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

Question

Graph each function over a two-period interval.

y= -1 + (1/2) cot (2x - 3π)

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 negative three plus 1/4 cotangent of three X minus two pi and we're given a blank graph. We have a vertical Y axis and a horizontal X axis. They come together at the origin. The range for what's drawn for our Y axis is from negative five to positive one. And the domain for what's drawn for our X axis is from zero up to 17 pi divided by 12. And that's marked off in increments of pi divided by 12. All right. So our function is given to us in the format of Y equals C plus A multiplied by cotangent of. And then we have that is multiplied by open bracket. B multiplied by the quantity of X minus D close parenthesis, close bracket. It's very close to that format that we're given. Except here, the B hasn't been factored out yet. So let's identify that we can see that our C is negative three plus our A is 1/4 multiplied by the cotangent of open bracket. We'll factor out what's in front of the X. So that B term is a three multiplied by the quantity of X minus. And then two pi divided by three is just two pi divided by three. So our B term is three and our D term is two pi divided by three. So when we're talking about cotangent, it's very important to identify the B and the D terms because we need to figure out the period and the phase shift. So for the period for cotangent, that's going to be pi divided by B, we said B is three. So it's pi divided by three. There's our period for the phase shift, that's important because we wanna next identify our vertical asymptotes. So typically with cotangent function, we'll have a vertical Asymptote at X equals zero. But here, because we have a phase shift, we have ad value that's anything other than zero. Our first vertical Asymptote is going to be at X minus D equals zero. We said that D is two pi divided by three. So X minus two pi divided by three equals zero, adding two pi divided by three to both sides. We have an Asymptote at X equals two pi divided by three. So let's draw that one. There's our first vertical Asymptote. Now, the next vertical Asymptote is one period later. So we'll take two pi divided by three and add one period which is pi divided by three, two pi divided by three plus pi divided by three is pi we have another vertical Asymptote at X equals pi. Now our third one, we do need a third vertical Asymptote because we're considering two periods and we only have one period periods worth of Asymptote so far. So again, let's take our pi and add a period of pi divided by three. That's going to get us four pi divided by three. There's our third Asymptote to give us two periods. OK. Now, the last thing that we need to do is we are going to divide those periods into four equal parts and then plot some points and then we'll have our graph, we connect to those points and we have our graph. So in order to divide a period into four equal parts, we have a period is pi divided by three, divide that by four, that's going to be pi divided by 12. And thankfully, the graph that we were given is already broken up into increments of pi divided by 12. So I'm going to make an XY table and for our first X value, we'll go to the right of that vertical Asymptote of X equals two pi divided by three. We're adding pi divided by 12 to that. And that's going to give us three pi divided by four, add pi divided by 12. To that, we get five pi divided by six, add pi divided by 12 to that, we get 11 pi divided by 12. Add pi divided by 12 to that. And we get our ver vertical Asymptote at pi. So I'm not going to write that one down. Add pie divided by 12 to pie and you get 13 pie divided by 12. Add another pie divided by 12. We get seven P divided by six and add our final pie divided by 12 and we get five pi divided by four. Now we're going to be plugging those into our cotangent function into negative three plus 1/4 multiplied by the cotangent of three X minus two pi to get our Y values. So a couple things to note with that, if you're using your calculator, you want to make sure that you are in radiance mode. And because we don't have a cotangent button, you're going to be using its reciprocal function tangent. So when you put that in the calculator, you'll keep the negative three plus 1/4 multiplied by and then you'll have one divided by the tangent of and then you put in the three X three multiplied by whatever X value is minus two pi. So when you plug in an X value of three pi divided by four into that, you're going to get a Y value of negative 11 divided by four, which is the same as negative 2.75. A little bit easier to see on our graph. When you plug five pi divided by six into that. If your calculator is like mine, you're actually going to get a domain error. And that's because this part down here for the tangent of three multiplied by five pi divided by six minus two pi that is going to get us a zero. Because for our cotangent, when we have this all evaluated, that's a zero. So it's negative three plus zero and it's just going to give us a negative three. But because we're using that tangent function, it thinks it's undefined, it thinks there's a domain error. So here's how you can get around that. Using your calculator. We recall from previous lessons that cotangent is the same as cosine divided by S. So you'll still have a negative three plus 1/4 multiplied by. And then in the numerator you'll take that cosine of three multiplied by your X value minus two pi that's divided by, in the denominator, you'll have the sign of three multiplied by. And this time it's five pi divided by six minus two pi and that'll give you a straight up answer of negative three without the domain error. OK? Four more points to go. You plug 11 pi divided by 12 in for your X value, you are going to get a Y value of negative 13 divided by four, which is the same as negative 3.25. And then that's one full period for the next period. Our Y values are just going to repeat. So for an X value of 13 pi divided by 12. That's negative 11 divided by four or negative 2.75. For an X value of seven pi divided by six, that's negative three. And for an X value of five pi divided by four, that's negative 13 divided by four or negative 3.25. Now let's graph these points. The first one three pi divided by four negative 2.75. Next 0.5 pi divided by six negative three. Next point negative th or excuse me, 11 pi divided by 12 negative 3.25. Next 0.13 pi divided by 12 negative 2.75. Next 0.7 pi divided by six negative three. And the last 0.5 pi divided by four, negative 3.25. Now we just need to connect the dots. So in our first period from X equals two pi divided by three to X equals pi, we can see that that's curving down for our Y values. So just to the right of our vertical Asymptote, X equals two pi divided by three. Our function is heading up toward positive infinity for the Y values. Then it's coming down and then it's going to go through each of our points. It's going to curve and go through the 0.3 pi divided by four negative 2.75 or 0.5 pi divided by six negative three. Our 0.11 pi divided by 12 negative 3.25. And then it's going to come up along the vertical Asymptote X equals pi and head down toward negative infinity for our Y values. And then during the next period, that same shape is going to repeat itself to the right of our vertical Asymptote X equals pi it's heading up toward positive infinity for the Y values. Then it's going to come down, then it's going to curve and it's going to go through our first point 13 pi divided by 12, negative 2.75. Next 0.7 pi divided by six, negative three. Next 0.5 pi divided by four, negative 3.25. Then it's going to come up along the vertical Asymptote X equals four pi divided by three and head toward negative infinity for the Y values. And there it is that is the graph of our function Y equals negative three plus 1/4 cotangent of three X minus two pi well done. We'll catch you on the next one.

Graph each function over a two-period interval.
y= -1 + (1/2) cot (2x - 3π)

Key Concepts

Period of Trigonometric Functions

Phase Shift in Trigonometric Functions

Vertical Transformations of Trigonometric Graphs

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