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 plane curve of x of t and y of t, these two parametric equations. I'm gonna get a bunch of x and y values. And to do that, I'm gonna have to make a grid or a table of three 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, zero and five and ten 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 gonna do here is I'm just gonna go ahead and pick some because they're not actually given to me. I'm just gonna pick negative two all the way up to positive two, incrementing by one. So that's five t values. Alright? And 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 two into the x of t equation, this says, take the negative of that t value and then add one to it. So when t is negative two, the negative of that is positive two, and then if I add one, that becomes three, and then so on and so forth. If you repeat this for negative one, zero, one, and two, 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 two. For t is zero, x is equal to one. And then for this one, this becomes zero. And then finally, my last value is negative one. 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 going to do the exact same thing here, and I'm just going to plug this into the y of t, equation. So for t equals negative two, if I plug this into my t squared equation, I'm gonna take the square of negative two, the positive or the negative sign drops, and you just get four. And if you do this for negative one, that's just becomes one. And for zero, it's zero. For one, it's one. And then for two, it's four. Alright. So now I have my columns filled out for five 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 three comma four, two comma one, so on and so forth. So what I want to do now is I'm just gonna plot all of these, these points. Got three comma four, two comma one, one comma zero, zero comma one, and then negative one comma four. 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 two to positive two. 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, and I'm gonna have to write them in next to the graph that give me the values or the the coordinates that I that I drew out. So for example, the coordinates three comma four was when I had a t value of negative t of negative two, and then so on and so forth. This is gonna be negative one. This is gonna be when t was zero. This is when t was positive one, and this was when t was positive two. 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? So 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 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? 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.

16. Parametric Equations & Polar Coordinates

Parametric Equations

Calculus

16. Parametric Equations & Polar Coordinates

Parametric Equations

Struggling with Calculus?

Join thousands of students who trust us to help them ace their exams!Watch the first video

Multiple Choice

Graph the plane curve formed by the parametric equations and indicate its orientation.
$x (t) = - t + 1$ ; $y (t) = t^{2}$
$negative 2 is less than or equal to t is less than or equal to 2$

Graph of parametric equations with arrows indicating orientation.

Verified step by step guidance

Step 1: Understand the parametric equations given: x(t) = -t + 1 and y(t) = t^2, where t ranges from -2 to 2. These equations describe the x and y coordinates of the curve as functions of the parameter t.

Step 2: Calculate key points by substituting values of t within the range [-2, 2] into the parametric equations. For example, when t = -2, x(-2) = -(-2) + 1 = 3 and y(-2) = (-2)^2 = 4. Similarly, calculate for t = -1, 0, 1, and 2.

Step 3: Plot the calculated points (x, y) on the graph. For instance, plot (3, 4) for t = -2, (2, 1) for t = -1, (1, 0) for t = 0, (0, 1) for t = 1, and (-1, 4) for t = 2.

Step 4: Connect the points smoothly to form the curve. The orientation of the curve is determined by the direction of increasing t. As t increases from -2 to 2, the curve moves from right to left and then back to the right.

Step 5: Indicate the orientation of the curve using arrows along the path, showing the direction of increasing t. This helps visualize the movement along the curve as t progresses.