so this is one of the topics in this class that uh most connects to math 111 if we just took math 111 last term and are kind of really up to speed on what we study in math 111 it makes this a topic like this in math 112 go much more smoothly if you uh sort of skipped math 111 because of placement testing um and didn't take it uh for you know maybe you you haven't studied that math for a while it'll just possibly be a little more painful studying uh this topic because you'll have to kind of memorize things rather than see them as just a kind of obvious a consequence of what you learned in 111 um i think it it as i'm suggesting it goes much smoother if you're familiar with the math 111 stuff so i you know emphasize these notes that i've got from math 111 you're encouraged to click on that link or uh well yeah if you access the electronic version if you haven't printed it you know if you access the electronic version of this document the link should work and you can get to those math 111 link notes and study the graph transformation sections maybe just brush up on it review that stuff what i've done here is basically copied and pasted the uh big conclusions we get in math 111 about uh graph transformations um so this is kind of the the after we've investigated and figured everything out we would then package all of the facts together in this box um there's a fair amount of information here you know it's a maybe a couple weeks of math 111 where you wander around with the different graph transformations and kind of end up with all these conclusions my plan today is to kind of literally think about these transformations with uh from the perspective of the sine and cosine functions um just that process might help us review some of that stuff the 111 stuff but our goal honestly is to end up just filling in the table in the bottom and once we've got this table filled in even if you have some some weaknesses or gaps i guess in your math 111 knowledge you can still do well in this class drawing the graphs of these functions or working with the graphs of these functions it just might feel a little bit like you've memorized stuff rather than you're just you're applying facts that you already understood from uh math 111 but by the end of uh this week um you i hope that everyone will feel comfortable uh graphing and working with um sinusoidal functions um whether or not you're uh you know you just took math 111 or not so um don't want to make anybody like scared about the math 111 stuff but it does go a lot smoother if you are familiar with the graph transformations we study um in math 111. so um this phrase sinusoidal this phrase sinusoidal functions sort of obviously is connected to the word sign um it's it's uh suggesting a a sign like function it's really it may be another more directly what it's saying is it's a transformation of this sine wave a sinusoidal is a distorted version of this sine function and it turns out that the cosine waves are also sinusoidal because what it means to be sinusoidal is that you can transform this sine wave into the graph that you're looking at so a sinusoidal function is just a transformed sine wave and we've noticed earlier or just i guess it had been last week when we defined sine and cosine and looked at their graphs we notice that the cosine graph could be understood as just a shift of the sine graph that they're very similar waves and they're just offset by pi over two shift so cosine is a shifted the graph of cosine is arguably just the graph of sine shifted horizontally so cosine becomes sinusoidal because cosine is a shift of the sine wave so whether we're building our function using sine or cosine we'll call them sinusoidal the graphs of such functions will um be just you know waves like this um the kind of standard wave shape we've seen in the graphs of sine and cosine they'll just be stretched and shifted and you know moved around a little bit but they'll look like these uh sine waves that we started seeing last class so we'll end up figuring out what the numbers involved in uh like we we should know already you know basically what sine is sine is the name of the function and assigns an inverse function and t is the input variable to use the input variable all of these other letters a w h do i have another color and k all of those numbers have predictable uh effect on the graph of the sine wave and so we're going to uh investigate what effect they have on the graph and end up filling in this table and once we have the table filled in we'll just be able to use the facts in this table to work with sinusoidal functions so i guess with that i'll go ahead and transition to page two i i wasn't sure what the best approach here i i thought about just sending you to the online notes for the preliminary stuff so and you may have already done that um this is different though than the online notes because it's uh i don't know it's a little more dynamic because we'd be in theory we'd be in class together working on this um so i thought we could kind of explore um the graphs of of uh adjusted versions of the sine wave so um the game here is that here we always start with one of the the primary sine or cosine functions and then we'll we'll consider a distorted version of that sine wave and in this case all that's different is the little 3 in front of the sine function and what we'd want to do here is just try to bank on what we studied in math 111 to predict the the effect that that 3 will have and when we multiply the outside of a function remember that we have uh t is the input um so the the the stuff when we affect the inputs will be affecting horizontal stuff we've already put a t into the sine function and gotten an output and then the three is multiplied by the outputs and what we learn in 111 is that when you multiply the outside of a function by a constant the graph will be vertically distorted so compared with the original graph of i called it f of t equals sine of t compared with this graph the adjusted function 3 sine of t is stretched when we multiply we stretch or compress by a factor the factor is really just representing how we're multiplying by three by a factor of three and i've kind of already made a little mistake like there's a a directional thing we want to conclude we want to stretch vertically we'll stretch in the vertical direction because the 3 is multiplied by the outputs not the inputs where if this we're not multiplying the t by three we're multiplying the sine of t the output value by three and what happens then is that every output is just tripled so when we had outputs that were at the height of one now those outputs will be at the height of three when we had outputs that were at the height of negative one those will be three times bigger a factor of three different than they were before so they'll be at the height of negative three so our uh maximal values that that used to be at one will now be three times bigger at three our middle minimal values that used to be negative one will be it now be at negative three and then the points that are on the midline right along the t axis these points when you stretch or multiply the output which is zero by three it's still zero so because zero times anything is still zero so when you triple the value of the outputs when they're right along the y the t axis you don't move the points at all and then we can piece our graph together and uh this is just the kind of beginning of this but you encourage you to kind of pay attention to how i draw these graphs i'll draw them in quarter waves so i have a kind of a chance of hitting my dots and getting the appropriate curvature and what happens then when we distort a graph at least this one this particular one the original midline was the t-axis and since we've stretched the function away from the t-axis the uh midline has not moved we have not distorted the midline all we've distorted is the amplitude now we stray three units above the midline and three units below the midline so the the amplitude is now three units after performing that vertical stretch and this turns out to be one of our most more straightforward uh transformations um because we when we multiply the outside by a constant like this the amplitude just becomes the constant or at least that's what it seems to be here i'm kind of overstating it the idea would be to go into desmos and check do some more investigating let's see if we change that number three to other numbers like a a generic number a and let a change to all sorts of numbers and see if we continue to get the amplitude value of whatever that number a is this is hyperlinked and it goes to uh a desmos file i'll have to switch sharing that you guys can access too um it's a saved file in desmos i'm not sure if it'll you know would be worthwhile doing right now while we're working together but if you ever