WEBVTT
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Language: en
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...Functions... a piecewise is a function that is defined in pieces.
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so "Piecewise"
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Okay, so if a function is in pieces, then that means we divided up what we do with the functions
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So we have something very simple..
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Can be very straightforward, like that.
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And people get very confused about this, saying "I don't know what to do"
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Well.. what this means is you always have this part first.
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This is the domain. The domain is chopped up. Which means it can be numbers less than zero, or greater than and equal to zero.
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And it says if you have a number less than zero, you use the function x + 2. And if you have a number greater than or equal to zero you use the function x - 2
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So you really, theres two things going on
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piecewise functions are going to show up a lot here and in calculus
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Okay, so what you can use to define things that apply to piecewise functions
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Like lets say
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Okay so I gave you a word problem because we love word problems
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We actually do love word problems
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One oh! Good question I forgot to mention that.
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So if you look on the course web page, which you can find from blackboard
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at the very bottom we have stuff about midterm one.
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So I put in there, five chapters from my trigonometry book that go over basic stuff
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Radians, how to find sine, and cosine and all that stuff.
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And then you will have other practice material going up on there in the next couple days
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Also I put the stuff the stuff from my book in the document section
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For my lecture, I don't have acess to the other lectures
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Okay so if you go to the course webpage at the bottom, it says something, stuff about midterm one. Click on that, you'll see a bunch of stuff from my book.
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We'll also put up practice problems, I don't know if we will have an actually sample exam but we'll have sample stuff
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A very good way to prepare for the exam is to go through the webassigns and the homework
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You want to get a feeling for the way professor Sutherland likes the questions
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okay so professor Sutherland likes the bulk of the exam, I just stand here and talk, he actually is his course
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It's kind of ours but he writes the exams, so you want to get a feeling for what the questions will be like look at those.
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Okay you also should know
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We do not grade your exams, the TA's grade your exam
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I do not have any strength to grade all of your exams
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I would just get a rubber stamp that says C minus
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It would really speed things up
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I meant A plus sorry
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Right, yeah, yeah
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This is an A plus right?
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No?
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Okay so that's where you can find a lot of stuff to help you prepare over the next week
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I also suggest that if you confused about something, you can watch the video over. We have the videos on the course webpage.
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You can watch me do my thing again for an hour, if you can stand it
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Or you have me, so we will have three or four minutes allow on each topic as well, okay?
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Put all that together
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Also next Monday and Wednesday will be review sessions.
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But I am warning you now..
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Wednesday is going to be right before the exam, the anxiety level tends to be very high
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Right? So Monday I'll do a lot of review. We will record that and it will be up again on Tuesday.
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I suggest you come to the review session, okay?
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That would be Monday, Wednesday will also be a review but a less of review because you guys will be just a wall.
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With my class you'll be nice arelax
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Today and Wednesday will be the review material and that is it.
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So one thing is,piecewise, how would you right this as a piecewise function?
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Piecewise, again, means you are breaking into pieces
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First you make one of these cool, thoroughly brace this is called
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You ever wonder how to make that, a long s and then a backwards s and put them together
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Did anyone ever show you that before?
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Thats just a way of saying, I'm using the following things
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10 dollars and hour for the first hour. So we say x is the number of hours
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Okay so its going to be 10 dollars times x. Assume x can be less than one hour, it can be a fraction
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That or zero up to 1 hour
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How bout more than an hour? Well now I charge 12 dollars and hour
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For the next 5 hours, so that's not exactly right, is it?
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Why is it 12x?
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Right, we need to plus that, exactly. It will be 12 times x minus 1 to subtract the first power, so i this was 3 hours it would be 12 dollars time 2 hours plus 10.
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And that would be until you get up to, including hour number 5.
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And then beyond that
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14 dollars times x minus 5 hours, and then lets see. For four hours i spend 12 dollars an hour, that's 48 bucks
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And for the first hour I spent 10 dollars, so thats 58 dollars.
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Again where does the 58 come from? 10 Dollars for the first hour
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Then four more hours, for 12 dollars an hour is a total of 58 dollars
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And then after that its 14 dollars times all the hours past 5 hours
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That's just an example on how you would do a piecewise function
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Yes?
