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 the graph of the given function Y is equal to negative 1/4 multiplied by sine of X divided by four. So we can begin this problem by identifying the equation of the reciprocal function. So we just need to take a closer look at our given equation which we can see is a sine function. So we will be talking about here, the reciprocal of sine. So we have one divided by sine. And so we just need to recall the Trigo metric identity which states that one divided by S sign is equivalent to cos. So with this identity, we can write our equation of our reciprocal function which is Y is equal to negative 1/4 multiplied by co of X divided by four. So now that we have identified our equation for our reciprocal function, we can begin graphing this equation here. So to graph this equation, we first want to go to our given graph where we first want to note all of the points for which our given function is zero. So all of these points are where our curve of the given graph intersects the X axis. So here we can see we have three different points. The first intersection is at X is equal to negative four pi, then we have another one at X is equal to zero. And lastly, we have one at X is equal to positive four pi. So at all of these points, since these are the points where our given sine function is zero, this tells us that these points are where our reciprocal function are infinity. So that tells us that these points serve as vertical asymptotes. So we need to take these points. And now on our sequent graph, we want to draw in our vertical asymptotes again at these points. So our first vertical Asymptote will be at X is equal to negative four pi. So we just draw in our dashed vertical line at this point, then we move on to where X is equal to zero. So our next vertical Asymptote is along the Y axis. And lastly our third vertical ACETO is at X is equal to four pi. So now that we have identified all of our vertical asymptotes and drawn them into our new graph, we want to look back at our original graph and take note of all of the points where we have the highest point and the lowest points on the curve. And the reason why we are taking note of these points is that they will serve as vertices on our sequent graph. So we just want to take these points and replicate them onto our sequent graph. So our first point is at negative two pi and positive 0.25. So we just need to draw in this point, then we have positive two pi and negative 0.25. And again, since both of these points serve as vertices, our next step is to just draw in our curves. So our first curve on the left side here will go from our point all the way till positive infinity. So we have a somewhat parabolic shape to our curve as they go up words and approach the asymptotes but never cross. And now with our second point, we have a similar curve but it is now downward facing as the lines approach the vertical asymptotes along the negative Y axis. So these lines approach or this curve approaches negative infinity along the Y axis. And it is important to note that the graph of a consequent function is periodic and it will oscillate between positive and negative infinity as X approaches zero. So that means that these curves will be replicated beyond this domain from negative four pi to positive four pi. So for example, our next point on the right side of our graph will be at six pi and positive 0.25. And then we will only see in this graph part of the curve where it approaches positive infinity along the Y axis. And then on the left side of our graph, we will have another point here which would be negative. So we would have the point at negative six pi and negative 0.25. And then the curve will approach negative infinity. So with this graph, we have again identified our function as Y is equal to negative 1/4 multiplied by sequent of X divided by four. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

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.

Key Concepts

Reciprocal Functions

Graphing Trigonometric Functions

Asymptotes in Trigonometric Functions

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