else is going on you can do it either way
00:13:15.420 --> 00:13:17.240
There are some of you that pay more tention, yes?
00:13:19.440 --> 00:13:24.420
So first i took this graph,, when I multiplied it by 2
00:13:26.200 --> 00:13:28.620
Now I went up 2 and down 2
00:13:29.180 --> 00:13:35.520
and then I subttract 1 from everything so this drops to one and that drops to negative 3
00:13:36.120 --> 00:13:40.400
So I took this x axis and I dropped it by 1
00:13:41.820 --> 00:13:43.180
so the whole curve comes down 1
00:13:43.840 --> 00:13:49.040
and how did I know that happened? Well thats what that x minus 1 does. F of x and subtract the 1 from
00:14:04.320 --> 00:14:16.660
Oh yes, right. So first you have to multiply by the amplitude, I did that last time, right. First you have to do the amplitude, multiply by 2 and negative 2 then you subtract 1
00:14:20.000 --> 00:14:24.080
Good question, what would happen if it was a negative 2 cosine
00:14:25.840 --> 00:14:28.160
Not an inverse, square upside down
00:14:28.800 --> 00:14:31.360
Inverse is coming in just a few moments
00:14:32.360 --> 00:14:34.640
Okay when you say inverse you think reciprocal
00:14:39.880 --> 00:14:42.120
Dont think we really covered that
00:14:43.400 --> 00:14:44.280
Probably not
00:14:45.720 --> 00:14:53.700
Cant make promises right now, I can give you better answers on monday, by monday we should have this test all ready to go. I hope we have it by friday
00:14:54.880 --> 00:14:56.080
Lose people today
00:15:25.840 --> 00:15:27.680
Lets do another one of these
00:15:54.760 --> 00:15:56.840
No we have three things going on
00:15:57.340 --> 00:16:01.340
I think 3 is a very high level of san, but some maybe in the mood
00:16:09.460 --> 00:16:12.660
y equals 2cosine x minus pi/4, plus 1
00:16:12.900 --> 00:16:14.980
So what do we have going on here?
00:16:16.800 --> 00:16:23.800
Now the question is do you do the x transformation first, or the y transformation first? I always do the x ones first
00:16:26.120 --> 00:16:28.520
So what I do is take one graph at a time
00:16:29.060 --> 00:16:33.380
If you really want to do these, say first you take the cosine graph
00:16:35.040 --> 00:16:35.840
Kind of looks like that
00:16:39.100 --> 00:16:41.080
Thats just a regular cosine graph
00:16:41.960 --> 00:16:44.260
and move it pi/4 to the right
00:16:48.580 --> 00:16:49.080
So now
00:16:50.820 --> 00:16:52.120
My graph kind of looks like that
00:16:54.200 --> 00:16:56.880
So instead of starting it at zero it starts at pi/4
00:16:57.540 --> 00:17:01.680
Instead of having the middle pi, the middle is now 5 pi/4
00:17:04.520 --> 00:17:09.040
And instead of 2 pi, its at 9 pi/4. And you keep going
00:17:09.040 --> 00:17:13.180
We are just taking a small piece of the graph, sine and cosines go on forever
00:17:31.060 --> 00:17:34.180
Now what do you do? You multiply everything by 2
00:17:34.760 --> 00:17:36.920
So me can sort of just make that a 2
00:17:39.540 --> 00:17:46.960
So we stretched it, instead of going from one to one it goes from two to two. And now we have to take the whole thing and move it up 1
00:17:55.460 --> 00:17:56.980
So its going to go up to 3
00:18:01.460 --> 00:18:02.900
and down to negative 1
00:18:06.820 --> 00:18:08.020
and thats pi/4
00:18:09.280 --> 00:18:10.380
thats 5pi/4
00:18:12.140 --> 00:18:13.420
and thats 9pi/4
00:18:16.620 --> 00:18:17.500
got that one?
