Graphing Parametric Equations: Study with Video Lessons, Practice Problems & Examples

Parametric equations express two variables, x and y, in terms of a third variable, t, allowing for the graphing of complex shapes. To graph these equations, create a table with columns for t, x(t), and y(t). For example, if x(t) = t + 1 and y(t) = 2t - 1, input values for t to find corresponding x and y pairs. The resulting points can be plotted and connected, illustrating the direction of increasing t values. This method highlights the orientation of the graph, which can represent various curves or lines.

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 are 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 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)=2t-1 equation? Well, basically, where it's the same exact principle, what What you're going to do here is you're going to need a bunch of x y pairs to plot, and what's going to happen here is you're going to have to make a table of values, but now since there are 3 variables, you'll just have 3 columns. You'll make a table of values, 1 for t, 1 for x, and 1 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 x y values. So these t values will either be given to you or you'll have to pick them, but in but 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) says this is just going to be t+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 sort 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. One 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.

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

$-2\le t\le2$

• A.
• B.
• C.
• D.

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

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

$t≥0$

• A.
• B.
• C.
• D.

Here’s what students ask on this topic:

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

Parametric equations express two variables, x and y, in terms of a third variable, t, known as the parameter. Unlike regular 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 graphing of more complex shapes and curves. For example, x(t) = t + 1 and y(t) = 2t - 1. By inputting values for t, you can find corresponding x and y pairs, which can then be plotted to form the graph. This method also highlights the orientation and direction of the graph, indicated by increasing t values.

How do you graph parametric equations?

To graph parametric equations, create a table with columns for t, x(t), and y(t). For example, if x(t) = t + 1 and y(t) = 2t - 1, input values for t to find corresponding x and y pairs. For t = 1, x(1) = 2 and y(1) = 1. Continue this for other t values to get more points. Plot these points on a graph and connect them to form the curve or line. The direction of the graph is indicated by arrows showing increasing t values. This method allows for the visualization of complex shapes and their orientation.

What is the role 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 creation of a set of points (x(t), y(t)) that can be plotted to form a graph. The parameter t helps in defining the orientation and direction of the graph, as the values of t increase. This is particularly useful for graphing complex shapes and curves that cannot be easily represented by a single equation relating x and y directly.

Can parametric equations represent shapes other than lines?

Yes, parametric equations can represent a variety of shapes beyond just lines. They can graph curves, parabolas, circles, ellipses, and more complex shapes. For example, the parametric equations x(t) = cos(t) and y(t) = sin(t) represent a circle. By varying the parameter t, you can create different shapes and visualize their orientation and direction on a graph. This flexibility makes parametric equations a powerful tool in graphing complex geometries.

Why is it important to indicate the direction of increasing t values on a parametric graph?

Indicating the direction of increasing t values on a parametric graph is important because it shows the orientation of the graph. Unlike regular equations, parametric equations do not inherently indicate the direction in which the graph is traced. By marking the direction with arrows, you can understand how the graph progresses as t increases. This is crucial for interpreting the behavior of the graph, especially for complex shapes and curves where the direction of traversal matters.