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 = 3 cot(x + π/2)

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Hello, today we're gonna be graphing two periods for the given function. So what we are given is Y is equal to five cotangent of X plus pi over six. Now, before we graph this function, let's take a look at the parent function which is Y is equal to cotangent of X. The para function is given to us with two asymptotes. The first one being located at zero and the second one being located at pie, the co Tai graph has an X intercept at pi over two and the shape of the Co Taio function has a snakelike structure that roughly looks like this. Furthermore, the period of this function is going to be pi. So what we want to do is we want to identify all of the transformations that are on our given function in order to graph the given function. So the first transformation is going to occur from the leading coefficient of five. Normally this would represent a change in amplitude. So we will want to take the absolute value of five which is just going to give us five. Now, one thing to note is that cotangent does not have an amplitude since its range is from negative infinity to positive infinity, what instead what is going to happen is that five represents a stretch of the graph. And this value is also going to help us find two points on the graph that represent the stretch. This value won't give us the X values for these two points. But it will tell us that the Y values are going to be plus or minus five. Now let's take a look at the second transformation. The second transformation occurs from the quantity X plus pi over six, X plus pi over six represents a horizontal shift. And a horizontal shift is normally represented by the quantity X minus H X plus pi over six is not in the standard form of X minus H, but it can be rewritten as X minus negative pi over six. So what this means is that H is going to equal to negative pi over six for our given function. And what this means is that every point on the graph is going to be shifted to the left pi over six units. So our X intercept is going to be shifted pi over six units to the left. But since we are shifting the X intercept, we must also shift the two asymptotes. So what we need to do is we need to find the new asymptotes. The first Asymptote is given to us as zero. And since we are shifting this value pi over six units to the left. We're going to be subtracting pi over six from the first Asma tote and zero minus pi over six is going to give us negative pi over six. The second Asom Tode is given to us as pie. And since we are shifting pie pie over six units to the left, we're going to be subtracting pie over six from pie pi minus pi over six is going to give us the value of five pi over six. And what this means is that the new Asy totes for the cotangent graph are going to be negative pi over six comma five pi over six. Now, one more thing to note is that the X intercept is also changing the X intercept for the traditional code tangent graph is given to us as pi over two. But since we are shifting this pi over six units to the left, we need to subtract pi over six from pi over two pi over two minus pi over six is going to give us the value of pi over three. And what this means is that the X intercept is gonna be located at pi over three comma zero. Now, since there is no change in period, the period is going to be remaining the same. So we're still going to have a period of pi between negative pi over six and five pi over six. Now that we have the period and we have the new atos and X intercept. Let's go ahead and graph this information. So if we draw an axis, we're going to label the first as tote at negative pie over six. And the second ay tote is gonna be located at five pi over six. We know that the X intercept for the code Tara is gonna be located at pi over three comma zero. And because we are not reflecting anything about the about the period, the shape of the graph is going to remain the same. So we're still going to have this snakelike structure. Now, recall from earlier, we stated that there's going to be a stretch of the graph. So what we need to do is we need to find two points on the cotangent graph that represent the stretch. We know that the Y value of the first point is going to be positive five. And we know that the Y value of the second point is going to be negative five. What we need to do is we need to find the X values for these two points. While this point lies halfway between the first ascent tode and the X intercept, that distance can be represented as negative pi over six plus pi over three. And since the point is located halfway between this distance, we're going to divide this distance by two negative pi over six plus pi over three is going to give us the value of pi over six and pi over six, divided by two is going to simplify to give us pi over 12. So the X value for the first point is going to be pi over 12. Now, we need to find the X value for the second point. The second point is located between the distance of the X intercept and the second ascent tope that distance can be represented as pi over three plus five pi over six. And we're gonna be dividing this distance in two. So pi over three plus pi over six is going to give us the value of seven pi over six and seven pi over six divided by two is going to give us the value of seven pi over 12. That means the X value for the second point of the stretch is going to be seven pi over 12. Now we have a fully labeled first period for the Cotangent graph but recall that the problem asks us to graph two periods of the tangent graph. So what we need to do is we need to identify a third ascent tote for the tangent graph. Now recall that the period remained the same. The period is still pi the distance from negative pi over 6 to 5 pi over six is a total distance of pi. So that means the third ASOM tote is gonna be located a distance of pi from five pi over six. So we need to add pi to five pi over six five pi over six plus pi is going to give us the value of 11 pi over six. And the second period of the graph is going to match the shape of the first period. So we're going to have a rough snakelike structure like this. And this is going to be the complete graph for the given cotangent graph. 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 = 3 cot(x + π/2)

Key Concepts

Cotangent Function and Its Properties

Phase Shift in Trigonometric Functions

Amplitude and Vertical Stretch

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