00:18:19.380 --> 00:18:28.160
and that one was y equals 2cosine x minus pi/4 plus 1
00:18:37.600 --> 00:18:42.560
Its okay if its not connected to axis. Because the truth is we moved the axis
00:18:43.740 --> 00:18:45.340
Whats really going on is
00:18:46.260 --> 00:18:47.700
this is the new origin
00:18:49.500 --> 00:18:53.300
we moved the origin pi over four to the right and up 1
00:18:53.760 --> 00:18:55.040
So the chip is there
00:18:59.540 --> 00:19:03.060
We subtract, x and x and y and the y axis
00:19:03.840 --> 00:19:08.760
Well you should take enough points to show us that you know whats going on in the transformation
00:19:15.880 --> 00:19:17.560
Now we have to do inverses
00:19:18.420 --> 00:19:20.420
So lets learn how to inverse things
00:19:24.800 --> 00:19:28.000
The last major topic thats going to be on the exam
00:19:28.900 --> 00:19:29.760
I know
00:19:30.380 --> 00:19:34.360
So does everyone have this copied down so I canerase it?
00:19:35.160 --> 00:19:36.800
yes? okay
00:19:47.020 --> 00:19:48.560
Turn my back and people leave the room
00:20:11.960 --> 00:20:14.880
So we can do a domain and a range, Okay?
00:20:15.520 --> 00:20:20.340
So an inverse function basically works whatever we are doing in verse
00:20:20.580 --> 00:20:23.060
But you have squares and square roots
00:20:24.120 --> 00:20:27.860
So 4 squared is 16, square root of 16 is 4
00:20:28.540 --> 00:20:32.140
So you are reversing the operation, called an inverse
00:20:33.560 --> 00:20:37.380
However, we can run into all sorts of interesting things when you do inverse
00:20:37.380 --> 00:20:44.060
The basic idea of an inverse, by switching..you start with a function and you take it and put in a domain a you get the range
00:20:44.060 --> 00:20:47.600
In an inverse function you put in the range and you get out a domain
00:20:48.760 --> 00:20:54.680
So you think of it as before you went through x,y
00:20:55.440 --> 00:20:58.720
and now you are going through y,x
00:20:59.620 --> 00:21:01.620
and thats what an inverse does
00:21:03.240 --> 00:21:08.680
Switches the number, x are the points in the domain and y are the points in the range
00:21:08.920 --> 00:21:10.200
It is a function of x
00:21:10.660 --> 00:21:13.740
So when you are doing an inverse you are reversing the operation
00:21:15.340 --> 00:21:18.780
So how does that show up practical?Well first of all
00:21:20.420 --> 00:21:25.320
YOu think about x and y, we go out x say you have y.
00:21:26.220 --> 00:21:29.680
Now you say you go out y and go up x
00:21:31.140 --> 00:21:34.020
and if you repeat that a lot, what happens is
00:21:34.980 --> 00:21:40.460
you are reflecting acorss the line y equals x
00:21:40.460 --> 00:21:45.260
So now you get a line y equals x. You sort of flip the graph over
00:21:45.260 --> 00:21:48.020
So if you had a graph that looks like that
00:21:48.980 --> 00:21:52.000
The new graph...will look like this
00:21:52.540 --> 00:21:58.760
And if this is a dotted line, this paper gets folded on the dotted line, this graph will go on top of that graph
00:21:59.840 --> 00:22:01.840
Okay and thats what inverse do
00:22:03.400 --> 00:22:13.000
So an inverse, you have a function thats the original function takes a point and take a point in the domain and get a range. Now you go the other way
00:22:17.760 --> 00:22:19.520
So now lets do a pack of them
00:22:19.600 --> 00:22:26.600
So one thing is, if you want to invert a function you literally take it and you reflex it over the line y equals x.