wanted to come back to this this the file is available to you you can still use that link i've saved this desmos application so what i'll do is i'll graph the sine function and that'll be in this blue color and then i'll graph a times the sine function and right now i've got a little slider that's forcing a to be one so it's the exact same the blue and the purple blue and purple are the same function the colors i'm not sure how this will look in video but i i purposely chose blue and purple because they're like related to each other so that we could try to see like these two similar colors how they the two two similar graphs behave the first thing we did was on the the paper we put a virtual paper we put three in and notice that the purple graph the purple one is going to be the distorted one the blue one's the original the purple one has now been vertically stretched by a factor of three and so our amplitude is the only thing that's been distorted the period is the same right it's still if you start at this dot here at the origin and do one and hump up one hump down you end up at two pi the same place the blue graph ends after one period we haven't distorted the graph horizontally so we shouldn't expect the period to change the only thing we've done is multiply it by a constant in the vertical direction so we basically should expect the amplitude to be distorted like that and what i can do now is return it to a and use the slider to let a get bigger and bigger and bigger and bigger and you see whatever number a is that's the amplitude of the function so if we jack it up to five the amplitude becomes five and it kind of shows how easy it is to manipulate these sine functions uh the sine function because the amplitude used to be one originally was the number one any amount we multiply one by uh the one turns into that number so we can get any amplitude we want by just choosing an a value that we want and notice if i go below one we compress vertically it just gets uh less amplified um the amplitude is smaller you right there the amplitude's one-half instead of one so we can make our amplitudes bigger or smaller but then one just by adjusting that value for a maybe i'll quickly go back to one hide the sine function just to make sure cosine's the same so cosine will be red i think throughout this activity and a times cosine is orange so when i release the a times cosine you see orange on top and if we let the a become a big number the orange function gets vertically stretched and all we do is distort the amplitude and so we hopefully are willing to come to a conclusion here on our handout that whatever this number a is that a vertically stretches the graph by a factor of a whatever number it is and since the vertical information on the graph original graph strays from 1 to negative 1 since the original output the original amplitude straight between one and negative one these perfectly manipulatable numbers one is such an easy number to work with because whatever a is you multiply one times a and you get a out and so these two graphs if we distort the sine and cosine functions by a factor of a like this they'll have amplitude a units where a is this a number not a word and just with that we can already get started this is gonna be a bit annoying flipping pages back and forth on a video but whoa especially if i do it like that but if we come back to page one at the bottom of the table now we know what the red number a does the red number a it represents the amplitude of the function so after we vertically stretch it by that factor of a the graph will have have an amplitude of a units so even if we weren't familiar with math 111 uh graph transformations it's not going to be that hard to sort of memorize this fact whatever numbers out in front of that cosine or sine function will be the amplitude it's just a lot nicer if you can understand it as a vertical stretch because then it's kind of kind of almost common sensical why you end up with the result you do now the next little chunk of examples is um related to this first one all i've done here is again used a multiplier on the outside multiplier on the outside the difference here is i've tried to use a negative number on the outside thus far we'd only talked about multiplying by a positive number on the outside now if we multiply by a negative number i'll try to squeeze stuff in here i fear i've really not given us enough room it's trying to be efficient with the size of the document so if we try to graph the function y equals negative 2 sine t it turns out that a reasonable way to interpret the number negative two as as being the product of two numbers negative one and positive 2. so we can sort of factor the negative number as the product of the kind of purest negative number negative 1 and the kind of size of the number two the two will do exactly what we just studied the two doing the two will stretch vertically by factor 2. every single point on this graph will be stretched vertically by a factor of 2 first before we do the negative 1 i'm going to do that this number negative two and two steps a sequence of two transformations so first i'll do the green one the green one will stretch everything by a factor of two so the amplitude initially will go to two all the values at one or negative one will become two or negative two and then the values along the midline don't move because when you stretch zero it's still zero and so i'll connect these dots as a preliminary step i we have not graphed the desired function yet we've just gotten started after we stretch by a factor of 2 in the vertical direction now we can go and do what the negative one does and what we learned in math 111 is that a multipl a negative multiplier will take all of the y values that used to be positive and make them negative and all of the y values that used to be negative it'll make them positive this causes a reflection about the horizontal axis in this case the t-axis so what will happen is that points um down here in the negative territory or all of this graph that's in the negative territory will flip up over the the t-axis and become positive outputs and all of the stuff that used to be positive will now be negative so we're going to flip the graph so this this orange dot will be flip over and become a red dot in the in the maximal position and this orange dot that used to be in a maximal position will now flip over and become a minimum value and the points on the midline again don't change because negative zero is still zero and so now we get this graph and this one the red one is like the combination of can i erase something in there the combination of green and red in a sense it's actually the final result almost want to erase the green one just so that we can really see the result maybe i'll do that to emphasize that that red one is the graph that we wanted to find and notice that because of the multiplication by 2 the amplitude is 2. so we don't really need to worry about the negativeness of negative 2 when we consider the amplitude the amplitude in fact it turns out to be the absolute value of that number a the amplitude the uh the amp is the absolute value of negative two which is the number two so that's how we we should have actually adjusted our result on the previous page i really should have wrapped that in absolute value bars just in case the value had been negative what the negative part of that number does is it flips you over the t-axis and so what we want to do is observe what that does to the graph and this reminds me of something that i probably haven't done a good enough job or haven't really had an opportunity yet to emphasize um like a crazy person like i intend to do i usually do notice that the original sign graph the blue one here um is there well let's just look at that original sign graph the original sine graph is going uphill as it passes through the midline along the t ax the y axis there the the original sine graph starts at 0 0 and goes up the original sine graph starts at 0 0 and goes up out of the midline that's something that i i strongly encourage you to internalize about the sine graph compare that with cosine just to real quick see how these are different cosine starts at its max and comes down from the max now when i say this concept start start isn't really a meaningful thing because these graphs go left and right forever they have domain all real numbers so there's really no starting place but it's natural reasonable kind of normal to think of the the y-axis as our starting spot it's our like friendliest place it's when um the input value is zero and so although the phrase start is a little bit false i encourage you to still think about the behavior of the function at the y axis sine starts at the middle and comes up cosine starts at the max and comes down now when we reflect the sine graph