INSTRUCTOR: This is going to be a very short video in the parametric section. The first day when we talked about parametrics, we did a little precalculus recap, looking at things that we really don't get into in many of our courses here but things that you should have a basic understanding of before you get into calculus. And there's only one type of problem that you will see on your quiz, and it's the only type of noncalculus question that also could show up on an AP test. And that is rewriting by eliminating the parameter. 

So we say that our t value is our parameter, but sometimes you also see theta used, like you see in the second one here. And our goal is to take parametric equations and write them in their rectangular or Cartesian form, meaning get it into xy notation and eliminate the two equations in the parameter. 

So there's two types of problems here. There's more of an algebraic problem, which is the one you're looking at on the left, and there's a trig-based problem which you're looking at on the right. So we'll do one of each. You're going to see one of each on your quiz. So make sure you're comfortable with this. 

When you're working with the ones that are algebraic, there is no trig in them, this is just a substitution method. Your goal is to take one of the equations and solve it for t and then to do a substitution into the other equation where you see the t So you get it all into x's and y's and eliminate that t value. You always want to work with the equation that's the easiest to rewrite. So I'm going to take the top equation because, to me, that's a lot easier because it is just linear. 

If I want to solve for t and get t by itself, I would have x plus 5 divided by 2 equals t. And again, I didn't show any work. I know I can do that in our head. And then you're going to take that and you're going to replace it into the other equation where you have the t. So you would have y equals x plus 5 over 2 squared over 3, plus 1. As far as simplifying, you really want to look at the choices because, a lot of times, this is a multiple choice question. You don't really want to leave it as a complex fraction no matter what. If it is a multiple choice, then you have to do a little bit more work at times to try to get simplified. 

What I would recommend here is I wouldn't fully square it out, but I would take the top and say, OK, that's x plus 5 being squared over 4, and then that's still over 3. And then when I do that-- and I still have this plus 1 here that I'm not really dealing with-- I'm going to rewrite it as x plus 5 squared over 12, because I'm going to have the 4 times the 3, plus 1. That's where I would advise stopping if it's not a multiple choice. 

Another question you might see is something like, well, what is the resulting curve? And you should be able to look at that and say, well, that's a parabola because I know it's going to end up being a quadratic. We had ones in class where we did this, and when you were done, you ended up with something that was linear. So you'd say, well, what does it look like? It looks like a line. Or we had the ellipses that we worked with in class. Those type of questions you should be able to answer. If you are allowed to use a calculator in something like this, you can graph it. You can graph the original equations in parametric mode and take a look at it. 

You can graph your final answer in the rectangular, or the calculator calls it function mode, and you should see a very similar picture. Sometimes it gets a little distorted with the mode and the screen view, but you should still see the same general shape. 

The second one is an example of a trig one. Now, when you do the trig ones, you do a little differently. You don't do a substitution because it gets really messy working with trig and inverse trig. Instead, you use your Pythagorean identity, which was the fact that sine squared plus cosine squared equals 1. And by using that and rewriting your parametric equations, we can replace it all in terms of x's and y's. So what I want to do with these is I want to take each equation and solve it for the trig function. So for the first one, for the x equation, sine of theta would be equal to x minus 3 over 2. 

In the second equation, I want cosine to be positive. So I'm actually going to move it to the right and move y to the-- or move it to the left and move y to the right. So cosine of theta is equal to 4 minus y. Now, I am doing a substitution here, but I'm subbing both of them into my Pythagorean identity. So in place of sine, I'm going to put x minus 3 over 2 and then square it. In place of cosine, I'm going to put 4 minus y and square it, equals 1. 

You don't have to go further with this. Some things that you might see is a lot of times, they will square out the constants. So let's say this is x minus 3 squared over 4 plus and then leave it as 4 minus y quantity squared. You obviously could go further and square it all out, although that is not something that's usually done. Usually, it's left in more of this form because as we move further into these and look at the types of shapes that occur, it is actually better writing it in a factored form. 

That's all you need to know from a precalculus standpoint. You should be able to rewrite algebraically. You should be able to rewrite parametric equations that are trig-based. You should have a decent idea of what the final result looks like, either from looking at the parametric equations or looking at the rectangular function at the end. Other than that, everything else that's going to be on the quiz is all going to be calculus-based parametric questions.