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That's why its x minus 5. Okay?
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Because theres 5.1 hours
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Okay? 5.1 minus 5
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Sure it wuld be 6.
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If they say what happens behind 5 and 6 hours, yes?
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Sure question?
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Well okay, it's 10 dollars an hour so we work for a half hour.
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You need 0 to get five dollars, so with a half hour you''l get 10 times a half
00:08:07.840 --> 00:08:09.200
Other wise i fire you
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x is the amount per hour. Also this isn't about real world example, Im just trying to provide a function
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This is the kind of thing, that you can shape with your boss. And you say you worked for 45 minutes and your boss says 3/4 at 10 dollars an hour. Wow 10 dollars
00:08:27.620 --> 00:08:31.820
And I just need 10 dollars to showing up and then he dont use you a full hour that's his bad.
00:08:32.080 --> 00:08:40.860
You can write that kind of function, it;s kind of more complex. Okay, it's not that complex, you can use something gthat is a less than function
00:08:43.220 --> 00:08:47.320
So where does the 58 come from..one more time.I have the first hour, that's 10 dollars
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Then I have four more hours at 12 dollars an hour
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That's 58.
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So at the end of 5 hours you've earned 58 dollars
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And then you get 14 dollars after that
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This shows up at real world stuff
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Because well you can use it with taxes, that's one variation
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Well we can go to a parking taxi meter, in a taxi in New York
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So much just to get in the cab and so much for every mile
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So these are all what this is all an example piecewise function
00:09:23.780 --> 00:09:28.860
So a more real world example, sorry the more mathy example. I can cover this?
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Nope why is it only four times 12?
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Oh my apologizes I made it up to hour number 5
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You are correct otherwise
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Thank you, that's what you were saying. Got it.
00:10:05.620 --> 00:10:07.380
Okay this is gonna allow it
00:10:14.020 --> 00:10:18.180
Absolute value Functions, Wait. I'll Let everybody copy that
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Absolute value of x
00:10:34.580 --> 00:10:37.500
Absolute value of x. If you are using a piecewise function
00:10:37.500 --> 00:10:39.580
So we love to test the absolute value of x.
00:10:40.320 --> 00:10:45.720
So what does the value of x mean? It means the distance from the origin
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MEans distance from zero
00:10:48.060 --> 00:11:00.260
We dont care about the sign, so the absolute value of 5 is 5 away from zero. which is 5. The absolute value of negative 5, is also 5 away from zer so it's also 5
00:11:02.200 --> 00:11:03.880
When you think about zero
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Those are called the distance of 5
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So we say absolute value, we just read and say how far away. We dont hear where the sign is
00:11:15.360 --> 00:11:20.240
So another way to think about absolute value is, as long as something is positive you dont have to do anything the the number
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But if something is negative, you switch the sign and the number. how do you switch a sign?
00:11:26.080 --> 00:11:27.280
You multiple it by negative 1.
00:11:33.600 --> 00:11:37.660
As long as x is greater or equal to zero, the absolute value of that number is the number. Right?
00:11:38.360 --> 00:11:41.820
Absolute value of 3, is 3, absolute value of pi is pi and so on..
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If x is a negative number
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All we do is switch the sign of x.
00:11:49.860 --> 00:11:54.420
So we will use absolute value over this class and first semester calc
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And we will expect you to think of it this way
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In a piecewise funtion
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The functions can be broken into two different pieces
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One more, my favorite types
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You get something like this.. What does that look like?
