This is Think Tech Hawaii, Community Matters here. It is Thursday afternoon folks, Ted Rawson here in our studio downtown Honolulu, Think Tech Studios with our show, Where the Drone Leads. And often you'll see on this table, you'll see rotorcraft of some kind, you might see a fixed wing aircraft, today you'll see something that is neither and that is both. And with it, we have also with us Josh Levy, Josh, thanks for coming on the show. Josh is the coordinator of unmanned systems in the Applied Research Labs at University of Hawaii. So we'll talk a little bit about this very unique table-spanning drone you've got here, Josh. Tell us a little bit about how it got here and what it's all about. Yes, it's actually a pretty fun story. So I could go back a long ways with this thing, but I guess... Where's the fun part? So the fun part is the family tie and the kind of very family-oriented connection that I have with this aircraft. So my brother works for a drone company over in LA called Flyway Aerospace. And so they actually built this drone and he physically built this drone himself. And he also physically came out and delivered it to us last week. And so you were out there the airfield with us. Personal service. Yeah, exactly, personal service to the door. Dan, thanks a lot, Dan. We got a shout out to Dan. Yeah, thanks, Dan. And yeah, so... Just some of your older brother, Dan, by the way. Older. And then, yes, you saw it when we came and opened it up. And he gave us a great tutorial that was very simple, very easy to use, kind of a quick checklist of how to set this thing up. And so... But the reason why this aircraft is so important, as Ted was saying in the beginning, is, you know, it's not a quad rotor. It's not a fixed wing. It's both. And so in that way, it can do all the things that both of those types of aircraft can do essentially at the same time using one more aircraft. So we're sitting at actually a traditional break in the technology from what we had in the past of just fixed wing or just rotorcraft and something that combines the benefits and values of both and has optimized in such a way that the user gets out of it more of what he asked either one of them for. And so we have a hybrid basically here. This is a combination and hybridization, to my understanding, took place in the mind of a guy named Dr. Trent Akazik at FlightWave. He was affected by trying to do some coral reef sampling in the Pacific Islands. He didn't have enough flight time on his quad rotors and was frustrated by that. He knew that wing-borne lift is five or six times, this one probably eight times more efficient than rotor-borne lift. And so in his mind, hybridization occurred. Probably not GMO hybridization, but plain old mechanical hybridization. So Trent has put together the vertical lift functionality and easy sky access and return of a rotorcraft, but the high performance of a fixed wing. Yeah, and so just to capitalize on that, so when we're out at our friend's house off of Kaava, you know how he has that property right on the beach there that has... That would be Craig. Exactly. Shout out to Craig. Shout out to Craig. So, you know, on that property, he has their telephone poles between him and the ocean and then there's a house on the other side, right? So you can't take off of the fixed wing there at all. You're going to run into some serious problems, but we were able to take off and land in a small area and go fly a large area of reef and come back and land without any chance of hitting anything pretty much. So you're not compromised in terms of operations anymore with this kind of a configuration? Exactly. You have the same footprint as the DJI product as an Inspire, the Matrice is very similar for landing pad footprint. And since this particular aircraft has in its DNA some testing on Lanai and testing on Maui and on Oahu prior to its current configuration, the concept of high winds and high turbulence, which we deal with all the time out here, is built into the design. Exactly. Yep. That's very stable in high winds, can sustain around 40-knot wind conditions and still be able to operate. And so looking at this thing from an aerodynamic perspective, as Trent would have done, he's put in here more aspect ratio than even a sailplane has in some cases, so a very low drag and very long endurance. And certainly a lot of brute power here in the vertical lift function, which we thank Ayam Boreh for helping a little bit with that out in Kailua. And so there's a lot of local knowledge in this, a lot of local experience built into this particular aircraft. And now it's at UH. And we're probably the first university user of this particular configuration or configurations like it. Yeah, I think so. I mean, so as you know, there are other hybrid systems out there but none are quite as effective as this, especially based on my experience and other peoples in UH's experience with other VTOL systems. And this one is a complex aerodynamic system, but it's been made simple to operate, about as simple as you can possibly make it to operate. Exactly. And that was the issue with some of those other hybrids that they were just too hard to fly. So people just kept them in the box, got them dusty, and that was all they wrote. But I mean, me as a, I've had a lot of experience operating multi-rotors, you know, quads and that kind of stuff, but I'd never flown fixed wings before. But when I started flying this, I was really surprised at how easy it was to move around, to operate, and then also most importantly, transition from the multi-rotor into the fixed wing and back in to multi-rotor again, into those different modes. I think that has a lot to do with how a trance approaches from a design perspective because the complexity of making that transition from wing-born to propeller-born or vice versa can't be something that the operator has to have to think through. It's got to be something baked into the design, baked into the flight management system so that it's relatively straightforward. At a certain speed, it transitions into vissue. Exactly. And that's what we've seen here and hats off to the designers, Trent and his team, for making that possible. Yep. So what we're also seeing, however, I believe, is a transition away from the popular quadrotor configuration which will remain popular in entertainment and in the hobbyist domain, that sort of thing. Getting something like this, we're talking about a sort of a professional level of operations. So you wouldn't want to, even though the systems have been simplified to the point that it fairly well takes care of itself transition-wise and such, there's still enough to consider here that you don't want to do it on a casual basis. Exactly, yeah.  