Understanding Parametric Equations

Parametric equations are a useful way to describe curves, especially when they are tricky to show just as functions of $y$ in terms of $x$, or the other way around.

In parametric forms, we write down the coordinates of points on a curve using a third variable, usually called $t$. This variable often represents time or any other quantity that changes.

For a flat curve, the parametric equations can look like this:

• $x = f(t)$
• $y = g(t)$

Why Use Parametric Equations?

Here are a few reasons why parametric equations are helpful:

• They allow us to show complicated curves more easily.
• They work well for curves that can’t pass the vertical line test, like circles and loops.
• In physics and engineering, they can show how something moves. In these cases, $t$ usually represents time.

Changing Parametric Equations to Cartesian Form

To switch from parametric equations to a Cartesian equation, we need to get rid of the variable $t$. Here’s how we can do that:

1. Solve for $t$: Rearrange the first equation ($x = f(t)$) to find $t$. This might mean reversing the function if $f(t)$ is one-to-one.
2. Substitute: Put