00:22:26.620 --> 00:22:29.180
So this way or this way it doesnt matter
00:22:31.600 --> 00:22:35.940
But notice something important, this function passed a vertical line test
00:22:36.620 --> 00:22:38.780
and this function passed the vertical line test
00:22:39.240 --> 00:22:40.920
Suppose I take a parabola
00:22:43.480 --> 00:22:46.760
I start a parabola and pass the vertical line test
00:22:47.440 --> 00:22:52.720
And now I flip it across the line y equals x and now I get this parabola
00:22:53.680 --> 00:22:56.220
That parabola is going to fail a vertical line test
00:22:56.780 --> 00:22:58.500
So thats not going to be a function of x
00:22:59.320 --> 00:23:02.760
and now if I draw a vertical line I'll get two answers
00:23:02.760 --> 00:23:07.720
For every x value. Remember for a function you plug in an x value and I'll get one y value
00:23:07.760 --> 00:23:15.520
Stony brook Student id number, you plug in one stony brook studen and you'll get one id number
00:23:15.520 --> 00:23:16.920
If you plug in the id number you only get one student
00:23:20.820 --> 00:23:23.940
You take that and you fold it and you get that, its not very hard
00:23:24.860 --> 00:23:25.940
Practice on a piece of paper
00:23:26.920 --> 00:23:30.820
Take a paper and turn it and literally look through it and see what the inverse would look like
00:23:31.020 --> 00:23:33.740
okay, you take two corners and you flip it
00:23:37.340 --> 00:23:40.060
Alright so if you have it flip it like that
00:23:41.320 --> 00:23:44.360
Now its switched to the functions inverse
00:23:45.540 --> 00:23:55.340
Thats the domain and range. So this is f of x equals x squared or y. This is basically x equals y squared or that means y equals x squared
00:23:56.360 --> 00:23:57.320
The problem is
00:24:00.440 --> 00:24:03.800
The first one is a function of x, and the second one is not
00:24:07.340 --> 00:24:10.860
So how would you know if you just looked at a function if it would fail a vertical line test?
00:24:11.160 --> 00:24:14.360
The inverse will fail the horizontal line test.
00:24:14.440 --> 00:24:16.360
You do a horizontal line test
00:24:16.440 --> 00:24:23.160
So you take an original function and take the horizontal line test, which means the horizontal line can not cut it twice
00:24:24.160 --> 00:24:28.460
If it fails a horizontal line test then the inverse will not be a function
00:24:30.440 --> 00:24:34.840
So if you take a function and pass the vertical line test thats nice
00:24:34.840 --> 00:24:39.920
but in order for its inverse of the function, it also has to pass the horizontal line testt
00:24:41.980 --> 00:24:46.540
and a function that passes both is called a one-to-one function
00:24:48.400 --> 00:24:51.260
So if you pass the horizontal line test
00:24:58.620 --> 00:25:00.980
will tell you if there inverse is a function
00:25:16.300 --> 00:25:17.960
and if f of x passes both
00:25:22.980 --> 00:25:25.300
vertical and horizontal line test
00:25:25.600 --> 00:25:27.520
Notice how to spell vertical
00:25:30.440 --> 00:25:32.560
i dont know know how people spell it without v
00:25:40.900 --> 00:25:43.540
Then it is called a one-to-one function
00:25:55.180 --> 00:25:57.560
A one-to-one function passes them both
00:26:00.040 --> 00:26:01.000
So far so good?