00:12:34.000 --> 00:12:35.700
You want to think of it as a piecewise function
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You say stuff, well as long as x is bigger than 5, the absolute value doesn't do anything.It is just x minus 5 over x minus 5
00:12:48.000 --> 00:12:52.440
as long as x is greater than 5, I dont have to do anything to the absolute value
00:12:54.840 --> 00:13:02.440
So if x is 11, 11 minus 5 is 6. If that says 5.1, 5.1 minus 5 is .1. So I'll always get a positive answer
00:13:03.620 --> 00:13:07.280
If x is less than 5, then things get interesting
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If x is less then 5, this is still x mins 5
00:13:21.520 --> 00:13:29.140
But this will now become negative, so say x is 1.. 1 is less then 5, if we plug it in you'll get negative 4. Then after that you get 4
00:13:29.580 --> 00:13:30.860
So what I do is I take
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The thing inside the absolute number bars and multiply it by a negative number
00:13:38.020 --> 00:13:39.780
Just the way I did over here
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So as long as I am greater than 5, that's the number. what's inside the absolute value bar is positive, you don't have to do anything to it.
00:13:49.780 --> 00:13:54.160
If less then 5, what's inside will be negative so I have to multiple it by a negative sign
00:13:55.540 --> 00:13:58.600
Okay but first I didn't put it equal to 5.
00:13:59.260 --> 00:14:01.180
So the problem here is in the denominator
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What's x minus 5 over x minus 5?
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1
00:14:14.720 --> 00:14:17.640
And the negative x minus over x minus 5, is negative 1.
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So this is just the function one as long x is greater than 5 and negative 1if x is less then
00:14:45.500 --> 00:14:47.020
If we were to graph that
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That's what it looks like. As long x is greater then 5, i have 1 and if x is less then I have negative 1. And nothing is going on in the middle
00:15:11.520 --> 00:15:13.640
Got the idea of piecewise functions? Let's do one more
00:17:22.380 --> 00:17:25.740
A pool, doesn't have to be a swimming pool because it will take a year to fill up
00:17:26.700 --> 00:17:41.500
A pool filled with water, with 12 gallons for the first 3 hours and 10 gallons an hour of water after that. Express the amount of water in the pool as a piecewise function
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So it fills with 12 gallons of water per hour for the first 3 hours.
00:18:02.380 --> 00:18:06.440
So f of x is the amount of water after x hours
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So for the first 3 hours it's 12 gallons
00:18:11.280 --> 00:18:19.200
That, as long as x is greater then or equal to 3. Has to be a positive number of course
00:18:23.100 --> 00:18:24.940
Then what happens after 3 hours?
00:18:25.720 --> 00:18:27.800
Well 3 hours times 4 is the pool?
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36 gallons after the first 3 hours
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Now, it will fill 10 gallons times x minus 3
00:18:40.240 --> 00:18:47.100
Because its been more then 3 hours, so you have to subtract 3 from the number of hours
00:18:51.920 --> 00:18:53.800
I guess in theory it can go to infinity
00:18:54.900 --> 00:19:01.300
So if we asked in domain for example, the domain will be all numbers greater than or equal to zero.
00:19:01.300 --> 00:19:03.520
Because you knwo how to fraction them out.
00:19:04.380 --> 00:19:12.260
Whole will be exactly hours, okay? Then it will be integers but every hour the gallons of the series just appear.
00:19:12.580 --> 00:19:17.080
If this was spilling smoothly we can have any number of hours
00:19:18.040 --> 00:19:19.680
you can't have negative hours
00:19:21.120 --> 00:19:23.120
You can have pi hours, sort of.
00:19:24.200 --> 00:19:31.200
I mean it would be a really weird way to do it, okay. So the domain will be all numbers greater then or equal to zero
00:19:31.980 --> 00:19:36.360
Alright, how much water is in the pool after 10 hours?
00:19:37.180 --> 00:19:51.940
Well we had 36 gallons after the first 3 hours and then you have the 10 hours. Well for the next 7 hours we got 10 gallons an hour thats 70.
00:19:52.320 --> 00:19:53.920
will give you 106
00:19:57.100 --> 00:19:59.460
Does that make sense? Have any trouble with that?
00:20:35.560 --> 00:20:37.480
Okay how did we do on this one?
00:20:38.060 --> 00:20:40.860
Do we understand the concept of what a piecewise function is?
00:20:41.340 --> 00:20:43.580
IT's really not that hard... yes?
00:20:48.840 --> 00:20:57.320
You can have 0 gallons and 0 hours. why is it greater than zero and not just 0.
