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

Below 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. 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 trying to take our given graph, which apparently our given graph is a graph for the function Y equals 4 multiplied by sine of 6 X, and we're trying to take this graph and graph its reciprocal function. So in order to do that, first off, we need to note that for all X intercepts, we need to draw vertical asymptotes through them. So I'm going to use a blue dotted line to draw our vertical asymptotes through all of our X intercepts. So we have an intercept at negative pi divided by 6, as you can see on our graph. Then we have another asymptote, a vertical asyote going through the vertical axis, the Y axis going through 00. Then we have another. Vertical acetote going through positive pi divided by 6. And finally, we have another vertical ascentote which is going through. Pi divided by 3, positive pi divided by 3, to be precise, fantastic. So that's our first step. So first step is you always want to look at all of your X intercepts, and then you want to draw in your vertical asymptotes going through the so going through these X intercepts, and we have to draw our vertical asymptotes. So our next step now is we need to draw smooth curves that pass through the vertices at the crests and troughs where the crest is like the peak. Of the mountain in the trough is like the valley of our grass. And we need to be sure to indicate on our graphs when we go to graph our reciprocal function that the that our curves are approaching these vertical asymptotes, but they are never going to intersect or touch them. So these vertical asymptotes are going to extend vertically to infinity, and the curves are going to get closer and closer and closer and closer. To these vertical asymptotes, but they're never going to intersect them, and they're never going to touch them. That is very important to note. So with that in mind, I've gone ahead and graphed the reciprocal function in my graphing software and made a copy of the graph. And as you can see, we have our different vertical asymptotes, which it looks like they opted not to draw one on the Y axis. They just used the bolded black line for the Y axis to denote an asymptote, but that is a vertical asymptote. It's very important to note. And as you can see, we have a point for one of our little curves, which is at negative pi divided by 12.4, noting that that it's in the format of X. Y. We are plotting our points. Then we have another curve at pi divided by 12.4, and then we have our third and final curve, which is at pi divided by 4.4. And as you can see, like we said before, we have our curves and they are going stretching out towards our vertical asymptotes, but they're not intersecting them and they're not touching them. So it's important to draw it as close as you can to the vertical asymptote without touching it because it's like, like we said, These curves will stretch following out like following upwards or downwards, following these vertical ascentopes forever and ever as they stretch out for vertically for forever, and these curves will be getting closer and closer and closer to these vertical ascentoes, but they will never intersect and never touch them. Awesome. So without a doubt, we could safely say that our reciprocal function is going to be equal to, so Y is equal to 4 multiplied by cosicant. Of 6 X. So that is our reciprocal function is Y equals 4 multiplied by cosicant of 6 X. And that's it. That is our reciprocal function. And with that, we've officially solved for this problem. Hooray! So this problem is very easy to solve for as long as you have a strong understanding of the background theory behind plotting trig functions. If plotting trig functions like this is confusing to you, I would highly suggest going through and watching more videos on this topic. I would also suggest reading up some more in your textbook. But as long as you know, like the most important points is that you always want to draw your vertical asymptotes going through your X intercepts, that's step one. Then once you have your vertical asymptotes drawn, that's when you can draw your curves. And when you draw your smooth curves that pass through the vertices at your crests and troughs. And it's also when you go to graph your graph, you need to make sure that you don't accidentally draw your curve as it's intersecting the vertical axis or touching the vertical, sorry, I said vertical axis, the vertical asymptote, because if you do that, you could lose points on an exam if you're trying to plot it by hand or on your homework assignment. So whenever you draw your curves, make sure that they never touch or intersect your vertical asymptotes. That is key. And with that, that is all the key tips you need to know for this particular 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.

<IMAGE>

Key Concepts

Reciprocal Functions

Graphing Trigonometric Functions

Vertical Asymptotes

Watch next