Graph the plane curve formed by the parametric equations and indicate its orientation. x ( t ) = − t + 1 x\left(t\right)=-t+1 ; y ( t ) = t 2 y\left(t\right)=t^2 − 2 ≤ t ≤ 2 -2\le t\le2

Welcome back, everyone. Let's get started with this practice problem here. We're gonna graph the plane curve that is formed by these parametric equations that are given to us here, and we're gonna indicate its orientation. Let's go ahead and get started. Remember the idea here is that to plot a plain curve of x of t and y of t, these 2 parametric equations. I'm gonna need a bunch of x and y values. And to do that, I'm gonna have to make a grid or a table of 3 columns, t x and y, because I have that new parameter. Alright? Now you might be thinking I could just start plugging in a bunch of t values, 0 and 5 and 10 and whatever just because they're not actually given to us But a lot of times what's gonna happen here is you're gonna actually have specific intervals that you'll have to restrict those t values to. For example, we're told here we can't just pick any t values We have to pick t values that are between negative two and positive two So, what I'm going to do here is I'm just going to go ahead and pick some because they're not actually given to me I'm just gonna pick negative 2 all the way up to positive 2, incrementing by 1. So that's 5 t values. Alright? Now remember, the idea here is I'm gonna take these t values, and I'm gonna use them as inputs to plug them into my x of t and y of t functions to get my coordinates of x and y, then I can plot them and graph them. Alright? So the idea here is if I plug in negative 2 into the x of t equation, this says, take the negative of that t value and then add 1 to it. So when t is negative 2, the negative of that is positive 2, and then if I add 1, that becomes 3, and then so on and so forth. If you repeat this for negative 1, 01, and 2, you're gonna see start to see a pattern. For negative one t, what you're gonna see here is that x is equal to 2. For t is 0, x is equal to 1. And then for this one, this becomes 0. And then finally, my last value is negative 1. Alright? So this is the x values. It's always just sort of easier to do one value at one sort of column at a time. And then I'm just gonna do the exact same thing here, and I'm just gonna plug this into the y of t, equation. So for t equals negative 2, if I plug this into my t squared equation, I'm gonna take the square of negative 2, the positives or the negative sign drops, and you just get 4. And if you do this for negative one, that's just becomes 1. And for 0, it's 0. For 1, it's 1. And then for 2, it's 4. Alright? So now I have my columns filled out for 5 chosen. I picked the values. And now what I can do here is I can go ahead and sort of see that I have my coordinate pairs here. I've got 3 comma 4, 2 comma 1, so on and so forth. So what I wanna do now is I'm just gonna plot all of these, these points. We've got 3 comma 4, 2 comma 1, 1 comma 0, 0 comma 1, and then negative 1 comma 4. So we can already start to see here what this graph is gonna look like, this plain curve. It's not gonna be a line. It's actually gonna be a parabola. Alright? Now remember, you might be tempted to sort of draw a line like this that connects all the points, but we can't go out all the way past, you know, past these these boundary points here because we're told the specific interval here was from negative 2 to positive 2. So what I have to do here is I have to sort of draw this. You can't go beyond these points here that we've graphed. Alright? So I'm just gonna connect the points, only the ones that we've drawn. Alright? That's how you're gonna see this stuff drawn out. So this is basically what your plane curve is going to look like. Now for the last part, we're supposed to indicate its orientation. So remember that that plane curves, unlike sort of normal graphs here, you can't just always assume that we're gonna read them from left to right. We can't always assume that these things are just gonna go this way from left to right. We have to sort of actually write out their t values because the orientation is along the direction of increasing t. So what I'm gonna do here is I'm gonna write the t values. I'm gonna have to write them in next to the graph and give me the values or the the coordinates that I that I drew out. So, for example, the coordinates 3 comma 4 was when I had a t value of negative t of negative 2, and then so on and so forth. This is gonna be negative 1. This is gonna be when t was 0. This is when t was positive 1, and this was when t was positive 2. You have to write them in because t doesn't have an axis on this graph. Alright? So clearly, we can see here that we draw the direction of this plane curve. Its orientation is gonna be along the direction of increasing t values. T actually increases as you go from right to left in this parabola. Alright? This is a little bit weird again, it's not exactly like a normal parabola, but we can see here that plane curves actually have a preferred direction, and it's not always necessarily left to right or right to left. You actually have to draw it out every time so you can figure that out. Alright? So that's how to draw this plane curve and its orientation. Let me know if you have any questions, and I'll see you in the next one.