in this video we're going to look at how we can start sketching parametric curves again keeping in mind that these are trajectories that evolve over time but of obvious interest would be the path traced out by these curves or by the particles they represent again parametric curves is simply a synonym for vector-valued functions they can be used interchangeably here's our first non-trivial function for vector valued functions and just looking at it it feels like it has some quadratic stuff happening in it but it's not at all clear how that's going to play out anytime we have new rules like here with a vector valued function a new system of looking at functions it makes sense to go back to first principles one of the first principles of looking at functions is that we can have formulas we can have graphs and we can have tables of values so if we start here with time as our base variable our independent variable and x and y as our dependent variable then we can imagine selecting a range of values for t and then seeing what the corresponding xy locations are coming out of that calculation since there's no domain issue here i can plug any real time value in here what we usually do is start with nice integers or whole numbers and because we can be negative it makes sense just to span across zero anyway and see what happens when we have negative two for our t remember that our x is equal to t squared by itself so that would be four and at the same instant at time minus two our y will be just equal to t so it'll be negative 2 itself filling in these columns it's actually easier to use one column at a time and then interpret it later as the pairs so we're going to go with the x's next all the way down at time negative 1 x will be negative 1 squared which is positive 1 at times 0 0 squared is 0 1 1 squared is 1 2 2 squared is 4. now over the y column no y is just equal to t so that's pretty straight forward whatever t is not x make sure you're jumping from the left hand column to the right hand column here we're going to end up with exactly the same values as the t's keep in mind though what we're going to plot is only the x and y values that is what is involved in the plot command of the plot process so here recognizing that the x's are all positive we can just focus on that side of the graph and let's do one two three four one two three one to negative three and we'll clean this up just a little bit here there we are so at time knight negative two but we're not going to show that we're going to be at the point 4 for x negative 2 for y down here and then at the next time point we'll be at 1 negative 1. that's here sorry that's positive 1 negative 1 for y and then 0 0 1 1 and then 4 2 then as with all graphing you've typically done before we make a little assumption that this graph is not too crazy in between the points that we picked and again we know linear functions and quadratic functions don't change that quickly so what we can do is infer though it is an inference we don't know for sure until we did more points but we infer that the graph looks like that however we do have that additional information so this is the trajectory but we also know that the point is not at all these or the the object we're following or tracing out is not at all these points at the same time it's at this point when t is negative two it's at this point when t equals one and so there's an implied directionality to this curve the particle will be tracing out this path from here to here and we're just seeing a trace of it on this graph and also there's that element of infiniteness that this curve would come from potentially all the way up from positive infinity and x here and come in and then hit the zero zero point and go back out again afterwards we put a bound on it here for practical reasons but the actual curve that is traced out by this particle is infinitely long in both directions and that's the essentials though we now have a new kind of function where we have t's and we define our x and y's as function of time this gives us a view a graphical perspective on what those functions mean and what we can see is that the y values are constantly increasing the x values looking left to right are going down first then bouncing back up and together they trace out this sideways parabola is this still a function you might have beef having flashbacks to the idea that this violates the vertical line test for functions absolutely this is a function of two variables x y is the outputs with one variable as the input for any single time value this particle is only in one position so we have to be careful about the kinds of rules that we had before for x y graphs and recognize what doesn't transfer over to this new kind of trajectory all right to sketch the graph defined by v of t is equal to 4 cos and minus 2 sine interesting so this has the same pattern as cos t sine t which we saw earlier which gave us a circle trajectory so what we'd expect with these small tweaks to it this four coefficient and negative two coefficient is that we're going to see something like a circle but slightly changed never hurts to guess what might have changed so think about that for a moment before we dive in but once you're ready we're going to follow the same process we did last time and build a table of values until we built some intuition it's hard to judge what the effect of these fours and negative two multipliers will have so this time we're going to set up a table of values from zero to two pi and what we're gonna do is we'll break it up a little bit more at the beginning and then we're just gonna go to the standard every pi over 2. so notice this gap from here to here is tinier this is the 45 degree jump 45 degrees and then 90 90 90 as we go around the rest of the circular-ish thing where does that leave us uh cos of zero is one and then we're gonna have one times four that'll be four sorry we should emphasize that this is the x-coordinate formula and that is the y-coordinate formula for y sine of zero is zero so that's going to be zero so right away we know we're in a slightly different territory in particular what we had before was a unit circle with radius one now what we're going to have is something obviously not radius one one two three four x and y we are starting at four zero what about the next point pi over four well i'm gonna do a little side here that cos of pi over 4 is 1 over root 2 and that's approximately 0.7 we can have more decimals if you like but when we're graphing on this resolution here that's going to be plenty accurate enough and so this value here is going to be around 4 times 0.7 or around 2.8 meanwhile sine of pi over 4 is conveniently 1 over root 2 as well and also around 0.7 so we're going to have here negative 2 times 0.7 and that's going to be around negative 1.4 so that puts our next coordinate at 2.8 it's close to 3 here's 2 and down instead of up and down around 1.4 there negative 2 and then as we keep going when we hit pi over 2 cosi pi over 2 is 1 sorry cosine pi over 2 is 0 and sine of pi over 2 is 1 0 1 times that multiplier out front is negative 2 so not too surprisingly we're going to continue this circular type trend we're just going to reach the most extreme y value here of negative two then we go to pi cosine