Use each graph to obtain the graph of the corresponding recipr...

Question

Use each graph to obtain the graph of the corresponding reciprocal function, cosecant or secant. Give the equation of the function for the graph that you obtain.

Accepted Answer

Hello, fellow physicists. 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. Using the key features of the given graph, draw the reciprocal function that corresponds to it and identify the equation of the resulting function. Awesome. So it appears for this particular problem, we're asked to use our graph that is provided to us to help us create a graph of our reciprocal function, and we're also asked to identify the equation of the resulting function. Awesome. So with that said, let's quickly look at our graph. So it turns out that our current graph that we're dealing with is for the function Y equals -4 multiplied by the cosine of 3 pi X. Awesome. Now as you can see, we have a nice cosine curve. Fantastic. And then it looks like it's at it intersects the X axis at -4.5.0. Then we have another point at -3.4. Then we have a point on the that intersects on the Y axis, which is 0.4. Then we have a point at 3.4. And a point at 4.5.0, and noting once again in case you've forgotten that this is in the XY format for our coordinate points. Awesome. So with that said, our first step is that for all the X intercepts, we need to draw vertical asymptotes through them. So meaning, so we have our first so all the X intercepts, so we have an intercept at -4.5.0, so we need to draw a dotted line. To denote our vertical asymptope. And then our second, where it hits the our ver so where it hits our X-axis again is we have another point, as you can see our curve, which is, I guess like a little bit bigger than 1. As you could see, So it's like at -1. something or other. And then our next. X intercept is once again on the opposite side of our cosine instead of it being like negative 1.something, it's gonna be at 1.something. And then our final X intercept is going to be at. Our point, which is at 44.5.0. Awesome. So I've drawn our dotted lines in blue to denote our vertical asymptotes. So now our next step is we need to draw our smooth curves that pass through the vertices at the crests and troughs where crests are like the mountain peaks and troughs are like the valleys. And we need to be sure to indicate that they are only, we need to make sure that we're indicating in our graph that these curves are approaching the asymptotes, but they will never intersect. They will never touch these vertical ascents, and they will extend these vertical ascents will extend forever vertically to infinity. So, with that said, when we create a graph, so I've gone ahead and plugged it in or plugged in our reciprocal function into our graphing calculator. And as you can see, we have our vertical acetoes which we drew to the left and with blue dotted lines, and we can see now that they're represented by a purple, green, blue, and orange dotted vertical lines for our asymptotes. And then we have our different curves, and as you can see, the curves, which are represented in red, will approach these vertical asymptotes, but they will never touch them. That's why it's very important when you're drawing your graph to indicate that. And then as you can see, we have our original points where we have -3.4 and then 3.4. And then we also have 0.4, and we could see how the curves have changed with our reciprocal function. And therefore, without a doubt, we can go ahead and safely write that our reciprocal function, which is the the required equation for this new graph, is going to be Y equals -4 multiplied by the. Cosicant of I divided by 3 multiplied by X. And that is our new graph. Hooray, we did it. We've solved for all parts of this problem. Hooray, we did it. So this problem is very easy to solve for as long as you have a strong understanding of the graphing tactics and graphing skills you need to graph trigonometric identities or trig functions. So it's really important, your first step is you need to make sure that you always draw your vertical asymptotes to correspond with your where the X intercepts are. That's your first step. And then once you've drawn these vertical asymptotes, you can now draw your smooth curves that will pass through the vertices at the crests and troughs, and then it's important when you draw your graph to make sure that your curves do not touch your vertical asymptotes. Awesome, 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.

Use each graph to obtain the graph of the corresponding reciprocal function, cosecant or secant. Give the equation of the function for the graph that you obtain.

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Key Concepts

Reciprocal Functions

Graphing Trigonometric Functions

Asymptotes in Trigonometric Functions

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