00:21:04.780 --> 00:21:07.320
In reality it doesnt make much of a difference
00:21:13.220 --> 00:21:16.700
Should I do another version of this? So we can understand piecewise?
00:21:20.560 --> 00:21:24.000
Let's talk a bit about transformation of a function
00:21:25.460 --> 00:21:28.980
This is going to show up a lot, especially in graphing
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Transformation if we were to take a function and change it into some way
00:22:03.840 --> 00:22:10.080
Whats happening when you transform a function, really what we are doing moving the x and y axis
00:22:10.980 --> 00:22:14.500
So originally the function is some place on the x axis
00:22:15.380 --> 00:22:16.100
So you have
00:22:18.260 --> 00:22:34.400
Some function sitting on the x axis and you sort of slide it. So let's say thats the origin. And you move it to the right and now it's out here. That's a transformation, we moved a funcion
00:22:36.440 --> 00:22:39.900
From there to there. So how do we do that?
00:22:40.300 --> 00:22:44.940
We take the original function, and sort move x to the right or the left.
00:22:46.380 --> 00:22:48.380
What is happening is
00:22:48.880 --> 00:22:56.440
Let's say this is the number 2 and this must be the number 5. We moved it 3 units to the right
00:22:58.840 --> 00:23:01.080
So another way to think about it is
00:23:02.020 --> 00:23:12.240
This point is f of 2. And if its at five and we moved it 3 to the right, we have to get the same value of what we got before when we had 2
00:23:13.260 --> 00:23:15.920
So you will have to have 5 minus 3
00:23:16.160 --> 00:23:21.000
When take away the 3 so we are also plugging 2 in here
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Does that make sense?
00:23:24.620 --> 00:23:32.480
So you guys probably di this in high school and never understood why when you subtract it you move it to the right. You wind up thinking thats move to the left
00:23:32.480 --> 00:23:33.340
Because you are taking away
00:23:34.300 --> 00:23:44.480
But what really happens is in order to get youre originally function value when you plug in a number you have to subtract whatever you shifted to get that back to that value
00:23:44.920 --> 00:23:50.960
okay so if this was f of x that one has to be f of x minus 3
00:23:51.920 --> 00:23:55.700
And that will shift the curve 3 units to the right
00:23:57.840 --> 00:24:01.120
So put it back where it started, you take away the 3
00:24:01.940 --> 00:24:09.300
When you want to move it 5 units to the right, you do x minus 5. So if you want to move it h units to the right, youll have x minus h
00:24:13.180 --> 00:24:15.580
You need a new rule to help you live by
00:24:17.140 --> 00:24:19.020
What to make sure you go the basic idea down
00:25:06.900 --> 00:25:12.860
To move it h units to the right, you have to do f of x minus h.
00:25:31.920 --> 00:25:40.900
What does that mean? Lets say I had y pr f of x would be better. If it was x squared
00:25:46.060 --> 00:25:48.640
That looks something like that, a basic parabola
00:25:52.220 --> 00:25:56.200
So f of x minus 5 squared,
00:26:00.940 --> 00:26:01.440
Yes?
00:26:20.400 --> 00:26:21.980
Now it will look like that.
00:26:27.700 --> 00:26:30.800
That's not very complicated, right?I moved it 5 units to the right
00:26:31.240 --> 00:26:40.340
So everything moved 5 to the right. Because now here when I plus in 0, I get 0 squared and here when I plug in 5, I get 5 minus 5 is 0 squared.
00:26:44.020 --> 00:26:45.600
Well the function is x squared.
00:26:46.920 --> 00:26:51.220
So this goes through lets say I plug in a 3 I get 9
00:26:52.700 --> 00:26:53.820
SO here I plug in 8
00:26:55.960 --> 00:26:59.100
8 minus 5 is 3, 3 squared is 9
00:27:01.020 --> 00:27:10.660
So the y value isn't changing. What' happening is the x value is changing. In order to get back to where I started I have to subtract the number
00:27:14.380 --> 00:27:19.640
You want to think about what's going on, in case it gets letters or pictures and things.
