six point four is transformations of trigonometric functions and again this is something that you did in grade 11 so we're going to I'm going to go over it carefully but if you if you find you you need a little more review you can look back at the transformations in trig in the functions eleven course so basically we're dealing with the same transformations that you've used over and over again to various parent functions so you have a function K X minus D plus C so if we apply that to the sine function this is what it would look like and we applied it to the cosine function it would look like this now you must always remember the K must be factored out of these two terms if it hasn't been and that's one little step that so many students get tripped up on don't forget isolate the K okay factor it out okay so let's talk about changes to the y-coordinate first the y-coordinate is the easiest one if I was going to do a map and girl what would I do to my y's you'd say well Y is going to be multiplied by a and then we add C so we get a y plus C now do you remember what the a does of course you do a means vertical stretch or compression right so it's a vertical stretch if a is greater than one so if a is greater than one and this is affecting let's write these in in color so you've got this this is an amplitude for my graph right the a value is amplitude in this for trig functions so if you have you have a vertical compression and you know what that looks like that means we're going to make a between 0 and 1 everyone else said well what about the negative the negative means a reflection right so that comes in another section here so we have a stretch or could be compression or third we can have a negative negative a mean reflection about I remember if you have trouble remembering which one you're affecting wise so the wise are going up and down this way so its reflection about the x-axis okay so we have negative and positive sine and cosine functions and we're going to take a quick look at that in a minute as well okay so what does the C do so C is your vertical shift up or down C units so in terms of our functions our trigonometric functions and we just write this down so I don't make a mistake vertical shift up or down SI units SI units and that's affecting all together now one two three that's the axis it tells you where the axis is going to be so if I have plus five here at the end your axis is going to be at five and then you're going to adjust your amplitude from that axis so amplitude for the a value and the axis amplitude and axis those are the two important changes to the y coordinate so let's take a look now at the X because everything's a little bit different in intrigued than it is and other transformations changes to the x coordinate so if I was going to ask for a mapping rule for my X you'd say I have to do X divided by K divided by KC plus D so X goes to X over K plus D so like I said in the grade-level course X's are weird this it's all backwards it looks like x a K - a D but you're going to divide by the K and add D will change the sign here okay so the K is important because it determines the the period of your trigonometric function so K helps to determine to determine the period period of function and this is a little calculation that you will need to do over and over again because if you remember y equals sine theta or cos theta has a period of 2 pi so if I want to know what the period is of y equals sine 2 theta the period is 2 PI over K okay so you have to always divide the period by the K value so in this case the K would be - whoops I should put a little sign there so the period would be PI right so 2 pi divided by 2 to PI over 2 pi ok so alternatively if you had the period let's say I had y equals sine 1/4 theta then the period would be 2 PI over 1/4 so I remember that's 2 pi times 4 which would be 8 pi so recall know that a 2 here are a number greater than 1 results in a compression and the fraction gives you the stretch right so if K if K is greater than 1 then you have a horizontal compression by one over K so in this case here we had a compression of 1/2 right so this one would be compression of 1/2 and this one would be a stretch times 4 right 1 over this so if K is between 0 & 1 in other words a fraction then it's a horizontal stretch and that's changing again it's changing the period of the function horizontal stretch by a factor of 1 over K again so you have to remember how to divide your fractions right okay so what else we have left to do here we need D right so we have d tells you this is the horizontal shift left and right horizontal shift left and right D units and then get up a red pen right now and right don't forget to factor I'll do an example in a minute don't forget to factor okay so the K helps determine the period and this little formula here period is 2 PI over K is very important the IP okay that formula you're going to use that lots to find the period of the function okay let's do an example it's over and we're going to do let's make up a good one for you pencil doing so we're going to say y is equal to 1/2 post x over 2 minus PI over 12 minus 1 your job is to graph this function state the transformations we can state them as we're doing them let's figure out what the mapping rule would be first we could do some key points you've done that before so you can either use key points or sometimes you can just eyeball these functions that's two kind of different methods but one you might feel more comfortable with them the other but I'll show you both okay so I have one half Coast x over two months PI over 12 so right away I see that I need to factor this this isn't X its X over two so I rewrite it one half Coase big bracket take out 1/2 that leaves me with X and please be careful when you divide minus PI over 12 by 1/2 