Use graphing software to graph the functions specified in Exer...

Question

Use graphing software to graph the functions specified in Exercises 31–36.

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Graph two periods of the function f (x) = 3 cot (x/2) + 1.

Accepted Answer

Hello. In this video, we are going to be drawing two periods of the given sign function below. Now, in order to approach this problem, let's just quickly recall what the general sine function looks like. We are given that the general form is F of X is equal to a multiplied by sine of B multiplied by X minus C. Plus D So the first thing we're going to want to do is we are going to want to rewrite our function in this general form. In order to do this, we are going to have to factor a 2 out from the given angle, and that is going to allow us to rewrite the original function to be F of X is equal to 3 multiplied by sine of 2 multiplied by X minus pi divided by 3. -1 Now, what we want to do is we want to look for the variables A, B, C, and D, because this is going to help us get the the values that we need for the function. So, A is going to be the coefficient in front of our sign function, and in this case here, A is equal to 3. B is going to be the coefficient in front of the angle inside the sine function, and that is going to be B is equal to 2. C is going to be the value that X is being subtracted by, and that is going to be pi divided by 3. And D is going to be the extra value that we are adding or subtracting to the sine function, and that is going to be -1. Now that we have all this information, let's go ahead and first start by talking about the midline. Of the function. Now, the midline is going to be a horizontal line that the function is centered around, and it is defined by the variable D. Now, in this case here, since D is equal to -1, we are going to have a midline located at Y is equal to -1. So let's go ahead and draw that on the graph. Now that we've identified our midline, let's go ahead and identify the amplitude. Now, the amplitude is defined by the variable A, and these define how high or how low we are going to be around the midline. Now, this is defined as the absolute value of A. And since A is equal to 3, the amplitude will equal to the absolute value of 3, which is going to give us positive and negative 3. So, in other words, the range of this function is going to be 3 units above and 3 units below the midline. If we draw this on the graph, we are going to have a maximum value of the graph defined by this horizontal line at Y is equal to 2, and the minimum values are going to be defined by this horizontal line located at Y is equal to -4. So we have our midline and we have our range set up. Let's go ahead and talk about the the period of the function. Now, the period is defined by the variable B. The standard period for the sine function is 2 pi, so the period will be defined as 2 pi divided by B. B is equal to 2, so we have 2 pi divided by 2, which is equal to 2. What this means is that a standard rotation of the sine function is going to have a length of not 2, because 2 pi divided by 2 is actually pi. So we have a period of pi, meaning that 1 full rotation has a length of pi. And finally, we need to talk about the phase shift. Now, the phase shift is defined by the variable C. Since C is equal to pi divided by 3, we have a phase shift of pi divided by 3. And what this means is that we are going to take the entire function and shift it over pi divided by 3. OK, so now that we've identified all this information, let's go ahead and first start by finding the endpoints of the first rotation. Now recall that the standard sign function. Starts at the origin 0. And at one full rotation ends on the X axis. Since we have a phase shift of pi divided by 3, and we have a period of pi, the end points to the first rotation are going to be pi divided by 3-1. And we are going to have an end point located at 4 pi divided by 3-1. So let's just go ahead and label those points. Honor graph. So, the start of the first period is at pi divided by 3-1, and the end of the first period is 4 divided by 3-1. But now we need to figure out what the behavior of the graph is going to be at the rest of the function. So, Recall that the main angles we like to focus on around the unit circle are 0, pi divided by 2, i, 3 pi divided by 2, and 2 pi. So what we need to do is we need to first figure out what the remaining angles are going to be. Now again, this is going to have oscillating value, so that means that the next value of the of the first rotation should be located at 2 on the Y axis. But what about the X-axis? In order to find the location on the X-axis, we are going to take the innermost angle from the original function 2X minus 2 pi divided by 3, and set this equal to the main angles of the unit circle. So, for the second point, we are going to take 2 X minus 2 pi divided by 3, set this equal to pi divided by 2, and solve. Solving will give us X's equal to 7 pi divided by 12, and the Y value for this point is going to be at 2. Repeating this process, we are going to take 2 X minus 2 pi divided by 3, and set this equal to pi. Solving for X is going to give us X's equal to 5 pi divided by 6, and this will have a Y value of -1. And finally, if we repeat this process one more time, the fourth point on the first rotation will be 2 X minus 2 pi divided by 3, set this equal to 3 pi divided by 2, and we get X's equal to 13 pi. Divided by 12 with the Y value of -2 or -4. So, let's just quickly plot these points. Now, we don't have to be super accurate, but we can determine that the next point will be located roughly around here. We have a value between the two endpoints located here, and the next value will be roughly located about here. And if we connect these points together, this is going to be the first rotation. Or the first period But the thing is, now we want to determine what the 2nd period is going to look like. Well, in order to determine the 2nd period, we need to determine the endpoints. Now, the 2nd period is going to start. At 4 pi divided by 3, which we know has a value of -1. So, in order to find the next endpoint, all we need to do is we need to add 124 pi divided by 3, and 14 pi divided by 3 plus pi. is going to give us 7 pi divided by 3. So we know that the final endpoint is going to be located at 7 divided by 3-1. Because the endpoint has to be located on the midline. And since we are, since the next period is just gonna mimic the behavior of the first period, we can roughly draw the second period to look like this. And this is going to be the total graph for the given sign function. So I hope this video helps you in understanding on how to approach this problem, and we will go ahead and see you all in the next video.

Use graphing software to graph the functions specified in Exercises 31–36.
Select a viewing window that reveals the key features of the function.

Graph two periods of the function f (x) = 3 cot (x/2) + 1.

Key Concepts

Cotangent Function

Transformations of Functions

Graphing Software

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