Graph two periods of the given cosecant or secant function.

Question

y = 3 sec x

Accepted Answer

Hello there. Today we're going to 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 Cosicant of X. Awesome. So it appears for this particular problem, like the problem itself states, we're trying to sketch a graph of this particular cosicant function using two periods. Awesome. So with that said, first off, let us sketch a graph of its reciprocal function, which its reciprocal function, as we should recall, will be Y equals 6 multiplied by cosine of X. And as we should be able to recall and note that from sine and cosine graphs, we should be able to recall that Y equals A multiplied by some trig identity, which is either gonna be T where T denotes either sign or cosine function. Multiplied by BX. And then we need to take BX and we need to add C. And then we need to add D to that expression. Where capital A in this case is the amplitude. And the period P, which we should recall that the period P is going to be equal to 2 pi divided by B, and C is the phase shift, and D is the vertical shift. So, when we consider our reciprocal function, which is Y equals 6 multiplied by the cosine of X, that will mean, and I'll draw an arrow, so that will mean that the amplitude A will be equal to 6. And we will be able to note that the period for this particular problem, so the period P, will be equal to 2 pi divided by 1, which is equal to 2 pi, and the phase shift, so the phase shift will be equal to 0, and D, the vertical shift will also be equal to 0. And we need to use the following domain in order to cover two periods. So let's write in red our domain. So 0 is less than or equal to X and X is less than or equal to 4, and this is to cover, once again, 2 periods. So we now need to evaluate Y at X values that are within the given interval of 0 is less than or equal to X and X is less than or equal to 4 pi. So when we do that, let's create a table. So I've gone ahead and created a table and ultimately provided a final graph of what our final answer should look like. Or our Y equals 6 multiplied by cosicant of X, but let's focus on our table first really quick. So looking at our table, we have different values for X in our top row, and then the bottom row is different values for Y. So when you plug in these values of X into your equation for Y, you will receive the following variables. So when X is 0, you'll get 6 for Y, etc. So when you plug in different variables for X, you will get different values for Y. And once again, all you're doing is you're just plugging in X into your original equation. So when we do that, we will be able to plot our points to make a sinusoidal wave that passes through these points, and let's label our graph as Y equals 6 multiplied by cosine of X, and we need to note that for all X intercepts, we need to draw vertical asymptotes through them. And then after that, we will need to draw smooth curves that pass through the vertices at the crests and troughs, where a crest is a peak and a trough is a valley. And be sure to indicate that they are only approaching the asymptotes, but they are never touching them because they extend vertically to infinity. So our curves will always be like very, very close to our vertical asymptotes, but they will never be touching them, and they'll never cross these vertical asymptotes. So as you can see in our graph, our asymptotes are denoted as blue dotted lines and green and purple dotted lines. So we have 4 different asymptotes. And we need to note that since we are only interested in the graph of the Cosicant function, we can remove the graph of Y equals 6 multiplied by the cosine of X. So I, to be completely honest, I did not graph the Y equals 6 multiplied by cosine of X on our graph because we were ultimately going to be removing it anyways, but if it helps. You, you can graph that first, especially if you're using graphing software and you could just remove it. You could like select the option on your graphing software to not show that particular graph. But if it helps you visualize what you're doing, and then I by all means do it if it helps you. But ultimately, we're ultimately trying to focus on graphing this cosicant function, so that's what I have presented to you here. And as you can see, we have our red curves, and as you can see when we plot our different values for X and Y, noting that we're using the following, you know, coordinate plane method for plotting our points, which is XY. if X is positive, you move to the right following the X axis, and if X is negative, you go to the left following the X axis. And if Y is positive, you go up following the Y axis, and if Y is negative, you have to go down following the Y axis. So as you can see, we have, looking at our points, we have one X equals 0 and Y equals 6, so 0.6, we have a we have our plot point. And we also have when X is 26, and then when 46636. So those are our Our different points, we have 1234, and 5. That's our 5 different coordinates coordinate points that we plotted. And as you can see, like, once again, it's really important to note that these curves, which are represented in red, will be like they're going to be extending and reaching very close to these vertical asymptotes which are denoted by these dotted lines. They're going to get very close to them, but they're never going to intersect or touch them. And that's it. We've officially solved for this problem. 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 = 3 sec x

Key Concepts

Secant Function and Its Properties

Amplitude and Vertical Stretch

Periodicity of Secant Function

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