And so we're in that transition zone, past that transition zone, into the commercial or professional operations area. In fact, you made an interesting point. I think Trent was out here a couple of years ago and we had a ship-side demo down at, with the Polar Star down in Pier 11. And the captain of the Polar Star said, I want this to operate at five years on my boat in 40 knots winds. I want to be able, as a Polar Star, was a Coast Guard icebreaker. I want to be able to fly out ahead and figure out where the ice is thick and thin and what the digital elevation model of the ice is and the structural bearing strength of it so I can figure where to push through it. So something like this would be a great asset. So that then brings up the point of the recording device, the camera, or the other sensors on it and the software that's tied to that. So speak to that for a little bit, Josh. How are we going to go forward in the world of the types of sensors that go on here and the software that does the analysis from those sensors? Right. And so that's still a little bit in the developing phase, especially with this aircraft. But the guys at FlyWave have definitely know what kind of trajectory that they want to take with this. And so what they've been doing is creating these adaptable nose cones where essentially you can swap payloads out essentially by just taking off the nose cone and replacing it with another one with a different sensor or different other payload type. And essentially just plugging it back in and then going off onto your next mission. So we have plug-and-play payloads that are balanced to keep the center of gravity where it has to be.  Exactly. And based on the configuration, the aircraft that is speaking to the ground control station should understand what the payload is and then what kind of various missions are capable of with those payloads. So whether it's mapping, whether it's search and rescue, whether it's something else in between, you can be able to do that with relatively little changing on the manual side. You know, that brings up another point that to me is most interesting. We had an experience last week and we're going to have it again this week with that grounded vessel off Waikiki. And the means by which you actually fly a mission to collect information either by conventional cameras or infrared or whatever it might be, that has to be properly structured consistent with the objective of the user who's going to get the information back. If that information is a software package that does a certain kind of analysis, then the collection has to be structured in kind with how the software works. So we're making that step into a more complex operational and analysis domain as well, which is interesting because the pictures that are taken to go on the TV news which show general scenes and general perspectives of what's going on aren't necessarily useful or usable in the analytical world where you want to know, is that ship damaged more? Is there a change in configuration? Is the hull twisting? How about the coral reef? Is there damage on a reef? How about the sedimentation from the crush coral? Where's all that going? That type of question and the answers associated that are necessary to answer it aren't going to come from just casual splitting about with the drone. Exactly. Yeah, it's not going to be the pilot that's going to be making those decisions. We have to be working with those end users to see what they need so we can then go collect data that they're going to be using to make decisions. So end user plus the software that's going to be used to generate their answers that is going to have to determine what the flight path is and actually the sensor mechanism itself. Exactly. So we work backwards from the desired end state to figure out what the steps are to get there. No longer just going out there and flying something and seeing what we got. Yup. So coming a method... methodological... And it's all at least the fact that you're our single guy doing this. You're our single point of failure, Josh. Hopefully that's not going to last forever. Well, what means we need to either clone you or we need to put an educational system and a training system in that has people start to recognize this. I think the training that we've seen today is principally associated with getting a 107 certificate a pilot certificate and that has almost nothing to do with how you want to use it and the very discussion we're having. And this is for all people involved in education. A structured end user oriented framed progressive way to teach this complicated process I think is your obligation as I recall from your job assignment. I think you're right. Okay. I think you're right. But you are getting in your own way. You're... The fact that you haven't done that has made it impossible to replace you at this point in time. Exactly, too busy flying. Too busy flying these things and doing the work to actually put the program together to instruct others to follow on behind you. I know, slowly. I think we're definitely making progress. There's a lot of interest, especially here at the community college level and even at UH as well, to get the stuff up and running. And so we're slowly going to be turning that cog that's pretty heavy as a lot of momentum. So it takes you energy to move it. What's really interesting, again, to me, a lot of things are interesting to me, but we see there's a step in technology that suddenly inspires a new whole new level of capability which requires the perception and the training and that runs along for a while. Then there'll be another step in technology and that'll move along. But this very configuration here, a hybrid configuration, so opens the envelope of utility for this kind of system that it really pushes it. The technology's here. Now we have to work on the other side of it, the applications and the training. And then every time the analytical software goes from the professional package to the free package, suddenly it gets broadcast to a wider audience and now we have a new level of capability on the analysis side. And that'll then turn into back into the equipment and define a new requirement for equipment. We have this cycle that's going to be jumping around as we go forward here. Yep, and it's going to make it really interesting. So, yeah. And again, this is a totally interesting point. Right here on the show, we're having this transition both in the analysis part and in the configuration part. And it's even properly configured and colored for Halloween being orange and black. So what a triple point has acquired here and arrived here thanks to Trent and I.M. and Phil Madoverie and others who were involved in this whole chain. So we'll come back and talk about where this future is going right after we get back from our break. Cool. This is Think Tech Hawaii, raising public awareness. We have this crazy thing going on today. I was just walking by and all these DJs and producers are set up all around the city. I just walked by and I said, what's happening, guys? They told me they were making music. It's a little talent and then sat down and kind of it's really nice. I saw it do it. You can be the greatest. You can be the best. It is still a new hour, folks. Jen Ralston here and Josh Levy at our studio downtown Honolulu Think Tech Studio with our show, Where the Drone Leads. And we're talking about the drone taking us in the direction of hybridization. Josh Levy again, who's the coordinator of unmanned air systems drones at UH. Josh, welcome on board again with your incredible find here, which is completely filling the table. Ray, we're going to need a new table, I think, in the future here on the show. Nice to have it at the beach here in Kailua. Are we Waimanalo today? I'm not quite sure. Looks like Waimanalo to me. Anyway, we're talking about this very incredible transition point in terms of a drone