Graph two periods of the given cosecant or secant function.

Question

y = 2 csc x

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Sketch the graph of the following cosicant function using two periods Y equals 6 multiplied by the Cosicant of X. Awesome. So first off, we need to note that we're ultimately trying to create a sketch of the graph for this following cosicant function using two periods. So that's what we're ultimately trying to do. So first off, in order to, you know, solve for this particular problem, let us once again emphasize the fact that we're trying to graph Y equals 6 multiplied by Cocicant. Of X, so cos he can of X. So first, let us sketch the graph of its reciprocal function Y equals 6 multiplied by sine of X. And as we should recall from sine and cosine graphs, let's make a note of this in blue, that we can state that Y equals A multiplied by Trigg Multiplied by. B X Plus C plus D, where A is the amplitude. The period is, and let's make a note of this at the period P. And I'll write this in black. So the period P is equal to 2 multiplied by pi divided by B. And C in this case is the phase shift, and D, the capital letter D, denotes the vertical shift, and trig, T R I G trig can either be a sign or cosine function. So let me graph, and I'll write this in. So when we graph. A sketch of Y equals 6 multiplied by sin. Of X, we will note that A, in this case the amplitude, is equal to 6. We will also be able to note that the period P will be equal to 2 multiplied by pi divided by 1, which is equal to 2 multiplied by pi. And we know that the phase shift in this case. So the phase shift is equal to so we said C denotes our phase shift, and our phase shift C is equal to 0, and our vertical shift. Which I'll call VS, our vertical shift is also equal to 0. So we now need to recall and note that we're gonna be using the domain of 0 is less than or equal to X and X is less than or equal to 4 multiplied by pi or 4 pi in order to cover two periods. So when we evaluate Y at X values that are within the given interval, which is 0 is less than or equal to X and X is less than or equal to 4, we will be able to put together the following table. So I've gone ahead and created a table and a sketch of our final Cosicant graph. So focusing on our table here really quick here, we could see that when X is 0, Y is 0, and when X is 2 when X is pi divided by 2 or Y is 6, and when i when X is pi Y is 0, and when X is 3 pi divided by 2, Y is -6, and when X is 2 pi, Y is 0. And when X is 5 pi divided by 2, Y is 6, and when X is 3 pi, Y is 0, and when X is 7 pi divided by 2, Y is -6, and when X is 4 pi, Y is 0. So when we plot these points and make a sinusoidal wave that passes through these points, we'll label our graph as Y equals 6 multiplied by sin of X, and we'll note that for, as we should recall that for all X intercepts, we will draw the vertical asymptotes through them, and we have 4 different vertical asymptotes. We have our asymptote that's and they're all dotted lines. We have a purple asymptote. We have a blue asymptote, we have an orange aspentote, and then we have a green asymptote. And as you can see, we have 123, so our first asymptote is at 3, and our second one is at 7. Or more like 6, and then we have one at 10, and then we have one about 1314, it appears. On the that it crosses through the X axis or the horizontal axis. So we need to draw smooth curves that pass through the vertices at the crests and tops where the crest is going, you know, is the is the bump up above, and then the tress or or the trout trouts are the ones that are go dig under, like the ones that are like a valley instead of a mountain. Crests are mountains, troughs are valleys. And we need to be sure to indicate that they are only approaching the asymptotes, but they are never touching the asymptotes. This is very important to note. So they are they these curves will approach the asymptotes, but they will never touch them while they extend vertically towards infinity. And since we are only interested in the graph of the Cosicant function, we can remove the graph of Y equals 6 multiplied by sin of X. So that's what my final graph that I plugged into my graphing software. Which I apologize. I totally didn't realize that I had. Our table was sticking out towards the end here. OK, it was off the same. So as you can see in my graph that I used in my graphing calculator, so this is our final graph for the cosicant function that doesn't have the graph of Y equals 6 multiplied by sin of x. So that has been removed and all we have is our cosicant function, which is the final answer that we're ultimately trying to solve for. So this problem is fairly easy to solve for as long as you have a strong understanding of the graphing principles behind a cosicant function and a sign function, and you need to be able to figure out how to like like sketch the graph following the Koican function using two periods, so you need to know how to use the two periods to graph your final sketch. And that's it. So basically, just as long as you have an understanding of like the graphing graphing methods behind a problem like this, like the background theory, it's gonna be really quick and easy to solve. And that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Graph two periods of the given cosecant or secant function.

y = 2 csc x

Key Concepts

Cosecant Function

Graphing Periodic Functions

Amplitude and Vertical Stretch

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