Stony Brook MAT 123 Fall 2015
Lecture 09: more graph transforms, inverse functions
September 23, 2015

Start   Half of sinx and you get 3sinx.
What does that do to the graph?
Because you can transform any of the graphs, with the rules of f(x) Now let's do it on the graph we know.. You can do the x squared You can do the square root of x So sin(x) Well regular sin(x) Kinda looks like that
0:30So 3 sin(x), Oh and this is pi and this is 2pi and that's minus 1 And that's 1.
That's one of my better sine graphs Didn't it come out okay?
You can come up and draw it for us So what's 3 sin(x) look like?
well It's going to look just like sin(x), because we stretched it vertically by a factor of 3
1:02so now... it'd be like that That's still pi still 2pi, now we go up to 3.
and down to negative 3
1:38okay now we can do other stuff with a graph I'll uncover that in a second
2:03I can do something like that So remember that's sin(x) All 3sin(x) graph is You took the sin(x) graph and you just stretched it So it's now 3 up and 3 down Well what is I do sin of x minus pi, over 4 Now I am going to take this sin(x) graph and I am going to move it pi/4 to the right We're going to transform it with a horizontal shift
2:33Horizontal means you can push it 4 tothe right Essentially all of these numbers are going to say pi/4 now. I'm going to cover that for a minute So now, the sine graph kind of looks like that Where that's pi/4 That's pi plus pi/4 This is five pi/4
3:01Thats 2pi plus pi/4 So it will be 9pi/4 But it still goes up to 1 and down to negative 1 Cause I took sin of x I could mutiply by 3 So that doesn't affect the x values It just makes it taller in both directions I could shift it to the right Which doesn't effect the y values It only effects where it shows up on the x axis
3:31And of course, what am I going to do next?
I am going to do both, okay?
We're going to do 3 sine x minus pi/4
4:07I get something like that Now it goes up to 3 And down minus 3 and moved pi/4 to the right How are we doing so far Loving this?
Not loving?
Ah good question! How many points should we show on the test?
Enough that it is clear to us that you know what you are doing
4:32So I would say at least 3 Somewhere between 3 and 5 Good the other two things you can show Where the maximum and minimum are Which you want to do is you want to demonstrate to us if you understand how to do the transformation You can lave the graph the same and change the numbers
5:02You can rearrange the graph We wont scale it, as far as I know we wont scale it I dont even know if there will be a graph We may give you a graph and say whats been done to it Which is the same idea or not You dont like that idea?
I am pretty sure we will have some sort of graph on the test I think so, I am giving you a non definite answer abou the exam
5:37Why dont we try one and see what happens?
if its sine or cosine theres more than one answer
6:01Okay thats a sign graph, what did I do to it?
Thats a 5 and thats a minus 5 That says pi over 6, 7pi over 6 and 13 pi over 6 Doing okay?
Well look what we did?
Its up 5 and down 5, so thats a multiple of 5 and I moved it pi over 6 to the right So its y equals pi sine
6:34x minus pi over 6. That's not so bad right?
Seems scarier then it was. Let's do a couple more
7:02Alright do that graph Ahh okay so We will do it here so you can see So what you do, you want to take the normal graph
7:30and you want to transform it This is zero This is pi, 2pi. Positive 1 and negative 1 This is the graph y equals sin(x) and now we are going to transform it to a new graph y equals one half, sin(x + pi/4) So what does the x + pi/4 do?
Move it to the left. How much?
The whole thing is shifted to the left by some amount right?
8:05By the way it keeps going We are just doing one curve So what happened? So this point zero moved over here So its now a negative pi/4 this point pi moved to the right how much? To the left, sorry. So you took pi and pi/4 to the left So you subtracted pi/4.
8:31You took pi and you took away pi/4 The other way you can think of it, you can say Same thing You can say x + pi/4 becomes pi and you can subtract Pi - pi/4. How do you find pi?
Remember its just fractions One minus a quarter, 1 minus a quarter is 3/4
9:02So thats 3pi/4. So thats this point And then 2pi also moves to the left So that one is 2pi minus pi/4 Which is 8/4 minus 1/4, which is 7pi/4 very good. Okay?
This is fractions
9:31This is why you learn fractions Uhh no that wouldnt work Okay I wont say it out loud essentially what you want to is you want to think of every point has this number being taken away from it Because you are taking the old access and moving it pi/4 to the left
10:04Now what happens to the amplitude, well instead of going up one and down one, it now moves up a half and down a half So we took this and we squashed it by a factor of a half
10:40Lets do a cosine graph By the way sine and cosine graphs are really the same thing of course What if I had that graph Y equals 2 cosine x minus 1
11:07Well lets figure out what cosine x looks like Cosine x 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
11:36and theres and pi and this is a pi over 2, ther Thats a 3pi over 2 So what does the 2 do?