what should your answer be please don't say PI over 24 it's going to be minus PI over 6 minus one okay because PI over 12 divided by one half is PI over is two PI over 12 or PI over 6 okay so be careful when you're dividing fractions don't make that mistake as it's kind of very sad to be doing that in grade 12 everybody including your teacher who won't want them even mark it okay so here's my mapping and not my mapping real about my transformation and I'm gonna state the mapping real now so I'm gonna say I take my X's and Y's and what do I do to my X's well let's do what you do the Y's first that's always easier 1 1/2 y -1 1/2 y -1 very easy why you should never make a mistake with and now this should you make a mistake with X because we have 1/2 so we're dividing by 1/2 that means multiplying by 2 so I have 2 X's and not minus PI over 6 but plus PI over 6 and there's a beautiful little mapping rule for you to use with some key points of the graph so I'm transforming a cosine function would probably be a good idea to have some points for the post function so I'm going to do X and Co sex so if I did 0 & 1 so I'm using just the basic cosine function here that you should be able to draw very quickly that's right quickly right over here so you go mmm like this start some ends here I remember this is PI over 2 here PI over sorry 2 pi this is PI this is PI over 2 and this is 3 PI over 2 okay make a quick sketch and now you have 1 2 3 4 5 points that you can use so PI over 2 and 0 I have PI and negative 1 I have 3 PI over 2 and 0 and 1/2 2 pi and 1 ok so those with these are key points of the parent function right y equal Costa that's all okay so I want to do these transformations to these points so I'm going to do 0 1 goes to and I'm going to plug in X is 0 here so that gives me PI over 6 and 1/2 minus 1 minus 1/2 okay and then we're going to do PI over 2 and 0 so PI over 2 times 2 is PI plus PI over 6 that's going to be 7 PI over 6 and minus 1 okay so what is art what's her period let's do that because that's really a good thing to know here so period is equal to 2 PI over K so that's 2 PI over my K is a hat off so that's 4 pi 4 PI for my period so I'm stretching it that's why all these points look like they're so far apart right okay so PI over two let's go to PI and minus one so PI that's two pi plus PI over six to PI is 12 PI over six and one more PI gives me 13 PI over 6 and minus one here for my white so minus 1/2 minus 1 is minus three-halves and then I have 3 PI over 2 and 0 and that gives me 3 PI over 2 times 2 is 3 PI that's 18 plus 1 is 19 PI over 6 and 0 gives me minus 1 and finally I have 2 pi and one I transform that to PI here this 4 PI 4 that's 24 over 6 plus one more is 25 over 6 25 PI over 6 you can see they're going up by 6 is here right pi 7:13 19:25 probably have done that and you had a little faster and 1/2 minus 1 yes minus 1/2 so you can see I've gone from minus 1/2 minus 1 minus 3 halves minus 1 minus 1/2 so it all looks really very pretty ok so what I want to do now is I want to graph this so I get up my trusty ruler here quickly I usually like to have these graphs already done now let's take a look first of all where the axis is remember this is your axis right here axis so I'm going to be below so I'm going to make sure when I draw my graph I don't have I'm not going to go off the page here so I want the axis to be at minus 1 so let's call this minus 1 right here and I didn't give you a domain but let's say it's between 0 & 2 pi which is what we've done okay so this is y equals minus 1 we've got our axis down here oh sorry I guess I couldn't see that and now I'm going to check out my amplitude the amplitude is 1/2 right we had 1/2 coasts so I'm going to be going up a half from here this is the highest point on my graph and the lowest point is going to be another half below this is always a good idea to do this I mean it just gives you a little better sketch minus three halves and you can see all my points go between minus 1/2 and minus three-halves so now i've got i have to go back up here to give me my my x-axis scale so I'm going to shift it to the right PI over 6 right so this said PI over 6 we shifted it to the right PI over 6 so let's say I'd make this 2 pi here I'm going to divide it nicely into quarters like I mentioned earlier so this is PI over 2 and my whole graph is going to be shifted PI over 6 to the right so that means I'm going to start PI over 6 this is PI over 2 so I want another so let's say this is PI over 6 right here all right over 6-1 6-2 6-3 6 it's about maybe about here okay so PI over 6 I'm gonna start my graph there now the other thing you should note is that we have this is a positive cost function yeah it was a positive coast so we're going to start at the highest point which is what we are at here so we're here and then 7 PI over 6 that's just PI over 6 away from PI and we're at minus 1 so that's here and then 13 PI over 6 so we have to stretch this right because we're going up 4 pi oops should draw my graph a little bit longer okay so wait PI over 6 7 PI over 6 2 pi is that far 4 pi is going to be this far quick sketch and we're going to end here at 25 PI over 6 my lowest point this is minus three-halves it's going to be at 13 PI over 6 13 PI over 6 that's 2 pi plus 1/6 so it's going to be right here my lowest point and I guess I should have made this y equals 1 way over here instead of right in the middle of the graph this one's going to be at 19 PI over 6 so that's 3 pi plus a little bit so that's my other minus 1 point so if you sketched all this together you'd have something going like this sometimes he grabs or almost flat don't be surprised if that happens okay so there's my graph now like I said you could make these points this is always a good thing to do know some key points do the transformations