configuration from just a rotor craft or just a fixed wing with their own mindsets and mentalities to something that combines the two of them. And suddenly the world is open in terms of the range of missions and functions that can be performed by something like this. So let's talk a little bit about what might be coming right around the corner here in terms of supporting our customers in Coast Guard Pacific Command and even the Department of Emergency Management here in Hawaii. Yeah, so we have a lot of projects coming up. And so one of which is the fishing vessel grounding that occurred off of Waikiki. So the Coast Guard is very interested in looking into using UAS to understand a little bit more about the condition of the vessel, the condition of the reef, all that kind of stuff we spoke about previously. And then something else that we're looking into is developing a mission for aircraft, surface and underwater vehicles to be operating all autonomously to help solve some kind of a problem. So in terms of the emergency management section and severe weather that the U.S. has been getting recently, that's a pretty hot topic. That's a good point. Certainly the global sea level rise and global warming is generating all kinds of storm energy and such in the Pacific. And a lot of inundation, a lot of islands that are being dragged on and that means populations are going to get displaced. They're going to get pressured anyway and displaced. Agriculture gets displaced. Cultural issues get threatened. Structural issues like ship moorings and such like that are going to become a challenge. So we need to collect information about that. But you can't necessarily put a person at risk to go collect it. And you may not even have the right perspective from a person in a boat or at low altitude or a person in an airplane at 10,000 feet. So something like a drone fits right in the middle here and can provide superb collection. Yep. So if we want to sort of the pictures that we put up. So I'm just showing you some of the imagery that we collected just a couple of days ago. Looking out from Craig's house is actually doing some test flights out there. If we keep going a little bit, we'll see. Aha. That's exactly what we're talking about. Okay, here we have the effects of sea coast erosion on Kamehameha Highway in Kauawa. And this perspective gives you a way to look at the land form or the ability of that land slope to hold on to itself or vice versa to the city and county to come in and do something to reinforce that road. So this perspective gives you incredible information. You can't otherwise get. Yep. So yeah, looking from above. So that's actually the initial image. The one that we saw previously was just zoomed in. I managed to find it in the bottom right-hand corner. So here we could look after a tidal wave or something or a tsunami. You could look at any bottom surface adjustment that made as a result of the way of washing ashore in terms of access, in terms of stuff in the water, obstructions, or any pollutants that might be entering in from some uncovered landside source. Exactly. And so that requires more than just having an aircraft collecting all this data, right? So we're going to have to have essentially a swarm of vehicles, of autonomous vehicles, air, surface, and underwater to be able to collect a full picture of what's going on environmentally during these situations. Then you've got to pull that information together and have it deconflicted and correlated and time-snapped and all these things that turn into something that somebody can use. And that brings up an amazing observation here. Once we start depending on systems like this and the analysis they produce, now we have a reliability requirement for the vehicles and the software systems to operate that way and operate that way consistently in the weather, as the age, and this sort of thing. So there's a design requirement that comes in from a function and reliability perspective driven by the dependence that we, as the public, put on these systems. Exactly. And especially as we're going to be using these systems more and more, right? I mean, not just us doing the single mission, but us as a society, right? Just depending on all these autonomous cars or that kind of stuff, we get this level of reliability that they just don't have yet. And that's the sort of thing that engineering gets into pretty well. But it doesn't require a large footprint like a Boeing factory or something like that next door. That kind of engineering can be done in the lab. So in terms of occupation for people in Hawaii where we don't have large aerospace facilities, that's an aerospace function that could occur here. That is the determination of analysis methods, the software, the analysis itself, software as a service. All those things are potential future careers for folks here in Hawaii after we graduate them through your course, once you build your course. Which will happen at some point. Okay, all right, and on we go. So once again, I keep repeating it here, but we're at actually quite a transition point here. This kind of technology available to us is going to so change the landscape of the future and push on issues such as collections, such as the software analytics and such. And then actually that will also turn to the incident command system where information is ingested and decisions are made. And we'll have to figure out how to pull this new form of complex information into that system so people can make decisions at the right time and with the assurance that the information being used is correct. Yep, exactly. So nothing like exercises to find that out. And I think every time we've done an exercise, which is probably 10 of them in the last year or so, we always find out something very elementary or simple that we could have thought of, but it didn't really show up until you tried the exercise. Yep, that's what testing's all about, right? There you go, and so we're going to try that next week. We will. Want to tell us a little bit about that? Sure, so essentially what we're going to be trying to do here is taking a page out of University of Porto's book. That's Porto in Portugal. Okay. Where they're already doing a pretty amazing job connecting aircraft, service and underwater vehicles to conduct operations. So what we're trying to do out here is have our own, create our own rapid environmental picture. Rapid environmental picture. R-E-P. Which means we're going to collect information about the environment consistently, commonly and rapidly and form a new picture that we haven't otherwise seen. Yeah, correct, to make some kind of decisions for whether it's emergency management after a big storm or some other environmental location that we don't really have that much information about to rapidly go and collect that so we can then go do whatever we need to do in that area. And the environment doesn't necessarily refer to the science of invasive species necessarily, which you often think of as environment, but this means what's the topographical environment? What's the axis look like? What are the water sources look like? What about health issues in terms of a polluted well or something like that? So anything that is in the environment, whatever it may be, is subject to be understood or needs to be understood in order to