Graphing Parametric Equations - Online Tutor, Practice Problems & Exam Prep

Parametric equations express two variables, x and y, in terms of a third variable, t, allowing for the representation of plane curves. To graph these equations, create a table with columns for t, x(t), and y(t), inputting t values to find corresponding x and y coordinates. Unlike standard equations, parametric equations have a specific direction indicated by arrows, reflecting the orientation of increasing t values. Understanding this concept is crucial for graphing various shapes, including lines and parabolas, and recognizing the relationship between the variables involved.

Introduction to Parametric Equations

So by now in this course, we've already seen how to graph equations with 2 variables, something like y=2x-3. But we're going to shift gears a little bit in this video and start talking about a new type of equation that you'll have to know called parametric equations because you'll have to understand and graph them. What I'm going to show you is that parametric equations are where you have 2 variables x and y that are written or expressed in terms of a third variable. That third variable is usually called t. So you'll see these equations written as x(t) and y(t). Alright? Now I know that sounds kind of scary because now there's multiple variables, 3 variables to keep track of, but all I'm going to show you here is that the way that you graph these types of equations is almost exactly like you graph something like 2x-3. So I'm going to walk you through it. We'll just jump right into this example. We're going to graph these equations. Let's get started. Alright? So these parametric equations express x(t) and y(t) in terms of a third variable called t, which is usually called the parameter. Alright? And, basically, the idea here is that the main difference between this type of these types of equations and 2 variable equations is that something like y=2x-3, you have one equation with 2 variables, right, y in terms of x. But in these types of problems, you have 2 equations x(t) and y(t), in which you basically have 3 variables going on x, y, and t. Alright? So how would I graph something like 2x-3? We've seen how to do this before. In order to graph anything, I'm just going to need a bunch of x and y values. So the idea here is I'm going to need a bunch of x values, and I'm going to use these x values, which are either going to be given to me or I would actually have to pick them. And I'm going to use these as inputs to plug into the 2x-3 equation, and then I'll get my outputs, my y values. And then we'll just get a bunch of coordinates to plot, like (2, 1), (3, 3), (4, 5). I would just plot those things and then connect them with a line or a curve, and that would be my equation. Alright?

How would I graph something like x(t)=t+1 and this y(t)=2*t-1 equation? Well, basically, where it's the same exact principle. What you're going to do here is you're going to need a bunch of xy pairs to plot, and what's going to happen here is you're going to have to make a table of values, one for t, one for x, and one for y. But the way that you do this is very similar. t, which is the parameter, is just a number. It's just basically a bunch of numbers that you'll input into the equations to get your xy values. So these t values will either be given to you or you'll have to pick them, but what you're going to do is you're just going to input these things into the x and y equations to get your points. So let's go ahead and do that. So x(t)=t+1 says this is just going to be t plus 1. So when t is equal to 1, that means that x is equal to 2. When t is 2, x is 3. t is 3, x is 4, so on and so forth. To get the y values, you just do the exact same thing. You just input 1 into this equation, and you solve. So if I input 1, I get 2 times 1, which is 2 minus 1, which is 1. And then if I do the same thing for 2, I'm going to get 3. Alright? You actually just go ahead and plug those in. So, basically, what's going to happen here is you actually end up with a bunch of points. And if you notice, we end up with the exact same points that we got when we just did this as 2x-3. So we just do the same thing. You're just going to plot these and connect them with a curve or a line. I've got (2, 1), (3, 3), (4, 5). And that's how you would graph something like these parametric equations over here. Alright? So these graphs of parametric equations, a couple of sorts of definitions here are called plane curves. That's just a fancy word that you might see. They don't necessarily have to be curves. In this case, they could be lines, but they can also be parabolas and other more complicated shapes. Alright? There's a couple of differences with these parametric equations. The first is that you actually have to, you'll often see the t values written alongside their corresponding coordinates. So for example, the t value that gave me the coordinate (2, 1) was when t equals 1. And then the one for this point over here was t equals 2. 1 for this point was t equals 3. T doesn't necessarily have an axis on this graph, so a lot of times, you can't just sort of assume that t increases from left to right or bottom to top or something like that. So that's why oftentimes you'll see them written in. Alright? And the other difference has to do with the direction or the orientation of these plane curves. Whereas something like 2x-3 kind of just goes off infinitely in both directions, parametric equations have a specific direction. And the way that you indicate the direction or the orientation is with these little arrows along the direction of increasing t values. That's why it's really important to write them in because you can see here that this graph increases, the t values increase as you're going from bottom left to top right. So a lot of times you'll see these little arrows written along this plain curve. Alright? So that's just a brief introduction to how you graph parametric equations. Let's go ahead and get some practice.

Graph the plane curve formed by the parametric equations and indicate its orientation.

$x\left(t\right)=-t+1$; $y\left(t\right)=t^2$

$-2\le t\le 2$

• A.
• B.
• C.
• D.

Graph the plane curve formed by the parametric equations and indicate its orientation.

$x\left(t\right)=2t-1$; $y\left(t\right)=2\sqrt{t}$

$t\ge 0$

• A.
• B.
• C.
• D.

Here’s what students ask on this topic:

What are parametric equations and how do they differ from standard equations?

Parametric equations express two variables, x and y, in terms of a third variable, t, known as the parameter. Unlike standard equations like y = 2x - 3, which directly relate x and y, parametric equations use two separate equations: x(t) and y(t). This allows for the representation of more complex shapes, such as curves and parabolas. Additionally, parametric equations have a specific direction indicated by arrows, reflecting the orientation of increasing t values. This directional aspect is not present in standard equations.

How do you graph parametric equations?

To graph parametric equations, create a table with columns for t, x(t), and y(t). Choose a range of t values, either given or selected, and input them into the x(t) and y(t) equations to find the corresponding x and y coordinates. Plot these (x, y) points on a graph. Unlike standard equations, parametric equations have a specific direction, indicated by arrows along the curve, showing the orientation of increasing t values. This method allows for the representation of various shapes, including lines and parabolas.

What is the significance of the parameter t in parametric equations?

The parameter t in parametric equations serves as an independent variable that both x and y depend on. It allows for the representation of more complex shapes and motions that cannot be easily described by a single equation relating x and y. The parameter t also provides a way to indicate the direction and orientation of the curve, as the values of t increase. This is crucial for understanding the behavior of the graph over a specific interval of t values.

How do you indicate the direction of a parametric curve on a graph?

The direction of a parametric curve on a graph is indicated by arrows along the curve. These arrows show the orientation of increasing t values. For example, if t increases from left to right, arrows will point in that direction. This is important because, unlike standard equations, parametric equations have a specific direction that must be represented to fully understand the behavior of the graph. Including the t values alongside their corresponding coordinates can also help clarify the direction.

Can parametric equations represent shapes other than lines?

Yes, parametric equations can represent a variety of shapes beyond just lines. They can describe curves, parabolas, circles, ellipses, and more complex shapes. This flexibility is due to the use of a third variable, t, which allows for more intricate relationships between x and y. By varying the equations for x(t) and y(t), you can create different types of plane curves, making parametric equations a powerful tool for graphing diverse shapes.