00:27:20.500 --> 00:27:24.260
But we also want to make different rules and use both ways
00:27:25.440 --> 00:27:27.360
Okay we are going to shift something to the left
00:27:28.760 --> 00:27:31.640
Well its the negative of that, so I would add
00:27:37.200 --> 00:27:48.720
We have a little app for this on the webpage. You can actually take a graph and move a slider back and forth and watch the graph slide back and forth
00:27:48.720 --> 00:27:54.520
Because what is happening is The curve is staying where it is and to make sure to move the axis
00:27:55.960 --> 00:28:02.220
You can even say the axis stand still and we are moving the curve. or the curve stays still and we move the axis, its kinda the same thing
00:28:06.820 --> 00:28:07.940
That make sense?
00:28:09.740 --> 00:28:10.300
To shift
00:28:12.680 --> 00:28:13.180
H units
00:28:15.020 --> 00:28:15.740
to the left
00:28:22.120 --> 00:28:23.920
use f of x plus h
00:28:29.700 --> 00:28:30.640
That should make sense
00:28:31.280 --> 00:28:34.460
Add the number the inside will make it move to the left
00:28:34.460 --> 00:28:36.520
Subtract the number and move it to the right
00:28:38.340 --> 00:28:41.620
When it says inside a parenthesis, suppose we had
00:28:48.040 --> 00:28:57.200
Then we are effecting what is happening to the x value so if I wanted to shirt this 10 units to the left
00:29:01.500 --> 00:29:04.740
This will become f of x plus 10
00:29:10.960 --> 00:29:12.900
Now we take x and replace it by x plus 10
00:29:20.000 --> 00:29:22.780
That make sense?This is why you can use simplify to take away the squareroot.
00:29:24.020 --> 00:29:30.740
To say wheres the difference from square root of x plus 2 and the square root o f x plus 10 plus 2 is a graph
00:29:31.540 --> 00:29:38.500
The first one is located subvise and the second one is 10 units to the left
00:29:38.500 --> 00:29:43.420
So all the y values will move over here, 10 units to the left of the other function
00:29:45.160 --> 00:29:51.520
The square root of x plus 2 kind of looks like this. Thats a negative 2.
00:29:54.660 --> 00:30:01.500
The square root of 10 plus 10 plus 2, is there. Same curve shifted 10 units to the left
00:30:03.220 --> 00:30:04.340
Who's confused?
00:30:06.260 --> 00:30:08.840
Not Bad, you guys should've seen this in high school
00:30:08.840 --> 00:30:11.460
I know everybody gets this in the regents and stuff like that
00:30:11.460 --> 00:30:13.660
Okay now we are going to move up and down
00:31:01.540 --> 00:31:06.740
So if we want to move a curve up, we will add k or move the graph down and subtract k
00:31:27.260 --> 00:31:29.340
So f of x equals x squared
00:31:34.480 --> 00:31:36.240
Looks something like that
00:31:37.180 --> 00:31:42.960
f of x equals x squared plus 2. Is now shifted up 2.
00:31:47.360 --> 00:31:53.440
So again you can think of this as we took the curve and moved it up too or we took the axis and you dropped them 2
00:31:54.840 --> 00:32:00.320
Its the same thing. Either the curve moved up or the axis moved down and theyll end up in the same place
00:32:03.860 --> 00:32:05.380
You guys see that over on that side?
00:32:15.200 --> 00:32:17.200
Now for the fun part, to do both
00:32:24.440 --> 00:32:26.360
What we call transformation
00:32:39.520 --> 00:32:43.540
Say I had the following graph
00:33:04.120 --> 00:33:07.200
So I have a graph that looks like that, and I'll call that f of x
00:33:08.340 --> 00:33:12.960
Lets think of what f of x minus 1 would look like
00:33:34.400 --> 00:33:36.320
So if I want to do f of x minus 1,
00:33:37.760 --> 00:33:40.320
Everything moves one unit to the right
00:33:40.580 --> 00:33:46.340
So one think you can do, is think of the actual coordinate.
