INSTRUCTOR: All right. In this video, we're going to talk about parametric curves. And one thing that parametric curves are good for are describing curves in the plane that aren't necessarily functions. So if your graph is y equals f of x-- maybe y equals sine x or something like that-- you'll get a nice graph that's a function. 

But the idea is, here, what we're going to do is, we're going to make our x-coordinate a function of a new variable we'll call-- usually, they usually denote it as t. And then the y-coordinate will also be a function of t. And a lot of times, what will happen in these cases-- one usefulness of parametric curves is to simply graph objects that maybe move around, all around in two dimensions. 

So this is one thing that parametric curves are going to be useful for. So anyways, let's talk about graphing some parametric curves. So suppose we want to graph the parametric curve x equals 1 plus square root of t, y equals t squared minus t. And then t is between 0 and 5. 

Just like when you normally make a T chart, sometimes what are called T charts, that's what we're going to do here though. Instead of making x and y, though, we'll pick a value for t. And then once we pick a value for t, we'll be able to calculate x, and we'll be able to calculate y. So notice if I plug in t equals 0 I'll get that x equals 1 plus square root of 0, or I'll get x equals 1. And then I'll get y equals 0 minus 0 or 0. 

And what that means is the point 1 comma 0 is on our graph. So here's our x-axis and our y-axis. So the point 1 comma 0 is on our graph. And that corresponds to a t value of 0. This point that corresponds to t equals 0 is what is called our initial point. 

And now in this case, t can vary between values between 0 and 5. So let's plug in some other values. Maybe we'll plug in t equals 1. If you plug in t equals 1, we'll get 1 plus 1 or an x-coordinate of 2. If we plug t equals 1 into the second, into our y value, notice we'll get 1 minus 4. 1 minus 4 is negative 3. So that means the point 2 comma negative 3 is also going to be on our graph. 

I'm going to skip and then take a t value of 4 because that's easy to take the square root of. And let's see. 1 plus square root of 4, that's 1 plus 2 or an x-coordinate of 3. And then if we plug in our t value of 4, we'll get 4 squared, which is 16, minus 4 times 4, which is 16. So we'll actually be back at 0. So once we're at the x-coordinate of 3, we'll be back at 0. 

And notice as t progresses, that's what's going to happen. The x-coordinate is going to get bigger. The y-coordinate will get a little smaller, but eventually, it'll start getting bigger as well. So here's our curve. I don't know, we'll just play connect the dots. We'll assume we don't really know the shape of it at. This point that corresponds to t equals 5 in this case, since t is between 0 and 5, that's what's called the terminal point. So maybe let's just sketch a couple other ones here too, just to get a basic feel for what's going on. 

Suppose in this case, we have that x equals square root of t and y equals 1 minus t. So in this case, we'll pick values for t, x, and y. Again, maybe I'll plug in values that are easy to take the square root of. Notice here, there's no restrictions on t that are given. But of course, t has to be greater than or equal to 0 since it's underneath the square root. 

So maybe we'll use 0, 1, 4, and then 9, just to see what's going on. So square root of 0 is 0. The square root of 1 is 1. Square root of 4 is 2. Square root of 9 is 3. And then 1 minus 0 is 1. 1 minus 1 is 0. If we plug t equals 4, we'll get 1 minus 4 or negative 3. If we plug in 9, we'll get 1 minus 9 or negative 8. So now I can simply start graphing these points. So at t equals 0, we'll be at the point 0 comma 1. Again, that'll be our initial point. 

So 0 comma 1. And again, this corresponds to t equals 0. At t equals 1, we'll be at the point 1 comma 0. At t equals 4, we'll be down here at 2 comma negative 3. And then at 9, we'll be at the x-coordinate of 3 and the y-coordinate of negative 8. So maybe we can just play connect the dots. It looks kind of like a part of a parabola, which kind of makes sense. The x-coordinate's getting bigger. The y-coordinate's going to get more negative. 

This is another important part about parametric curves too. Parametric curves are curves that have a direction associated with them. So notice as t increases, the coordinates would actually move off to the right and down as your parameter t increases. So the direction of this curve, the curve is moving-- the particle, if you want to think about this as a particle at any particular time-- and x and y are formulas for its coordinates. Notice the curve is just moving down and to the right. 

And this brings up one other little important idea. You can actually do what's called eliminating the parameter. Notice we have x equals square root of t and y equals 1 minus t. Notice if I solve x equals square root of t, if I square both sides, I'll simply get that x squared equals t. And then if I substitute that into my other equation, I'll get that y equals 1 minus x squared. And this is now what's called eliminating the parameter. We have now eliminated the parameter t. 

So we've eliminated the parameter. And if you think about the graph of 1 minus x squared, well, what does 1 minus x squared look like? It's a parabola flipped upside down that is moved up one unit. So when you eliminate the parameter, you have to be careful, though, because in this case, our graph isn't the entire parabola. It's only the righthand side of the parabola. 

But once you've eliminated the parameter, it basically says your curve is going to at least be part of the graph that you've produced. So in this case, I know that when I sketch x equals square root of t, y equals 1 minus t, I know that the graph of that is going to look at least like a portion of the graph of 1 minus x squared. And in fact, that's what's happening. 

So to determine what part of the graph you get, then you have to go back and think about domains on t, the domain of your x and y-coordinates, what t values are acceptable to use. And from that, you'll often be able to deduce what portion of the graph you're getting. So here's some little basics on parametric equations. I'll do some more graphing and also talk about tangent lines and more complicated things in another video. 

Fee free to look around YouTube. I will also have links to all of these on my website, along with a lot of other videos, so feel free to take a look there too.