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 = sin (x − π/2)

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Given the function Y equals of X minus 3 pi, identify the amplitude, period, and phase shift from the options below. Then sketch its graph by considering only one period. Awesome. So it appears for this particular problem we're asked to solve for 4 separate things. So we're trying to figure out the amplitude is our first answer, the period is our second answer, the phase shift is our 3rd answer, and our 4th and final answer is we're trying to sketch a graph of this specific function by considering only one period. So with that in mind, let's read off our multiple choice answers to see what our final answer pair or answer set should be. And note that we're gonna read the amplitude first, then the period, and lastly the phase shift. So A is 12 pi and negative 3, B is 12 and 3 pi. C is 32 and 3, and finally D is 3 and 3 pi. OK. So, first off, it's important to note and recall that our given function is written in the following form. It's written as Y equals A multiplied by sine of BX minus C. Where the absolute value of A, where A denotes this capital A denotes our amplitude, and the amplitude in our particular problem is given to us as one. So the absolute value of one is one. And we're also told the P and the period P is equal to 2 I divided by B and we're told that B is 1, because X all by itself means there's like an invisible one out in front. So that means that our period is equal to 2 pi divided by 1, which is equal to 2 pi. Fantastic. And then our phase shift, we get this by taking C divided by B, and our phase shift is going to be equal to 3 pi divided by 1, which is equal to 3 pi. Awesome. So in order to help us graph this particular function, we need to recall and note that for the graphing function, we need to start with the phase shift. So when we start with the phase shift, you can say that X1 is equal to 3 pi. So let's create, so when I say X1, I mean X subscript 1. So let's create a couple different points for us to work with. So let's solve for X1, 234, and 5. So we know that we need to divide, so let's make a little note here off to the side in blue. We need to note that we need to divide the period into 4. So we need to take the period P and divide it by 4. So that means that our period P It's going to be equal to 2 divided by 4, which is equal to pi divided by 2. Fantastic. So, considering that piece of information we can now solve for X2, because we know that X1 is equal to 3 pi. So X2 is equal to X1 plus P divided by 4. Which is gonna be equal to 3 pi. Plus pi divided by 2, which is going to be equal to 7 pi divided by 2. So now that you know how this is working, so for 3, so 3 X, it's similar, so it's gonna be 3 X is equal to X2 plus P divided by 4, and I'm gonna skip writing it out, but this will ultimately equal 4 pi. And moving on to 4X, so 4 X or X to the subscriptive 4. So 4 X is equal to 3 X. Plus P divided by 4, which is equal to, as we should recall. From our previous adding method, it's gonna be when we plug that into our calculator, 9 pi divided by 2. And finally, for X, so for 5 X 5 X is equal to X4 plus P divided by 4, which is going to be equal to 5 pi. So now we need to take our different values for X, and we need to plug it into a table to help us solve for our different values of Y so we're going to be plugging in our value for X into our equation, that's solving for Y to get our diff so basically to help us get our coordinate points that we're gonna plot on our graph. So I've gone ahead and created a table of values that we will use to help us plot our graph, and it's important to note, in case you've forgotten that our standard plotting method is going to be X. Y. So you go down the X axis first either to the left or right, and then you go up or down for Y. So it's important to note, like using our phase shifts that we found together or the different values for X, we know that when X is 3 pi, Y is 0, when X is 7 pi divided by 2, Y is 1, when X is 4 pi, Y is 0. And when X is 9 pi divided by 2, Y is -1, and when X is 5 pi, Y is 0, and when we plot these different points, we have 1234, and 5. When we plot these 5 different points on our graph, we will obtain the following graph. As you can see, because we can using the easy one, so using our X equals 0, so we're just staying on the horizontal X axis, and then for, so we gotta go to the right 5 pi. So we go to the right 5 pi and then we don't go up or down any since Y is 0, so it's gonna be at 5.0. And then when we go X equals 9 pi divided by 2, we go over 9 pi divided by 2, and then we go down 1 because it's -1 or -1 for a Y. And then for when X equals 4 pi, we go over to the right positive 4 pi, and then we and then we plot at 0. Her lie And then when X equals 7 pi divided by 2, we go to the right, 7 pi divided by 2, and then we go up 1 means it's a positive one. And finally, when X equals 3 pi, we go over to the right 3 i and then we plot at Y equals 0, so we don't go up or down and so it's just at 0. And as you can see, this is 1 period, as you could see because we have one mountain and one. Valley is what I call it, so that's like when it goes up and down, that is one full period. And that's it. We've officially solved for this problem. So when we look at our multiple choice answers, our correct answer without a doubt has to be multiple choice answer letter B, which states that the amplitude is 1, the period is 2 pi, and the phase shift is 3 pi. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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

Key Concepts

Amplitude

Period

Phase Shift

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