00:33:47.160 --> 00:33:49.520
So here I am at negative 2 and 2
00:33:51.020 --> 00:33:54.940
So now I'm gonna move one to the right and get negative 1 and 2
00:33:56.900 --> 00:34:09.240
Here at negative 1 I am at zero. So now it is zero zero. zero 2 moves to 1,2 and so on
00:34:17.720 --> 00:34:24.720
The graph hasn't moved up or down, so the y values stay the same. All that happened with the graph is we picked up the graph and shifted it
00:34:28.800 --> 00:34:31.520
One unit to the right. Everyone see that?
00:34:38.640 --> 00:34:39.760
Nice and simple?
00:34:41.260 --> 00:34:48.140
So lets take the same graph and now lets shift it down
00:35:08.160 --> 00:35:09.520
We will have x minus 2
00:35:10.840 --> 00:35:17.920
So notice what happens, when you have something inside the parenthesis you change the x values.
00:35:18.080 --> 00:35:23.380
So the marking on the x axis are shifted if you want to think of it that way
00:35:25.060 --> 00:35:31.420
When I do f of x minus 2, I already found f of x now I am subtracting and effecting the y values not the x values
00:35:31.640 --> 00:35:33.880
You can think of it as y equals f of x
00:35:34.800 --> 00:35:39.180
So now we take this curve and everything will come down too
00:35:39.700 --> 00:35:49.580
So with negative 2, I'll now be at 0. At negative 1, I'll be down here at negative 2
00:35:53.780 --> 00:35:55.320
With the origin I do that to there
00:36:01.800 --> 00:36:03.000
So I took the graph
00:36:09.480 --> 00:36:10.820
And I am going to drop the graph 2
00:36:12.560 --> 00:36:15.440
Does that make sense? You know whats coming
00:36:19.100 --> 00:36:21.100
Okay lets see if you guys can do
00:36:29.340 --> 00:36:30.720
Take a minute and see what that graph looks like
00:36:33.060 --> 00:36:34.940
It looks just the way it did before
00:36:35.440 --> 00:36:43.540
So first what happens is basically the axis moved or the function moved you can think of it either way
00:36:46.580 --> 00:36:51.220
Theres a couple more motions you can learn, I can stretch and shrink it
00:36:54.560 --> 00:36:55.400
How did we do on this one?
00:36:57.340 --> 00:37:04.600
Yes, no? Everybody who loved this and continue to love this even if you had nightmares in high school the nightmares back
00:38:07.160 --> 00:38:14.500
So what do we mean stretch? We literally mean make the graph longer and skinner.
00:38:14.500 --> 00:38:17.100
It gets a little tricker when we mean the word stretch
00:38:44.080 --> 00:38:46.400
It will look something like this..
00:38:46.860 --> 00:38:50.660
So I have f of x equals x squares and 3x squared
00:38:50.660 --> 00:38:54.120
Because what am I doing? I am taking each of the y values and dividing it by 3
00:38:55.000 --> 00:38:55.800
For example
00:39:00.700 --> 00:39:12.800
This goes through 1 comma and 2 comma 4. Now I multiple things by 3 to make it up to 1 I am at 3 and when I get to 2 I am at 12
00:39:15.500 --> 00:39:18.300
So I took the graph and kind of stretched it
00:39:19.700 --> 00:39:31.320
Stretch is easy. mathematicians don't really want to use a word like stretch. What we did was you took it in the reverse direction and you spread out the numbers
00:39:32.760 --> 00:39:34.220
By multiply it by 3
00:39:34.420 --> 00:39:40.500
And to get it to compress, go in horizontally I multiply it by 3
00:39:40.500 --> 00:39:42.260
Why does that compress it?
00:39:43.040 --> 00:39:44.320
Or it thined again?
00:39:51.700 --> 00:39:55.720
What is the difference between f of x and f of 4x.
00:39:57.700 --> 00:40:04.020
Well here when I plug in 4, I get f of 4. Here when I plug in 1 I get f of 4.
00:40:05.200 --> 00:40:10.800
Here when I plug in 8 I get f of 8 and here when I plus in 2 I get f of 8
00:40:11.400 --> 00:40:12.900
I need much less, right?
