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 sec πx

Accepted Answer

Welcome back. Everyone. In this problem, we want to sketch the graph of the function Y equals negative one third multiplied by the second of two pi multiplied by X using two periods. Now, what do we already know? Well, recall that the sea can, is the reciprocal of the cosine function. And what that means is that if I were to draw a graph with the cosine function, then the second would look like it's reciprocal look something like this. Now you might be saying that doesn't look exactly like the multiplicative inverse or, or the reciprocal of the function. But that's because the C can function only works when the cosine is not equal to zero. And that makes sense because if the cosine were to equal to zero, then the C would be one divided by zero, which would be undefined. So at those points that are undefined or a second forms an Asymptote, it tends towards infinity but never touches the line. So if we want to draw the graph of the C can we can go ahead draw the cosine graph and then find those undefined points plot our sequent and remove our cosine graph So the cosine graph can be like a crutch to help us to figure out the graph of the C A. So let's go ahead and do that. So let's set Y to negative one thirds of the cosine of two pi X. Now recall that for a trigonometric graph. So if we let Y it's usually in the form a multiplied by the cosine of BX minus C plus D where A is the amplitude or the coefficient of the trick function. In this case, it would be negative one third. B is equal to the coefficient of X, which in this case would be two pi C would be the phase shift which is zero and D would be the vertical shift, which is also zero. Now using this information, this tells us that for our graph, the amplitude which equals A would have to be negative one third. Likewise the period which equals two pi divided by B would now be two pi divided by two pi which equals one. So from our equation, it tells us that our graph has an amplitude of negative one third. In other words, we're going to make it a third of its size and invert it and it has a period of one. In other words, its length will be one unit. So I'm going to go ahead and plot the, this graph on my grid that's given here. Now, if you look on the graph, you'll realize that that is a plot of Y equals negative one third multiplied by the cosine of two pi X. If you're not sure how we got this graph, then you'll probably need to revisit one of the previous lessons on sketching uh the graphs of sine and cosine. But now that we have our graph, we can identify the peaks and the valleys on our curve because those are the points that the graph of the country will touch. So that means it will come at a half when X is a half, when X is one, when X is three halves and when X is two. Now, where will our graph be undefined? Well, it will be at the points where the cosine value equals zero, the cosine of X equals zero. So I can also identify those points with my vertical line because here, notice that it passes through this value on our X axis. OK? I think I can draw that pretty accurately here. Another value that passes through right here. OK. And I feel like I want to make these lines pretty straight. So yes, that would be here. And if I continue, I can identify all the points at which it passes through the line at the or whenever the cosine of our value equals zero. So now that we have this information, we can go ahead and plot the second of our graph. So that means it will in our first uh section between those values where the X, where the peak is that X equals a half, the next one where X equals one, it would also be where X equals three halves the other peak. And then finally, where X equals two, and this one would be a, basically a half of our complete curve. Likewise, we would also have one for between zero and, or point on the graph. So no, all of those points would be the, the, the curve for our second. So I can go ahead remove my sand graph on my call sand graph and put those points in. Because notice that at those points when X is zero, let me write it right here. As a matter of fact, let me rub this one out. OK? When X is zero, oops and let me put it in red. So finally, when X is zero, Y is negative one third, when X is a half, why is one third? When X is one, X equals one Y equals negative one third? And when X is three halves, Y equals one third. And finally, when X is equal to two Y equals negative one third. So those would be all the points on our sketch. And that would be the sketch of the graph Y equals negative one third multiplied by the C count of two pi X. Thanks a lot for watching everyone. I hope this video helped.

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

Key Concepts

Secant Function and Its Properties

Amplitude and Vertical Stretch/Compression

Period of the Secant Function with Horizontal Scaling

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