Sketching the Graph of a Trigonometric Function In Exercises $55-66,$ sketch the graph of the function.

$$y=\csc 2 \pi x$$

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to asks us to sketch the graph of the function below. Why equals Cosi kin of two pi X. Now, before we can scratch this or sketch this graph, we should break it down to really see what it means. Well, we know the coast seeking is the same as one over sign so we can write this function in the alternate form one over sign of two pi X. Now we want to find the period of our CO c can function because we need that if we're gonna sketch it. So looking at sign, we know that the period is always the same as B. Oops did that wrong? Excuse me? The period is the same as two pi divided by the absolute value of B. And in this instance, B equals two piles. So our period will be two pi divided by two pi which equals one. And now, although I said that was for sign, that extends over to our co c can function. So we know that the period of our graph is gonna be one now, a really interesting or cool technique you can use to graft. Cose Eakin is by first initially graphing um, sign. So we're gonna make the graph of sign of two pi X. We know that the amplitude is equal to one because there's nothing ah, trailing at the beginning of our checking a metric function. So that is gonna make this a lot simpler. Um, so first I'm gonna begin by in blue graphing sign of two pi X. It's just a sketch. So right, the general idea down goes like this, all right? And we know that at each of these, uh, that's one That's two. That's me. I would say that this is Thank you, Warren, and so on. So each one of these is a period of our function, all right, Now, because we're supposed to be graphing Coast seeking to pi X, everywhere that are signed function equals zero. There's going to be an ass motto, and so we could see that easily by just looking at where our sine function intersex the X axis. So I'm gonna come through and draw these yellow dash lines everywhere There's going to be on asked meto which is really nice, really good visual, way off. Like seeing how to graft. Cassie can sense it makes it really obvious where our function is undefined everywhere the same crosses or equal zero. And I'll do that for their that much. Now, in green. I'm gonna come in here and I'm gonna draw secret graph. Now, everywhere that are sine function has a maximum value. That's gonna become a new minimum value, all right, approaching the values and they are touching right there. A positive one, which is in the negative one. And this continues this periodic behavior. Since it's a periodic function, we'll continue forever in the positive and negative directions. Uh huh, Yeah. And just know, they thes maximum values right here. All at y equals one. And then these other values don't hear all that. Negative one. Okay. And so there you have it. That is a sketch of the graph of CO seeking to pi X.

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