Determine the amplitude, period, and phase shift of each funct...

Question

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function.

y = cos(x + π/2)

Accepted Answer

Welcome back, everyone. Given the function Y equals the cosine of X plus three halves of pi identify the amplitude period and phase shift from the options below. Then sketch its graph by considering only one period A says the amplitude is three halves, the period is two pi and the phase shift is negative pi divided by two which we can see it's a graph on the diagram B says the amplitude is three halves, the period is pi and the phase shift is a half of pi C says the amplitude is one, the period is two pi and the phase shift is negative three halves of pi. Again here is the diagram and the D says the amplitude is one, the period is two pi and the phase shift is three halves of pi. Now let's go back to our equation. OK. And for our equation, let's pick it apart. OK. And we can do that by asking ourselves, what do we know about cosine functions in trigonometry? We recall that the general form of this trigonometric function is equal to A plus BX minus C. OK. Where A is the amplitude, OK. We can find a period, I should have, let me write that in the other, let me write that the other way around. So the amplitude is A, OK? Are the absolute value of air. The period is equal to two pi divided by B and the Fears shift. OK. The fears shift is equal to C divided by B. Now, do we have all of those values from our trigonometric function? Well, for Y being equal to the cosine of X plus three pi divided by two, if we compare it to the general form, OK, then we can notice that the coefficient of the cosine which is a would be equal to one, the coefficient of X. The term inside the cosine B is equal to one and our phase shift C OK is going to be equal to negative three halves of pi because notice here we can rewrite X plus three pi three pi divided by two as X minus negative three pi divided by two which will more closely fit our general form. Know that we have the values for A B and C. That must mean the amplitude is going to be equal to the absolute value of one, which is one, the period is going to be equal to two pi divided by one which equals two pi and the phase shift is going to be equal to C negative three halves of pi divided by B which is one which would still leave us with negative three halves of pi thus, we have the amplitude period and phase shift. Now that we have this information, let's try to use it to sketch our graph. OK. So I'm gonna just copy this here. OK. I'm gonna bring it down with me where I have a little bit more space. Let me use right here. OK. And now for us to continue for us to plot this graph, we'll first need to start with this phase shift. OK. So let's let X one be equal to negative three halves of pi. That's where we're starting in our sketch. Next, let's divide the period by four. OK. Now that will help us to better sketch our graph. So that's gonna be two pi divided by four which equals a half of pi. So now it makes sense for us to go up in increments of a half of pi. OK? Now let's put a table OK? Of X and Y values first to keep track of what points are going to be on our sketch. OK? And as we said, we're starting at negative three halves of pi and increasing by increments of half of pi. So that's gonna be negative pi OK. Then negative half of pi OK, then zero, then a half of pi. Now what are the Y values going to be? Well, remember we're increasing by values for half of pi here. So all we're going to do is substitute these X values into our function. Now, we could go ahead and do that. But for the sake of time, I'm going to leave you to do that on your own. And when you do, you should find that when X equals negative three house, sorry, negative three halves of pi then Y equals one. When it's negative pi Y is zero. When it's negative pi divided by two, it's negative one when X is zero, Y is zero. And when X is pi divided by two Y is one, now we have all the values, we need to plot our graph. OK. So I'm gonna come down here just gonna go right below it. Uh I think, yeah, I think this is a good spot and now let's just set up our graph with our X and Y values and let's split them up. So now we know we have negative three pi divided by two negative pie, negative pi divided by 20. OK? And then pi divided by two. And then we're gonna have our Y values of 10, negative one zero. OK? And then a half of pie. So now again, this is just a sketchy probably is gonna look way better than mine, but it should look something like this. That's what our graph should look like. So with this amplitude and with this sketch, which one of the answer choices would be correct? Well, we're right there. Answer choice C you probably notice that. And when we go down to our graph. OK. Notice that. Well, in, in my own little way, this graph looks similar to our sketch. Therefore, c is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function.
y = cos(x + π/2)

Key Concepts

Amplitude of a Trigonometric Function

Period of a Trigonometric Function

Phase Shift of a Trigonometric Function

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