INSTRUCTOR: So we want the equation for this tangent graph. All right. The first thing I like to look for for the tangent graph is the midline because the midline is based upon the initial value. What we mean by the initial value is the value at 0. So what I do is, I look for, where does the graph at 0? It's right here. So I always look at the y-axis and find that dot. That finds the midline for my graph because a tangent has the dots from a sine graph. 

Anyways, but because it's related to sine, the y-intercept is the middle or midline of your graph all the time. So let's draw our midline in. So there we go. There is my midline. 

Now let's pause and think. OK. Where would be the end of my period from there. Well, OK. If I look at that, well, won't it be right there? Because the period starts and ends at the same place, and then you see it repeat. So if it goes like this and like this and then it repeats, that right there is the end of your period. So that right there is your period. Then we look between our asymptote and our first dot and find the middle of that. 

And then from that middle, we go up and down until we hit the graph, which is right here. In the middle of the asymptote and the value on the other side, we go up until we hit the graph. And that's our other dot. And if you look at this, can't you see a sine graph right there? Which that's what we're looking for. 

Those dots right there help us see our midline, our vertical stretch. It's not a amplitude. I said that wrong. And our period. So let's write out our midline, vertical stretch, and period. Our midline for this one is y equals 0, that blue line. Our vertical stretch, not our amplitude, is-- let's see. How far down did we go and up? 

Well, that's 1. Isn't that 1/2? Aren't we going a half of a vertical stretch? And if you notice about tangent, it's going upside down. So this also has a flip action. It's upside down. So there's going to be a negative a value coming up. And then our period-- well, our period is right here. So our period is right there. So it's pi/2. 

So now we want to make an equation. But before we do the equation, our equation doesn't want the period. Our equation wants our b value. So let's first find our b value. And then from our b value, you can make our equation. So our b value, our b value. OK. Our b value is going to be pi divided by our period. b is always pi divided by period, which would be pi times the reciprocal. 

Let's put a 1 underneath there. Those cancel, leaving you 2. So your b value is 2. So let's piece it all together. y equals-- you do your a value or your vertical stretch. So it's going to be a 1/2 negative. All right. So that's going to be a negative 1/2. We have the word tangent because this is a tangent graph. 

Next, b, not our period, this is what's going to be multiplied by the x. So that's going to be 2x. And lastly, we put the midline, which is nothing. So we're done. We don't need to add a 0 at the end. It's not necessary. And this would be the equation for this graph. 

I want to write the equation for this graph. You hopefully notice it's a tangent again. I always start at the y-intercept. First thing I want to do is find the y-intercept, which is right there. And then I go across. That right there is my midline. And then as I go across, this right here is your period distance. 

With that said, that is your period. And then we go halfway between your initial dot right here and your asymptote. And you go up until you hit the graph. Then halfway between your dot right here and the asymptote, you go down, and you hit your graph. And this distance right here is 2, which is your vertical stretch. 

So let's piece it all together. Our midline is going to be y equals negative 1. Our vertical stretch is going to be the 2. That's how high you went here. There is no flip. It's going up. That's normal for tangent. Our period is pi because it starts and ends and then repeats. So our period is pi. 

But to make our equation, we want b. We don't want pi. We want b. So we go pi divided by pi, which is 1, not 0. Be careful. That's 1. All right. So let's make the equation. Y equals-- we start with the vertical stretch or the a value, which is 2. We have tangent. 

Next, we take-- I'm going to use theta this time. You can use theta or x. And that's it. OK. You can put the 1 if you want. If you want put 1 theta, go for it. It's not necessary. And then we put our midline of minus 1. That right there is our equation. Again, you can use theta or x. You don't need that 1. It's not necessary. You could even put f of x in the front if you want. Or f of theta would be this one, f of theta.