 This lesson is on an Introduction to Parametric Equations. You will find this introduction in your book on pages 148 through 150. It's very small there, so if you could find a pre-calculus book that has parametric equations in it, I would refer to that. So let's go on. Well, the best way to introduce you to parametric equations is to do a little problem-solving. A jungle and wildlife preserve extends 80 miles north and 120 miles east of a ranger station. A ranger leaves from a point 100 miles due east of the station and travels 0.6 miles north and 0.5 miles west every minute, 6-tenths of a mile north, 5-tenths of a mile west every minute. A lion leaves the west edge of the preserve 51 miles due north of the station and at the same time the ranger leaves that station. Each minute the lion travels 0.1 miles north and 0.3 miles east or 1-tenth of a mile north and 3-tenths of a mile east. So let's graph this first. We can mark off increments of 10. The ranger leaves 100 miles due east. So this would be east and this one would be north. So 100 miles would put the ranger right here. The lion leaves the western edge at 51 miles. So this is the ranger, this is the lion. The ranger travels 0.6 miles north and 0.5 miles or 5-tenths of a mile west each minute. Now you can create a regular function for this but we are breaking this up into its x component and its y component. So let's look at the ranger. Starts 100 miles due east. That's an x component. So we say x is equal to 100 and it moves 5-tenths of a mile back for every minute. So that's negative 0.5t and it moves 6-tenths of a mile north for every minute. So that's 0.6t. So we've made an x component and a y component for our ranger. That is a set of parametric equations. The lion goes a tenth of a mile north. So that's your y component and it starts 51 miles north so it's 51 plus 0.1t. And it goes 0.3 miles west so x is equal and that's a positive direction. So it's 0.3t. So we have another set of parametric equations for the lion this time. So ranger lion. Well let's see how these would graph on a calculator. So let's go to our calculator. Let's go down here to parametric. Now when I graph these I want them to be graphed simultaneously. So I've also checked on simultaneous. So when we put our y equal to up we get x is equal to and y is equal to. Now let's put our equations in. 100 minus 0.5t for x and 0.6t for y. Now our wildlife preserve has certain dimensions. I'm just going to graph this one for now so let's do our window. Our window starts with t which is that independent variable that we are using for x and y so we have to address that just as we did with poles with the theta. So in this case we can start t at 0 and maybe make t 10 to see how far they get in 10 minutes and put the t step at 1. X min at 0 since the wildlife preserve extends out 100 miles at least so we can put 120 for that with the scale of 10 as I did on my graph. Y min again is 0. Y goes up to at least 51 so we can make it extend past 51 to 70 for our graph and again the scale is 10. So let's graph this and we see that the ranger only traveled a little bit in those 10 minutes. So let's go on and change our window to maybe 100 minutes and see how far he goes. So he goes much farther and it looks like he will be near to where the lion will be at some point so let's keep that up and put our lion down. So the equations for the lion are 0.3t and 51 plus 0.110 and let's graph both of these. Now they don't touch so let's continue on and change our window to be larger than 100 let's say 150 and when this graphs you will see a simultaneous graphing of both the lion's path and the ranger's path. And watch to see if they cross at the same time and obviously they don't so they do not come near each other. So the next question is after we looked at these two paths how close do they get because we know they do not cross each other at the same time fortunately for the ranger. So let's see how we can do this. So if we create another line here where y is equal to some number let's say 10 and we use our distance formula second square root of x of 2 minus x of 1 quantity squared plus y sub 2 minus y sub 1 quantity squared we can find out how close these two get. So let's do that. And the way we do this is to go to vars the y vars we want parametrics and we want x2 so we want 3 minus vars y vars parametrics again x1 and we're going to square that and we're going to add to it y sub 2 minus y sub 1 again from the vars and y sub 4 minus 2 square that and when we graph this should be included in our graphing and we see a horizontal line coming towards the y axis and we see how close it gets and it stops seems to stop but it really doesn't stop what happens it gets close to the axis as the distance gets closer and then it moves back out as the distances get further and further away. Let's find out how close the lying gets in order to have some lunch today see if it gets anywhere near that