Transforming sinusoidal graphs: vertical & horizontal stretches

we are asked to graph the function y is equal to negative two point five cosine of one-third X on the interval 0 to 6 pi including the endpoints so let me do my best attempt at graphing that and to start off I'm going to graph with the simplest function or the simplest version of this or the kind of the root of this which is just cosine of X so let me just graph and eventually you can kind of let me just let me graph cosine of X so this is my y-axis and I want to have some space here so I can eventually graph this entire thing so let me so let's say that this is negative 1 this is negative 2 this is positive 1 this is positive 2 and let's say that this right over here is this right over here is 2 pi 2 pi and then of course that could be pi right over there now the first thing I'm going to let me copy this because I could use it later I could use it later to graph the whole thing so so let's start off so I'm just going to graph y is equal to cosine of X so when X is equal to 0 and I'm just going to do it between the interval 0 & 2 pi obviously it's a periodic function we'll keep going in the negative and the positive directions so what happens when X is equal to 0 what is cosine of X well cosine of 0 is 1 what about when what about when X is equal to PI what is cosine of PI well cosine of PI is negative 1 cosine of PI is negative 1 and then what's cosine of 2 pi well that's 1 again we get back we've completed in a period or we completed an entire cycle and 2 pi is the period of cosine of X so this is one cycle right over here I could keep going if I wanted to but the whole point I just wanted to graph this one cycle between 0 and 2pi now what I want to think about is what happens to this graph instead of graphing instead of graphing y equals cosine of X and draw some graph paper again instead of drawing y is equal to cosine of X I'm going to draw Y is equal to cosine of 1/3 X so the only difference in that and that is now I'm multiplying the X by 1/3 what's going to happen to the graph over here how is this going to change instead of being an X if it's a 1/3 X what's going to happen over here and now I'm going to do it over the entire interval between 0 & 6 pi so let me just make sure I have enough space so that's 3 PI 4 PI 5 PI and 6 PI what's going to happen to this graph well there's a couple of ways to think about it the easiest might just to be say well to complete an entire cycle we're going to we're going to go 1/3 as as fast or we're going to go 3 times slower or if you just want to think about the period here what's the period of cosine of 1/3 X well the period the period is going to be 2 pi divided by the absolute value of this coefficient right over here so it's the absolute value of 1/3 which is just 1/3 so the period is 2 PI over 1/3 which the same thing as 2 pi times 3 which is 6 pi which is which gels with the intuition that's going to take what it's going to take three times as much time to get whatever we input into the cosine function to get back to 2 pi because whatever we take X we're taking 1/3 of it so instead of we to get to 2 pi you can't just have x equals 2 pi X now has to equal 6 pi to get 2 pi input it into the cosine function so the period is now 6 pi at X is equal to 0 1/3 times 0 is 0 and the cosine of 0 is 1 when X is equal to 6 pi you have 6 pi divided by 3 is 2 pi cosine of 2 pi is equal to 1 and if you want to go in between over here we to go in between we tried pi but over here we could try 3 pi when X is 3 PI you have cosine of 1/3 of 3 PI that's cosine of PI cosine of PI is negative 1 so when X is equal to three PI we have cosine cosine of 1/3 times 3 bi is negative 1 so it's going to look something like this it's going to look something something like this drawing my my best attempt best attempt to draw it so it's going to look something like this so you see to go from coast y equals cosine of X to cut y equals cosine of 1/3 X it essentially stretched out it stretched out to this function by a by a factor of three or you can see this period is three times longer the period here the period here was pot 2pi period here was 2pi all right well there's only one more transformation we need in order to get to the function that they asking us about we just have to instead of having a cosine of 1/3 X we just have to negative two point five cosine of 1/3 X so let's try to draw that so let me let me put my axis here again and let me let me label it so that's cosine 2 pi 3 pi 4 PI 5 PI and 6 PI and our goal now is to draw the graph of y is equal to and we're just doing it over this between 0 & 6 pi here we only did it between 0 & 2 pi here obviously they're they're all periodic they all keep going on and on they all keep going on and on but now we want to graph y is equal to negative two point five times cosine of 1/3 X so given this change we're now multiplying by negative two point five what is going to be well actually let's think about a few things what was the amplitude in the first two graphs right over here so what was the amplitude amplitude in these first two graphs well there's two ways to think about it you could say the amplitude is half the difference between the minimum and the maximum points in either of these case the minimum is negative one maximum is one the dip - half of that is one half of that is one or you could just say it's the absolute value of the coefficient here which is implicitly a 1 and the absolute value of one is once again one what's going to be the amplitude for this thing right over here well the amplitude the amplitude is going to be the absolute value of what's multiplying the cosine function so the amplitude in this case I'll do it in green the amplitude is going to be equal to the absolute value of negative two point five which is equal to two point five so given that given that how is multiplying by negative two point five going to transform going to transform this graph right over here well let's think about it if it was multiplying by just a positive two point five you would stretch it out each at each point it would go it would go up by a factor of two and a half but it's a negative two point five so at each point you're going to stretch it out and then you're going to flip it over the x-axis so let's do that so when X was zero you got through one in this case but now we're going to multiply that by negative two point five which means you're going to get to negative two point five so let me draw a negative two point five right over there so that's negative two point five that would be negative they make it clear this would be this would be negative three right over here this would be positive three so that number right of there is negative two point five and let me draw a dotted line there it could serve to be useful now when when our function is is it when cosine of one-third X is zero it doesn't matter what you multiply it by you're still going to get zero right over here now when cosine of one-third X was negative one which was the case when X is equal to 3 PI when X is equal to 3 PI what's going to happen over here well cosine of 1/3 X we see is negative one negative one times negative two point five is positive two point five so we're going to get two positive two point five which is right over here we're going to get to positive 2.5 which is right over there and then when cosine of 1/3 X is equal to zero doesn't matter what we multiply it by we get to zero and then finally when cosine of 1/3 X when X is at 6 PI cosine of 1/3 X is equal to 1 what's that going to be when you multiply it by negative two point five what's going to be negative two point five so we're going to get back over here so we're ready to draw our graph it looks something let me do that magenta color since that's what the color I wrote this in it will look like this it will look let me draw I can draw it as a solid line so it will look like like that so you saw what happened by by putting this one-third here it stretched out the graph it increased the period by a factor of three and then multiplying it by negative two point five if you just multiply it by two point five the amp you you would have you would just have the the you just kind of you would just multiply that out a little bit but now it's a negative so not only do you increase the amplitude but you flip it over so it is indeed the case that the amplitude here is two point five we very two point five from our middle position or you could say that the difference between the minimum and the maximum is five so half of that is two point five but it isn't just multiplying this graph by two point five if you multiply this graph by two point five you'd get something let me be a little neater you would get something that looked something like like that but we actually because we had a negative we had to flip it over the x-axis and we got this year so this amplitude is two point five but it's a flipped over version of this graph