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

Question

In Exercises 29–44, graph two periods of the given cosecant or secant function.y = sec x/3

Accepted Answer

Welcome back. I am so glad you're here. We're asked to graph the following C can function using two periods. Our C can't function is Y equals the CC of X divided by two. And then we're given a graph, we have a vertical Y axis, a horizontal X axis, they come together at the origin. And then our domain for our given graph is from negative two pi divided by 2 to 8 pi and for the range it's from negative 5 to 5. All right. So how do we graph AC can't function? Well, we need to start off by graphing the reciprocal function. So we're going to spend quite a bit of time graphing the reciprocal function which is Y equals the cosine of X divided by two. And we can see that this is in the format of Y equals a multiplied by the cosine of B X. We can identify our A and our B A is being multiplied by the cosine. Well, there's nothing written there. So our A is just one and then B is being multiplied by the X, that's a one half being multiplied by our X. And now we can use this, we recall from previous lessons on graphing the cosine function. And from the steps in the book that the first thing we want to do is identify our amplitude our period. And if there's a phase shift, so for our amplitude, that's going to be the absolute value of A and A here is one. So the absolute value of one which is one, our amplitude is one, our phase shift, there is no C. So this begins for cosine, it's going to begin at 01. And then for the period, our period is going to be two pi divided by B and our B here is one half. So we have two pi divided by one half, which is the same as two pi multiplied by two. So it's four pi that's our period. Alright, now that we have those identified step two is we're going to identify the five key X values which are going to be our X intercepts and our maximum in minimums. And so we can start with X sub one, as we said, there's no phase shift. So this is going to start right at 01 for our cosine function. So our X value X sub one is at zero. Now to find all of the subsequent X values, we are going to add our period divided by four. So a quarter period, our period here is four pi and four pi divided by four is pi so we're going to add pi to each X value. So our X of two is zero plus pi. That's pi our X sub three is pi plus pi. That's two pi our X sub four is two pi plus pi. That's three pi and our X sub five is three pi plus pi that's four pi which makes sense because that is one period. So that's the first five key X values. But we're asked to graph two periods. So we can keep going one more cycle. So our X sub 64 pi plus pi is five pi our X sub +75 pi plus pi is six pi our X sub 86 pi plus pi is seven pi and our X sub +97 pi plus pie is eight pi there's are two periods, right? Step three is we're going to find the matching the corresponding Y values. So we know that this begins at 01. So our Y sub one is one and now our Y sub two, we can always find these Y values by going back to our Y equals the cosine of X divided by two plugging in our X values for X and we'll get the Y values. So you can always do it that way. But we can also discern a lot of this just by knowing what's going on with our graph. So our, for the cosine our Y sub two and our Y of four along with our X sub two. And our X sub four, those are the X intercepts. So our Y values at Y sub two and Y sub four are going to be zero. And then for our Y sub three and our Y sub five, we started off our Y sub one. That is our maximum, that's the amplitude above our X intercept. Our amplitude here is one. And so that's at a maximum of one above our X axis. And so for after our maximum, we're gonna have a minimum, our Y sub three is going to be a minimum. It's going to be the amplitude below our X axis. So that's going to be a negative one. And then for Y sub five that cycles back up to where we started, that's going to be at the amplitude above our X axis, which is going to be back up to a positive one. And now for the second period, this just continues the cycle. So our Y sub six is going back down to zero, our Y sub seven down to negative one. Then it turns around our Y sub eight back up to zero and our Y sub nine is back up to where we started at one. So now we can graph all of these nine points. But keep in mind we're not trying to graph the cosine, we're trying to graph the C can. So we'll get this graphed and then we have to switch gears to the C can function. So our first point is at 01 from the origin, we are going to go up one space. Our next point pi zero from the origin, we go to the right pi units. Our next point is at two pi negative one from the origin. We go to the right two pie. Oh I drew that wrong, didn't I? OK. Pi is two of these lines. OK. So graphing the point pi zero from the origin, we go to the right pie units which is here. And for the next 0.2 pi negative one from the origin, we go to the right two pi units and down one unit, the next 0.3 pi zero from the origin, we go to the right three pi next 0.4 pi one from the origin. We go to the right four pi up one, the next 0.5 pi zero from the origin. We go to the right five pi the next 0.6 pi negative one, we go from the origin to the right six pi and down one, the next 0.7 pi zero from the origin, we go to the right seven pi and the last point from the origin we with the point is eight pi one. We go to the right eight pie and up one. And now we can try to connect these as smoothly as possible and there is our cosine but we're not trying to graph the cosine. We're trying to graph the CCA so for every X, no, I don't need that X intercept. There we go X intercept of our cosine function that's going to turn into a vertical as to, for our C can't function for every maximum from our cosine function that's going to turn into a minimum four hours, you can't function. And for every minimum for our cosine function that's going to turn into a maximum for our cca function. So let's do that. Our first thing. Let's first start with the X intercepts turning into vertical asom totes. Our first X intercept is at zero or at, excuse me, it's at pi zero. So we can draw a vertical asom tote going through X equals pi our next X intercept is at X equals three pi so we can draw a vertical asom tote and label that X equals three pi our next X intercept is at five pi we can draw vertical asom tote and label that at X equals five pi and then our final X intercept is at seven pi so we can draw vertical as and tote and label that X equals seven pi. All right. Now turning our maximums into minimums and so on. So we have a maximum at 01 from our cosine function for our C can, that's going to be our minimum. So we start at 01 and then we're going to curve upward toward positive infinity for our Y values hugging up along our vertical ascent tote of X equals pi. And now we're moving to the right of our vertical Asymptote X equals pi. And now we're going to turn the minimum from our cosine function into a maximum for our cca function. So our line is going to be to the right of our vertical Asymptote X equals pi heading down toward negative infinity for the Y values hugging up along that Asymptote. Then it's going to touch our cosine functions minimum at two pi negative one. Turn around head back down toward negative infinity for the Y values along our vertical Asymptote X equals three pi. Now our next line is to the right of our vertical Asymptote, X equals three pi this one's positive for the Y values heading up toward positive infinity for the Y values along our vertical Asymptote coming down, touching our cosine functions maximum at four pi one and then that becomes our C camp functions minimum. And then it turns heads back up toward positive infinity for the Y values along our vertical asom tote of X equals five pi two more lines to draw. Now to the right of our vertical Asymptote, X equals five pi our line is heading toward negative infinity for the Y values along that vertical Asymptote, it comes up and has a maximum at our cosine functions minimum of six pi negative one touches, that turns around comes down toward negative infinity for the Y values along our vertical Asymptote of X equals seven pi last line to the right of our vertical Asymptote X equals seven pi up in the positive values for the Y axis. It'll be heading toward positive infinity to the right of that Asymptote. And we draw it coming down and it is going to touch our cosines maximum at eight pi one. And that is all we are drawing for now. So there is our C can function of X divided by two. 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 = sec x/3

Key Concepts

Secant Function

Period of a Trigonometric Function

Graphing Trigonometric Functions

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