We're request to determine theamplitude and the period of y equals negative1/2 cosine the 3x. For this reason the first thingwe have to ask oneself is, what doesamplitude also refer to? fine the amplitude ofa periodic role is just half the differencebetween the minimum and maximum values it bring away on. For this reason if ns were to attract aperiodic role like this, and it would just go backand forth in between two-- allow me attract it alittle little neater-- that goes back and forthbetween 2 values choose that. So between thatvalue and also that value. You take it the distinction betweenthe two, and fifty percent of the is the amplitude. Another means of thinkingabout the amplitude is how much does it swayfrom its center position. Appropriate over here,we have actually y equals an adverse 1/2 cosine of 3x. Therefore what is going to bethe amplitude the this? Well, the straightforward wayto think about it is just what is multiplyingthe cosine function. And also you could do the same thingif it to be a sine function. We have actually negative1/2 multiply it. For this reason the amplitudein this instance is going to be the absolutevalue of negative 1/2, which is equal to 1/2. And you could say, well, whydo I no care about the sign? Why perform I take it theabsolute value of it? Well, the an unfavorable justflips the duty around. It's no going tochange exactly how much the sways in between its minimumand best position. The other thing is,well, just how is it simply simply the absolutevalue the this thing? and also to establish they, girlfriend just need to remember that a cosinefunction or a sine function varies between positive1 and an adverse 1, if it's just a basic function. Therefore this is just multiplyingthat positive 1 or negative 1. And also so if normallythe amplitude, if friend didn't haveany coefficient here, if the coefficient waspositive or negative 1, the amplitude would simply be 1. Now, you're changingit or you're multiply it by this amount. For this reason the amplitude is 1/2. Now let's thinkabout the period. So the an initial thingI want to ask you is, what walk the periodof a cyclical function-- or periodic function,I have to say-- what walk the period ofa periodic role even express to? well let me attract some axes onthis function right over here. Let's say the this rightover below is the y-axis. That's the y-axis. And also let's just say, forthe sake of argument, this is the x-axisright end here. For this reason the period of aperiodic role is the length of thesmallest expression that includes exactly one copyof the repeating sample of that routine function. Therefore what perform they mean here? Well, what's repeating? So we go down and also thenup just like that. Then we go downand then us go up. So in this case, the lengthof the smallest interval that consists of exactly onecopy the the repeating pattern. This can be one of thesmallest repeating patterns. And so this length in between hereand right here would it is in one period. Then we can go in between hereand below is one more period. And there's multiple--this isn't the only pattern that you can pick. You can say, well, I'm goingto specify my pattern beginning here going up and also thengoing down choose that. For this reason you can say that'smy the smallest length. And also then girlfriend wouldsee that, OK, well, if you walk in thenegative direction, the next repeatingversion of that pattern is right over there. However either method you're goingto get the same size that that takes torepeat the pattern. So given that,what is the duration of this functionright end here? Well, to figure out theperiod, we just take 2 pi and divide the bythe absolute value of the coefficientright over here. For this reason we divide it by the absolutevalue of 3, i m sorry is just 3. So we get 2 pi end 3. Currently we need to thinkabout why go this work? Well, if friend think aboutjust a classic cosine function, a classic cosinefunction or a timeless sine function, it hasa period of 2 pi. If girlfriend think aboutthe unit circle, 2 pi, if you start at0, 2 pi radians later, you're earlier towhere you started. 2 pi radians,another 2 pi, you're ago to wherein you started. If you go in thenegative direction, girlfriend go an adverse 2 pi, you'reback to wherein you started. For any type of angle here,if you go 2 pi, you're ago to whereyou to be before. You go an adverse 2 pi, you'reback to whereby you to be before. Therefore the periods forthese are all 2 pi. And also the reason whythis makes sense is the this coefficientmakes you acquire to 2 pi or negative-- inthis case 2 pi-- it's walking to make you getto 2 pi every that lot faster. And so it gets-- your periodis walk to be a reduced number. The takes less length. You're walking to gain to 2pi 3 times together fast. Currently you can say,well, why space you taking the absolute worth here? Well, if this wasa an unfavorable number, that would gain you to negative2 pi every that lot faster. Yet either way, you're goingto be completing one cycle. So through that outof the way, let's visualize these two things. Let's actually drawnegative 1/2 cosine of 3x. So let me attract my axes here. My best attempt. For this reason this is my y-axis. This is mine x-axis. And then let me draw some--So this is 0 ideal over here. X is same to 0. And let me draw x isequal to positive 1/2. I'll attract it appropriate over here. For this reason x is equal to hopeful 1/2. And we haven't change thisfunction up or under any. Then, if we want to, wecould add a consistent out here, outside of the cosine function. Yet this is hopeful 1/2, or wecould just write that together 1/2. And then down here, let's saythat this is an unfavorable 1/2. And also so permit me draw that bound. I'm simply drawingthese dotted lines so it'll becomeeasy because that me come draw. And also what happens when this is 0? well cosine of 0 is 1. But we're going come multiplyit by an adverse 1/2. So it's walking to it is in negative1/2 right over here. And then it's goingto start going up. It have the right to only go inthat direction. It's bounded. It's walk to begin goingup, climate it'll come earlier down and then the will gain backto the original point right end here. And the inquiry is,what is this distance? What is this length? What is this length going come be? Well, we recognize whatits period is. It's 2 pi over 3. It's walk to gain tothis allude three times as rapid as a traditionalcosine function. For this reason this is goingto be 2 pi over 3. And also then if girlfriend giveit another 2 pi over 3, it's going to obtain backto that same point again. So if you go one more 2 piover 3, for this reason in this case, you've now gone 4 pi over 3,you've completed one more cycle. For this reason that length rightover over there is a period. And also then you couldalso do the exact same thing in the an unfavorable direction. So this ideal over below would benegative, negative 2 pi over 3. And to visualize the amplitude,you watch that it can go 1/2. Well, there's 2 waysto think about it. The difference in between themaximum and the minimum point is 1.

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Half of that is 1/2. Or you could say that it'sgoing 1/2 in magnitude, or it's swaying 1/2 awayfrom its middle position in the hopeful or thenegative direction.