It changes it and stretches up 2 and down 2 And now what about that take away 1?
What does that do?
The whole thing moves down one, perfect
12:01So one thing you can do to make your life easier If its shifted down one Draw a dotted line at negative one The whole graph just drops 1.
And then it goes up 2 and down 2 It goes up to positive 1 and then down to negative 3 Okay because before it was going up 2 and down 2 and drop everything by 1.
12:30Everybody see that?
So you go to the middle and go up 2 and dow 2 and it looks like that What did we do to this graph? well we dropped it and then we stretched it. or we stretched it and dropped it You can do it either way As long as nothing else is going on you can do it either way
13:15There are some of you that pay more tention, yes?
So first i took this graph,, when I multiplied it by 2 Now I went up 2 and down 2 and then I subttract 1 from everything so this drops to one and that drops to negative 3
13:36So I took this x axis and I dropped it by 1 so the whole curve comes down 1 and how did I know that happened? Well thats what that x minus 1 does. F of x and subtract the 1 from
14:04Oh 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 Good question, what would happen if it was a negative 2 cosine Not an inverse, square upside down Inverse is coming in just a few moments
14:32Okay when you say inverse you think reciprocal Dont think we really covered that Probably not 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 Lose people today
15:25Lets do another one of these
15:54No we have three things going on I think 3 is a very high level of san, but some maybe in the mood
16:09y equals 2cosine x minus pi/4, plus 1 So what do we have going on here?
Now the question is do you do the x transformation first, or the y transformation first? I always do the x ones first So what I do is take one graph at a time If you really want to do these, say first you take the cosine graph
16:35Kind of looks like that Thats just a regular cosine graph and move it pi/4 to the right So now My graph kind of looks like that So instead of starting it at zero it starts at pi/4 Instead of having the middle pi, the middle is now 5 pi/4
17:04And instead of 2 pi, its at 9 pi/4. And you keep going We are just taking a small piece of the graph, sine and cosines go on forever
17:31Now what do you do? You multiply everything by 2 So me can sort of just make that a 2 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 So its going to go up to 3
18:01and down to negative 1 and thats pi/4 thats 5pi/4 and thats 9pi/4 got that one?
and that one was y equals 2cosine x minus pi/4 plus 1
18:37Its okay if its not connected to axis. Because the truth is we moved the axis Whats really going on is this is the new origin we moved the origin pi over four to the right and up 1 So the chip is there We subtract, x and x and y and the y axis
19:03Well you should take enough points to show us that you know whats going on in the transformation Now we have to do inverses So lets learn how to inverse things The last major topic thats going to be on the exam I know
19:30So does everyone have this copied down so I canerase it?
yes? okay Turn my back and people leave the room
20:11So we can do a domain and a range, Okay?
So an inverse function basically works whatever we are doing in verse But you have squares and square roots So 4 squared is 16, square root of 16 is 4 So you are reversing the operation, called an inverse
20:33However, we can run into all sorts of interesting things when you do inverse The basic idea of an inverse, by start with a function and you take it and put in a domain a you get the range In an inverse function you put in the range and you get out a domain So you think of it as before you went through x,y and now you are going through y,x and thats what an inverse does
21:03Switches the number, x are the points in the domain and y are the points in the range It is a function of x So when you are doing an inverse you are reversing the operation So how does that show up practical?Well first of all YOu think about x and y, we go out x say you have y.
Now you say you go out y and go up x
21:31and if you repeat that a lot, what happens is you are reflecting acorss the line y equals x So now you get a line y equals x. You sort of flip the graph over So if you had a graph that looks like that The new graph...will look like this And if this is a dotted line, this paper gets folded on the dotted line, this graph will go on top of that graph Okay and thats what inverse do
22:03So 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 So now lets do a pack of them 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.
So this way or this way it doesnt matter
22:31But notice something important, this function passed a vertical line test and this function passed the vertical line test Suppose I take a parabola I start a parabola and pass the vertical line test And now I flip it across the line y equals x and now I get this parabola That parabola is going to fail a vertical line test So thats not going to be a function of x and now if I draw a vertical line I'll get two answers
23:02For every x value. Remember for a function you plug in an x value and I'll get one y value Stony brook Student id number, you plug in one stony brook studen and you'll get one id number If you plug in the id number you only get one student You take that and you fold it and you get that, its not very hard Practice on a piece of paper Take a paper and turn it and literally look through it and see what the inverse would look like
23:31okay, you take two corners and you flip it Alright so if you have it flip it like that Now its switched to the functions inverse 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 The problem is
24:00The first one is a function of x, and the second one is not So how would you know if you just looked at a function if it would fail a vertical line test?
The inverse will fail the horizontal line test.