or you could have just stretched it out by 4 PI shifted it all over PI over 6 and done everything by quarters I'm going to do that in the next example that I'm going to do for you okay so the second one this is from 8c from your homework in case your teacher gives it to you you already have it done or at least an idea of how to do it so this one they've already factored out the 2 here so this is nice it didn't have to factor it out and I'm going to I'm going to sketch it just by just by sketching it so let's figure out what the period is first period the axis axis is easy y equals 2 remember right here right there staring you right in front and this is it's a negative sine function now let's go back to the period so it's 2 PI over K and in this case it's going to be 2 PI over 2 which is pi so we've compressed it horizontal compression by a factor of 1/2 shifted to the left PI over 4 axis is at y equals 2 it's a negative sine function so quick sketch here's your positive sine function right so this is a negative sine function so it's going the other way negative sine theta okay so I'm going to write that in red here so you could see you really have four graphs to think about positive and negative coasts and sine functions so should look like that when I'm done right okay so let's let's draw it without doing a mapping rule you can do both if you want I'm just going to sketch it like this okay so my axis is at +2 so first thing you want to do is make a +2 here so we write that on right away kind of axis y equals 2 now my period is only PI and I'm shifting it to the left PI over 4 so where should I start to the left PI over 4 so I want to start at minus PI over 4 the period is PI so that means this is a quarter of a pi is another quarter here's I should have made it a little farther apart maybe so we're only going to PI right so here's PI here let's call this pi and from here here to here that's half a pie that's a quarter of a pie just straighten this up a bit got some red red ink came through on this and moved my quarter down just a bit so it looks more exact this is a quarter pie half pie 3/4 of a pie pie over this is PI but one complete cycle is going to end here right because from here to here is PI and my period is pi okay now because I have this y equals two and I'm looking at the axis here the amplitude is 2 so I'm going to go up 2 from that so this is a B 4 and I'm going to go down 2 so this is 2 plus 2 2 minus 2 right that gives me the total amplitude of my function and not amplitude but the range so the amplitude is 2 so I'm starting at a negative sine function so you always start a sine function on the axis don't start it down here this is your axis here so you want to start it right here okay start it on your axis there's another thing I get students that try to put it down here and then they don't know what to do so I have this is my maximum height this is gonna be my minimum height right so it's a negative sine function it's starting at minus PI over 4 and it's going to end here that's one full cycle if your teacher asks you for two complete cycles well then you're just going to draw another cycle afterwards cut and paste it okay so if I go from minus PI over 4 to 3 PI over 4 where is the lowest part on this graph so 1 2 3 4 1 2 right here right this is gonna be looks that's the midpoint between the two maximums and it's going to be right here in the graph so I'm going from this way like this because it's going negative sign it's going to go whoops what did I do wrong here I have to end at the I have to end at 3 PI over 4 and I have to go up and down between them so PI over 4 this is let's see 3 PI over 4 this is going to be PI over 2 PI over 4 0 so it's going to have to go through here this is gonna be my lowest and then it's going to go PI over four here my highest point is going to be here so you're dividing it into quarters and back down to here so you have a starting in an endpoint right this is PI units so go PI over two units and that's going to be back on the axis so we've axis three points the highest point at PI over 2 the lowest point at zero and there's your sketch now you could keep going here right keep going like this depends on how far your teacher wants you to draw it but you've got I do this for you okay so there you go there's your negative sign top 2 X PI over 4 plus 2 I've got my axis I'm up to down - I started on the x axis it was negative sign so I went down first and there's your negative sine function now you could go back and say let's just write up the map and rule I won't do it all for you because it's getting a little long here but let's say we have x and y go - what do we do - the x's so we have x over 2 minus PI over 4 and minus 2 y plus 2 that's my y coordinate C minus 2 y plus 2 this divided by 2 X divided by 2 subtract PI over 4 and then you would use all the key points on the graph of the regular sine theta so I'll just write them out quickly here at 0 0 PI over 2 and 1 PI and 0 3 PI over 2 and minus 1 and 2 pi and 0 so you get all your points and then you apply all these get all the points using these transformations ok so we might as well just finish it here - 4 PI over 4 and to the goes to zero and zero and then PI over 4 + 2 PI over 2 + 4 + 3 PI over 4 + 2 so 3 PI over 4 + 2 so here we go we're all in the right spot so you could do this is using points key points key points and mapping rule and the other one is just kind of eyeballing it it's not this one wasn't that hard to do sometimes you might want to use this to check it or whatever you have time to do ok so we're going to get into some more the word problems that are associated with this in two more lessons the next lesson I think is reciprocal trig functions and then there's a lot of modeling so we'll do a lot of a lot of that work then okay bye for now