ranger. So look at our calculator and we find out in the calculation portion that we can only do values we can do dy dx dy dt or dx dt we cannot calculate a certain point in question. So what we need to do is just trace and if we just start at t is equal to 1 and continue on and trace along 3 is the one we are really looking for keep going 49 56 57 this is all in minutes and if we check our x's around here I go back it's 9.7 7 6 1 9.7 6 5 so when time is 119 minutes the x coordinates are 9.7 6 1 apart and the y is always at 10 so they are around 9.7 6 miles away from each other and so the lion doesn't even get close to getting lunch he needs to be a lot smarter than that. So this is a nice introduction to parametric equations let's go on and do more with them. Suppose we want to set up just any formulas in parametrics let's say we have x is equal to t and y is equal to t squared what would this look like well let's go to our calculator one more time and put in x is equal to t and y is equal to t squared I don't think we need as big a windows we had before so let's just do a zoom 6 on this and see what happens hmm we get half of a parabola well let's look at that window and we see the windows t min is 0 and the t max is 6 that means it is just graphing with a t usually time but in this case just a t is 0 and only goes to 6 let's change this to go from negative 10 to 10 and see what happens this time from the negative t's we get the negative part of our graph so we knew it was going to be a parabola because just by looking at the equations you can see that if we substitute in x for t in y's we get y is equal to x squared but again when you are graphing these on your calculator make sure you think about doing negative t's as well as positive t's and in this case it would be the negative x's as well as the positive x's that come into play let's try another one if x is equal to t y is equal to sin t what will this look like well again if we substituted in we get y is equal to sin x which would be a sine curve let's put that on our calculator and if we graph that again I am in zoom 6 and we see we get a nice sine wave now let's change things a little bit let's put in x is equal to cosine t and y is equal to sine t this time it's a little bit different try that on our calculator sine t there I'm going to change my window that part is okay for now and let's just do a negative 4 to 4 negative 4 to 4 and graph that and lo and behold we get something that looks almost like a circle if I do a zoom square we might get a better-looking circle and again we have to think about what angles in this case would create the circle so we should really try 0 to 2 pi so let's change this from 0 to 2 pi see if that works and it does the worm ended when the graph ended so we know that's what we needed to make our circle let's see if I can make that circle just look a little bit better and we can do that by changing our t-step let's do a slower t-step maybe 0.05 and graph that it's a little slower don't know if it makes a better curve but remember as your t-steps get smaller and smaller a it takes longer to graph b you're graphing more points let's look at one other formula let's look at this cosine and sine business again how does this work with Cartesian coordinates well if we square x we get x squared is equal to cosine squared t if we square y we get y squared is equal to sine squared t add the x squared and the y squared up which means we add cosine squared and sine squared up and of course cosine squared t plus sine squared t is 1 so we get x squared plus y squared is equal to 1 and this was certainly a circle centered around the origin with a radius of 1 let's go to another interesting parametric equation x is equal to 3 cosine cubed t and y is equal to 3 sine cubed t let's put this on our calculator so we have 3 and cosine t cubed and then 3 sine t cubed let's graph that and we get what is known as a hypocycloid and in this case the endpoints are at the threes on the different axis and this is one of the curves that are used a lot in parametrics and for calculus so let's look how we can develop its rectangular coordinate equation x is equal to 3 cosine cubed t so let's take the cubed root of both sides so we have x over 3 to the one-third is equal to cosine t and y over 3 to the one-third is equal to sine t and of course if we square both of these and add them together we get x over 3 to the two-thirds plus y over 3 to the two-thirds is equal to 1 and that is your rectangular form for your hypocycloid what do you need to know when it comes to parametrics you need to know how to graph them you need to know the meaning behind the x's and the y's we will be coming back to these when we do vectors because vectors and parametrics are very very close together and make sure you know what the components are doing the x component and the y component this concludes your lessons on an introduction to parametric equations