In Exercises 29–44, graph two periods of the given cosecant or...

Question

In Exercises 29–44, graph two periods of the given cosecant or secant function. y = 1/2 csc x/2

Accepted Answer

Welcome back. I am so glad you're here. We're asked to sketch the graph of the following coy can function using two periods. Our Cosic can function is Y equals one third coi can of X divided by three. So how do we graph CO C can? Well, what we want to do is actually graph the reciprocal function of Cosic can. So that's gonna be sine, we wanna graph Y equals one third sign of X divided by three. So we're going to spend a lot of time working on that first. So how do we graph a sign function? Well, we see that this is in the format of Y equals a sign of B X where A is being multiplied by the sign. So A that's in front of the sign, that's one third and B is being multiplied by the X. So that's actually also one third X divided by three is one third multiplied by X. So now we use that information to help us graph, we recall from previous lessons and from the steps in the text book that we want to start off identifying the amplitude, the period, the phase shift and figuring out where this begins. And because we're not given a graph to start off, we also want to figure out the domain and range so that we can start to graph. So one, let's start off with the amplitude. The amplitude is the absolute value of A and the absolute value of A is the absolute value of one third, which is one third. So our amplitude is one third next, the period, the period is two pi divided by B well B is also one third. So it's two pi divided by one third, which is the same as two pi multiplied by three. So this period is six pi. Now for the phase shift, there is no C term here, there is no phase shift. And because this is a sign function, it's going to begin at the origin at 00. And so we want to figure out our domain and our range. So we know how to draw our graph for the domain. We're beginning at 00 and then we're going over two periods. So one period is six pie, two periods is 12 pie. So from 0 to 12 pie, now the range, the range for our sign function is going to be from the amplitude above the X axis to the amplitude below the X axis or so the other way around. So we start with the negative first. So it's going to be from negative one third to positive one third. However, we're not just graphing the sign function, we also need to graph the co C camp function. After that, it's going to have the same domain. But the range for our co C camp function is going to be from negative infinity to positive infinity. So we need to keep both of those in mind as we draw our graph. So let's start with that drawing our graph, we're going to have a vertical y axis. And because our domain is just from 0 to 12 pi the Y axis can be pretty far to the left on our graph. And then we have a horizontal X axis right in the middle of that. And so for our why values we can label we can have a unit at positive one third above the origin, we can have another unit at positive two thirds above the origin and another unit at positive one. And then below the X axis along the y axis, we can have a negative one third unit, a negative two thirds and then a negative one. So those can be the labels for our Y axis. Now, for the X axis, we've got that domain from 0 to 12 pi. But when we start to mark all of the key values along the X axis, those are going to be marked by taking the period and dividing it by four a quarter period. So the period is six pi dividing that by four. So six pi divided by four, simplified is three pi divided by two. So we can put this in units of three pi divided by two. So going along the X axis to the right of the origin, we'll start off with positive three pi divided by two, adding three pi divided by two to that is a positive three pi add another three pi divided by two is nine pi divided by two. Add another three pi divided by two is six pie. Add another three pi divided by two is 15 pi divided by two, add another three pi divided by two. And that's nine pi add another three pi divided by two. And that's 21 pi divided by two and one more pi divided by two added is 12 pi. So there's our two periods and we've got our graph made. Now we need to plot all of our points for our starting with our sign function. So step two in the graphing process, we want to identify the five key X values. And so X sub one is going to be where this begins. And we said this sine function with no phase shift begins at 00. So X sub one equals zero X sub two. This is where we said we're going to be adding quarter periods to figure out each X value. So our X sub two, that's going to be zero plus three pi divided by two, which is three pi divided by two And so we've already figured out all of these X values because we're just adding three pi divided by two. So we'll add 33 pi divided by two, gets us to three pi X sub four is nine pi divided by two X sub five is six pi. So that's one period, but we're going two periods. So X sub six is 15 pi divided by two X sub seven is nine pi X sub eight is 21 pi divided by two and X sub nine is 12 pi. So there's step two. Now, first step three is figuring out all of the corresponding Y values. There's a couple ways to go about doing this. We can plug our X values into our sine function. Y equals one third sign of X divided by three. And we'll get the corresponding Y values as our solutions to that. So Y sub one, if you plug zero in for X, you're going to get a Y sub one value of zero. But we also know these just