00:40:13.140 --> 00:40:21.720
Im going to have 1, f of 1 and I have 4 and whereever f of 4 is.
00:40:22.840 --> 00:40:23.340
Here
00:40:24.820 --> 00:40:26.180
Adds up... I'm sorry
00:40:29.940 --> 00:40:30.500
That is 8
00:40:34.580 --> 00:40:36.420
Sorry I have to rewrite this
00:40:44.080 --> 00:40:49.140
Here at 4, I have f of 4. At 8 I have f of 8
00:40:50.260 --> 00:40:58.220
And here at 1 I have f of 4, and 2 I have f of 8
00:41:02.060 --> 00:41:10.800
So the graph squeezes in, 4 times 2 is 8. Thats why you get a horizontal compression
00:41:12.720 --> 00:41:17.320
With this amaze function, I can do a bunch of things to it
00:41:18.640 --> 00:41:22.480
If I was to write an exam question, I would ask you to do that
00:41:22.480 --> 00:41:23.380
And may or may not
00:41:53.540 --> 00:41:55.060
So Lets make a function
00:42:08.740 --> 00:42:10.260
So lets say this is f of x
00:42:37.200 --> 00:42:41.120
2 f of x minus 1. So the 2 is going to stretch it
00:42:41.460 --> 00:42:43.940
So shove it in the verticla direction
00:42:43.940 --> 00:42:46.220
and the minus 1 is going to move it 1 to the right
00:42:47.540 --> 00:42:48.400
So that means its going down
00:42:56.200 --> 00:42:59.400
What does 2fx do? You will double all the y values
00:43:00.900 --> 00:43:02.840
And then subtract 1 and you'll move everything down 1
00:43:02.840 --> 00:43:07.160
So first you double everything. You should do it in pieces that will often help
00:43:08.060 --> 00:43:11.360
If you want to do 2f of x make, just take the original function and double everything
00:43:12.360 --> 00:43:14.340
So it looks sorta
00:43:16.980 --> 00:43:17.620
Like that
00:43:21.000 --> 00:43:25.440
Instead of going up to 4 you now go up to 8
00:43:27.420 --> 00:43:29.420
So you took the original graph
00:43:31.560 --> 00:43:32.920
And you stretched it
00:43:35.080 --> 00:43:37.640
Okay and heres IF I want to move it down 1
00:43:38.640 --> 00:43:40.480
I just have to take the whole thing and shift it down
00:44:31.620 --> 00:44:37.140
Well Im doing it in two sets, that part is just the true f of x, this is the 2f of x minus 1
00:44:55.280 --> 00:45:00.340
Right away you can think this is a composite function because the f of x is now inside a new function
00:45:04.920 --> 00:45:05.420
Yes?
00:45:09.420 --> 00:45:13.160
Does it matter if you stretch or compress it before you shift it? Try it and see what happens
00:45:17.140 --> 00:45:24.980
Generally you should do the 2f of x first before you then add or subtract a number
00:45:24.980 --> 00:45:33.140
And I always tend to do everything in the parenthesis before I do the outside of the parenthesis
00:45:35.140 --> 00:45:38.580
What would happen if you first added or subtracted 1
00:45:43.200 --> 00:45:50.920
Well what happen is you take the 4 and subtract it by 1 and get a 3. and if you doubled it you'll get 6. You wouldn't end up in the same spot
00:45:55.680 --> 00:45:56.640
PEMDAS
00:46:00.860 --> 00:46:02.940
It shouldve, Its the bad artist
00:46:14.140 --> 00:46:16.620
Alright lets do one thats everything
00:46:19.900 --> 00:46:20.880
Start off with that same graph
00:46:27.340 --> 00:46:28.900
Now I want to do everything at once
00:46:39.580 --> 00:46:40.460
Lets try that
00:46:44.500 --> 00:46:47.780
Support system tricks will make this easier to do
00:46:50.060 --> 00:46:52.380
So the one thing is to do it in pieces
00:46:52.600 --> 00:46:55.840
So what does f of 3x mean? Shrinking by 3
00:46:58.960 --> 00:47:02.880
So we are going to do one step at a time. First we are going to define f of 3x.