of pi goes to pi reach for your calculator if you need to is negative one oh wait but it's times four so we're gonna get negative four and sine of pi is zero and i think you can start to see some of the patterns emerging here we started at four zero then half a cycle later we're at negative four zero not too surprisingly when we go to the next step sorry we should add that to our graph just for thoroughness here when we go to the next step of our graph at 3 pi over 2 we're going to be at cos of 3 pi over 2 which turns out to be 0 and sine of three pi over two which turns out to be negative one but it's not just negative one that's negative one times that leading negative two out front and so we get positive two and so we see again that alternating zero minus two to zero positive two four zero to minus four zero and that puts us exactly where we'd expect to be if this was a circular type pattern and last but not least at two pi we're going to know exactly where we started so again filling in the blanks here with our intuition and knowing that sine and cos we've captured the most important ingredients about sine and cosines change we expect to see this kind of shape and if we add the arrows of directionality here we would be traversing this trajectory in a clockwise manner here based on this change of the sine coefficient to being negative two that flips things vertically somehow on us instead of going counter clockwise we end up going clockwise it takes a little more evidence than we have here but this is in fact an ellipse with axes of four long and too long starting off with our circular pattern is simply cos t and sine t we can expand it or stretch that in both the x and y direction separately and get an ellipse with the axes dimensions that we like so it becomes a very easy way to construct more interesting shapes than simple circles but using circles as a platform to build off of finally for the series let's involve some more interesting functions like the square root function and the one over the reciprocal function and what arises in this particular example that wasn't in the previous ones is the issue of domain in the last case let's just flip back quickly there was no t value that is a problem for cosine cos is defined for every value of t likewise so is sine however root t is only valid or only defined for ts greater than or equal to zero zero is okay square root of zero is fine but negative anything is bad and one over t is only valid for t bigger than zero and t less than zero t can't be equal to zero i'm not allowed to divide by zero and if that's the case what we have for the domain of s is the intersection if you like of those two ingredients so i can have zero or larger and i can have strictly larger than zero or negative well the negative doesn't work with this so we roll it out and the equals doesn't work because i can't do one over t for the equals zero so we end up with t strictly greater than zero well that informs our choice of values for our table of values so what we might want to do here is be a little more careful about our selection of t's again remembering that the x coordinates are going to be given by root t and the y coordinates by one over t so focusing in our domain of t larger than zero we could start just past zero which would be like 0.0001 well let's take a more straightforward choice around point one then basically we're going to center ourselves around one i think just to see what's happening so let's go 0.2 0.5 and then one and then 2 and then 5 something like that actually we have square roots so let's make our lives a little bit easier and go 2 4 and 9. then we'll get some nice values for some of the coordinates so x and y one of these is straightforward let's do it first in the interest of faster calculations one over t is our y value so if t is point one type in your calculator or do fractions we're going to get 10. t is 0.2 1 over 0.2 is 5. 0.5 is a half 1 over a half is 2 and 1 1 over 1 is one and then it gets even easier because it's just one over two one over four and one over nine done all right square roots again start with the easy ones ones you can do by inspection square root of t square root of 1 is 1 square root of 2 is square root of 2 but for graphing purposes that's around 1.4 square root of 4 is 2 square root of 9 is 3. now these other ones we definitely do need our calculator for so we'll pull out our casio 991s and we get square root of 0.1 is around 0.3 and again we don't need precision here because we're not able to graph that accurately we're simply looking for trends at this stage i'll put 0.44 there should be okay i can't let it go it'd be 0.45 if we round it correctly and square root of 0.5 is going to be 0.7 and that provides us enough information and notice everything here is positive so this time it makes sense to build our axes in the vertical direction or sorry in the first quadrant so our x's and y's go like so and we have to go up to ten oh my goodness and three two four six ten there we go so ten and ten we start off at y is ten and x is point three so this is one that's one so we would be up here then point four five and five the y drops quickly one two three four five but it's still in that same zero to one x range then we get a little closer to one over here and then we're at one one then about one point four and a half then two and a quarter and here it gets kind of fuzzy three and one-ninth so the implied trajectory through these points shows up like so again it has a directionality as time increases we're going to be heading down this curve and out towards these extremes we aren't going to cross that axis we're going to stay above it given the structure we have for our formulas here but we're going to get that kind of trajectory now what we can feel is that somehow the speed here is changing and this is a little less clear because we didn't use even time points but just as a quick example we're going to go back to matlab and just use these formulas for our position functions and we're going to use this kind of range for our time values and we'll see how that plays out in an animation what we have here is a modification of the earlier animation script for the circular motion and the only real changes we had to make were the square root changes and the one over t changes the other ingredients that were helpful are setting the x limb and the y limits so we actually get the same dimensions the same range as we had in this plot here so if we run this script what we see is a very quick drop and then thinking about the time values it took to one full second to go from one one to two one half or root two and a half another second to get to root three and another like seven seconds to get all the way out to nine seconds or x equals three we're just going to run that again and so what we see is very fast motion followed by slower and slower and slower motion again the trajectory is one thing but just showing that hides some of the other important information about the particle moving on that trajectory which is the the speed and velocity and acceleration all those other important physical quantities that are best seen through an animation are also seen through an animation procedure fast motion then slower motion in this case here