Graph the functions in Exercises 13–22. What is the period of ...

Question

Graph the functions in Exercises 13–22. What is the period of each function?

sin (x/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. Graph the function Y equals sign of 2 X and determine its period. Consider only 1 period when graphing. Awesome. So it appears for this particular problem we're asked to graph this function of Y equals sin of 2 X. Awesome. So now that we know that we're trying to graph this function, and that's our final answer we're ultimately trying to solve for, we need to recall and note that the given function is in the form of Y equals. A multiplied by. Sign of B. X minus. See Plus D, where the absolute value of A where A denotes the amplitude. And 2 pi divided by B is equal to the period. And C divided by B is the phase shift. And capital DD is the vertical shift or the translation. Fantastic. So let us not for this particular problem, the amplitude, the absolute value of A, which means anything inside of the absolute value symbols will become a positive value. So A is equal to the absolute value of 1, which is equal to 1. The period, which is 2 pi divided by B, which in this case is going to be equal to 2 pi divided by 2 is our value for the For B in this case is 2, which is gonna be simplifying to simply pi when we perform that eval when we perform that. Math for it. So when we take 2 pi divided by 2, we'll just get we'll get simply pi. And then the phase shift, which is C divided by B, is going to be equal to 0 divided by 2, which is simply going to be equal to 0, and the vertical translation D is simply going to be equal to 0. And now we need to divide the period into 4 equal parts, so we need to take the period P divided by 4. So when we do that, we'll get pi divided by Or So let us note that at X subscript 1, so X1 is gonna be equal to 0. And let us also note that X2 is going to be equal to X1 plus pi divided by 4, which is going to be equal to I divided by 4. And let us note that X3 is going to be equal to X2's value, so the value for X2. Plus pi divided by 4, which is going to be equal to pi divided by 2, and let us note that X to X4. So we're gonna take for X4 is going to be equal equal to the value of X3 plus I divided by 4, which is gonna be equal to 3 pi divided by 4. And finally, like 5 or 5 so like X5, so X subscript 5 is going to be equal to X4, the value of X4. Plus pi divided by 4, which is gonna be equal to simply pi. So now we need to evaluate the function at these specific X values. So with that said, we need to take, and let's draw a line to separate so we don't get confused. So for Y1 is equal to Y of X1. Which is gonna be equal to Y of 0, so we're plugging in a value of 0, N for X. So once again, we need to note that our original function. is going to be simply Y equals of 2 X. So if we plug in a value of 0 in for X, so sign of 2 multiplied by 0 is gonna be sin of 0, so that's going to be simply equal to 0. So I make a, I'll make a quick little note here. So note that our original function is Y equals sign of 2 X. So what we're doing now is we're plugging in different values for X. So in this case, when we plug in a value of 0, we'll get sin of 0 is equal to 0. So now we need to solve for Y2 and Y2 is going to be equal to Y of X2, which is going to be equal to Y of I divided by. Or Fantastic. And I said that for X to the power, so X subscript 3, I meant to say that you're going to get an answer of pi divided by 2. So let's quickly, let's take a moment here to pause. So for X1 is 0, and then X2 is going to be pi divided by 4, then pi divided by 2 for X3, and then X4 is going to be 3 pi divided by 4, and then X5 is going to be equal to pi. So now, going back to solving for Y2, since we know that X2 is equal to pi divided by 4, it's gonna be equal to drum roll. Simply one, when you plug in the value for I divide by 4. So in an effort to save time, since you know the pattern, I'm just going to simply write what the values for Y3, Y4 and Y5 are following this pattern. So Y3 is simply going to be equal to 0. Y4 is. going to be equal to -1 and Y5 is going to be simply equal to 0. So now we need to plot the points 0.0, which is the origin points divided by 4.1. We also need to plot. I divided by 2.0 in parentheses. We also need to plot the point. 3 pi divided by 41, and we also need to plot 1.0 in parentheses. So we're going to be plotting these points. We're going to plot the 12345. We're plotting these 5 points and we're connecting them with a smooth curve. So when we plot these 5 points and we connect them with a smooth curve, We should produce the following graph. And as you can see, we have a curve. Represented in blue, and then we have our five different points represented by yellow dots. So we have our point at 0.0, at pi divided by 4.1, pi divided by 2.0, 3 pi divided by 4.1, and 0. And as you can see, it starts at the origin point. It goes up, and then it goes down to cross through. The X axis and then it dips down and then it curves back up to touch the X axis a second time. And this once again is a like a shape of a standard sign graph. And that's it. That is how you plot the graph, and we've solve for our final answer. Hooray. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Graph the functions in Exercises 13–22. What is the period of each function?

sin (x/2)

Key Concepts

Period of a Function

Transformation of Functions

Graphing Trigonometric Functions

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