Graph each function over a one-period interval.
y = cot ...

Question

Graph each function over a one-period interval.

y = cot (3x)

Accepted Answer

Hey, everyone in this problem, we're asked to graph the following function and to consider only one period. So the function we have is Y is equal to cotangent of six X. However, given this blank coordinate plane graph to graph this on, so we wanna consider only one period. So the first thing to do is to figure out what a period of this function is. And what is the period of this function we have Y is equal to cotangent of six X. Now, when we're dealing with the period, we want to recall that it's related to the B value of our function. And that B value is the numerical coefficient in front of the X inside the brackets. OK. So that number that's multiplying our X in the brackets. In this case, that's going to be six. Now, our period is related to B through the regular period of this function. So we think of cotangent of X instead of cotangent of six X code tangent of X has a period of pi and so the period of the function we have here is going to be that regular period pi divided by our B value which in this case is going to be pi divided by six. So we have this new period for our function and we wanna graph just one period. So we're gonna graph from zero all the way up to pi divided by six. All right. Now, some important things to note here, we have no negative out in front. So we have no reflection. So this is gonna look like a regular cotangent curve. It's not going to be reflected and we have no vertical or horizontal shift. There's nothing being added or subtracted at the end, there's nothing being added or subtracted in the um brackets. So nothing is going to be shifted. OK? So this is gonna look very much like a regular cotangent curve. It's just going to be stretched, compressed a little bit. OK? So let's start by finding the asymptotes. We know that with cotangent, we have asymptotes on our graph. Let's find where those are gonna be. And we're just gonna call these VA S or vertical acids. Now, we have the function Y is equal to cotangent of six X. We recall that cotangent is the reciprocal of tangent. And so we can write this as cosine of six X divided by sign of six X. And when we do that, it's easier to see where the vertical asymptotes come from. They're gonna come from the denominator being equal to zero. If the denominator is zero, this function is undefined. So we get vertical asymptotes when sign of six X is equal to zero. Hm. When do we get sign of something equal to zero? Well, if that's something is some integer multiple of pi, OK. Sign is zero when the is zero, theta is pi theta is two pi so we have some integer multiple of pi, right? Well, we don't wanna know when six X is going to be zero. We wanna know when X is going to be or sorry, not 16 X is equal to K pi one X is equal to whatever it's going to be equal to. OK. So we're gonna divide both sides by six. We had that X is equal to K multiplied by pi divided by six. OK? So we have these in any integer multiple of pi divided by six is where we are going to get an Asymptote for this function. Now, on the interval we're dealing with, OK? From zero to pi divided by six. Where do these asymptotes occur? Well, we get an Asymptote when X is equal to zero. OK. That corresponds to K being equal to zero. Now we get another one when X is equal to pi divided by six. A corresponding to K equals one. And so we have these two asymptotes that we're gonna grasp and we're gonna do that in green so that they stand out and we're gonna draw them as dotted lines. So X equals zero rayed along that Y axis we have a vertical Asymptote and then at pi divided by six, we have the same. All right. So we have our asym toes, we know what we're the period. So we know the points we're graphing between, what we're gonna do is just take a look at the points that lie between those two asymptotes to get an idea of what this function is doing. So we have our X values, our Y values, we're gonna do a table of values and sort this out. So we have X equals zero pi divided by 24 pi divided by 12 pi divided by eight. And finally, pi divided by six. Now zero and pi divided by six, we know we have our vertical isom. OK? So we don't need to worry about what our Y value is doing there. We know we have a vertical Asymptote and from these other points, we're gonna be able to infer which direction our graph is going at those asymptotes. Now, when X is pi divided by 24 our function is going to become code tangent of six, multiplied by pi divided by four or sorry, pi divided by 24 which just gives us pi divided by four. OK. So code tangent of pi divided by four. Well, cosine of pi divided by four, divided by sine of pi divided by four. This just gives us one. OK. All right. Now when X is pi divided by 12, our function becomes code tangent of pi divided by two, a cotangent of pi divided by two. Well, that's just going to be equal to zero. OK? Because cosine will be zero there and sine will not be pi divided by eight. We're looking at code engine of three pi divided by four. And in this case, we get that this is going to be negative one. All right. So let's go back to our diagram. We're gonna plot this thing in blue. So at pi divided by 12, we know that our function is at zero pi divided by 24 our function is gonna be at one and pi divided by eight. It's going to be a negative one. OK? So we can see that on the X equals zero Asymptote, our function is going towards positive infinity. We're gonna connect these points together and then it's gonna go down to negative infinity as we go towards that pi divided by six Asymptote in our function code tangent of six X is going to look like that. Thanks everyone for watching. I hope this video helped see you in the next one.

Graph each function over a one-period interval.
y = cot (3x)

Key Concepts

Period of Trigonometric Functions

Cotangent Function Properties

Effect of Horizontal Scaling on Graphs

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