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more function visualizations so let's say you have a function it's got a single input T and then it outputs a vector and the vector is going to depend on T so the X component will be T times the cosine of T and then the Y component will be T times the sine of T this is what's called a parametric function and I should maybe say one parameter parametric function one parameter and parameter is just kind of a fancy word for input parameter so in this case T is our single parameter and what makes it a parametric function is that we think about it as drawing a curve and its output is multi-dimensional so you might think when you visualize something like this oh it's got you know a single input and it's got a two-dimensional output let's graph it you know put those three numbers together and plot them but what turns out to be even better is to look just in the output space so in this case the output space is two-dimensional so I'll go ahead and draw a coordinate plane here and let's just evaluate this function at a couple different points and see what it looks like okay so I might maybe the easiest place to evaluate it would be 0 so f of 0 is equal to and then in both cases it'll be 0 times something so 0 times cosine of 0 is just 0 and then 0 times sine of 0 is also just 0 so that input corresponds to the output you could think of it as a vector that's infinitely small or just the point at the origin however you want to go about it so let's take a different point just to see what else could happen and I'm going to choose PI halves of course the reason I'm choosing PI halves of all numbers is that it's something I know how to take the sine and the cosine of so T is PI halves cosine of PI halves then you start thinking okay what's the cosine of PI halves what's the sine of PI halves and maybe you go off and draw a little unit circle while you're writing things out oops so the problem with talking while writing sign of PI halves and you know it's if you go off and scribble that little unit circle you say PI halves is going to bring us a quarter of the way around over here and cosine of PI halves is measuring the X component of that so that just cancels out to zero and then sine of PI halves is the Y component of that so that ends up equaling one which means that the vector as a whole is going to be zero for the X component and then PI halves for the Y component and what that would look like you know the Y component of PI halves it's about one point seven up there and there's no X component so you might get a vector like this and if you imagine doing this at all the different input points you might get a bunch of different vectors off doing different things and if you were to draw it you don't want to just draw the arrows themselves because that'll clutter things up a whole bunch so we just want to trace the points they correspond to the output the tips of each vector and what I'll do here I'll show a little animation let me just clear the board a bit an animation where I'll let T range between zero and ten so let's write that down so the value T is going to start at zero and then it's go to 10 it's going to go to 10 and we'll just see what values what vectors does that output and what curve does the tip of that vector trace out so it goes all of the values just kind of ranging zero to ten and you end up getting this spiral shape and you can maybe think about why this cosine of T sine of T scaled by the value T itself would give you this spiral but what it means is that when you you know we saw that zero goes here evidently it's the case that ten outputs here and a disadvantage of drawing things like this you're not quite sure of what the what the interim values are you know you could kind of guess maybe one go somewhere here to go somewhere here and kind of hoping that they're evenly spaced as you move along but you don't get that information you lose the input information you get the shape of the curve and if you just want you know an analytical way of describing curves you find some parametric function that does it and you don't really care about the rate but just to show where it might matter oh I'll animate the same thing again another function that draws the same curve but it starts going really quickly and then it slows down as you go on so that function is not quite the T cosine T et sine of T though I originally had written and in fact it would mean that let's just erase these guys when it starts slowly you could interpret that as saying okay maybe well actually it started quickly didn't it so one would be really far off here and then two it could have zipped along here then three you know still going really fast and then maybe by the time you get to the end it's just going very slowly just kind of seven is here it is here and it's hardly making any progress before it gets to ten so you can have two different functions draw the same curve and the fancy word here is parameterize so functions will parameterize a curve if when you draw just in the output space you get that curve and in the next video I'll show how you can have functions with a two-dimensional input and a three-dimensional output draw surfaces in three-dimensional space