Graph the Tagent Function Fast!

all right so in this video I want to show you how to graph the tangent function fast because I think a lot of times when students need to identify or look at the tangent function or maybe even graph the tangent function with Transformations they don't know what the parent graph looks like they always forget it they confuse it with cotangent or what exactly the shape is and where are the points so let's just cover the basics of how you can graph tangent very very quickly but also understand it and remember it and even if you do forget it you always have a great way to go back to it so it all comes into knowing two things one the relationship of tangent with the unit circle all right now I know I probably lost maybe some of you once I started saying the unit circle and maybe even gave you some PTSD you're like oh no I hate the unit circle I can't remember the unit circle I always forget the points but ladies and gentlemen can we just agree that these are the easiest Four Points to remember on the unit circle right because again the unit circle basically just means from the center of from the origin right you're just going out one unit so we have 1 Z 0 1 negative 1 0 and 0 comma 1 now when we're dealing with points any points that are on the unit circle the tangent function has this relationship for whatever angle that we are dealing with all right so if you understand these four points on the circle and you remember the tangent relationship right tangent of theta of the angle is y/x then you can go and graph any you can graph the tangent function very very quickly so when we're graphing the tangent function what we're doing is we're graphing it on an XY plane or at least in this case it's going to be a Theta y plane okay so just remember Theta is going to be your input variable it's going to represent the angle that we're looking at but also so a lot of times when we use it with X or use Alpha you can use anything as your input variable and then Y is representing your output which is going to be on this vertical line all right let's go go ahead and pick some easy values and again you can use like a table of values and actually you know what let's let's actually do that let's do a table of values so you can see how you can graph this quickly now obviously the video is titled graph tangent fast so a lot of this what I would do is just actually do this in my head but I am slowing it down a little bit just so you guys can see the process of what I'm doing in my head that you can do rather quickly on your own as well okay so we're going to have our input variable and our output variable so let's go and pick a point that is going to be easy what about zero so if we have an angle that's zero that's going to be right here right so we have the Y over the X right because again we have x coordinate and a y coordinate right so 0 over one is obviously going to be zero so we know that this point is going to work here on the graph the next Point here we could say is pi Hales or 90° I prefer radians so we'll do PI halves so at Pi halves we're at 1 over zero right so let's go here well guys we know one over Zer you can't divide by zero right so therefore that is going to be undefined that is where our ASM toope is now before I get to going into continue doing more and more values or more angles remember we can also go in the negative direction right cuz typically if you're familiar with what the tangent graph looks like you recognize the initial period is going to be right here on the origin so if I go negative piales what I want you to recognize here is that is also a undefined now I started graphing I was like whoa whoa whoa I didn't explain this yet so let's get to this here in just a second so how do we know does the graph open like this right does the graph look like this does the graph look like that how does this graph look is that parabas like what what does it look like so again let's look at our coordinate point here if I had any point XY now you might be familiar you might remember what the coordinate points are in that first quadrant one thing I want you to recognize is it doesn't even matter like yeah you can use them you can use a calculator and find the coordinate point and then plot them but all I want you to understand is in the first quadrant you have a positive point over a positive point so I know whatever my ratio is going to be in positive whereas over here you're dealing with a coordinate point that is going to be a positive and a negative right the xord positive the y coordinate is negative so that means over to the left here for all negative angles to negative pies you're going to be dealing with negative points that is why we get the graph that's going to look like this so again you hopefully you guys can see here you're going to have your asmt toes at Pi hals negative pi Hales you're going to have the cor point0 and then based on the ratio or the understanding of tangent you can see the shape of the graph how it's going to look now all I want you to understand is again all this is doing is just repeating itself right here you're going to have an asmt then you're going to have another zero then you're have another ASM toope and at your point Z so that's why you're going to get this graph that's just going to continue right this is just what we call the initial period but then you can go ahead and repeat this because what's happening is every Pi is going to be your period right because so the next Point here is going to be at pi and then the next ASM toope is at Pi Hales to 3 pies and I got a little wider but that's okay and then you can see you can also go in this negative Direction but that is how you graph the tangent function fast once we start adding in some Transformations still you're going to want to start with this initial period and then you just go from there with your um transformation so hopefully this video was helpful for you hopefully you remember this and remember that and you'll be all set cheers