over the horizontal axis notice that it no longer starts at the middle and comes up the red one starts at the middle and goes down starts at the middle and goes down if you reflect a sine wave you shift or change the way it quote starts quote quote starts it's not really starting but the concept starting the behavior along the y-axis a reflected sine wave starts on the midline just like the sine function does but it travels down that's the way it's different from sine sine starts at the midline and travels up negative sine when the a value is negative starts such a function starts at the midline and travels down instead of up now if we go and consider the cosine situation the cosine graph we were given or the function we were given was y equals negative 4 cosine t same idea here i'll uh break up the number negative 4 into a product of the simplest negative number negative one and the sort of interesting number involved in negative four which is the four the fourness of negative four and we learn that the four stretches vertically by factor of four and so we could maybe get that one out of the way right right now we could stretch every uh output value by a factor of four so this output's at the uh height of one so it'll end up at the height of four heights of negative one will end up at the height of negative four so we get our maximal points plotted at the heights of four and negative four and then our midline values don't budge at all because they start at zero when we're essentially multiplying zero by four and so it's still zero so step one we could think of as giving ourselves a stretched cosine wave but that's not the graph of this function because it only involves the for this we haven't done the negative oneness this will reflect about the horizontal axis and so now we take all of our points that used to be at a minimum and flip them over and make them at the maximum values and the points that used to be maximums flip over and become minimum values maximum values minimum values and the points along the midline don't budge because we're taking negative of zero essentially and we can connect these dots and we get a reflected cosine the 4nis it determines our amplitude is 4 but the negativeness changes the cosine behavior from starting looking at this little the original one in black there it starts at the peak and comes down from its peak after a reflection it will start at its minimum start at its minimum and come up from the minimum a reflected cosine wave starts at a minimum and travels up from that minimum this is only useful if you have already uh absorbed the idea of uh cosine starting at the peak and coming down like that's the sort of memorized idea about cosine in comparison with sine starting at the middle and coming up and then if you reflect you have just these exact opposite sort of behaviors all right um i don't know if there's a there's a nice place to put that in our table just yet so let's just go ahead and move on to the next little type of transformation this one's a pretty straightforward one that middle section is a little bit confusing because we're involving this additional concept of a negative multiplier but now we're just back to adding or we're going to look at adding we haven't looked at adding yet we're going to compare plain old cosine with cosine of t plus 2. when we learn in math 111 is that if we add the constants to the outputs of the function this isn't a t plus 2 we're not adding it on the inside to the input value we're adding it to the outside after we've cosined the t value we've gotten a cosine value then we're adding the two so we're adding it to the outputs and when we add stuff or do anything to the outputs we should expect a vertical consequence and in this case compared with the original the graph of the adjusted version y equals cosine again this is just uh building on 111 so it's only uh sensible they don't think this way if you you know currently you know remember the math 111 stuff but what we learned in 111 is that plus 2 there on the outside of the function it will i need to try to figure out if i need a verb there i think i should say is the graph is shifted up to units this would mean every single point on the graph shifted up 2 units turns out when you're trying to do this you can have pretty good success if you just focus on the easy points like the i want to say corner points but there's no corners on these waves they're all curvy and smooth but if you focus on the places where it intersects the midline those are kind of core uh easy to locate or important to locate i guess points and then the max and the mins and the maxes and the mins those are key points if we move all of those dots into the appropriate locations and then connect them we should have a pretty good graph so i'll just literally shift every single dot up two units hopefully i'm doing this right and then connect this is such a flat wave like there's so little sort of amplitude that it's a little hard to get a nice graph i did it like this so that we could shift it up we could have room to shift it up and still fit in the coordinate plane um and i didn't uh only leave room up instead of bound i wanted it to not be obvious which which direction we need to shift i don't really like this one so if we shift it all up it's a kind of a weak graph it's a gives us the idea every single point has shifted up and uh since we've shifted it not stretched it you know the amplitude is still the same we're not straying more vertically we're just start of starting higher vertically it turns out that the thing that's moved here the only thing that moves is that original midline the original midline was y equals zero so certainly that will move up two units because every single thing will move up two units when we do this so after we shift the graph up two units the midline moves from y equals zero to y equals two exactly two units bigger than it used to be otherwise the period is the same the amplitude is the same we're not doing any other kinds of distortion other than shifting it up and what we learn now that when we shift up we change the midline and lucky for us in a sense the midline just becomes that number this number 2 and this number two are the same and we sort of get lucky like that it's not really luck but we get lucky like that because the original function had a very smart there was a smart decision with the original function by giving it a midline of zero any amount of shifting we do will the that amount of shifting is the midline right the midline of our shifted graph will be y equals that number two that we shifted and it turns out this is this will be be a consistent behavior if we shift over to desmos and close these out and slide down to this little section where i've got the plus k and the plus k so we just worked on was a cosine situation so we could start there if we draw the graph of cosine and then look at the shifted up version of cosine but k right now is zero so we haven't shifted at all if we shift up two units uh we get a graph that is every single point shifted up two units and so we can see what the midline is actually you know what it wouldn't have been such a probably would have been smart to have graphed y equals k uh oh i haven't made a k yet because i put a 2 there right so if you put this k back in so if you draw draw the midline it kind of helps you follow what's what's going on if i let k get bigger and bigger and bigger notice it just moves rigidly up and it takes the midline with it whatever k is if i stop k at five you see that the midline is five um and we can make k a negative number and then we'll just have a a negative uh midline k is negative three the midline is negative three so again these these this functions are pretty easy to work with or relatively tame because of some of the decisions that were made when they were defined having a zero midline makes shifting very easy to interpret however much you shift vertically that is your new midline and it be having an amplitude of one made stretching vertically easy because no matter how much you stretch that is your midline i'm sorry your amplitude so we could also look at the sine graph it's the same thing if we bring it up to zero the color here isn't as good because the graphs are blue but if we let uh i the purple one on top has the plus k so if i let k be a positive number it just rigidly shifts up and or if k is negative it rigidly shifts down so we see that whatever that value k is is our midline and so if we look generally about at cosine plus k or sine plus k the these graphs will have a midline of y equals k whatever that number is whatever the vertical shift is becomes the midline and we could scroll up to the table and recognize this new fact the value that we add to the outside of a sine or a cosine function becomes the midline midline y