In Exercises 59–62, sketch the graph of the given function. Wh...

Question

In Exercises 59–62, sketch the graph of the given function. What is the period of the function?

𝔂 = cos πx/2

Accepted Answer

Welcome back, everyone. In this problem, we want to determine the period T and sketch the graph of the function Y equals the cosine of pi X divided by 4. Here we have an empty graph on which we're going to sketch the graph of Y. Now, how can we sketch our graph? What do we know? Well, notice here that we have a cosine function. And recall, OK, that the general form. The general form. Of a cosine function is Y equals A + B multiplied by X minus D, OK. Plus C. Where A represents the amplitude, B affects the period of the function, C represents the vertical shift, and the D represents the horizontal shift. Now, if we compare the general form, OK, to the equation that we have, Y equals the cosine of pi X divided by 4, OK, then we should notice that A. Equals one, OK. B is gonna be equal to 14th of pi because it is what is multiplying X. C, the vertical shift equals 0, and D, sorry, also equals 0. Now, I, I spoke about C earlier, but what does A, B, and C mean? Well, A represents the amplitude. So in this case, the amplitude of our function is 1. B will help us to find the period of the function because the period of the function T is equal to 2 pi divided by the absolute value of B. So now in this case. That means T is gonna be 2 pi divided by the absolute value of pi divided by 4, which is pi divided by 4, and the 2 pi divided by pi divided by 4 is going to be equal to 8. Therefore, our period T is 8 units. Next, like we said earlier, C represents the vertical shift. So since C equals 0, there is no vertical shift and D represents the, sorry, represents the horizontal shift or the phase shift. Since D equals 0, there is no horizontal shift, so the function completes one full cycle from X equals 0 to X equals 8 and repeats every 8 unit. And since there's no vertical shift, that means the range of the function is from -1 to 1 inclusive. So now for us to determine, well, now that we've found the period T, OK. For us to sketch the graph of the function, we'll need some points. So let's use our equation to help us find some X and Y coordinates for our graph, OK? Now, when X equals 0, for starters, OK, then Y is going to be equal to the cosine of. Pi X divided by 4, so that's divided by 4 multiplied by 0. That's going to be the cosine of 0, which equals 1. And this is the maximum point in our graph. Now, when X equals 2, OK, then Y is going to be equal to the cosine of a half of pi, which equals 0. When X equals 4. Y equals the core sign of pi, OK, which is going to be -1, the minimum value on our graph. When X equals 6, Y equals the cosine of 3 halves of pi. Which is 0 And when x equals 8, Y equals the cosine of 2 pi. Which is equal to one. So no, from these values, the points on our graph are going to be 01. 20 Or some points in our graph for -1. 60 and 81. So all we need to do now is to plot these points on our graph and then plot Y equals the cosine of pi X divided by 4. So now, if we go ahead and plot here, like we said, we're gonna have 01. 20 4-1. 60 And 81, OK? And now we know that our cosine graph is symmetric about the Y axis. So the corresponding points over here will be at -2, -4, -6, and -8 respectively. Now we can go ahead and plot. So now we know it's gonna look something like this. OK, oops, oh, I just missed that. Let me put it in a red for us to see it a little bit there so. You know, here, this is gonna be our graph. Let's try our best. Let's try our best here so you can try your best to make sure that you have a smooth curve. OK, I'm trying to make mine as smooth as possible, but it starts to get a little messy on the left, right? But basically, this is what our curve should look like, OK? So this is going to be the graph of Y equals the cosine of pi X divided by 4. Thanks a lot for watching everyone. I hope this video helped.

In Exercises 59–62, sketch the graph of the given function. What is the period of the function?

𝔂 = cos πx/2

Key Concepts

Period of a Function

Graphing Trigonometric Functions

Transformations of Functions

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