put the right picture together so that reaction can occur. So I want you to describe that experiment that we're looking at next week. Sure, so we're hoping to conduct some test runs off of working with Billow's Air Force Base over in Waimanalo. They've been extremely kind of receptive for this idea and actually pretty excited about seeing all this stuff come together, right? Because they want this data almost as much as we do. They're looking forward to seeing what we're going to be collecting with this aircraft. A surface vehicle called a WAMV, which we decided today is a wave adaptive modular vehicle. That's off to Zach and his gang. Zach from the UH. And then a small ROV with just an RGB camera. And so using all three of these aircraft simultaneously to collect different types of information about a certain area, we start piecing together a pretty interesting picture about the terrestrial topography, the underwater bathymetry, and then also collecting water samples and other kinds of information that are going to help us understand a little bit more about our environment. And that's, again, is going to be a good test and a challenge of the system of bringing information from multiple sources that didn't ever necessarily work together before together into a common picture. So the whole issue of expression and visualization and extraction, they are the next phase of discovery and development that has to occur promoted and pushed by the functionality and capability of something like this. And it's underwater equivalent and it's surface equivalent. So now we're going to get flooded with information. And so artificial intelligence, virtual reality, a lot of modeling and simulation, those are the areas of development that this technology is going to push on. Exactly. Yeah, so this is kind of that first step in this whole situation. I mean, the big data problem and the AI, all that kind of stuff is still a really interesting task that people are working on. Now that we have the platforms to be able to collect that amount of data and that quality of data, so everything's going to start to move pretty quickly, I think. And so we're talking about, at the end of the day, we're talking about big data. These things are makers of the pieces that become big data. So I hope that Noah Haffner is watching this and is feeling guilty about what he's going to have to perform in the future here. It's funny you should say that. So I got a text from our boss, Mark Redwood, this morning with Noah CC saying, on average, how much data does a UAV collect during an hour? So this is the exact situation that we're thinking about right now. And converting that data into something useful is at the end of the day what we have to think about. The data by itself doesn't mean much. Now I'll just go through a quick analysis, years and years ago, like way, way back when the Ice Sage had just left it, I was a young engineer. We had a new airplane called the MD-11 coming around, which was the first of Douglas's electronic airplanes. And I had to stage a dinner between the chief pilot and a bunch of engineers to see if I can get conversation going because pilots think one way, engineers think another. And finally at the right time in the dinner, it showed up and one of these young engineers said, oh, now we got all this electronic stuff coming to the cockpit, we can paint the sky with data. How much data do you want? What kind of data can I get for you? And his chief pilot looked down at this young engineer and said, you know, I want to know just two things. The house is getting bigger, which means I'm heading for a house or the house is getting smaller. That's all I care about. I don't care about your data. I want to know, am I at risk of hitting a house? That was, he was using that a bit from an expressive point of view, but that was the point he was making. Hey, data doesn't matter. What matters is, is there a risk here that I have to attend to? So that's what we have to think and we need to get Noah going in that direction. Exactly, and that's that ties right back into the end user needs, right? Exactly. I don't care how well this flies. I don't care how well the camera collects data. I don't care how much. And they shouldn't care about that. That shouldn't be their job to worry about that. That's my job, unfortunately. Exactly, right? That's about the fourth job you've defined for yourself, your job. Exactly, yep. Okay, well, anyway, this brings us to the conclusion of this exciting phase here. I guess we've concluded the transition phase. We're now in this new domain of unmanned air systems operations, and it's really exciting to see here. Thanks to your special pursuit the last year or two, working with Trent and I am in such and pushing it forward. I am out there, Trent, and others who've been helping push this particular technology forward. This is going to be a game changer, absolutely. And so, Josh Levy, thanks for coming on, and we'll get some air under this guy and go do some good work next week. For sure. Thanks a lot. Thanks.


Question from the homework. Page 289-ish. Yo, 10. Love to. Okay? Often what they're going to do is they're going to give you part of a graph. So can you all look at number 10 and I think I like this question. Okay? You want to have in your mind, Victoria, this bad boy, there's sign and this one I always have a tougher time drawing. This bad boy, there's cosine. Okay?  I said this is what we need to know because when I glance at this, I think they've given me a fraction of one trig wave, not a whole wave. And I think starting up high, I think it's cosine. So the first thing that I would be saying is, even if they hadn't said cos right here, I'd be going, it's cos. What's my amplitude? So I would say this for starters, 5 cos. Now I want to find B. Remember B is 2 pi over the period. And what I need to do is figure out what the period is. What fraction of a cosine wave have they given me here? I think half a wave, that much. See it? And that's pi by 9. Half the wave is pi by 9. How long is the whole wave? Don't overcomplicate it. Half the wave is 1 pi by 9. How long is the whole wave? 2 pi by 9. Is that okay? Yeah? So I'm going to say this. B is 2 pi over 2 pi by 9. How do I divide by a fraction? Flip it and multiply. Yeah. Dividing is not times. It's flipping and multiplying. So this is going to end up being 2 pi times 9 over 2 pi, which is quite convenient. The 2 pi's cancel. And in fact, B ends up being 9. I guess that's what it should say in the back. Okay? Similar for this one, by the way, for number 9, what fraction of the sine wave is this right here? I think half the wave. So I can get the period. Amplitude I can certainly get. Okay? Any others? Yep. 12 B. Any before 12? Earlier stuff went good. Did I sign 12 B? Yeah, I did. Okay.  And we want to make sure we're in radians. And you got 12 A, okay, yes? You said it was going to be equal to y equals 8 sine of. The period is 3. The period is 12 T. Now, remember I said that B is 2 pi over the period. What's the period here? If you're really lazy and you really want to do this quickly, you can just put a 2 pi over 12 there, because we're going to go to our calculator. Now, what is 2 pi over 12 in lowest terms? Okay. On your test, if this is multiple choice, the answer to pick from would be pi by 6. But I'm less interested in A. For part B then, here's what I'm going to do. I'm going to go to my calculator and I'm going to put a 4 in for T. I meant as why I said I'd get my calculator out. I'm going to go like this. 8 sine. Oh, it opened the bracket for me quite conveniently. 