Graph each function over a two-period interval.
y = tan(...

Question

Graph each function over a two-period interval.

y = tan(2x - π)

Accepted Answer

Hey, everyone in this problem, we're asked to graph the following function to consider two periods. And the function we're given as Y is equal to tangent of four X minus pi and we're given this blink coordinate plane that we can add our graph to. So we're after graph two periods. First thing we're gonna do is figure out what the period of this function is. OK. So we're giving Y is equal to the tangent of four X minus pi. Now, when we're talking about the period, recall that the B value of our function is going to determine the period. OK. So the B value is that coefficient associated with the X inside of our tangent. OK. So in this case, the B value is four that coefficient on the X. OK. So we have our B value. Now we have to remember how that relates to our period. OK. Now the parent function we're dealing with is tangent. If we just have Y is equal to tangent of X, the period is pi OK. So that's the regular period for our function. And so the new period, the period of this transformed function we're gonna call that T is going to be that basic period of pie divided by RB value. So in our case, that's gonna be pi divided by four. All right. So the period of our function is going to be pi divided by four. And so for two periods, we're going to graph a distance of pi divided by two on our X axis, right? All right. So we're gonna be graphing pi divided by two units. We now want to find our vertical asymptotes. OK. We're dealing with tangent. We know there are asymptotes. So let's get the asymptotes drawn on our graph. Now, we're given some X values along our graph. So figuring out where those asymptotes are, is gonna help us determine which two periods we want a graph. OK? And then they're gonna help determine the rest of the graph as well and what this function is going to look like. So to find our vertical asymptotes, what do we do? Well, we're gonna write out our function again. Why is equal to tangent four X minus pi I want to recall that tangent is sine divided by cosine. And when we're looking for vertical asymptotes, the denominator matters. So if we can break tangent up and write it in terms of sine and cosine, it's gonna be more obvious to see those vertical asymptotes. So our function can be written as Y is equal to sine of four X minus pi divided by cosine of four X minus pi. Now, we can clearly see that denominator and recall that our vertical asymptotes occur when that denominator is equal to zero. If the denominator is zero, the function is undefined at those points. And that's where we get those asymptotes. So our vertical asymptotes are going to occur when cosign A four X minus pi is equal to zero. Now, this looks a little bit messy because we have this four X minus pi in the bracket. But don't get confused by that. We're looking for points where cosine of something is equal to zero. We know that if that's something which in this case is four X minus pi is equal to pi divided by two. OK? Or some odd multiple of pi divided by two. So pi divided by two plus pi K or K is an integer. The cosine of that thing is going to equal to zero. OK. All right. So we have this equation and we want to figure out what this means for the X value. What is the condition on the X value? So we're gonna solve this equation for X, we can start by adding pi. So we have four X is equal to pi divided by two pie pie. OK? Now we can simplify this. OK. Right now we have pi divided by two plus pi plus pi K, we can wrap that plus pi into our plus pi K. We can create a new variable if we want we can say that four X is going to be equal to pi divided by two plus pi multiplied by say N where N is an integer and N will just be one more than K. OK. So when K is one, we have pi plus pi, that's the same as if N is two, we get two pi. And so because it's that perfect, even multiple of pie, we can combine those together. Now, the last step is to divide by four on both sides. So we get the X is going to be equal to pi divided by eight plus pi divided by four. And OK. So we have this general equation. Now for vertical as the X values where they occur, we wanna look for specific vertical asymptotes on the interval with which we're given in our graph. So let's start with if K if N is one or zero, sorry, if N is negative, then X is going to be negative. Our graph does not go into the negative X value. So we're gonna start with N being zero. OK. If N is zero, then we get X is equal to pi divided by eight. If N is equal to one, we get X is equal to pi divided by eight plus pi divided by four which is equal to three pi divided by eight. If N is equal to two, we get X is equal to pi divided by eight plus pi divided by two which gives us five pi divided by eight. And you can see that each vertical Asymptote is broken up by a distance of pi divided by four. OK. So that's an entire period. So while we draw these asymptotes on our graph, we're gonna divide it up into these two periods that we're interested in. Those are the two periods we're gonna choose to graph. Can you just based on the X values we were given in our graph? All right. So we found these vertical asymptotes at pi divided by eight. We're gonna draw a dashed vertical line there at three pi divided by eight and up five pi divided by eight. And again, you can see that it breaks this up into these two periods. So we're gonna be graphic from pi divided by eight all the way up to five pi divided by eight. That's gonna be a distance of pi divided by two units or those two periods. OK. So we have our asym totes, we know the period, we know where we're graphing. Let's go ahead and use a table of values to figure out where are some of the points are between these subjects? What's going on between them? Now, we only need to do this for one period because we know that the next period, it's just a re repeat. OK. That's how these periods work. That's how trigonometric functions work if this periodic behavior that repeats every single period. OK. So we only need to do this for the first period, then we're able to copy it for the second. So we have X values Y values, we're gonna start on the left hand side. So we have plied vitamin E three P divided by 16 pi divided by four, five pi divided by 16 and three pi divided by eight. Now pi divided by 83 pi divided by eight. We know this is where our vertical asymptotes are. So we don't need to worry about the values there. We know we're going to positive or negative infinity. And the other values are gonna help us determine which one it is whether it's positive or negative. So three pi divided by 60. If we substitute three pi divided by 16 sx into our function, we get tangent uh three pi divided by four minus pi OK. This gives us tangent of negative pi divided by four which is going to be equal to negative one. We do the same for the next X value. If X is pi divided by four, we get tangent of pi minus pi, this is just tangent of zero which is equal to zero. And finally, if X is five pi divided by 16, we get tangent of five pi divided by four minus pi. This gives us tangent of pi divided by four and we get positive one, we have these three points we can add to our graph and three points is really all we need to draw out a tangent curve. Once we know our asymptotes, we add these points. So pi divided by 40 at three pi divided by 16 were at negative one. And at five pi divided by 16, we're a positive one and we can connect these knowing it looks like a tangent curve. Yes, we're gonna connect them and it's gonna look something like this. OK. So as we approach pi divided by eight, that vertical ascent toe from the right hand side, we're approaching negative infinity. As we approach three pi divided by eight from the left hand side, we're approaching positive infinity. OK? So that's our first period and then we can do the same for the second. So those points are going to match and the curve is going to match as well. We're just repeating it in that second period. And so this is two periods of our graph for Y is equal to tangent of four X minus pi. Thanks everyone for watching. I hope this video helped you in the next one.

Graph each function over a two-period interval.
y = tan(2x - π)

Key Concepts

Period of the Tangent Function

Phase Shift in Trigonometric Functions

Asymptotes of the Tangent Function

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