equals k so even if you didn't know about vertical shifting you could certainly learn to work with these facts but it's nice when it's just a sort of natural application of math 111's material so sliding down then uh into the top of page three another type of transformation is when we multiply on the inside by a number we have not looked at multiplication on the inside have this number 2 on the inside that doesn't appear here what does that 2 do well since it's on the inside since it's multiplying the t the input variable we know it must have a horizontal effect on the graph we're distorting horizontal information and so we should expect there to be a horizontal consequence and the it's a multiplication not an addition or a subtraction when we multiply we stretch or compress by factors so we should expect this graph to be stretched or compressed in the horizontal direction by a factor that's associated with the number two it turns out that this is one of these graph transformations that tends to uh strike students as being the opposite of what it's uh supposed to be or something like that people often memorize inside horizontal transformations as being the um opposite of what they sh that the common sense would have led us to think so the two inside there uh naively we might think this graph is doubled in the horizontal direction and it's like stretched horizontally so the period might get stretched out and doubled but it turns out that that doubling inside does the opposite so it causes a compression so compared and we'll obs we'll observe this on a desmos in a second but compared with the original graph the graph of y equals cosine 2t this number 2 is the key term there that 2 will be stretched or compressed in this case it happens to be compressed is maybe there should be over is compressed horizontally or horizontally compressed might be a nicer phrase it's compressed horizontally by a factor of 1 over 2. the inside multiplier behaves reciprocally the factor of stretch is at the reciprocal of the number that you see inside the function let's just go ahead and draw it and then you know maybe we can you know continue to talk about what that's going to why that's the case and that sort of thing although really this just comes comes to one of those things that is a you know a challenge in math 111 um but at this point we kind of like to try to just you know use what we've already learned so what i'll do is all with that phrase compressed horizontally compressed horizontally means that every point is sort of half as far away from the y-axis the y-axis is the center of the compression we compress toward the y-axis the phrase is honestly a little bit ambiguous if you from which perspective do you compress it's that y-axis perspective that we compress and so we take a point that is at pi over two and move it to pi over four half is close a point that's at pi gets half as close so it's at pi over two three pi over two leaps over to three pi over four uh two pi ends up at pi 5 pi over 2 is 5 pi over 4 that's a little bit harder to find on there now when you take this point here on the y-axis and you stretch it toward compress it it's toward the y-axis by a factor of one-half when you take the x-value involved there the t-value sorry the t-value involved there and multiply it by one-half the t-value right here is zero one half of zero is still zero so this point doesn't move at all and then these points crunch closer to the t axis half is closer than they used to be and these are the points we were able to move so then we can connect our dots and observe what's left over you know what kind of graph did we end up with well the amplitude hasn't changed right the amplitude is this uh amount of vertical strain that the graph achieves the blue graph and the pink graph both have an amplitude of one unit nothing's changed vertically of course it hasn't we haven't stretched it or shifted it vertically at all all we've done is compressed it horizontally and again since we haven't done anything vertically we haven't moved the midline so the midline's still the same thing what happens when we horizontally compress or also it would happen if we stretched it what changes is the period in this case our period is now half as big as it used to be the period here is the distance between peak to peak that's a way to measure the period this is now only pi units while the original cosine wave had a period of two pi units the period the period of the function is a horizontal distance it's the amount of space uh between um sort of the start of the period and the end of a period in the horizontal direction it's a horizontal interval the length of a horizontal interval so when you compress the graph horizontally by a factor of one-half you will definitely com compress the period by that same factor because the period is horizontal information so the period used to be 2 pi cosine comes for free with a period of two pi that period of two pi has been compressed by a factor of one over the number in that function and then we can do this little math problem and find that the period is pi units as we see on our graph so let's try then to check out desmos for this idea so that's i have it i have a mid liner in there that's these these uh graphs here did i okay so i started with cosine it seems so there's our original cosine and this will be our uh horizontally stretched or compressed cosine wave the orange one right now w is one notice if i change w to the number two it's compressed this graph has been compressed by a factor of two notice that uh this is hard to show i'm not sure what you can see but the the orange graph does two full waves in the space of one wave for the red graph the orange graph is achieving periods twice as quickly as the red graph that's actually a way to understand the two two compresses the graph in this kind of physical sense it's half as big physically but there is something that's doubled there is a doubling and that's why there's a two in there what's doubled is the pace at which behavior is occurring we're getting periods twice as quickly as we used to get periods so there is a doubling that happens it's kind of like the uh what's doubled is the the the pace at which things are happening um so then the graph appears compressed because we're getting more information and sort of less t values and fewer t values um i'll go ahead and put w back in oh by the way w is is uh in our uh kind of formal resources w is used because it's it's not w it's omega it's the greek letter omega in math 111 we we'd use i think it's usually like b at least in our book i think uses the letter b um it doesn't matter what letter you use in there but for some reason uh um trig a context use that uh symbol w um we you know there's a or it's actually again not the symbol w it's really the greek letter omega but the the letter we use is really kind of meaningless um it's just that whether whatever numbers multiplied on the inside there and if i let that number get big notice that the graph accordions in on itself it gets compressed in when the multiplier on the inside is a big number we don't stretch the graph we compress it and if we go in the other direction uh uh if we i've got to get to the number one i'm gonna get to one ever we get to one if we let w be a small number maybe it's a little easier to think about it being like one half up one half one half would if if w is the number one half you'd flip that number in the reciprocal of it and get two the graph is horizontally doubled notice that the orange graph is very slowly completing periods it takes four pi units to complete one period after we uh when we use a w value of one half i think by doing that i lost the slider ability though so it's kind of cool to just play with it when you're stressed out just come play accordion with sine waves okay and the same thing happens with that was a cosine but same thing happens with sine if we let it become one now as w gets bigger we compress the graph we get more periods in the shorter space of time and when w gets small we get fewer periods we're stretching it out it's taking longer to happen so that's a way to think about it if the numbers on the inside like 0.5 we we say the graph is being stretched horizontally and often that causes confusion because you think well 1.5 should make something smaller not bigger what's getting sort of smaller is the speed that stuff is occurring in look how slow the purple one is you know when you start at zero the purple one goes up and down it takes two pi units just to get up and down while the the blue one did a whole period in two pi units so when you have a small w that value being less than one suggests sort of a slowing down um so we can see it something has gotten smaller but it's more it's time that's going slower rather than uh physical space being