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 = 2 sec(x + π)

Accepted Answer

Welcome back, everyone. In this problem, we want to sketch the graph of the function Y equals four, multiplied by the second count of two X minus pi using two periods. Now, what do we already know will recall that the second is the reciprocal of the cosine function. And what that tells us is that if the cosine function that looks something like this, then our second is going to look like the reciprocal of that the inverse of that in a way which would look something like this. Now, I know you're probably saying that doesn't look like an inverse but it actually is because the C can function only works when the cosine of X is not equal to zero. And think about it. That makes sense because if the cosine of X is zero, then our would be one divided by zero, which would have been undefined. So at that point where our graph is undefined, we have our asym tote tending to a vertical line. Therefore, if we are going to draw the graph of our ac, we can first draw the graph of the cosine get the points for that undefined value of the C that is the points at which extending to infinity and then remove and plot the, remove the cosine graph and plot the graph of our C. So let's draw the cosine graph first for Y equals four multiplied by the sequence of two X minus pi. So that would be Y equals four. Let me put it in a different color. Let me put it in red. OK. Let's let Y be equal to four, multiplied by the cosine of two X minus pi. Now recall as well that for a trigonometric function that's written in the form a multiplied by the cosine of BX minus C plus D. OK? Where in this case, A equals four, the number multiplying the cosine B is the coefficient of X which is two C is the opposite value of the constant in the parenthesis for the cosine. So that would be positive by, and we don't have any value of DD would be zero. Then our amplitude would be equal to the value of A which is four or period equals two pi divided by B which in this case has two pi divided by two, which is pi or face shift equals C divided by B which would be pi divided by two. And of course, we don't have oops pi divided by two. We don't have any vertical shift of, as we said, D equals zero. So what these values tell us is that we're going to make our graph four times as big with a period of pi and a phase shift of a half of pie. In other words, we're going to shift our graph a half of pie units to the right. So let me go ahead and first draw that graph of four multiplied by the cosine of two X minus pi. Now here's my graph of four multiplied by the cosine of two X. I haven't done the phase shift yet. And if you're not sure how we got that graph, then I'd recommend watching a, that, that previous lesson on sketching cosine graphs. But no, I have drawn it with an amplitude of four and a period of pi. So I just need to do my phase shift of a half of pi. And what that means is that I'm just going to take my curve and I'm going to move it by a half of pi units to the right. When I do that, this is where my curve would end up give or take. Now, before I draw the graph of my cosine, let's identify the vertical lines for that assim to that is where those values of the cosine equals zero. Notice that the cosine equals zero at 3/4 of pi so a vertical line will run through that point. It also does. So for five fourths of pi 74. So by OK, so you're looking at where the X axis equals zero along this line here and also for nine fourths of pi. Now, before we plot, let's also identify those some pretty important points which would be the peaks and the valleys for our cosine curve, our cosine graph and know that we have those, we can go ahead and plot the graph of our C card. So the graph of our would look something like this. So now we're going to draw it, tending towards our vertical line but never touching it. And it's also going to pass through the points. OK? So it would look something like this. Please forgive me if you don't think this is the best one. I know I can do better. I'm trying. So that's what our would look like. It would be the line in the blue. I'm just going to finally remove my cosine graph. It was such a big help to us, but we don't need it anymore. And no, I can just identify those key points. So let me zoom in for you for you to see that a little better. OK? And at the point when X is a half of theta Y is 46 point half of theta four at pi why is negative for at three halves of pie? Why is four at two pi Y is negative four and at five halves of pi Y is four. And those would be our values for the sketch of the graph Y equals four multiplied by the second of two X minus pi 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 = 2 sec(x + π)

Key Concepts

Secant Function and Its Properties

Phase Shift in Trigonometric Functions

Amplitude and Vertical Stretch

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