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 = 1/2 sin(x + π)

Accepted Answer

Below 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 1/4 multiplied by sign of X + 2 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 answers. First, we're trying to solve for the amplitude, then the period, then the phase shift, and then our fourth and final answer we're trying to solve for is we're trying to create a sketch of this graph for this specific function considering only one period. So with that in mind, let's read off our multiple choice answers to see what our final answer might be. Noting that we're going to read the amplitude first, then the period, then the phase shift. So A is 42 pi and pi divided by 2. B is 42 pi and negative pi divided by 2. C is 1/4, 4 pi and pi. And finally, D is 1/4, 2 pi and negative 2. OK. So first off, let us note that our function that's given to us by the problem itself is in the following form. It's in the form of Y. Equals a multiplied by sign of. B X minus C. So it's in this following format, and let us note that the absolute value of A, where capital A denotes the amplitude. In this case, it's going to be equal to the absolute value of 1 divided by 4 or 1/4, which is equal to. 14th And we're also told the P, which I'll call the period P once again is equal to 2 pi divided by B. And in this case, it's going to be equal to 2 pi divided by 1 means 1 for B in the case of our problems, since it's BX, there's a 1 in front of the X. So that means that our value for B is going to be 1. So that means our period is going to be equal to 2 pi. And then our phase shift, which is C divided by B, is going to be equal to -2 pi divided by 1, which is equal to -2 pi. Fantastic. So, when we bring up our graphing software or our graphing calculator, so when we start to graph the function, we need to start with the phase shift. So we'll say that X1. Is equal to -2 pi. Awesome. And we need to divide the period into 4 is our next step. So when we do that, we will find that the period divided by 4 is equal to 2 pi divided by 4, which is equal to pi divided by 2. Fantastic. And as we should recall a note, we could go ahead and write that once again that X1, where I, when I say X1, I mean X subscript 1. So X1 is equal to -2 pi. And similarly, we could write, as we should recall, that X2 is equal to X1 plus P divided by 4, which is equal to -2 pi. Plus pi divided by 2, which is equal to -3, so it's gonna be equal to -3 pi divided by 2. Fantastic. And then next we have to do X3, and X3 is equal to X2 plus P divided by 4, which is equal to -3 pi divided by 2, plus pi divided by 2, which is equal to negative pi. Fantastic. So now we need to do X4, and X4 is equal to X3 plus. P divided by 4 where once again, in case we've forgotten, P is our period, and this is equal to when we plug in our nons, negative pi plus I divided by 2. Fantastic. So then this is going to be equal to negative pi divided by 2. And then lastly, X, so 5 X is equal to 4 X plus P divided by 4, which is all equal to negative, I divided by 2, plus pi divided by 2. Which is equal to 0. Fantastic. So now using these values, since we're using X once again means as we should recall, X is helping us determine our our phase shift, as we should recall. And since we're trying to start with the phase shift, so if we plug these values into a table, so let's scroll up. So now when we can, we can basically what we're going to be doing is we're going to be taking these phase shift values that we found for X, and we're going to be using them to plot our points on our graph to help us create our sketch. That is essentially what we're doing. So we found our different values for X. So as we can see looking at our table. Which I put together a table and I copied and pasted a sketch of a graph that I created in my graphing calculator or graphing software for this particular function. So as you can see, when X equals -2, Y equals 0, and when X equals -3 pi divided by 2, Y equals 1/4, and when X equals negative pi, Y equals 0, and when X equals. Negative pi divided by 2, Y equals -1/4, and when X equals 0, Y equals 0. And as you can see, these points are plotted on our graph, which is our sketch of our graph. So as you can see, we have a point at 00, which is the origin point. Then we have a point at negative pi divided by 2, which is at so it's at negative pi divided by 2, 1/4. Then we have a value at X equals negative pi and Y equals 0. Then we have at X equals -3 divided by 2 at 1/4, so it's -3 divided by 2.14. And then finally we have a point at -2 for X, and then our y value is 0. So X.0. So that's what you should remember for your formatting for your graphing points. It's X. Y. It's how we plot, and as you can see, they are all lined up on the X horizontal axis. As you can see, and we have our little sign curve along these points. And that's it. We've officially created this graph, or sketch of a graph. So that's our final final answer that we're ultimately trying to solve for. So what we did is we plotted the points and connected them to create a smooth curve. So looking at our multiple choice answers, the correct answer, without a doubt has to be multiple choice answer letter D, which once again is the amplitude is 1/4, the period is 2, and the phase shift is negative 2 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 = 1/2 sin(x + π)

Key Concepts

Amplitude of a Sine Function

Period of a Sine Function

Phase Shift of a Sine Function

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