https://youtu.be/Kd3XF4LZoFE

In this video I go over the definition of parametric equations and how they can be used to graph curves that are not possible with regular y = f(x) functions. Parametric equations involve a third parameter, t, in which both the x and y are functions of t. This makes it possible to graph functions that fail the Vertical Line Test (I will go over this test in more detail in later videos) because at any specific x value there could be multiple y values. The best way to visualize parametric curves is by thinking of a particle that moves through time so at each given time, the particle's x and y coordinates will change based on what time it is. This is a very interesting topic because it allows for graphing some very remarkable curves and shapes. I will go over some cool examples in later videos so stay tuned!

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# Parametric Equations and Curves

Image that a particle moves along the curve through time as shown below:

![](https://steemitimages.com/DQmRd2tTNcBfpWPa8g3rDHkU3vH9s3uC7YNZyn8Ptotph5d/image.png)

It is impossible to describe this curve by an equation of the form y = f(x) because this curve fails the Vertical Line Test.

![](https://steemitimages.com/DQmS1hNpN2z7854ccve5peTSMfrgFYkPwRKoVzYnvQ625SY/image.png)

But since the particle is moving through time, the x and y coordinates of the particle are also functions of time, and so we can write x = f(t) and y = g(t).

Such a pair of equations is often a convenient way of describing a curve and gives rise to the following definition.

Suppose that x and y are both given as functions of a third variable t (called a parameter) by the equations:

![](https://steemitimages.com/DQmZJU55E3p4r16p7dpE8ezxoL1zFFKBWxyerngTQ6FKwRV/image.png)

Where these are called parametric equations.

Each value of t determines a point (x, y), which we can plot in a coordinate plane.

As t varies, the point (x, y) = (f(t), g(t)) varies and traces out a curve C, which we call a parametric curve.

The parameter t does not necessarily represent time and, in fact, we could use a letter other than t for the parameter.

But in many applications of parametric curves, t does denote time and therefore we can interpret (x, y) = (f(t), g(t)) as a position of a particle at time t.