 Over the last several videos in our lecture series, we've been learning how we can transform sine and cosine functions based upon vertical reflections, stretches and compressions, shifting, that is translation. So how does one decode that if you're given a trigonometric function a modified sine or cosine? Well the idea is this is the template you should be looking at. If it's a sine wave it's going to look like y equals k plus a sine of b times x minus h or if it's a cosine wave the generic form is going to look like y equals k plus a cosine of b times x minus h which you can assume b is a positive number because of the symmetry identities if b were negative you could you could deal with it using symmetry of course. So if your function if your sine or cosine it could always be put in this standard form right here and then once it's in the standard form you can start looking at the characteristics of the function. The coefficient of the sine and cosine this is going to give you the amplitude at least if we take the absolute value the amplitude is going to be this number if the number happened to be negative right here that means you reflected across the x-axis if a was positive no reflection happens okay this number k that you're adding to either sine or cosine it's outside of the function it's not inside it's not affecting the angle it's just it's outside there this will be a vertical shift you're moving up by k units or if you're subtracting that's actually equivalent to moving things down. Now inside the function if you're looking at the things that affect the angle you're now stepped into the horizontal zone do-do-do-do-do-do-do in the horizontal zone you have to be very careful because things might not work the way you think they do in the horizontal zone it's very important that things be factored if things aren't being factored it turns out your intuition the learning you're getting right now actually could it could be contrary to what you expect if you factor things things life will be much simpler you don't have to worry about phase shifts or anything like that so if you factor it so you have b times x minus h this number h here will represent a horizontal shift do notice you expect a negative sign in the horizontal zone but you expect a plus sign in the outside of the horizontal zone right so subtracting h moves things to the right adding h actually moves things to the left and this number b right here affects the period it'll change how frequent the graph is oscillating affects the frequency there so p equals 2 pi over b and so if you if you can remember these principles then you could graph any sine or cosine wave and let's do a few examples of that right now so consider the function y equals negative 3 minus 2 sine of pi x so you have to first check the horizontal zone if there's no horizontal shifting going on there's nothing being added or subtracted from the x we we shout for joy because that makes things a lot easier you don't have to worry about factor anything the only thing we check for is is the coefficient of x positive it is so we move on we're now in our standard form so let's start looking at things this coefficient of negative 2 right here it affects the amplitude we see that amplitude is going to equal 2 and we see there is a reflection that'll take place on the graph there's a reflection what does this negative 3 do this negative 3 represents there's going to be a vertical shift so we're going to shift up i guess i should if we shift up by negative 3 but i should say it's much easier to avoid a double negative let's shift down by 3 in particular our midline is going to take place at y equals negative 3 so pay attention to that well what about the pi that's inside of this so if that that coefficient of pi affects the period the period is going to be 2 pi divided by that number which itself is pi cancels out we get a period there of 2 and so with this information about the midline the amplitude reflection in the period we're now ready to graph our function which we're going to graph it on the grid lines we can do right here if i'm graphing a a trigonometric function whether it's sine or cosine the first thing to do is to consider the midline i like to draw it as a dashed line so i recognize it's not actually part of the graph but the midline should go in there it takes place at negative 3 okay so then i want to graph one cycle of the function just one standard cycle so we're going to go from zero to two and mark that up if you need to we need to go from zero to two to complete one cycle of the graph and in that one cycle i pay attention to the five major points the the five points that coincide with the original quadrangle angles zero pi halves pi three pi halves and two pi and now as you start changing the period those that can go a little bit more complicated but basically you take your standard period and you slice it into five equal parts like we have right here as you go from zero go from zero to one to two and then this we get one half right here and we get a three halves right here we have those five equal parts going on right there so remember our graph i'm going to write it down so we can see it so we don't have to move it all the time so if we were to write it down here the original graph y equals negative three minus two sine of pi x okay so we changed it so that because of the period being changed we're going from zero to two now right so we still have these five equal parts the standard sign would go like excuse me sine starts at zero then to one then to zero then the negative one to zero again so we're gonna have that same basic shape except that it has been reflected across the x axis so we are going to start on the midline sine always starts on the midline and every time you have an x intercept that means you should be on the midline so zero one and two will be points on the midline but because of reflections we're actually going to go down first and then we're going to increase later when we change the amplitude as well the amplitude is now two so that means we should go all the way down to negative five when we're at x equals one half and we should go to above the midline which is actually going to be negative one and so if we draw one one cycle of the graph we're going to get this picture right here so that's all they have to do to draw one cycle and then we're just going to repeat this process for for the rest of it I mean the graph goes from negative four to four so it turns out we actually have to draw four cycles of the graph just copy and paste this thing over and over again right that's all that one has to do to draw the rest of this graph so we're just going to repeat the cycle like so okay we'll just repeat it and so we're going to get a point right here just copy and paste it I'm drawing these points to help me make sure I draw it correctly and just connect these dots and the typical sinusoidal way like so and now we have a pretty good graph of this our function y equals negative three minus two sine of pi x and if we were to zoom out a little bit I can compare this picture to a computer generated picture which we can see it's you know my my drawing's a little bit sloppy but I'd say it's pretty good really it's good let's look at another example here let's consider the function y equals four cosine of two x minus three pi over two so this now we have to be a little bit more careful because in this situation the horizontal zone is not factored we have to factor the horizontal zone so this this format right here is incompatible with what we want to do so we need to factor the two away from