2 pi times T4 and then divided by 12 because there's a 12 in the bottom. That actually tells you how high the tide is. How high the sea level is. 6.9 meters. That's it. The other thing they can do is instead of giving you an X value, because this is sitting where our X normally is, they could give you a Y value. We'll talk about how you'll handle that. I think we'll be talking about that today if there wasn't one in the homework. Any others?  I said I actually just draw it with the axis with no numbers and then I just adjust the scale, the numbers. Unfortunately, that starts to break down as a trick in today's lesson. Can you please turn to lesson 9, graphing trig functions or transformations of trig functions part 2. All we looked at yesterday was expansions and reflections. Expansions compressions. Stretches and shrinks and flips. And I said to you, by the way, when it comes to trig, we don't do horizontal flips very often at all and you'll see why in a while. In this lesson, we're going to consider graphs of sine and cosine that look like this. In particular, we're going to start to have slides, but we have to give them special names. So in the first part, we're going to concentrate on C and D. Describe how this graph here compares with this graph. How has this graph been moved? Troy, 30 degrees to the right. How has this, don't write that down. How has this graph been moved? Is the two inside next to the X? How would I show that it was inside next to the X if it was? Okay, we're going to put, we're going to be a little fussy, we have to be in brackets. It's not, so it's vertical. Two what? Two up. Okay. What about here? 60 degrees left, one down. What about here? This one first. 45 degrees right and not 45 degrees, but 45 up because it's next to the Y and everything's backwards. We're going to give these different names, which is why I said not to write that down. We call a horizontal slide a phase shift. We would actually say this has a phase shift of 30 degrees. Skip the second one for a second. The third one has a phase shift of negative 60 degrees or you could say 60 degrees to the left. I'll take either or. Or you could say 30 degrees to the right, but we're calling it a phase shift. This has a phase shift of 45 degrees to the right, so positive 45 degrees. Having said that, we're going to get tired of writing the words phase shift, I'm going to use PS as my abbreviation, if that's okay. What did you say this one was? Two up. Okay. When you move a trig function, a periodic function up or down, we call that a vertical displacement. This has a vertical displacement. I don't feel like writing vertical displacement. D period, D period. Okay. D period, D period of negative one. Or you could say justine one down. I'll take both. Vertical displacement of negative 45. Positive 45. Okay. So let's look at this little chart here. Let's do the horizontal phase shifts first. What's the phase shift here? It's a trick question. None as a number, please. Zero. None as a number is zero, right? What's the phase shift here? Pi by four. What that does is it takes your letter S, the sine curve, and moves it pi by four to the right. What's the phase shift here? Zero. What's the phase shift here? Negative three pi by two. And now we're going to try and generalize this. What's the phase shift here? See, what's the phase shift here? Still see. What you're going to have to watch out for again is an alarm bell if this B is not factored out. It's C as long as you have this lovely little set of brackets right there. If the B is not outside the brackets, we'll have to do our little alarm bell. What's the vertical displacement here? Zero. How about here? Zero. How about here? Five. How about here? Negative pi. How about here? D and D. That one I kind of like because I think vertical displacement D works for me. A for amplitude works just great for me. B for period. Well, B and the letter P kind of rhyme. Start. C for phase shift? I got nothing, but it's phase shift. Okay. It says, would you expect something similar for cosine? Yep. What about for tangent? Ugly cousin? Turn the page. So this page here is one of those lovely bookmark posted star asterisk pages. This is a summary of all of the graphing of sine and cosine. It says this, changing the parameter C on sine and cosine as long as it's all factored results in a horizontal phase shift to the right. If C is positive to the left, if C is negative. Changing the parameter D will do the following. A vertical displacement up if D is positive. A vertical displacement down if D is negative. You can also get the vertical displacement from a graph using this equation. Now this doesn't, this looks kind of confusing. You'll see it does make sense once you have a picture in front of you. Ah, here's the money page, right here. That's the one that summarizes all of our graphing. Let's see if we can understand all of this. First of all, for sine and cosine of Y equals A, sine or cosine of B bracket X minus C plus D, the amplitude we said highest minus lowest divided by 2. How far apart are they? Divide that by 2. That's your amplitude. Period. 360 over the absolute value of B or 2 pi over the absolute value of B. Why absolute value of B? Because we said that period is never what? Negative. Oh and amplitude. Why did I say absolute value of A? Because amplitude is never what? Negative. The phase shift is C and it's the same stuff as last unit. If you have a minus and then the positive number right there it's to the right. If you have a plus and a positive number right there it's to the left. The vertical displacement D up or down and you can find D by doing that. We'll talk about how that works in a second. We're gonna add one more thing by the way over here. If B, sorry, if the period is 2 pi over B what's B? 2 pi over the period. I always write that in because if they want me to find the equation they'll tell me the period and I need to find B. If they give me the equation they'll tell me B. I don't need to find the period. And both of those are totally fair game.  Amplitude, all reels. A technically represents a vertical expansion or a vertical compression but we really aren't gonna deal with it all that much. Period is not 360 or 2 pi over B. It's 180 or pi over B. Period is pi. Yeah okay you can have a phase shift and a vertical displacement but we really don't deal with that much either. We will deal with this a bit because this here and a period change will move the asymptotes and the domain around instead of the gaps being at 90 pi by 2. If you slide it sideways the gaps move with it and that to me will be a good question. I might say hey where are the new asymptotes? Or what's the new domain? Remember for tangent the domain was what X couldn't be and the asymptotes were take the domain and instead of not equal to equal to or that was where the asymptotes were. Let's try some of these but this is a good page to you know keep track of. Example 3. Consider equations of that form there where A and B are 1 so we're not gonna add an amplitude or a period change yet. Write an equation which represents a cosine function having a horizontal phase shift 75 degrees to the right. Well Y equals what's the amplitude here? Oh they told me 1.  