Graph each function over a two-period interval.
y = 1 - ...

Question

Graph each function over a two-period interval.

y = 1 - cot x

Accepted Answer

Hey, everyone in this problem, we're asked to graph the following function and to consider two periods, the function we're given is Y is equal to five minus cotangent of X. So first things first, we as graph two periods. So let's figure out what the period of this function is and we have Y is equal to five minus cotangent of X. Now, when we have a trigonometric function like this, the B value of this function is going to determine that period. Now recall that the B value is the coefficient on the variable inside of our trig function. OK. So it's the coefficient on this X instead of our cotangent. Now, in this case, RB value is just one and because our B value is just one, the period is going to be the exact same as the regular period for just cotangent of X. And if we had Y equals cotangent of X, we know that the period is pi OK. In this case, our B value is one and so the period again is going to be pi now this makes sense, we have cotangent of X, we're just taking five minus cotangent of OK. So we have some vertical shifting, we have a reflection but otherwise we have that same cotangent curve. OK. So in terms of the period, it's not changing, we're only changing things vertically. In this case, right now, we wanna graph two periods. If we take a look at the blank curve, we've been given, we have some labels already on the X axis, we can see that they go from zero up to two pi. So we're gonna take that as our two pi interval that will have two periods. OK. All right. So we're gonna graph from zero to two part. And if you weren't given this blank curve, you could choose any two pi interval, right in order to graph this. But because we have this graph already, we're gonna choose to graph from 0 to 2 pi. Now we're dealing with cotangents. So we know there's gonna be vertical asymptotes. So let's go ahead and find out where those are gonna be. We can add those to our graph and that's gonna help us get an idea of the shape of this curve. So we are going to find our vertical asymptotes or VA S, we have the function Y is equal to five minus cotangent of X. And if you're dealing with cotangent or tangent and you're looking for vertical asymptotes, we wanna remember that we can break those up into sine and cosine. OK? When we do that, it's gonna be more clear what the denominator of our function is and that's gonna tell us where those vertical asymptotes are. So we can write our function as five minus cosine of X divided by sine effects. OK. So we've written code tangent as cosine divided by S. Now we want to put this into one fraction. So we're gonna find a common denominator. So on our five, we're gonna multiply numerator and denominator by sin of X. And then we can write this as five sign of X minus cosine of X all divided by sin of X. And now we can see why we've done this. Remember that the vertical asymptotes are gonna occur when the denominator of our function is zero, the denominator of our function is zero. The function is undefined at that point. And so we get these vertical asymptotes, right? So when we've written it like this, it's very clear now to see that the vertical asymptotes are going to occur when the denominator which is sin of X is equal to zero. When does sin of X equal zero, we know that sin of X equals zero if X is some integer, multiple of pi, OK. So X is K pi or K is an integer. Now, on our interval, where are these vertical asymptotes? So we're graphing from 0 to 2 pi if K is zero, we're gonna get that X equals zero as an Asymptote. If K is one, we get X equals pi as an Asymptote. And if K is two, we get X equals two, pi is an Asymptote. These are all on the interval. We're graphing on OK. If we get, go to K values that are larger or K values that are smaller, we're gonna be outside of our intervals. So we don't need to worry about those. But we have these three asymptotes that are gonna be in the interval. We're graphic. So let's go up, We're gonna add these to our diagram. We're gonna draw them as dotted lines vertically but X equals zero, we have one doing it in green. Hopefully, we can see it over top of that axis, but there is a vertical Asymptote there at zero. Then we have another app pie and then we have another at 25. And you can see that these vertical asymptotes kind of break our interval up into those two periods that we want to grasp. Hey, been broken up. All right. So let's think about this function again. We mentioned that this is a regular cotangent function it's reflected because it has the negative right out front of it. And then there's a vertical shift by five units. OK? Because there's this additional five. So that's what we expect to happen. And we can use a table of values to find exactly where these points are gonna be between our asymptotes, what's going on between the asymptotes. OK? All right. So we have our X values and Y values and we wanna know what's going on. Yeah, we're gonna do the table of values just for the first period because we know the second period is just a repetition of the first. OK. That's how these periodic functions work. They repeat each period. So we have X equals zero pi divided by four pi divided by two, three pi divided by four and pi now at zero and pi these are our vertical asymptotes. So we don't need to worry about the Y value there. We know it's going to be going off to positive or negative infinity. And we're gonna infer that from the rest of our table of values. No, if X is pi divided by four, our function becomes why minus cotangent of pi divided by four. Now co tangent of pi divided by four is just one. And so this is going to be five minus one which gives us a Y value of four. If X is pi divided by two, we get five minus code tangent of pi divided by two cotangent of pi divided by two is just zero. So we have five minus zero which gives us a value of five. And finally, if we have X equals three pi divided by four, we have five minus cotangent of three pi divided by four. Let's make it clear that's a minus sign a cotangent of three pi divided by four is negative one. And so we have five minus negative one which gives us six. OK. So now we have three points in one of our periods and we can add to our graph when X is pi divided by four, why is going to be four? When X is pi divided by two, Y is going to be five when X is three pi divided by four, Y is going to be six. And we know that this follows the typical cotangent curve because we're not doing anything horizontally to it. And so we can connect these in the way that the cotangent curve goes just like this. Now, we can see that as we approach our Asymptote at X equals zero from the right hand side, we're approaching negative infinity as we approach our Asymptote pi from the left hand side, we're approaching positive infinity. Mhm All right. So that's one period we're gonna repeat for the second period. So we can mirror this exact same thing. OK. So we have our three points, we connect them with something that looks like a cotangent curve or a tangent curve and there we have it. OK. So this is two periods of our function Y is equal to five minus cotangent of X. Thanks everyone for watching. I hope this video will see you in the next one.

Graph each function over a two-period interval.
y = 1 - cot x

Key Concepts

Cotangent Function

Transformations of Functions

Graphing Over an Interval

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