in this video we're going to continue our illustration of how to eliminate the parameter from a vector value function and see how that might help us sketch the graph here we have x is equal to 3t minus 2. i'm going to leave out the as a function of time because we're actually going to be playing with that and it's easier to do that without those extra symbols in the way and our second equation is 4t minus 3 and this feels like it should be linear and it probably will be in the relationship between x and y as well as x to t and y to t but let's see what the details are in particular things like slope and intercept can we get those out of the x y relationship since we're most familiar with y equals formulas let's get rid of the uh x t relationship here and let that go into equation two afterwards so for now let's take our x and add two to it and then we're going to divide through by 3 x plus 2 over 3 equals t if we take that equation and we sub it into equation 2 then we're going to have y equals 4t minus three but we're going to replace that t with the relationship over here a bit more room x plus two over three that's t and then minus three so this whole conglomeration here is our t value now that isn't super obvious but we can see that the patterns of a straight line formula are arising here we have four thirds positive four thirds x then we have eight thirds and we have minus three and so if we summarize that we're going to get eight thirds is almost nine thirds almost three minus three we're going to end up with negative one-third and that we can read off because it's a form that we're familiar with it has slope positive four-thirds and an x-intercept of negative one-third for y and so we can quickly throw that together and get a curve that looks like four thirds for every three we go we're gonna go up four there we go a little steeper than one there we are and that would all be great if i put the intercept in the right spot so just arrange that conveniently and tada we're gonna call that intercept here negative one-third that's our x that's our y and again we don't have a directionality at least from this form but if we go back to the original form up here what we can see is that there's a number of different ways to do this but i think here looking at the x value we can talk about increasing and decreasing relationships as time increases here then our x value increases as well me see whatever t is as t goes up x is going to go up that's going to happen all the time and so we can add the directionality to this trajectory which is the particle will be moving along this line from the bottom left towards the upper right let's take a look at a more challenging example next we've seen something that looked like an ellipse earlier and so it makes sense to kind of revisit that idea some students may not have seen this before many of you have but if you take a quick look at this formula you see this x squared plus y squared equals one formulation and remembering that x squared plus y squared equals one is the unit circle centered at the origin circle pardon me and with that unit circle we can do a number of different adaptations the first is we can change the right hand side and that gives us the radius of the circle here it happens to be 1 for the unit circle radius 1 but if we wanted a radius of 2 we'd put 2 squared on the side here this looks fairly simple for some reason putting things as a fraction often makes the formulas look more complicated but if you just take a quick look at the relationship here this x squared plus y squared equals r squared being a circle centered at the origin with radius r it can be converted by dividing through all the terms by r squared and we get x squared over r squared plus y squared over r squared equals one and so that's completely equivalent to the more standard formula up here this just happens to be nicer no one likes dealing with fractions unless we have to well now we have to because if we have an ellipse we have different radii so if you imagine this formula here how do we capture a different radii well we don't have to if we want to change it to have different x and y radii we simply use different values in the denominators here where we have an x component or an x contributor to the overall circle ellipse formula and a y contributor so x squared over a squared plus y squared over b squared equals one is going to give us an ellipse centered at the origin and depends on the values but this length here is going to be a or this coordinate here is going to be a and the coordinate in the y direction is going to be b and the extents of course are going to be out to minus a and minus b in the x and y negative directions respectively as well just like a circle would go to negative 1. so this is an alternative form of an ellipse and what we want to do is just make sure that the vector value formula that we saw a little while ago that involved something like that also satisfies that ellipse formula so remember what we had for this example was x equals 4 cos t and y is equal to negative 2 sine t and when we did a quick sketch of that we ended up finding out the particle or the trajectory would start here and this one went downwards and around like so and what we'd like to do is confirm that that actually is an ellipse form and what we are going to be able to do to achieve that is to try to turn our given vector valued function this one here into a form that matches this by eliminating the parameter so that's going to be our next step now it is definitely trickier to eliminate the parameter in this kind of context here because we don't really want to solve for t we're trying to get a relationship with both x and y in it that's going to work out a special way so what we can do is square both sides of each equation because we have a destination in mind rather than just trying to discover something new what we can do is get x squared equals 16 cos squared t and y squared equals positive 4 sine squared t by squaring both sides of both equations once we do that what we can try to do is see how these ingredients could be made to look like the equation in the ellipse so it's certainly easiest if you recognize if the form that we're aiming towards which is x squared over a squared plus y squared over b squared equals one so what we can do with that is just take an example of x squared plus y squared and see what we get well that would be 16 co squared t plus 4 sine squared t realize okay well that's not that's just a mix of trig functions with different coefficients so this is experimentations we just look at that value we do realize though that if we just had co-squared plus sine squared if we just had those as a goal then what we could do is co-squared plus sine squared by themselves add up to 1. well how can we achieve that what we're going to do is divide this term by 16 and this term by 4. you can't just go ahead and do that so we have to make sure that our starting point also has that construction in it so what we're going to do is look at that relationship instead x squared over 16 plus y squared over 4. again this is inspired by the destination we know we have in mind co squared t over 16 4 sine squared t over 4 well that just adds up to co squared t plus sine squared t and that through the trig identity is just one and so what that tells us is that all points on u of t satisfy the ellipse formula which is x squared over 16 plus y squared over four equals one and we know how to draw that ellipse once we have that formula it's an ellipse with radius in the x direction of square root of sixteen so four and square root of four which is two and we know it is now an exactly an ellipse rather than just something that might maybe kind of look like one because it has the cosine signs in it now we have the conformation by limiting the parameter of exactly the shape we've constructed