In Exercises 25–28, use each graph to obtain the graph of the cor...

Question

In Exercises 25–28, 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

Hey, everyone here we are asked to use the key features of the given graph to draw the reciprocal function that corresponds to it and identify the equation of the resulting function. So here we are given a function and its graph where the function is Y is equal to 1/4 multiplied by cosine of four pi X. So our first step here is to identify the equation of our reciprocal function. So we just need to look at our given equation which we can see is a cosine function. So we just need to take note here that we are talking about the reciprocal of cosine which is one divided by cosine. And with this, we just need to recall the trigonometric identity which states that the reciprocal function or the reciprocal of cosine. In other words, one divided by cosine is equivalent to sin. So with this information, we can go ahead and write our reciprocal function which is Y is equal to 1/4 multiplied by second of four pi X. So now that we have identified the equation of our reciprocal function, we want to now work towards graphing it. So the first step is to look at our given graph, the function of our given graph here. And take note of all of the points where our given function is zero. So all of these points are where the curve on the given graph intersects the X axis. And we see we have eight different points here where the first point is that X is equal to seven negative seven eights and then we increase I 1/4. So each point lies 1/4 to the right. So our final rightmost point is that X is equal to seven eights. And the reason why we are taking note of all of these points is because again, that is where our cosine function is zero. So that means that at these points, the reciprocal function is at infinity. So that means that all of these points serve as vertical asymptotes for our second function. So we just want to now draw in all of these vertical asymptotes for our second functions graph. And so we start with our leftmost vertical Asymptote which again is that X is equal to negative 7/8 we then move on to X is equal to negative five eights. Then we have X is equal to negative 3/ then we have X is equal to negative 18. Now moving on to the positive X axis, we have the same asymptotes but they are now all positive. So we have positive 1/8 then we have X is equal to positive 3/8 then positive 5/8 and lastly positive 7/8. Now with all of our vertical asymptotes drawn in, we can move on to our next step where we need to take note of all of the lowest points and highest points on our given curve. And all of these points will now serve as vertices for our second function. So we just want to begin by writing or drawing in all of these points on our second function graph. So we can begin with the first point on the left side which is at negative 0.75 and negative 0.25. Then we have the next point at negative 0. and positive 0.25. Then we have the point negative 0.25 and negative 0.25. Then our next point intersects the Y axis. So we have the 0.0 and positive 0.25. Then we have the next point at 0.25 and negative 0.25 then 0.5 and 0.25. And lastly, we have the 0.0 0.75 and negative 0.25. So now all of our points drawn in again, these points serve as vertices. So our last step here is to draw in our smooth curves from the vertices to infinity. So starting with our leftmost point, we will have a downward facing curve here which follows a somewhat parabolic shape. So we will have, we will go from our point to negative infinity along the y axis. So our curves approach the vertical asymptotes but never cross them. And then for our next point, we will face upwards this time. So we will go from our point to positive infinity along the y axis. And then we repeat this process where the next point we are downward facing again and then the fourth point we are now upward facing. So we just follow this same pattern for all of our vertices. And we can take note here that our graph, all of these curves will continue along this pattern to infinity along the X axis. However, we are just looking at a little snapshot of our graph. So with this, we have our curve Strawn our vertices identified as well as the vertical s coats identified. So we have finished graphing our function which we identified as Y is equal to 1/4 multiplied by second of by X. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

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In Exercises 25–28, 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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