Graph the plane curve formed by the parametric equations and indicate its orientation. x ( t ) = 2 t − 1 x(t)=2t-1 x ( t ) = 2 t − 1 ; y ( t ) = 2 t y(t)=2\sqrt{t} y ( t ) = 2 t <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"> t ≥ 0 t≥0 t ≥ 0

If you haven't already done so, go ahead and pause the video and see if you can work this one out on your own. We're supposed to graph the plane curve that's formed by these parametric equations here and indicate its orientation, just like we've done before. Let's get started here. So we've got our parametric equations, x of t equals 2 t minus 1, y of t equals 2 times the square root of t. Alright? Now, again, we're not given any t values. We're gonna have to build a little table of values for t x and y. We're not given any, but we are told that we're supposed to pick values that are greater than or equal to 0. The reason for that is that you can't plug in negative values into this y of t equation because you'll get imaginary numbers. Alright? So we're just gonna have to pick any number that's positive or 0. Alright? So what I'm gonna do here is I'm just gonna go ahead and pick small numbers because those are gonna be the easiest to work with. So I'm just gonna go ahead and pick a couple, and I'm gonna I'm gonna do 0, 1, 2, 3, and 4, so I have 5 values. Okay? The idea here, remember, is that these are all inputs, and we plug them into our x of t and y of t equations to get our x and y values. If I plug 0 into 2 t minus 1, I get negative 1. If I plug in 1, 2 times 1 is 2 minus 1 becomes positive 1. And then, basically, each of these things are gonna increment by 2. So this is just gonna be 3, this is gonna be 5, and this is gonna be 7. You can just go ahead and double check that and sort of make sure that that works. Alright? Now now to do the same exact thing for the y values, I'm gonna plug this these t's into this equation over here. So if I take 2 times the square root of 0, that's just 0. So 2 times the square root of 0, 2 times the square root of 1, this just becomes 2 times the square root of 1, which is just 2. What I've got here is 2 times the square root of 2, which, by the way, if you go ahead and plug this into your calculator just to get an approximation, this is gonna be about 2.8. This is gonna be 2 times the square root of 3. The square root of 3 is, like, 1.7. So this is gonna be about 3.4, and then this is gonna be 2 times the square roots of 4, which, by the way, is just gonna be 2 times 2, which is 4. Alright? So, basically, these are gonna be our y values over here that we'll go ahead and plot. Alright? So, now, we've got all of our values. So let's just go ahead and plot those coordinates. My first one is negative 1 comma 0, so in other words, we're gonna be right over here for that first point. The next one is gonna be 1 comma 2. So it's gonna be 1 comma 2 over here. This one's gonna be 3 comma 2.8 ish. So it's gonna be 3. We're gonna go all the way to 2.8, and let's see. We're gonna see that's right about there. This next one over here is gonna be an x value of 5, y value of 3.4. So So we have x5 and then y value is 3.4. It's going to look something like this, halfway between 34. And then finally, what we're gonna have is 7 and a y value of 4. So it's gonna look something like this. So clearly, we can see here it's not a line. It's gonna be something that sort of curves as it's sort of going off in the x direction, which actually looks a lot like a square root of x function. Okay? So let's take a look at how we can graph these things, but first, let's write out the t values. This was when t is equal to 0. This is when t was equal to 1. This is 2. This is 3, and this is 4. So those are all my t values and if I connect them with a curve, I basically am to look I'm gonna get something that kind of does look like a square root function. Alright? You can see it goes this way. And then the orientation of this is gonna be along the direction of increasing t's. So in other words, the arrows, I'm gonna have to draw them this way. That is the graph of these parametric equations. Build out the little table, draw out the points, connect them, and then indicate its orientation. That's it for this one, folks. Let me know if you have any questions, and I'll see you in the next one.

16. Parametric Equations

Graphing Parametric Equations

Precalculus

16. Parametric Equations

Graphing Parametric Equations

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Multiple Choice

Graph the plane curve formed by the parametric equations and indicate its orientation.
$x(t)=2t-1$ ; $y(t)=2\sqrt{t}$
$t≥0$

Verified step by step guidance

Start by understanding the parametric equations given: x(t) = 2t - 1 and y(t) = 2√t, with the condition t ≥ 0.

Determine the range of t values to plot the curve. Since t ≥ 0, you can start with t = 0 and choose several values of t to see how the curve behaves.

Calculate the corresponding x and y values for each chosen t value. For example, when t = 0, x(0) = 2(0) - 1 = -1 and y(0) = 2√0 = 0.

Plot the points (x(t), y(t)) on the graph for each t value. Connect these points smoothly to form the curve.

Indicate the orientation of the curve by drawing arrows along the curve in the direction of increasing t values. This shows the path traced by the curve as t increases.