smaller so i left more room here because i want to try to emphasize this idea of computing the period so w it turns out okay the w the multiplier on the inside has everything to do with the period the only thing that gets affected by that multiplier on the inside is the period but the w value that number in the function is not the period notice that in our first example here the number was 2 but the period is pi the the number in the function is not the period of the function the period does not appear in the formula for a sinusoidal function for example let's start with our simplest sine function sine of t this has period 2 pi units but there's no two pi anywhere in the function when there's a one here when there's a one here you get a two pi for free the period is two pi when there's a one in there if there's a different number in there that number will distort the intrinsic period of the function it'll distort 2 pi so if we have a kind of generic number in there inside of inside multiplier these functions uh let me start with a little bit of text kind of the graphs of these will sorry will be horizontally stretched or compressed it just depends on whether or not omega the or w is more bigger or smaller than the number one will be horizontally stretched or compressed by a factor of one over that number since the period is horizontal info the period will be stretched or compressed in the same way by the factor one over w whoa i hit page down the period will be compressed by a factor of one over w therefore the period of these functions by that phrase what i'm referring to these functions this one and this one when we put a w an omega value inside of cosine or sine the period of those functions will be p equals the original period of cosine and sine which is 2 pi if you use sine or cosine you just can't avoid their intrinsic properties they have properties whether you like them or not one of their properties is a period of 2 pi if you multiply the input value by a va a constant like this omega that number will distort the period by a factor of 1 over w i would suggest or prefer just leaving it like that a lot of people simplify this formula and call it 2 pi over w of course that's the same thing but for me this boxed expression jives with math 111 this is just applying what we already know you know when if assuming you know we already know it it's not a new thing it's just a direct consequence of what we already know so i leave it in that way if there's a going back to our first page if there's a multiplier on the inside that multiplier will distort the period of the function the period you don't have a choice if you write sine or cosine you're already starting with a period of two pi that period will be distorted by a factor of one over that number in the function so that gives us the period and i think with that well there's obviously more to do the next thing is horizontal shift horizontal shift and then the hardest thing of all which is combining stretching and shifting in the horizontal direction or stretch compress and shift in the horizontal direction so we've started our investigation of horizontal stuff horizontal distortions this is the first one multiplying on the inside horizontally stretches or compresses and that distorts the period and next we'll want to look at shifting and then combining shifting and horizontal distorting okay so in this next set of examples we're going to consider what happens when we add or subtract on the inside of a function we're adding or subtracting these numbers uh on the inside we're affecting the input variable t and whenever we affect the input variable we should expect a horizontal consequence on the graph and adding versus multiplying always results in shifting adding and subtracting causes shifts multiplying causes stretching or compressing so we should expect to shift these graphs a rigid number of units like the entire the shape of the graph won't change we'll just slide it left or right and this is one of those things that we confront in math 111 as being somewhat standard kind of naive intuition leads you to the wrong assumptions i guess so we could see subtraction inside there and the concept subtraction tends to make us feel like we're going to go to the left direction because left is where negative stuff appears but it turns out that this kind of adjustment inside there will cause us to shift certainly horizontally because it's an inside change but rather than using a generic word like horizontal we can use the directional term like right or left to specify the type of horizontal shift so we'll shift right pi over 3 units it turns out that the the minus is on a deceptive i guess and the amount of shift is this positive number on the other side of the minus sign and we can look at desmos to confirm this but hopefully you do recall that from 111 that this kind of shift will cause us to go to the right not the left and so every single dot will shift to the right pi over three if you study the scale here this is pi over two that's um pi over two is there so this must be a third of pi over two this must be pi over six i'm not going to be able to successfully put these in very well but just to suggest what we've got in there the first three ticks are pi over six pi over three pi over two if we were in degrees that's 30 then 60 the 90 30 degree multiple uh uh the sticks are a space the 30 degrees apart or pi over six radians apart and so it's that second tick here that's going to represent pi over it's two ticks two ticks is pi over three units how about if i say like that two ticks in the horizontal direction is pi over three units and so what we're going to want to do is shift every single point in this graph to the right two ticks basically this point that used to be at or is at zero zero now will be slid to the right and b at pi over three zero and this tick that's at pi over two one will be shifted to the right two ticks it'll be a little harder for us to do the math here and figure out exactly where that dot landed because we're adding a pi over three to a pi over two and it just doesn't you know might not be on the tip of our tongue how far we've shifted but if we take every dot and shift it to the right two units every dot and shift it to the right two units right two units right not two units but two two scale ticks two scale ticks will give us the pi over three shift and connect our dots and as we probably expected we're getting the exact same shaped graph which just slid over a little bit so we're still getting a sine looking graph but it's just slid over in fact that idea of where the sine wave starts after shifting to the right pi over three units the graphs quote starts at pi over three it starts literally at pi over three and comes up from the midline at pi over three actually i just said it starts literally at pi over three there is no literalness about the starting concept it's just a uh an idea that helps me kind of keep track of behaviors i learn that the sine function quote quote starts at the origin and comes up and so if i'm supposed to shift to the right pi over 3 units i'll go over to pi over 3 and have that exact same behavior happening at pi over 3 that used to happen at zero some the way that i'll encourage us to keep track of this information as we'll see as we progress now in this situation there's a plus plus in kind of naive intuition would be to go to the right instead it ends up being shifting to the left so this will cause us to shift left pi over 4 units and so we look on our scale and in this case the scale's a little different there's only two ticks before pi over two so each tick is a pi over four sized interval like a 45 degree equivalent 90 degrees as we go around the circle if we think of it in degrees so we're just going to want to shift every blue dot to the right one tick so the key tick is the one at uh the north i mean not the north pole this is this isn't a circle at the the maximum value for the cosine graph up there at the peak that point will shift to the right to pi over four where pi over four is aligned the t value pi over four and then we'll have our cosine behavior starting quick starting at the pink dot instead of at the black dot we'll come down from the max starting to oh did i just do all that backwards is anybody telling me that no so i went to the to the right but of course i was just testing you seeing if you were paying attention we have to go to the left negative pi over 4 will be the t value if we shift to the left pi over four units and our cosine wave will start at negative pi over four instead of starting at zero and we can just proceed to just shift every point to the left one tick one tick one tick one tick one tick tick and then just connect our dots when all we do is horizontally shift our graph we get a parallel wave the wave is exactly