What trig function do they want me to write? Let's write the word cos. What's B here? 1 so what's the period not 1? 2 pi okay bracket 75 degrees right X minus 75 degrees okay. And by the way nice review of transformations as well. A sine function having a horizontal phase shift of 3 pi by 5 radians okay. Y equals we said amplitude is 1 sine bracket. 3 pi by 5 radians left X plus 3 pi by 5 and then a vertical displacement 4 units up plus 4. So a good multiple choice question would be something like number 4 where I give you an equation and I ask you to tell me probably two of the four. I'll say tell me the phase shift and the vertical displacement or the phase shift and the period. I can make lots of nice wrong answers because there's four different numbers in there for you to pick from. Good little multiple choice question. So let's list everything here. It wants us to find the amplitude, the period, the phase shift and the vertical displacement. What's the amplitude? Sorry? 2. What's the period? It's not 3. 2 pi over 3 remember period is 2 pi over B, B is 2 pi over the period. So 2 pi over 3 and if that reduced I would but I'm good with that. What's the phase shift? You can either say negative pi or you can say pi left. How do I know it's in radians? Not just because of the pi if I could be degrees how would I show it was pi degrees? There's no degrees symbol okay. You'll notice I was very fussy here I put this I put the degrees on the 75. Vertical displacement negative 4 or 4 down B amplitude period phase shift vertical displacement. What's the amplitude? 2 over 3 or negative 2 over 3? 2 over 3 almost certainly if I gave you a question for amplitude on the test I'll make it negative just to see if who will pick the negative one and I'll have that as an answer to pick from. What's the period? Well I got to do some arithmetic. The period is going to be 2 pi over a quarter. How do I divide by a fraction? 2 pi times 4 over 1 and now it's top times top bottom times bottom. I guess the period is 8 pi which makes sense to me because what we would have said last unit was horizontal expansion by 4 so instead of 2 pi 8 pi 4 times around the circle. Phase shift pi by 12 you can write to the right if you want to. Vertical displacement positive 3 or you could say 3 up okay. Now before we turn the page or move let's go back to this one here. What's my vertical displacement 4 down? So remember the cosine graph which started up high you would move it the whole thing 4 down. What's my amplitude 2 if I was sketching it from once I moved everything 4 down I'm not going to go one up one down one up one down what's my amplitude 2 up 2 down 2 up 2 down I would keep that in my mind as well. Phase shift I would move everything 2 pi by 3 to the right so instead of starting right on the y-axis up high I'd be starting 2 pi by 3 over. I'd have to do something with a period. We're going to talk about how you sketch these but we're going to do that mostly later on a bit this class mostly next class. Alright example 5 says find the amplitude the period the phase shift and the vertical displacement says y equals 2 sine bracket 3x plus pi minus 4.   Phase shift is not negative pi I need to factor this 3 out I need to rewrite this which is why I left the space as 2 sine bracket factor the 3 out bracket bracket bracket minus 4. What the heck is going to go in here well every when I factor a 3 out of a 3x I'll get an x when I factor a positive out of a positive it'll stay positive. When you're factoring you're dividing pi over 3 so what's my amplitude 2 what's my period 2 pi over 3 2 pi over b what's my phase shift negative pi over 3 or pi over 3 to the left but certainly not pi vertical displacement negative 4. Is there an alarm bell in B? Do I need to yell? Try B on your own. Fractions? Don't panic. Are you becoming mediocre at fractions now? I like it. I got those. I'm willing to bet that at least five or more of you are just bursting to ask me where the heck that came from so if you're not sure you can ask I don't mind explaining it maybe everyone yes okay when you're factoring you're dividing so when I factor a 2 out of here and out of here it's the same as dividing by 2 which is why there's no more 2 and dividing by 2 dividing by 2 means putting an extra 2 in the denominator there's already a 2 down there you'll end up with a 4 down there so this has been a 3 6 it's been a 7 14 okay see we are becoming mediocre at fractions turn the page tougher much tougher much more challenging is writing the equation if they give you the graph if I give you the equation and say tell me about the graph all you have to do is memorize what the letters stand for if I give you the graph and say tell me the equation you really got to follow along and so I'm going to slow down I'm going to say if you're tired I did a yell a little while ago wake up think up yes area you too okay so here's what it says graphs a through d represent the same trig function and they do somebody I think it was in this class last day said something they said mr. duick I notice sign and cosine look the same you could slide sign over and it looks just like cosine as it turns out any sine graph I can write that as a cosine equation if I want to any cosine graph I can write that as a sine equation mathematically if I want to and not only that any positive sine graph I can write that as a negative sine graph if I want to and I mean negative cosine graph I can write it as a positive cosine graph any graph has at least four equations any trig function you can write it at least four different ways and that's without going all co-terminal like that's without going phase shift all code there's an infinite number of answers so this graph here we're going to write four different equations and we're going to use the instructions to kind of tell us our starting point says this it wants us to write this as a sine graph where a is positive so it wants us to write this as a positive sine graph now a positive sine graph over in the little margin right up here do this that's a positive sine graph except what I'm going to do is I'm going to erase that line there because now we're sliding graphs back and forth what I'm going to say is a positive sine graph is where this graph starts in the middle and goes up so I'm going to look along here and now it's in the middle in terms of its height right there but what direction is it heading down the first point that it's in the middle and starts going up is right there I'm going to put a little red dot right there that's where I'm going to sort of start my stopwatch as it were just that's where I'm going to start to count this graph from I could start all over the place which is why you can get so many different equations but since they wanted positive sine I know sine looks like this can you see it okay write it in this form y equals what's my amplitude okay total distance is 8 half of that for sine the period is 2 pi so B is 1 they didn't put a B here here's what I want you to notice if I want to draw this as if I want to write this as a positive sine graph what's my phase shift I gotta think how many squares make up pi count 