Welcome back, everyone. So in this example, we have a couple of parametric equations. We've got x(t)=t−3, y(t)=1t−5. That's a rational equation. And we're going to go ahead and graph the plane curve. And then later on, we're going to write the equivalent rectangular equation. Let's get started here. We know how to do this before. All we need is just a bunch of x and y pairs over here. And, normally, we would just go and start picking a bunch of t values, but they're actually already given to us just to make things a little bit easier. The one thing we're told here is that t can be pretty much anything except for 5. If that seems like it's a really specific number, it's actually just because, if you take a look at this rational equation over here, y=1t−5, you just can't have any t values that are going to make the denominator 0 because then you would end up with something that's undefined. So t can be anything as long as it's not 5. We've got a couple of values here. Let's go ahead and just get started. So we're just going to use these t's as inputs. We're going to plug them into the x and y equations to get our coordinates. Alright?
So for t equals 0, x(t)=t−3, so this just ends up being negative 3. You're just going to subtract 3. So you just go down the x column. So if you go down the x column, this is going to be negative 3. And when t is 3, you just get 0. When t is 4.8, you get 1.8, 2.2, and then you just get 4. Alright? So these are my coordinates. Now for the y values, it's a little bit sort of more challenging or difficult, but it's the same principle. You just plug this into this formula and then just go ahead and evaluate. So if I plug in t equals 0 into 10−5, I just end up with 1−5, which ends up being negative 0.2. All right. So I'm just going to write these as decimals. They'll be easier to visualize on a graph. When t is equal to 0, I'm going to end up getting or sorry, when t is equal to 3, what I end up getting here is 13−5 becomes negative 2. So that's negative one half. So this is just negative 0.5. Keep going here. I'm going to get 14.8−5. So this actually ends up being 1−0.2. I am dividing by a decimal, so I am going to get a higher number, and this ends up being negative 5. So, if I do 5.2, I get the exact same thing except it is going to be 10.2, that's positive. So I just end up getting positive 5. And then finally, 17−5 becomes 0.5. So again, I've written these all as decimals just to help sort of visualize these points.
Let's actually go graph them. So my coordinates are going to be negative 3 comma negative 0.2. So it's going to be something that is really, really, really close to the x-axis all the way on the left. And I've got 0 comma negative 0.5, so that's going to be something over here. I've got 1.8 comma negative 5. So it's going to be close to 2 but a little bit to the left and then all the way down at negative 5. Then I've got positive 2.2 and then positive 5. So, basically, what happens is this point ends up being all the way up here. And then finally, I get 4, comma, 0.5. So I get 4, comma, 0.5 that looks something like this. Notice how there's not really a pattern that emerges. Like, that's clear here. It sort of looks like it's going to be something like this, but that wouldn't resemble any sort of normal function. What we can see here is that you sort of gain this out. If you start getting t values that are really close to 5, you're going to end up getting increasingly larger and larger numbers. So this is actually going to start to look like a rational equation, and that makes sense because the y function is or the y equation is a rational equation. So I'm going to go ahead and sort of draw a sketch, a little curve that sort of resembles what this might look like, and it's going to look something like this. I would just. Yeah. It's let's draw something like this. All right. This is going to look something like that, and it's going to continue on like this. And then I'm going to have values that come in all the way up here, and then I'm going to have that sort of flatten out like this. Or I can clearly see that this t this x equals 2 is going to be a vertical asymptote. Alright? I can't actually cross that line because I can't have t equal 5. Alright? So, this is going to be our parametric equation. And, what happens is if I write out my t values, this is going to be t equals 0. This is equal to t equals this is when t was 3. This is when t was equal to 4.8. This is when t was equal to 5.2. And then finally, this is t equals 7. So, in other words, what we can see here is that the orientation of this plane curve is it sort of goes off to the right these values. Alright? So this would be the orientation of our parametric equation. So why did we get something like this? Well, let's take a look at our equivalent rectangular equation. What we're going to do is we're going to have to eliminate the parameter. Remember, we're going to solve one equation for t and then plug it into the other. Usually, it's easier to solve the x equation and plug it into the y. Not always. But in this case, what's going to happen here is, if you were to try to solve the t equation for the y equation, you're just going to have more work because you're going to have to flip them, and you're going to have to insert that nasty fraction inside of the other. Whereas it's much easier to solve this equation for t and then plug it into the thing that's already rational. Right. So that's what I'm going to do here. So I've got x=t−3, which means that x+3=t. So, therefore, what happens is when I go to my y equation, y=1t−5. Remember, the whole idea here is that you solve for t. So you're going to get some expression for x, and then you're going to plug that in for t to eliminate the parameters. This ends up being 1x+3−5. When you simplify this, this actually ends up being 1x−2. I've eliminated the t's and I get an equation that's just y with x's. And we can see here that this is a rational equation, which is exactly what we expected. And also, what we can see here is that this is a rational equation that's going to have a vertical asymptote at x equals 2. So remember, these types of equations, you're going to get a vertical asymptote at x equals 2. Alright? So it's going to be the opposite of whatever that number is, and that's exactly what we got. We got a rational function in which we couldn't sort of cross that x equals 2 boundary. Alright. That's it for this one, folks. Let me know if you have any questions.