In Exercises 17–24, graph two periods of the given cotangent f...

Question

In Exercises 17–24, graph two periods of the given cotangent function. y = 1/2 cot 2x

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Hello. Today we're gonna be drawing the graph of two periods of the given function. So what we are given is Y is equal to one third times co tangent of three X. Now, before we take a look at this graph, let's go ahead and take a look at the pair function code tangent of X cotangent of X is given to us with a period of pi it has two ascent totes, one located at zero and one located at pi and the graph of cotangent has X intercept at pi over two comma zero. And the rough shape of co tangent X is gonna have a snakelike shape in this direction. So what we want to do now is we want to identify all the transformations that are occurring on the given equation and compare that to the parent function. Now, the first transformation we can identify is the coefficient of one third. Normally when you have a coefficient that's not one in front of a trigonometric function that indicates that there is a change in the amplitude. However, cotangent does not have an amplitude since its range is from negative infinity to positive infinity. What we can do with one third is we can use this to identify two points that are located on the cotangent graph. We're not sure what the X values are going to be. But this does tell us that the Y values for the two given points are gonna be plus or minus 1/3. The next transformation is given to us in the form of three in front of the X term, the coefficient in front of the X term is not one. So that means that the cotangent graph is going to have a phase shift. And what a phase shift means is that the period of cotangent is going to change. So how do we identify the new period while the new period is given to us as pi over B and B represents the coefficient in front of the X term. In this case, here we are given three. So B is equal to three, which means that the period is going to be pi over three. And that means that the new ascent totes for the cotangent graph are gonna be located at zero comma pi over three. Now that we have the two asymptotes and we have the location of the two points. We want to go ahead and identify the X intercept for this cotangent graph. The X intercept is given to us as the period divided by two. The period in this case is pi over three. So that's going to be pi over three divided by two. And simplifying this quantity is going to give us an X intercept of pi over six comma zero. So now that we have the X intercept, we have the ascent totes of the first period and we have the locations of the first two points. Let's go ahead and graph this new information. So if we draw an axis, we have the first ASOM tote which is located at zero and the next ASOM tote is going to be located at pi over three. Now keep in mind the problem asks us to identify two periods of the cotangent function. So that means we need to draw the location of the next period. Since we know that the distance of the period is pi over three, we just need to add another measure of pi over three to the second ascent tode pi over three plus pi over three is going to give us the second ascent tode located at two pi over three. So let's go ahead and graph the first period of the cotangent. So we know that the X intercept is located at pie over six comma zero. And since we don't have a negative sign in front of the cotangent graph, the shape of the cotangent graph is gonna remain the same. Now we want to go ahead and identify the two points that are located on the cotangent graph. Now recall that we don't know what the X values for these points are, but we do know that the Y values are going to be 1/3 and negative 1/3. So how do we find the X values for these points? Well, keep in mind that this point is located halfway between the first ascent tode and the X intercept. So what we can go ahead and do is we can add the distance between zero and pie over six and divide that distance by two in order to get that X value. So zero plus Pi over six is going to give us pi over six and pi over six divided by two is going to give us the value of pi over 12. So the first point on the cotangent graph is going to be pi over 12. Now, in order to get the X value for the second point, we want to take the distance between pi over six and the second and tote which is pi over three and divide that distance by two. So what we get is pie over six plus pi over three. And we're dividing that distance in half pi over six plus pi over three, gives us the value of pi over two and pi over two divided by two is going to give us pi over four. So we know that the location of the second point is going to be pi over four, a negative 1/3. Now that we have the fully labeled first period, the second period of the function is going to mirror the first period and the shape of the second period is roughly gonna look like this. And with that being said, this is going to be the graph for the given cotangent function. 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.

In Exercises 17–24, graph two periods of the given cotangent function. y = 1/2 cot 2x

Key Concepts

Cotangent Function and Its Graph

Period of a Trigonometric Function

Amplitude and Vertical Scaling

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