6 so what's each square worth pi by 6 how many squares over to the right has this graph been moved x minus 2 pi by 6 and I'll draw a little arrow I'd write that as pi by 3 but just in our notes so we know if the heck we did we count that's that graph as a positive sine function what if they want me to write it in part B now in part B they still want me to write it as sine but Alex they want a to be negative they want me to write this is a negative sine function again I'm gonna do a little sketch negative sine looks like that and again I'm gonna get rid of this axis just to remind myself that's the shape that I'm looking for so I'm gonna look around and find where this graph is middle-high but heading downwards and I think the first point is that guy yeah y equals amplitude for oh heading downwards negative sine bracket x what's my phase shift now how many squares for left or right plus 4 pi by 6 is because that's what each square is worth by the way what I write 4 pi by 6 what would I write if this is on a test if I gave this to you as a multiple choice 2 pi by 3 I'll do a little arrow 2 pi by 3 okay Dylan here's what this really means if you graph that on your graphing calculator is y1 and you graph this as y2 only one graph would show up there same graph oh by the way back to here if I had wanted to I could have actually started counting right there is that also where it's in the middle and goes up so you could have Jesse phase shifted one two three four a bunch of more squares over let's decide from now on we're gonna try and keep our phase shift to the smallest easiest one I had a student about five years ago who delighted in phase shifting about 5,000 off the page every time and making me double-check that his mouth was right was Willie his math was right and after about three quizzes it was stopped being funny and just started being annoying because I had to go do all this arithmetic you can phase shift back and forth just so all of our answers look the same closest phase shift to zero please not only can I write this as positive sign not only can I write this as negative sign I can write this as a positive cosine equation write this as a cosine where a is positive now again I'm gonna do my little sketch positive cosine looks like that but I'm gonna get rid of this I'm looking for where this graph is up high it's up high in two locations here or here this is the closest one to zero isn't it I think justine I'm gonna put a big red dot there that's where I'm gonna start counting this graph from if I do that as my reference point it's gonna be y equals positive what's my amplitude still for cosine what's my period still to pie so what's be one bracket X what's my phase shift this time Tyler left or right so what am I gonna put here minus five pi by six could also have done plus seven pi by six and that would start right here same graph this is the weird thing about periodic functions Jesse's there's an infinite number of equations that can be drawn because they repeat themselves which makes it either should be an infinite number of equations that you can do because you could just keep moving over until it starts to repeat again and you kind of start your stopwatch right there so to speak hey I can write this as a negative cosine graph as well now a negative cosine graph instead of starting up high we'll start where yeah it's gonna look sort of like that so I'm gonna find somewhere you know what I think I'm gonna pick this one it's almost no phase shift at all and in fact what I'll teach you is whenever possible if you notice that right on the y-axis you're either up high or in the middle or down low use a phase shift of zero it's way easier and adjust your positive or negative sign or cosine to it this is gonna be y equals negative coasts amplitude still for cosine x plus pi by six all the same graph for different equations which ones right they all are usually I'll give you a hint I'll say write it as a sine function or write it as a cosine function not always usually one will be way easier I'll be honest if I had a choice I might have picked D because it's a nice small phase shift I find the mistake kids make the most is they count squares wrong because they're in a rush that's me why on the page okay let's put the whole package together ready here is a trig function I can write this as sine or as cosine or as negative sine or as negative cosine it's all gonna depend if I wanted to write it as sine where I start my stopwatch or as cosine or as negative sine or as negative coast but we're gonna have to have some things in common what's the amplitude let's see how high does this graph go one how low negative five total distance six amplitude three what's the period I find the easiest way to count most of these is from either a peak to a peak or a trough to a trough yes those are the actual fancy words how many squares apart is that count carefully 12 is it 12 yeah what's one square worth look at your scale how'd you get that how many squares make up pi three of them so one square is pi by three so if I hear you the period is 12 pi by threes oh this one I will tidy up what is 12 pi by three and lowest terms 4 pi okay I don't want the period right next to the period in the margin here what's B because that's what it's gonna show up in our equation what's B what was the expression for B 2 pi over okay let's write that down 2 pi over the period which is gonna be 2 pi over 4 pi oh what's B I'm gonna write down here B equals one half that's what's gonna show up in my way now we're gonna come back to the phase shift in a second what's my vertical displacement now the vertical displacement is where the center vertically of this graph is how high are we one how low are we negative five find halfway between them now while you're thinking that look up that's what this meant find halfway between the top and the bottom that's how you find actually there's an easier way there's an easier way did we already get the amplitude Alex then go three down from there or go three up from there but that's your middle what is the vertical displacement not to negative to you know what all of you right now put a little dotted line right there I always do that the vertical displacement is negative to and the reason that's so important is that's where the y-axis the x-axis has been moved to that's the middle of my graph that's where sign is gonna begin sign starts in the middle and positive sign starts in the middle and goes up that's the smallest possible left right phase shift for positive sign what is the phase shift negative pi by three or pi by three to the left now I can write the equation but I need to write big so I can't fit it in here I'm gonna write it right here y equals three amplitude which trig function did we commit to here sign be a half bracket x phase shift is pi by three to the left or to the right plus pi by three close bracket close bracket minus two that's that equation B sorry nope B is positive periods always positive B is gonna be positive that's why I said we're not gonna deal with the horizontal reflection very much at all hey what if I wanted to write it as a negative sign graph I'm gonna change colors I'm gonna go blue negative sign still starts in the middle but instead of heading up which way is ahead you know what right there folks the equation's