00:26:02.380 --> 00:26:02.880
So
00:26:05.920 --> 00:26:12.180
Thats what an inverse is graphically , now lets find how to find one algebraically
00:26:22.520 --> 00:26:29.180
Suppose I have the function f of x equals 5x mins 4
00:26:29.480 --> 00:26:33.040
Nice, Im going to do some nasty.. over 7
00:26:33.040 --> 00:26:36.880
Not very hard and I want to find the inverse of that function
00:26:47.180 --> 00:26:47.980
How do I do it
00:26:49.360 --> 00:26:50.640
Well remember what
00:27:01.220 --> 00:27:03.220
Id say you were 98 percent
00:27:04.540 --> 00:27:08.260
So If you take your time you will get the right answers. If you didnt hear her thats okay
00:27:08.260 --> 00:27:09.680
So we will show you anyway
00:27:10.840 --> 00:27:16.680
Remember what I say what was going on when we were finding the function verse the inverse
00:27:16.680 --> 00:27:21.140
You are going from x comma y to y comma x. You are switching the x and y coordinates
00:27:21.140 --> 00:27:23.920
So one of the easiest things to do it, write this
00:27:24.600 --> 00:27:27.240
A y equals 5x minus 4 over 7
00:27:30.020 --> 00:27:31.140
Switch x and y
00:27:38.240 --> 00:27:40.040
Step one, switch x and y
00:27:40.700 --> 00:27:45.060
2. Solve for y
00:27:47.540 --> 00:27:54.240
and then switch back. Well in a minute tell you where the signs. Its an interesting short cut
00:27:55.320 --> 00:28:04.600
Okay you switch x and y. And you got x is 5y minus 4 over 7
00:28:11.760 --> 00:28:12.800
very good, okay
00:28:13.240 --> 00:28:14.840
Now I want to solver for y
00:28:16.220 --> 00:28:17.660
So I can multiply for 7
00:28:22.400 --> 00:28:23.300
add 4
00:28:26.740 --> 00:28:29.940
7x plu 4 equals 5y.
00:28:30.460 --> 00:28:31.420
and divide by 5
00:28:44.520 --> 00:28:46.440
Its just x and y and solve for y
00:28:46.440 --> 00:28:48.340
and then you just call that inverse of x
00:28:48.560 --> 00:28:51.200
and your notation is for the negative 1
00:28:55.980 --> 00:28:57.820
Hope everybody can see that
00:29:00.420 --> 00:29:06.900
So f inverse of x is 7x plus 4 divided over 5, The negative 1 means the inverse, it does not mean 1 over
00:29:08.980 --> 00:29:13.140
Not that you thought it meant one over, it does not mean one over
00:29:13.980 --> 00:29:15.560
Thats something totally different
00:29:16.520 --> 00:29:21.560
The reciprocal of multiple inverse, Do you know what a multiple inverse is?
00:29:23.120 --> 00:29:27.540
Something that you multiply by something else, to get a multiple of identities or another words in english
00:29:28.420 --> 00:29:29.860
You multiply and get 1
00:29:30.220 --> 00:29:36.100
So the multiple numbers of 3 is a third so you take 3 and multiply it by a third, you get 1
00:29:37.920 --> 00:29:43.680
We call it reciprocal because regular arithmetic reciprocal is the multiple inverse
00:29:43.940 --> 00:29:45.380
That was my math vowel
00:29:46.180 --> 00:29:51.160
So witch x and y, solve out for y and call it f inverse of x
00:29:52.080 --> 00:29:57.220
Sometimes webassign will ask you for f inverse of y
00:29:58.340 --> 00:30:04.420
All that means is when you put your answer in the box put it in terms of y instead of terms of x
00:30:04.420 --> 00:30:06.080
write 7y plus 4 over 5
00:30:06.720 --> 00:30:08.480
I dont know why it does that
00:30:10.220 --> 00:30:16.820
So sometimes the webassign, pay attention because youll get it wrong and say I dont understand anything professor Kahn said to do
00:30:16.820 --> 00:30:20.360
I hate my life, Im not going to be a doctor, Im crying blah blah blah
00:30:21.760 --> 00:30:23.760
Just change the letters okay?
