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 = 2 cot x

Accepted Answer

Hello, today we're gonna be drawing the graph of the first two periods of the given function. So what we are given is Y is equal to four cotangent of X. Now, before we draw the graph of this function, let's go ahead and take a look at the pair function for cotangent of X. The parent function is given to us with a period of pi it has two asymptotes for the first period which is going to be between zero and pay the graph has an X intercept at pi over two comma zero. And the Co Tara has a snakelike structure which roughly looks like this. So let's go ahead and take a look or identify at all the transformations that are happening on four cotangent of X. So comparing the two graphs, there's only one transformation that is happening and that is the coefficient of four in front of the cotangent function. Normally, when you have a co fission, that is not one in front of a trigonometric function that indicates a change in the graph's amplitude. However, cotangent does not have an amplitude because the range is from positive infinity to negative infinity. So instead this four can be used to help us identify two points on the cotangent graph. This doesn't tell us the X value for those two given points, but it does tell us that the Y value is going to be plus or minus four. Now, since there are no other changes to the graph, the period is still going to be pi for the given function and the X intercept is going to be unchanged. So it's going to be located at pi over two comma zero. So let's go ahead and graph the new version of cotangent of X. So we know that the ascent totes are located at zero. And that pie we know that the X intercept is located at pi over two. And the rough shape of the cotangent graph is going to remain the same. Now with the new coefficient, we can identify two new points on the cotangent graph. Again, we don't have the X values for these two points, but we do know that the Y values are going to be positive four and negative four. So how do we get the X values for these two given points? Well, notice that the first point lies halfway between zero and the X intercept pi over two. So what we can go ahead and do is add the distance between zero and pi over two and divide that distance in half zero plus pi over two is going to give us pi over two and pi over two divided by two is going to give us the value of pi over four. That means that the first point is gonna be located at pi over four come a positive four. Now, in order to get the X value for the second point, we're going to take the distance between power two and pi and divide that distance in half. So doing so gives us pi over two plus Pi divided by two pi over two plus pi gives us the value of three pi over two and three pi over two divided by two, gives us three pi over four. So we know that the X value of the second point is going to be located at three pi over four. So what this means is that four cause the horizontal stretch of the cotangent graph. This is the fully labeled version of the first period of the graph. But keep in mind that we are asked to graph two periods of the graph. So we're going to need to identify another ascent to to represent the second period in order to get the location of the second ascent tote, we're going to add another period to the first ascent to. So adding Pie to Pie is going to give us the second ascent at two Pie and the second period is going to mirror the image of the first period. So the second period is roughly gonna look like this and this is going to be the graph for the given cotangent function. And with that being said, I hope this, 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 = 2 cot x

Key Concepts

Cotangent Function and Its Properties

Amplitude and Vertical Stretch

Period of the Cotangent Function

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