gonna be nearly identical look up just watch for a second the y is not gonna change the amplitude is not gonna change the sign is not gonna change the B because the period is not gonna change and the vertical displacement is not gonna change what is gonna change two things negative in my phase shift how many squares to the right how many squares to the right always counting from the y-axis right from zero one two three four to the right minus five pi by threes sorry what is that a question see is this the same equation same is the same graph as the previous one thread it is cosine everything but the phase shift is gonna be the same amplitude three period four pi B one half vertical displacement negative two now positive cosine starts where up high remember the little sketch that we did right there yes you could start over here but it wants us to do the smallest phase shift you could start over it wants us to do the smallest phase shift how many squares to pi by threes so let's write it as a positive cosine y equals equals what three cosine B is a half bracket x we're to the right minus two pi by three minus two what if I want to write this in part D as a negative cosine graph well okay y equals negative three coasts that's not gonna change I'll put a negative in front of the three the one half the period is not gonna change the x is not gonna change the minus two way over here is not gonna change where would negative cosine start bottom right there yes Miguel or right there but that's further how many squares Miguel left or right plus four pi by threes if you were to graph all four of those equations very very carefully you'd get one graph okay almost done on the page says consider sine and cosine and there's a b c and d a b c and d which of these parameters a b c or d if you change them affect the domain well what is the domain of sine and cosine all reels hey all you guys like this because you need to stretch anyways everybody like this don't hit someone and stretch while you're at it feels good here's my domain if you change the amplitude is that gonna change your domain we do this is that gonna change the domain at all no what about what's the next one be if I change the period now that might make it happen more often but will that still make it all reels even if I squish it or stretch it okay so that's not gonna change the domain what about the phase shift if I move it left and right will that change the domain no what about the vertical displacement if I move it up and down will that change the domain you know what the domain of sine and cosine always is all reels I probably won't ask you that boring range what's the range normally between sine and cosine how low how high okay so everybody all of you put one hand like this there's negative one there's positive one there's your range okay if I change the amplitude will that change the range ah okay let's write that down hey okay back to our normal range if I change the period that's horizontal isn't it I don't think that's gonna change my range okay what about if I change the phase shift if I move this back and forth is that changing the distance between them no oh D vertical displacement if I move this whole thing up or down will that change the range yeah because instead of between negative one and positive one the distance will be the same but you might be between negative two and zero okay so D does three is more obvious which one changes the amplitude a which one changes the period me which one changes the zeros the what the x intercepts okay now I got to think a little bit more here when in doubt because it takes all of one second okay if I change the amplitude will that change the zeros what about if I change the period by making this not as long or not as short will that move these three points yeah I think the period will so I'm gonna add say B what about the phase shift if I move that whole graph back and forth yep what if I move that thing up or down yeah in fact I can move it so high that it has no zeros um oh what if I moved it so high that it had no zeros but then put a huge amplitude in front of it could it get the zeros again turns out actually the amplitude if there's a phase shift can also change the zeros no phase shift amplitude won't do a thing sorry I said patient vertical displacement vertical displacement if there's a vertical displacement well then the amplitude can also affect the zeros because you could be so high that you have none put an amplitude of 10,000 in front of it suddenly you do have a zero it'll reach down that far state the highest and lowest values of the function in terms of a b c and d so they want high values that's x that's x I'm not even gonna look at those they're talking about vertical here what they're really saying is tell me the range okay for sign and for cosine here's the range the first thing you do is you move it up or down the vertical displacement so let's pretend the vertical displacement boom to there d high first and then from there you go amplitude down and amplitude up you say that again say that again to find the range Dylan algebraically and I probably might give you an algebraic one with no numbers there's your vertical displacement that's the center of your graph and from there you go up amplitude and you go down amplitude the lowest you get is start d high and go down that far less than or equal to y less than or equal to start d high and go up from there that's your range the algebraic one is tougher I'm just thinking right now what I should have done is skip b gone to c first and then d b would have made more sense but let's look at c it wants us to find the range let's see if we can do this in our head what's the vertical displacement negative four so the whole graph has moved four down look up four down we're right here negative four what's my amplitude that means from negative four I'm gonna go three up and three down I go three down from negative four what's my lowest point gonna be negative seven three up from negative four and now that we did that maybe that makes a bit more algebraic sense through a lot at you today okay hopefully it ties in sort of to the transformation stuff that we've learned already except stacy because these graphs are periodic phase shift vertical displacement and the fact that there's multiple answers for an equation because they repeat themselves number one is good number one good good good because that's not in brackets alarm bell good alarm bell okay and then we're gonna practice getting the equations so number three or Victoria this is very similar to the one that you asked me this is only a fraction of a wave okay it wants you to graph it as a positive cosine where does positive cosine start you'll have to figure out how long one whole wave is but again Victoria I think they gave you a half wave because I think the whole cosine graph looks like that so I think half the period double it there's your whole period and now you can figure out B okay five is good which graph haven't I talked about ugly cousin I'm gonna assign six with the expectation I'll probably have to talk about it but see you can get it okay remember what I taught you about tangent seven I'm gonna say how about a and D eight nine no ten there you go graphing trig functions


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