00:30:24.220 --> 00:30:26.540
That means your going to be a doctor
00:30:26.740 --> 00:30:28.020
Its going to be good
00:30:49.380 --> 00:30:51.580
So lets practice with something slightly harder
00:30:58.440 --> 00:31:00.360
Lets find the inverse of that
00:31:00.940 --> 00:31:08.540
Think of this as y equals 2x cubed plus 7 over 11
00:31:12.900 --> 00:31:14.320
Switch x and y
00:31:14.320 --> 00:31:19.900
Some people who learn this in high school switch x and y at the end. Its generally easier to do it at the beginning but it doesnt really matter
00:31:26.000 --> 00:31:33.200
at this state you technically done the inverting but now we want to put it in the form of x inverse of x eqauls something
00:31:33.400 --> 00:31:38.160
So now we just have to isolate y. So we crss multiply
00:31:38.280 --> 00:31:44.480
and you get 11x equals 2y plus 7
00:31:47.340 --> 00:31:48.660
Subtract 7
00:31:53.400 --> 00:31:54.540
Dived by 2
00:31:59.180 --> 00:32:01.180
and take the cubed root
00:32:03.380 --> 00:32:04.500
the cubed root of
00:32:05.020 --> 00:32:07.920
11x minus 7 over 2
00:32:08.580 --> 00:32:10.720
and that is the inverse. Yes?
00:32:20.680 --> 00:32:29.540
You mean here? Because Ill have to take the 2 times y equals cubed of x. Its simpler just to say y cubed isolated before you do the cubed root
00:32:31.060 --> 00:32:37.540
I mean technically you can take the cubed root itd be messy so you really want to have it by yourself
00:32:37.540 --> 00:32:41.220
How did we do on this one? You should then write this as f inverse of x
00:32:42.020 --> 00:32:48.480
is the cubed root of 11x mins 7 over 2
00:32:48.480 --> 00:32:51.680
How do we know thats an inverse? Well lets see what happens
00:32:51.800 --> 00:32:59.980
Initially you take x the first thing you do and cube it then you multiply it by 2 and minus by a 7 and finally divided by a 7
00:33:00.940 --> 00:33:05.340
Now here you take x, remember the last thing you do is divide it by 11
00:33:05.340 --> 00:33:07.620
So the first think you do here is multiply by 11
00:33:08.200 --> 00:33:11.400
Here you add 7, so here you are going to subtract 7
00:33:11.400 --> 00:33:14.560
Here youre gonna want to multiply by two and here you divide by two
00:33:14.560 --> 00:33:18.540
The first think was cubed and the like thing is find the cubed root
00:33:18.540 --> 00:33:22.020
So you are versing the operation and you are reversing the order
00:33:24.780 --> 00:33:31.780
Did I do that to fast, everyone understood what I did? You dont have to be able to do that. But thats why we know it is an inverse
00:33:32.480 --> 00:33:38.560
Everything was the opposite and the opposite work. Kind of like we were rewinding the video
00:33:40.820 --> 00:33:41.700
Thats the dvr
00:33:47.840 --> 00:33:51.380
Alright how bout one more of these?
00:34:17.100 --> 00:34:19.260
Okay very similar to the last one
00:34:23.280 --> 00:34:24.720
find that inverse of x
00:34:29.680 --> 00:34:36.700
Rewrite this as y equals 6x to the fifth plus 3 all/4
00:34:37.360 --> 00:34:39.040
well go this way this time
00:34:40.820 --> 00:34:42.980
First thing we do is switch x and y
00:34:43.920 --> 00:34:48.020
You get x is 6y to the fifth plus 3/4
00:34:48.020 --> 00:34:51.420
See this is the kind of algebra stuff you just have to be good at
00:34:51.420 --> 00:34:57.220
You just have to comfortable with all this multiplying dividing and cubing and cube rooting and all that
00:34:57.800 --> 00:34:58.840
Cross multiply
00:35:06.260 --> 00:35:07.340
Subtract 3
00:35:12.760 --> 00:35:13.820
Divide it by 6
00:35:21.140 --> 00:35:23.860
and last but not least take the fifth root
00:35:25.180 --> 00:35:31.640
So y is the fifth root of 4x minus3 over 6
00:35:31.740 --> 00:35:33.260
Therefore the inverse
00:35:34.480 --> 00:35:41.100
Is the fifth root of 4x minus 3over 6
00:35:41.100 --> 00:35:47.840
and as I said about webassign pay attention if it ask for f inverse of y, just change this letter y
00:35:49.580 --> 00:35:53.580
Otherwise youll get it wrong and you start losing the points
00:35:53.580 --> 00:35:56.760
and then you wont get enough points for webassign and what does that mean?