00:47:12.520 --> 00:47:15.140
First I am going to use f of 3x.
00:47:16.320 --> 00:47:19.880
F of 3x takes the whole graph and moves it in by 3
00:47:23.900 --> 00:47:35.780
So where I have 1, I now have a 1/3. Where I had 3, I have 1 and Where I had 5 I have 5/3. So I take all the x values and divided them by 3
00:47:36.600 --> 00:47:39.800
And squeezing that graph in by a factor of 3
00:47:45.300 --> 00:47:46.100
Thats f of 3x
00:47:50.300 --> 00:47:51.260
So far so good?
00:47:52.640 --> 00:47:55.500
Now, Im going to multiple everything by 2
00:47:57.360 --> 00:48:00.140
So I need to do 2 f of 3x
00:48:12.920 --> 00:48:19.080
Okay so now the graph, Is still similar to before, again my artwork. But it is now twice as tall
00:48:19.940 --> 00:48:22.180
It sort of looks like it is tinier.
00:48:22.620 --> 00:48:24.620
It gets a little tricky, okay?
00:48:27.020 --> 00:48:30.780
It has to get up to 8 in the same amount of time it would have take before to get up to 4
00:48:32.060 --> 00:48:34.980
Its going to be longated. When you stretch something you can make it thin
00:48:35.680 --> 00:48:37.280
And now I am going to add 1
00:49:03.760 --> 00:49:05.280
Howd you do on that one?
00:49:06.020 --> 00:49:07.700
Not as bad as you thought?
00:49:10.580 --> 00:49:17.580
Do you have to make it look skinner? That's an excellent question. When I draw these graphs, I just change the numbers around
00:49:17.580 --> 00:49:19.440
But the graphs kind of all look the same
00:49:21.380 --> 00:49:28.380
That is fine. So what you are doing really, is you are changing the scale and the axis rather than the actual graph
00:49:37.460 --> 00:49:45.920
Can you keep the scale the same way it would be. Yes you can do it either way. You can pinch the graph or the scale the way it would be in the same place
00:49:46.580 --> 00:49:47.940
Okay lets do one more
00:49:49.960 --> 00:49:51.080
Make sure everybody can do this
00:50:29.460 --> 00:50:31.540
Okay that is our original graph
00:50:49.160 --> 00:50:52.200
Look at that you have three transformations.
00:50:53.120 --> 00:50:56.200
So 4f of x minus 3 minus 1
00:51:05.480 --> 00:51:10.800
I am just making this up, this isnt the function, id take a lot of work to figure out this function
00:51:11.420 --> 00:51:13.160
Okay so theres three things.
00:51:13.160 --> 00:51:17.340
First we are going to shift it 3 to the right, so we go 3 to the right
00:51:18.260 --> 00:51:20.860
Thats f of x minus 3
00:51:29.180 --> 00:51:30.540
Kinda look like that
00:51:41.560 --> 00:51:47.940
Okay now we can move 3 to the right and it still goes up to 3 and still goes down to negative 2
00:51:48.800 --> 00:51:50.640
Okay thats your first step!
00:51:51.300 --> 00:51:54.320
So x minus 3 moves everything 3 to the right
00:51:55.060 --> 00:51:56.780
So thats our first strength of many
00:51:58.820 --> 00:52:01.700
Our second transformation is multiply by 4
00:52:04.800 --> 00:52:07.640
So now you are going to take that picture and multiply it by 4
00:52:15.100 --> 00:52:18.080
So instead of it coming down to negative 2 its going to come down to negative 8
00:52:18.160 --> 00:52:20.720
Instead of going up to 3, it goes up to 12
00:52:35.220 --> 00:52:39.780
So thats what multiply it by four does it takes it and stretches it by 4
00:52:40.620 --> 00:52:44.320
and the last thing is to subtract by, subtract 1
00:53:25.600 --> 00:53:26.960
Okay we got the idea?
00:53:28.640 --> 00:53:32.920
Alright so we will do more of this on wednesday and we will do some other stuff