You do a horizontal line test So you take an original function and take the horizontal line test, which means the horizontal line can not cut it twice If it fails a horizontal line test then the inverse will not be a function
24:30So if you take a function and pass the vertical line test thats nice but in order for its inverse of the function, it also has to pass the horizontal line testt and a function that passes both is called a one-to-one function So if you pass the horizontal line test will tell you if there inverse is a function
25:16and if f of x passes both vertical and horizontal line test Notice how to spell vertical
25:30i dont know know how people spell it without v Then it is called a one-to-one function A one-to-one function passes them both
26:00So far so good?
So Thats what an inverse is graphically , now lets find how to find one algebraically Suppose I have the function f of x equals 5x mins 4 Nice, Im going to do some nasty.. over 7
26:33Not very hard and I want to find the inverse of that function How do I do it Well remember what
27:01Id say you were 98 percent So If you take your time you will get the right answers. If you didnt hear her thats okay So we will show you anyway Remember what I say what was going on when we were finding the function verse the inverse You are going from x comma y to y comma x. You are switching the x and y coordinates So one of the easiest things to do it, write this A y equals 5x minus 4 over 7
27:30Switch x and y Step one, switch x and y 2. Solve for y and then switch back. Well in a minute tell you where the signs. Its an interesting short cut Okay you switch x and y. And you got x is 5y minus 4 over 7
28:11very good, okay Now I want to solver for y So I can multiply for 7 add 4 7x plu 4 equals 5y.
28:30and divide by 5 Its just x and y and solve for y and then you just call that inverse of x and your notation is for the negative 1 Hope everybody can see that
29:00So f inverse of x is 7x plus 4 divided over 5, The negative 1 means the inverse, it does not mean 1 over Not that you thought it meant one over, it does not mean one over Thats something totally different The reciprocal of multiple inverse, Do you know what a multiple inverse is?
Something that you multiply by something else, to get a multiple of identities or another words in english You multiply and get 1
29:30So the multiple numbers of 3 is a third so you take 3 and multiply it by a third, you get 1 We call it reciprocal because regular arithmetic reciprocal is the multiple inverse That was my math vowel So witch x and y, solve out for y and call it f inverse of x Sometimes webassign will ask you for f inverse of y All that means is when you put your answer in the box put it in terms of y instead of terms of x
30:04write 7y plus 4 over 5 I dont know why it does that So sometimes the webassign, pay attention because youll get it wrong and say I dont understand anything professor Kahn said to do I hate my life, Im not going to be a doctor, Im crying blah blah blah Just change the letters okay?
That means your going to be a doctor Its going to be good
30:49So lets practice with something slightly harder Lets find the inverse of that
31:00Think of this as y equals 2x cubed plus 7 over 11 Switch x and y 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 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
31:33So now we just have to isolate y. So we crss multiply and you get 11x equals 2y plus 7 Subtract 7 Dived by 2 and take the cubed root
32:03the cubed root of 11x minus 7 over 2 and that is the inverse. Yes?
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
32:31I mean technically you can take the cubed root itd be messy so you really want to have it by yourself How did we do on this one? You should then write this as f inverse of x is the cubed root of 11x mins 7 over 2 How do we know thats an inverse? Well lets see what happens 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
33:00Now here you take x, remember the last thing you do is divide it by 11 So the first think you do here is multiply by 11 Here you add 7, so here you are going to subtract 7 Here youre gonna want to multiply by two and here you divide by two The first think was cubed and the like thing is find the cubed root So you are versing the operation and you are reversing the order 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
33:32Everything was the opposite and the opposite work. Kind of like we were rewinding the video Thats the dvr Alright how bout one more of these?
34:17Okay very similar to the last one find that inverse of x Rewrite this as y equals 6x to the fifth plus 3 all/4
34:37well go this way this time First thing we do is switch x and y You get x is 6y to the fifth plus 3/4 See this is the kind of algebra stuff you just have to be good at You just have to comfortable with all this multiplying dividing and cubing and cube rooting and all that Cross multiply
35:06Subtract 3 Divide it by 6 and last but not least take the fifth root So y is the fifth root of 4x minus3 over 6
35:31Therefore the inverse Is the fifth root of 4x minus 3over 6 and as I said about webassign pay attention if it ask for f inverse of y, just change this letter y Otherwise youll get it wrong and you start losing the points and then you wont get enough points for webassign and what does that mean?
You wont be a doctor
36:16Alright so thats inverse, oh wait we can do more inverses
36:42I have y of sine of x Okay now I want to find the inverse Well you switch x and y and now you isolate y, you do not divide by sine. Sine is a function. Its not sine times y, its sine of y
37:05So the inverse is called arcsine of x On your calculator you'll see it written like this Okay those mean the same thing Inverse sine, arcsin Whats the problem with writing it this way?