00:35:57.780 --> 00:35:59.060
You wont be a doctor
00:36:16.820 --> 00:36:20.580
Alright so thats inverse, oh wait we can do more inverses
00:36:42.100 --> 00:36:43.760
I have y of sine of x
00:36:45.600 --> 00:36:47.840
Okay now I want to find the inverse
00:36:49.220 --> 00:36:50.920
Well you switch x and y
00:36:54.500 --> 00:37:05.060
and now you isolate y, you do not divide by sine. Sine is a function. Its not sine times y, its sine of y
00:37:05.600 --> 00:37:11.100
So the inverse is called arcsine of x
00:37:11.560 --> 00:37:14.920
On your calculator you'll see it written like this
00:37:16.640 --> 00:37:18.640
Okay those mean the same thing
00:37:19.220 --> 00:37:21.220
Inverse sine, arcsin
00:37:21.540 --> 00:37:24.420
Whats the problem with writing it this way?
00:37:24.420 --> 00:37:29.040
Some people will say, doesnt that mean one over, when you raise something to the negative one?
00:37:29.040 --> 00:37:30.780
So they get a little confused
00:37:30.780 --> 00:37:36.200
By why in the calculator do they use sin rather then arcsin...it takes up less space
00:37:37.320 --> 00:37:39.400
Real reason, it fits on the keys
00:37:40.520 --> 00:37:46.920
So most people in math world and of course you all are future mathematicians are going to use arc
00:37:46.920 --> 00:37:50.260
Okay it stands for the same thing, it means working backwards
00:37:51.000 --> 00:37:55.620
So why is arc sin x over here, its something tricky. Remember I told you about the horizontal line test
00:38:00.080 --> 00:38:02.000
Thats what sin of x looks like
00:38:02.440 --> 00:38:04.880
Correct so if invert it
00:38:06.800 --> 00:38:09.120
You something that looks like that
00:38:10.560 --> 00:38:13.460
Thats going to be a problem, its going to fail the vertical line test
00:38:14.100 --> 00:38:17.740
So what we do, we restrict sine of x
00:38:18.360 --> 00:38:22.660
The arcsine of x will narrow down the sine of x
00:38:31.540 --> 00:38:38.220
We only use this much, from pi over 2 to minus pi over 2
00:38:38.580 --> 00:38:40.660
When the arcsin
00:38:44.180 --> 00:38:45.220
looks like this
00:38:49.760 --> 00:38:54.500
So when pi over 2, sine will equal to 1. So 1
00:38:55.380 --> 00:38:57.140
arcsin is equal to pi over 2
00:38:58.560 --> 00:38:59.600
Then negative 1
00:39:00.120 --> 00:39:01.800
equals negative pi over 2
00:39:05.800 --> 00:39:09.440
Why do we do that? We can pick any piece of sine we want
00:39:09.440 --> 00:39:17.380
We can cut any s shape piece we want, this piece so that we get a one-to-one function
00:39:17.380 --> 00:39:20.060
So we find the inverse to define one place
00:39:21.920 --> 00:39:25.260
Thats going to be arcsin of x
00:39:26.140 --> 00:39:27.180
Lets do a cosine
00:39:32.740 --> 00:39:38.800
if we said the graph inverse sine, that would be a perfect, kind of perfect graph
00:39:38.800 --> 00:39:41.840
To get the point across, I've done better graphs in my life
00:39:46.920 --> 00:39:51.020
Ah so cosicant of x equals 1 over x
00:39:51.160 --> 00:39:53.360
Arcsine of x is backwards so..
