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

Question

Graph each function over a two-period interval.

y = 1 - 2 cot [2(x + π/2)]

Accepted Answer

Welcome back. I am so glad you're here. We are asked to graph the following function consider two periods. Our function is Y equals four minus two cotangent of four, multiplied by the quantity of X plus pi divided by four. And then we are given a blank graph. We have a vertical Y axis, a horizontal X axis. They come together at the origin in the middle. The range for what's drawn for our Y axis is from negative 10 to positive 10. And the domain for what's drawn for our X axis is from negative five pi divided by 16 to positive five pi divided by 16. And those are all in increments of pi divided by 16. All right. So our cotangent is given to us in a standard form of Y equals C plus a cotangent of B multiplied by the quantity of X minus D. And that allows us to easily identify our A's and B's and C's and D's. But the ones we're going to focus on here first are our B and our D, we can see that B is equal to four and D is equal to a negative pi divided by four. And that's important because when we are graphing cotangent, we want to start off by looking for the period, the period of cotangent is going to be found as pi divided by B, we said our B is four. So our period is going to be pi divided by four. Now, our D is important in helping us determine our vertical asymptotes. Typically with cotangent, we're going to have a vertical Asymptote right at X equals zero. However, it changes when there's a phase shift or in other words, when there's ad value here, we do have a devalue, we have a phase shift. So our first Asymptote is going to be found by taking X minus D equals zero. So we said D is negative pi divided by four. So we have X minus negative pi divided by four, well minus a negative. So that's gonna be plus. So X plus pi divided by four equals zero, we subtract pi divided by four from both sides and we get X equals a negative pi divided by four. That's our first vertical Asymptote. The next one will occur one period later. So we'll take negative pi divided by four and add one period which is pi divided by four negative pi divided by four plus pi divided by four equals zero. So we have another Asymptote at X equals zero and then to get a third one because we are asked to find two periods worth of information here. We're going to take our zero and add another period, another pi divided by four which will equal pi divided by four. So we have a third vertical Asymptote at pi divided by four. And we can draw those in first one, X equals negative pi divided by four. Next one is it X equals zero. So thats a long, the Y axis and the last one is it X equals positive pi divided by four. There, we can see our two periods set up. All right. The next thing we're going to do is we are going to figure out our X and Y values, plot those points and connect the dots. So for our X and Y values, we need to first divide our period into four equal parts. So if our period is pi divided by four, we're going to divide that by four which will get us pi divided by 16. So we want this in increments of pi divided by 16. Thankfully, the graph that we're given is already drawn with that. So that's fantastic. So we can just look at the graph as well. So for our first X value, we're going to the right of our vertical Asymptote X equals negative pi divided by four. We're adding pi divided by 16 to that, that's a negative three pi divided by 16 and pi divided by 16. To that, we get a negative pi divided by eight and add another pi divided by 16. To that, we get a negative pi divided by 16 and we do need to figure out two periods worth of information. So going to the right of our vertical Asymptote X equals zero. We'll add pi divided by 16. That's pi divided by 16, add another pi divided by 16. That's pi divided by eight, add another pie divided by 16, thats three P divided by 16. OK. So now just figure out those Y values we've got our X values. What we're going to do is plug our X values into our function Y equals four minus two, multiplied by the cotangent of four, multiplied by the X value plus pi divided by four. So when we're doing that, you want to make sure a couple of things on your calculator, you want to first make sure that you are in radiance mode. And next, because we have cotangent here, we recall from previous lessons that its reciprocal function is tangent. Most calculators don't have a cotangent button. So you can put this into your calculator with the tangent. You'll put in four minus two, multiplied by. And then here you want to have one divided by the tangent of and then the rest of our function, the tangent of four multiplied by put in whatever that X value is plus pi divided by four. So for the first X value that's going to be a negative three pi divided by 16, you plug that in for the X value and you're going to get a positive two. Now, if you plug in negative pi divided by eight for your X value, if your calculator is like mine, you're going to get a domain error. And that's because this tangent value here is going to end up being a zero. We do have, it's going to evaluate for this whole cotangent. It's going to be a zero, it's going to be four minus zero. And our answer is going to be four. But there is a way to avoid getting that on getting that domain error on the calculator. We recall from previous lessons that the cotangent is not only the inverse function of the tangent, it's also found by taking the cosine of our X value divided by the sign of our X value. So what you can do to put this into your calculator is you'll put in four minus two, multiplied by and then in the numerator you'll have the cosine of four multiplied by our X value negative pi divided by eight plus pi divided by four. Close those parentheses divided by, in the denominator, the sign of four multiplied by the quantity of the X values negative pi divided by eight plus pi divided by four. Close those parentheses and you'll get a straight up four. All right. So for that next X value or for the next Y value you'll put in the X value of negative pi divided by 16 and you'll get a Y value of six. Now, the great thing here is that's one period for the second period, those Y values just repeat themselves. So for the X value of pi divided by 16, that's going to be a two. Again, for the YX value of pi divided by eight, the Y value is four. And for the X value of three pi divided by 16, the Y value is six. So now let's plot those points. First one, negative three pi divided by 16 two. Next point negative pi divided by 84. Next point negative pi divided by 16 6. Next point pi divided by 16 2. Next point pi divided by 84 and the last 0.3 pi divided by 16 6. So now we just need to connect these smoothly. We can see that this is curving upward. So we're looking at the first period from X equals negative pi divided by four to X equals zero. It's going to start off with our function heading toward negative infinity along the vertical Asymptote, X equals negative pi divided by four. And then it's going to come up along that vertical Asymptote, then it's going to start to curve and then it'll go through our points first one at negative three pi divided by 16 2. The next one at negative pi divided by 84, the next one at negative pi divided by 16 6. And then it'll come up along that vertical Asymptote X equals zero and it'll head toward positive infinity for the Y values. All right, in our next period, it's going to repeat that same shape just to the right of our vertical Asymptote X equals zero. It's heading down toward negative infinity. But then it's gonna come up along that Asymptote and then it'll start to curve and it'll go through our three points. It'll first go through the point pi divided by 16 2, then the point pi divided by 84, then the 0.3 pi divided by 16 6 and then it'll curve and it'll head up toward positive infinity for the Y values along our vertical Asymptote X equals pi divided by four. And that's it. That is our graph of Y equals four minus two, multiplied by the cotangent of four, multiplied by the quantity of X plus pi divided by four. Well done. We'll catch you on the next one.

Graph each function over a two-period interval.
y = 1 - 2 cot [2(x + π/2)]

Key Concepts

Cotangent Function and Its Properties

Function Transformations (Shifts and Scaling)

Period of a Trigonometric Function

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