What are Parametric Equations?

###$\underline{\textbf{What are Parametric Equations?}}$

$\textbf{Parametric Equations}$ are equations that depend on another independent variable, conventially $t$. Parametric equations are used to define parametric functions. To get a better understanding of what parametric functions entail, let us first discuss the features of a non-parametric equation. Parametric equations often describe motion using vectors that can be written and visualized.

###$\underline{\textbf{How are Non-Parametric Functions Defined?}}$

First and foremost, a $\textbf{non-parametric}$ function is essentially an equation that relates a dependent variable, $y$, to an independent variable, $x$.

An independent variable is a variable that $\textit{does not}$ depend on the value of any other variable, while a dependent variable $\textit{does}$ depend on an another variable.

For instance, take the function, $$y = x^{2}$$ The variable $x$ in this case could be any value, it does not depend on any other value in order to define it's value. The value of the $y$ variable, however, does depend on another variable, which is $x$. $y$ will always be equal to $x^{2}$. If $x = 2$, then $y = 4$, If $x = 5$, then $y = 25$, and so on and so forth. Given this, $x$ is the independent variable and $y$ is the dependent variable.

###$\underline{\textbf{How are Parametric Functions Defined?}}$

Given what we now know about how non-parametric functions are defined, we can now be able to pinpoint the crucial way in which parametric functions $\textit{differ}$ from non-parametric functions. As stated earlier, parametric equations are equations that depend on another independent variable, $t$.

Say we have the functions, $$g(t) = t^2 + 2t + 4$$ $$h(t) = t^3$$ The output of both of these functions depends of the value of $t$, the independent variable; it itself does not depend on another value. When we set the $x$ value of the parametric function equal to $h(t)$ and the $y$ value equal to $g(t)$, we get these two relationships: $$y = t^2 + 2t + 4$$ $$x = t^3$$ These parametric equations are what define a parametric function.

Though, the independent $t$ variable is present in both equations representing $x$ and $y$, parametric functions are typically written using two $\textit{separate}$ equations, because of the fact that the $x$ and $y$ variables each depend on the value of $\textit{different}$ functions of $t$.

In other words, the value of our $x$ value now depends on the value of the function $g(t)$, a function that itself depends on the independent variable $t$, and the value of our $y$ variable now depends on the value of the function $h(t)$, which again itself depends of the value of $t$.

(Video) Parametric Equations

by Professor Dave Explains

This video by explains what parametric equations are and when you would use them.

Summary

So far, we've learned a lot about different types of equations where $y$ relates to $x$. You may also be familiar with polar equations, where $r$ relates to $\theta$. This lesson covers parametric equations, where $x$ and $y$ depend on another variable.

$x$ and $y$ are calculated in separate equations, and each is dependent on some other variable. We see this concept a lot with two-dimensional motion of objects, such as projectile motion. Consider when you throw a ball into the air. The $x$ and $y$ motion are independent of each other (how high the ball goes is independent of how far the ball goes). This concept is more complicated, as there is both horizontal and vertical velocity, but you get gist of it. In these types of equations, it makes sense to have an equation that models the $x$ coordinate as a function of time and a separate equation that models the $y$ coordinate as a function of time.

$$x=f(t)$$ $$y=g(t)$$

These are two separate functions that depend on the same variable, $t$.

In terms of graphing, we can either graph these equations parametrically or normally. If we want to graph these types of equations in parametric form, we solve for $x$ and $y$ based on set values of $t$, and then we graph the corresponding $x$ and $y$ points. If we want to graph normally (i.e., in rectangular form), we can use substitution to get rid of the third variable, $t$.

Parametric Equations

In this next section, we are going to learn about parametric equations.

Parametric Equations

Parametric equations are continuous functions that have an independent variable, also called a parameter. We most commonly use parametric equations when dealing with trigonometric functions of $x$ and $y$ that employ a parameter, $t$. You will see parametric equations with many common physics problems.

The above image is a graph of the set of parametric equations, $x(t) = 3 sin (t)$ and $y(t) = 3sin(t)$. These two equations are centered at the origin. Important points on the curve are labelled.

Once parametric equations are introduced, the next step is parametrizing a curve. To compare parametric equations with standard forms, you can convert them by finding rectangular equations from parametric equations.