00:39:53.740 --> 00:39:55.580
The sine of pi over 6
00:39:56.140 --> 00:39:56.700
is a half
00:39:57.920 --> 00:39:59.600
The inverse sine of a half
00:40:00.240 --> 00:40:04.600
Is pi over 6. So there arcsine of a half is pi over 6
00:40:05.880 --> 00:40:09.600
So the sine of pi over 6 is a half
00:40:09.940 --> 00:40:15.340
So that means we know that if I know something is a half, whats the angle
00:40:15.480 --> 00:40:19.220
Id say the arcsin of a half
00:40:20.360 --> 00:40:21.080
is pi over 6
00:40:22.320 --> 00:40:29.760
Okay but here you give me the angle I give you the sine, here you give me the sine I give you the angle
00:40:31.960 --> 00:40:33.480
Its working backwards
00:40:34.700 --> 00:40:38.480
The cosecant and cosecant is one over tangent
00:40:38.700 --> 00:40:41.260
Are the reciprocal and are very useful
00:40:41.260 --> 00:40:44.360
for when we have sine in the denominator of a function
00:40:44.360 --> 00:40:49.560
and you want to have the function in the numerator, you dont have it in the denominator, you use cosecant
00:40:53.240 --> 00:40:54.680
What about cosine of x
00:40:56.300 --> 00:41:02.560
Well cosine of x looks like that and I'll have a horizontal line to test the problem again
00:41:05.180 --> 00:41:08.060
Im only going to use the piece from zero to pi
00:41:09.160 --> 00:41:10.040
If I invert it
00:41:12.140 --> 00:41:13.000
It looks
00:41:14.860 --> 00:41:15.500
like that
00:41:22.940 --> 00:41:29.500
To think of it another way, I only used the first and second quadrant. So if you go in your calculator
00:41:29.780 --> 00:41:34.160
If you put in, find the inverse of cosine of a half
00:41:34.820 --> 00:41:37.480
It will give the answer of pi over 3
00:41:37.480 --> 00:41:39.800
Or 60 degrees, depending on what mode its in
00:41:40.900 --> 00:41:44.420
And if you did it a negative half. It has some choices
00:41:44.460 --> 00:41:46.700
It uses the second quadrant angle
00:41:46.940 --> 00:41:51.260
You can use a different angle, but uses the sceond quadrant angle
00:41:51.920 --> 00:41:56.700
It doesnt really make sense why we pick those but you can really pick anyone you want
00:41:58.000 --> 00:42:03.020
So inverse sine, use the first quadrant angle or we use the fourth quadrant angle
00:42:03.100 --> 00:42:04.620
So lets right that down
00:42:06.540 --> 00:42:09.000
Got to find a new place to right that, it will go on this board
00:42:32.780 --> 00:42:37.880
By doing the inverse sine, sorry, or arcsine
00:42:44.580 --> 00:42:49.140
Im using arcsine of x, or arccosine of x. Can you read that over there?
00:42:49.880 --> 00:42:50.600
Not really
00:42:50.600 --> 00:42:52.660
Can you guys see that over there?
00:42:53.500 --> 00:42:56.860
Alright Ill move this, Ill put it over here instead
00:42:59.240 --> 00:43:00.520
only takes a second
00:43:00.520 --> 00:43:03.200
That psychology exams not going anywhere
00:43:13.140 --> 00:43:15.240
So if I have arcsine of x
00:43:17.120 --> 00:43:18.240
Or arc cosine of x
00:43:21.240 --> 00:43:22.360
and x is positive
00:43:24.440 --> 00:43:26.760
and we use the first quadrant angle
00:43:27.280 --> 00:43:29.840
So if I say what is the arc sine of a half?