the x so that there's an x right here but if we factor a two away from the x we have to also factor the two away from the three pi halves which then leaves behind a three pi force like so and so now that it's factored we can correctly diagnose everything we need here so we see for example the amplitude here is going to be a four there's no reflections there's no shifting going on this two right here affects the period we see the period is going to equal two pi over two which says the period is equal to pi and then what does this three pi fourths do to us this three pi fourths is going to be a horizontal shift to the right so we see that h is going to equal three pi fourths like so and so with this information we can then start graphing our function okay we want to grab a standard period a standard cycle of some kind for which we can see that since the cycle is pi the period is now pi we would a standard cycle would be from zero to pi like so and we're trying to graph cosine after all right remember it was for cosine of the of the two times x minus three pi fourths that's what we're trying to graph right here it's cosine so cosine starts at its maximum then goes down kind of something like this of course i've changed the period right here that's the basic shape of cosine so we're going to start at a maximum but because we have this shift going on here let's think of the first original quadrantal points there so the amplitude is a four so we should have something like this and then it comes down to an x intercept and then it comes down here then here and here these points that i drew in blue right here they would give us the non-shifted cosine this has the amplitude changes the period change but doesn't have to shift so if three pi fourths is over here we just need to move everyone three points down on our grid lines here so we move this one over we're going to move this one over so we get a point right here we move this one over so we get a point right here this one needs to move over and this one needs to move over so you get all these points there if you don't like the little blue marks if you feel like it's just too cluttered it's okay to remove them now because those were just supposed to be helper points so we have these ones right here and now you can draw this graph in the typical cosine manner so you get something like this it didn't specify domain so we should just fill in the grid lines that are provided to us so just filling in the rest of it we get a picture that looks something like this like so and so notice i'm doing this every other every other tick mark is i'm going to get an x intercept and then between them i get a maximum which goes all our or minimum right i have to go up to four and down to negative four and we get something like this all right so this gives us the graph of the function we provided when you look at this graph you'll notice that the graph passes through the origin is this a cosine function or is it a sine function well our original our original instructions were it was a cosine but when you look at that thing it kind of looks like it's a sign and because of the co-function theorem it turns out that every sinusoidal wave is a sine or a cosine it turns out you could represent the same graph in multiple ways this is a result of trigonometric identities all right let's look at one last example of graphing this one and this one's going to be the whole inch a lot it's got everything going on right here so we see we have to graph y equals four fifths sine of two x plus pi halves minus one it's not factored so we do need to factor that thing so we're going to get a negative one plus five over four sine we're going to factor out the two and that leaves behind x plus pi force pi force excuse me like so and so this is now in our standard form and so the things we can see here is we have to graph a sine function all right there's going to be a shift by negative one so the midline the midline is going to take place at negative one the amplitude is going to be five force and then we see that the period the period is going to be two pi divided by two so it's going to be a pi and then finally there's this horizontal shift of h by negative pi force it's just going to be a shift to the left so now with this information let's graph our function here okay so let's think about the standard sine function standard sine function we should think of as going from like here to here to here to here to here like so we need to modify this according to the to the instructions we have now but unlike the previous examples you'll notice there's no grid line whatsoever I excuse me there's no markings on the grid lines we see we see the lines x and y axis we see the little tick marks but there's no labels here this actually is good for us because we can actually make the label be whatever we want as we've learned before graphing a sine or cosine wave it's important to pay attention to the five special points that normally coincide with zero pi halves pi three pi halves and two pi because the thing isn't labeled I'm going to take these five points right here so one two three four five I'm going to make that just one period this is going to be one period of my graph right here and I'm just going to take that my period's now a pi so I'm going to label this point right here as pi in which case then this would be pi halves we have pi fourths right here we have three pi fourths right there that's great that is going to be our standard just our standard period right there and then we can continue on of course you're going to end up with a three pi halves right there and then the last point on the on the line there would be a two pi so that's nice we can go label the other portions if we want to the next thing I would consider is the midline where should that be it's going to be below the x axis somewhere it makes sense we can just put it right here I could be negative one and that actually seems quite good with me so I'm just going to keep the y scale what it is so we're going to have here a one and a two this would be negative one this would be negative two and so the midline would then be this point right or this line right here and let's do it the other side as well and so then our amplitude is going to be five fourths so if we were drafting sign we would start on the midline we would then come up to our amplitude which is five fourths that's going to be a little bit above the x axis then we come back down then we're going to come down here and so we get a point a little bit below negative two then come back up that's going to be a standard sign which this has the correct period change and the correct amplitude but it doesn't have the shift we need to shift things over by to the left by five fourths but that's exactly this mark right here this unit is pi four so we need to move everything over by one point right there so let's do exactly that so it's like we started here so we get this point this point this point this point and this point and so that gives us one one complete cycle of our graph then from here we just copy the picture we just do it over and over and over and over and over and over again we just draw as many points as we need and just repeat the pattern given that this sign is just a periodic function so we just switch between points above points on points below the midline like we're doing and we just connect the dots it's very soothing very relaxing it's like watercoloring drawing a sine wave and so this demonstrates how we can graph a sine function or cosine function using all these transformations that we've encountered so far