the same shape and trajectories at every moment in time kind of but it just slid it's off offset from the original and again what i'll emphasize is trying to locate the shift by going to the t value where um the original uh maximum point for cosine now occurs and drawing a wave that quote starts there at t equals negative pi over four finding out where the quote starting spot is will be helpful i don't know exactly what's on our desmos file for this um [Music] gotta hide these old grass so if we uh look at the sine graph before we we complicate things if all we do is shift we could shift in making using this purple one and initially it had pi over three there and you see that the purple one is offset from the blue one shifted to the right when we have a minus a constant we go to the right and if we replace this now with h we can use a slider to let it slide to the right and see that however far we oh it even like i just got a little distorted like after you get to a certain range you might like it's hard to tell which period you're you know you're looking at this just keeps repeating and repeating and repeating but as we as we slide h to get h becomes bigger and bigger our purple graph nudges itself to the right we'd have to have allowed ourselves to go into the negative territory if if you have a natural minus sign inside there h is going to have to be a negative number so that if h is negative and there's a negative in there it would look to a kind of normal person it would look like a plus um so when i use a negative h i'm i'm getting the same result as having a plus a sine of t plus a number and you see you slide to the left when that happens and if we go to the cosine graph cosine is the orange one i want to standardize our shifting so when i have a i mean that's the red one and the orange one is going to be the shifted version strange it's not like on top of the order is wrong oh is that because of maybe this is order um so if i uh let h become a positive number there's a minus in there so when h becomes positive it'll be t minus a positive number and we'll shift to the right if we let h become negative it would look like t plus a number and we're going to go to the left we had it looking like plus pi over four and you see the orange one nudges itself to the left a little so pluses go left minuses go right and we see that horizontal shifts are pretty easy to deal with uh except for this slight uh potential confusion about the direction just got to convince herself that the sig n in here uh sort of manifests itself in the opposite way standard uh intuition might lead us to to think um it is shifting in the correct direction but it's a it's fair if you prefer to memorize it kind of learn it as thinking that horizontal things are always like wrong you know they're always the opposite of what they quote should be um in the long run though it is exactly what it should be it's got it it's got to be what what it is anyway the challenge with horizontal stuff the biggest challenge with horizontal stuff comes when we combine stretching and shifting in that same horizontal direction so we're finally there an exam in the last example on this page to bring the uh different horizontal uh transformations together so you can see that in both of these situations there's a a number in the multiplication of the inputs inside multiplier there's a 2 in the inside multiplier here and there's a 2 in the inside multiplier here and there's a pi over 3 in that sort of shift position where it's like a minus pi over 3 and a minus pi over 3 they both involve the same numerical values in the roles of or at least seemingly same numerical values in the roles of w that's the blue one right the stretching factor horizontal stretching factor that's what we call w and h we tend to call that horizontal shifting value the green one they're both use the same numbers so we might make an initial observation that says something like both the the function p y equals p of t and y equals q of t um involve the same uh horizontal stretching compressing factor that's that number two and the same shifting maybe horizontal shifting constant and that's that number pi over three so we might expect them to have exact the exact same graph let's go ahead and try to investigate that uh this link to desmos will be to a different application a different file i've created that's also linked you're about you know you can access this if you if you want all right so um what we'll try to do is figure out what the differences are between these functions p and q you can see they're ready to be graphed here we'll start by graphing the uh cos2t function because it turns out that that uh the two t component of these functions the fact that there's a two in that w spot um is just uh unavoidable right that number is going to have the same effect no matter where we put it with regard to those parentheses so it's um going i i think it's going to be helpful to kind of think of q and p as being a horizontal shift of this red graph so suppose we first deal with the the two factor and compress and that we create this red one so i've already got the two in there and then we want to see what the the pi over three does to these graphs so let's see what happens when i uh expose q expose q we get a graph that i don't know it might be exactly what we expected right because um we see here inside of q a horizontal shift to the right pi over three units and you can see that this this red dot that i've plotted has moved exactly pi over three units to that green dot that red dot seems to have shifted exactly to that red that that red dot has shifted to that green dot and it's exactly pi over three units to the right of where it used to be so the green that this q of t does seem to be affected by the two in the in a you know the the way that the period is is compressed and then it seems to be shifted to the right pi over three units now let's see what happens when we expose the uh p of t which also seems to have the same uh horizontal stretching factor and also seems to have the same horizontal shift the difference here is where those parentheses are the fact that the the t minus pi over 3 is not factored out of the two is going to make a difference in fact here i'll just go ahead and let you see so there's p and you could see that the the function p its sort of key point that max maximum point that point is at the point uh pi over six not pi over three so this purple graph p of t it simply does not look like a function that has been shifted to the right pi over three units even though there's a pi over three in the formula the visible amount of horizontal shift is that shift between the red dot and that well now it's gray that purple dot so it looks like the the purple one p seems to be have been shifted pi over 6 units to the right not pi over 3 units to the right it turns out that in order to make use of the stuff we just reviewed in the last couple pages in order to be able to predict where the cosine wave is going to quote start based on the horizontal shift information we need to have it look like q we need to factor the inside of the function notice if i attempt to factor the inside of p if i pull the two off of the t and then try to figure out what needs to be replaced here two times what is pi over three well two times half of pi over three two times pi over 6 is pi over 3. if we factor the inside of the cosine function so that it appears like this now the number that you see in the horizontal shifting spot is literally how far that peak point has moved that peak point has moved to the right pi over six units it has not moved pi over three units what this uh situation is getting at is the uh by far i think the most complicated component of graph transformations which is that the order matters specifically the order you apply transformations that affect the graph in the same direction so both the 2 and the pi over 3 or the pi over 6 however you look at it these numbers are horizontal transformations they affect the horizontal component of the graph and when two transformations both affect the same sort of direction they can interfere with each other and in fact they very much do in this case it turns out that we have a choice when we look at the purple version of p and the black version of p they're both correct right they're both giving us if i hide some of these i don't know if that's helpful but we hide them there's only one graph here right the the purple graph and the black graph are just the same exact graph there's just two ways of writing the same algebraic expression whether we factor the inside or not now if you don't factor the inside and you want to do follow the graph transformations it turns out that what you would need to have done is first shifted it to the right pi over three units after you do the shift then