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


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