Some people will say, doesnt that mean one over, when you raise something to the negative one?
So they get a little confused
37:30By why in the calculator do they use sin rather then takes up less space Real reason, it fits on the keys So most people in math world and of course you all are future mathematicians are going to use arc Okay it stands for the same thing, it means working backwards So why is arc sin x over here, its something tricky. Remember I told you about the horizontal line test
38:00Thats what sin of x looks like Correct so if invert it You something that looks like that Thats going to be a problem, its going to fail the vertical line test So what we do, we restrict sine of x The arcsine of x will narrow down the sine of x
38:31We only use this much, from pi over 2 to minus pi over 2 When the arcsin looks like this So when pi over 2, sine will equal to 1. So 1 arcsin is equal to pi over 2 Then negative 1
39:00equals negative pi over 2 Why do we do that? We can pick any piece of sine we want We can cut any s shape piece we want, this piece so that we get a one-to-one function So we find the inverse to define one place Thats going to be arcsin of x Lets do a cosine
39:32if we said the graph inverse sine, that would be a perfect, kind of perfect graph To get the point across, I've done better graphs in my life Ah so cosicant of x equals 1 over x Arcsine of x is backwards so.. The sine of pi over 6 is a half The inverse sine of a half
40:00Is pi over 6. So there arcsine of a half is pi over 6 So the sine of pi over 6 is a half So that means we know that if I know something is a half, whats the angle Id say the arcsin of a half is pi over 6 Okay but here you give me the angle I give you the sine, here you give me the sine I give you the angle
40:31Its working backwards The cosecant and cosecant is one over tangent Are the reciprocal and are very useful for when we have sine in the denominator of a function and you want to have the function in the numerator, you dont have it in the denominator, you use cosecant What about cosine of x Well cosine of x looks like that and I'll have a horizontal line to test the problem again
41:05Im only going to use the piece from zero to pi If I invert it It looks like that To think of it another way, I only used the first and second quadrant. So if you go in your calculator If you put in, find the inverse of cosine of a half
41:34It will give the answer of pi over 3 Or 60 degrees, depending on what mode its in And if you did it a negative half. It has some choices It uses the second quadrant angle You can use a different angle, but uses the sceond quadrant angle It doesnt really make sense why we pick those but you can really pick anyone you want So inverse sine, use the first quadrant angle or we use the fourth quadrant angle
42:03So lets right that down Got to find a new place to right that, it will go on this board
42:32By doing the inverse sine, sorry, or arcsine Im using arcsine of x, or arccosine of x. Can you read that over there?
Not really Can you guys see that over there?
Alright Ill move this, Ill put it over here instead only takes a second
43:00That psychology exams not going anywhere So if I have arcsine of x Or arc cosine of x and x is positive and we use the first quadrant angle So if I say what is the arc sine of a half?
43:30The answer is going to be 30 degrees because the whole sine of a lot of things is a half Sine 30, sine 150 and so on and so furth So they do that in radiants, Its always going to be the first quadrant answer So remember theres an infinite number of places where sine is equal to a half I only care about the first quadrant angle for negative Theres an infinite number of answers for arccosine I use the second quadrant angle
44:01and for arcsine, I really use the first quadrant, i really use the negative angle. So if I said arcsine of negative a half the answer is negative pi over 6 What I do is go down to there, okay?
44:52So do you guys understand what I mean, to use the first quadrant angle You can get all the answers you want. So if I say this sine of what angle is equal to radical 2 over 2.
45:02You say where theres a lot of that, okay Give me the first quadrant angle If I say the sine is equal a negative radical 2 over 2 Again theres a lot of answers, I just the negative fourth quadrant, negative pi/4 and i shouldve gotten the graph down when I did that
45:43Ill remind him to post it on blackboard
46:20So lets jut do a little more of this and then Ill stop and youll all go and eat
46:53f of x is 3sin x plus pi/4. how do we do the inverse Well first we switch x and y and we write this 3sine x plus pi/4
47:06And then you make it x equals 3sine y plus pi/4 So what do you do? You copy o the person next to you And you hope thats the right answer and heres the right answer divide by 3
47:30and now you dont divide by sine You do the inverse of sine, so you get arc sine of x over 3 of x Equals y plus pi/4, you just subtract pi/4 Arc sine x over 3 minus pi/4 is y, the inverse function I would never ask you to graph that
48:03Not sure I want to graph that How are we doing on these?
Do you think the inverse is bad? not so bad?
48:31Guys get the concept?
The transformation stuff? Cause its the last class before we go into review mode Let me make sure I covered everything You all are packing up Im pretty sure thats all we are going to ask ypu about inverse If theres new stuff on monday Ill hit you with it