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

Question

Graph each function over a two-period interval.

y = cot (3x + π/4)

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 cotangent of five X plus pi divided by two. We're given this blank coordinate plane to draw our graph form. Now to start, we're asked to draw two periods of this graph. So let's figure out what the period of this function actually is. OK. So we have our function Y is equal to code tangent of five XP pi divided by two. OK. So from our cotangent, it looks like we have some sort of compression. OK. So a horizontal compression and then we have a phase shift as well. OK. You know, this is not being shifted vertically up or down, it's not being reflected, there's no coefficients out in front of our cotangent. So everything is kind of happening in the horizontal direction. Now, when we think about our periods, OK, we wanna look at the B value of our function. Now we're called the B value is going to be that coefficient on the X inside of the brackets. OK. So the coefficient multiplying our variable in the brackets and in this case, that's going to be five. OK? And let me just make it clear that that's not the X, it's just the five there. OK? So B is five. Now, the period of our function is going to be related to the period of regular cotangent function. OK. If we just have Y is equal to cotangent of X and this B value now, if we just have Y is equal to cotangent of X recall that the period is pi so when we calculate the period of the function, we have, we start with that regular period of cotangent which is pi and divide it by our B value. OK. All right. RB value is five. And so the period of this function is going to be pi divided by five. OK. So we know what the period is. We wanna graph two periods. So we're gonna be graphing two pi divided by five. And if we look at our graph, we can see that the far left end is labeled negative pi divided by 10. The far right is labeled three pi divided by 10. OK. That's a span of four pi divided by 10 or two pi divided by five. And so that is gonna cover two full periods. And so that's where we're gonna graph, we're gonna graph from negative pi divided by 10. Took the pi divided by 10 right. And this is a trigonometric function. It's going to repeat each period is going to repeat. So we could technically graph this anywhere, any choice of X values you want that cover two periods. But based on the graph we've been given, that's the X values we're gonna choose for our two end points. OK. So we have our period. Now, let's find your asymptotes. We have a cotangent function. We know that there are going to be vertical asymptotes. So let's locate those. So we're gonna go ahead and we are gonna find are vertical asymptomat va. Now our function Y is equal to cotangent of five X plus pi divided by two. Recall that code tangent can be written as cosine divided by sine. So what we have here is cosine of five X plus pi divided by two divided by sign A five X plus pi divided by two. When we write it this way, it makes it a little bit easier to see where those vertical asymptotes come from. They recall that a vertical Asymptote is gonna come from the denominator being equal to zero. If the denominator is equal to zero, the function does not exist at that point. That's where we get our asymptotes. So now that we've written like this, we can see that the vertical asymptotes are going to occur when the denominator which is sign A five X plus pi divided by two is equal to zero. OK. So now we have this condition and we want to figure out what this means for the X values where, which X values do we have asked them to attend? So we have that sign of something is going to equal zero. Well, we know that that's true if that something is some integer multiple of pi, OK. So if five X plus pi divided by two is some integer multiple of pi, so it's equal to K pi or K is an integer. The sign of that thing is going to equal zero. So now we have this little equation that we need to be satisfied in order to have our vertical asymptotes. And again, we want the X value, we want the condition on X. So we're gonna rearrange our equation to solve for X when we do that. And we're gonna start by subtracting pi divided by two from both sides. Five X is going to be equal to K pi minus pi divided by two. And then we can divide both sides by five to get that X is going to be K pi divided by five minus pi divided by 10. Uh So now we have our general equation that represents all of the vertical asymptotes of this function. Let's take a look at specific K values to get specific asymptotes within the interval that we're breathing. OK. So if we take K is equal to zero, then we get that X is going to be equal to negative pi divided by 10. Now this is the left end of our graph. Remember we're starting on the left hand side. A negative pi divided by 10. So that's within our interval. That's an Asymptote that we're gonna be interested in. Let's move to the next asym that's gonna occur when K is equal to what we get that X is going to be equal to pi divided by five minus pi divided by 10, which gives us positive pi divided by 10. Again, this is still within our interval. Let's do the next K value K is equal to two. We get that X is going to be equal to two pi divided by five minus pi divided by 10, which gives us the pi divided by 10. OK. And this is the far right of our graph. So that's as far as we need to go. So on the interval, we're graphing, we have three asymptotes, vertical asymptotes and let's go ahead and add those to our graph right now. So we're gonna go to our graph. We're gonna do this in green. So that's easy to see. And we found vertical asymptotes at X equals negative pi divided by 10. We're just gonna draw a dotted vertical line there. Uh Pi divided by 10 and that's the pi divided by 10. And you can see that these vertical ay totes kind of break up those two periods that were graphic. All right. Now, we said initially that we have some phase shift in this function because we're adding pi divided by two inside of the brackets. Let's go ahead and take a look at a table of values that's gonna be the easiest way to figure out what this function is doing in between our Asymptote. So we have a table of values, we have our X value and Y values and we know that at negative pi divided by 10 and at pi divided by 10, we have our asymptotes in between. We have negative pi divided by 20 zero and pi divided by 20. OK. So we're just looking at the first period. Once we move to the second period, it's going to be the exact same. We're just repeating the exact same function. OK. That's why we have this periodic behavior. All right. So let's do it. So at negative pi divided by 10, it's a vertical Asymptote at pi divided by 10, it's a vertical Asymptote at negative pi divided by 20. What do we have? Well, we end up with code engine of negative pi divided by four plus pi divided by two, which is gonna be cotangent of pi divided by four, which gives us positive one. When X is zero, we get cold tangent of P of automa too. She is just going to be equal to zero. And pi divided by 20 we get cotangent of pi divided by four plus pi divided by two. So cotangent of three pi divided by four and this gives us negative one. So if we take a look at our graph. OK? We can plot these points at negative pi divided by 20. We're gonna be a positive one at zero. We're gonna be at zero at pi divided by 20. We're gonna be at negative one. And now we can connect these like a cotangent curve approaching this Asymptote. So we know from the left we're gonna be approaching this X equals negative pi divided by 10. OK. Coming from the right hand side, we're gonna be approaching up to positive infinity and then approaching our pi divided by 10 Asymptote approaching negative infinity. And so that's one period of our cotangent curve. And now we're gonna repeat the same for the next period. Now you could do another table of values for these as well. But we know it's gonna be the exact same. So in the middle at pi divided by five, we're gonna be at zero at three pi divided by 20 we're gonna be a positive one and at pi divided by four, we're gonna be at negative one and we repeat that same first and that's it. That's going to be the graph for that function Y is equal to cotangent of five X plus pi divided by two. Thanks everyone for watching. I hope this video helped see you in the next one.

Graph each function over a two-period interval.
y = cot (3x + π/4)

Key Concepts

Cotangent Function

Phase Shift

Vertical Stretch/Compression

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