Graph each function over a one-period interval.
y = ½ co...

Question

Graph each function over a one-period interval.

y = ½ cot (4x)

Accepted Answer

Hey, everyone in this problem, we're asked to graph the following function and to consider only one period, we're given the function Y is equal to one quarter code tangent of eight X. And then we're given this blank graph to draw on. Now we're after, after just one period. So let's think about what the period of this function is. Now the function we have Y equals one quarter cotangent of eight X. And to determine the period, we wanna think about what the B value is and we're called the B value is going to be the coefficient or the number multiplying the X inside of the bracket. OK. So in this case, we have cotangent of eight X. So our B value is going to be the eight, five B is eight. Now, how is this related to the period? Well, it's related through the period of just a regular code tangent function. OK. So if we just had code tangent of X, the period would be pi and so the period of R function T is going to be that regular period pi divided by RB value. OK. So in this case, it's going to be pie divided by eight. If we look at the graph, we were given, we have this graph, it goes from 0 to 5 pi divided by 32. So we're gonna choose to graph the period from zero two pi divided by eight. OK. That's gonna give us one period and that's the period we're gonna choose to graph. Now, looking at our function, we have a positive number out in front, there's no negative. So we have no reflection. OK. So in terms of the shape, we expect it to be the typical shape of cotangent. OK. Stretched and compressed a little bit vertically and horizontally, but the same general shape and there's no horizontal or vertical shifts either, right. So we expect it to be kind of centered around that period that we've found. All right. Now, when we have code tangent, we have vertical asymptotes. So let's think about finding the vertical asymptotes for this particular function. So we want to find the vertical asymptotes. Now we're gonna rewrite our function. So we can see it Y is equal to one quarter code tangent of eight X. Now, cotangent recall is the reciprocal of tangent. OK. So we can write cotangent of cosine divided by S so our function becomes one quarter coign of X divided by sign of eight X. And if we write it like this, it makes it much more clear where these vertical asymptotes come from. OK. When we have a function, the vertical asymptotes are gonna occur where the denominator is zero. If the denominator is zero, then the function is undefined, we get a vertical Asymptote. So in this case, we're gonna have vertical asymptotes when sign of eight X is equal to zero. OK. Well, when is sign of something equal to zero, sign of something is equal to zero. If that's something in this case, A X is some integer multiple of pi, OK. So if we have pi multiplied by K or K is an inch, OK. So if the something is equal to zero, pi two pi et cetera sign will be zero. Now we've written this in terms of eight X, we wanna know what the condition is for X, what X values will give us this condition. So we're gonna divide both sides by eight to isolate for X and we get that X is gonna be some multiple A pi divided by eight. OK? Some integer multiple of pi divided by eight. So now we have the general condition for asymptotes, we wanna look at this particular period we're graphing. OK. So on the interval from zero to pi divided by eight, where are these asymptotes? Well, if K is equal to zero, we get X is equal to zero as a vertical asom to, if K is equal to one, we get X is equal to pi divided by eight. OK. So the two asymptotes on our interval are X is equal to zero and X is equal to pi divided by eight. If we choose any other K value, we are gonna be outside of our interval. So let's go to our graph and we are gonna add those asymptotes to it. We're gonna draw them as dotted vertical lines. So we have one at X equals zero and we have another at pi divided by eight. OK. That one at X equals zero is there, it's a little bit hard to see because of the Y axis but it's there. All right, we know our period, we have our asymptotes. We have a general idea of what the function is gonna look like. Let's go ahead and write a table of values for some of these points in between our asymptotes to get a better idea of what this function is doing. We're gonna do a table of values with our X and Y values. OK. We're gonna choose the one shown in the graph. So we have X equals zero. Pi divided by 32 pi divided by 16 three, pi divided by 32 and pi divided by eight. Now pi divided by uh sorry, X equals zero and pi divided by eight, we know that these are our vertical asymptotes. So the function is going to be approaching positive or negative infinity. And when we take a look at the other X values and the corresponding Y values, it's gonna give us an indication of which one is approaching positive which one is approaching negative infinity. OK. All right. So a pi divided by 32 what happens to our function when we get one quarter multiplied by code tangent of pi divided by four. Now code tangent of pi divided by four is just one. And so our function is just going to be equal to one quarter. When X is pi divided by 16, we get one quarter, code tangent of pi divided by two, a co tangent of pi divided by two is going to be zero. And so the entire function is going to be equal to zero. And then when X is three pi divided by 32 our function simplifies to a quarter code tangent of three pi divided by four. When we have cotangent of three pi divided by four, that's going to be equal to negative one. And so our function becomes negative the core. OK. So now we have our vertical asymptotes three points in between. Let's go ahead and graph using this information. So at pi divided by 16, we know our function is at zero or Y value pi divided by 32 it's going to be at one quarter and at three pi divided by 32 it's going to be a negative one quarter. OK? So we can connect these with a nice smooth line that looks like a code tangent function. OK. From the left hand side as we get closer and closer to the X equals zero Asymptote. Our function is going up to positive infinity. And as we approach, the X equals pi divided by eight Asymptote, our function is going down to negative infinity. And that is a graph for this function one quarter cotangent of eight X. Thanks everyone for watching. I hope this video helped you in the next one.

Graph each function over a one-period interval.
y = ½ cot (4x)

Key Concepts

Cotangent Function and Its Properties

Period of Trigonometric Functions

Amplitude and Vertical Scaling

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