you would apply the horizontal compression by a factor of one over that number two if you shift twice as far as you're supposed to because it's really only pi over six shift if you shift twice as far as you're supposed to and then after you shift you compress horizontally you will compress the shift and move it to the place it's supposed to be so we could have first shifted too far way over here to uh pi over three comma one and then compress that dot and moved it back to where it's supposed to be but if you try to learn the math that way if you try to do it in that order it's just going to be kind of miserable to try to draw graphs we're going to want to know where the the key points in this case that peak point at 0 1 where it moves so you can see that if we focus on uh the two now factored form so i'm i'm deleting the the one that was typed on the notes ignoring the one that isn't factored if you factor the inside of your function then this number here is literally where you go and you quote start one of these kind of functions so notice that the green one starts quote starts it has at a peak value when the x value and the input value is pi over 3. that original peak value at 0 1 is moved to pi over 3 and you can see pi over 3 in the formula for the function compare that now with the black one if you factor the inside like this you can literally see pi over 6 in the the graph pi over 6 is how far that red dot moved so the the moral here is going to be always factor the inside of the trig function always factor the inside of the trig function you have to factor it or you can't use the kind of theory that i'm trying to set us up for using so um i don't know if there's really room to write much more in here but um note that we focus on uh p p of t equals twice no i'm sorry i put the two in the wrongs button cosine twice i'm oh i'm sorry okay i want to write it literally like what we see there cosine two t minus pi over three if we take this expression the inside of that expression and we factor the two off of the t then when we recover what's got to be left over two times t is two times t and two times what well it turns out to be pi over 6 is pi over 3 and we'll still need a minus if we factor our function like this then we can see that the cosine wave starts the cosine wave quote quote starts at t equals pi over six it simply does not start or have any linkage with pi over three so we need to factor the inside of our horizon of the the um cosine or sine function in order to properly interpret the horizontal shift i feel like i've lost track no we're still okay i was thinking of lost track of our table on the front page it's actually right now where we can fill in that last section i left more room there because it's kind of an involved situation so we want to talk about this amount of horizontal shift but the only way we can wrap this up in a and a tidy formulaic sort of approach that will always work is if we factor the number omega the stretching factor off of the horizontal shift if if the input that's the the entire expression inside the sine and cosine function if the input for sine or cosine is factored as shown in the uh templates i don't know what the right word is for that maybe we can clarify that in these uh generic representations of sinusoidal functions if you factor as shown there specifically the inside the w and the h factored in the way that it's shown then we can quote quote start because there is no starting in our ending our sine or cosine wave at t equals whatever the h value is if you don't have it factored the function won't actually quote start at the number that's visible in the function maybe i can one one more time go back to desmos to sort of prove that or whatever enhance that once more time one more time so let's look at the two originals uh not the factored one but notice the purple one when we don't factor the inside we see the number pi over 3 popping off the formula right you can't miss it there's there's a pi over 3 in the formula but then when we go and look at our purple graph we go and look at the purple graph the key cosine information cosine starts at the maximum and comes down that starting sp point has moved here which is pi over six comma one it is not pi over three comma one so unless you factor the inside of the function the number that you see in here is not going to jive with the result of the graph the final you know graph that you'd need to draw if you factor it then the number that you see here jives perfectly with exactly how far that key point has shifted so it's going to be crucial that we factor the inside of our sine or cosine functions so i think we should go ahead and push forward and do at least this one draw a sinusoidal graph and so you can see what um how i'll encourage you to to bring all this stuff together so if we were doing this in math 111 we'd um start and we're not going to do this here because it's just going to be too much problem we could start with a a graph the sine wave right remember the sine wave just uh goes uh is this the graph of a sine yeah so we'll draw the sine the sine wave ah i'm not being very accurate the sine wave would look kind of like that one period of it then what we could do with this sine wave is we could stretch it vertically by a factor of two we could shift it up three units we could figure out what this stuff does of course we're going to have to factor that input expression to deal with what you know i've just been trying to discuss but we could create the appropriate horizontal compressing to draw the graph and we could crea shift the graph appropriately but what that would require us to do is have like four iterations of graph each step we're getting closer to the the actual graph we'd have to do four distinct transformations the problem with that is that these graphs are curvy and uh in some cases the periods will change so dramatically based on the amount of horizontal stretching that if you start with this graph of the sine wave that i've drawn here with the this scale it'll be almost impossible to draw an accurate graph of the resultant one that's honestly not the case in this particular circumstance but i'm trying to suggest that what you probably did in math 111 what i would have had students do in math 111 which would have been to like go through a sequence of steps to sort of first vertically stretch it then uh vertically shift it then horizontally stretch it then horizontally shift it if you do a sequence of four steps like that it's just going to be too hard to end up with a reasonable graph so what we'll do instead is not even start with the sine graph and and like sort of literally distort it instead we'll predict the key facts for this function we'll determine what the period is and then when we draw a graph we'll just make sure our period is correct we'll determine what the midline is and then when we draw a graph we'll just make sure we draw a graph with the proper midline we'll determine what the amplitude is and when we draw a graph we'll just make it the correct graph you know to have the correct amplitude so we're going to determine the key facts the key features of this wave and then draw a wave with all the correct features okay so the uh some of the information we can pluck off immediately but the uh inside factorization is always going to be an issue for us so it's not a bad idea to just confront that right away so the first thing i'll do is just rewrite the function but i'll factor the horizontal stretching factor off of t so this little one half here that's multiplied by the t i'll pull it off and then i'll try to see what i need to recover to get the same expression that we see in there in here so one-half times what is one-half t well that one's pretty easy we know we're gonna have to put a t there to get the one-half t out and then one-half times what is pi over 4 so that we can recover the information that was in the original expression now some some of these uh sort of arithmetic games are admittedly challenging and you can kind of get yourself mixed up i mean we have to do some kind of halving with pi over four and so you might kind of confuse yourself on which direction to go do we go to pi over two or pi over 8. you know maybe you think oh i'll do pi over 8 because that's half a pi over 4 for example let's say you tried this example test it just undo it do those orange things do the orange multiplying untangle it and see if you get back to the original so if it was in this format if we tried pi over eight one half of t will be one half of t that's good one half of pi over eight well that's pi over 16 half of pi over 8 is pi over 16 it's not pi over 4. so that was a wrong choice pi over 8. it turns out we had to go in the other direction if we do pi over 2 half of pi over 2 is pi over 4. so i'm suggesting a bit of a guess