from knowing what's going on with our sign function. So we know that Y sub one, Y sub three and Y sub five and their corresponding X values, those are all X intercepts. So the Y values will be zero. Now for Y sub two and Y sub 41 will be a maximum and one will be a minimum. Our a value is positive. So the first one, our Y of two is going to be the maximum. So that's going to be the amplitude above the X axis. The amplitude is one third. So our Y sub two value is one third which you can also get by plugging in the X sub 23 pi divided by two in for X in the function Y sub four is going to be the amplitude below our, below our X axis. So that's gonna be negative one third. And now for these last four Y values, the cycle just continues. So for Y sub six positive one third, Y sub seven back to zero, Y sub eight will be a negative one third and Y sub nine will be back to zero. All right. Now we can plot these points starting with 00. That's at the origin. Then we have three pi divided by 21 3rd. So we go to the right from the origin three pi divided by two and we go up one third for the 0.3 pi zero from the origin. We go to the right three pi for the 0.9 pi divided by two, negative one third, we go from the origin to the right nine pi divided by two and down one third of a unit. For the next 0.6 pi zero from the origin. We go to the right six pi for the next 0.15 pi divided by 21 3rd from the origin. We go to the right 15 pi divided by two and up one third for the next point from nine pi or 49 pi zero from the origin. We go to the right nine pi for the next 0.21 pi divided by 21 or negative one third from the origin, we go to the right 21 pi divided by two and down one third. Oops I put that in the wrong place right here. And then for the final 0.12 pi zero from the origin, we go to the right 12 pi. Now we try to connect these points as smoothly as possible. And maybe yours will be more. Oh my goodness, maybe yours will be more smooth and connected than I can do mine. Not so smooth or connected. You might need to use your imagination. OK. So there's our sign function, but we're not graphing the sign function. We're graphing the coy can function. So how do we do that? Well, everywhere that we have an X intercept for our sign function that is going to be a vertical ascent tote for our co C camp function everywhere have a maximum for our sign function. That is going to be a minimum for our co C camp function. And everywhere we have a minimum for our sine function, that's going to be a maximum for our co C can function. So let's start off drawing our vertical asymptotes at each X intercept. So the first one is at 00. So we'll draw a vertical Asymptote along the Y axis where X equals zero. The next vertical Asymptote is the next X intercept at three pi zero. So we draw a vertical Asymptote through three pi and label that X equals three pi the next X intercept is at six pi zero. So we draw vertical asom tote and label that X equals six pi the next X intercept is at nine P. We draw vertical asom tote and we label that X equals nine pi the last X intercept is at 12 pi zero. We draw a vertical asom tote and label that X equals 12 pi. Now for the maximums becoming minimums and vice versa to the right of our vertical Asymptote X equals zero, we are going to use that maximum at three pi divided by 21 3rd and that becomes a minimum of our CO C can function. So our line is going to be heading up to positive infinity for the Y values along our vertical Asymptote X equals zero. We draw it coming down along that Asymptote and it's going to touch the sine functions maximum at three pi divided by 3rd that becomes our co C camp functions minimum and it curves heads back up toward positive infinity for the Y values along the vertical asym tote X equals three pi. Now to the right of the vertical Asymptote X equals three pi we're going to be in the negatives for the Y values. So along that vertical Asymptote heading toward negative infinity for the Y values, we draw it coming up and it's going to encounter our sine functions minimum at nine pi divided by two negative one third that becomes this coy can functions maximum turns around heads down toward negative infinity for the Y values along the vertical ascent tode X equals six pi. Now to the right of the vertical Asymptote, X equals six pi we're heading up to positive infinity for the Y values along that Asom tote. And then we draw it coming down. It's going to encounter the maximum of the sine function at pi divided by 21 3rd that becomes the Cos Can's new minimum. Turns around heads back to positive infinity for the Y values along the vertical Asymptote of X equals nine pi last line to the right of the vertical Asymptote X equals nine pi we're in the negatives for the Y values heading toward negative infinity. But we draw it coming up along that vertical Asom tote. Then it's heading toward the sine functions minimum at 21 pi divided by two negative one third touches, that becomes a new maximum for our CO C can function turns around heads back toward negative infinity for the Y values along the vertical Asom tote X equals 12 pi and that's it. That is our CO C can function Y equals one third, co can of X divided by three. Well done. We'll catch you on the next one.

In Exercises 29–44, graph two periods of the given cosecant or secant function. y = 1/2 csc x/2

Key Concepts

Understanding the Cosecant Function

Effect of Transformations on Trigonometric Graphs

Period of the Cosecant Function

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