Parametric Equations

In standard algebra, we usually describe a curve using a single equation relating $x$ and $y$, such as $y = f(x)$. However, sometimes it is more useful to describe both $x$ and $y$ in terms of a third, independent variable called a parameter (usually denoted as $t$).

Instead of one equation, a parametric curve is defined by a pair of equations: $x = f(t)$ $y = g(t)$

As the parameter $t$ changes, it generates a set of $(x, y)$ coordinates that trace out a curve. This is especially useful in physics for modeling projectile motion or circular paths, where $t$ represents time. Parametric equations tell us not just where an object goes, but when it gets there.

Eliminating the Parameter

To understand the shape of a parametric curve, it often helps to convert it back into a standard rectangular (Cartesian) equation containing only $x$ and $y$. This process is called eliminating the parameter.

Example: Eliminate the parameter from $x = 2t + 1$ and $y = t^2 - 3$.

Step 1: Solve for $t$ in one of the equations. It is usually easiest to pick the simpler linear equation. Let's solve the $x$ equation for $t$: $x = 2t + 1$ $x - 1 = 2t$ $t = \frac{x - 1}{2}$

Step 2: Substitute this expression for $t$ into the other equation. $y = \left(\frac{x - 1}{2}\right)^2 - 3$

Step 3: Simplify. $y = \frac{(x - 1)^2}{4} - 3$

By eliminating the parameter, we can easily see that this curve is a parabola opening upwards.

Parametric Equations for Circles

Parametric equations are incredibly powerful for defining shapes that fail the vertical line test, like circles. We can model a circular path using trigonometric functions.

The general parametric equations for a circle centered at the origin $(0,0)$ with radius $r$ are: $x = r \cos(t)$ $y = r \sin(t)$ (where $0 \le t \le 2\pi$ to trace the full circle once)

Example: Write parametric equations for a circle of radius $4$ centered at the origin.

Simply plug the radius $r = 4$ into the general formulas: $x = 4 \cos(t)$ $y = 4 \sin(t)$

We can verify this represents a circle by eliminating the parameter using the Pythagorean identity $\cos^2(t) + \sin^2(t) = 1$:

1. Isolate the trig functions: $\cos(t) = \frac{x}{4}$ and $\sin(t) = \frac{y}{4}$.
2. Substitute into the identity: $\left(\frac{x}{4}\right)^2 + \left(\frac{y}{4}\right)^2 = 1$ $\frac{x^2}{16} + \frac{y^2}{16} = 1$ $x^2 + y^2 = 16$

This matches the standard rectangular equation for a circle of radius 4!