00:43:30.440 --> 00:43:33.940
The answer is going to be 30 degrees because the whole sine of a lot of things is a half
00:43:33.960 --> 00:43:37.180
Sine 30, sine 150 and so on and so furth
00:43:37.300 --> 00:43:42.340
So they do that in radiants, Its always going to be the first quadrant answer
00:43:43.580 --> 00:43:45.500
So remember theres an infinite number of places
00:43:45.740 --> 00:43:46.700
where sine is equal to a half
00:43:47.540 --> 00:43:49.400
I only care about the first quadrant angle
00:43:50.620 --> 00:43:51.500
for negative
00:43:54.160 --> 00:43:57.600
Theres an infinite number of answers for arccosine
00:43:57.600 --> 00:43:59.400
I use the second quadrant angle
00:44:01.540 --> 00:44:07.960
and for arcsine, I really use the first quadrant, i really use the negative
00:44:09.120 --> 00:44:11.320
angle. So if I said
00:44:13.400 --> 00:44:15.160
arcsine of negative a half
00:44:16.580 --> 00:44:18.660
the answer is negative pi over 6
00:44:20.460 --> 00:44:23.000
What I do is go down
00:44:24.660 --> 00:44:25.700
to there, okay?
00:44:52.740 --> 00:44:56.620
So do you guys understand what I mean, to use the first quadrant angle
00:44:56.620 --> 00:45:02.740
You can get all the answers you want. So if I say this sine of what angle is equal to radical 2 over 2.
00:45:02.740 --> 00:45:04.820
You say where theres a lot of that, okay
00:45:04.820 --> 00:45:06.260
Give me the first quadrant angle
00:45:07.060 --> 00:45:11.420
If I say the sine is equal a negative radical 2 over 2
00:45:11.440 --> 00:45:17.400
Again theres a lot of answers, I just the negative fourth quadrant, negative pi/4
00:45:19.900 --> 00:45:23.260
and i shouldve gotten the graph down when I did that
00:45:43.940 --> 00:45:46.500
Ill remind him to post it on blackboard
00:46:20.900 --> 00:46:26.140
So lets jut do a little more of this and then Ill stop and youll all go and eat
00:46:53.140 --> 00:46:57.700
f of x is 3sin x plus pi/4. how do we do the inverse
00:46:57.740 --> 00:47:04.440
Well first we switch x and y and we write this 3sine x plus pi/4
00:47:06.000 --> 00:47:11.340
And then you make it x equals 3sine y plus pi/4
00:47:13.320 --> 00:47:17.060
So what do you do? You copy o the person next to you
00:47:17.060 --> 00:47:20.060
And you hope thats the right answer and heres the right answer
00:47:20.660 --> 00:47:21.620
divide by 3
00:47:30.960 --> 00:47:33.640
and now you dont divide by sine
00:47:33.640 --> 00:47:37.140
You do the inverse of sine, so you get arc
00:47:38.280 --> 00:47:41.020
sine of x over 3 of x
00:47:43.420 --> 00:47:46.700
Equals y plus pi/4, you just subtract pi/4
00:47:48.180 --> 00:47:57.780
Arc sine x over 3 minus pi/4 is y, the inverse function
00:47:58.240 --> 00:48:00.480
I would never ask you to graph that
00:48:03.340 --> 00:48:05.180
Not sure I want to graph that
00:48:26.260 --> 00:48:27.940
How are we doing on these?
00:48:28.060 --> 00:48:30.860
Do you think the inverse is bad? not so bad?
00:48:31.480 --> 00:48:32.920
Guys get the concept?
00:48:34.060 --> 00:48:40.660
The transformation stuff? Cause its the last class before we go into review mode
00:48:40.660 --> 00:48:43.040
Let me make sure I covered everything
00:48:43.940 --> 00:48:45.380
You all are packing up
00:48:47.440 --> 00:48:53.140
Im pretty sure thats all we are going to ask ypu about inverse
00:48:54.520 --> 00:48:57.640
If theres new stuff on monday Ill hit you with it