and check or you know make an educated stab at it and check it and then if it didn't work kind of make an educated response you know adjust sensibly based on you know what kind of a results you get let me cover the information there so we have factored the inside of our sinusoidal functions so now we can trust what we've learned about shifts but we'll go through a the sequence of uh facts so one of the easiest things to figure out is the amplitude whatever the number in front of the the vertical the outside multiplier is the amplitude another uh really relatively easy term to work with is the plus three on the outside there that's going to shift the graph up three units which will move the midline the period is really our other you know famously named thing right then the period comes from this number one half but remember i kind of went crazy earlier in this uh session um saying how the period does not appear in the formula for a sinusoidal function remember y equals sine of t the kind of original sine function has period 2 pi units but there's no 2 pi in the formula so you should not expect to see the period in the function unlike the amplitude in the midline the period isn't just leaping off the page like uh kind of giving you no choice but to find it you know the period's a little harder the period you have to do a bit of calculation now the period the period is dependent on the function at the root of this situation right we're working with sine sine has a period of two pi you can't avoid that so the period here is going to have a connection to sine's period which is two pi but after you sort of start with the sine function then you distort it by this factor of one-half the factor of one-half is multiplying the period by one over it it's the reciprocal of that number is the way that the period is affected i kind of wish i had a allowed the one half to stand out in red because that's what's being plucked out of the formula from the function pluck out of the form m of t's formula is only the one half all of the other pieces of that little formula the black pieces are coming based on um facts about uh horizontal stretching and then the intrinsic period of the sine function and we do a little calculation so 2 pi a reciprocal of a fraction is just the flip of the fraction and so we're going to get a period of 4 pi units so we know our amplitude is two our midline is three and our period is four pi four pi four pi four pi it's twice as big as the sort of typical period the traditional period of the sine function that's because this one half stretches it out by a factor of two it slows it down all right now the last piece of information to sort out is where are we going to start quote start drawing a graph that has these three key features and that comes from that work we did factoring the input to the trig function because we see a uh another color let's try orange this this input value along with the fact that this is a sine function tells us that we need a sine wave starting and this is a fake idea it's not a real starting spot but it's the behavior that were used that we're familiar with happening at the y-axis that behavior will now occur at t equals negative pi over two remember when there's a plus you shift to the left direction the op yeah shift to the left and so our sign behavior will start left at pi over two and now we just want to piece all this stuff together in a graph looks like i can barely see the facts it's great and uh what if you know it's i've already started with a scaled graph to make this a little more realistic you'd have just a blank grid you know and you'd have to decide how to scale it in this case our midline is three that's a good thing to kind of get on there quick because uh the amplitude is like strays away from the midline so once you get your midline of y equals three you can then uh address the fact that the amplitude is two the amplitude really isn't like a lines on the graph but it's not a terrible idea to shift up two units and draw a a horizontal line and shift down two units and draw a horizontal line because what this creates for you is sort of a set of like railroad tracks that the sine wave will bounce between so you're just kind of outlining an amplitude of two remember the amplitude is the number of units above and the number of units below the midline that the function strays and now what we can do is go find our key starting spot our starting spot is negative pi over 2 so we go find negative pi over 2 and what we know is that sine quote quote starts at the midline right remember uh let's go back somewhere so here underneath here is a sine wave it's a terrible picture but notice that the the sine the original sine function starts at the middle and comes up out of the middle that's how sine traditionally behaves at the midline at the at the y-axis this is now going to start to the left at pi over 2 so i'll go to pi over 2 and i need it to be at the midline and coming up from the midline but it's offset to the left pi over 2 units and before we actually draw our wave let's give ourselves a like sort of dots that will give us an appropriate period so the period is supposed to be four pi units that means four pi units later we'd finish a period um we've shifted left one tick so if i shift left one tick from four pi that will give me four pi units between my starting tick and my last tick of a period so i'll definitely get the appropriately sized period and then we know how sine works halfway between first half of the period it does sort of the the upper hump so we just got to find the middle of these uh blue dots honestly it could be as sort of silly as going one unit to the right one unit to the left one into the right one into the left when it's the right when you're the left one and find the middle of it you know i've noticed that this would be i think four ticks and let's see that's four and four so i need to go five ticks like if i go to there and go to there like you can kind of just sort of physically find the middle or you could do math and subtract the two blue dots the distance between here is 4 pi units so we need to go to 2 pi in either direction and that's gonna put you there notice it's one two three four ticks in that direction and one two three four ticks in that direction to just try to make sure you're in the middle and halfway between uh the each quarter period sign reaches the peak in a quarter and after this and then this next quarter it'll reach the trough it'll reach its bottom and so you give yourself a key quarter period dots and then you can just connect your dots and there's one period of the sine wave if we need to make uh additional periods if we're interested in drawing additional periods we don't need to be as uh it's not as uh it's easier now because we can just notice a pattern it looks like every two ticks we go from like immediate middle to top two ticks middle two ticks bottom two ticks middle you know you don't even have to like think about your scale very accurately in the same way because you just uh you figured out the the pattern for a period you know two ticks to the left it should be at the bottom and so you can kind of flush out your graph and give it some more uh draw some more of it and you see with with uh this approach we only drew the graph once you know we didn't drew it draw it in iterative steps we didn't first vertically stretch it then vertically shift it then horizontally stretch it then horizontally shift in theory you could absolutely do it that way but these graphs have uh i don't know their kind of nuances make it um you're just very unlikely to get a very good graph if you try it that way and there's going to be some situations where it's it's essentially impossible because the scale of the graph will change so dramatically that if you're stuck with this particular scaling you'll just never be able to draw an accurate graph so you're better off just determining the key facts and then drawing a graph that satisfies all those key facts if that makes sense and so um what's kind of nice about this is that if you do have any kind of uh concerns about your math 111 background regarding graph transformations it doesn't need to sort of ruin your chances in at this topic because you can just learn how learn the approach we just went through here where we hopefully understand for example that the amplitude is two because two is a vertical stretcher and we're stretching uh the amplitude that used to be at one unit by a factor of two so it'll become two units i mean this stuff becomes hopefully um very uh understandable based on what you know from 111 but if you um aren't aren't familiar with that until you have a chance to review that you can just sort of learn the facts in the box on page one and draw graphs accordingly you know that have the appropriate features