In Exercises 1–4, graph one period of each function.

y = 2 tan x...

Question

In Exercises 1–4, graph one period of each function.

y = 2 tan x/2

Accepted Answer

Hey, everyone. Here we are asked to draw the graph of the function for one period. Here we are given the function Y is equal to five multiplied by tangent of X divided by three. Here we are given a blank graph where along the Y axis, we range from negative 20 to positive 20. Then along the X axis, we range from negative three pi halves to positive three pi halves. So to begin this problem, before we begin finding the components of our graph, we first want to recall the formula for a tangent function which is Y is equal to a multiplied by tangent of BX minus C. And now we want to use this formula to compare it with our function to first identify our values for A B and C. So starting with a, we see our value is five, we then have our value for B as one third. And lastly, our value for C is just going to be zero. Now, with the parts of our function clearly identified, we can move on to finding the components of our graph by starting off with finding our two consecutive asymptotes. So to find our asymptotes, we need to recall two expressions where first we have BX is equal to positive pi haves. And then we also have BX is equal to negative pi haves. And now by solving both of these equations for X, we will be able to find our vertical asymptotes. So recalling that our value for B is one third, we can now have our two equations where first we will have one third X is equal to pi halfs. We then have one third X is equal to negative pi halves. Now again, we need to solve both equations for X. So all we need to do for each equation is multiply both sides by three. And in doing this, we find our two asymptotes. So one of our asymptotes will be X is equal to positive three pi halves. And then our second as is going to be negative three pie halves. So now with both of our asymptotes identified, we can move on to our next step where we now want to find the X intercepts by using the midpoint between our two asymptotes. So here, since Y is going to be going to remain zero, we just need to find the midpoint of our X values. So using the values of our asymptotes, we have three pi halves plus negative three pi halves, all divided by two. And now simplifying we see in the numerator we sum to zero. So we have zero divided by two and simplifying, we just get zero. And so now this tells us since our X value comes out to zero, that our X intercept is going to be at the 0.00. Or in other words at the origin. And now that we have found our X intercept, we can move on to finding two other points which our graph crosses through or the line of our function crosses through. So for this, we can utilize the value for a which is five. And since five determines the steepness of our graph, we know that the graph, the line of the graph will pass through the points on the Y axis of positive five and negative five. So we just want to find the points whose Y coordinate is equal to negative five and positive five. So we just need to use the X value given by our X intercept as well as the values given by our asymptotes. So first, we will have negative three pi haves plus zero all divided by two. And simplifying we get negative three pi divided by four. And so this tells us that one of our points will be at the point negative three pi divided by four and negative five. So now repeating this process, we will have positive three Pi Hals plus zero, all divided by two. And now simplifying we get positive three pi divided by four. So our second point will be at three pi divided by four and positive five. So with all of our components of our graph, we now just want to begin graphing our information. So we can start with our point at the origin at the 0.00. Then we also have another point at negative three pi divided by four and negative five. And then we also have the opposite of that. We have the point at positive three pi divided by four and positive five. Next, we have our vertical asymptotes which are at negative three pi hands and we also have one at positive three pi ha. And now our last step is to just draw our line that goes from negative infinity along the Y axis to positive infinity after crossing through both points. So we see that the line approaches both, both vertical asymptotes but never crosses. So with this, we have drawn the graph of our given function. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

In Exercises 1–4, graph one period of each function. y = 2 tan x/2

Key Concepts

Tangent Function

Period of a Function

Vertical Stretch

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