In Exercises 13–16, the graph of a cotangent function is given. S...

Question

In Exercises 13–16, the graph of a cotangent function is given. Select the equation for each graph from the following options:y = cot(x + π/2), y = cot(x + π), y = −cot x, y= −cot(x − π/2).

Accepted Answer

Hello, today we're gonna be finding the equation for the given graph. Now, one thing to note is that this graph is in the form of a cotangent graph. So in order to identify the graph, let's go ahead and draw the parent function for cotangent of X co tangent of X has a period of pie. And it also has the ascent totes between zero and pie. It contains an X intercept at pi over two and it has a rough shape just like this. And for one full rotation, the next Asymptote for the para function is gonna be located at two pi and the next intersection is going to be at three pi over two. So this is going to be the parent function of cotangent X between zero and two pi. The reason why this is important is because it's going to help us identify the graph for the given function. So one thing to note is that the graph that is given to us is reflected about the X axis when compared to the parent function. So what this means is that we have a negative sign in front of the cotangent graph that is going to be the first transformation of the graph. Next, the given graph has X intercepts at negative pi over six. Now, if we were to extend the parent function to one region behind the X axis, the next ascent will be located at negative pi and the X intercept at this point is going to be at negative pi over two. So what this means is that we have a shift of the cotangent graph. Now how do we indicate find what type of shift that has occurred? Well, let's go ahead and take the X intercept of negative pi over two. We're going to be adding some value to negative pi over two. And this is going to give us the new X intercept of negative pi over six. So if we solve for X, that is going to tell us the type of horizontal shift that has occurred for the following cotangent graph. So in order to solve for X, we're going to be adding the value of pi over two to both sides of the equation that is going to leave us with X is equal to negative pi over six plus pi over two negative pi over six plus pi over two is going to give us the value of positive pi over three. So since we have X is equal to positive pi over three, what this means is that there has been a horizontal shift of the graph pi over three units to the right, the way that we represent a horizontal shift is by using the quantity of X minus H. And in this case, here H is going to equal to pi over three. So we can represent the horizontal shift as X minus pi over three. And if we combine this transformation with the transformation, we identified earlier, the graph that we are left with is Y is equal to negative cotangent of X minus pi over three. This is going to be the equation for the given graph. And with that being said, the answer to this problem is going to be D. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

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In Exercises 13–16, the graph of a cotangent function is given. Select the equation for each graph from the following options:y = cot(x + π/2), y = cot(x + π), y = −cot x, y= −cot(x − π/2).

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