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 = 3 sin(πx + 2)

Accepted Answer

Below there. Today we're going to 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 5 multiplied by sign of i multiplied by X + 4. 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 figure out what the amplitude is. Second, we're trying to figure out what the period is. Thirdly, we're trying to figure out what the phase shift is, and lastly, we're asked to create a graph only considering one period of our specific function. So with that in mind, let's read off our multiple choice sensors to see what our final answer set might be, noting we'll read the amplitude first, then the period, and the phase shift last. So A is 52, and -4 divided by pi. B is 52, and 4 divided by pi. C is 42 and 5 divided by pi, and finally, D is 42 and 5 divided by pi. OK. So, first off, let us recall and note that our given function is written in the following form. It's written as Y equals A multiplied by sin of B X minus C. So where capital A denotes the amplitude and the amplitude is given to us, so we need to note that we need to take the absolute value of the amplitude. So the amplitude A is equal to 5, so the absolute value of 5 is 5, as stated in our problem itself with the function itself. And then we're told the period, which we'll call the period P is equal to 2 pi divided by B, and we know that B in this case, is going to be pi, so we need to take 2 pi divided by pi. Which is equal to 2. And finally, our phase shift, we could find that by taking C divided by B. So our phase shift is going to be equal to -4 divided by pi. Awesome. So with that in mind, we now need to start focusing our attention on the graphing the function. So in order to graph the function, we need to start with the phase shift. So when we start with our phase shift, we will find that X1, or X subscript 1 is equal to -4. Well, so -4 divided by pi. So -4 divided by pi. So -4 divided by pi to be precise. So now our next step is we need to divide the period into 4. So when we do that, and let's make a note of this off to the side here. So when we say the period P divided by 4, we will say that this is equal to 2 divided by 4. Which is ultimately, when we simplify going to be equal to 1/2. Fantastic. So now that we know that our X1 value is equal to -4 divided by pi, let's solve for X2 X3, X4, and X5. So we can have 5 different points for us to plot on our curve, or on our graph to get a nice curve going, so we have a nice graph to work with. So we have X2, and X2 is going to be equal to X1 plus. P divided by 4. So this is going to be equal to -4 divided by pi plus 1 half, which is gonna be equal to or rather it's gonna be approximately equal to -0.773. Fantastic. So, similarly, so I'm not gonna write it down each time, so using this method, so we're gonna be, so X1 equals -4 divided by pi, and then X2 equals X1 plus the P divided by 4. So similarly, X3 is going to be equal to X2 plus P divided by 4. So instead of writing out the expression, I'm just gonna give you the Answer. So when you plug in those values into your calculator, it's going to be approximately equal to negative 0.273, and then X4 is equal to X3 plus P divided by 4, which is approximately going to be equal to 0.227. And finally, so 5 X. It's going to be equal to X4 plus P divided by 4, which is approximately going to be equal to 0.727. Awesome. So now we could take these different values for X and we can plug them into a table to help us plot our graph. So I've gone ahead and created a graph in my graphing calculator or you can use whatever graphing software that makes the most sense to you. So I've gone ahead and plugged in our different values for X into our table. So as you can see, when X is approximately equal to -1.273, Y will be equal to 0, because when you plug in the value of X into our function of Y, you will get 0 as a result. And then when X equals -0.773, Y will equal 5, and when X is approximately equal to 0.273, Y will be equal to 0. And when X is equal to 0.227, it be Y will be -5. And finally, for when X is equal to 0.727, Y will be equal to 0. So when we plot these points, it's important to note that we're using the following format that it's X, Y, so we're either going to the left or right. If it's negative, we're going to go to the left for X, and if it's positive, we're going to go to the right on the horizontal axis. And if we're dealing with Y, if Y is positive, that means we will go up, but if Y is negative, we will go down. The y axis. So with that said, so when we have our different plots, when you plot our table onto our graph, we can see that our red dots denote our different plots. You could see that we have one crest and one trough or one valley, so one mountain in one valley, so that represents one period, because that's how long it takes to do one period. So we have like a nice curve here for our sign graph here. And that's it. So this problem is very easy to solve solve for as long as you have a a strong understanding of how to graph a problem like this. So the background theory behind graphing sine and cosine graphs. In this case, we're just focusing on sign graphs. So with that Said, since we've officially solved for this problem, looking at our multiple choice answers, the correct answer has to be the letter A, which the amplitude is equal to 5, the period is equal to 2, and the phase shift is equal to -4 divided by pi. And that's it. 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 = 3 sin(πx + 2)

Key Concepts

Amplitude of a Sine Function

Period of a Sine Function

Phase Shift of a Sine Function

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